Numerical Analysis and Simulation of Plasticity

Numerical Analysis andSimulation of Plasticity

J.C. SimoDivision of Applied MechanicsDepartment of Mechanical EngineeringStanford UniversityStanford, CA 94305, USA

HANDBOOK OF NUMERICAL ANALYSIS, VOL. VINumerical Methods for Solids (Part 3)Numerical Methods for Fluids (Part 1)Edited by P.G. Ciarlet and J.L. Lions© 1998 Elsevier Science B.V. All rights reserved

183

Contents

PREFACE 187

CHAPTER I. The Classical Models 191

1. Notation and brief summary of some standard results 1912. Classical rate-independent plasticity: Local evolution equations 1963. The rate form of classical plasticity: Elastoplastic tangent moduli 2004. Some specific models of classical plasticity 2045. General rate-independent multisurface plasticity 2116. Dissipation: Interpretation of the model of associative plasticity 2137. Rate-dependent plasticity: The viscoplastic regularization 2168. Example: Viscoplastic regularization of J2-flow theory 2189. The weak formulation of dynamic plasticity 221

10. Contractivity, uniqueness, and dissipativity of the elastoplastic flow 224

CHAPTER II. Integration Algorithms 229

11. Local integration of rate-independent plasticity: An overview 23012. Motivation: The perfectly plastic model of J2-flow theory 23413. Backward difference and implicit Runge-Kutta methods: Basic results 24014. Generalized backward difference return mapping algorithms 24415. Generalized implicit Runge-Kutta return mapping algorithms 24716. Algorithms for the computation of the closest-point projection 24917. The consistent algorithmic elastoplastic moduli 25318. Examples: Closed-form return mapping algorithms 25419. Practical accuracy assessment: Iso-error maps 26120. Accuracy analysis of return mapping algorithms 26421. Extension of return mapping algorithms to viscoplasticity 26922. Return mapping algorithms for general models of viscoplasticity 27123. The algorithmic initial boundary value problem 27324. Nonlinear stability analysis: Uniqueness and dissipativity 27725. Spatial finite element discretization: An illustration 28326. Application: A class of mixed methods for incompressibility 29127. Illustrative numerical simulations 296

CHAPTER III. Nonlinear Continuum Mechanics 303

28. Basic kinematic results in nonlinear continuum mechanics 30429. Stress tensors and alternative forms of the equations of motion 31130. Objective transformations and frame invariance 31331. Elastic constitutive equations and isotropy group 31732. Volumetric/deviatoric uncoupled finite elasticity 32233. Isotropic elasticity formulated in principal stretches 327

185

34. Multiplicative plasticity at finite strains: Basic concepts 33035. Elastic response and free energy for multiplicative plasticity 33536. Plastic flow evolution equations for multiplicative plasticity 33637. Volumetric/deviatoric uncoupled finite plasticity: J2 -flow theory 34138. Rate form and variational inequality for multiplicative plasticity 34639. Variational formulation: Weak form of the momentum equations 35140. The total and incremental weak forms of momentum balance 35541. Initial boundary value problem: Dissipativity and a priori estimate 357

CHAPTER IV. The Discrete Initial Boundary Value Problem 361

42. The Galerkin projection: The spatially discrete problem 36343. The linearized (semidiscrete) initial value problem 36544. Matrix form of the semidiscrete initial value problem 36945. Mixed finite element discretization: An illustration 37346. Exponential return mapping algorithms for multiplicative plasticity 38347. Exponential return mappings for isotropic elastic response 38748. Implementation of exponential return mapping algorithms 39049. Linearization: The exact algorithmic tangent moduli 39450. A generalization of J 2-flow theory to finite strains 39851. Two-stage projected exponential return mapping algorithms 40252. Closed-form of the two-stage projected IRK for elastic isotropy 40853. Global time-stepping algorithms for dynamic plasticity 41154. Remarks on the implementation of return mapping algorithms 41455. Representative numerical simulations 416

CHAPTER V. The Coupled Thermomechanical Problem 433

56. Integral, local and weak forms of the general conservation laws 43557. Constitutive equations for multiplicative thermoplasticity 43958. Formal a priori stability estimate and conservation laws 44759. Time integration of the coupled problem: General considerations 45160. Monolithic and staggered schemes: Product formula algorithms 45661. Model problem: The coupled system of linearized thermoelasticity 46062. Generalization: A staggered scheme for nonlinear thermoplasticity 46963. Representative numerical simulations 475

186 J.C. Simo

REFERENCES 489

Preface

Efforts aimed at understanding the behavior of metals under loading beyond theelastic regime provided the starting point, at the turn of this century, for the devel-opment of a body of constitutive theories for deformable solids commonly referredto as classical plasticity. By the mid-fifties, the foundations of the subject were well-established within the context of the infinitesimal theory as summarized, for instance,in the classical book of HILL [1950] and the review article of KOITER [1960]. Becauseof its inherent nonlinear character, only a few elementary problems in classical plas-ticity are tractable via analytical methods. The application of the theory to problems ofengineering interest, metal forming in particular, required drastic simplifying hypothe-ses involving, for instance, the assumption of rigid plasticity. A revolution in the fieldtook place with the advent of the computer and the development of the finite elementmethod. This made possible the numerical solution of complex problems by utilizingthe full theory of plasticity without introducing unduly restrictive assumptions.

The objectives of this article are to provide a comprehensive overview of both thecontinuum mechanics and the numerical analysis relevant to the large scale simula-tion of problems involving plastic deformations. Large scale simulations of this type,ranging from crash analysis of vehicles to metal forming and forging processes, arebecoming routine and constitute a significant portion of current scientific computingin solid mechanics. The goals of this exposition are to present in a unified settingthe classical models of plasticity, emphasizing the unifying mathematical structure,and address in detail the algorithmic issues involved in the solution of the resulting(highly nonlinear) initial boundary value problem. An overview of the topics coveredin this article is given below.

Chapter I provides a concise formulation of rate-independent plasticity and (rate-dependent) viscoplasticity, with attention restricted to the case of linearized kinemat-ics. Most of the results described there are fairly classical. Emphasis is placed on aformulation of the initial boundary value problem suitable for the subsequent algorith-mic treatment. This includes, in particular, the formulation of the loading/unloadingconditions of rate-independent plasticity as Kuhn-Tucker optimality conditions, theinterpretation of viscoplasticity as a (penalty) regularization of the rate-independenttheory as well as the formulation of plasticity as a variational inequality. The chapterconcludes with an overview of the properties of the initial boundary value problemthat play a central role in the analysis of the discretized problem.

Chapter II addresses the central topic of interest in this article, namely, the de-sign and analysis of algorithms for plasticity and viscoplasticity. The key observation

187

exploited in the design of numerical schemes is the interpretation of the constitu-tive equations of rate-independent plasticity as a differential algebraic system. Thispoint of view allows the generalization of classical schemes, known as return map-ping algorithms and typically restricted to first-order accuracy, to methods possessinghigher-order accuracy. A number of results presented in this chapter provide new in-sights into the algorithmic treatment of plasticity. These include the formulation ofhigher-order return mapping algorithms, a precise accuracy analysis that exploits thegeometric structure of classical plasticity and a detailed analysis of numerical stabilitythat exploits the contractive structure of the continuum problem.

Chapter III undertakes the generalization of the classical models of plasticity de-scribed in the first chapter of this article to the finite deformation regime. To make thepresentation self-contained, a comprehensive review is given of those aspects of non-linear continuum mechanics relevant to the problem at hand. Motivated by microme-chanical considerations, touched upon only briefly in the course of the exposition,the extension of classical plasticity described in this chapter relies on a multiplicativedecomposition of the deformation gradient. The rate form of the resulting theory isshown to be in a one-to-one correspondence with certain formulations of finite strainplasticity based on a rate formulation of the elastic constitutive equations, which havebeen widely used in past algorithmic treatments of the subject. The chapter concludeswith a detailed presentation of the total and incremental initial boundary value prob-lems.

Chapter IV describes a generalization of the algorithms presented in Chapter IIto the full finite deformation problem. For the isotropic theory, which includes theimportant case of J2-flow theory as a particular case, it is shown that the classicaland generalized return mapping algorithms carry over to the finite deformation regimewith essentially no modification. The exposition emphasizes the design of algorithmsaimed at preserving key qualitative features inherent to the continuum models, such asframe invariance and certain conservation laws. A number of representative numericalsimulations are presented to illustrate the effectiveness of these techniques in a wideclass of problems.

Chapter V is concerned both with the extension of the continuum models of clas-sical plasticity to the full thermomechanical regime and the numerical approximationof the resulting coupled initial boundary value problem. The different time scalesinvolved in the coupled thermomechanical problem have motivated the widespreaduse of staggered schemes involving the sequential solution of a mechanical phasefollowed by a thermal phase. Approximations of this type, however, typically lead toschemes that are, at best, only conditionally stable. The numerical analysis relies onan interpretation of staggered algorithms as fractional step methods constructed via anoperator split of the problem of evolution. This interpretation is shown to lead to thedesign of alternative staggered schemes which retain the computational convenienceof conventional methods while achieving unconditional stability. The performance ofthese schemes is again illustrated in a number of representative simulations.

I am grateful to a number of colleagues, former students and visitors at StanfordUniversity for their collaboration on the subject of this article over the past years. Inparticular, Dave Bamnett and Thomas J.R. Hughes at Stanford, Robert L. Taylor and

188 J.C. Simo

Preface 189

Jerrold E. Marsden at U.C. Berkeley and Peter Wriggers at Darmstad for many fruitfuldiscussions. I am particularly indebted to Francisco Armero and Christian Miehe forour joint work on the topics covered in Chapter V; without their collaboration, manyof the results described there would not have been possible. I am also indebted to TodLaursen for generously contributing to the simulations involving contact and frictiondescribed in Chapter IV, to Steve Rifai for his help with a number of figures drawn bycomputer, as well as to Sanjay Govindjee for useful remarks on the stability analysisof return mapping algorithms. Finally, my sincere thanks to Carlos Agelet, OscarGonzalez, Alecia Chen and Charles Taylor for their careful reading of the manuscript.

It is a pleasure to acknowledge the support of the Air Force Office of Scientific Re-search and the Lawrence Livermore National Laboratory while writing this manuscript,as well as the support of the National Science Foundation.

CHAPTER I

The Classical Models of

Rate-Independent Plasticity and

Viscoplasticity

This chapter summarizes the equations governing classical rate-independent plas-ticity, and its viscoplastic regularization. The presentation is restricted to an outlineof the mathematical structure of the field equations relevant to the numerical solutionof initial boundary value problems and the analysis of numerical algorithms.

To set the stage for the basic theory described below and the numerical analysisresults presented in Chapter II, a summary of the notation adopted along with somebasic results on continuum mechanics is first given. Attention is restricted here tolinearized kinematics and the purely mechanical theory. Further details can be foundin standard textbooks, e.g., SOKOLNIKOFF [1956] or GURTIN [1972]. The basic struc-ture of rate-independent plasticity is then outlined within the classical framework ofresponse functions formulated in stress space, as in HILL [1950] or KOITER [1960].Special attention is given to the proper formulation of the loading/unloading condi-tions in the so-called Kuhn-Tucker form. These are the standard complementarityconditions for constrained problems subject to unilateral constraints. This form ofthe loading/unloading conditions is, in fact, classical and has been used by severalauthors, e.g., KOITER [1960], MANDEL [1965] and MAIER [1970]. Because the algo-rithmic elastoplastic problem is typically cast into a strain driven format, throughoutthis discussion the strain tensor is viewed as the primary (driving) variable. This isthe standard point of view adopted in the numerical analysis literature, starting fromthe pioneering work of WILKINS [1964]. Alternative stress space frameworks havebeen explored by several authors; e.g., JOHNSON [1977] and SIMO, KENNEDY andTAYLOR [1988].

1. Notation and brief summary of some standard results

The scheme of notation adopted in this work is fairly standard in modern textbooks oncontinuum mechanics, such as CHADWICK [1976], GURTIN [1981] or CIARLET [1988],and follows widely accepted conventions as well as standard abuses in notation. A very

191

brief summary of the conventions used herein is given below.Throughout this work italic letters are used to designate scalars and real-valued

functions and lowercase bold face letters are used to designate vectors and vector fieldsin Rn , while both lower- and uppercase bold letters are used to designate tensors andtensor fields of any order. Second-order tensors are viewed as linear transformationson IRn, n 2, elements of the vector space L(R', Rn). Fourth-order tensors areviewed as linear transformations on the Cartesian product RI x RI. We designateby {eij, 1 < i < n, the standard basis in Rn and assume throughout the standardsummation convention on repeated indices. Accordingly,

a = aiei, A = Aijei ej, D = Dijklei 0 ej ® ek ® ei, (1.1)

where the symbol 0 denotes the standard tensor product of two vectors, i.e., the lineartransformation defined by the relation (a ® b)d = (b d)a, for any a, b and d in ]Rn.Here a b = aibi is the standard Euclidean dot product with associated norm denotedby lal = a a. We use the symbols

1 = ijei ® ej and I = [ik6jl + il 8jk]ei ®( ej 0 ek 0 el, (1.2)

to designate the second-order and fourth-order symmetric tensors, respectively.The vector space of second-order tensors, with dimension n x n, is equipped with

an inner product and an induced norm, also denoted by a dot and two vertical bars,respectively, defined by

A B = tr [AT B] = AijBij and IAI = A, (1.3)

for any A, B E L(IRn, Rn). Finally, we denote by S c L(In, Rn ) the subspace ofsymmetric second-order tensors with dimension n(n + 1)/2.

Following a standard convention, GL+ (n) C L(IR n , Rn) will stand for the subsetof positive definite, n x n real matrices. This set is a compact Lie group undermultiplication. Finally, the subset of symmetric, positive definite, n x n real matriceswill be denoted by S+ C GL+ (n).

(A) Local form of the momentum equations. Let 1 < ndim < 3 be the space dimen-sion, 1[ = [0, T] C R+ the time interval of interest, and Q c R" ?d m the referenceplacement of a continuum body. We assume that 2 is an open and bounded set withsmooth boundary aQ and closure 2 = Q U Q2.

We denote by po: 2 -- R+ the reference density and by u: x I -t IRndi thedisplacement field at time t E Il of material points x E 2 in the reference placement,with displacement gradient Vu(x,t) = uije 0 ej and velocity field v(x,t) =

it(x,t). In what follows we use the conventions (),j = (.)/xzj and (-) = 0(-)/atfor spatial and time derivatives, respectively. The infinitesimal strain field in the bodyat time t is the symmetrized displacement gradient

E[u] = sym[Vu] = [Vu + (VUt,)T],

192 J.C. Simo CHAPTER I

(1.4)

with component expression E[u] = (ui,j + uj,i)ei ej relative to the standard basis.In addition, we denote by

or: x with ar(x, t) = ijei ej (1.5)

the (infinitesimal) stress tensor field on the body at time t. We assume that the boundarya2 of the body is divided into two parts Pr and r,, such that

FU£r= a2, and Fr,£P= 0, (1.6)

where the displacement field and the traction vector are respectively prescribed as

u = i on r x and on = t on r, x 1I. (1.7)

Here t(., t) and t(-, t) are given functions and n: a2 - S2 is the outward unit normalfield on a0. Under these conditions, the local form of balance of linear momentumfor a prescribed body force field f: 2 x I ]R

9ndim takes the form

div[ in 12xIE, (1.8a)po = div[tr] + f

where div[or] = aij,jei denotes the spatial divergence of the stress field. Recall thatbalance of angular momentum results in the symmetry condition or = rT is assumedat the outset. This system of partial differential equations is supplemented by theboundary conditions specified by (1.7), subject to the restrictions (1.6), together withthe initial data

u(*,t)jto = uo(-) and v(.,t)lt=o = v0 (.) in 2, (1.8b)

where uo(.) and vo(-) are prescribed functions in Q. Equations (1.8a,b) together withthe boundary conditions (1.7) yield an initial boundary value problem for the displace-ment field u(x, t) when the stress field o-(x, t) is related to the displacement fieldu(x, t) through a constitutive equation. This problem is linear only if the constitutiverelations are linear.

EXAMPLE 1.1. The simplest model for a constitutive equation is provided by a hy-perelastic material, in which the stress response is characterized in terms of a storedenergy function W: 2 x S It R such that

(x, t) = 0aW(x, E(x, t)), i.e., aij = aW/laij. (1.9)

The second derivative of the stored energy function defines the linearized elasticitytensor

C(, t) = 82W(x,E(, t)), i.e., Cijkl = (2W.aEij1aE,

SECTION The classical models 193

(1.10)

which possesses the symmetries Cijkl = Cklij = Cijlk = Cjilk. For the infinitesimaltheory, C is positive definite restricted to S in the sense that

C4· C = (ijCijklkl i> aC 2, (l.l)

for some a > 0 (depending on x e Q), and any C E S. This condition, also known aspoint-wise stability (see, e.g., MARSDEN and HUGHES [19831, Chapter 3) is equivalentto demanding that the stored energy function W be a convex function of the infinites-imal strain tensor e e . If W does not depend on x E 2 [that is, aW = 0] thematerial is said to be homogeneous. Finally, if W is rotationally invariant the materialis said to be isotropic. If, in addition, C is constant the material is said to be linearlyelastic and one has the classical result

C = A1 X1 + 2I = nl ® 1+ 2[I - 1 1], (1.12)

where A, p are the Lame constants and ;r = A + 2p/ is the bulk modulus.

REMARK 1.1. Convexity is an unacceptable restriction for the full (geometrically)nonlinear theory, see, e.g., CIARLET [1988]. A restriction on C which is significantlyweaker than positive definiteness is the strong ellipticity condition

(a b) C(a X b) ola121bl2 , (1.13)

for some a > 0 (depending on x E Q) and any a, b E R nd m. It can be easily shownthat (1.11) implies (1.13) but not conversely; see MARSDEN and HUGHES ([1983],Chapter 3). The strong ellipticity condition, on the other hand, is equivalent to therequirement that wave speeds in the material be real, i.e., the so-called Hadamardcondition on the acoustic tensor.

(B) Weak form of the momentum equations. The existence theory and the numericalimplementations of linear elasticity are most easily formulated in terms of the weakform of the local equations (i.e., the virtual work principle). For example, the numericalsolution of the preceding initial boundary value problem by finite element methodsrelies on the weak formulation, see CIARLET [1988].

The weak form of the momentum equations is constructed as follows. Let St bethe solution space for the displacement field at (frozen) time t E 1i defined as

St = {u(.,t) Wl'P(2)ndim: u(., t) = Ut(, t) on ru}, (1.14)

where the appropriate exponent p 2 in the Sobolev space W P() nd im is dictated bytechnical considerations. For elasticity, the choice of p is dictated by growth conditionson the stored energy function; see, e.g., CIARLET [1988] or MARSDEN and HUGHES([1983], Chapter 6). For instance, p = 2 is the appropriate choice if the stored energy


is quadratic in the strains. We shall let V denote the space of admissible displacementvariations (test functions) associated with St and defined in the standard fashion as

V = 7 E Wi'P(2)dd i : 7 = 0 on rF}. (1.15)

If, with a slight abuse in notation, the symbol (, .) is used to denote the standardL 2 (2)-inner product of either functions, vectors, or tensors depending on the specificcontext, the weak form of the momentum equations (1.8a) takes the form

(pov, ri) = (pOU, )(pov, ) + E[])= (f, o) + V)rV, (1.16a)

subject to the initial conditions

(u(., 0), r) = (uo, ) and (v(.,O),i ) = (vo, 77), Vqr e V. (1.16b)

Formally, one arrives at this result by taking the L2-inner product of the local form(1.8a) with any admissible variation and using integration by parts. Result (1.16a)holds regardless of any constitutive assumption.

By specialization of the weak form via a specific choice of test function, one obtainsa basic result known as the mechanical work identity. Consider the case in which theessential boundary conditions are time-independent, i.e., au/at = 0 on Fr. Underthis assumption, for fixed but otherwise arbitrary time t GE I, the velocity field v(x, t)is an admissible test function, i.e., v(-, t) E V. Setting (.) = v(-, t) in (1.16a) andmaking use of the elementary identity

(poi, v) = (pov, v)

yields the following fundamental result

dK(v) + Pint(U, v) = Pext(v) for all t E , (1.17)

where K(v) is the kinetic energy of the system, Pit(o, v) is the stress power andPxt(v) is the external power of the applied loading, respectively defined by

K(v) = (pov, v) kinetic energy,Pint(a, v) = (, E[v]) stress power, (1.18)

Pext() = (f, v) + (t, v)r external power.

Both the weak form (1.16a) of the momentum equations and the preceding result areindependent of the specific form adopted for the constitutive equations.

SECTION 2 The classical models 195

2. Classical rate-independent plasticity: Local evolution equations

With the preceding notation in hand, we summarize below the governing equationsof classical rate-independent plasticity within the context of the three-dimensionalinfinitesimal theory. Throughout our discussion, if no explicit indication of the ar-guments in a field is made, it is understood that the fields u, E, ' and so on, areevaluated at a point x E 2 and at current time t G , where the time interval 1 ofinterest is often taken as the entire 1R+ for convenience. In addition, we shall denoteby r - E, (x) = E(x, ), where T E (-oo, t], the strain history at a point x E Q upto current time t E R+. Typically, one assumes that this mapping is C°. Frequently,we shall omit explicit indication of the spatial argument and write Tr - (r) or simplyuse the symbol e,, for r C (-o, t].

From a phenomenological point of view we regard plastic flow as an irreversibleprocess in a material body, typically a metal, characterized in terms of the historyof the strain tensor and two additional variables: the plastic strain eP, and a set ofnint > 0 internal variables generically denoted by e and often referred to as hardeningparameters. Accordingly, in a strain driven formulation, plastic flow at each pointx E Q up to current time t C ]R+ is described in terms of the histories

T C (-o, t] - {E(x, T), P(X, r), ( T)}. (2.1)

In this context the stress tensor is a dependent function of the variables (e, EP) throughthe elastic stress-strain relations, as discussed below. This leads to a strain space for-mulation of plasticity. Although we regard (2.1) as the independent "driving" variables,the response functions in classical plasticity, i.e., the yield condition and the flow rule,are formulated in stress space in terms of the variables

r E (-o, t] H~ {a(x, ),q(x, )}, (2.2)

where the stress tensor ot and the internal variables q are functions of (EP ) . Inthe following discussion of classical plasticity we shall adopt this point of view andformulate the response functions in stress space. Nevertheless, implicitly we alwaysregard (2.1) as the independent variables.

The basic assumptions underlying the formulation of phenomenological models ofclassical plasticity, leading to a set of local evolution equations for the plastic straineP and the internal variables , can be summarized as follows.

(A) Additive decomposition of the strain tensor. One assumes that the strain tensorE(x, t) =- E[u(x, t)] can be decomposed into an elastic and plastic part, denoted bye and eP, respectively, according to the relation

E= Ee + EP . i.e., Eij = E + j. (2.3)Cu 2.

I Within a thermodynamic framework, q are viewed as the "fluxes" conjugate to the affinities .


The classical models

Since is regarded as an independent variable, and the evolution of EP is definedthrough the flow rule (as discussed below), Eq. (2.3) should be viewed as a definitionof the elastic strain tensor Ce = - P.

(B) (Elastic) stress response. The stress tensor and the stress-like hardening variables(o, q) are related to the elastic strain and the strain-like hardening variables (e, E)by means of a free energy function T : 2 x S x Rflnit -- R according to the potentialrelations

o = kae (x, e, ) and q= -oT1(x, e, ). (2.4a)

The free energy function is independent of x if the material is homogeneous.Furthermore, in most applications, l is uncoupled and has the form

!(E, ) = W(Ee) + 7-/(S), (2.4b)

where W(.) is the elastic stored energy function and () is the potential functionfor the (stress-like) hardening variables. For linearized elasticity, both W(-) and 7-(.)are quadratic forms in the elastic strains e and the strain-like hardening variables ,respectively, i.e., W = Ie C ee and 7-t = 1 · He where C is the tensor of elasticmoduli and H is the matrix of plastic moduli both assumed constant. Equations (2.4a)and (2.3) then imply

r =C[ - P], i.e., oij Cijkl(Ekl - k), (2.5)

q = -HE, i.e., qi = Hijj.

We observe that Eqs. (2.4a) and the decomposition (2.3) are local. Therefore, althoughthe total strain is the (symmetric) gradient of the displacement field, the elastic strainis not in general the gradient of an "elastic" displacement field. Note further that EPand, consequently, Ee are assumed to be symmetric at the outset, i.e., EP E S . Thus,the notion of plastic spin plays no role in classical plasticity.

The essential feature that characterizes plastic flow is the notion of irreversibility.This basic property is built into the formulation as follows.

(C) Elastic domain and yield condition. We define a function f:S x Rnint _ IRcalled the yield criterion and constrain the admissible states (, q) E x RTin instress space to lie in the set E defined as

E = {(o, q) E $ x ]lin': f(a, q) < O}. (2.6)

One refers to the interior of E, denoted by int(E) and given by

int(E) = {(a, q) E S x R'nt: f(a, q) < 0(,

SECTION 2 197

(2.7)

as the elastic domain; whereas the boundary of E, denoted by E and defined as

aE = {(a, q) CE x l1'Ti: f(or, q) _ 0}, (2.8)

is called the yield surface. For fixed internal variables q the admissible stress fieldslie in a set denoted by E. Note that states (, q) outside E are nonadmissible, andare ruled out in classical plasticity.

(D) Flow rule and hardening law Loading/unloading conditions. The key notion ofirreversibility of plastic flow is introduced by means of the following (nonsmooth)equations of evolution for (P, )

P = yr(,q) and = yh(a,q), (2.9)

which are referred to as the flow rule and hardening law, respectively. Herer: x -Rni' S and h: S x lTn' R IR n t are prescribed functions which definethe direction of plastic flow and the type of hardening. The parameter y > 0, referredto as the consistency parameter, is assumed to obey the following Kuhn-Tucker com-plementarity conditions

Y > 0, f(a, q) < 0, and yf(a, q) =0. (2.10)

In addition to conditions (2.10), y O0 satisfies the consistency requirement

yf( , q) = 0 if f(a, q) =0. (2.11)

In the classical literature on plasticity, conditions (2.10) and (2.11) go by the nameof loading/unloading and consistency conditions, respectively. These conditions playa fundamental role in convex programming (see, e.g., KARMANOV [1977] or CIARLET[1989], Chapter 9) and, in the present context, replicate our intuitive notion of plasticloading and elastic unloading.

The Kuhn-Tucker complementarity conditions provide a compact statement of thedifferent regimes possible in a model of classical plasticity. To illustrate the alternativesituations that can arise in a complex loading program, consider first the case in which(a, q) int(E) so that, according to (2.7) f (a, q) < 0. Therefore, from condition(2.10)3 we conclude that

yf = 0 and f <0 y =0. (2.12a)

From (2.9) it then follows that P = 0 and 0 = O. Thus, (2.3) yields E = e, and therate form of (2.5) leads to

dr = C = Cue. (2.12b)

In view of (2.12b) we refer to this type of response as instantaneously elastic. Nowsuppose that (a, q) G aE which, in view of (2.8) implies that f(a, q) = 0. Condition

198 J.C. Simo CHAPTER


dE,s (f = 0)[NON-ADMISSIBLE]

f>0

FIG. 2.1. Illustration of the elastic domain in stress space.

(2.10)3 is then automatically satisfied even if -y > 0. Whether y is actually positiveor zero can be concluded from condition (2.11). Two situations can arise.

(i) First, if f(a, q) < 0, from condition (2.11) we conclude that

yf = 0 and < 0 == y =0. (2.13)

Thus, again from (2.9) it follows that P = 0 and : = 0. Since (2.12b) holds and(a, q) is on aE, one refers to this type of response as unloading from a plastic state.

(ii) Second, if f(a, q) = 0 condition (2.11) is automatically satisfied. If y > 0then &P 0 and i y 0, a situation which is referred to as plastic loading. The casey = 0 (and f = 0) is termed neutral loading.

To summarize the preceding discussion we have the following possible situationsand corresponding definitions for any (a,, q) E E.

f < 0 (cr, q) E int(E) ==, y = 0 elastic response,

0= and 7y > 0 plastic loading.

We observe that the possibility f > 0 for f = 0 has been excluded from the analysisabove. Intuitively, it is clear that if f(a, q) > 0 for some (a, q) E aE at some timet E R+, then the condition f < 0 would be violated at a neighboring subsequent time(see Fig. 2.2). A formal argument is given in the following.

THEOREM 2.1. Let T - {a-,, q} for T E (-o, t] be the history in stress space upto current time t C R+. Set

f(t) = f(aot, qt), (2.15)

and assume that (at, qt) is on IE so that f (t) = O. Then the time derivative of f(t)cannot be positive, i.e.,

If f(t) = 0 at t e R+, then f(t) < O.

SECTION 2 199

(2.16)

f(r)

77�Yf(t) > 0

f(t) = ---

f(t) <o

FIG. 2.2. Illustration of the consistency condition f(t) = 0 => f (t) < 0.

PROOF. Assuming that f(.) is smooth the result follows from elementary considera-tions. In fact, for C > t by Taylor's formula

f() = f(t) - t]f(t) + - t]f(t) + 01 - t 2, (2.17)

where, by definition, OIl - t12/[( - t] -* 0 at --- t. Now since we must have

f() 0< and f(t) = 0, dividing (2.17) by [C - t] leads to the inequality

/(t) + 9- t] < 0. (2.18)K - t]

The result then follows by taking the limit of (2.18) as [ - t] - 0.

The consistency condition (2.11) is used to relate the plastic multiplier y > 0 tothe current strain rate in the rate form of the constitutive equations.

3. The rate form of classical plasticity: Elastoplastic tangent moduli

The constitutive theory outlined above can be cast into an evolution equation thatrelates the stress rate to the total strain rate via a constitutive matrix of elastoplastictangent moduli. The derivation of this rate equation makes crucial use of the con-sistency condition (2.11) as follows. The time derivative of f at (, q) C aE isfirst evaluated by making use of the chain rule together with the rate form of thestress-strain relations (2.5), and the flow rule and hardening law in (2.9), leading to

f =af C +aqf i

= af- C[ - &P] + aqf ii

= a,f - C- [,f'.Cr + aqf. Hh] < 0. (3.1)



FIG. 3.1. Plastic loading at (a, q) aE takes place if the angle 0 defined by cos = *af · C/[raaf ·Ca, f]j/ 2[ · C0]1/2 is such that 0 < 7.

To carry the analysis a step further, an additional assumption on the structure ofthe flow rule and hardening law in (2.9) is needed to preclude "excessive softeningresponse". Explicitly, the following hypothesis is made.

ASSUMPTION 3.1. The flow rule, hardening law, and yield condition in stress spaceare such that the following inequality holds

G = [af . Cr + aqf. Hh] > 0, (3.2)

for all admissible states (, q) E.

It will be shown below that this assumption always holds for associative plasticityeither in the hardening or perfectly plastic regimes. An interpretation of Assump-tion 3.1 will be given in the context of a one-dimensional problem in a subsequentsection. With such an assumption in hand, it follows from (2.11) and (3.1) that

f = 0 and f = 0 -y = G-l]af .CE[, (3.3)

where ]x[= ½[x + -xi] denotes the ramp function. In view of (3.2) and (3.3) we alsoconclude that

forf= 0 andf= O, y 0 >afCfE>O. (3.4)

This relation provides a useful geometric interpretation of the plastic loading andneutral loading conditions in (2.14) which are illustrated in Fig. 3.1. Plastic loadingor neutral loading takes place at a point (a, q) E aE if the angle in the inner productdefined by the elasticity tensor C between the normal a,f(a, q) to E at (, q) andthe strain rate E is less than or equal to 90 ° .

The computation of the tangent elastoplastic moduli is completed by using therelation

a = C[E - EP] = C[E - ],

SECTION 3 201

(3.5)

which follows from (2.5) and (2.9). Substitution of (3.3) into (3.5) then yields the rateof change of o in terms of the total strain rate as

a= ceph, (3.6)

where the tensor CeP of elastoplastic tangent moduli is given by the expression

Cep~ { if 'Y ~O, (3.7)CP{ C-G - l [Cr] [c , ] if y > 0. (3)

Note that ceP is generally nonsymmetric for arbitrary r(o, q), except in the case

r(a, q) = f(o, q), (3.8)

which has a special significance and is referred to as an associative flow rule.The derivation leading to expressions (3.3) and (3.4) relies crucially on As-

sumption 3.1. This assumption is also necessary in order to establish the equiva-lence between the Kuhn-Tucker complementarity conditions and the classical load-ing/unloading conditions in strain space, which are essentially equivalent to (3.4).

REMARK 3.1. While the elasticity tensor C is always positive definite (point-wise sta-ble), the matrix H of plastic moduli need not be. The case H = 0 corresponds toperfect plasticity whereas the case in which H is either positive or negative definiteis referred to as hardening and softening plasticity, respectively. For either hardeningor perfect plasticity, Assumption 3.1 always holds provided that the flow rule is as-sociative. This conclusion is the direct result of the positive definiteness of C since(3.2) reduces to

G = af . Caf + aqf Haqf > QCEIaf 2 + ZHlaqf 2 > 0, (3.9)

where aOE > 0 is the ellipticity constant and aH Ž> 0 both for perfect and hardeningplasticity. Recall that positive definiteness of C holds for any symmetric tensor C E S,in particular, for ( = daf.

The Kuhn-Tucker form of the loading/unloading conditions admit an alternative, butentirely equivalent formulation which will prove particularly useful in the formulationof return mapping algorithms. Define the trial elastic stress rate as

tr = C, (3.10)

and declare a process to be plastic whenever

f(o,q)=0 and f(. ,q)' t r >0.

202 J. C. Simo CHAPTER I

(3.11)


trial := C:

FIG. 3.2. Interpretation of the loading/unloading conditions in terms of the trial elastic stress rate dr.

TABLE 3.1Classical rate-independent plasticity.

(i) Uncoupled, linear elastic stress-strain relations:

= EW( e )= C[E - P] and q =-aH(~ ) =-H~

C = 2eeeW(ee) and H = a?2 E() (both constant)

(ii) Elastic domain in stress space (single surface):

E = {(o, q) E S x Rnin': f(oa, q) 0}

(iii) Flow rule and hardening law:(iiia) General nonassociative model

p = yr(, q) and E = yh(a, q)

(iiib) (Particular) associative case

E = ya-f(oa, q) and 4 = yaqf(o, q)

(iv) Kuhn-Tucker loading/unloading (complementarity) conditions:

y 0, f(-, q) < 0 and -yf(o, q) =0

(v) Consistency condition:

(f = 0) = yf(a,q) = 0

The fact that this condition is equivalent to the Kuhn-Tucker conditions follows atonce from (3.10) and (3.3) by noting that, for f (a, q) = 0,

f = 0 = = G-'af · &.tr

SECTION 3 203

(3.12)

Since G > 0, it follows as a result of Assumption 3.1 that

ry>0 0= a,f &r >0, for = f = 0, (3.13)

and the equivalence between (3.11) and the Kuhn-Tucker conditions follows. Thesimple geometric interpretation of tr = CE should be noted, and is illustrated inFig. 3.2. We observe that Ce is the rate of stress obtained by "freezing" the evolutionof plastic flow and internal variables (i.e., by setting P = 0 and 0 = O) hence thedenomination "trial elastic stress" rate.

The notion of a trial elastic state arises naturally in the context of the algorithmictreatment of the elastoplastic problem. In the computational literature, use of the algo-rithmic counterpart of the trial stress rate condition goes back to the pioneering workof WILKINS [1964] on the now classical radial return algorithm for J 2-flow theory.The notion was subsequently formalized independently by MOREAU [1976, 1977] whocoined the expression "catching-up" algorithm. We remark that the preceding form ofthe loading/unloading conditions is equivalent to the so-called strain space form ofthe loading/unloading conditions discussed in NAGHDI and TRAPP [1975] or CASEYand NAGHDI [1981, 1983a,b].

For the convenience of the reader and subsequent reference we summarize the basicgoverning equations of general rate-independent plasticity Table 3.1.

4. Some specific models of classical plasticity

We illustrate the general structure of classical plasticity outlined above with a numberof representative examples.

(A) One-dimensional plasticity. Consider a one-dimensional bar with unit cross sec-tion A = 1 occupying a closed interval Q = [0, L]. The stress, total strain, and plasticstrain then become one-dimensional fields {r, E, EP}. Consider further ni,nt = 2 inter-nal hardening variables ~ = (, ) and specify the uncoupled free energy functionT(E', E) = W(Ee ) + 7-t() using

W(Ee) = lE'EE, -()= . Ht where H = O JHere E > 0 is the Young modulus and K, H are referred to as the isotropic and kine-matic hardening moduli, respectively. The (generalized) elastic stress-strain relationsare

= E(e-EP), q =-Kf and q=-H. (4.1)

The yield criterion in stress space is specified as

(4.2)f(, q) = - + q - y 0.



elastic domainin stress space

-cry cry

FIG. 4.1. Elastic domain in stress space for a one-dimensional model of plasticity, with linear isotropichardening, corresponding to K > 0 and H = 0. The ordinate is = -q/K > 0.

Accordingly, the interior of the elastic domain int(E) = {(or, q): f(a, q) < O} instress space is an open set centered at r = q with half length (ay - q) > 0. Thecase in which K > 0 and H = 0, illustrated in Fig. 4.1, is referred to as isotropichardening. The rate equations for the internal variables (, ) (the hardening law)dictate the evolution of the center of the (one-dimensional) elastic domain and its"size" in time. An associative flow rule yields

gP = ya f((, q) = y sign[ - ],

1 ra~i(·,4i= ?~ _,,.~-,) (4.3)f= qf(a, q) = ' { -sign[a - ]

The first component of (4.3)2 defines a linear isotropic hardening law. The idealizedstress-strain curves for the case H = 0 and both K = 0 (perfect plasticity) andK > 0 (isotropic hardening) are illustrated in Fig. 4.2. The second component of(4.3)2 is referred to as the kinematic hardening law, often credited to PRAGER [1956]with further improvements by ZIEGLER [1959], and is motivated by the experimentalobservation that the center of the yield surface experiences a motion in the directionof the plastic flow. This hardening behavior is closely related to a phenomenon knownas the Bauschinger effect. The stress-like internal variable , which defines the centerof the yield surface, is known as the back-stress.

The incremental stress-strain relations are given by the rate equations

E if3' = O,u= EP where E = E[H+K] (.

E+[H+K] if > 0,(4.4)

which are easily obtained by specialization of (3.7). Ee P is the one-dimensional elasto-plastic tangent modulus.

SECTION 4 205

I

CHAPTER I

Cry I

--oy

FIG. 4.2. Idealized one-dimensional stress-strain curve for perfect plasticity and (nonlinear) isotropic strainhardening corresponding to a yield criterion f(cr, q) = Icrj + g(q) - uy.

a

HI>0

ftening)

ftening)

FIG. 4.3. For one-dimensional plasticity, Assumption 3.1 places a limit on the amount of allowable softeningin the model.

REMARK 4.1. In the present one-dimensional context, Assumption 3.1 has a simpleinterpretation. Substitution of (4.3) into (3.2) along with a straightforward computationreveals that

G = af. Cr + a3qf. Hh = E + [H + K] > 0. (4.5)

The physical significance of this condition can be easily appreciated by inspectionof Fig. 4.3. Condition (4.5) places a restriction on the amount of allowable softeningin the sense that [H + K] > -E A> EeP > -oc where Ee P is defined by (4.4).It should be noted that the classical condition of nonnegative second-order workdensity (see HILL [1958]) implies that E + H > 0, which is much stronger thanthe restriction implied by Assumption 3.1. Even if this assumption holds, models ofplasticity exhibiting softening response present considerable mathematical challengeswhich do not arise in conventional hardening models. For instance, in the presence of

206 J.C. Simo


softening response the initial boundary value problem can be shown to always exhibitsolutions possessing strong discontinuities, see SIMI, OLIVER and ARMERO [1993]and references therein. These topics are currently an active area of research.

(B) General quadratic model for three-dimensional plasticity. Let A E R9X9 be asymmetric (positive definite) matrix and, as before, let ay be the flow stress. Considera yield criterion of the form

f(a, q) = (a) + q - y 0,

((ot) = -.A o, (4.6)

q = -K.

As before I stands for a strain-like hardening variable which models isotropic hard-ening, with evolution defined below and dual variable q = -KE, where K is the(isotropic) hardening modulus. If K < 0 we speak of strain softening response. Theevolution of eP and is defined by the associative rules

AaP= yaf = yV and =yaqf = , where V(rr) =

(4.7)

with y > 0. Observe that = /P ·A-l&P. The elastoplastic moduli are obtained byspecialization of the general expression given in (3.7). Using the preceding relations,an easy manipulation yields the result

C if Ty=0,Cep= C CV&(r) CV(r) if y > 0. (4.8)

Observe that this expression gives symmetric elastoplastic moduli, in agreement withthe associative character of the model under consideration.

REMARK 4.2. The quadratic form S(r) defined by (4.6) is obviously a convex functionwhich is homogeneous of degree one, in the sense that 0((a) = (a) for any ( > 0.By Euler's theorem, this property implies the result

a,&(t) ·r = q(o'), for all E S, (4.9)

which is easily verified. Practically all of the yield criteria used in plasticity enjoy thepreceding property.

REMARK 4.3. The extension of the preceding model to account for other types ofhardening is straightforward. For instance, consider a yield criterion f(ao, q) of theform

f(a, q) = (o - ) + q - y ~< 0,

SECTION 4 207

(4.10)

where q defines the center of the yield surface, and define the hardening potential as7-/() = - H where

( ={j } and H =[ HI' (4.11)

so that q = -H. The associative hardening evolution equations

= aq'yf(a, q) q = 'HV4(oa - O) and = - K, (4.12)

then furnish a generalized model of combined kinematic/isotropic hardening of thetype described in the context of the preceding one-dimensional example. All the resultsdescribed above carry over to this more general setting without essential modification.

(C) Plane strain and three-dimensional J2 -flow theory. The preceding generalquadratic model includes a number of plasticity models widely used in practice. Inparticular, by suitably restricting the quadratic form that defines the yield condition,one recovers the anisotropic criterion of HILL [1950], the general anisotropic yieldcondition of TSAI and Wu [1971], and the von Mises-Huber yield criterion both inplane strain and plane stress.

As an example, a widely used extension of the classical Prandl-Reuss equations ofperfect plasticity that incorporates hardening is obtained by considering linear isotropicelastic response, with elasticity tensor C = AXl 1 + 2/I, and the following pressureinsensitive, isotropic, Huber-von Mises yield condition

f(a, , q) = Idev[ ] - l- [Oy-] 0,- (4.13)

where dev[a] = ao - tr[o ]l is the deviatoric part of the stress tensor and ry is theone-dimensional flow stress. Here q is the back stress and q is the isotropic (stress-like) hardening variable, assumed to be related to the strain-like hardening variables

= ({, E) via the relations

q=-K'( ) and q=-sH4, (4.14)

which arise from the hardening potential 7-() = K() + HItg 2 . It is implicitlyassumed that tr[] = 0, i.e., the back-stress is a deviatoric tensor. Denoting by n =(dev[a] - q)/dev[r] - qJ the unit normal field (in S) to the von Mises cylinder, theassociative evolution equations can be written as

gP = af = 7n, -= 7qf = -'n and a = ?aqf = By. (4.15)

Relation (4.15)1 is known as the Levy-Saint Venant flow rule. Equation (4.15)2 alongwith (4.14)2 define the evolution equation for the back stress as q = 72Hn; a relationknown as the Prager-Ziegler kinematic hardening rule already discussed above; K"(E)


and H are referred to as the isotropic and kinematic hardening moduli, respectively.Since IP I = y, relation (4.15)3 implies

,(xt) = 3P(x, T) d, (4.16)

which is in agreement with the usual definition of equivalent plastic strain. Alterna-tively, one may use the notion of equivalent plastic work; e.g., see NAGHDI [1960],KACHANOV [1974] or MALVERN [1969], to characterize hardening. In applicationsto metal plasticity it is often assumed that the isotropic hardening is linear so thatK(() = K 2 . The following form of combined kinematic/isotropic hardening lawsis widely used in computational implementations

H = (1 - f)H, K'(() = /3H, p E [0, 1] and H = constant. (4.17)

More generally, nonlinear isotropic hardening models are often considered in whicha saturation hardening term of the exponential type, as in VOCE [1955], is appendedto the linear term, i.e.,

K'() = PHf + (Koo - Ko) [1 - exp(-6S)], (4.18)

where H > 0, Koo > Ko > 0 and 6 > 0 are material constants. Finally, theelastoplastic tangent moduli in plastic loading are obtained by particularization of(3.7) as

CeP= 1 X1+2P [Ip- 1& n for y > 0. (4.19)L 1 + H+K"J

(D) Plane stress J2 -flow theory. In this example classical J2-flow theory with com-bined kinematic and isotropic hardening is cast into the general format of the quadraticmodel described above under the assumption of plane stress. Recall that the planestress subspace is defined by Sp = {o E S: c13 = 23 = 33 - 0}. Since Sp isisomorphic to Ri3, we introduce the following vector notation:

= [11 22 c12]T and q = [qll q22 ql 2]T. (4.20)

Note that 33 = -(qll + q22) since the back stress tensor qij is deviatoric. Followingstandard conventions, we collect the components Eij of the strain tensor E S invector form as

E = [1 22 212]T and EP = [EP EP2 2 P2]T (4.21)

The standard convention of multiplying the shear strain component E12 by a factor oftwo allows us to write the stress power as rijij = aTg. To formulate the plane stress


version of J2-flow theory directly in Sp, observe that the von Mises yield conditioncan be written as

f(O, , q) = -/[TY - q] < O,(4.22)

2 - 06P= -1 2 0,

0 0 6

where = r - q is the relative stress. Noting that the mechanical dissipation can bewritten in terms of the vector notation introduced above as D = TkP + qT + q J,the specialization to plane stress of the basic equations of three-dimensional J2-flowtheory takes the form

= Y7P3, = -yP,3 and =y 3 /23Tp/3,

where we have replaced y > 0 by y/ /3TP/3. The generalized stress-strain relationstake the form

O = CIE - EP], q =-H and q = -K'((). (4.23)

Here C and H are the elastic and plastic moduli in plane stress which, with thepreceding conventions, take the following form for the isotropic case:

C 1 = 2 V 1 0 and H= 2 H 1 ,1 (4.24)(- 0 0

where E > 0 is the Young's modulus and v E [f, ] is the Poisson's ratio. Theelastoplastic moduli can be easily obtained either by specialization of the general resultin (3.7) or via a direct computation. To facilitate comparison with future algorithmicdevelopments, the explicit result is quoted below under the assumption of plasticloading:

CP/ 3® CP/Cep C C P C (4.25)/ T[P(C + H)P + 2K"( )P] D

Equations (4.22) and (4.23) are now in a form which is ideally suited for the applicationof the general return mapping algorithms discussed in the next chapter.

REMARK 4.4. For the case of isotropic elasticity the constitutive matrices C, H andthe projection matrix P have the same characteristic subspaces and, therefore, can besimultaneously diagonalized (H is already diagonal); see SMO and TAYLOR [1986]for the explicit result. Since P and C have the same eigenvectors, it follows that P,C and H commute.


REMARK 4.5. It should be noted that the strain components 33, 33, E3P3 and q33 donot appear explicitly in the formulation. These are dependent variables obtained fromthe basic variables (, EP, o), the plane stress condition, and the condition of isochoricplastic flow. For the case of isotropic elasticity we have

E33 = - (El + E22),

(4.26)EP3 = ( +P2) and q33 = -(qll +q22) ·

The total strain E33 then follows simply as E33 = E3 + EP3.

5. General rate-independent multisurface plasticity

In many application of interest, the boundary aE of the elastic domain E is nonsmooth.A classical example is furnished by the Tresca yield criterion of metal plasticity, seeHILL [1950], which can be visualized as a hexagon on the 7r-plane. Yield criteriawidely used in solid mechanics, such as the Drucker-Prager condition or the so-calledcap models introduced in DIMAGGIO and SANDLER [1971] and SANDLER, DIMAGGIOand BALADI [1976], are constructed as the intersection of several yield surfaces lead-ing, therefore, to a nonsmooth boundary aE. An overview of possible applications ofthese in geomechanics is given in DESAI and SIRIWARDANE [1984]. Multisurface plas-ticity also arises in a class of phenomenological models for metal plasticity known ascomer theories, see, e.g., CHRISTOFFERSEN and HUTCHINSON [1979], which attemptto replicate experimentally observed results not explained by classical J 2-flow theory.A modified J 2-flow theory that partially accounts for these effects is given in SIMO[1987]. Multisurface plasticity also plays an important role in formulations of crys-talline plasticity due to the presence of multiple slip systems, see the review articleof ASARO [1983].

In this more general setting of multisurface plasticity, the elastic domain E is aconvex set defined in terms of smooth convex functions f,:S x IRin't -i R, with, C { 1,... , m}, as the constrained set

E = { = (, q) S x Inint: f(X) < 0, forp = 1,..., m}. (5.1)

It is implicitly assumed that the constraints f,(Z) are qualified; a condition whichholds if either the constraints are affine or if there exists a point ± E S x IRnT ' such thatf () < O0 for = 1,... ,m, see CIARLET ([1988], p. 345) for further details. Thistechnical requirement is satisfied by all elastic domains used in classical plasticity.

For convenience, in what follows we shall refer to Z = (, q) E as the gen-eralized stress. The smoothness hypothesis on the functions f,(Z) along with theconvexity assumption imply the following standard result (see, e.g., CIARLET [1989],p. 243)

(5.2)


f,(T) - f,(-) >, [T - _] Vf,(), T,.E c x R'i'",.

f

FIG. 5.1. Illustration of the convexity property for a smooth one-dimensional function f: IR -- R (ndim = ).

An illustration of this property is given in Fig. 5.1. The associative form of theflow rule and the hardening law for multisurface plasticity is given by the followingevolution equations which go back to KOITER [1960] (see also MANDEL [1965]):

r=l m

Ep= 'yPa.f',(oq), ( =aq),

(5.3)

yU > 0, f,(a, q) 0 and yf,( , q) = 0, for =1,...,.P=1

As before, y* > O0 are the plastic consistency parameters and relations (5.3)3 definethe Kuhn-Tucker form of the loading/unloading conditions. A deeper motivation forthese evolution equations is deferred to the following section.

The elastoplastic tangent moduli in the incremental relation = Cepe are obtainedby a generalization of the consistency argument described in Section 3. This gener-alization requires a precise definition of the active set of constraints during plasticloading. For this purpose, let

JIt = {t E {1, *, m}: f(a, q) = 0} (5.4)

denote the set of indices containing the constraints which have the possibility of beingactive at (a, q) aE and set mtr = dim[Jt]. The subset J C Jt' of active constraintsis clearly defined by the condition

= E Jt': f?(r, q) - O. (5.5)

Our goal is to provide an explicit characterization of this set. Using the chain rule andthe (generalized) stress-strain relations, it follows that

(5.6)


f, = af4 · C[i - P] - aqfo Hi.

Now define the mtr(m tr + 1)/2 coefficients G,, = Gm by the relation

Gm , = a,f, · Caaf, + aqf, Haqfv, /' E Jtr. (5.7)

Observe that the matrix G = [G,,] is symmetric and positive definite for eitherperfect or hardening plasticity, with inverse denoted by G- 1 = [GIm]. For softeningplasticity, the requirement that this matrix remain positive definite is analogous toenforcing Assumption 3.1. Inserting the evolution equations (5.3) into (5.6) and settingotr = C gives

f = afl t r- G>f ,,' for J. (5.8)

By enforcing the consistency condition (5.5), the plastic multipliers are explicitlycomputed as

Y7 = G °af, . tr for Jtr, (5.9)veJ

tr

and the set J of active constraints becomes J = {/ E Jt: 7y > 0}. From this resultand (5.9), the elastoplastic tangent moduli are computed as

cep = C -- E E GI"'[Ca,f,] [Caf]. (5.10)l~EJ vEJ

The preceding developments have an algorithmic counterpart. The interested readeris referred to SIMO, KENNEDY and GOVINDJEE [1988] for further details.

REMARK 5.1. The set Jtr depends on the specific point under consideration. The ad-ditional assumption that the constraints are independent at each point on the boundaryaE, understood in the sense of linear independence of the vectors a,f, is crucial forexpression (5.10) to remain valid. Without this hypothesis the multipliers y/' > 0 neednot be unique and the matrix [G,,] is generally noninvertible. The importance of theseconditions does not appear to have been appreciated in the literature on plasticity. Fora discussion of these issues in the general context of nonlinear optimization and, inparticular, the role played by the technical hypothesis that assumes the constraints tobe qualified, see KARMANOV [1977] and CIARLET ([1988], Chapter 9).

6. Dissipation: Interpretation of the model of associative plasticity

The associative form of the flow and the hardening laws arise when the functionsr(oa, q) and h(a, q) coincide with the gradients of the yield criterion. A more funda-mental interpretation of this choice arises when the notion of dissipation is introduced.Classical plasticity defines a dissipative dynamical system in the sense that a localfunction of the state variables, called the dissipation function, does not decrease withthe evolution of plastic flow. Given a general elastic domain, among all possible


choices of flow rule, the associative form is that leading to maximal dissipation in thesystem. In essence, this is the contents of a classical result going back at least to VONMISES [1925] (see HILL [1950]) and known as the principle of maximum dissipation.This result is described below after motivating and defining the appropriate notion ofdissipation for classical plasticity.

We shall be concerned with a general model of plasticity in which hardening pro-cesses are characterized by nint internal (stress-like) variables q(x, t). It proves expe-dient to introduce the following notation:

_ = (, q), EP = (EP, ), and E[u] = (E[U],O). (6.1)

We consider a general free energy function 1(x, e, E), convex in (es, 5) and a generalelastic domain defined by a closed, convex set IE C S x RIIn ~ with boundary OlE notnecessarily smooth. The specific examples described in the preceding section providean illustration of this general setting. Here, attention is focused on the general case ofmultisurface plasticity corresponding to

E = C x 'i: f() 0, for ti= l,...,m}, (6.2)

where f,: x in' - 1 are smooth convex functions.Within this general framework, the local internal dissipationfunction D is defined

as the difference between the stress power and the rate of change of free energy, i.e.,

D = a E[v] - dT(Ee, ,), in x . (6.3)

Carrying out the time differentiation via the chain rule and using the generalizedstress-strain relations along with e = E[V] - EP yields the expression

D = - P + q = - EP . (6.4)

The local form of the second law requires that the dissipation inequality D) > 0 holdin any admissible process in the material body. Since the internal variables EP arenot present in elasticity, it is clear that D) O0 for an elastic material, a propertywhich motivates the term "perfect materials" coined in TRUESDELL and NOLL [19651.It follows that classical elasticity trivially satisfies the dissipation inequality.

EXAMPLE 6.1. A more interesting example is furnished by associative plasticity withelastic domain E defined by (6.2) where the convex functions f,(Z) are of the form

fl(I) = (7) -rryf, p = 1,2,. ., m. (6.5)

Here cry > 0 are constants and the functions 5,(-) are convex, homogeneous ofdegree one, so that VO,(Z) · = 5,(L') where Va, = (,, q,). Using this

214 J. C. Simo CHAPTER I

property and the associative flow rule (5.3), the dissipation function becomes

m m

v= y VO . = E 1' [(b - orY) + ay,]A=1 /=1

7 [ + ayl] (6.6)L=1

Hence D = yl A tuyuo> ) 0 since yr/ > 0 and 2=l yf. = 0, as a result of theKuhn-Tucker conditions. The model, therefore, obeys the dissipation inequality.

Let Z denote the actual stress field in 2 for prescribed strain rates E[v] and EP.The evolution of plastic flow is said to obey the local principle of maximum dissipationif, for any (x, t) E Q x II, the following inequality holds:

[T-X] -EP = ( - () P + (p - q) O 0VT = (r,p) E E. (6.7)

The significance of this postulate lies in its intimate connection with the classicalmodel of associative plasticity, as the following result shows.

THEOREM 6.1. The principle of maximum dissipation is equivalent to the associativeflow rule with loading/unloading conditions formulated in Kuhn-Tucker form, i.e.,inequality (6.7) is equivalent to

T

ip = Z'Y"Vf.(Z),tI=l

(6.8)

y" 0, f (Z) I 0, and E yfb() = 0,/z=l

with E defined by (6.2) and an arbitrary convex free energy function @(x, e, E).

PROOF. Suppose that maximum dissipation holds and assume that the elastic domainis given by (6.2). According to standard results in optimization, this means that theactual state is a stationary point of the (unconstrained) Lagrangian

(T, y ) =-T EP + f,(T). (6.9)p1=1

The standard optimality conditions for this saddle-point problem are (see, e.g., CIAR-LET [1989], Chapter 9)

aT(T, Y')T= = -EP + E y7Vfpi(Z) = 0, (6.10)fi=l


along with the Kuhn-Tucker complementarity conditions. Hence maximum dissipationimplies the associative flow rule (6.8). Conversely, if (6.8) holds, contracting the flowrule with [T - ] yields

i[T-] * E· =Z yatVfp(L) *( [T VT E . (6.11)

Using the convexity assumption on the functions f(E) and enforcing the Kuhn-Tucker conditions we conclude that

aUVfi,(z ) [T - Z] 4 -[ f()]/1=1 =1

= 7~f,(T) < 0, (6.12)

since y" > 0 and f,(T) < 0 for any admissible T E. By combining (6.12) and

(6.11) we arrive at the dissipation inequality (6.7).

7. Rate-dependent plasticity: The viscoplastic regularization

The inclusion of rate-effects in classical plasticity leads to the so-called viscoplasticmodels considered subsequently. We restrict our discussion to those mathematical

aspects which are essential for our subsequent algorithmic developments. In particular,

emphasis is placed on the interpretation of viscoplasticity as a penalty regularization ofthe local dissipation function. This point of view, pioneered in the work of DUVAUT andLIONS [1976], is the most useful one in an algorithmic sense. For general treatmentsconcerned with the physical aspects underlying models of viscoplasticity we refer to

PERZYNA [1971] and LUBLINER [1972, 1973], among others.

The construction of viscoplastic models described below, inspired in the approachadvocated in DUVAUT and LIONS [1976], is very closely related to modern algorithmic

treatments of plasticity based on the notion of closest-point projection of the gener-alized stress onto the convex domain E described in the next chapter. For simplicity,attention is restricted to the case of a quadratic free energy function defined as

(E'e ,) = 2e * Ce + 2a.HE, (7.1)

where both the elasticity tensor C and the ni,,n x nin matrix of plastic moduli H are as-

sumed to be constant and strictly positive definite on S and ]R'" T x lR`T , respectively.

In other words, attention will be restricted to hardening plasticity. It therefore fol-

lows that the complementary Helmholtz free energy function, defined via the standard

Legendre transformation as

7(Z) = 2 +C-lo-+ q*-H q,2( )

216 JC. Simo CHAPTER I

(7.2)

is a strictly convex function on S x Rn int. Under the preceding hypothesis 2(T()defines a norm on the finite-dimensional vector space S x Inin' equivalent to thestandard Euclidean norm.

The key feature exhibited by viscoplastic models is that, in sharp contrast with rate-independent plasticity, admissible (generalized) stresses Z are no longer constrainedto lie in the elastic domain E. Stress states may lie outside of the elastic domain as aresult of rate effects and, as time progresses, may relax to the yield surface, often atan exponential rate if the loading is held constant. Plastic flow then becomes a rate-dependent process. To incorporate this fundamental effect in the model one proceedsas follows.

Step 1. Consider an arbitrary stress state Z E $ x Rnint, not necessarily lying in E,and define the functional J: S x ]Rni - by the constrained minimization problem

J(Z) = min{/2 2( -T), for all T E}. (7.3)

It follows that J(Z) gives the distance measured in the complementary Helmholtz freeenergy (.) between a given point 7 and the convex set E. Clearly, J(Z) > 0 forany Z E x ]Rin t, and J(r) = 0 iff £ E IE. In view of the preceding considerationswe have

J(') = V/23(- ), X c IE, (7.4)

where X. denotes the closest-point projection of Z onto E in the norm 2(.). Bythe projection theorem (see, e.g., CIARLET [1989], Chapter 8), this point is unique.

Step 2. Let y+:IR - IR+ U {0} be a nonnegative, C'-convex function with theproperty

y+ (x ) > and y+(x) = x = 0. (7.5a)

Consider the following viscous regularization of the dissipation function D = E .EP:

-D = -z. EP + 1 -+ (J(r)), (7.5b)

where E (0, o) is a regularization parameter with dimensions of time, referred toas the characteristic relaxation time in what follows. A possible choice for y+(-),widely used in nonlinear optimization, is

_ {x 2 if x O, d( ={ ifx 0 (77+

'( X) = 2 0, so that +() ifX 0 (7.6)

0 if < 0, if X<0


By standard results in convex optimization, see, e.g., LUENBERGER [1984], it followsthat the problem of minimizing minus the regularized dissipation -D over all un-constrained stresses E S x IRn'tn is simply the penalty regularization of the classicalconstrained principle of maximum dissipation, with dissipation function D.

THEOREM 7.1. The optimality conditions associated with the unconstrained maximiza-tion of the regularized dissipation D, yield the following viscoplastic flow rule

= { g(J(17))G-' [a - 1 if J() > (77)0 if J(17) = 0,

where g(x) = dy+(x)/dx and G = DIAG[C, HI are the generalized moduli.

PROOF. The result follows from a computation of the gradient of J(-) that uses relation(7.4). Further details are omitted.

In summary, as a result of the preceding construction a model of associative vis-coplasticity is completely defined by specifying the elastic domain E, the free energyfunction or, equivalently, the complementary Helmholtz free energy function _(X)and the function g: i -- ]R+ that characterizes the rate-dependent flow rule. Observethat the Kuhn-Tucker optimality conditions of rate-independent theory are now re-placed by the conditions in (7.7). The two-step construction outlined above will beillustrated in the context of the classical model of J2 -flow theory described in thepreceding section.

8. Example: Viscoplastic regularization of J2-flow theory

Consider the model of J2-flow theory described in Section 4 incorporating bothisotropic and kinematic hardening, assumed to be linear for simplicity. In the presentcontext, the complementary Helmholtz free energy function is given by

2 (lH) -= I2 trj]| 12 l (8.1)

where > 0 is the bulk modulus, bM > 0 is the shear modulus, K is the isotropichardening modulus, and H is the kinematic hardening modulus. The yield conditionis the von Mises yield criterion

f(E) = dev[ ] - l [- [ y-- q < 0. (8.2)

Our goal is to carry out explicitly the two steps in the construction outlined aboveleading to the regularized flow rule in (7.7).



FIG. 8.1. Geometric interpretation of the closest-point projection of a stress point onto the elastic domainfor J2 -flow theory in the absence of hardening mechanisms (perfect plasticity). The notation s = dev[ar]

is used.

Step 1. Given Z = (a, q, q), the projection X. = (, , q,) onto E and thedistance J(Z) to E are computed from the optimality conditions of the Lagrangian

£(27, X,) = S( - 2,,) + A.f('*). (8.3)

Figure 8.1 provides an illustration of the significance of 7 and J(Z) in the simplercase of perfect plasticity. Since

dev[,] - q/af( ) = -af(E.) = n, = dev[u*.] - (8.4)

with In. I = 1, a straightforward calculation determines the optimality conditions as

dev[a,,] - dev[o] = -2/ A. n.,

* - = 3 2H An*, (8.5)

q.-q = -V-K A,

along with A,, > 0 and A*f('*.) = 0. We assume the nontrivial case A* > 0 so thatf(2*) = 0. Clearly, the point 7.* is completely determined from the multiplier A*and the unit vector n,.

(A) Determination of n* and A* > 0. From Eqs. (8.5)1,2 we conclude that

[Idev[a] - ] n = [dev[a*] - q* 1 + A* (2/ + H)] n, (8.6)

where n is the unit vector also given by expression (8.4)3 but with (ar, q*) replacedby the given (a, q). This relation determines n, as

dev[a]- (n = n dev[] - (8.7)

Idev[o,] - 41'

SECTION 8 219

Using this result, relations (8.6), (8.5)3 along with the von Mises criterion (8.2) yield

f(*) = f(Z) - (2t + H+ K)X,. (8.8)

Therefore, by enforcing the condition f(17) = 0 one arrives at the following explicitexpression for the multiplier A,:

)* = f (Z)/(2p + K + H). (8.9)

The projected point , is then defined by relations (8.5) along with (8.7) and (8.9).

(B) Computation of the distance J(Z). From the optimality conditions (8.5), thecomplementary Helmholtz free energy function of the difference .7 - Z,, is easilyevaluated as

2;( - 1.) = (2 + K + ~H)A2.. (8.10)

Inserting (8.9) into this expression yields

J(Z) = /2 - ( - ) = ()/ /(2 + K + H). (8.11)

Step 2. Adopting for simplicity expression (7.6) for the function y+(), the vis-coplastic regularization of the dissipation can be written as

D,r = -. EP - [f(_)] 2 , for f(7) > 0, (8.12)

where 7 > 0 is a viscosity coefficient (with dimensions [stress] x [time]) defined interms of the relaxation time r as

1 = (2 + K + H) . (8.13)

Finally, for the present model problem the general viscoplastic flow rule takes thefollowing explicit form:

ep= 1(f(Z))n,

I .1(8.14)=--g(f())n, and = -g(f()) , (8.14)

37 37

where g(x) = [x + z[x] is the ramp function.

We observe that the evolution equations (8.14) can be obtained from their coun-terparts (4.15) in the rate-independent theory merely by providing the constitutive


equation y = g(J())/rl for the consistency parameter and omitting the Kuhn-Tucker conditions. This idea appears to have been suggested by PRAGER [1956], seePERZYNA [1971]. From well-known results in penalty methods for constrained opti-mization (see, e.g., LUENBERGER [1984] and BERTSEKAS [1982]), one concludes thatthe rate-independent model can be recovered as the limit of zero viscosity ( - 0);an obvious property from a physical standpoint.

REMARK 8.1. Early algorithmic treatments of plasticity often exploited the view ofviscoplasticity as a penalty regularization of classical plasticity as a means of de-vising numerical schemes for the rate-independent problem, as in ZIENKIEWICZ andCORMEAU [1974], CORMEAU [1975], HUGHES and TAYLOR [1978] and PINSKY, ORTIZand PISTER [1983]. This approach, however, has the disadvantage of inheriting thecharacteristic ill-conditioning experienced by penalty methods for very high values ofthe penalty parameter and are considered no longer efficient.

REMARK 8.2. From an algorithmic standpoint the advantage of the class of modelsdescribed above, which are based on the regularization suggested in DUVAUT andLIONS [1976], is that the characteristic ill-conditioning experienced by penalty meth-ods can be completely circumvented. As observed in SIMO, KENNEDY and GOVIND-JEE [1988], the key idea is to compute the viscous (rate-dependent) solution fromthe rate-independent solution and not vice versa. The numerical approximation tothe rate-independent problem is obtained via return mapping algorithms described insubsequent chapters.

9. The weak formulation of dynamic plasticity

The weak form of the momentum equations, briefly described in Section 1, alongwith the constitutive equations formulated as a variational inequality comprise theweak formulation of the initial boundary value problem for classical plasticity andviscoplasticity. The goal of this section is to provide an account of those mathematicalaspects involved in the weak formulation of the problem which are relevant to itsnumerical approximation. Throughout this section attention is restricted to the case ofa quadratic free energy function defined by (7.1). All the results are easily extendedto an arbitrary convex stored energy function.

The weak form of the constitutive equations for associative plasticity, the situationof interest here, is simply the global formulation of the maximum (plastic) dissipa-tion inequality (6.7) over the entire domain Q2 of the body. Denote by (ij,pi) thecomponents of T = (r, p) relative to the standard basis and let

T = {T = (r,p): $ X T/x "int: Tij G L 2(Q) and pi L2(0)}. (9.1)

By eliminating the generalized plastic strains via the constitutive relations, inequality(6.7) for the rate-independent problem can be written as

(r-a)., e[v]-(r-a) .C- - (p-q) .H-'<0 V(T, p) EE. (9.2)


CHAPTER I

/

trial := (C: E[t], 0)

.E

FIG. 9.1. Geometric interpretation of the variational inequality A( t r - T, ,T- E) 0, for E =

{E: f() < 0%

Motivated by this expression, one defines the bilinear forms a(-, ) and b(-, ) as

a(,r) = J C-' lQ, b(q,p) = q HpdQ. (9.3a)

Since the elastic moduli C are positive definite on and the plastic moduli H areassumed positive definite on ni t, it follows that the bilinear forms a(., ), and b(-, )are coercive. Consequently, the bilinear form A(., ) : T x T - defined by

A(, T) = a(o, r) + b(q, p) (9.3b)

induces an inner product on I. The square of the induced norm, A(, E), of anygeneralized stress field Z E IE is precisely twice the integral over the body of thecomplementary Helmholtz free energy function E(L). With this notation in hand, theglobal version of the maximum dissipation inequality (9.2) for the rate-independentproblem takes the form of the following variational inequality:

A(~tr- , T- ) 0, v E EnT,

where ttr= (Ce[Ui], 0). (9.4)

Recall that Ztf is the trial stress rate about X! E E, obtained by freezing plastic flow(i.e., by setting P = 0 and = 0).

The geometric interpretation of inequality (9.4)1 is illustrated in Fig. 9.1. The actualstress rate Z is the projection in the norm defined by the bilinear form A(-, .) of thetrial stress rate ±tr onto the tangent plane at Z E aE. Convexity of E then impliesthat the angle measured in the inner product defined by A(-, ) between [tr -_ I]and [T - ] is greater than or equal to 90° for any T E E; a condition equivalent to(9.4)1.

222 J.C. Simo


The weak form of the momentum equations (1.16a) together with the dissipationinequality (9.4) give rise to the following weak formulation of the initial boundaryvalue problem for rate-independent plasticity:

Problem Wt: Find t E Il H4u C St and t e I1 - = (, q) E E n T such that

(p(v - it), n) = O V E V,

(pov, 1 ) + (,e E[]) - (f,r) - (t,7)r = 0 Vr E V, (9.5a)

A('t - , T - E) 0 Vr E n T,

subject to the initial conditions

(u(-, O), 7) = (Uo, 7) and (v(.,O), ) = (vo, q), V77, V. (9.5b)

REMARK 9.1. For the rate-dependent problem inequality (9.4) is replaced by the weakformulation of the maximum dissipation inequality associated with the regularizeddissipation (7.5b). The dissipation over the entire domain is

J Dd =A( ,y+ (J())d > 0, (9.6)

and the assumption of maximum dissipation implies the following inequality for vis-coplasticity:

A( tr - ,T- ) 1 r [y+(J(T)) - y+ (J(Z))] d, (9.7)

for all r c T not necessarily contained in E.

The weak form (9.5a) of the dynamic elastoplastic problem involves the velocityand the generalized stress field. In the subsequent analysis it is convenient to use thenotation

X = (v, Z) C Z where Z = V x [TnE I] (9.8)

is the space of admissible velocities and admissible generalized stress fields. Thespace Z is equipped with a natural inner product induced by the kinetic energy andthe complementary Helmholtz free energy function, which is denoted by ((., .)) anddefined by

((XI, X2)) = ((POrl, r12)) + A(T, T2). (9.9)

The associated natural norm is denoted by |Il -| || = ) and is interpretedas twice the sum of the kinetic and Helmholtz energies of the elastoplastic body. It

SECTION 9 223

will be shown below that the norm | | *)) is the natural norm for theelastoplastic problem.

REMARK 9.2. The variational formulation of plasticity given by equations (9.5b) goesback to the pioneering work of DUVAUT and LIONS [1976] and has been consideredby a number of authors, in particular, JOHNSON [1976a, 1978]. The assumption ofhardening plasticity, i.e., the presence of the bilinear form b(., ) defined by (9.3a)2,is crucial for the functional analysis setting outlined above to remain applicable.

REMARK 9.3. The situation afforded by perfect plasticity is significantly more com-plicated than hardening plasticity since the regularity implicit in the choice of Stin (1.14) no longer holds. The underlying physical reason for this lack of regular-ity is the presence of strong discontinuities in the displacement field, the so-calledslip lines, which rule out the use of standard Sobolev spaces. For perfect plasticitythe appropriate choice appears to be St = BD(S2), i.e., the space of bounded de-formations [displacements in L2 (2) with strain field a bounded measure] introducedby MATTHIES, STRANG and CHRISTIANSEN [1979], and further analyzed in TEMAMand STRANG [1980]. See MATTHIES [1978, 1979], SUQUET [1979, 1981], STRANG,MATTHIES and TEMAM [1980], and the recent comprehensive summary in DEMENGEL[1989] for a detailed elaboration on these and related issues. Very little is currentlyknown for the case of softening plasticity, other than the solutions must exhibit strongdiscontinuities (see SIMO, OLIVER and ARMERO [1993] and references therein). Thesediscontinuities are intimately related to the formation of shear bands.

10. Contractivity, uniqueness, and dissipativity of the elastoplastic flow

Two properties of the initial boundary value problem are examined below in somedetail which are of fundamental importance in the formulation and analysis of algo-rithms. First, the dissipative character of the problem, which translates into an a prioristability estimate for the solutions. Second, the contractivity property of the solutionsrelative to certain norm, called the natural norm of the problem, which motivates thenotion of algorithmic stability employed in the subsequent analysis of return mappingalgorithms.

The following contractivity property inherent to problem Wt identifies the normII · III induced by the energy inner product (9.9) as the natural norm and plays,

therefore, a crucial role in the subsequent algorithmic stability analysis. Let

Xo = (vo, o) Z and Xo = (iio, ) E Z (10.1)

be two arbitrary initial conditions for the problem of evolution defined by (9.5a) anddenote by

tC]Ik-X=z(v,L)CZ and t E:1j X- = (0, .) Z


(10.2)


the corresponding flows generated by (9.5b) for the two initial conditions (10.1),respectively. For the quasistatic problem, the following result is implicitly containedin MOREAU [1976, 1977]; see also NGUYEN [1977].

THEOREM 10.1. Relative to the norm || 1 II induced by the inner product (9.9) thefollowing contractivity property holds:

IIllx(-,t)-(-,t)lll < IllX0-xoll, VtE . (10.3)

PROOF. By hypothesis, the flows in (10.2) satisfy the variational inequality (9.5a) 3. Inparticular,

A(±, s- ) < A(±Lr, S - 5),1(10.4)

-A(X, - ) < -A(tr, - )

Adding these inequalities and using bilinearity, along with definitions (9.3a)l and(9.4)2 yields

A(- , L- ) < A( - ,H, L - ) = (e[v- i], - ). (10.5)

Now observe that, for fixed (but arbitrary) t E 1, v(., t) - (., t) E V. Using thelinearity of (-,-) and the fact that the flows (10.2) satisfy the momentum equation(9.5a) 2 gives

(a, E[V - ]) - (, ey - ]) = -(po - , - = -dK(v - ), (10.6)

so that the right-hand side of (10.5) equals minus the rate of change in the kineticenergy of the difference v - . Combining (10.5) and (10.6), along with the definitionfor the natural norm, yields

dK(v - ) + ((E- , - )) = allX- X il 2 < 0, (10.7)

which implies dlHX - tl[j/dt < 0. Therefore, for any t E 1[,

I lixo- - XI - IIx0 - x0ol1 = ~ iIx( ,- ) - y(', T)II dr 0, (10.8)

which proves the result.

An immediate consequence of contractivity is the following uniqueness result forthe initial boundary value problem of infinitesimal elastoplasticity.

THEOREM 10.2. The stress and velocity fields in the solution of the initial boundaryvalue problem of elastoplasticity are unique.

SECTION 10 225

PROOF. Suppose that X = (v, 27) and X = (, ±) are two solutions of the systemassociated with the same initial data Xo. Inequality (10.3) then yields 0 < lIX-Xl I 0 for all t C , which implies that X = X since Il I III is a norm.

The contractive property of the elastoplastic problem relative to the natural normIlI Il motivates the algorithmic definition of nonlinear stability employed subsequently.

REMARK 10.1. An identical contractivity result holds for classical viscoplasticity. Thecorresponding initial boundary value problem is obtained by replacing inequality(9.5a) 3 with (9.7). The flows t C I - 27 EC T and t E II - C E associatedwith the two initial conditions (10.1) then satisfy

A(±,2 -) < A( -tr, - + [g(J()) - g(J(2))] drQ

(10.9)

-A(Z,2 - 2;) <-A(, - )-7 / Q [g(J(Z))- g(J(2;))] dQ.

Adding these inequalities yields again (10.5) and the contractivity result for viscoplas-ticity follows exactly as in the proof of result (10.3).

A fundamental property of the elastoplastic flow closely related to contractivity isprovided by the notion of dissipativity, which translates into an a priori energy-likestability estimate for the solutions of the initial boundary value problem. The totalinternal free energy in the continuum body at any time t C II is given by the integral

Vint(, (E ) = / TI(x, E, ,) d2. (10.10)

By integration over the reference placement of the local (instantaneous) mechanicaldissipation SD at any (x, t) cE 2 x II one obtains the total dissipation in the continuumDr2 at time t as

DQ = Pint(v) - dtVint (10.11)

where Pi,, is defined by (1.18)2. The inequality DQ2 > 0 for all times t GE 1 is acceptedas a basic principle (see TRUESDELL and NOLL [1965] for a detailed discussion).A mechanical system for which D• -0 for all t G II is said to exhibit no dissipation.Otherwise, the system is called dissipative. From the foregoing discussion it is clearthat classical plasticity and viscoplasticity are dissipative systems while elastodynam-ics is not.

If the external loading is conservative with potential Vext: St , the power Pextof the applied loads satisfies the relation

Pext(v) = - Vext() = -VVext(U) V. (10.12)dt~~~~~~~~~~~~~~~1.2



The total potential energy in the system is then given as the sum of Vext and Vit. Thetotal energy E in the system, obtained as the contributions of the potential energy andthe kinetic energy K defined by (18.1)1, then satisfies the following a priori decayestimate.

THEOREM 10.3. The total energy E(u, v, se, E) = Vext(u) +Vint(E, 5) +K(v) decaysat the rate

d E(u, v, e,) = -D < 0 Vt E1. (10.13)

Thus, the total energy is a nonincreasing function along the flow.

PROOF. This result is an immediate consequence of the mechanical work identity(1.17), since

dE(uV, e, ) = [K(v) + Vext(U)] nt(e,dt ~dt .)-et(u)] + dt

Fd?-Pi(, v)+ d~,(ee, () (10.14)

The result follows from the definition of DQ in (10.11).

REMARK 10.2. For elastodynamics Dr 0 and (10.13) reduces to the familiar lawof conservation of the total energy. Inequality (10.13) is a formal a priori energyestimate on the solution of the initial boundary value problem that captures its in-trinsic dissipative character. The contractive and dissipative properties of the floware key qualitative features of the dynamics that should be inherited by any mean-ingful algorithmic approximation. Recent work in numerical analysis of dynamicalsystems is aimed at constructing numerical algorithms that preserve key qualitativefeature of the dynamics. See SIMO and TARNOW [1992] and the review in SCOVEL[1991] for the Hamiltonian case; FOIAS, JOLLY, KEVREKIDIS and Trri [1991], STUARTand HUMPHRIES [1993], and SIMO and ARMERO [1993] for representative numericaltreatments of dissipative systems.

SECTION 10 227

CHAPTER II

Integration Algorithms for Classical

Plasticity and Viscoplasticity

The numerical solution of nonlinear initial boundary value problems in solid me-chanics involving inelastic response is based on an iterative solution of a discretizedversion of the momentum balance equations arising from finite element/finite differ-ence procedures. In most of the computational architectures currently in use, the weakform of the stress divergence term is evaluated via numerical quadrature and the evo-lution equations defining the inelastic constitutive response are enforced locally, asa system of ordinary differential equations at each quadrature point. The integrationin time of this local system, typically stiff and subject to algebraic constraints, maybe regarded as the central problem of computational plasticity as it corresponds tothe main role played by constitutive equations in actual computations. A key featureof this problem, illustrated in Fig. 10.1 and justified below, is that it can always beregarded as strain driven in an algorithmic context in the sense that the state variablesare computed for a given deformation history.

The role played by the local integration of the plastic flow evolution equations isillustrated in Fig. 10.1.

The evolution equations of classical elastoplasticity, as summarized in Table 3.1,define a stiff differential algebraic system subject to unilateral constraints. By applica-

Strain Increment Aen(x)

RETURN({ n(z) En(), An(Z) } > MAPPING { n+l(Z), EPn+l(Z), °"n+ () }

ALGORITHM

FIG. 10.1. The role of elastoplastic return mapping integration algorithms. The stress tensor a is viewedas a dependent variable, defined in terms of ee by means of elastic strain relations, which (a) is computed

only at quadrature points and (b) is computed for prescribed total strain.

229

tion of a suitable integrator in time, the numerical integration of this system reducesto a constrained optimization problem governed by discrete Kuhn-Tucker optimalityconditions. Below we describe the structure of this discrete problem, the fundamentalrole played by the discrete Kuhn-Tucker conditions, and the geometric interpretationof the solution as the closest-point projection in the energy norm of the trial elasticstate onto the elastic domain. In particular, because of the contractive and dissipativeproperties of the initial boundary value problem, conventional higher-order methodsdo not necessarily lead to improved numerical performance. These and related issuesinvolved in the design of numerical algorithms are motivated within the context ofthe classical model of J2-flow theory, subsequently generalized to viscoplasticity. Thechapter concludes with a complete nonlinear stability analysis of the algorithmic prob-lem, which exploits in a crucial manner the contractive and dissipative properties ofthe continuum problem, along with a brief exposition of mixed finite element methodsfor the elastoplastic problem.

The developments in this chapter provide a unified treatment of a number ofexisting algorithmic schemes which, starting with the classical radial return algo-rithm of WILKINS [1964], have been largely restricted to J2 -flow theory. Represen-tative algorithms include the extension of the radial return method in KRIEG andKEY [1976] to accommodate linear isotropic and kinematic hardening, the mid-point return map of RICE and TRACEY [1973], the extension of these ideas tothe plane stress problem in SIMO and TAYLOR [19861, and alternative formula-tions of elastic-predictor/plastic-corrector methods given in KRIEG and KRIEG [1977].To a large extent, return mapping algorithms of the type described below, or"catching-up" algorithms in the terminology of MOREAU [1977], have replacedolder treatments based on the elastoplastic tangent modulus, as in ARGYRIS [1965],MARCAL and KING [1967], NAYAK and ZIENKIEWICZ [1972] or HINTON andOWEN [1980]. Because of the significant industrial applications, these methodshave received considerable attention over the last decade in the engineering litera-ture.

The now standard return mapping algorithms for classical plasticity are based onthe backward Euler method. Attempts to generalize these techniques to higher-ordermethods have found mixed success, see, e.g., ORTIZ and POPov [1985], SIMO andGOVINDJEE [1988] and the follow-up work in CORIGLIANO and PEREGO [1991]. Thenew schemes described below generalize these algorithms within the context of theextremely convenient backward difference methods or ideas emanating from projectedimplicit Runge-Kutta methods widely used for stiff problems.

11. Local integration of rate-independent plasticity: An overview

To appreciate the role played by numerical integration of the evolution equations ofclassical plasticity and viscoplasticity, an informal outline is given below of the keysteps involved in a numerical solution of the elastoplastic initial boundary value prob-lem. The local form of the problem of evolution to be integrated numerically is sum-marized next to set the stage for the general class of algorithms described in the subse-quent sections. The feature that differentiates this problem from a conventional system

230 J.C. Simo CHAPTER II

Integration algorithms

of ordinary differential equations is the presence of an algebraic constraint, which isdefined by the Kuhn-Tucker conditions, leading to a differential algebraic system.

(A) The role of return mapping algorithms. The weak form of the momentum equa-tions, supplemented by the history-dependent constitutive equations, comprise theinitial boundary value problem for classical plasticity. Current solution strategies incomputational inelasticity adopt a local formulation of the evolution equations thatdefine the constitutive model as a system of ordinary differential equations at eachpoint x e Q2. Solution schemes based directly on a weak formulation of the consti-tutive equation, although not common, are also possible; see, e.g., JOHNSON [1976b,1977] or SIMO, KENNEDY and TAYLOR [1988]. The iterative solution procedure forthe elastoplastic problem involves two phases as described below.

Phase I. Spatial discretization of the domain of interest Q by a finite elementtriangularization Tt. The solution space St and the space V of test functions areapproximated via finite-dimensional spaces to arrive at the discrete counterpart of theweak form of the equilibrium equations. In doing so, one is confronted with evaluatingthe finite element approximation to the weak form of the divergence term. The keyobservation is that, at this stage, the stress field is prescribed via the computationperformed in Phase II.

(i) Consider the contribution of a typical element Qe computed via numericalquadrature. From the quadrature rule it follows that knowledge of the stress field is required only at the quadrature points of the element. This information is providedby the local return mapping algorithm outlined below and allows the evaluation ofthe element contribution to the weak form.

(ii) The contributions of all the elements are assembled via a standard assemblyalgorithm into a global residual vector which depends nonlinearly on the displacementfield. If the norm of this residual vector is not below a specified tolerance, the systemis linearized about the current iterate and a correction to the displacement field iscomputed.

Phase II. Time discretization of the interval of interest = Un= 0 [t", t,+l]. Withina typical time step [t, tn+l] one is given as initial data, at the quadrature points ofa typical element Q2, the internal variables (eP , ~n). The key observation is that, atthis stage, the displacement field is prescribed at the current iterate found in Phase I.

(i) Consider a specific quadrature point x. of a typical element Q2e and evaluatethe current strain field E[uh+l] at x. E •?e via the chosen finite element interpolation,for prescribed nodal displacements.

(ii) Evaluate at x. the stress field ,n+l (xz) and the internal variables via the returnmapping algorithms described below. The problem then reduces to the integration forprescribed strain field of the differential algebraic system defined by the flow rule andthe hardening law at the quadrature points of each element in the triangulation.

Phase I is accomplished by the standard finite element method, see, e.g., CIAR-LET [1978] or JOHNSON [1987], and typically remains unchanged even if modelsdifferent from those described in the preceding chapter are employed. Phase II, onthe other hand, depends crucially on the particular constitutive model and constitutes

SECTION 1 231

the central objective of this chapter. A conceptual algorithm that combines these twophases proceeds as follows:

Within each interval [t, t+l] the nodal displacements ds of the triangularizationand the internal variables (Pn, E,) at the quadrature points of each element Q2e aregiven. We then proceed with the following scheme to determine the nodal displace-ments dn+l at time t,+l.

Step 1. Initialization: Define an initial guess of the nodal displacement field d+and compute the associated stress field at quadrature points (Phase II).

Step 2. Residual evaluation and convergence test: Compute the finite element ap-proximation to the weak form for given iterate dn+ l and test the norm of the residualfor convergence (Phase I).

Step 3. Linearization and displacement update: If convergence has not been attained,linearize the residual about current iterate d+l to compute an increment Adk+l.Update the solution as d+l = d+l + d+l(Phase I).

Step 4. Local integration of plastic flow: Compute the stress field ak+l at the quadra-ture points of each element in the elements of the triangularization via a local returnmapping algorithm. Compute algorithmic moduli for Step 3 above (Phase II). Returnto Step 2.

Convergence of the preceding incremental solution procedure can be easily estab-lished under the assumption of convexity. These issues have been recently addressedin MARTIN [1988] and COMMI and MAIER [1989].

(B) The local strain driven differential algebraic system. To provide a concise state-ment of the local problem of evolution to be treated numerically, it proves convenientto make use of the compact notation introduced in the preceding chapter and summa-rized below for convenience:

X= (, q), E= (e, 0),

EP (EP, ), and G =DIAG[C,H]. (11.1)

For simplicity, attention will first be restricted to the case of a single surface elasticdomain E = {2: f(X) < O}. Let

N

X-- U [tntn+II1r0n=O

be a partition of the time interval of interest and xa: G a fixed, but arbitrary pointin the reference placement, identified with a quadrature point in a finite elementtriangularization. Assume that at time t, E and at points x E 2 (to be subsequentlyidentified with the quadrature points in a Galerkin finite element discretization) one


is given the internal variables EP = (eP, ,) and the strain history {ek(x): k =0, 1,..., , n + 1}. The problem to be addressed is the integration (for prescribed E) ofthe system

= G[E - EP],

EP= fyVf(), (11.2)

f(Z) < 0, y > 0, and -yf( )= O.

The simple appearance of this problem is deceiving. The key feature that rendersmatters nontrivial is the appearance of the additional algebraic constraint defined by theKuhn-Tucker conditions. Problems of this type are referred to as differential algebraicsystems. The monograph of BRENAN, CAMPELL and PETZOLD [1989] and the book ofHAIRER and WANNER [1991] provide excellent introductions to this extensive subject.

A first approach to the solution of (11.2) is to eliminate the multiplier y > 0 bydifferentiating the constraint. This yields

Vf().· GEly Vf(Z) . GE if Vf. GE > 0, (11.3)

-f(V) GVf()

i.e., for plastic loading, and y = 0 otherwise. Substitution of this result then yields theconventional rate problem already considered in the preceding chapter, which can betreated by conventional methods. This is, in fact, the approach taken in early treatmentsof elastoplasticity. The well-known drawback is that the algorithmic solution fails tosatisfy the constraint. Matters worsen considerably as time progresses.

Since the constraint need only be differentiated once to eliminate the multiplier, thisdifferential algebraic system is said to have index one; see BRENAN, CAMPBELL andPETZOLD [1989]. Two basic techniques for the solution of stiff differential algebraicproblems are

(i) Linear multistep methods and, amongst them, the so-called backward differencemethods first applied to differential algebraic systems in GEAR [1971]. The lowest-order backward difference method is the backward Euler method: a very stable scheme(it is simultaneously A-stable, B-stable and L-stable; see HAIRER and WANNER [1991]).Backward difference methods are widely used for stiff differential algebraic systems.

(ii) Implicit Runge-Kutta methods. These are one-step methods which, in sharp con-trast with linear multistep methods, are not subject to the so-called Dahlquist barrierswhich limit to two the order of accuracy of A-stable linear multistep methods. In fact,there exist implicit Runge-Kutta methods of arbitrarily high order which are A-stable;see, e.g., HUNDSDORFER [1985] and HAIRER, NORSETT and WANNER [1993]. In spiteof their excellent stability properties, implicit Runge-Kutta methods are cumbersometo implement and rather expensive compared to linear multistep methods.

As alluded to above, the key issue in the numerical integration of (11.2) is relatedto the enforcement of the constraint. To gain insight into the issues involved in thenumerical approximation of (11.2), we consider a model problem that plays a centralrole in classical plasticity.

SECTION 12 Integration algorithms 233

12. Motivation: The perfectly plastic model of J2-flow theory

To motivate the general class of algorithms presented in the subsequent sections, threespecific schemes are described in detail below within the setting of the perfectly plas-tic model of J2-flow theory. The goal here is to identify a number of key featuresspecific to the problem at hand, and to gain insight into a number of possible algo-rithmic strategies. Moreover, the early work of WILKINS [1964] and MAENCHEN andSACKS [1964] on this classical model of perfect plasticity, leading to the developmentof the radial return algorithm, constitutes the point of departure for modem treatmentsof classical plasticity. It is for this reason that this setting is adopted as a modelproblem to motivate alternative algorithmic schemes.

Set s = dev[oa], R = avy, and n = s/Isl to write the perfectly plastic model

of J2 -flow theory as

s = str _ 2yn,(12.1)

'y , f()= sl - R 0, and yf(ao)=0.

Here, tr = 2 /i dev[] where dev[] is the strain rate deviator and t > 0 is the shearmodulus. Consider the following strain history, linearly increasing in time

dev[e(t)] = et for t > 0, with e = constant. (12.2)

In what follows we will compare the performance of alternative algorithms developedbelow to the exact solution of the differential algebraic system (12.1) for the strainhistory (12.2). We remark that this solution also satisfies the full initial boundary valueproblem under the simplifying assumptions of homogeneous deformation, negligibleinertia forces, zero external loads, and suitable displacement boundary conditions.

(A) The exact solution. Assume plastic loading so that f = 0 and st` * n > 0. Sincef = 0 we enforce the consistency condition f(a) = 0 and conclude that -y = eon > 0.The system (12.1) then yields the incremental problem

s = tr - (r · n)n where tr = 2/te. (12.3)

Define the unit vector mo = e/lel and set = cos- (m 0 · n). It is easily concludedfrom (12.3) that the evolution of s takes place on a circle in the plane defined bymo and the unit normal no = n(t)lt=o (see Fig. 12.1). Taking the dot product of(12.3) with m0 yields, after use of the identity sin2 (0) = 1 - cos2 (0), the differentialequation

-- 2/ lel sin(O),(12.4)

19(0) = 00 = os- ' (no mo).

234 J.C. Simno CHAPTER 11


nfs,$so

n+l

(t) (it)

FIG. 12.1. Perfectly plastic J2-flow theory. (i) The radial return method, an example of a backward difference

method, and (ii) The two-stage mid-point rule, an example of an implicit Runge-Kutta method.

Integration by quadrature yields the closed-form expression

tan (9) = tan( 90) exp [Is - so /Rl , (12.5)

where str = so + (2/te)t and so = s(O) is the initial stress at time t = 0 assumed tobe such that Isol = R. Setting to = m - cos(9o)no, the stress field s(t) becomes

s(t) = 2-tlel [cos(9 - o)no + sin(0 - i9o)to]. (12.6)

Two key properties of this solution should be noted:(i) As Is tr - sol - oo we have - 0. Equivalently, n becomes parallel to mo

as time t coc. In view of (12.6), the stress field

s(t) -so = Rmo as Is r - sol - o. (12.7)

(ii) Recall that the instantaneous dissipation is D = -· &P. Using the flow rule,this gives D = 'ys * n = Ry, since by assumption Is ] = R for t E [0, o). Noting thaty = e n = leln · mo, the total dissipation in a monotonic plastic loading processfor t C [0, T] becomes

D[O,TJ = Ddt = Rlel cos(d) dO = Rlel [sin (o) - sin ((T))],0 9(T)hence, lim D[O,T] = D[o,o] = Rjel sin(0o). (12.8)

Since the complementary Helmholtz free energy during plastic loading remains fixedat its initial value ,2 /2pu, the stress power is converted entirely into mechanicaldissipation leading, therefore, to a strongly dissipative process.

SECTION 12 235

S.=

1_111~m

From the preceding properties one concludes that soo defined by (12.7) is a steady-state equilibrium point that attracts all the trajectories associated with the loading e(t)regardless of the initial condition so at time t = 0. This property is consistent withboth the dissipative and contractive properties of the elastoplastic flow, described inthe preceding chapter.

(B) ALGO 1: Backward Euler method. Consider the approximation of (12.1) bythe lowest-order backward difference method, the backward Euler scheme, and setAe = Ate to obtain the algebraic system

Sn+1 = s + 2 Ae, (12.9Sn+l = Sn+ 1 -2/1 Ay nn+l,

where nn+l = sn+l/Sn+l and the multiplier Ay > 0 is to be determined by enforc-ing the constraint f(Un+l) = ±Sn+lI - R = 0. To do so, observe that (12.9)2 can bewritten as

[In+ll -I 2[tA'y]n,+i = s+ 1. (12.10)

Since A-y > 0, Isn+1 = R, and f(en+l1) = Is,+l - R > 0, from this expression weconclude that

nn+l =Sn+ l Stn+ II and 2tAy = St+II- R>0. (12.11)

Substitution of these expressions into (12.9) yields the following result:

Sn+l= [R/Ist+ll]s+l, (12.12)

which defines Sn+l as the radial scaling of the trial stress sn+l onto the von Misescircle sI = R; see Fig. 12.1. This is the classical radial return method of WILKINS[1964] and MAENCHEN and SACKS [1964]. This algorithm is of course consistent, i.e.,Is, - s(tn)I - 0 as At --+ 0 and first-order accurate. In spite of its restriction tofirst-order accuracy, two features make this algorithm remarkable:

(i) Let s, = so be the initial data, denote by sat = Rnat the stress computedwith the radial return method defined by (12.12) in a single step At = t and letOat = cos- I (nAt m0 ). From (12.12) it is easily concluded that

lim O9At = 0 and lim Sat = Rmo = s. (12.13)

At-oo At-oo

Therefore, the radial return method yields the exact solution as At - oo.(ii) Let DAt be the total dissipation in a single step At during plastic loading. In

view of expression (12.8) and (12.13)1 one concludes that

lim DAt = Rlelsin(i9o) - lim= Rel sin()] sin(00).At-oc At-oo


(12.14)


Therefore, the radial return method yields the exact value of the total dissipation inthe process as At - oo.

These properties imply that the long-term behavior of the radial return method isidentical to that exhibited by the exact solution, since the exact steady-state equilibriumpoint s, is also an attracting point for the algorithmic solution. This example illus-trates that the radial return method inherits the dissipative and contractive propertiesof the continuum problem, a fact rigorously proved for the general problem below.

(C) ALGO 2: Two-stage mid-point rule. To improve upon the first-order accuracy ofthe backward Euler method, consider the approximation of (12.1) via the implicit mid-point rule. To account for the constraint this single-step method is cast into the form ofa two-stage projected implicit Runge-Kutta method, as follows (see Fig. 12.1). Set

Sn+1/2 = Sn + 2[2 2 e], Sn+ = Sn + 2/Ae, (12.15a)

and consider the two-stage formulae

Sn+1/2 = Sn+1/2- A'Y 2tnn+l/2 where nn+l/2 = Sn+/2 (12.15b)S'+ = + -t y2 2unn+/2 iSn+1/21

with the multipliers Ayl, Ay 2 to be determined by enforcing the constraints

f(o'+1/2) = sn+l1/21 - R = 0 and f(on+l) = 1sn+1I - R = 0. (12.16)

This ensures that the intermediate and final stages lie on the manifold lE = {s: Is -R = 0}. Since the first stage is identical to the backward Euler method with half thestep-size At, the preceding analysis gives

nn+l/2 = Sn+l/2/lsn+l/21 and 2/A/ = I Sn+1/21 - R, (12.17)

provided that f(s~l/2 ) > 0. Otherwise, set A-yl = 0, nn+l/2 = s t+l/lss+i and

proceed to the second stage to determine AY2. For the case f(sn+l/2) > 0 we de-termine A'Y2 as follows. Take the dot product of the second stage in (12.15a) withnn+l/2 to obtain

[(Sn+l - n) nn+l/2 - 2]LAy 2] = (Sn+l - Sn) nn+l1/2 (12.18)

Then observe that the second stage in (12.15a) along with (12.17) implies that nn+l/2

is parallel to (sn+l + sn), since

sn+l + Sn =2[sn + (2,uAe)] - 2/LAy 2 nn+l/ 2

= [21Sn+1/2 - 2Y 2] nn+t /2 -

SECTION 12 237

(12.19)

Finally, use the identity (s+l - s,,) (sn+i + sn) -[sn+-l 2 -]s,, 2 ] to concludethat the right-hand side of (12.18) vanishes, thus arriving at the following expressionfor the multiplier Ay2 in the second stage:

tr tr [2/tAy2 (sn+l - Sri) Str+1/ 2/IS +l/21 (12.20)

In summary, Eqs. (12.17) and (12.20) provide an explicit solution of the two-stagemethod (12.15b). This scheme is obviously consistent, second-order accurate and en-sures that the final stage s,+ l is on the admissible domain. It would appear, therefore,that the final stage s,+I computed with this scheme gives an improvement over ALGO1. Remarkably, this is not the case if one is interested in the long-term behavior ofthe system as the following observations show.

(i) Again take Sn = so as the initial data and denote by sat the algorithmicsolution computed with the two-stage mid-point rule in a single step At = t. Property(A) of the backward Euler method implies that limsAt/2 = Rmo. However, sinceSL+1 · +1/2 = Sn · Sn+1/2, the second stage of the algorithm yields

lim sat · so = R2 cos(2090) while s, · so = R2 cos(19(). (12.21)At

This means a 100% error in the angle between the final stage sat and the initial dataso. The intermediate stage sat/2, however, yields the exact solution.

(ii) Similarly, denoting again by DAt the total dissipation in a single step At com-puted with the two-stage mid-point rule via the exact formula (12.8), in view of (12.21)one obtains

lim DAt = Riel [sin(o0) - sin(-00)] = 2RLel sin(90o), (12.22)

i.e., a 100% error relative to the exact solution.The formulation of this algorithm as a one-stage method was originally introduced in

RICE and TRACEY [1973] in the context of J2 -flow theory and subsequently generalizedin ORTIZ and PoPov [1985].

(D) ALGO 3: Projected mid-point rule. The second stage in ALGO 2 involves anoblique projection of the trial state st+,+ onto the von Mises surface. However, suchan oblique projection cannot be guaranteed to exist for arbitrarily large time steps ifa general model of plasticity is considered. An alternative scheme that circumventsthis difficulty, while retaining the second-order accuracy of the preceding algorithm,can be constructed as follows (see Fig. 12.2).

Step 1. Define the first stage of the method exactly as in ALGO 2; i.e.,

str+/2 = s, + [2IAe] S+1=n2[ with ni+~l/2 - ,~n+1/2' (12.23)

S+1/2 _ n+-l/2 -- ATY 2n+/2 n+1/21

238 J.C. Simo CHAPTER 11


(t)(tt)

FIG. 12.2. Perfectly plastic J2-flow theory. (i) Illustration of the projected mid-point rule, (ii) Long termasymptotic behavior as At - oc.

AYi 1 > 0, f(oa,+l/2) = ISn+l/2l - R < 0, and AyIf(On+ll/2) = 0.

As before, this ensures that the intermediate stage lies in the elastic domain E ={s: FSI - R O}.

Step 2. Extrapolate the solution Sn+l/2 in the first step via the mid-point rule for-mula s,+l = [2Sn+1 /2 - Sn], set s+, = sn+l and define s,+l as the closest-pointprojection of this extrapolated value onto the elastic domain; i.e.,

St+ = 2

Sn+1/2S - with n,+l = ,+

sn+l = n+l -A'y2 2nn+ sn+l I (12.24)

Ay72 > 0, f(-.+l) = Is+,I - R < 0, and Ay 2f(On+l) = 0-

Again, this ensures that the final stage s,+l lies in the elastic domain E.

For the von Mises yield criterion, both steps can be solved in closed form. Thesolution to Step 1 is given by formulae (12.17), under the assumption of plasticloading, while the solution to Step 2 is given by

rl l/ +l and 2,tA 2 . (12.25)

In contrast with ALGO 2, stage 2 in the extension of ALGO 3 to general modelsof plasticity is guaranteed to always have a solution for arbitrarily large time steps.However, the long-term behavior of ALGO 3, although improved relative to ALGO 2,is not optimal when compared with that exhibited by ALGO 1. Figure 12.2 shows the

SECTION 12 239

eI

II

asymptotic solution obtained by letting At --+ oo. Since the justification of this resultuses the same analysis as in ALGO 2, further details are omitted. ALGO 3 was intro-duced in SIMO [1992] improving upon earlier work of SIMo and GOVINDJEE [1988].

In summary, if one is interested in short term accuracy, the second-order accurateALGO 3 is optimal. If, on the other hand, long-term is all that matters, the less accurateALGO 1 becomes optimal. These results, although obtained in a rather simple setting,are representative of the actual performance to be expected in more complex situations.In KRIEG and KRIEG [1977], for instance, iso-error maps obtained for a wide range ofstrain increments show consistently superior performance of the radial return methodover schemes possessing higher accuracy, as measured by the local truncation error.Similar results, obtained by SCHREYER, KULAK and KRAMER [1979], YODER andWHIRLEY [1984] and others, are reviewed below.

REMARK 12.1. The striking conclusion to be drawn from the preceding analysis isthat, for dissipative systems, higher accuracy as measured by the local truncation errordoes not necessarily imply improved long-term performance. For the incompressibleNavier-Stokes system, for instance, SIMO and ARMERO [1992] have recently shownthat the backward Euler method preserves the dissipativity of the dynamics, as charac-terized by the presence (in two dimensions) of absorbing sets and a universal global at-tractor, see, e.g., TEMAM [1988] for a comprehensive overview of infinite-dimensionaldissipative systems. In fact, for the incompressible Navier-Stokes equations, the back-ward Euler method again exhibits optimal long-term properties while the higher-orderaccurate mid-point rule does not.

13. Backward difference and implicit Runge-Kutta methods: Basic results

As a prelude to the generalization of the ideas presented in Section 12 a brief summarywill be given of a number of stability results on classical backward difference andimplicit Runge-Kutta methods.

Backward difference methods are a particular class of linear multistep methodswidely used in the numerical integration of stiff ordinary differential equation sys-tems, originally introduced in CURTISS and HIRSCHFELDER [1952] and popularized inthe work of GEAR [1971] on stiff differential algebraic systems. The brief overviewgiven below is aimed at summarizing the main properties relevant to the algorithmictreatment of plasticity. Consider the standard initial value problem

z = f(Z,t),

Zt=o = Z0.

Let {z,,+l-j: j = 1, .. , s} denote s given approximations to the solution z(t) of(13.1) at times t,+l-j, j = 1, .. ., s. Recall that a linear multistep method of order sdefines the algorithmic approximation z,+l to the solution z(t,+l) by the formula

S 8

EjZn+,-j = AtEZjf(zn++ljtnlj) (13.2)j=0 j=0


The coefficients (aj, /3j) E R x IR define the specific method. Associated with (13.2),one defines the companion one-leg method by the formula

ajZn+l-j = Atf( E jZn+l-j, tn+l-j (13.3)j=0 o j=o j=o

The two methods (13.2) and (13.3) are identical for linear problems. For nonlinearproblems, there is a one-to-one mapping between the solutions computed via (13.2)and (13.3) (DAHLQUIST [1975]). It is conventional to introduce the following notationfor the polynomials defined by the coefficients of the above methods

s S

p(() = Z j s - j and )= / 3j(c - j for E C. (13.4)j=o j=o

The algebraic equation p(() -/ a() = 0 for p C C is known as the characteristicequation of the method. In general this equation has k < s roots i(z) E C withmultiplicities mi(/p) > 1 such that EkI= mi(u) = s. The set

S= cC: (i(L) 1 if mi(l) = 1 and I i(p)I < 1 if mi(I) > 1}(13.5)

is known as the (linearized) stability region of the method for reasons summarizedbelow. An example of a linear multistep method is the trapezoidal rule (for cl =-a2 = 1, 31 2 = 32 =), with the implicit mid-point rule as the companion one-legmethod.

An s-step backward difference method is a one-leg method (or a multistep methodfor that matter in view of the definition given below), with coefficients defined ac-cording to the formulae:

1o = O, 30 = 1, ojZn+l-j = DJz,+l, and fj = 0, j > 1,

(13.6)

where Dj denotes the backward difference operator in time, as defined by the follow-ing recurrence relation:

DJ+lZn+l = DjZ,+ - DJz, where D°z = z. (13.7)

From the preceding formulae, one concludes that the one-step and two-step backwarddifference methods are given by the following explicit expressions:

s = 1: Zn+ - Zn = Atf(zn+l, tn+1),

S = 2: 2Zn+l - 2z, + 2Zn- = tf(z+lI tn+l). (13.8)


The feature that in the present context renders backward difference methods partic-ularly attractive is the need for only one evaluation of the right-hand side of (13.1).It will be shown below that this property results in an extremely convenient proce-dure for the enforcement of the algebraic constraint in classical plasticity. An s-stepbackward difference method has order of accuracy p = s and possesses the followinglinearized stability properties.

(i) A-stability (DAHLQUIST [1963]). Recall that a method is A-stable if the algo-rithmic solution {z,, E C: n E N} in the complex plane of Dahlquist's model equation

z = )z, A G C, has the property

lim IZnl = 0 for Re(A) < 0. (13.9)

A-stability is equivalent to the requirement that the left-half complex plane be con-tained in the stability region S defined by (13.5). It can be shown that both the first-and second-order backward difference methods are A-stable. Higher-order backwarddifference methods cannot, of course, be A-stable since Dahlquist's second barrier(DAHLQUIST [1963]) limits to p = 2 the maximum attainable order of accuracy ofan A-stable linear multistep method. Backward difference methods progressively losetheir good stability properties with increasing order of accuracy. It can be shown thatfor s > 7 the methods are unconditionally unstable.

(ii) L-stability. Recall that a one-step implicit Runge-Kutta method is said to beL-stable if it is A-stable and the algorithmic solution R([t) to Dahlquist's modelproblem z = Az, known as the stability function of the method, has the propertylimd0 R( ) = 0, where = AAt. To generalize this property to linear multistepmethods one writes the algorithm for Dahlquist's model problem as

Z,+ = A(Pt)Zn with Z, = [zn zn-! ... +t (13.10)

where A is the amplification matrix whose characteristic polynomial coincides withthe characteristic polynomial p() - cr( ) of the method. Let ps[ A(/)] be thespectral radius of A(Xi). Then, an A-stable linear multistep method is L-stable iflim.o ps [A(/l)] = 0. L-stability implies the asymptotically annihilating propertylim o,, zn+l/z = 0. Both the backward Euler method (s = 1) and Gear's two-stepbackward difference method (s = 2) can be shown to be L-stable.

Strictly speaking, the preceding two notions of stability are formulated within therealm of linear problems. A possible extension of the notion of numerical stability tononlinear problems is based on the concept of contractivity. Suppose that z(t) andz(t) are two solutions of the initial value problem (13.1) corresponding to two initialdata zo and o, respectively. The initial value problem is said to be contractive if,relative to a certain norm 1- called the natural norm of the problem, the followingestimate holds

Ilz(t 2 ) - (t2)j % J|z(t)- (tl)||, Vtl,t 2 et with t2 t,.


(13.11)


This notion is especially relevant to elastoplasticity since this problem is contractiverelative to the norm induced by the complementary Helmholtz free energy function,a fact proved in the preceding chapter.

(iii) B-stability. Suppose that the initial value problem (13.1) is contractive in thesense of the estimate (13.11). Let {zn} and {,} be two sequences generated by agiven algorithm for initial data zo and io, respectively. The algorithm is said to beB-stable if it inherits the contractivity property (13.11) relative to the natural norm

* Il, i.e.,

IIZn+- n -z -Z Vn e N. (13.12)

It can be shown that the backward Euler method is B-stable but other backwarddifference methods, and in general all linear multistep methods, are not. The notion ofB-stability was introduced by BUTCHER [1975] in the context of implicit Runge-Kuttamethods. A complete characterization of contractive Runge-Kutta methods is due toBURRAGE and BUTCHER [1979, 1980] and CROUZEIX [1979]. It can be shown that,for Runge-Kutta methods, B-stability implies A-stability, but not conversely.

(iv) G-stability. The notion of B-stability is too restrictive for linear multistepmethods and, historically, was motivated by the concept of G-stability introducedin DAHLQUIST [1975]. An s-step linear multistep method defined by (13.2) is saidto be G-stable if the companion one-leg method, defined by (13.2), inherits the con-tractive property (13.3) relative to an algorithmic norm induced by a suitable s x spositive definite matrix G. Accordingly, if the natural norm is induced by the innerproduct (., .), the algorithmic G-norm is given by

S s

IZ 1 = S gi j(Zn+l-i, Zn+l-j) (13.13)i=l j=1

for some positive definite matrix G = [gij], and the G-stability condition for a linearmultistep method takes the form

I1Zn+1 - Zn+1IlG < IIZn - ZnlG, Vn E . (13.14)

The backward Euler method (s = 1) is obviously G-stable since it is B-stable andtherefore contractive relative to the natural norm. The two-step (s = 2) backwarddifference method can be shown to be G-stable with

G = [ 12]; (13.15)

see HAIRER and WANNER ([1991], p. 332) for a direct verification of this result.Remarkably, in spite of the broader scope of the notion of G-stability, a fundamentalresult of DAHLQUIST [1978] shows that G-stability is equivalent to A-stability (underthe mild assumption that the polynomials p(() and a(() have no common divisor).Therefore, G-stable linear multistep methods can be at most second-order accurate.

SECTION 14 243

14. Generalized backward difference return mapping algorithms

The first generalization of the now classical return mapping algorithms for classicalplasticity described below is based on the use of backward difference methods. Fromour preceding discussion, two schemes are of interest for the problem at hand: (i) Thebackward Euler method (s = 1), a first-order accurate, B-stable and therefore A-stable,L-stable scheme and (ii) Gear's two-step method (s = 2), a second-order accurate,G-stable and therefore A-stable, L-stable scheme. Backward difference methods oforder s > 2 are no longer A-stable and therefore of little interest, while methods oforder s > 7 are unconditionally unstable. Accordingly, only schemes of order s = 1and s = 2 are considered in what follows.

Suppose that at time t the history {E ,... , EP+l_s} of the internal variables isgiven. Then, by applying a backward difference method for prescribed strain fieldE,+l = (en+l, 0), one arrives at the nonlinear algebraic problem

DjEP+ = AyVf (En+') where En+t = C [En+ - E+], (14.1)j=l

supplemented by the following discrete counterpart of the Kuhn-Tucker conditions

f(n+l) < 0, A-y > 0 and A-yf(nl) = 0. (14.2)

As in the continuum case, the Kuhn-Tucker conditions (14.2) define the appropriatenotion of loading/unloading. These conditions may be reformulated in a form directlyamenable to computational implementation by introducing the trial elastic state. Con-sider first the case s = 2 and rewrite (14.1)1 as

(E+l - EP ) - (E - En ) = A/Vf(+l). (14.3)

Multiplying this equation by and noting that it is permissible to redefine the multi-plier Ay as 2A-y for the case s = 2, the algorithmic flow rule becomes

s = 1: En,+ - E = Vf(-E,+),(14.4)

s=2: E+ - E = A -Vf(X,,+l) + (E - Epn

Then define the trial elastic state by the formula

E,+n = EP, + (s-1)[EP - E n I] f- -12.14+E4 = 3+ G l)Enl - E i for s = 1,2. (14.5)t 1 = G[En+1 - E]



With this definition the algorithm can be written in a unified form, valid both fors = 1 and s = 2. Relations (14.1) and (14.2) then imply

.n+l = t.+1 - AYGVf( 1n+l),(14.6)

f(n+l) < 0, Ay > 0, and Ayf( +l) = 0.

From a physical standpoint the trial elastic state is obtained by freezing plastic flowduring the time step. Observe that only function evaluations are required in definition(14.5). From an algorithmic standpoint, a basic result is the fact that loading/unloadingconditions can be characterized exclusively in terms of the trial state, provided that theyield criterion is convex. This is the key result needed for the computational statementof the loading/unloading conditions.

THEOREM 14.1. (i) If f: S x RInn -- Rt is convex then f(G +l) > f(Z+l), more-over; (ii) loading/unloading is decided solely from f (Z+l) according to the condi-tions

f(n+l ) <0 elastic step Ay = 0,(14.7)

f(LZ+1) >0 plastic step X Ay > 0.

PROOF. (i) The convexity assumption on f(.) implies

f(E 1+,) - f((n+l) > [+ l - n+l]Vf(n+). (14.8)

From (14.6) we have n+1 = Z'+ -AyGV7f( +l). Substituting this relation into(14.8) yields

f (~+,) - f (n+,) > y [Vf (En+,+) GVf (En+)] 0, (14.9)

since Ay > 0 and G is assumed positive definite.(ii) First, if f(Z+) < 0, then by the last result it follows that f(E,+1) < 0. The

discrete Kuhn-Tucker condition A-yf(Zn+l) = 0 then implies Ay = 0. Thus,

El = En + (S-1 ) (EP-EP )

for either s = 1 or s = 2, and the process is elastic. On the other hand, if f (,r+ 1) > 0,then etr , cannot be feasible, that is, ', + E'+. Thus, we must have A-y + 0. SinceAy cannot be negative it follows that Ay > 0. The discrete Kuhn-Tucker conditiontAf( En+,) = 0 then implies f(,+ 1) = 0, and the step is plastic.

The geometric interpretation associated with algorithm (14.6), under the assumptionthat the elasticity tensor C is constant, is entirely analogous to that associated withthe continuum problem and provides the key to its numerical implementation.

SECTION 14 245

CHAPTER II

FIG. 14.1. Geometric illustration of the backward difference closest-point projection algorithm for perfectplasticity and s = 1 (backward Euler).

THEOREM 14.2. The solution xZ,+1 of algorithm (14.6) is the closest-point projectionof the trial state ~n+l onto the boundary aE, of the elastic domain in the norminduced by the metric G. Equivalently, the minimization problem

min [Zn+t - ]. G- l [nr'+ - ]: for E, (14.10)

yields the solution Z+1 of the backward difference algorithm (14.6) for prescribed

x +I computed via (14.5).

PROOF. The result follows merely by noting that the Lagrangian associated with theconstrained minimization problem (14.10) is

£(, 7y) = [r+- - ] + G-yfQ() (14.11)

The corresponding Kuhn-Tucker optimality conditions are given by

a = G -. + + n+] + ATyVf(En+I) =0,

(14.12)f(,n+i) < 0, Ay > 0, and Ayf(~,+i) = 0,

which coincide with the algorithmic equations (14.6).

To summarize the preceding developments, the steps involved in the local integra-tion of the constrained differential algebraic system in a typical interval [t, tn+l ] viaa projected backward difference method are summarized in Table 14.1.

246 J.C. Simo

TABLE 14.1

General backward difference return mapping algorithm.

(1) Given the history {EP,..., EPn+l_s} (s = 1, 2) for the generalized internalvariables (at quadrature points) compute:

Et+ = EP + 3(s- 1) [E p -

EPI] for either s = 1 or s = 2.

(2) For prescribed strain En+l = (E+l, 0) compute the trial state via thegeneralized stress-strain relations and evaluate the yield criterion:

tr+ = G [En+l-E pt r ] and f+= f ( +l)

(3) Test the trial state and perform a closest-point projection:If fr+ I < 0 then set E,+ = Er+l and EP+I = Ep t

Otherwise, compute the closest-point projection:Find E.n+ C E such that J(E+l) = min{J(X): E C E},where J(E) = - ES] G- [E+r -+ .

Update generalized strains as EP+ I = En+ -G-lEn+l

15. Generalized implicit Runge-Kutta return mapping algorithms

The second generalization of the classical return mapping algorithms for classicalplasticity described below uses ideas from projected implicit Runge-Kutta methods.Although it is possible to construct one-step methods with arbitrarily high order ofaccuracy by considering several intermediate stages, attention will be restricted toschemes possessing at most second-order accuracy. The goal is to extend ALGO 3,introduced in the context of J2 -flow theory, to general models of classical plasticity.

Suppose that one is given at time t, the generalized internal variables E p =(EPn, n), the strain field e,, and a strain increment AE so that En,+ = E,, +Ae. Moti-vated by the structure of ALGO 3 in Section 12, consider the following interpretationof the generalized mid-point rule:

Step 1. By definition, the generalized stress at time t, is ,n = G[En - EP]. SetAE = (Ae, 0) and define the intermediate step at time t+ by the formulae

.Etr+ = E, + 9GAE,

n+@ = n+ - A'yGVf (n+), (15.1)

A'Y > 0, f (E.+) < 0, and A-yf(E,+4) = 0.

Here 9 E (0, 1] is an algorithmic parameter.

Step 2. Extrapolate the generalized stress E,+o at time t,+, to time t,+l by thegeneralized mid-point rule formula

£T+I = [n+9 - (1 - ()1n] /


(15.2)

CHAPTER II

FIG. 15.1. Geometric interpretation of the return mapping algorithm based on the projected generalizedmid-point rule.

TABLE 15.1Projected generalized mid-point return mapping algorithm.

(1) Given the generalized internal variables Ee = (e', -n) (at quadraturepoints) and the strain increment AE = (Ae, 0), compute:

E = GEn, Eent+ = Ee + OAE, and r+,, = GEe '9

(2) Perform a return mapping (s = 1) to compute the intermediate stage:If f(~Et+o9)

<0 then set ,n+ot = Et7+ o.

Otherwise, compute the closest-point projection:Find .Z,+o e E such that J(E,+O) = min{J(LZ): te E},where J(E) = [Ent+ - E] *G-' [9+, - J.

(3) Define tr+l [Tn+, -(1-iO) n ]/O by linear extrapolation, and performa return mapping (s = 1) to compute the final stage:

If f(0tr+) S O set E,+ = and E e+l = rG- t+l.Otherwise, compute the closest-point projection:

Find r7,~+ t E such that J(F,+) = min{J(/): E E E},where J(_E) = [tr +1 - ] · G-[Etr+l - ].

Update generalized strains as En+ = G-1,+l.

The linearly extrapolated value Z,+ does not, in general, lie within the elastic domainE. In fact, the convexity assumption on f(-) implies that

f( ,n+l) [f(Z+o) - (1 - )f(Zn)]/. (15.3)

Hence, f(Zn+l) > 0 in sustained plastic loading since f(Z,) = 0 and f(Z+;) = 0.

Step 3. Define the final stress state ,+1 exactly as in Step 1, with trial state theextrapolated value 7,+l; i.e.,

248 J.C. Simo

tr= -n+l,

,+l =Et+r -- AyGVf (7, +1), (15.4)

Aj' > 0, f(Ln7+l) < 0, and Aj'f(.n+I) = 0.

This ensures that X,+1 lies in the elastic domain E = {Z: f(27) < O}.

A detailed accuracy and stability analysis of this algorithm is deferred to the lastsection of this chapter. Here, we remark that second-order accuracy is attained if 19 =and B-stability holds for 9 . The scheme can be interpreted in terms of closest-point projections exactly as the backward difference methods described above (seeFig. 15.1). A summary of the steps involved in the implementation of the algorithmis given in Table 15.1. For the subsequent extension of this scheme to the finite straintheory we have replaced EP by the elastic strain e = e - EP and introduced thealternative set of generalized history variables E e = ( e , _-)*

16. Algorithms for the computation of the closest-point projection

The key step in the implementation of the class of algorithms described in the pre-ceding two sections lies in the computation of the closest-point projection of the trialstate onto a convex set, the elastic domain E, in the metric induced by the general-ized moduli G- 1. This is the standard problem in convex optimization. Two possiblesolution strategies are outlined below.

(A) The general closest-point projection algorithm. This procedure, first introducedin SIMO and HUGHES [1987], boils down to a systematic application of Newton'smethod to the system of equations (14.6) to compute the closest-point projection fromthe trial state onto the yield surface. A geometric interpretation of the iteration schemeis contained in Fig. 16.1. To accommodate the algorithms summarized in Tables 14.1and 15.1, it is implicitly understood that 9 = 1 for the first- and second-order backwarddifference methods, whereas 19 (0, 1] for the projected generalized mid-point rule.

Step 1. Initialization. Assume plastic loading, i.e., f(n+o9) > 0 - Ay > 0. Set

k = 0, Z( = ,9, and Ay(0) = 0.

Step 2. Residual evaluation and convergence test. For current values (k) andAy(k) at iteration k > 0 evaluate the residual and the yield condition as

R(k) -G-Tstr _(k) -is (k) f _V (k) ),n+19G- - y n+

(16.1)

~f = f (ere+) O

n+If f < TOL, where TOL is a prescribed tolerance, convergence has been attained.Then set (,+9,AT) = ( k)+, -(k)) and terminate the algorithm.


CHAPTER II

FIG. 16.1. Geometric interpretation of the closest-point projection algorithm in stress space. At each iterate(.)(k) the constraint is linearized to find the intersection (cut) with f = 0. The next iterate (.)(k+l),

located on level set f(kL+ l) = 0, is the closest-point of that level set to the previous iterate ()(k) in themetric defined by the complementary Helmholtz free energy function.

Step 3. Linearization. If fko > TOL, linearize the residual about the current iterate

(,(k9, Ay(k)) to obtain the constrained linear problem

-DEk ( (k (k))Vf( )] = (0,

(16.2)

Vf (+f ) * + D o + n+f =

where d(+k = [G- + D(A-y(k))V2f(ZUnk+)]_ is the tensor of algorithmic moduli.

Step 4. Solution of the linear system and update. Solve the linear problem by takingthe inner product of (1 6 .1)1 with Vf(4k,+ ) and using (16.2)2 to obtain

f(k) f+ Vf( k)) . (k) R(k)D(Ay(k)) = n+0 n+ _ n+f n+f

n+ n+ 7+)

(16.3)DP(") ~(kr) [R - D (-

Then update the current iterate by setting

(k+l) = , (k) D r(k and A? ( k+l) = A/y(k) D(A (k)).n_ 3 n+, + D n+, and + D (16.4)

Increment the counter k = k + 1 and return to Step 2.

250 J.C. Simo


Once X,+,g and A-y are computed, the internal variables are updated by invertingthe stress-strain relations, as indicated in Tables 14.1 and 15.1. Observe that thebackward difference formulae enter only in the computation of the trial state E+, forthe internal variables. The rest of the algorithm is independent of the specific schemeadopted for the computation of the trial state. For purposes of comparison with thealternative scheme described below, it is useful to summarize the basic characteristicsof the preceding Newton iteration.

(i) This scheme is an implicit procedure that involves at each step the solution ofa local system of N x N equations, where N = 4ndim(ndim + 1) + nint.

(ii) The normality rule (i.e., the associative character of the plastic flow) is alwaysenforced at the final (unknown) iterate.

REMARK 16.1. Convergence of the method is guaranteed since the problem is convex.The scheme outlined above is trivially modified to include a line search procedure thatrenders the iteration globally convergent, see, e.g., LUENBERGER [1984]. Given thesmall size of the problem, Newton's method becomes extremely competitive relativeto alternative iteration schemes.

(B) The cutting plane algorithm. The main drawback associated with the closest-point iteration procedure described above is the need for computing the first andsecond gradients of the yield criterion. This task may prove to be exceedingly laboriousfor complicated plasticity models. The objective of the iterative algorithm describedbelow, originally proposed in SIMo and ORTIZ [1985] and further analyzed in ORTIZand SIMO [1986], is precisely to circumvent the need for computing these gradients.The algorithm falls within the class of convex cutting plane methods for constrainedoptimization, see LUENBERGER ([1984], Sections 13.6 and 13.7).

Assuming plastic loading, i.e., f( ,+ ) > 0 so that Ay > 0, the key idea is toapproximate the closest-point projection by a gradient flow constructed as follows.

Step 1. Initialization. Set k = 0 and (0 O) 9 Z.t

Step 2. Steepest descent. Consider the hypersurface S = {(E, z): z = f ()}.Given Ek+ the intersection of the hyperplane Pk +l tangent to S at (k, with the

normal section of S at EX(k) is the straight line of steepest descent on the tangenthyperplane pk+ 1 The parametrization

x() = n+ - (GVf (Y ),n+

z(,) J- f((k) + Vf ) _ , k](k) (16.5)Z(¢)-- sn+) n+ Vs\z +) . 9c

defines a point (,(¢), z(¢)) on the line of steepest descent on Pk+l forC ( [, oo).

SECTION 16 251

CHAPTER II

ElasticDomain

FIG. 16.2. Geometric interpretation of the cutting plane algorithm in stress space. At each iterate ()(k) theconstraint is linearized about ()(k). The intersection of the plane normal to f(k) = 0 with the level set

f(k+l) determines the next iterate ()(k+l).

Step 3. Convergence check and update. The intersection of the steepest descent linedefined by (16.5) with the plane z = 0 is obtained for the following value of theparameter ~ which defines the updated iterate:

(k+1)and = U((+). (16.6)((k) (n+).[Vf(Z+) ° GVf(~o)]=

If f(Lk+ 9)) > TOL, where TOL is a prescribed tolerance, set k = k + 1 and return to

Step 2. Otherwise, set n+9 = nk+, and terminate the algorithm.

Again, once Xn+Z is computed by this algorithm, the generalized internal vari-ables are updated via the inverse stress-strain relations. It should be noted that theconvergence of this algorithm towards the final value Zn+ is obtained at a quadraticrate. A geometric interpretation of the algorithm is given in Fig. 16.2. The path thatreturns the trial state onto the elastic domain can be viewed as a sequence of straightsegments on the space S x ni't of generalized stresses. To some extent, the character-istics of this algorithm are opposite to those of the closest-point projection algorithmas summarized below.

(i) This scheme is an explicit procedure that involves only function evaluations.No equation system needs to be solved to update the sequence of iterates.

(ii) Normality is enforced at the initial (known) and not at the final state. Thealgorithm is clearly consistent.

REMARK 16.2. The simplicity of the cutting plane algorithm leads to a very attractivecomputational scheme for large scale simulations. However, our computational ex-periments indicate that, in sharp contrast with the closest-point projection algorithm,significant errors can result for large time steps. This suggests that the cutting planealgorithm is best suited for explicit transient simulations, where the allowable timestep is severely restricted by the Courant condition, while the closest-point-projection

252 J.C. Simo

tt

r - ... -swa


scheme is appropriate for implicit calculations where large time steps are typicallyused. Further evidence on the suitability of the former scheme for explicit rather thanimplicit transient simulations is provided by the fact that an exact closed-form lin-earization of the algorithm, leading to the notion of algorithmic elastoplastic modulidescribed below, does not appear to be possible.

17. The consistent algorithmic elastoplastic moduli

The closest-point projection algorithm described above can be exactly linearizedin closed form, leading to the notion of consistent-as opposed to continuum-elastoplastic tangent moduli. The former are obtained essentially by enforcing theconsistency condition on the discrete algorithmic problem, whereas the latter notionresults from the classical consistency condition on the continuum problem. This dis-tinction, first made in SIMO and TAYLOR [1985], is essential if the global algebraicproblem arising from a temporal and spatial discretization is to be solved by New-ton's method. Use of the algorithmic elastoplastic moduli produces the exact Hessianfor the discrete problem, thus preserving the quadratic rate of convergence of New-ton's method, while use of the continuum elastoplastic moduli destroys this quadraticconvergence.

REMARK 17.1. In the iterative solution of the initial boundary value problem dis-cussed subsequently, the algorithmic moduli are always computed at time t+l forthe backward difference methods summarized in Table 14.1, whereas for the implicitRunge-Kutta scheme in Table 15.1 the algorithmic moduli are always computed attime t+o. To accommodate these two cases in a unified setting, it will be understoodin what follows that En,+ = (, + 9As, 0) with = 1 for backward differencemethods.

The derivation of the algorithmic moduli mimics that of the continuum elastoplasticmoduli, but performed on the algorithmic problem. Differentiation of the generalizedelastic stress-strain relations and the discrete (algorithmic) flow rule yields

dXn+ = Gn+o [dE,+o - dEP+o],(17.1)

dEn+g = Ay [V 2 f (E+O)] d~7+o + d(A-y)Vf(X'n+).

Denoting by G+e the generalized elastic algorithmic moduli defined as

G~+o = [G + ATy[V2f(Xn+o9)]], (17.2)relation (17.1) can be written as2)

relation (17.1) can be written as

dn+g = G,+ [dEn+o - d(A-y)Vf( + -)].

SECTION 17 253

(17.3)

To determine d(Ay) one differentiates the discrete consistency condition f (Zn+) = 0to obtain Vf(Zn+) d~Z+o = 0. Using this condition on (17.3) gives

d(AT) = Vf(,+o) . G+ dE,+d(A Vf(y) n * Gn± Vf(Ln+ti) (17.4)

Vf( ,,) - G.+gV f(7,+Eo)

Finally, let P = DIAG[I, O] denote the projection operator defined by the relationP[] = a and, likewise, P[E] = e. Expression (17.3) then implies

dn+ = Cn+9 [dEn+o - d(Ay)8af(,+ )]

where C,+og = [PG,+gP], (17.5)

since a,f(Z) = PVf(2U). Combining (17.5) with (17.4) yields the result

Cn9= c, - n+i ( n+L97

(17.6)

nn+o = I -

This derivation shows that to obtain the algorithmic tangent moduli all that is neededis to replace the elastic moduli C in the expression for the continuum elastoplasticmoduli by C,+ = PG,+gP, where G,+o is defined by (17.2).

18. Examples: Closed-form return mapping algorithms

As an illustration of the preceding ideas, the case of J2-flow theory incorporating kine-matic and isotropic hardening will be examined in detail. In particular, new second-order accurate radial return algorithms are described based on the backward differencereturn mapping algorithms presented in the previous section. For three dimensions,the simplicity of the von Mises yield condition-a hyper-sphere in stress deviatorspace--enables one to obtain essentially a closed-form solution of the closest-pointprojection algorithm. To illustrate the applicability of the general scheme, the case ofplane stress is considered subsequently.

(A) Radial return mapping algorithms for J2-fiow theory. Recall that for J2 -flowtheory the plastic internal variables are EP = (EP,1 , ), with conjugate generalizedstresses E = (s,q, q), where tr[EP] = 0 and s = dev[acr]. To arrive at a compactformulation of the algorithm, it proves convenient to define a relative stress as thedifference between the stress deviator s and the back stress . Accordingly,

3 = s-q, whereq=-_2H and q=-K'(().


(18.1)

Note that a general nonlinear isotropic hardening function K(() is assumed. It istherefore convenient to use S in place of q as the independent variable in the returnmapping algorithm and write the von Mises yield criterion as

f(, ) = 11 - [y + K'(()] < 0, and let n = P/IP1. (18.2)

Consider the application of the backward difference scheme in Table 14.1 to thepresent example. Within a typical time step [t, t+l], the trial state for prescribeden+l = dev[en+l] and given En is then defined by the following formulae:

Sn+I = 2[en+l - (ep + (s - 1) (eP - en1))],

n+l = -H[n + (s - 1)(&n - n-l)], (18.3)

C.+1 = n + 3 (S -)( n-)

In the present context, the general equations defining the return mapping algorithmreduce to

Sn+ = Sn+ - 2Aynn+l,1

qn+l = qn+l + 3HA-y nn+l, (18.4)

n+ = n +l + AT.

An extension of the argument described above shows that the solution of (18.4) reducesto the solution of a scalar equation for the consistency parameter Ay. To see this,observe first that 3n+l can be written as

3n+ = (Sn+- +1) - [2 + H] 'Ynn+l. (18.5)

Next, to arrive at the algorithmic counterpart of the consistency condition we notethat by definition, 3 n,+1 = 1n,+l lnn+l. Hence, from (18.5) the unit normal nn+l isdetermined exclusively in terms of the trial elastic stress tn+ as

n,+l,= t+l /l1t+l i where 03+1 = +, - r+l (18.6)

By taking the dot product of (18.5) with nn+l we obtain the following scalar(generally nonlinear) equation that determines the consistency parameter Ay:

(A'Y) =- [ay + K'((n+1 + A1y)]

+ + -A [2u + H] = 0. (18.7)

The solution of Eq. (18.7) may be effectively accomplished by a local Newton iterationprocedure since g(A-y) is a convex function, and convergence of the Newton procedure


TABLE 18.1

Consistency condition. Determination of Ay.

(1) Initialize: A( 0 ) = 0 and ()+ = 0,n.

(2) Iterate on k: Do until g(A-y(k))j < TOL(2.1) Compute iterate A-y(k+l):

(ay k) y [ y +K' ((l)] + +P$ I -t -A [2s+ 2 H]

Dg (A)) =A-2p 1 H +2 K1,1 (l)

Ay(k + ) A( k) _g(A) )Ug(Ay(k))

(2.2) Update equivalent plastic strain:

A,+l) = ,tr + Ar

(k+l

is then guaranteed to occur. Details pertaining to the local Newton procedure aresummarized for convenience in Table 18.1.

REMARK 18.1. If the kinematic/isotropic hardening law is linear, then Eq. (18.7) isamenable to closed form solution that results, for s = 1, in the generalization of theradial return algorithm in KRIEG and KEY [1976]. Set

fn = 13+l [Y + K+l] (18.8)

where oy > 0 is the flow stress in pure tension, K is the isotropic hardening modulus,and H > 0 is the kinematic hardening modulus. Substitution of (18.8) into (18.7)yields

A7 = (18.9)2/ + 2K + H'

The update is completed by substituting (18.9) into formulae (18.4).

For convenience, a step-by-step description of the algorithm discussed above hasbeen summarized in Table 18.2. The geometric interpretation of this algorithm isshown in Fig. 18.1. Next, we proceed to compute directly the consistent algorithmicelastoplastic tangent moduli by linearization of the two-step return mapping algorithm.These moduli relate incremental strains and incremental stresses and play a crucialrole in the overall solution strategy of the boundary value problem.

We conclude this example by providing the exact linearization of the radial returnalgorithm leading to a closed-form expression for the elastoplastic tangent moduli.



TABLE 18.2

Radial return algorithm. Nonlinear isotropic/kinematic hardening.

(1) Compute trial elastic stress for prescribed en+l = dev[en+l]:

Sn+l = 2 -[(en+ -e -( (e )]

tn+r =- 2 [n 3 3 l (51)( n -n-1) ]

+ = + + 3( - 1)(n -n-).

(2) Check yield condition with 3n+l = s +l - 4t+l

fn+ = 0-n+4 - [nY + (+ )]

If fn+l > 0 then perform the following steps:(3) Set n+ +l = 1n+I +II and compute A-y as

A'y = defined by Table 18.1 (nonlinear hardening)

ft+A 2 2y (K + H) (linear hardening).

(4) Update back-stress, plastic strain, and stress:

qn+l = /n+l + HAfnn+l

+1 .tr - 2Ayn+ l

eP+l = e+l - sn+1/2p

+n+I=r,*+ A.

(5) Compute consistent elastoplastic tangent moduli:

eCn+ = l S 1 + 2Cn+l [I - 3I ®g 1] -21n+lnn+l ® nn+,

(n 2/A'y

2/.t

2p, + 2 (K"(, + 1) + H)

Rather than particularize the general result (17.6) to this specific example, we proceed

directly by differentiating the algorithmic stress-strain relations defined by n+l =

K(tr[,+i])1 + sn+l - 2[Aynn+l to obtain

don+l = C dEn+l - 2 [d(A-y)nn+l + Ay dnn+l]

= c- 2/-tnn+ a -+ - 2UAy aE n+l dEn+l, (18.10)

258 J.C. Simo CHAPTER II~~~~~~~~~~~~~.. al-1

FIG. 18.1. Geometric interpretation of the return mapping algorithm for the von Mises yield condition andisotropic/kinematic hardening.

where C = il1+2,t(I- 1®1) is the elasticity tensor. To carry out the computationfurther use is made of the following result.

THEOREM 18.1. The derivative of the unit normal field n(/3) = /3/l/3 is given by theformula

3n/a3 = [I - n n]/31. (18.11)

PROOF. The result easily follows with the aid of the directional derivative. First wenote that for an arbitrary vector h C R6 we have

d / + hl = h/l = n h (18.12)

By the chain rule it then follows that

d I n(3 + h) = h - (n h)n/131 = [I-n X n ] h3 , (18.13)d =0

so that (18.11) holds.

As in the general case, the term ATy/e,,+l appearing in (18.10) is obtained bydifferentiation of the scalar consistency condition (18.7). Accordingly,

aeysn+l tu Ko (n+l)+ Ha(. in o ,. (18.14)

Substitution of (18.11) and (18.14) into (18.10) produces, after some manipulation,

the expression summarized in Table 18.2. This expression should be contrasted withEq. (4.19) for the "continuum" elastoplastic moduli. As a result of the radial return



algorithm the shear modulus pI enters in the "consistent" tangent moduli scaled downby the factor ,,+l. Observe that I+l 1 so that, for large time steps, st+, maylay far out of the yield surface and (,+l may become significantly less than unity.In addition, since <(n+1 = Ay + (n+ - 1, we have the bound Ay - 1 < 5(+l < Ay.Therefore, for large time steps, the algorithmic tangent moduli may differ significantlyfrom the "continuum" elastoplastic tangent. However, as At -- 0 and Ay - 0, thealgorithmic and continuum tangent moduli coincide. This result is a manifestation ofthe consistency between the algorithm and the continuum problem.

REMARK 18.2. According to the algorithm in Table 18.1, the values () are cal-culated based on the converged values (-), at time step t = t. The (nonconverged)values ()k,+l at the previous iteration play no explicit role in the stress update.

REMARK 18.3. The adaptation of the projected generalized mid-point rule in Ta-ble 18.2 to J2-flow theory is immediate from the preceding results. The intermediatestage is computed via the algorithm in Table 18.2, with s = 1 and e±+l = e, + Aereplaced by e,+, = e, + 9Ae. The final stage is computed again via the same algo-rithm, with s = 1 and the trial state replaced by the linearly extrapolated values asindicated in Table 15.1.

(B) Backward difference algorithms for plane stress J2-plasticity. For the plane stresscase, a simple radial return would violate the plane stress condition and thus isno longer applicable. The basic idea in the algorithm proposed in SIMO and TAY-LOR [1986] and described below is to perform the return mapping directly in theconstrained plane stress subspace so that, by construction, the plane stress conditionis identically satisfied. In the plane stress subspace, however, the von Mises yieldcondition is an ellipse and not a circle. Thus, when solving the discrete problem(14.6) one is confronted with the general problem addressed in Section 16; namely,the computation of the closest-point projection in the complementary Helmholtz freeenergy.

Recall that for plane stress J2 -flow theory with combined kinematic/isotropic hard-ening the generalized strain-like internal variables are EP = (EP, ~, 6). Using thebackward difference formulae, it proves convenient to define trial generalized strainsvia the expressions

EPn+l = 3 (s- 1) (P - n-

4n+ = 4n + '(S - )(4n - n 1)I (18.15a)

tr+ = + (S - 1)(4n -n-I),

where s = 1 for the backward Euler method and s = 2 for Gear's two-step backwarddifference method. The trial elastic state tn+1 is defined for a prescribed strainincrement E,n+l via the generalized stress-strain relations as

(18.15b)tr = and q+l =-H +'n+1 = C [EI - HP

SECTION 18 259

where C and H are given by (4.24) for the isotropic case, and qt+l =-K'( ) isthe internal variable controlling isotropic hardening. In addition, from Eqs. (4.22) thegeneralized backward difference return mapping algorithm takes the following form:

an+l =+1 - A'y CP3n+1,

Qn- n+ t + AYHP/3n+l, (18.16)

,+n =+ /n + A-Yfn+l,

where /3,+1 = n,+l - qn+l is the relative stress and P along with f are defined by(4.22) as

2 -1 0+1 /fn = P+l 3n+1 and 3 -1 2 0 (18.17)

0 0 6

Instead of applying directly the closest-point projection algorithm described above, itproves more efficient to combine (18.16)1 and (18.16)2 to obtain, assuming the non-trivial case of plastic loading (Ay > 0), the following system of equations involvingonly the relative stress and the multiplier AA:

rn+ = [I + Ay(C + H)P] =3+,,

(18.18)f 2(y) = i n+2- [K'( + 2l)] 0,

where fn+l is defined by (18.17)1. This system can now be solved by Newton'smethod for Ay > 0 and ,n+.

REMARK 18.4. The algorithmic elastoplastic tangent moduli consistent with the pre-ceding integration algorithm are developed by linearization of the preceding formulae.This is an exercise in the application of the chain rule from which only the relevantresults are quoted. Differentiation of the system (18.18) gives the linearization of theconsistency parameter as

nfiPCn+l d,+ld(Ay) = +l (18.19a)/3n+l[P(Cn+ +Cn+ilCH)P+ 2K"P] n+I

where C,+1 are the effective algorithmic tangent moduli given by

C,+1 = [C - 1 + y(I + C-1H)P]-. (18.19b)

Substitution of (18.19a) into the differentiated version of the algorithmic stress-strain relation (18.15b)1 yields the desired result.



REMARK 18.5. For the case of isotropic elasticity, the system (18.18) has a particularlysimple form since the matrices P, C and H all commute. By introducing a spectraldecomposition, this system can be reduced to four scalar equations exhibiting minimalcoupling. The interested reader is referred to SIMO and TAYLOR [1986] for a detailedaccount of these developments. More recently, SIMO and GOVINDJEE [1988] noted thatfor linear kinematic hardening and certain forms of isotropic hardening the system(18.18) can be further reduced to a quartic equation for the consistency parameter Aywhich can then be solved in closed form.

Although the preceding examples are restricted to J2 -flow theory, the same method-ology applies with essentially no modification to other classical models of plasticity.In particular, first-order accuracy for Drucker-Prager and other cap models for geo-materials, as described in SANDLER and RUBIN [1979], LORET and PREVOST [1986],SIMo, JU, PISTER and TAYLOR [1988] and others, are extended in a straightforwardfashion to achieve second-order accuracy via backward differentiation formulae.

19. Practical accuracy assessment: Iso-error maps

A practical assessment of the accuracy of the proposed class of algorithms can bemade by examining iso-error maps, developed on the basis of a strain controlledhomogeneous problem. The procedure has been employed by a number of authors,e.g., KRIEG and KRIEG [1977], SCHREYER, KULAK and KRAMER [1979], IWAN andYODER [1983], ORTIZ and PoPov [1985], ORTIZ and SIMO [1986], SIMO and TAY-LOR [1986] and LORET and PREVOST [1986]. The construction of iso-error maps willbe outlined taking as an example the return mapping algorithm for plane stress de-scribed above. For simplicity, attention is restricted to the conventional backward Eulermethod (s = 1). A rigorous accuracy analysis is presented in the following section.

Three points on the yield surface are selected which are representative of a widerange of possible states of stress. These points, labeled A, B, and C on Fig. 18.2correspond to uniaxial, biaxial, and pure shear stress, respectively. To construct the

E = 30,000 Ksi

v = 0.3

-a1 ¢ = 1/Vjllsl-R=0

R = av1r2= 0

FIG. 18.2. Plane stress yield surface. Points for iso-error maps.

SECTION 19 261

YT

_

&FI/EiY

FIG. 19.1. Iso-error map corresponding to point A on the yield surface.

iso-error maps we consider for each selected point on the yield surface a sequenceof specified normalized strain increments. The stresses corresponding to the (homo-geneous) states of strain prescribed in this manner are then computed by applicationof the algorithm. At each point the normalization parameters are chosen as the elasticstrains associated with initial yielding. Without loss of generality, the calculation isperformed in terms of principal values of the strain and stress tensors; i.e., it is as-sumed that el 2 = 0. Results are reported in terms of the relative root mean square ofthe error between the exact and computed solution, which is obtained according tothe expression

"(-1 aC) (O - "*)6= t ) ( ) x 100. (19.1)

Here, is the result obtained by application of the algorithm, whereas * is theexact solution corresponding to the specified strain increment. The exact solution forany given strain increment is obtained by repeated application of the algorithm withincreasing number of sub-increments, The value for which further sub-incrementingproduces no change in the numerical result is regarded as the exact solution.

The iso-error maps corresponding to points A, B, and C are shown in Figs. 19.1to 19.3, respectively. The values reported here were obtained for a von Mises yield con-dition with no hardening and a Poisson ratio of 0.3. Observe that Figs. 19.2 and 19.3


E2Y


2yE25

a E,/E1

FIG. 19.2. Iso-error map corresponding to point B on the yield surface.

ae2E2y

FIG. 19.3. Iso-error map corresponding to point C on the yield surface.

exhibit a symmetry which may be expected from the location of points B and Con the yield surface. From these results, it may be concluded that the level of errorobserved is roughly equivalent to that previously reported in the literature for other

SECTION 19 263

5

return mapping algorithms. As a rule, good accuracy (within 5 percent) is obtainedfor moderate strain increments of the order of the characteristic yield strains. It is alsonoted that exact results for any strain increment are obtained for radial loading alongboth symmetry axes, as expected. Iso-error maps for higher-order methods, such asthe two-step backward difference scheme (s = 2) and the projected mid-point rule(9 = ½), confirm the superior accuracy of these methods for small time steps overthe backward Euler method but reveal a significant degradation in performance formoderate and large time steps. These observations are consistent with the analysis pre-sented in Section 12 which identifies the backward Euler return mapping algorithmas the optimal scheme for very large time steps (i.e., optimal long-term dissipativebehavior).

20. Accuracy analysis of return mapping algorithms

The accuracy analysis of general return mapping algorithms described below exploitsthe following geometric framework. We regard the boundary BE of the elastic domainas a smooth hypersurface in S x IRn'mI with Riemannian metric induced by G-1 (andendowed with the standard Levi-Civita connection). From this point of view theequations of rate-independent plasticity possess an intrinsic geometric interpretationindependent of the parametrization. Within this context, higher-order rates of changeof the plastic flow at a point are naturally measured in terms of higher-order covariantderivatives. To illustrate the main ideas involved, we choose as a model algorithmthe generalized projected mid-point rule in Table 15.1 and show that this scheme issecond-order accurate only if 9 = . A completely analogous analysis, the detailsof which are omitted, confirms that the two step (s = 2) backward difference returnmapping algorithm in Table 14.1 is also second-order accurate.

(A) The local geometry of the plastic flow. For convenience, let us denote in whatfollows the local inner product induced by G as (, )G = () G - ' (.) and use thenotation I - IG = V , )ci for the associated norm. Then define the normal field N ata point X E aE by the expression

GVf(z)N = IGVf(Z) so that INIG = 1. (20.1)

For given strain rate history t - Et, the map t - ±r = GEt defines a correspondingrate in stress space. In general ±t. will not be tangent to the yield surface aE. Theprojection of 7Ztr onto the tangent plane at Zt aE, denoted by P(±t,), is given by

() = r - (r, Nt)GNt if (tr , Nt)G > 0, (20.2)- _F__r ~ otherwise,



so that, in view of the definition for the elastoplastic tangent moduli, the evolutionequation describing the plastic flow t Et can be written as

= ?( r) = = (GEt). (20.3)

Plastic loading will be assumed throughout so that ('7, Nt)G > 0. The vector fieldt - ±t is therefore tangent to the yield surface. To compute its time derivative denotedby ±t, observe that (20.3) implies (7, Nt)G = (t, Nt)G since (Nt, Nt)G = 0 asa result of the unit length constraint (20.1). Using this observation, time differentiationof (20.3) then gives

_t = tr _ (A, N)GN t - (tr, Nt)G t, - (, Nt)GNt, (20.4)

where Xtr = GEt. In general, however, Xt is not tangent to the yield surface alE. Thecovariant derivative of ±t, denoted by D/Dtt, is precisely the projection (in theG-inner product) of t onto the tangent plane at Zt and defines, therefore, a vectorfield tangent to the yield surface (see, e.g., THORPE ([1979], p. 45) for an introductoryexposition of these ideas). Using again the fact that (Nt, Nt)G = 0 one obtains

-t,4 = iX, (r, Nt), - (Zr,Nt)GNt, (20.5)

since the normal component of the acceleration is given from (20.4) as t, Nt)G =

-(t,, Nt)c. From (20.1) the time derivative of the unit normal field is

GV2 f(L't)Nt = tt ±- (Dtt, Nt)G Nt where f2t = IGVf( ()t (20.6)

The bilinear form (, Ot )G at Xt induced by S2t is easily shown to be self-adjointand is known as the Weingarten map. In summary, expression (20.6) along with (20.4)translate into the classical result

(t, Nt)G = -(t, Nt)G , = t S)G = KN(L't) l tl , (20.7)

where rtN(Xt) is known as the normal curvature of the surface aIE at the point Xtin the direction of ±t; see THORPE ([1979], p. 62). It follows from (20.7) that thenormal acceleration of the plastic flow at a point on the yield surface is proportionalto the normal curvature of the yield surface at that point.

SECTION 20 265

(B) The discrete local problem: Accuracy analysis. In the two-stage implicit algo-rithm summarized in Table 15.1 the generalized stress at each stage is defined viaa closest-point projection of two trial states. To assess the accuracy of this schemeassume the nontrivial case of plastic loading and consider a generic stage written inthe form

Zh = -h - YhGVf( h) with Yh > 0 so that f(Zh) = 0. (20.8)

Here Zh and Z'~ are interpreted as the final and trial states, respectively, of either thefirst- or second stages of the algorithm. The subscript h [0, At] is used to emphasizethat the computed algorithmic solution in each of the stages depends parametrically onthe time step. We assume that h' h=O = hIh=O and define the algorithmic moduli

Gh = [G-1 + hV2f (h)] , (20.9)

where Yh > 0 is computed by enforcing the condition f (Zh) = 0 via the closest-pointprojection algorithm in Section 16. The same argument described in Section 17 forthe derivation of the algorithmic elastoplastic moduli now yields

Vf(ah) Gh(G - 1 dxh)

Vf(oh) · Gh Vf (zVh)(20.10)

dh = Gh (G - dh) - dYh GhVf(Lh).

Assuming that the mapping h Z' is prescribed, (20.8) defines an algorithmic flowh - Zh with initial data Zh I h=O = lh h=O Observe that by setting

GhVf(Zh) GhV 2f (Zh)h , Nh = and h O h = - (20.11)

IGhVf (Zh) IGh IGhVf (oh) lIh

where I 1Gh is the algorithmic norm associated with the inner product (, )h induced

by Gh, result (20.10) can be written as

dZh = Gh(G-' dU) - (Nh, (G 1 dZy))GhNh. (20.12)

This algorithmic differential equation is of the form dZh = I h[Gh(G-' d ' hr)], whichis identical to (20.3) with the Riemannian metric G - l replaced by the algorithmic

metric Gh . Using the notation d(-)o = d(.)Ih=o/dh we have the following basicresults.

266 J. C. Simo CHAPTER

THEOREM 20.1. The first derivatives of the functions h, Gh and Zh are

dYo = (No, dsr)G/lGVf (o)lG,

dGo = -G-'oG-(No, d)G, (20.13)

dlo = dX0f - (dl'4, NO)G No.

PROOF. Relation (20.13)1 follows from (20.9) and (20.10) by noting that ?YhIh=0 = 0.Relation (20.13)2 follows from (20.13)1 and (20.9). Relation (20.13)3 follows fromthe preceding two and (20.10).

The second-order accuracy assessment of the one-parameter family of algorithmsin Table 15.1 hinges on the following result.

THEOREM 20.2. The second derivative of the mapping h F-+ Zh at h = 0 is given by

d2 o = d2 r - (d2 , No )GNo

- 2(No, dT)GdNo - (dL'o, dNo)GNo, (20.14)

where dNo = od~o - (od~o, NO)G No-

PROOF. This is an involved but otherwise straightforward application of the chain ruleto expression (20.10) that uses the relations in the preceding lemma. The details areomitted.

EXAMPLE 20.1. We consider here the conventional backward Euler return mappingalgorithm, obtained as a particular case of either the backward difference methods inTable 14.1 for s = 1, or the projected implicit Runge-Kutta methods in Table 15.1for 9 = 1. Within a typical interval [t,, t,+l], this algorithm is recovered from (20.8)by setting

Z = , XT = n + G[En+l-En] and h = +l . (20.15)

Since d = G = , consistency of the algorithm (i.e., first-order accuracy)follows immediately from (20.13)3, which becomes identical to (20.3). Noting thatd2 r, = GE = GE , = , expression (20.14) reduces to

d2 = tr -(', Nn)GNn - 2(t, Nn)GNn - ( n, n)GNn, (20.16)

which differs from expression (20.4) for the exact flow at time t = t, by a factorof two in the third term. Hence, the conventional backward Euler return mappingalgorithms are not second-order accurate. Observe, however, that the curvature term(d2Z, NO)G = -(d2o, Qo d0o)G is second-order accurate.

Consider next the application of the preceding results to the two-stage scheme inTable 15.1 with 0 < < 1. One comment may be appropriate here. The result below


may seem obvious at first glance but it is not. Many different options are possible inthe design of the second stage. The one recorded in Table 15.1 is precisely that onecapable of retaining second-order accuracy.

THEOREM 20.3. The two-stage, implicit, projected mid-point rule method in Table 15.1achieves second-order accuracy for ?9 =

PROOF. The proof involves the sequential application of the result in the last theoremto each of the two stages of the algorithm.

Stage 1. The first stage of the algorithm is recovered from (20.8) for

0 = n, E = .n + 9G[[En+ l - E] and Zh = n,+3. (20.17)

As a result, dEor = VGE = 9 which, in view of (20.3), gives dL 0 = 0L,' wheninserted in (20.13)3. Since d2So r = 09L, application of (20.14) then yields

d2 z 0 = o[E - n Nn)GN]

- 202(1X, Nn)GNn - 09(,, N,)GNn. (20.18)

Therefore, the first stage is first-order accurate with the exact solution at time t = t,+o.

Stage 2. The second stage of the algorithm in Table 15.1 is recovered from (20.8)by setting

Zo = Xn.+o,(20.19)

ht = [,+ - (1 - ),,]/0 and Xh = 7,+1.

The composite two-stage method is consistent since the preceding formulae and(20.13)3, along with the condition (n, Nn)G = 0, imply

d J t = 1 dh . = E(n) = i7, so that dZo = .. (20.20)

Next observe from (20.19) that 2 d2L'r = d2 _f,+O/dh2Ih=O, which is given by ex-pression (20.18) computed in the first stage. Therefore, since dZ 0 = En, result (20.14)specialized to the second stage takes the form

d2o= [: (, Nn)GNn] - 2(, Nn)GNn

- a(n, Nn)GN, (20.21)

which agrees with (20.4) if and only if = []

REMARK 20.1. A similar analysis shows that the algorithm ALGO 2 described inSection 12 for the Prandl-Reuss model is also second-order accurate. The extension



of this scheme to the general case appears to be questionable since it involves thecomputation of an oblique projection. By contrast, ALGO 3 and its generalizationoutlined in Table 15.1 involves the computation of two closest-point projections which,by the classical projection theorem, are guaranteed to exist if E is convex. From thepoint of view of implementation, it should be emphasized that the two stages arecarried out with an identical algorithm. The only difference between them lies in thedefinition of the trial values. Finally, we remark that the accuracy analysis of backwarddifference methods involves the same ideas although it is considerably simpler. Furtherdetails are omitted.

21. Extension of return mapping algorithms to viscoplasticity

The general class of algorithms developed in Sections 3 and 15 can be easily ex-tended to viscoplasticity by regarding this latter model as a viscous regularization ofthe rate-independent model, in the sense described in the preceding chapter. By adopt-ing the regularization procedure described in Section 6, which can be viewed as anextension of the technique first proposed in DUVAUT and LIONS ([1976], p. 234), thesevere ill-conditioning exhibited by early treatments of viscoplasticity (e.g., HUGHESand TAYLOR [19781 and the references in Section 6), can be entirely circumvented.We consider first the extension of the general backward difference return mappingalgorithms and then examine the important case of J2 -flow theory as a specific exam-ple.

To motivate the key idea underlying the algorithmic treatment of viscoplasticityintroduced in SIMO, KENNEDY and GOVINDJEE [1988], we shall consider first thecase of linear viscosity. The general evolution equations (7.7) yield the followingviscoplastic regularization of the rate-independent problem (11.2):

P = G-[7T -- .] with = G[E - EP]. (21.1)

Recall that > 0 is the relaxation time and , is the closest-point projection ofX onto the convex elastic domain E in the complementary Helmholtz free energy,as described in Section 6. Consider an algorithmic approximation of (21.1) via thes-stage backward difference method in Table 14.1 with either s = 1 or s = 2. Setting

E+, --- E + 3- 1) (En - En-l,(21.2)

n+l = G [E,,+- En+l]

and denoting by Z,+1. the closest-point projection of the (unknown) generalizedstress field .,+1 onto E, the backward difference approximation to (21.1) can bewritten as

At,n+l = t['+ 1 ,- -n+1*], At = [1 - (s - 1)]At. (21.3)~+;~~~~ 3 7~~--

SECTION 21 269

Multiplying both sides of this equation by T/Ats and collecting terms yields theequivalent result

Zn+l = +l* + T/AJ[, - nl*]j (21.4)

Clearly, .n+ -I + , * as T/At, -- 0 and one recovers the rate-independent solutiondefined as the closest-point projection of Z+1 onto E. Denoting this projection byF+1,, it follows that E,+l, = n+,1* and (21.4) reduces to

n+l n = t+1* + l+i- [ 7+/At*] (21.5)+ -/Ats [r-

The algorithm implied by this result reduces to the following three-step method thatrelies crucially on the solution of the rate-independent problem:

Step 1. Compute the trial state exactly as in the rate-independent problem via for-mulae (21.2).

Step 2. Compute the rate-independent solution +,* via the general closest-pointprojection algorithms described in Section 16.

Step 3. Compute the regularized, rate-dependent, viscous solution by applying theexplicit formula (21.5).

The remarkable property that makes this algorithm rather useful from a practicalstandpoint is the well-conditioning of formula (21.5) for any value of the relaxationtime rT [0, oc). In particular, one can set = 0 in (21.5) to recover exactly therate-independent limit. Observe that for s = 2 the algorithm is second-order accurate,retains the property of A-stability and is G-stable. This method appears to be new.

REMARK 21.1. The linearization of the preceding algorithm is trivial. Let C ? denotethe algorithmic elastoplastic tangent moduli consistent with the closest-point projectionalgorithm described in Section 16. From (21.5) it immediately follows that

= Ce - + '/At 8 -[C- 6*] (21.6)1 + T/At8

are the algorithmic elastoplastic moduli associated with the three-step method outlinedabove.

REMARK 21.2. An identical construction applies to each of the two stages in thegeneralized projected mid-point rule algorithm summarized in Table 15.1.



22. Return mapping algorithms for general models of viscoplasticity

The algorithmic treatment introduced above for linear viscosity, the only case con-sidered in DUVAUT and LIONS [1976], is easily extended to accommodate the generalmodel of viscoplasticity, with evolution equation now defined by

p = g(J())G [ - ,], (22.1)T

where the function g(.) has the property g(z) = 0 if and only if x < 0, and g(x) > 0for x > 0. The algorithm (21.5) now reads

[ ,, - .+,] - g(J(+l)) [,+ 1l - =r+l*] = 0, (22.2a)

At5

where the distance J(,n+l) between En+1 and the convex set E remains to becomputed to complete the algorithm. Remarkably, this computation involves only thesolution of a scalar equation, as the following result shows.

THEOREM 22.1. The determination of J(Z,+l) is accomplished by solving the fol-lowing nonlinear scalar equation:

J(LZ+l) [T/At, + g(J(n+l))] = T/Ats J(ln'+l), (22.2b)

with J(T+l) prescribed from the solution of the rate-independentproblem.

PROOF. Adding and subtracting n,+l in (22.2a) yields

- + g(J(L'n+t))] (r.+i - Xn41*) = k-(L - n l*) (22.3)[At, At, n~ l*, + (22.3

Recalling that the complementary Helmholtz free energy is given by .E(Z) = ZG- 1Z, Eq. (22.3) implies that

At,

The result follows from definitions (7.3) and (7.4) for the distance J(Z). D

We remark that the three-step algorithm outlined above, with (21.5) now replacedby (22.2a,b), inherits all the properties described in the case of linear viscosity. Inparticular, it is well-conditioned for any E [0, oo) including r = 0. An identicalconstruction also applies to each of the two stages in the projected version of thegeneralized mid-point rule algorithm given in Table 15.1.

SECTION 22 271

The preceding strategy is illustrated below in the specific context of J2-plasticitywith combined linear isotropic and kinematic hardening. The algorithm presentedbelow provides a unified treatment of a constitutive model widely used in applicationswhich includes both the (linear) viscous case as well as the inviscid limit.

EXAMPLE 22.1. Consider the model of J 2-flow theory treated in Section 18, withquadratic potential K(() = Kt2 for isotropic hardening and a constant kinematichardening modulus H. As noted earlier, under this assumption, the radial return algo-rithm described in Section 18 then becomes amenable to a closed-form solution. Let

= (a, /, q) and denote by In+l, = Zt[+l, the rate-independent solution com-puted with the algorithm in Table 18.2. Consider for simplicity linear viscosity, withcharacteristic relaxation time r > 0, and rewrite the inviscid radial return algorithmin the following form which allows a literal application of formula (21.5). Set

Vf (Zn+1) +t =I N+ti, (22.5)

where n,+l = n+/ 3 nf+1l and observe that the generalized elastic moduli G aregiven by

G= 1+ O 1G= [O H1 0 . (22.6)

0To0 T K

In terms of this compact notation, the rate-independent solution given in Table 18.2takes the form

tr

_r AyGNU l where y (22.7)n+l* r+l-y* GNtr+ where A7* = 2/ + (K + H) (22.7)

Inserting this expression into formula (21.5) gives

n+l = ntr+l - y*GNr+, + 1 /At AAT *G N ~ +In~l 1 + L/AtA'y(.

= tr - At GN t ' (22.8)

By comparing (22.7) and (22.8) we arrive at a remarkable result from a computa-tional standpoint. To incorporate linear viscosity in J2-flow theory only two trivialmodifications are needed in the algorithm in Table 18.2 (with K" = K constant):

(i) Redefine the consistency parameter Ay by the formula

AY (=1 + (22.9a)(1 + )[2 + (K + H)](22.9a)

Azt,,r 3 '1


(ii) Redefine the coefficient ~n+l in the linearization of the algorithm as

I ( 1 + - (1 - p2+1). (22.9b)[2/t + (K H)]

It should again be emphasized that if expressions (22.9a,b) are adopted in place ofthose given in Table 18.2, the rate-independent model can be recovered exactly withoutintroducing any ill-conditioning in the algorithm merely by setting r = 0.

23. The algorithmic initial boundary value problem

A discretization in time using the generalized return mapping algorithms described inSections 14 and 15, along with a suitable time discretization of the inertia term in theweak form of the momentum equations, reduces the initial boundary value problem ofdynamic plasticity to a boundary value problem within a typical time step [t., tn+l].The goal of this section is to provide a concise statement of this problem suitable forthe nonlinear stability analysis addressed in detail subsequently.

The first step in the weak formulation of the initial boundary value problem for dy-namic plasticity is the reformulation of the return mapping algorithm as a variationalinequality. Before doing so, we recall briefly the functional setting for the problemintroduced in Section 9 under the assumptions of (i) hardening plasticity, with gener-alized plastic moduli H assumed constant and positive definite, and (ii) linear elasticresponse with point-wise stable, constant elastic moduli C. The space T of generalizedstress states is defined by

T = {T = (,p): ? $ x nintht Tij L 2 (2) and Pi £ L 2 (2)}. (23.1a)

The space T is equipped with the natural energy inner product induced by the symmet-ric bilinear form defined by twice the complementary Helmholtz free energy functionas

A(, T) =/ C.- d2 + Jq . H-lpdQ = a(, r) + b(q, p).

(23.1b)

The solution space St for the displacement field in the body S2 is the set defined as

St = {u(.,t) E Hl(2))ndi: u(.,t) = (.,t) on Fu}. (23.2)

The associated vector space of admissible displacement variations V consists of vectorfields on 2 satisfying the homogeneous form of the essential boundary conditions,i.e.,

V = H()ndim'

: = O on Fu}.


(23.3)

Assuming the essential boundary conditions are time-independent, then for fixed timet E II the velocity field v(., t) is in V. As in Section 9, we use the notation X =(v, IF) Z where Z = V x [T E]. The space Z is equipped with the natural innerproduct ((., .)) induced by the sum of twice the kinetic energy and the bilinear formA(-, ), i.e.,

((Xl, X2)) = (Pol, 72) + A(T, T 2) VX,X2 C Z. (23.4)

The associated norm is denoted by I I II = 7 -)) and provides the natural normfor the elastoplastic problem since, relative to this norm, the initial boundary valueproblem for dynamic plasticity is contractive.

Let II C IR+ be the time interval of interest and consider an arbitrary partition

II = U: 0o[t, t+l]. Our first objective is to develop the variational form of the two-step generalized return mapping algorithms in Table 15.1 within a typical time step[tn, tn+l] C II for given initial conditions X, = (v,, XZ) E Z and u, C S,. Letu,+l C Sn+l be the algorithmic approximation to the displacement field at time t,+land let Au = u,+l - u, E V be the displacement increment. Consistent with thelocal algorithm in Table 15.1, define the trial stresses at the intermediate and finalstages as

Z+.14 = (a, + CE[Au], qn)(23.5a)

and Z+ = [n+ - (1 -)]/

respectively. The solutions E7,+ and IE,+l of the intermediate and final stages,defined locally as the closest-point projections onto E in the metric G - l , can becharacterized as the optimality conditions of the variational inequalities

A( + - +1 - -.n+ ) < 0(23.5b)

and A(Z.+ - Zn+, T - n+ 1) < 0

for all admissible stresses T E E n T. It should be noted that this characterizationof the two-stage algorithm is rather general and holds even when the boundary ]Eof the elastic domain is nonsmooth. The following result, a generalization of (14.10),furnishes the algorithmic counterpart of (6.8).

THEOREM 23.1. If the elastic domain is the convex set defined by (6.2), i.e.,

E = {X E $ x nit: f() < O, for / = 1, ... , .m}, (23.6)

then the local optimality conditions associated with (23.5b) yield the two-stagealgorithm


mEn+@ = -X+" A-yPVf(X+ +),

p=I

(23.7a)

17n+ = Z,' -a7ey+ Vff(Z+ ),1=l

where A-yL > 0 and Aj" > 0 obey the Kuhn-Tucker conditions

m m

EA'y1f(n+ o) = 0 and ZA¢yif,(X,+l) = 0. (23.7b)L= f t=1

PROOF. Suppose that the optimality conditions (23.7a) hold. Taking the G-inner prod-uct of (23.7a)l with (T - 7n+o) and using the convexity assumption on the functionsf,(.) along with condition (23.7b) 1 yields

(T --En+9) G-' (1X.+ - 17,+o)

= -(T -n+0) * Vf,'(+,)/z=l

= A'y [ff(T) f (n+)]: = A'yf,(T) < 0, (23.8)

since Ay, 0 and f,,(T) < 0 for any T E E. Integrating (23.8) over Q gives(23.5b)1. An identical argument holds for (23.5b) 2. The converse result is provedexactly as in (14.10) by replacing the single constraint in the Lagrangian definedby (14.11) with the m constraints associated with the convex set (23.7a).

REMARK 23.1. For the viscoplastic problem a similar argument shows that the varia-tional inequalities characterizing the algorithmic treatment in Section 23 become

A(17~'nr+ og - 7+0:, T- X,+ Lt) < /[g(J(T)) -g(J(1X+n ))] dO,

(23.9)

At+ - Xn+1T- n+1) < A l [g(J(T))-g(J(17n+,))] d?,

for all admissible stresses T E T which are no longer constrained to lie in E.

The unknowns to be determined in the incremental algorithmic problem are Ln+1 ET, u,+l E S,+l and v,+l E V for prescribed forcing function f(., t) and prescribedboundary conditions t(-, t), t(., t) in [t, t,n+l]; see the statement of the problemgiven in (9.5b). As pointed out above, for the continuum viscoplastic problem thegeneralized stresses 1 E T need not lie within the elastic domain E; a fact alsoreflected in the algorithmic inequality (23.9). On the other hand, inequalities (23.5b)


ensure that in the algorithmic version of rate-independent plasticity both Z andZn,+l, as well as the intermediate stage n+0o, are in E n T.

Viewing the return mapping algorithm as a generalization of the conventional mid-point rule suggests enforcement of the algorithmic counterpart of the momentumequations precisely at the intermediate stage. This leads to the following algorithmwhich is consistent with the generalized mid-point discretization of the initial boundaryvalue problem:

Stage I: Find Au and X,+o = (v,+o, n+o1) such that

I (poAu, ) - (vn+O9,i ) = 0,

t( Po(un+o - v),) + (+.o, L [7]) = (fn+, ) + (n+l, r)r, (23.10)

A(X+ - Z+ T - Zn+ ) < 0,

for all (, T) E Z. The variational equations (23.10) determine the velocity andgeneralized stress Xn+2 = (vn+o, Z+) e Z, along with the displacement incrementAu E V. Finally, the displacement and velocity fields are updated consistent with ageneralized mid-point rule approximation while the generalized stress is computed byenforcing consistency.

Stage IIa: Update Un and v, via linear extrapolations as

Vn+ = v+9 + I - v n,VI'I = + j ;1 Vn,(23.1 a)

Un+l = Un + Au.

Stage IIb: Compute Zn+l for given Z"+l via the return mapping algorithm

A(E+, -Zni'.T- n+i) O VT EnV,(23.1 lb)

where Z+, = [Xn+ - (I - )X.n]/V.

Computational issues involved in the implementation of this problem are deferred toChapter IV where they are addressed within the context of the full nonlinear theory.A brief account of mixed finite element methods suitable for the spatial discretizationof the elastoplastic problem is given in the last three sections of this chapter togetherwith sample numerical simulations.

REMARK 23.2. For viscoplasticity, Eq. (23.10)3 is replaced by (23.9). In general, thestress a,,+l does not satisfy the equilibrium equation at t+l unless the forcing termsare linear in time since the momentum equation (23.10)2 is enforced at the generalizedmid-point configuration at time t,,+o.



24. Nonlinear stability analysis: Uniqueness and dissipativity

This section provides a complete nonlinear stability analysis of the preceding algorith-mic problem and addresses some of the issues involved in a numerical implementation.The notion of stability used in the analysis is dictated by the contractive property ofthe continuum problem. In short, this notion implies attenuation of arbitrarily largeperturbations in the initial data relative to the natural norm of the problem, which is de-fined by the complementary Helmholtz free energy. The nonlinear approach describedbelow should be contrasted with other approaches found in the literature which typi-cally employ linearized notions of stability (see, e.g., HUGHES [1983] and referencestherein).

Let Xn E Z and n G Z be two initial conditions at time t. The precedingalgorithm then generates two sequences {Xn}ncN and {Xn,}nEN. Nonlinear stabilityholds if the algorithm inherits the contractive property of the continuum problemrelative to the natural norm III 111; a property also known as A-contractivity orB-stability. Equivalently, finite perturbations in the initial data are attenuated by thealgorithm relative to the natural norm if the B-stability property holds. To carry out thestability analysis the following preliminary result for the second stage of the algorithmis needed. This property is an immediate consequence of the projection theorem (e.g.,CIARLET ([1988], p. 268).

THEOREM 24.1. Let A7+ 1 = +l - ~-+1, where Znr+l C T and t+1 Tare arbitrary. Then, the second stage of the return mapping algorithm, defined by thevariational inequality (23.11 lb), does not increase the energy norm in the sense that

A(An+i, A,,+l ) - A(A~r+,, I An'+l)

-<-A(An+ - A. +l,r.rn+l, A,,- A'[+,) < 0, (24.1)

where AI7+ = 7n+1 - n+1l is the difference in the solutions of (23.11b).

PROOF. Applying inequality (23.1 lb) to n+l and Zn+, with T E EnT respectivelychosen as T = L',+1 and T = En+1, yields

A(-t+l - v'+1, +1-I +1) < 0,(24.2)

A (±nt+ 1- ±n+ 1, 0-Yn+ - n+ ) < 0.

Adding these two inequalities gives, after adding and subtracting a term, the result

A(A_+ I- Al - 1, [ 1 + + An+ I]

+ [An+l - +1]) 0. (24.3)

Estimate (24.1) then follows from the bilinearity property of A(., .). []

SECTION 24 277

The B-stability property of the two-stage algorithm defined by (23.10) and(23.1 1a,b) is contained in the following result.

THEOREM 24.2. The algorithm defined by the variational problem (23.10) [Stage I]and the update formulae (23.11a,b) [Stage II] is B-stable for > , i.e., the followinginequality holds:

Illxn - xnlll < IIixo - olil, Vn e N, if >. (24.4)

PROOF. (i) First, consider Stage I of the algorithm defined by (23.10) and introducefor convenience the following notation for any 09 C [0, 1]:

AZn+, = n+g - +o and Av,,+ = v~,+ - v,+. (24.5)

By hypothesis, the two sequences generated by the algorithm satisfy the problem(23.10). Choosing successively T = Z,+g and T = ~Z+, using the definition ofE+7t and the notation introduced above yields

a(Ce[Ai4 ao,+ - c&,+) + A( - ,+, AnZ+9) 0,(24.6)

-a(VCE[Au], c,+ - &,+o) - A(Zn - Zn+o, An+)) < 0.

Adding these two inequalities and using (24.5) yields the result

A(A27,+9 - An,, A2]n+O) < Ja (CE[Au - Au], n,,+ - a,+). (24.7)

Since the difference Au - Ai is in V, use of definition (23.11a)1 along with thealgorithmic weak form of the momentum equations yields

a(Ce[Au - Aiu], r - &+,9)

= (a,+g[ E[Au-Aii]) - (,+v, E[u - A])

t (P0(Av,+ - Av), Au- Au)

= -t(Av + l - Av )A Au -Aii), (24.8)

where we have used again the identity 9[Av+o - Av,] = Av,+l - Av, implied bythe linear extrapolation formulae (23.11 a). On the other hand, the variational equation(23.10)1 along with a simple algebraic manipulation yield the local relation

t [Au - Au] = Av,+o = Av,+l/2 + (0a - )[Av,+ - Avj. (24.9)

Inserting this expression into (24.8), and using the identity

K(Av,+t) - K(Av,) = (p0Avn+1/ 2, AVn+ -Avn)


(24.10)


gives, after a straightforward manipulation, the result

a(CE [Au] - AU], c,,+B - n+T)

= -[K(Avn+l) - K(Avn)] - (219- 1)[K(AVn+ - Av,)]. (24.11)

Combining (24.7) and (24.11) one obtains the following estimate:

K(Av,+l) - K(Avn) + -A(AXn+o - A., A+o)

- (V - )2K(Av,+l - Avn). (24.12)

(ii) Now consider Stage II of the algorithm defined by formulae (23.11a,b). Theextrapolation formula (23.1 1b) 2 that defines the trial state implies the identity

An+ = [An+l + AXt'T+] + ( - 2) [AZ"+1 - AX',] (24.13)

along with AZ,+o - Aln = 9[At+, - ALn']. Using the bilinearity of A(., ) wearrive at

1A(AE~n+o - AZ, AZn+9)

+ 2 [A(A.tr+l, A.-r+) - A(An An)]. (24.14)

Combining (24.12) and (24.14) gives the estimate

[2K(Avn+,) + A(A,+ 1, A.+l 1 )] - [2K(Av) + A(An, AZ)]

< -(9- ) [2K(- An+l- AV)

+ A(A2?+ - A7, A+ 1 - AX)]. (24.15)

Since the right-hand side of (24.15) is nonpositive for 9 - 0 if follows that thelinearly extrapolated state is contractive for > . To complete the proof we need toestimate A(AX,+i, AZ,+1) in terms of A(A A,'+, A2+l) using the second stage.This is precisely the result stated in the preceding theorem. Thus, adding inequalities(24.15) and (24.1) gives

IIIx + - - x+1112 - IIX - 1112

= [2K(Avi+l) + A(An+,, AEl)] - [2K(Av) + A(A, An)]

< -(0- 2) [2K(Avn+l- Avn) + A(A?+l1 - A, 7+ - A2)]

- A(ATl+1 - .+ +, - An+l1) (0 for 9 , (24.16)-- A~~~~n~l, A~],~+i (~~~~24.6

which in conjunction with a straightforward induction argument implies (24.4) andcompletes the proof of the contractivity result. [1

An argument entirely analogous to that given above also shows that the incrementalalgorithmic problem for viscoplasticity is A-contractive relative to the natural normII - Ill. In summary, the analysis for dynamic plasticity and viscoplasticity describedabove proves nonlinear stability in the velocity and the stress field X = (v, Z),provided that '>9 2.

REMARK 24.1. Uniqueness of the solution to the algorithmic problem defined by(23.10) and (23.11a,b) follows immediately from the preceding contractivity result.Suppose that X, and X, are two solutions of the problem for the same initial data.Setting Xo = Xo in (24.4) gives 0 < I IX,,-Xn l I 11 0, which implies IIX, l-X,, I = 0and therefore X, = , since * 11I is a norm.

The dissipative property of the initial boundary value problem for dynamic plasticity,encoded in the a priori stability estimate (10.13), is also inherited by the algorithmicproblem defined (23.10) and (23.11a,b) provided that 9 Ž 2. To prove this resultobserve first that the internal energy Vint(e, ~), defined by (10.10), coincides with thetotal complementary Helmholtz free energy function SE2 (I) under the assumption ofa quadratic free energy function, i.e.,

Vint(Ee , ) = ,Q(7) - = A(, ), (24.17)

where A(-, ) is the bilinear form defined by (23.lb). The total energy of the systemthen becomes a function of X e Z defined by the expression

E(X) = K(v) + _En() + Vext(u) (24.18a)

where K(v) is the kinetic energy. For simplicity, attention will be restricted to deadloads, with potential energy given by

Vext(U)= (f,u) + (t, u)r. (24.18b)

Next, consider the general model of multisurface plasticity with elastic domain definedby (23.6) and assume that the functions f,: S x REin -- R are of the form

fl(1) = ,(17) - yf, for = 1, 2,. . , m, (24.19)

where , (.) is convex, homogeneous of degree one. As shown in Example 6.1,from Euler's theorem for homogeneous functions it follows that the model obeys thedissipation inequality Do > 0 in 11. Under these hypotheses we have the followingresult.


THEOREM 24.3. For > 2 the algorithmic initial boundary value problem defined by(23.10) and (23.1 la,b) inherits the dissipative property in the sense that the followingestimate holds

E(Xn+l) - E(Xn) 0 for any n E N (24.20)

where E: Z -- IR is the energy function defined by (24.18a).

PROOE First write the change in complementary Helmholtz free energy as

'(-n+l ) ( ) EDX -(n) (n+) - (+l )]

+ ['Q (nr+l) - 72 (xn)]. (24.21)

The first term on the right-hand side of (24.21) is estimated merely by noting thatEQ (Z) is a quadratic functional. Definition (24.17) together with an elementary iden-tity then yields

[Qn l- 2Q(n+,li)] = A(n+l,±+ l t- )- .EQ(an+ - 1tr)

< A (,+l, 17+ - Znr+1) (24.22)

For the second term on the right-hand side of (24.21), an elementary manipulationtogether with the extrapolation formulation in the first phase of Stage II of the algo-rithm, defined by (23.1 la), yields the following expression for the second term on theright-hand side of (24.21):

,F- ( )- (n) = A(X.r+, + En, + - ,)

= A(Xn+o9, En+l - n,) - (29 - 1) E( _+1 -- )

~< WA(wn+, .,+ - n), (24.23)

provided that 9 Ž 2. Since +l = , + 9(CE[Au], 0), setting ,+Z = r+, +[17+o - n7+g] in the first term of (24.23) and using bilinearity gives the followingestimate valid for > 1:

( +l- () - 1 () A(,n+, .n+ - ,+g)

+ (Z+go, e[Au]). (24.24)

For > , the change in kinetic energy during the time step is easily estimated bymeans of the extrapolation formula (23.1 la), the same identity repeatedly used above


and Eq. (23.10)1 in Stage I leads to

K(v+l) - K(v,,) = - v,v,,+v ) - (20 - I)K(v1,+,)

1< V(t(vn+V - vr, Au,+9). (24.25)

Finally, we observe that under the assumption of dead loads, the change in the potentialenergy of the loading can be written as

Vet(U7,+l ) - Vext(Un) = (f, Au) + (t, Au)F. (24.26)

According to (24.18a), the total energy with the time step [tT, tn+l] is obtained byadding the contributions arising from (24.21), (24.25) and (24.26). By inserting (24.22)and (24.24) into (24.21) and cancelling terms using the momentum equation (23.10)in Stage I we arrive at the following estimate valid for 19 > 1:

E(xn+l ) - E(Xn)

• A(Zn+i, n+l - '+l) + A(Z.n+, ,+ - ) (124.27)

The proof is completed by noting that the algorithmic flow rule (23.7a), arising in theapplication of the algorithm to multisurface plasticity, together with Euler's theoremand the Kuhn-Tucker conditions imply

rn+A(Zn~v, Zn~l-v ) =-E (A jYVfp('ivni), Znvi)

=- - hoJ_ Af(+ ) d? < 0, (24.28)

and similarly A(Zn+i, -n+l - E+!) = 0; hence the result. 1

REMARK 24.2. The preceding result can be easily generalized to the case of an arbi-trary free energy function T (, ~), not necessarily quadratic, but convex in its twoarguments. The required modifications are (i) the closest-point projection and the lin-ear extrapolation formula must be written in terms of the elastic strains and (ii) theproof is carried out directly in terms of Vit(Ee, ~) rather than the complementaryHelmholtz free energy function En (Z). Since this is the setting adopted in the fullnonlinear theory, further details will be omitted.



25. Spatial finite element discretization: An illustration

It is by now well established that displacement based finite element methods maylead to grossly inaccurate numerical solutions in the presence of constraints suchas incompressibility, or nearly incompressible response, see, e.g., HUGHES ([1987],Chapter 4) for a review and an illustration of the difficulties involved in the con-text of linear incompressible elasticity. As first noted in the NAGTEGAAL, PARKS andRICE [1974], the classical assumption of incompressible plastic flow in metal plastic-ity is the source of similar numerical difficulties. Finite element approximations basedon mixed variational formulations have provided a useful framework in the contextof which constrained problems can be successfully tackled. A large body of litera-ture exists on the subject, which has its point of departure in the pioneering work ofHERRMANN [1965], TAYLOR, PISTER and HERRMANN [1968], KEY [1969], and NAGTE-GAAL, PARK and RICE [1974]. Review accounts of several aspects of this exponentiallygrowing area can be found in several textbooks, e.g., CIARLET ([1978], Chapter 7)and more recently in GIRAULT and RAVIART ([1986], Chapter III), JOHNSON ([1987],Chapter 11), ZIENKIEWICZ and TAYLOR ([1989], Chapter 12). A comprehensive reviewof the subject is given in BREZZI and FORTIN [1991].

The goal of this section is to provide an illustration of the implementation of aparticular class of mixed methods, which retain the simplicity and computational con-venience afforded by strain driven return mapping algorithms and, at the same time,properly account for nearly incompressible response. These methods, often referredto as assumed strain methods, has gained considerable popularity in recent years.Direct precedents of this methodology can be found in the reduced and selective-reduced integration techniques originally introduced mostly in an ad hoc fashion inZIENKIEWICZ, TAYLOR and Too [1971], DOHERTY, WILSON and TAYLOR [1969] forthe nearly incompressible problem. The seminal work on the subject appears in thepaper of NAGTEGAAL, PARKS and RICE [1974]. This approach is subsequently gener-alized in HUGHES [1980] where a related methodology commonly referred to as theB-bar method is suggested. A unified approach that identifies these techniques as aparticular class of mixed finite element methods, both for linearized and finite strainelasticity and plasticity, is described in SIMo, TAYLOR and PISTER [1985]. Within thiscontext, for instance, B-bar methods are shown to arise from finite element approxi-mations constructed on the basis of a three-field variational formulation. The fact thatgeneral assumed strain methods can be made consistent with a three-field variationalformulation of the Hu-Washizu type was first pointed out in SIMO and HUGHES [1986].

To develop the class of assumed strain methods considered in this section, the al-gorithmic flow rule and hardening law, as well as the Kuhn-Tucker form of the load-ing/unloading conditions, are enforced locally rather than via the variational inequalitydescribed in the preceding sections. Accordingly, the internal variables {Pn+ n+l}

are determined locally at the quadrature points via the closest-point projection algo-rithm with the strain field regarded as the primary "driving" variable. This point ofview proves particularly useful in the generalization of the techniques described hereinto the finite strain regime. To emphasize this latter aspect, we shall adopt throughoutthis presentation a general stored energy function W(ee) not necessarily quadratic in

SECTION 25 283

the elastic strain tensor. Following a common usage in the finite element literature,in the exposition given below tensor notation is replaced by the following matrix andvector notation:

(i) Rank two tensors in three-dimensional Euclidean space are mapped into col-umn vectors according to the following convention which distinguishes the stress fieldfrom the strain fields:

2eP2 2.2 (22

EP= eP3 E[1]= 3,3 and o'= 0233l (25.1a)2eP2 U,2 -n 2,1 . 12 '

2eP3 U1,3 + U3,1 0 132eP3 U2,3 + 3,2 023

(ii) Rank-four tensors are mapped onto matrices. Of particular importance is theelasticity tensor C = V2 W(e) which in matrix notation takes the explicit form

-C1 11 C1 1 2 2 C1 1 3 3 C1 11 2 C1 1 1 3 C1 1 2 3

C2222 C2233 C2212 C2213 C2223

C33 33 C3 3 12 C3 313 C3 32 3C =_~ (25.lb)C1212 C1312 C2312

sym C1313 C2313

C2 3 2 3

(iii) According to the preceding conventions, the rank-four unit tensor I and therank-two unit tensor 1 become

I = DIAG[l,1, 1, 1, l, 11 and 1=[1 1 1 0 0 O]T . (25.1c)

(iv) Contraction between rank-two tensors is replaced by dot product, and appli-cation of a rank-four tensor to a rank-two tensor reduces to a matrix transformation.Similar conventions apply to two-dimensional problems including plane strain, planestress and axisymmetry.

For simplicity, attention will be restricted to the quasistatic perfectly-plastic case(i.e., no hardening) and the three-dimensional problem ndim = 3 will be assumedthroughout. The class of mixed finite element approximations of interest is constructedfrom the following weak formulation of the elastoplastic problem:

t~("--~.iol sfll~dn 1 .q~dP= 0, (25.2a)( a " + - E[7/]- pof r/) d - t d = (25.2a)

/2' (E[Un+l] - n+l) df2 = 0, (25.2b)

C o [- '+l + VW(En+l - En+)] d2 = 0.


(25.2c)


These variational equations hold for any admissible displacement test functions 7 EV, any stress field r E ;T = L2 (S')6 , any strain field E V = L2(S2)6, andare supplemented by the local algebraic constraints that define the return mappingalgorithm, i.e.,

En+I = En + AyVf (VW(E+ - - (S- 1)[Pn - EP _])), (25.2d)

AY > O, f(VW(En+ - Pn+,) 0,(25.2e)

Ayf (VW(En+ - E)n+l)) = 0.

Observe that in this formulation of the return mapping algorithm for single-surfaceperfect plasticity the (assumed) strain tensor En+l E V,, and not the stress tensor is thefield that plays the role of the independent variable. The treatment of viscoplasticityfollows the same lines as rate-independent plasticity and, therefore, further details willbe omitted.

(A) Discontinuous stress and strain interpolations. The finite element formulationdiscussed below is based on discontinuous interpolations of stress and strain over atypical element S2? C RIndim of a discretization • U=di •2e. Our goal is to recovera displacement-like finite element architecture. We start by introducing a conformingfinite-dimensional approximating subspace Vh c V for the space of test functions,along with a finite element space Th for the admissible stress fields defined as

Th = rh f2 -, R6: Irhjf2 = S(X)ce for c, E Rm}. (25.3)

Here, S(x) is a (6 x m) matrix of prescribed functions, generally given in terms ofnatural coordinates. Explicit examples will be given below. Similarly, one introducesa strain finite element approximating subspace Th defined as

h = {Ch: R2 -- J 6: ChI,2 = F(x)ae for a, E Rm , (25.4)

where F(x) is a (6 x m) matrix of prescribed functions. In general, S(x) 4 F(x).The specific form taken by these prescribed functions will be specified subsequently.In what follows, for notational simplicity, the superscript h will be omitted. Since theapproximations (25.3) and (25.4) are discontinuous over the elements, the variationalequations (25.2b,c) hold for each element •2?. By substituting (25.3) and (25.4) into(25.2b) and solving for the element parameters a, GC R one obtains a mappingun+l E+1 = ~[Un+l] defined as

e[un+l]lC2e = F(x)H-1 ST(x)E[Un+lI] d, (25.5)

SECTION 25 285

which defines the element strain field E,+l n in terms of the displacement fieldu,+ll2e in a typical element S2, where

H=J ST(zx)F(x)d. (25.6)

Similarly, by substituting (25.3) and (25.4) into (25.2c) we find the following finiteelement counterpart of the elastic constitutive equations for the stress field within atypical element F2,:

r,- ,+ = S(X)H-T rT(X)VW(E:[u,±tI d2. (25.7)

It follows that, as a result of the discontinuous (assumed) strain and stress inter-polations defined by (25.3) and (25.4), one obtains a discrete approximation to thesymmetric gradient operator [-.] defined by (25.5), which we shall denote by [.] insubsequent developments, along with an algorithmic constitutive equation defined by(25.7). Remarkably, the following result shows that only the discrete gradient operatore [.] appears in the final expression for the discretized weak form (25.2a). The algorith-mic constitutive equations (25.7) do not enter explicitly in the variational formulationof the problem and are needed only in the stress recovery phase.

THEOREM 25.1. The momentum balance equation in the assumed strain finite elementmethod outlined above has identical form as in a displacement model, with gradientoperator E[-] replaced by the discrete gradient operator [-] defined by (25.5), i.e.,

G(un+jleP,,;n r; ,X W(t[un,+ -P +1) * E[q] dV

- Gext(ij) =0, (25.8)

for all r E Vh, where Gext(.) is the virtual work of the external loads defined by

Gext() = Pof r2dQ + J t. dF. (25.9)

PROOF. Making use of (25.5), (25.6) and (25.7), the variational equation (25.2a) re-stricted to a typical element Qe can be rewritten as follows:

e[rt] a,+ dQ

=J [[i ] [S()H-T / FT(1x)VW([u+i] -E+i) dQ d

= / VW([u n + l] - EP+) [F(x)H - ' J ST (x)e[J] dS dQ

286 J.C. Simo CHAPTER i

= J~ Vw(([un+1 ] - En+J) [/] dQ, (25.10)

which implies the result in the theorem.

Expression (25.8) becomes completely determined once e+ is defined in termsof En+-s for either s = 1 or s = 2, and the discrete strain field en+l = [Un+l].This is accomplished by solving at each quadrature point the local equations (25.2d,e)that define the closest-point projection algorithm. For this purpose the general closest-point projection iteration described in Section 16 can be employed, reformulated instrain space, with the strain field evaluated by means of the discrete gradient operatoras En+l = [un+l]. For completeness, a step-by-step summary of the computationalprocedure for the case of ideal plasticity is contained in Table 25.1. A derivation ofthe expression for the consistent discrete tangent stiffness matrix is given below.

TABLE 25.1

Closest-point projection algorithm in strain space.

(1) Compute the trial stress )I = VW(E[Un+l - En (s3 - 1)[en -

If f(0+ 1) < 0. The quadrature point is elastic.

Set P = EP and terminate the algorithmOtherwise, the quadrature point is in the plastic regime

Set A-y(°) = 0 and proceed to Step 2.(2) Return mapping iteration algorithm (closest-point projection)

(2a) Compute stress residuals for k = 0, 1,...

W = VW (E[u-n1J - e (k))l n+l n+l 1

'(k) &(k)-1

f '(

rn+1 n+l ( nV3,r+l

(2b) Perform the following steps while IIr(kl 11 > TOL:

Compute consistent (algorithmic) tangent moduli

C(k = 2W ([U+l] _ En(k)

C(k)- [C-l +A ()Vf(&kn+l)

Update plastic variables with kth-increments

,(k) rvf &) T~k) (k)Ay(k+

l)= A(k) +!+l + + n

[vf(k) ]T-(k) Vf(k)

p(k+l) ep (k) +C(~d( k l[At7(k) Vf(l -(k)]n+I n+1 n-sl n+ - rn'lj


REMARK 25.1. The algorithm summarized in Table 25.1 is performed at each quadra-ture point. Consistency and loading/unloading are therefore established independentlyat each quadrature point of the element. For the von Mises yield criterion, either inthree dimensions or in plane strain, convergence is attained in one iteration and thealgorithm reduces to the radial return method of WILKINS [1964].

(B) Linearization. Consistent tangent operator. It remains to determine the expres-sion for the tangent stiffness operator associated with the reduced residual (25.8). Here,the notion of consistent tangent moduli, introduced in Section 17 and obtained by ex-act linearization of the closest-point projection algorithm, plays an essential role. Theclosed-form expression for the linearization of the residual (25.8) is easily obtainedin view of the following observations:

(i) The discrete gradient operator g[u.+l] defined by (25.5) is a linear function ofthe displacement field u,+l.

(ii) The plastic strain En+1 and the consistency parameter A7 are nonlinear func-tions of en+l = [un+l] and the plastic strain Ep defined by the algorithm in Table25.1. Since EP+,-. for either s = 1 or s = 2, is a fixed (given) history, the onlyremaining independent variable is displacement field u+ l.

From the preceding two observations it follows that the reduced residual defined by(25.8) becomes a function of the displacement field u,+l. Setting dun+l = Au anddifferentiating expression VW([u,+ - P+l) in the reduced residual (25.8) gives

d [VW ( [,+ - EPn+l])] = Cn+l ([Au] - d+,), (25.11)

where Cn+l := V2 W(g[Un+ -EP +) is the tensor of elastic moduli. As in Section 17,we define the algorithmic elastic moduli C,+l by the expression

Cn+ = [C+ 1, +AyV 2f(VW( [u+I]- E+],))] . (25.12)

Proceeding exactly as in the derivation described in Section 17, de,+1 is determined bydifferentiating the return mapping algorithm and enforcing the algorithmic consistencycondition, to arrive at the following result:

Cn+ ([u+] -+ +l - de+ 1 n+ ()= n i,+) [A],

where nn+l: Cf+l]f+ (25.13)

[Vfn+i]TCn+i Vf+i

The linearization of the reduced equilibrium equation (25.8) in the direction of theincremental displacement Au G V is now easily obtained via a straightforward appli-cation of the chain rule. The result is easily shown to take the following form:

dG (u+ , E,+ 1; 77) [I .

(Cnl -nn+l nA'n+!)£[Au] d. (25.14)



Observe that result (25.14) has an identical structure as a displacement model withthe gradient operator e[-] replaced by the discrete gradient operator ~[].

(C) Matrix expressions. To illustrate expressions (25.8) and (25.14) for the residualand tangent operator in a familiar context, let Na(x), a = 1,..., nnode, denote theshape functions of a typical element e2 C In di m with nnode nodes, so that

nnode

uh = Na () da. (25.15)a=l

Denoting by Ba the matrix expression of the symmetric gradient of the shape functionsNa, the preceding interpolation together with (25.5) imply

nnode nnode

E[UI]h] = ZBada and g[uh]jnc = EBada, (25.16a)a=l a=l

where

Ba = F(x)H- 1 f ST(X)Ba dr. (25.16b)

The contribution of node a in element Qf2 to the momentum equation (25.10) isthen given by

ra =j Bai{VW([Un+] - En+,)} d, (25.17a)

where EPn+ is computed from the algorithm in Table 25.1. Finally, the contribution tothe tangent stiffness matrix associated with nodes a and b of element Q2, is obtainedfrom (25.14) as

kab = B (Cn+l - n+l ® nf+l)Bb d, (25.17b)

where the algorithmic elasticity tensor Cn+l is defined by (25.12).

(D) Variational consistency of assumed strain methods. The foregoing argumentsshow that the preceding three-field mixed variational formulation is equivalent to ageneralized displacement method in which the standard discrete gradient matrix Bain (25.15)2 is replaced by an assumed matrix Ba defined in the present context by(25.16a,b). We shall be concerned here with the converse problem, and consider theconditions for which an assumed strain method with Ba given a priori, not neces-sarily defined by (25.16a,b), is variationally consistent. The question of variationalconsistency is relevant because of the following two reasons:

(i) The convergence analysis of assumed strain methods is brought into corre-spondence with the analysis of mixed methods for which a large body of literature

SECTION 25 289

exists. In particular, the convergence of certain assumed strain methods, such as theB-method for quasiincompressibility discussed in the next section, is readily settledonce the variational equivalence is established.

(ii) By exploiting the variational consistency of the method, one can develop ex-pressions for the stiffness matrix which are better suited for computation, a pointwhich will be illustrated in the next section.

To simplify our presentation, attention is focused in what follows on linear elas-ticity. The results, however, carry over to the nonlinear situation by straightforwardlinearization. Thus, consider an assumed strain method in which strains and stressesare computed according to the following expressions

Ehls2 = Bde and oh e= CBde, (25.18)

where B is given a priori, and we have employed the following matrix notation

B = [BIB 2 ... Bnod] (25.19a)

(de) = [(del (de) . * (dn,. d.) (25.19b)

Assumption (25.18) leads to admissible variations Ch E h and 'rh CE Th given by

(hs = Bae and rh| = CBae , (25.20)

for arbitrary ae E RT" "

XdeXd im, where ndim < 3 is the spatial dimension of the problem.Substitution of (25.18) and (25.20) into the Hu-Washizu variational equations using(25.2) yields

Ge =ae. [I BTCB dQjd -G.XI JQe (25.21a)

0=ae. T BTC[B - B] dQ] de, (25.21b)

o = ae .J T [ Cde + Cde] dS2. (25.21c)

The third equation is satisfied identically. The second equation on the other hand, issatisfied if the following condition holds

BTCB dF = BTCB dQ. (25.22)

This orthogonality condition, first derived in SIMO and HUGHES [1986], furnishes therequirement for an assumed strain method to be variationally consistent. Note thatthe right-hand side of (25.22) yields an equivalent expression for the stiffness matrixwhich is better suited for computation than the standard expression for assumed strain


methods furnished by the left-hand side of (25.22) since B is usually a fully populatedmatrix whereas B is sparse. Further details on implementation are discussed below.

26. Application: A class of mixed methods for incompressibility

As an application of the ideas developed above we consider a generalization of theideas presented in NAGTEGAAL, PARKS and RICE [1974] within the context of a three-field mixed finite element formulation, following the approach described in SMO,TAYLOR and PISTER [1985]. The resulting class of mixed finite element methods can becast into a format suggested in HUGHES [1980] and leads to methods currently widelyused in large scale inelastic computations, see, e.g., HALLQUIST [1984]. Because ofits practical relevance an outline of this methodology is given below along with twoalternative implementations.

The basic idea is to construct an assumed strain method in which only the di-latational part of the displacement gradient is the independent variable. The mainmotivation is the development of a finite element scheme that properly accounts forthe incompressibility constraint emanating from the volume preserving nature of plas-tic flow. The situation is analogous to that found in incompressible elasticity.

(A) Assumed strain and stressfields. According to the preceding ideas, one introducesa scalar volume-like variable E L 2(Q) and then considers the following assumedstrain field

E = dev[e[u]] + 091, (26.1)ndim

where, as usual, dev[] = () - tr()l/ndim denotes the deviator of the indicatedargument. Similarly, one introduces a pressure-like variable p L2 ( Q2) so that theassumed stress field takes the form

r = dev[VW(E - eP)] + pl. (26.2)

For subsequent developments, it proves convenient to rephrase (26.1) and (26.2) inan alternative form in terms of projection operators as follows. Define the rank-fourtensors

Pdev = I---1 1 and Pvol, 1 1. (26.3)ndim ndim

The matrix P defines an orthogonal projection which, therefore, satisfies the followingstandard properties:

PdevPvol = PvolPdev = 0 and Pdev + Pvol = I, (26.4)

together with P2ev = Pdev and PVo1 = Pvo1. It follows that Pdev and Pvo 1 are orthogonalprojections that map a second rank tensor into its deviatoric and spherical parts,

SECTION 26 Integration algorthms 291

respectively. In terms of these projections, the assumed strain and stress fields (26.1)and (26.2) can be written as

E = Pdev [[U]] + - 91 and a = Pdev [VW(e - P)] + pl. (26.5)ndim

In the present context, the variational structure of the assumed strain method takes thefollowing form. We regard {un+1, n+t, p,n+l } as the set of independent variables inthe variational problem and, as before, assume that the plastic strains EPn+ are definedlocally in terms of X,+l and the history of plastic strain En+p _,8 for either s = 1 ors = 2, by the local equations (25.2d,e). Then, we have the following result.

TIEOREM 26.1. For the assumed strain and stress fields defined by (26.5) the weakforms (25.2a,b,c) reduce to:

(E[r] dev[VW(E +l - En+l)] + pdiv[t]) dQ2 - Gext() = 0, (26.6a)

s q(div[un+l] - ,ldQ 0, (26.6b)

" +n + di tr [vw(,+l -Ep+)] dQ 0, (26.6c)

for any rl E V, q E L2(Q) and 0 E L 2((Q).

PROOF. The proof follows by application of properties (26.4). In particular, fromexpressions (26.3) and (26.5) we observe that

Pdevl = 0 and Pdev [E[Un+t] -En+l] = 0. (26.7)

By making use of these relations along with properties (26.4) it follows that

r ([Un,, l - En+,+)

= (Pdev7) ([Un+l] - En+l) + (PvoIr) (E[Un+l] - En+l)

= T * (Pdev[E[Un+l] - En+l]) + I (tr[r])1 (E[u,+l] - e,+l)ndim

= (tr[r]) (tr [e[u,+1] -en+l])ndir

= q(div[un+l] - 9n+1), (26.8)

where q = (l/ndir) tr['r]. This proves the variational equation (26.6b). A similarargument holds for (26.6a) and (26.6c). [



As in the preceding section, we consider discontinuous interpolations of the assumedstress and strain fields, and introduce the following approximating subspace both forthe volume and the pressure fields:

Tvol = {h E L2(/2): Fn( = Fr(x)&e; Oe i}, (26.9)

where FT(x) = [Fl(x),..., m(x)] is a vector of m-prescribed local functions.Setting

H = J F(x)FT(x) dQ, (26.10)

the substitution of these interpolations into the variational equations (26.6b,c) thenyields

+ = di +] =rT(x)H ' F(x)div[un+l1 df2, (26.11a)

Pn+l = r )H 1 Jr(x) -- tr [V(E+ - l) ] df2. (26.1 lb)e J F ~x) dim

On the basis of these finite element approximations to the volume element zn+l andthe pressure field Pn+, two alternative implementations of the method are possible.To alleviate the notation in the exposition of these to approaches, we shall omit super-script h. All the fields involved are understood to be approximated via the precedingfinite element discretization.

(B) First implementation: Assumed strain method. Within the present variationalframework, the first implementational procedure hinges on the following alternativeexpression for the term involving the pressure field in the weak form (26.6a):

j pdiv[rq]d /2 = I tr [VW(n+l - p+)]jdv[7] dQ2, (26.12a)fsiv~rlldn= k ~ ndim

En+ = dev[E[un+]] + (dv[un+l]), (26.12b)ndim

div[un+lt] = T(x)H - ' F(x)div[un+,] d. (26.12c)

This result is easily verified by proceeding exactly as in the proof of expression(25.8). To complete the implementation, we use matrix notation and set

nnode n.ode

div[u,+l] = adn+, and div[u+l] = bda+l, (26.13)a=l a=l

SECTION 26 293

where the vector ba is defined by

bT = FT(X)H- J F(x)bTdQ. (26.14)

Now define assumed strain nodal matrices Ba via the expression:

Ba = [Ba - Bao + B where B=al 1 ( ba anda dimn

Ba 1 ( ba (26.1.5)Tdim

Noting that properties (26.4) imply the relation

B,{dev[VW(en+I - P+)] } = Ba{dev[VW(En+i -~ P) }, (26.16)

by making use of (26.4) it follows that the contribution of node a of element £Q tothe finite element approximation (26.6a) to the equilibrium equation can be written as

re = B{ VWE([u+] - En+I) d2, (26.17)

which is the same result obtained in the preceding section, see Eq. (25.17a). Therefore,the tangent stiffness matrix is again given by expression (25.17b).

(C) Second implementation: Standard mixed method. An alternative implementationof the present assumed strain method can be developed directly from the mixed for-mulation (26.6a,b,c). As will be shown below, the main advantage over the previousimplementation is that computations are performed with the standard discrete gradientmatrix B,, which is sparse, instead of the Ba-matrix which is fully populated. Themain steps involved in the present implementation are as follows:

(i) In addition to computing div[un+l], one also computes P,+I by using (26.12b).(ii) The contribution to the momentum balance equation of node a of a typical

element Qe is now computed by setting

re = B a { pn + l + Pdev [W([unIFj] - eP+ )] } dR, (26.18)

(iii) Setting CP = C - Tl, the contribution to the tangent stiffnessmatrix of nodes a, b of a typical element Q2 is computed according to the expressionas

kah =J Ba [PdevCePPdev] Bb dQ + T(ndim12 ba g b] (1T C ePl) d2

+ J {BaT [PdevC ePl] bb + ba [iT ePdev] Bb} d2. (26.19)

294 J.C. Sino CHAPTER 11

This result is easily verified by noting that Ba = PdevBa + (l/ndim)l ba and usingproperties (26.4).

The advantages of this second implementation become apparent in the common caseencountered in most applications for which the elasticity is linear, with uncoupledpressure/deviatoric response, and plastic flow is isochoric. The reason is that thepressure p,+l is trivially evaluated once 9,f+I is computed from (26.1 lb),, since

Pn+l = Ki9n+1 where K = bulk modulus. (26.20)

Note that no additional computations are involved since only the evaluation of bais needed, which is also required in the first implementation. Furthermore, since thedeviatoric response is uncoupled from the bulk response, no coupling terms arise inthe calculation of the stiffness matrix. Consequently, in this situation we have

C eP1 = 1 PdevCePl = 0, (26.21)

and result (26.19) reduces to

keab = BT[CeP /1 ( 1]Bbd + f [ba (bb]dQ (26.22)

Finally, because of the linearity of the pressure/volume elastic response, the calculationof the second term in (26.22) reduces to a mere rank-one update. This observationfollows from (26.15)1 and definition (26.10) of H, since

;J , [ba ® bb] d

= n [ (x) 0 ba df2T H-' [ F(x) bb dQJ. (26.23)

This is the implementation adopted in the numerical simulations described below.

EXAMPLE 26.1. The finite element method outlined in this section includes, as a par-ticular case, classical mixed methods for which a rather complete accuracy, stabilityand convergence analysis currently exist. Two standard examples are:

(i) Four-node quadrilateral element with bi-linear isoparametric interpolation func-tions Na for displacements, and F = [1] constant over Q2e. Essentially, this is the meandilatation formulation advocated by NAGTEGAAL, PARKS and RICE [1974]. This is awidely used element known not to satisfy the BB condition. However, applicationof a pressure filtering procedure renders the discrete pressure field convergent; seePITKARANTA and STERNBERG [1984].

(ii) Nine-node element with bi-quadratic isoparametric interpolation functions Nafor displacements, and F = [1 x y]T, where (x, y) are defined in terms of natural co-ordinates by means of the standard isoparametric mapping. This is an optimal elementknown to satisfy the BB condition, see BREZZI and FORTIN [1991].


The convergence properties of the class of assumed strain methods described inthis section are the direct consequence of their variational consistency, and follows atonce from the convergence characteristics of the corresponding mixed finite elementformulations.

27. Illustrative numerical simulations

A number of numerical simulations are presented that illustrate the performance of thereturn mapping algorithms and the practical importance of consistent tangent operatorsin a Newton solution procedure. These simulations exhibit the significant loss in rateof convergence that occurs when the elastoplastic "continuum" tangent is used in placeof the tangent consistently derived from the integration algorithm. The overall robust-ness of the algorithm is significantly enhanced by combining the classical Newtonprocedure with a line search algorithm. This strategy has been suggested by a num-ber of authors, e.g., see DENNIS and SCHNABEL [1983], or LUENBERGER [1984]. Thespecific algorithm used is a linear line search which is invoked whenever a computedenergy norm is more than 0.6 of a previous value in the load step (see MATTHIES andSTRANG [1979]). Computations are performed with an enhanced version of the generalpurpose nonlinear finite element computer program FEAP, developed by R.L. Taylorand the author, and described in ZIENKIEWICZ and TAYLOR [1989]. Convergence ismeasured in terms of the (discrete) energy norm, which is computed from the residualvector R(d+l ) and the incremental nodal displacement vector Ad,+ as

AE(d(') = [Ad+)] TR(d) +l). (27.1)

Alternative discrete norms may be used in place of (27.1), in particular the Euclideannorm of the residual force vector. In terms of the energy norm (27.1) our terminationcriteria for the Newton solution strategy takes the following form

AE(d() ) < 10-9AE(d()l ). (27.2)

While it would appear that this convergence criterion provides an exceedingly severecondition difficult to satisfy, it will be shown through numerical examples that criterion(27.2) is easily satisfied with a rather small number of iterations when the consistenttangent operator is used.

All the numerical simulations described below employ a four-node quadrilateralwith bilinear isoparametric interpolations for the displacement field and discontinuouspressure assumed and volume fields assumed to be constant within a typical element.The results for the stress field and the internal variables reported below are computedvia an L2-projection of the values at the quadrature points of a typical element onto thenodes that uses the bilinear interpolation functions. Related "smoothing" proceduresare discussed in ZIENKEWICZ ([1977], Section 11.5, and references therein), and areoften used as a device for filtering spurious pressure modes of this particular element(e.g., LEE, GRESHO and SAM [1979]).



(A) Thick wall cylinder subject to internal pressure. An infinitely long thick-walledcylinder with a 5 m inner radius and a 15 m outer radius is subject to internal pressure.The properties of the material are E = 70 MPa, v = 0.2. The isotropic and kinematichardening rules are of the exponential type, defined according to the expression:

K() = K - [K - Ko] exp[-b] + H,

~K' ~~~({~) ~= ~K (), ~(27.3)H() = (1 - 3)K((), EC [0,1].

The values i3 = 0 and 3 = 1 correspond to the limiting cases of pure kinematicand pure isotropic hardening rules, respectively. The simulation is performed withclassical backward Euler radial return method described in Section 18 (correspondingto s = 1), with the parameters in (27.3) chosen as

Ko = 0.2437 MPa, K, = 0.343 MPa,(27.4)

6 = 0.1, H=0.15MPa, 3 = 0.1.

The internal pressure is increased linearly in time until the entire cylinder yields.The finite element mesh employed in the calculation is shown in Fig. 27.1(a). The sizeof the time step is selected as to achieve yielding of the entire transverse section intwo time steps involving plastic deformation. The stress distribution in the transversedirection (i.e., all) at time t = 0.0875, corresponding to an elastic-plastic solution,is shown in Fig. 27.1(b). The calculation is performed with both the "continuum"and the "consistent" elastoplastic tangent, and the results are shown in Table 27.1. Inspite of the better performance exhibited by the "consistent" tangent, no substantialreduction in the required number of iterations for convergence is obtained except in

STRESS 1Min = -2.85E-01Max = -1.24E-03

PUMPZ -2.45E-01

(a)

(b) .

-2.04E-01

-1.64E-01

-1.23E-01

-8.24E-02

-4.18E-02

Current View0 Min =-2.85E-01

X = 5.OOE+00Y = .OOE+00

X Max = -1.24E-03X = 1.50E+01Y = .OOE+00

FIG. 27.1. Thick-wall cylinder. (a) Finite element mesh consisting of 40 elements (Q/PO). (b) Contours ofthe stress component Arl at time t = 0.0875 (backward Euler return mapping).

I111! I HHH H II HN HTHII HI

SECTION 27 297

CHAPTER 1

TABLE 27.1

Simulation (A). Iterations for each time step.

Step 1 2 3 4 5

State el el-pl el-pl pl pl

Continuum 2 6 9 10 6

Consistent 2 5 7 5 3

FINITE ELEMENT MESH (780 nodes, 722 elements)

FIG. 27.2. Plane strain strip with a circular hole. Finite element discretization. Stabilized mixed method(Qi/PO) with backward Euler return mapping.

the fully plastic situation. This is due to the extreme simplicity and well-posedness ofthe boundary value problem at hand: essentially one-dimensional. The next examplewill confirm this observation.

(B) Perforated strip subject to uniaxial extension. We consider the plane strain prob-lem of an infinitely long rectangular strip, of dimensions 10 x 12m, with a circularhole of radius R = 5 m in its axial direction, subjected to increasing extension in a di-rection perpendicular to the axis of the strip and parallel to one of its sides. The elasticproperties of the material are taken as E = 70MPa, v = 0.2, and the parameters inthe saturation type of hardening rule (27.3) are K0 = Kc = 0.243 MPa, H = 0, anda = 1 (perfectly plastic behavior). Loading is performed by controlling the vertical dis-placement of the top and bottom boundaries of the rectangular strip. The finite elementmesh is shown in Fig. 27.2. For obvious symmetry reasons, only 1 of the strip is con-sidered. The contours of shear stress component r12 are shown in Fig. 27.3 after 5 stepsof At = 0.0125, computed with the classical radial return algorithm (backward Euler).

298 J.C. Simo


STRESS 4Min = -3.72E-02Max= 1.74E-01

-7.04E-03

2.32E-02

5.34E-02

8.36E-02

1.14E-011.44E-01

Current ViewO Min = -3.72E-02

X = 6.18E-01Y = 4.96E+00

X Max = 1.74E-01X = 6.05E+00Y= .OOE+00

FIG. 27.3. Plane strain strip with a circular hole. Contours of shear stress after 5 steps of At = 0.0125(backward Euler return mapping).

STRESS 5Min = .OOE+00Max= 6.08E-02

|0zqle -ii ,, 5.50E-025.52E-02

5.54E-02

5.56E-02

5.58E-02

5.60E-02

Current View< Min = .00E+00

X = 4.84E+00Y = 2.66E+00

X Max= 6.08E-02X = 6.16E+00Y= 3.41E+00

FIG. 27.4. Plane strain strip with a circular hole. Elastic-plastic boundary obtained in single time step ofAt = 1.5 with backward Euler return mapping.

SECTION 27 299

CHAPTER II

TABLE 27.2

Simulation (B). Iterations for each time step.

Step 1 2 3 4 5

State el el-pl el-pl el-pl el-pl

Continuum 2 13 23 23 22

Consistent 2 5 5 4 5

FINITE ELEMENT MESH (765 nodes, 704 elements)

FIG. 27.5. Plane stress strip with a circular hole. Finite element mesh. Standard Galerkin finite elementmethod.

The calculation is performed with both the "continuum" and the "consistent" tangentoperators, and the number of iterations required to attain convergence is summarizedin Table 27.2. The superior performance of the "consistent" tangent is apparent fromthese results. This example also provides a severe test for the global performance ofthe Newton solution strategy. Although the calculation is completed successfully witha time step of At = 0.0125, for twice this value of the time step the iteration proce-dure diverges. However, when the Newton solution procedure is combined with a linesearch procedure, as described in MATTHIES and STRANG [1979], global convergenceis attained in a single time step of size At = 1.5. The elastic-plastic interface is shownin Fig. 27.4.

(C) Extension of a strip with a circular hole in plane stress. The geometry and finiteelement mesh for the problem considered are shown in Fig. 27.5. A unit thickness isassumed and the calculation is performed by imposing uniform displacement control

300 AC. Simo


STRESS 5Min = 6.82E-02Max= 2.11E+00

1 .; : ~9.99E-019.99E-019.99E-011.OOE+001.OOE+001.OOE+00

Current View0 Min = 6.82E-02

X = .OOE+00Y = 8.06E+00

X Max= 2.11E+00X = 4.71E+00Y = 1.68E+00

FIG. 27.6. Plane stress strip with a circular hole. Elastic-plastic interface after two time steps of At = 0.06.Backward Euler return mapping algorithm.

on the upper boundary. For obvious symmetry considerations only one-quarter of thespecimen need be analyzed. A total of 164 4-node isoparametric quadrilaterals with bi-linear interpolation of the displacement field are employed in the calculation. It shouldbe noted that for plane stress problems no special treatment of the incompressibilityconstraint is needed. A von Mises yield condition with linear isotropic hardening isconsidered together with the return mapping algorithm outlined in Section 18 for s = 1(classical backward Euler method). The elastic constants and nonzero parameters inhardening law (27.3) are E = 70, v = 0.2, K0o = K = 0.243, H = 2.24, and /3 = 1.

The problem is first solved using prescribed increments of vertical displacement onthe upper boundary of 0.04 followed by three subsequent equal increments of 0.01.The resulting spread of the plastic zone is shown in Fig. 27.6. Note that spread ofthe plastic zone across an entire cross-section is achieved in the third load increment.Only seven iterations per time step are required to reduce the the H-energy normof the residual to a value of the order 1 x 10 -3 3 at an asymptotic rate of quadraticconvergence. No line search is required for this step-size. Finally, as in the precedingexample, the robustness of a solution procedure that combines Newton's method withline search and the exact algorithmic moduli is demonstrated by solving the problemin a single time step of size At = 1.5. Although the entire specimen is in the fullyplastic regime during the first two Newton iterations, the solution procedure is ableto produce a solution of the problem for this extremely large time step.

SECTION 27 301

CHAPTER 11

STRESS 3Min = -435E-02Max= 3.61E-01

1 .43E-02

7.21E-02

1.30E-011.88E-01

2.46E-01

3.03E-01

Current View0 Min = -4.35E-02

X = 2.78E+00Y = 4.16E+00

X Max= 3.61 E-01X = 5.00E+00Y= .OOE+00

FIG. 27.7. Plane stress strip with a circular hole. Shear stress contours for a solution obtained with twotime steps of At = 0.06. Backward Euler return mapping.

STRESS 5Min= 2.15E-01Max= 1.22E+01

- -; 9.90E-01

9.92E-01

9.94E-01

9.96E-01

9.98E-011 .OOE+00

Current ViewO Min = 2.15E-01

X = .OOE+00Y = 6.75E+00

X Max= 1.22E+01X = 1.00E+01Y = 7.50E+00

FIG. 27.8. Plane stress strip with a circular hole. Elastic-plastic interface for a single time step of At - 1.5.Backward Euler return mapping.

302 J.C. Simo

CHAPTER III

Nonlinear Continuum Mechanics and

Multiplicative Plasticity at Finite

Strains

While the formulation and numerical analysis of classical plasticity are now wellestablished, extensions to the finite strain regime have been the subject of some con-troversy only settled, at least partially, in recent years. Up to the mid-eighties, conven-tional schemes for finite strain plasticity exploited a formulation of the elastic stressrelations directly in rate-form, based on the notion of a hypoelastic material (seeTRUESDELL and NOLL [1972]). See, for example the review articles of NEEDLEMANand TVERGAARD [1984] and HUGHES [1984]. More recently, computational approachesdirectly based on mechanically sound models employing a multiplicative decomposi-tion have received considerable attention in the literature. The idea of exploiting themultiplicative decomposition in a computational setting is suggested in ARGYRIS andDOLTSINIS [1979, 1980], within the context of the so-called naturalformulation, butis subsequently abandoned in favor of hypoelastic rate models; see ARGYRIS, DOLTSI-NIS, PIMENTA and WiUSTENBERG ([1982], p. 22). Independently, the first successfulcomputational approach entirely based on the multiplicative decomposition appears inSIMO and ORTIZ [1985] and SIMO [1986].

A fairly complete account of the current status of numerical analysis issues in finitestrain plasticity will be given in the following chapter. The objective of this chapteris to outline the continuum basis and the mathematical structure underlying modelsof plasticity at finite strains based on a multiplicative factorization of the deformationgradient. The material in this chapter is organized as follows.

First, to make the exposition self-contained, a number of basic results on nonlinearcontinuum mechanics used in the subsequent treatment of nonlinear plasticity arebriefly summarized. This account is intended to provide only a brief overview ofthose aspects of the nonlinear theory relevant to the problem at hand and shouldnot, by any means, be considered exhaustive. Detailed expositions of the subject canbe found in the classical treatises of TRUESDELL and TOUPIN [1960], TRUESDELLand NOLL [1965], and the more recent accounts of GURTIN [1981], OGDEN [1984],

303

and CIARLET [1988]. For detailed expositions of the geometric structure underlyingcontinuum mechanics see MARSDEN and HUGHES [1983], and SMO, MARSDEN andKRISHNAPRASAD [1988].

A detailed formulation of micromechanically based constitutive models of finitestrain multiplicative plasticity is considered next. The last part of this chapter thensummarizes the weak formulation, in the Lagrangian setting, of the quasilinear ini-tial boundary value problem arising in nonlinear solid mechanics and describes theformulation of dynamic plasticity at finite strains. Topics treated there include a (for-mal) a priori stability estimate that demonstrates the dissipative character of the initialboundary value problem along with a description of the structure of the incrementalor rate version of this problem.

28. Basic kinematic results in nonlinear continuum mechanics

We shall denote by Q C IRnd ' , 1 < ndim 3, the reference placement of a continuumbody with particles labelled by X E 2. For present purposes it suffices to regard Q2as an open bounded set in RIndim with smooth boundary a2. A smooth deformationof the continuum body is an embedding of the reference placement into R3, i.e., aninjective, orientation preserving map qo: - I 'i m. The set S = Wo(Q) is referredto as the current placement of the continuum body, with points designated by x CS. The deformation gradient F: 2 - GIL+(ndim) is the Frechet derivative of thedeformation o. We use the standard notation

F(X) = DWp(X) with J(X) = det [F(X)] > 0 (28.1)

and, following a standard convention, often omit the explicit indication of the argu-ment X. The orientation preserving condition J(X) > 0 models the local impenetra-bility of matter. The right and left Cauchy-Green tensors are mappings from Q2 intoS+ respectively defined as

C= FTF and b= FFT. (28.2)

According to the polar decomposition theorem, a special case of the singular valuedecomposition, at any X G Q2 the deformation gradient can be decomposed as

F(X) = R(X)U(X) = V(X)R(X), (28.3)

where R(X) G SO(ndim) is a proper orthogonal tensor, called the rotation tensor,and U(X) E $+, V(X) S+ are symmetric positive definite tensors called the rightand left stretching tensors, respectively. We shall denote by IA (A = 1,.. ., nldim) theprincipal invariants of either C or b. For din = 3 the principal invariants are definedby the standard expressions

I, = tr[C], = (I12 - tr[C]2 ) and 13 = det[C].I7

304 J. C. Simo CHAPTER III

(28.4)

Nonlinear continuum mechanics

Since both C and b are symmetric and positive definite at each X E Q2, a standardresult in linear algebra implies the existence of the spectral decompositions

ndim ndim

C = A2N(A) ® N(A) and b=E S2n(A) n(A), (28.5)A=l A=1

where A > 0 are the eigenvalues of either C or b, while N(A) and n(A) are therespective unit eigenvectors as defined by the eigenvalue problems

CN(A) = A2N(A) and bn(A) = A2 n(A) (A = 1,.. ., ndim). (28.6)

The polar decomposition theorem together with the unit length normalization of theeigenvectors imply the relation

b = RCRT so that n(A) = RN(A) with IN(A) I = In(A) I = 1. (28.7)

For ndim = 3 the triad {N(1), N(2 ), N(3 )} defines the so-called Lagrangian axes orprincipal directions at a point X E Q2 in the reference placement. On the other hand,the triad {n(l), n(2 ), n(3)} defines the so-called Eulerian axes located at x = qo(X).The spectral decompositions of the deformation gradient and the rotation tensor thentake the form

fndim ndim

F = E AAn(A) N(A) and R = E n (A) ® N(A). (28.8)A=l A=1

The eigenvalues AA (A = 1,.. ., ndim) are also known as the principal stretches at apoint, which are directed along the principal material directions N(A). The squaresof the principal stretches (i.e., the eigenvalues of either C or b) are the roots of thecharacteristic polynomial defined, for ndim = 3, as

p(A2 ) = ,6 _ I 114 + 22 _ 3. (28.9)

Finally, from (28.6) the spectral decompositions of the right and left stretching tensorare given by

3 3

U = A . AN (A ) N() and V = E An(A ) n(A) (28.10)A=1 A=1

The preceding decompositions play a crucial role in the closed-form numerical im-plementation of the polar decomposition discussed below.

First, the roots of the characteristic polynomial can be computed in closed form viathe well-known solution for a cubic equation, as summarized in Table 28.1. Second,

SECTION 28 305

the rank-one matrices of principal directions can also be computed in closed form viathe following result. (The case ndim = 3 is assumed throughout).

THEOREM 28.1. The rank-one tensors of principal directions associated with b aregiven by:

Case 1. Three different roots, 1 #) A2 A3:

[B Al LC AJ

where A f B C denotes a cyclic permutation of the indices { 1,2, 3}.

Case 2. Two equal roots, A: = A1 A2 # A3:

3® n L3)[1--A - (3) n(3)] = [A2 A2]. (28.11b)

Case 3. Three equal roots, A: = Al A = A32 = : b = A21. (28.1Ic)

PROOF. Consider first Case 1. The spectral decomposition of b implies

3

[b - Al = ] (A2 -2 )n(A) n(A) (28.12a)A=-A#B

and

[b- A21] = (A - 2)n(A) O n A). (28.12b)A=lAOC

Result (28.11a) then follows by multiplying (28.12a,b) and noting that n(A) n(B) =6AB. Similarly, for Case 2, the spectral decomposition of b now gives

[b - A21] = (A2 - A2 )n( 3 ) n(3) (28.13a)

and

[b - 321] = (A2 - A2 ) [1-n() X ®(3)] (28.13b)

which implies (28.11b). Case 3 is obvious since b is a spherical tensor.

Note that the formula for the principal direction associated with a single root remainsuniformly valid in the two cases above since squaring (28.13a) gives (28.11 la). The

306 IJ.C. Simo CHAPTER II


preceding formulae can be used to obtain explicit expressions leading to a closed-form algorithm for any isotropicfunction of C and b; see MORMAN [1987] and SIMOand TAYLOR [1991]. These expressions, although explicit, depend on the number ofrepeated eigenvalues. However, for the case of the square root, the preceding formulaecan be combined into a unified, singularity-free expression which encompasses allthree different cases. In fact, U can be written as (see HOGER and CARLSON [1984a,b],and TING [1985])

U = -- i [ -3 [-c 2 + (i - i2 )C + ili31], (28.14)

where iA (A = 1,2, 3) are the principal invariants of U. A similar expression canbe derived for the inverse tensor U - as is given in Table 28.1 where the closed-form algorithm for the polar decomposition is summarized. The derivation of theseformulae involves a systematic use of the Cayley-Hamilton theorem.

(A) Motions. The Lagrangian description. Next, we summarize a number of resultspertaining to the motion of continuum bodies. Recall that a motion of a continuumbody is a one-parameter family of configurations indexed by time. Explicitly, let I1 CR+ be the time interval of interest. Then, for each t G II, the mapping pt : Q2 -- ]Rndimis a deformation which maps the reference placement Q onto the current placementSt = pt(Q) c IR3 at time t. We write

x = t(X) = (X,t) (28.15)

for the position of X cE Q at time t. In the Lagrangian (or material) description of themotion, the coordinates that define the position of particles in the reference placementQ of the continuum body are taken as the independent variables. Accordingly, thematerial velocity, denoted by V(X, t), is the time derivative of the motion while thematerial acceleration is the time derivative of the material velocity, i.e.,

V(X,t) = atqp(X,t) and A(X, t) = atV(X,t) = aW(X,t). (28.16)

For fixed time Vt = V(., t) and At = A(, t) define the material velocity andacceleration fields at time t E II, respectively. Thus, the motion, the material velocity,and the material acceleration fields are associated with material points X E 2, andhence parameterized by material coordinates. In components we have

Va(XA, t) = (a(XA,t) and Aa(XA,t)= a (28.17)at at2

where {XA} = (XI, X2 , X3 ) are the Cartesian coordinates of a material point X G Q2relative to an inertial frame, and {za} = (, X2, X3) are the coordinates of a pointx St. When the material coordinates {XA} are the independent variables onespeaks of the Lagrangian or material description of the motion. Note that the mappingt I 1- qt(X)]x=fixed gives the trajectory of the material point X in the timeinterval II.

SECTION 28 307

TABLE 28.1

Algorithm for the polar decomposition.

(i) Compute the squares of the principal stretches 12 (A = 1, 2, 3) (the eigenvalues of C) by solving(in closed form) the characteristic polynomial as follows:

Define the coefficients

b = 12 - I/3,

c -27I+ -I3.

If (bl < TOL) then set

XA = -C/3.

Otherwise, if (bl > TOL) then set

m=2-b/3,

3cn=

mb'

t= arctan [¥1-/n2/g] /3,

XA = m cos [t + 2(A - 1)r/3].

The eigenvalues are then

IA = A + 11/3.

(ii) Closed form evaluation of the stretch tensor U.Compute the invariants of U

il = 11 + 12 + 13,

i2 = 1112 + 1113 + 1213,

i3 = 111213.

Set

D = ili2 - 3 = ( + 12 )(11 + 13)(12 + 13) > 0.

U=- - C2 + (i2 - i2)

C + i 31],

U- l

-[C- iU+i21 ].23

(iii) Closed-form evaluation of the rotation tensor R.

R= FU 1 .

* The function datan2 (n, / - n2) = arctan (/1 - n2/n) should be used instead of the arccosine

function dacos (n) in either FORTRAN or C implementations to avoid the ill conditioning near theorigin.

308 J.C. Simo CHAPTER III


(B) The Eulerian description of the motion. The Eulerian or purely spatial descrip-tion of the motion is obtained from the material description by changing the inde-pendent variables from fixed material coordinates (of a particle) to current positionsin Euclidean space. At any time t G the spatial velocity and acceleration fields,respectively denoted by v(x, t) and a(x, t), are obtained via the change of variables

V(X, t) = v (o(X,t),t) and A(X,t) = a(qo(X,t),t). (28.18)

Since Vt is a deformation for all t, the condition J = det[Dot] > 0 in 2 insures thatSot is invertible and the preceding relations become

v=Voqo, 1 and a=Ao ot 1. (28.19)

The material time derivative of a spatial field, denoted by a superposed dot, is thetime derivative holding the particle fixed. The material time derivative of the spatialvelocity field is the spatial acceleration field. By the chain rule, one has the well-knownrelation

i = a = atv + (v V)v. (28.20)

The tensor Vv(x, t) with components a Va(xb, t)/aXb is referred to as the spatial ve-locity gradient. Here the notation at (.)(x, t) implies taking the partial time derivativeof the spatial field holding the current position x fixed. In general if a (x, t) is aspatial tensor field its material time derivative, denoted by &(x, t), is defined by theformula

a& = [ (a o t)] o p'. (28.21)

With a slight abuse in notation, a superposed dot will often be used to designate boththe material time derivative of a spatial field and the time derivative of a material

field. Thus, for a material field (.)(X, t), the notations at(.) and (.) are equivalent.

(C) Classical rate of deformation tensors. The time rate of change of the deformationgradient F = D t is obtained from (28.1) as atF = F = GRAD[V]. One refers toGRAD[V], with components av, /aXB, as the material velocity gradient. By the chainrule

F = GRAD[V] = GRAD[V o ot] = (Vv o ot) Do, = (Vv o tp,)F. (28.22)

This relation implies the expression Vv = FF - o -' for the spatial velocitygradient. The symmetric part of the spatial velocity gradient, denoted by d, is knownas the rate of deformation tensor whereas its skew-symmetric part, denoted by wv, isreferred to as the spin tensor. Since the Lagrangian description of the motion is almost

SECTION 28 309

universally used in solid mechanics, we append a composition with the motion to theusual definitions of d and iv, and set

d = I [Vv + (Vv)T] o Wt and :ii = [v - VVT] Pt (28.23)

The spatial vorticity field w is defined as twice the axial vector of the Eulerian spintensor i o pt

1. Recall that the axial vector of a skew-symmetric tensor is uniquelycharacterized by the relation ih = w x h, for any h G IR3. This definition yieldsthe standard result w (x, t) = curl[v(x, t)] for the spatial vorticity field. The followingrelation will be useful in our algorithmic treatment of plasticity. From (28.22) and(28.2) one concludes that

= TF + FTF = FT [(Vv o t)T + Vv o Pt] F = 2 FTdF. (28.24)

The result C = 2FTdF justifies the name material rate of deformation tensor oftengiven to 2C.

(D) The rotated rate of deformation tensor. The local configuration obtained byapplying the rotation tensor to a neighborhood Ox of a point x in the current con-figuration St is known as the rotated configuration. Such a configuration is local inthe sense that the "rotated neighborhoods" do not "fit together" unless the deforma-tion of the body is homogeneous. The rotated rate of deformation tensor is defined asDR = RTdR. An alternative expression can be derived using the polar decompositionas follows:

C = 2URTdRU < DR = RTdR = C 1 /2CC-L/2. (28.25)

The rate of change of the rotation tensor is given in terms of a skew-symmetric tensorQ according to the conventional expression

= RRT with + hT= 0. (28.26)

Since n(A) = RN(A), the vector field 1? associated with the skew-symmetric tensorfield Q, defined by the relation ha = x a for any a i R3, gives the relative angularvelocity between the Eulerian and Lagrangian triads. This result can be verified asfollows. Denote by W the angular velocity of the Lagrangian triad, so that

N(A) = W x N(A) and set = RW. (28.27)

Time differentiation of the relation n(A) = RN(A) and use of (28.27) together with(28.26) and the well-known property Q(al x a2) = Qal x Qa2, for any Q E SO(3),then gives

i:(A) = ? X n(A) + R(W x N(A)) = ( + Q-) x n(A)

310 JC. Simo CHAPTER TI

(28.28)


It is clear from (28.27) that 2 gives the velocity of the Lagrangian axes expressedin the Eulerian triad and, therefore, measures a physical quantity that has little to dowith the (spatial) vorticity field.

REMARK 28.1. Likewise, the skew-symmetric tensor h does not coincide with thespin tensor iv. In fact, a direct computation gives

F = RRTRU + RUU lRTF = [ + (R&U -)RT]F. (28.29)

Combining (28.23)2 and (28.29) results in the expression

w = D + Rskew[UU-']RT , (28.30)

where skew[-] indicates the skew-symmetric part of the rank two tensor []. Thephysical significance of this result agrees with the fact that i, and h measure differentphysical objects. While the spin tensor gives the instantaneous angular velocity of thetriad defined by the eigenvectors of the rate of reformation tensor d, the tensor wmeasures the relative velocity of the Eulerian axes n(A) (obtained by "freezing" theLagrangian axes). Expression relating wii and i2 to different kinematic tensors can befound in MEHRABADI and NEMAT-NASSER [1987].

29. Stress tensors and alternative forms of the equations of motion

We denote by o the Cauchy stress tensor, a symmetric tensor field defined on thecurrent configuration St of the body. In addition, we denote by P the nonsymmetricnominal stress tensor, also known as the first Piola-Kirchhoff tensor. This tensor isrelevant to a Lagrangian description of continuum mechanics. We have the standardrelations

= PFT and r=Jo o t. (29.1)

The stress field , parameterized by material coordinates but defined on the currentplacement, is known as the Kirchhoff stress tensor field and differs from the Cauchystress tensor field by a factor of J and a composition with the motion. The convectedsecond Piola-Kirchhoff stress tensor field S, parameterized by material coordinatesand defined on the reference placement, is defined by

S = F-'P = F-lTF- T. (29.2)

In index notation one has the component expressions Tab FaASABFbB = PaAFbA.All the stress tensors introduced above are conjugate to associated rate of deformationtensors through the following important stress-power relations

7Pint = r · d = P · P= 1S ·-.

SECTION 29 311

(29.3)

In index notation this relation reads Pint = Tab dab = PaAFaA = SABCAB

REMARK 29.1. The nonsymmetric stress tensor defined by the relation T = RTP =US and called the Biot stress, is a stress measure preferred by a number of authors,see, e.g., OGDEN [1984]. To find its conjugate strain measure one introduces the polardecomposition to obtain

P F=P (RTRU + RU)

- PF T + RTP = * + sym[T] · U, (29.4)

where sym[. ] indicates the symmetric part. Since r is symmetric and ? = RRT isskew-symmetric, it follows that r · = 0 and P. F = sym[T] U. Therefore, the(symmetric part) of T is conjugate to the right stretching tensor. The skew-symmetricpart of the Biot stress is a reaction force in the sense that it does not enter in theexpression for the stress power. Other stress tensors and conjugate strain measurescan be introduced by a formalism due to Hill; see OGDEN [1984].

The two forms of the local equations used in numerical treatments of continuummechanics problems are the Lagrangian and the Eulerian description. In the Lagrangianformulation-the primary description used in solid mechanics-the motion itself isthe primary variable in the problem, and the density function p0: •2 - R+ in thereference placement is a part of the given data. In the Eulerian description the spatialvelocity field and the current density Pt: St x --- R+ are the primary variables inthe problem.

(A) Lagrangian description. Let f2? denote the boundary of Q. Assume that thedeformation is prescribed on r, c a0 while the nominal traction vector tN isprescribed on the part of the boundary FT C 2, with unit outward normal N,as

so = on Fr x and tA = PN = t on FT x , (29.5)

where r,.n rT = 0 and F U rFT = an2. The local equations of motion in theLagrangian description then reduce to the first-order system

at(p~o,) = poV in Q2 x I, (29.6)at(poV) = DIV[P] + B

where B(., t) is the nominal body force per unit of reference volume and DIV[.] is thedivergence operator in material coordinates, i.e., DIV[P] = PA/XA. Conservationof mass in the Lagrangian description is merely the statement that atpo = 0 in Q2.



(B) Eulerian description. In the purely spatial or Eulerian description, the counterpartof Eqs. (29.6) is formulated on the current placement St = Wt(2) of the continuumbody, with the motion qo and its time derivative V replaced by the spatial density Ptand the spatial velocity field v as the independent variables. Equation (29.6)1 is thenreplaced by balance of mass, obtained by time differentiation of the basic relationpo = J(Pt o ot) and use of the chain rule leading to

atpt + (Vpt) v + ptdiv[v] = 0 in St x . (29.7)

In arriving at this result, use is made of the well-known relation t J = J div[v] ocpt forthe time derivative of J = det[Dot]. Observe that, according to (28.21), the first twoterms in (29.7) comprise the material time derivative of the spatial density functionpt(x) = p(x, t). The Eulerian form of the equation of motion (29.6)2 is obtainedfrom (29.6)1 by the change of variables X - x = Wot(X). Using the well-knownPiola transformation DIv[P] = Jdiv[to] o Wot and expression (28.20) for the spatialacceleration field, the continuity equation (29.7) and spatial version of the momentumequation (29.6) yield the following first-order system written in conservation form:

atpt = -div[ptv] in St x II, (29.8)Or(ptv) = -div[ptv ® v] + div[a-] + bt

where bt = (B/J)o o 1- is the body force per unit of volume in the current placement

and div[-] is the divergence operator in spatial coordinates, i.e., (div[a])a = aab/aXb.

REMARK 29.2. The spatial or Eulerian form of the equations is not suited for theformulation of the initial boundary value problem in solid mechanics in the presenceof anisotropic constitutive response (this notion is briefly described below). The reasonbeing that it is not possible to characterize the constitutive response of an anisotropicsolid solely in terms of information on its current placement St. Such a characterizationis possible only if the material is isotropic. The assumption of isotropy has wideapplicability in fluid mechanics, the linear viscous Navier-Stokes model being theclassical example, but is unduly restrictive in solid mechanics. For this reason, theEulerian description of the motion is abandoned in all of the subsequent developments.

30. Objective transformations and frame invariance

As a first step in the generalization of the elastoplastic constitutive equations to thefinite strain regime we consider the formulation of classical finite strain elasticity. Theprinciple of objectivity plays a central role in this subject.

Given a motion Wpt: Q2 - RIdim with t e I, a superposed rigid body motion of thecurrent placement St = Wt((2) is a map It: St --E ]Rdi-m defined as

x E St H- XZ = ibt(x) = r(t) + Q(t)x cG ndi, (30.1)

where t - r(t) E I Tndim and t H Q(t) E SO(ndim) are arbitrary functions of time. Inwhat follows we consider the case ndim = 3. The superposed motion is called rigid

SECTION 30 313

because for any two given points xl, x2 St, since Q(t) SO(3) is orthogonal wehave

x + - x = Q(t)[I - 2 ] :1 ! - 2 1= - X - 2, (30.2)

where Ixl - 212 = (x 1 - x2 ) (xl - x 2) is the square of the Euclidean distance.Thus, (30.1) preserves distances and, therefore, defines an Euclidean isometry.

A spatial tensor field is said to transform objectively under superposed rigid bodymotions if it transforms according to the standard rules of tensor analysis.

EXAMPLE 30.1. The total motion obtained by composition of (30.1) with the givenmotion is

zt = iP+(X, t) = + o o(X, t) = r(t) + Q(t)o(X, t). (30.3)

The deformation gradient for the total motion then becomes

F + = D + = Q(t) D~9(X, t) = Q(t)F. (30.4)

Therefore, the spatial velocity gradient is given by

V+v+ = F+ (F+)- = Q(t)VvQT(t) + Q(t)QT(t), (30.5)

which does not transform objectively due to the additional skew-symmetric termQ(t)QT(t). However, its symmetric part, the rate of deformation tensor, does trans-form objectively since, from (30.5) and (28.23)1, we have

d+ = Q(t)dQT(t). (30.6)

Note that the spin tensor, defined by (28.23)2, transforms according to the nonobjectiverule

i+ = Q(t)IiVQ T(t) + Q(t)QT (t). (30.7)

REMARK 30.1. Convected objects, that is tensor fields on the reference configuration,remain unaltered under spatial superposed rigid body motions. For example from(30.4) we have

C + = (F+)TF+ = FTQT(t)Q(t)F = C. (30.8)

Likewise, C+ = C is unaltered by spatial superposed rigid body motions. To under-stand this result, Fig. 30.1 may prove useful.

Given an objective spatial tensor field its material time derivative will not, in gen-eral, retain the objectivity property. It is precisely this lack of objectivity of the material

314 J.C. Simo CHAPTER Ill


FIG. 30.1. Reference and current placements. Illustration of superposed rigid body motions.

time derivative of objective spatial fields that motivates the introduction of objectiverates. To see how this lack of objectivity arises, consider the Cauchy stress tensorand use the definition of the material time derivative together with the chain rule toevaluate as

dr = [at(a O Pt)] o t 1 = a0 t + (V - V)U. (30.9a)

In component form this result reads

(5'ab(Xt) = aOb(X, t) aO-ab(X, t) 0Pc(X, t) (30.9b)at ax, at

Now assume that ar transforms objectively so that o + = Q(t)aQT(t). From (30.9a)it then follows that

a+ = Q(t)rQT (t) + [Q(t)QT (t)] a+ - a+ [(t)Q T (t)] (30.10)

which is clearly nonobjective. Objective rates are essentially modified time deriva-tives of the Cauchy stress tensor constructed so that objectivity is preserved. A largebody of literature has been concerned with the development of objective rates which,remarkably, extends to recent dates. Concerning the practically infinite number of pro-posals made we remark, following TRUESDELL and NOLL ([1965], p. 404), that "...Despite claims and whole papers to the contrary, any advantage claimed for one suchrate over another is pure illusion."

SECTION 30 315

In fact, one can show that any possible objective stress rate is a particularcase of a basic geometric object known as the Lie derivative; see TRUESDELL

and TOUPtN [1960], MARSDEN and HUGHES ([1983], Chapter 1), So and MARS-DEN [1984] or ARNOLD [1989]. A few widely used proposals found in the literatureare recorded below.

(i) The Lie derivative of the Kirchhoff stress tensor, also known (up to a factorof J) as the Truesdell stress rate, is defined as

£vr = {Fat[F-17F-T]FT } = {F[atS]FT }. (30.11)

Technically speaking this definition gives £v o oLt. However, in the present contextthe notation in (30.11) is preferred since in solid mechanics one is almost exclusivelyinterested in a Lagrangian description of the motion. Using the well-known resultat(F-l) = -F- PF-t for the derivative of the inverse of a matrix, the definitionof the material time derivative and the spatial velocity gradient imply

£Er = - (Vv)r - r(Vv)T. (30.12)

One can easily show that (30.12) is objective (in fact, £vr meets the much strongercondition of covariance, see MARSDEN and HUGHES [19831).

(ii) The Jaumann-Zaremba stress rate of the Kirchhoff stress is essentially a co-rotated derivative relative to spatial axes with instantaneous velocity given by the spintensor. One finds

V = -- Wr + (. (30.13)

(iii) The Green-Mclnnis-Naghdi stress rate of the Kirchhoff stress is defined by anexpression similar to (30.11), but with F replaced by R, i.e.,

r= {ROt [RTrR] RT}. (30.14)

Recalling that R = DR, we find

r = - r + r f. (30.15)

Although (30.15) is similar in structure to (30.13), we remark that ~ ii unlessU = 0 (i.e., an instantaneously rigid motion), see (28.30). This objective rate wasbrought into prominence in DIENES [1979] and is further discussed in JOHNSON andBAMMANN [1984].

For a catalog of the many proposed objective stress rates, up to the early sixties,see TRUESDELL and TOUPIN ([1960], Section 48, p. 151).



31. Elastic constitutive equations and isotropy group

Attention will be restricted here to classical nonlinear elasticity; see TRUESDELL andNOLL ([1965], §17-19) for a detailed discussion in a general context. The constitutiveequation for a hyperelastic material is defined in terms of a stored energy functionW(X, F(X, t)), depending locally on the deformation gradient, such that

P(X,t) = aFW(X,F(X,t)), (X, t) E FQ x . (31.1)

The stored energy function W(X, F) is said to be objective or frame invariant ifthe following condition holds. Let t - Wot be an arbitrary motion, and consider asuperposed rigid body motion of the form (30.1), with deformation gradient F+ =Q(t)F, where Q(t) SO(3). If no additional restrictions are placed, in generalW(X, F+ ) need not coincide with W(X, F). The requirement of objectivity is thecondition

W(X, QF) = W(X, F) VQ(t) E S0(3). (31.2)

Restriction (31.2) then implies that the stored energy function can depend on F onlythrough C = FTF, i.e.,

W(X, F) = W(X, C). (31.3)

From this reduced constitutive function along with (31.1), one obtains the followingclassical constitutive equations for elasticity

S = 2acW, P = 2F[acW] and r=2F[acW]FT . (31.4)

For clarity, explicit indication of the argument has been omitted. We shall often followthis practice in subsequent developments. Objectivity of the stored energy functionis closely related to the balance of angular momentum; i.e., to the symmetry of theCauchy stress tensor. In fact, it is easily shown that W(X, F) is objective (i.e., (31.2)holds) if and only if the balance of angular momentum condition PFT = FPT holds(with P given by (31.1)).

(A) Hyperelastic rate constitutive equations. The rate form of the hyperelastic consti-tutive equations play a central role in the incremental formulation of plasticity. Someof the key results are summarized below. First, time differentiation of relation (31.4)1gives

S = C , i.e., SAB = CABCD1CCD, (31.5)

where C(X, C) is the material elasticity tensor given by

C = 4 0ccW(X, C), i.e., CABCD = 4CABCCD (31.6)acAacCD

SECTION 31 317

Recalling that £vr = [FSFT], along with the relation C = 2FTdF, expression(31.5)1 leads to

(£vr)ab = FaASABFbB = [FaAFbBFcCFdDCABcD]dcd = cabcddcd, (31.7)

where with components Cabcd is the spatial elasticity tensor related to C by the(push-forward) transformation

Cabcd = FaAFbBFcCFdDCABCD. (31.8)

Note that (31.7) results in the spatial rate constitutive equation

£v = d, i.e., (CvT)ab = CabCddCd (31.9)

Also note that analogous expressions can be derived for any objective stress rate otherthan the Lie derivative. The derivation of hyperelastic rate equations for other stressand strain measures constitutes an exercise (often nontrivial) involving the applicationof the chain rule (see, e.g., OGDEN [1984], Chapter 5, and the remark below). Here,we shall simply quote one further result useful in numerical implementations. From(31.1), we find

P = AF, i.e., PaA = AaAbBFbB, (31.10)

where A is called the first elasticity tensor, and is given by

A = aFW(X,,F), i.e., AaAbB -OFaFbB (31.11)

Alternatively, starting from (31.4)2 we arrive at (31.10), along with the followingimportant relation connecting A and c

Aa BcD = FBb1[Cabcd + Tacbd] F· (31.12)

REMARK 31.1. Starting from the properly invariant hyperelastic relations (31.4), onecan derive spatial rate constitutive equations of the form (31.9) which are also properlyinvariant. This equation can be expressed in terms of any objective rate; for instance,in terms of the Jaumann stress rate, on account of the relation

Evr = - - (d + -) - (d + ~)T = 7 -dr - rd, (31.13)

Eq. (31.9) can be written as

V7 = ad where aabcd = Cabcd +

6ac~bd + 6bdTac(


(31.14)


Conversely, given any rate constitutive equation of the form (31.14) the questionarises as to whether a stored energy function exists such that -r is given by (31.4)3.The answer to this question is, in general, negative. In addition to the full symmetryof the moduli a a set of compatibility relations must hold for a rate equation of the

Vform r = ad to be derivable from a stored energy function. We refer to TRUESDELLand NOLL (1965], Chapter IV) for a detailed account of the relevant results.

Rate equations of the form (31.14) which are not derivable from a stored energyfunction lead to the notion of hypoelasticity. We recall two basic results.

(i) In general, hypoelastic materials produce nonzero dissipation in a closed cycle.See TRUESDELL and NOLL ([1965], p. 401) for the precise statements.

(ii) The assumption of r = ad with a being the constant isotropic tensor of theinfinitesimal theory of elasticity is in general incompatible with hyperelasticity (SIMOand PISTER [1984]). It should be noted that such an assumption is typically made inphenomenological theories of plasticity.

EXAMPLE 31.1. To illustrate the preceding ideas in a concrete setting, consider thefollowing stored energy function (SIMo and PISTER [1984], CIARLET [1988])

W= A(J2 - 1) - (A + )log[J] + ml'(tr[C]- 3), (31.15)

where A, u > 0 can be interpreted as Lame constants. Using the relation acJ =JC - 1, the constitutive equations (31.4) become

s= 1A(J2 -1)C 1 I+,(1 - C- 1),(31.16)

= X(J2 1) + l(b - 1).

In view of relations (31.6) and (31.8) the spatial elasticity tensor is given by theexplicit expression

c = A J21 ® 1 + 2u[ - (J2 1)A//]I, (31.17)

where I with components labcd = [acSbd + badSbc]/2 is the rank-four symmetricunit tensor. The following four important properties should be noted. (i) As J -- 0or J -- 00oo we have W -- oo. (ii) Both W = 0 and r = 0 for F = 1, i.e., atthe reference state. In addition, c reduces to the elasticity tensor of the linear theory.(iii) W can be written as W = U(J) + ½,/(tr[C] - 3) where

U"(J) = A(1 + 1/J2) + l/J 2 > 0, for J C (0, oo). (31.18)

It follows that, U(J) is a convex function of J. (iv) The stored energy W is a poly-convex function of F. Thus the only known global existence results for elasticitybased on the existence of minimizers of the total potential energy for poly-convexstored energy functions apply to this model (see, e.g., CIARLET [1988], Chapters 4, 7,and MARSDEN and HUGHES [1983], Chapter 6).

SECTTON 31 319

(B) The notion of isotropy group. Isotropic elastic response. The isotropy groupat a point in the reference placement characterizes the invariance properties of thematerial constitutive response under superposed rigid body motions on the referenceconfiguration. The notion of isotropy should not be confused with the notion of frameindifference. Whereas the former notion refers to a particular property which thematerial response may enjoy, the latter notion is a fundamental principle of mechanics,which holds for all possible response functions.

Let X EC be a point in the reference placement Q of an elastic body. Consider asuperposed rigid deformation on Q, i.e., the isometry

X E Q2- X + = b(X) = r + QX , for Q E SO(3), (31.19)

which maps Q onto Q+. Since p+ o = fpt, the chain rule implies

F+ = FQT so that C + = QCQT and b+ = b. (31.20)

It follows that under superposed rigid body motions of the reference state 2 thedeformation gradient and the Cauchy-Green tensors transform according to (31.20).It is clear that, in general, the values of the stored energy function at X CE 2 evaluatedat C and C + will be different. The isotropy group at X CE 2 is precisely the set ofproper orthogonal transformations which leave the stored energy function unchanged;i.e.,

Gx = {Q E SO(3): W(X, QCQT ) = W(X, C)}. (31.21)

It can easily be shown that Gx C SO(3) is indeed a group. Note that Gx is asso-ciated with each point X GE 2, unless the material is homogeneous (i.e., unless Wis independent of X). Furthermore, the isotropy group Gx is defined relative to aparticular reference configuration. If Gx = SO(3) the material is said to be isotropic(relative to 2, at the point X E Q). Otherwise, the material is said to be anisotropic.

The condition of isotropic response places strong restrictions on the admissibleforms of the response function. Here we shall recall only one important result whichwill be used below.

From our preceding discussion, a function f: -- R of symmetric tensors isisotropic if and only if

f(QHQT ) = f(H) VQ E SO(3) with H E S. (31.22)

We may think of f as the free energy or any other response function (e.g., the yieldcondition as discussed below). Depending on the context, H E $+ C $ will denotethe right Cauchy-Green tensor or any other symmetric tensor (e.g., the Cauchy stressfield, a tensor field in S). The following well-known result is a basic fact on isotropictensor functions.



THEOREM 31.1. Representation theorem for isotropic functions. A function f : - Ris isotropic if and only if f (H) depends on H S through its principal invariants;i.e., if and only if there exists a function f : R3 - I>R such that

f(H) = f(1 1 (H),I2(H),13(H)) VH E 5, (31.23)

where I (H) = tr[H], I2(H) = (I (H) 2 - tr[H 2]) and I3(H) = det[H] are theprincipal invariants of H.

This result is an immediate consequence of the spectral theorem for symmetrictensors; see, e.g, GURTIN ([1981], p. 230).

EXAMPLE 31.2. For homogeneous isotropic elasticity, the stored energy is a functiononly of the principal invariants of C = FTF; i.e.,

W(C) = W( 1 , 2, 13), (31.24)

where IA (A = 1, 2, 3) are given by (28.4). An example of an isotropic stored energyfunction was given in the preceding example. Making use of the relations

acI1 = 1, aci2 = Il - C and acI3 = I3 C - , (31.25)

the constitutive equation (31.4) for the symmetric Piola-Kirchhoff tensor becomes

S=2[aW+ II] 1 - 2 C+2 13C- (31.26)all a'2 a 2 a 3

Using the relation r = FSFT one obtains the following constitutive equation for theKirchhoff stress tensor:

awi rawaw 1r = 2 -I31+ 1 + h] b- 2 W-b2, (31.27)

ai13j ai 1 DI 2 a2

where b = FFT is the left Cauchy-Green tensor.

REMARK 31.2. For isotropic response, and only for this case, the stored energy func-tion is a function of the left Cauchy-Green tensor. This result is an immediate conse-quence of the fact that Gx = SO(3). Since the constitutive equations are local it ispermissible to chose Q = R in (31.21), where R is the rotation tensor in the polardecomposition of the deformation gradient. This gives

W(X, C) = W(X, RCRT ) = W(X, b), (31.28)

since b = FFT = RUURT = RCRT; hence the result.

SECTION 31 321

REMARK 31.3. The obvious way to automatically insure satisfaction of frame indiffer-ence without precluding anisotropic response is to formulate the constitutive responsefunction in terms of objects associated with the reference configuration. For elasticitythis amounts to formulating the constitutive response in terms of the right Cauchy-Green tensor C and the second Piola-Kirchhoff stress tensor S. The formulationof elasticity in terms of (S, C) is called the convective representation of elasticity.That use of convected coordinates (the convected representation) automatically in-sures frame indifference was known to ZAREMBA [1903]; see TRUESDELL and NOLL([1965], p. 45). Convected coordinates are also used in HILL [1978], where the ter-minology "embedded system" is favored, and most notably throughout the classicalbook of GREEN and ZERNA [1960]. For a discussion of the convected representation(including the corresponding Hamiltonian structure) see SIMo, MARSDEN and KRISH-NAPRASAD [1988].

REMARK 31.4. In a computational setting, the convective representation in terms of(S, C) (or, equivalently, the use of convected coordinates) has been used by severalauthors, see, e.g., NEEDLEMAN [1982], for applications to plasticity theories.

32. Volumetric/deviatoric uncoupled finite elasticity

Within the context of the infinitesimal theory, elastic materials possessing a generalstored energy of the form

W(E) = U(tr[E]) + W(dev[e]) where dev[e] = E - tr[El, (32.1)ndim

are said to exhibit uncoupled volumetric/deviatoric response. An example of such amaterial is provided by the linear isotropic elastic solid for which U(tr[E]) = ½r(tr[E])2

and W(dev[s]) = /i(dev[E] dev[E])2 , where nr > 0 is the bulk modulus and p > 0 isthe shear modulus. The generalization to the geometrically nonlinear of the uncoupledstored energy function (32.1) first appears in FLORY [1961] for isotropic materials andis systematically developed in SIMo, TAYLOR and PISTER [1985] both for elastic andplastic materials. The idea is to introduce the volume preserving and volumetric partsof the deformation gradient F = Dop, respectively defined by

J = det[D(] and F = J-/dm Dq. (32.2)

By the properties of the determinant it is clear that this definition yields det[F] = 1.Expression (32.1) is then generalized by considering an uncoupled stored energy func-tion of the form

W(C) = U(J) + W(C),


(32.3)

where C is the volume preserving right Cauchy-Green tensor and E denotes thevolume preserving Lagrangian strain tensor respectively defined by

= TF and E= (C-1). (32.4)

As in the geometrically linear theory, the functions U and W define the volumetricand volume preserving contributions to the stored energy function, respectively. Animportant application of uncoupled models with stored energy function of the form(32.3) arises in the context of metal plasticity, as described in a subsequent section.The stress response is obtain by evaluating the partial derivatives of the stored energyfunction (32.3) with the help of the following result:

THEOREM 32.1. The partial derivatives of C relative to C and J are given by

[ac6 ] =J- 2 /ndmi [I- C C-] and 3cJ = JC - '. (32.5)ndim

Moreover, the material deviator of any rank-two (convected) tensor field H on thereference configuration 2, defined by the relation

DEV[H] = J2 /ndim [ac6]H = H - [H C]C- , (32.6)ndim

has the following properties:

dev[FHFT ] = FDEV[H]F T ,(32.7)

DEV[H] C - l = dev[FHFT ] . 1 = 0,

where dev[h] = h- (/ndim) tr[h]l is the usual deviatoric part of any (spatial) tensorfield h in the current configuration S = (Q2).

PROOF. Consider a one-parameter family of right Cauchy-Green tensors of the form

C, = C + EH, with H = HT and E > 0. (32.8)

Clearly, for E > 0 sufficiently small we have det[C,] > 0. Setting J, = det[Cand using the standard formula for the derivative of the determinant it follows that

d J =o 2J2tr[C- d C ] = Jtr[C-IH]. (32.9)dE Jd2

SECTION 32 Nonlinear continuum mechanics 323

Differentiation of the defining relation C' = J 2/ndimC, and use of (32.9) then yields

|d j=J-2 /ndiH - -- tr [C-H]C]

de =0 ndim

J-2

/dim [I - 1C1 C] H; (32.10)ndim

a relation which holds for any H = HT. By the chain rule, it follows that

d r e= [acc dE CE = [cC]H. (32.11)

This result, together with (32.9) and (32.10), implies relation (32.5) in the statementof the theorem. Relation (32.7) follows from (32.10) and (32.11) while properties(32.7) are a direct consequence of (32.6) and the definition of the right Cauchy-Greentensor. El

Observe that definition (32.6) for the deviator of a convected (contravariant) stresstensor is consistent with the geometric setting of the convective representation ofnonlinear elasticity. Recall that within this context the right Cauchy-Green tensor isinterpreted as a (flat) Riemannian metric on the reference configuration 2, the traceof a contravariant tensor is given by its contraction with the metric and its deviatoricpart is defined by (32.6).

Using the results in the preceding theorem, the stress-strain relations associated withthe uncoupled stored energy function (32.3) can be easily computed by specializationof the general hyperelastic stress-strain relations S = 2DcW(C), cast in terms of thesecond Piola-Kirchhoff stress tensor and the right Cauchy-Green tensor, by makinguse of the following relation implied by the chain rule:

cw(C) = ,U'(J) acJ + [acc]O~w(C). (32.12)

First, by combining (32.12) with (32.5) the constitutive equation for the second Piola-Kirchhoff stress tensor becomes

S = JU'(J) C-I + J-2/ndim2DEV[DcW(C)]. (32.13)

Second, from the relation r = FSFT and properties (32.7), expression (32.13) yieldsthe following constitutive equation of the Kirchhoff stress tensor:

r =Jpl +dev[2FaaCW(C)F T and p = U'(J), (32.14)

where F is the volume preserving part of the deformation gradient defined by (32.2)2.Thus the uncoupled stored energy function (32.3) produces the uncoupled stress-strain relations (32.14) in which the hydrostatic pressure response, defined by theterm p = U'(J), is uncoupled from the deviatoric stress response.

324 J.C. Simo CHAPTER I1


To complete the foregoing developments, it only remains to compute the elasticmoduli associated with the uncoupled stored energy function (32.3). The elastic moduliin the convective description, associated with the volume preserving part W(C),denoted by C, are defined as

C = 4~a4 aw(Ca)C 4O4 CW(C), i.e., CIJKL = 40 J(32.15a)

while spatial moduli associated with W(C) are defined via the standard push-forwardtransformation as

Cijkl = FiIFjJFkKF1LCIJKL. (32.15b)

The components of both C and c obey the usual symmetry conditions associatedwith the existence of the potential function W(C). It proves convenient to cast expres-sion (32.15a) directly in terms of partial derivatives relative to the volume preservingright Cauchy-Green tensor C, which is the argument of the function W. This taskis accomplished by application of the chain rule. Using the results in the precedingtheorem, a somewhat involved but otherwise straightforward manipulation yields thefollowing result:

2 2C = Cdev - DEV[S] C - 1

_- C-l ® DEV[S]ndim ndim

+ 2 J -2/ d im (a W. C) [IC- C - C- ], (32.16)

ndim C ndim

where DEV[S] = J-2/ndimDEV[a0W], IC-, is the fourth-order tensor with components

(I-I)IJKL = - [CIKCJL - CIL CjK], (32.17)

and Cdev is the fourth-order tensor of deviatoric tangent moduli defined by the ex-pression

Cdev=J-4/ndm"'[O3W - [ WI Cc-,C-1 ]

Cdev - 4Jc4/(ndim)-- + ( )2

C [c ] c

i [C-l ® [ W]C- [aL W]Cc -l]c . (32.18)ndim Ca Ca

The preceding notation is motivated by the property CdevC = 0, which is an easyconsequence of expression (32.18), and shows that Cdev is a fourth-order deviatorictensor. The complete expression for the elastic moduli is obtained by adding to C

SECTION 32 325

the contribution O2ccU(J) associated with the volumetric part of the stored energyfunction. Using (32.5)2 the result can be written as

C = J 2 U"(J)C l® C-l + JU'(J) [C-' 0 C-'- 2Ic , + C. (32.19)

The corresponding expressions for the elastic moduli in the spatial description are eas-ily obtained from the preceding results by using the transformation (32.15b) togetherwith properties (32.7). From (32.16) one obtains

2 2C = Cdev - dev[r] 1 - 1 dev[T]

ldim ndim

+ ditr [2a ' T ] (I- 1 1) (32.20)

where

(Cdev)ijkl = FilFjJFkKFlL (dev)zJKL. (32.21)

Similarly, expression (32.19) together with properties (32.7) yield the following resultfor the spatial elastic moduli:

c = J2 U"(J)1 1 + JU'(J)[1 X 1 - 21] + 6. (32.22)

The example below provides an illustration of the preceding results in the spatialdescription.

EXAMPLE 32.1. The simplest example of stored energy function of the form (32.3)is given by the following extension to the compressible regime of that Neo-Hookeanelastic material:

U(J) = (J 2- 1 - logJ) and W(C) = (tr [I] - ndim) (32.23)

The derivates of the compressible part U(J) of the stored energy function are

U'(J) = J - and U"(J) = ll + J2 (32.24)

Since U"(J) > 0 for J E (0, oo) it follows that U(.) is a convex function for > 0with U(1) = 0 and U"(1) = . Noting that aaW(C) = 1 expression (32.14) reducesto

'r = (j2 1)1 + dev [FFT ] .


(32.25)

Finally, since a-s W(C) = 0, the tensor c of spatial elastic moduli defined by (32.22)becomes

C = [J21 1 - (J2 - 1)I] + 2 1 tr [FFT](I- I 1 1Tndim ndim

2 2-- dev[r] ® 1 - 1 0 dev[T]. (32.26)

ndim ndim

Observe that c particularized at the stress-free reference configuration (F = 1, J = 1and r = 0) gives the usual elastic moduli of linear isotropic elasticity.

33. Isotropic elasticity formulated in principal stretches

The stored energy function of an isotropic material can only depend on the principalinvariants of the right (or left) Cauchy-Green tensor. Equivalently, the stored energyfunction can be formulated as a function of the principal stretches. This latter represen-tation, although equivalent to the former, turns out to be often much more convenientin applications. A notable example is furnished by so-called Ogden materials, whichhave proven quite useful in the modeling of the response of rubber like materials(see OGDEN [1984], and references therein). For isotropic finite strain plasticity, theformulation of the elastic response in terms of principal stretches leads to a remark-able simplification of return mapping algorithms (SIMO [1992]) which are describedsubsequently.

The computation of the stress response for an isotropic elastic material with storedenergy

W(C) = W(AX, A2, A3), (33.1)

given as a function of the principal stretches, relies on the following result.

THEOREM 33.1. The derivatives of the principal stretches are given by the followingformulae:

(i) Three different roots. If A1 7 A2 7 )A3 then

acA 2 = N(A) ® N(A) (A = 1,2,3). (33.2a)

(ii) Two equal roots. If A := = : 2 3 then

a0c2 = 1 - N(3) N(3 ) and aC, 2 = N(3) N(3). (33.2b)

(iii) Three equal roots. If A = A) = A2 = A3 then aCA2 = 1. (33.2c)

PROOF. The proof of these results is straightforward. For the case of three differentroots, for instance, differentiation of the eigenvalue problem (28.6) gives

dCN(A) + C dN(A) = d(A2)N(A) + AA dN(A).


(33.3)

Taking the dot product of this relation with N(A) and using the orthogonality conditiondN( A )N (A ) = 0, which follows from the normalization IN(A) I = 1, gives

d(A2 ) = N(A) . dCN(A) = tr [dC(N(A) ® N(A))]. (33.4)

Since tr[-, ] defines an inner product in the linear space L(3) of 3 x 3 matrices, thepreceding relation along with the directional derivative formula yields (33.2a). Theother relations are proved similarly. [

(A) Elastic constitutive equations. The elastic constitutive equation for the secondPiola-Kirchhoff stress tensor S is computed by the chain rule as

s = W(C) =i 1 Ow( 1, X 2, 3) O( )s = 2acw(c): IA a A (33.5)A=I

From the results (33.2) one concludes that (33.5) is merely a restatement of the spectraldecomposition of S, i.e.,

ndim w

S = E SAN (A ) 3 N(A) with SA = a (33.6)A=1 AA A(

This expression is consistent with the restriction to isotropy, which implies that S andC commute or, equivalently, that S and C have the same characteristic subspaces.Using relation r = FSFT Eq. (33.6) gives

ndim awr = X TAn(A) n(A) with TA = AA (33.7)

A=1 AA

This expression furnishes the constitutive equation for the spatial Kirchhoff stress.Next, a closed-form expression is given for the material elastic moduli.

(B) The material tangent elastic moduli. Recall that the material elastic moduli C aredefined by the relation S = C C. An easy way to compute C exploits the followingobservation (OGDEN [1984]). The time derivative of the eigenvectors of C can beexpressed as

?Zdim

N ( A ) = W x N(A) = > WABN(B), (33.8)B=lBAA


where WAB = -WBA. Inserting this result into the time derivative of the spectraldecomposition of C gives

ndim

= , t (2 ) N (A) ® N(A)A=I

ndim dim

E E 2 A _AB)WABN(A) N(B).A=l B=1

BOA

Similarly, time differentiation of the spectral decomposition (33.6) of S and use of(33.9) gives

ndim ndim i as

S= = E E aa >32 2 \(A) + N(A)

A=1 B=

ndim ndim S - SB 2 +E E SA - t A-AX2B)W A BN(A) N() (33.10)

A=l B=1 2 A BB4A

By inspection of (33.9) and (33.10) it is easily concluded that the tensor C of elasticmoduli is given by the explicit formula

ndi SA B N ( A )X N ( B )

X [N(A) ® N ( B) + N ( B ) 0 N(A)]

A=1 B=1 A BBOA

?dim ndim r 1 a

A=1 B=1 AB aB AA aA

The expression for the spatial moduli C follows immediately from (33.11) and (31.8).

(C) The spatial elastic moduli in terms of logarithmic stretches. For the purpose ofcharacterizing the elastic response of models of multiplicative plasticity, as well asthe formulation of appropriate integration algorithms, it proves convenient to writethe stored energy function in terms of principal logarithmic stretches as

w(A,A, A 3) = f(e1,E2, 3 ) where EA = log(AA). (33.12)

In this situation, the closed-form expression for the spatial elasticity tensor c can bederived from the preceding results as follows. First, using the chain rule we have

1 a 1 a W] = I [ 2 a(31AB _AB w -2 =- 2AB · (33.13)

-B a-B A A w A aEA--B EB


Second, using definition (31.8) for the spatial elasticity tensor together with the relationFN(A) = AAn(A) and expression (31.8), we arrive at the following result

ndim dim [ a2-c L a n0A) n (A) ® n(B) + g (33.14a)

A=IB=1

where the nonzero components IJKL in the Eulerian axes n(A) } of the tensor gare given by (I, J = 1,..., ndim)

giII = -2 ,T and g.jjjj = gIjjj = T2 - ZJ for I J. (33.14b)A2 - X2

Note that the principal values of the Kirchhoff stress are given from (33.7)2 in termsof the logarithmic stretches as r = w/SeE.

REMARK 33.1. From a computational standpoint, the advantage of the closed-formexpression (33.14) for the elastic moduli is that it remains well-conditioned in thepresence of repeated principal stretches. A conceptual algorithm for the computationof the stress and the elastic moduli for stored energy functions defined in terms ofprincipal stretches proceeds as follows.

Step 1. Given C compute the Lagrangian axes by solving the eigenvalue problemCN(A) = A2 N(A).

Step 2. Use the preceding formulae to evaluate S and C (or r and c).

The key step is the computation of the orthogonal matrix Q that orients the La-grangian triad, i.e., Q = [N(O) N(2) N(3 )]. Although closed-form solutions are possi-ble, currently we favor an iterative solution using Jacobi's algorithm to avoid possiblenumerical difficulties associated with repeated roots. For the symmetric eigenvalueproblem it is well-known that convergence of Jacobi iteration holds regardless of themultiplicity of the eigenvalues.

34. Multiplicative plasticity at finite strains: Basic concepts

This section addresses the extension to the finite strain regime of the classical modelof infinitesimal plasticity described in detail in Chapter I. A key feature of the modeldescribed below is the replacement of the additive decomposition of the infinitesimalstrain field by a multiplicative decomposition of the deformation gradient into elasticand plastic parts. This multiplicative decomposition is motivated by a well-understoodmicromechanical picture for single crystal metal plasticity. Comprehensive expositionsof the micromechanical description of single crystal metal plasticity can be found inthe review article of ASARO [1979, 1983] and the recent monograph of HAVNER

[1992]. The basic ideas go back to the fundamental work of TAYLOR [19381 and TAY-

LOR and ELAM [1923, 1925], subsequently expanded upon in HILL [1966], HILL andRICE [1972], ASARO and RICE [1977], HAVNER [1971, 1982], NEMAT-NASSER [1983]



and others. The emphasis in the exposition below is placed on those mathematicalaspects of the model relevant to its numerical treatment, as described in detail in thefollowing chapter.

(A) Motivation: The key assumptions of the infinitesimal theory. To motivate thestructure of the finite strain theory, it proves useful to recall the key assumptionsmade in the formulation of the classical model of infinitesimal plasticity, as describedin Chapter I.

(Al) The total infinitesimal strain field E(X, t) at (X, t) EG Q x [ is decomposedadditively into elastic and plastic parts as E = Ee EP.

(A2) The stress response is defined in terms of a free energy function T(Ee, (a) viathe potential relations (2.4), where {(} are a set int (strain-like) internal variablesconjugate to the (stress-like) variables {q'} in the sense that q' := -jTf.

(A3) The stress a and the set of nint stress-like internal variables {qO} that providea phenomenological characterization of microstructural hardening mechanisms areconstrained to lie in a convex set E c S x Rnin, called the elastic domain E. Ingeneral, E is defined by (23.6).

(A4) In the simplest version of classical plasticity, the evolution of the internalvariables {eP, ,} is specified by postulating maximum dissipation in the system,i.e., via the inequality (6.7). This assumption yields the evolution equations (6.8) ofassociative plasticity.

A critical examination of these assumptions reveals two basic issues arising in ageneralization of this classical model to the finite strain regime.

First, the infinitesimal strain tensor is not an invariant measure of deformation in thefinite strain regime; i.e., it does not transform properly under superposed rigid bodymotions. Assumption (Al) must therefore be revised. Although a number of alternativeoptions are possible, the use of Lagrangian strains being the obvious one (GREEN andNAGHDI [1964, 1966]), the option adopted must be motivated by micromechanicalconsiderations.

Second, a specific stress measure must be adopted in the definition of the convexset E within the finite strain regime. In this regard it should be noted that the obviouschoice, the Cauchy stress tensor a, restricts the theory to isotropy. To justify thisassertion consider the simplest situation in which nnt = 1 and E is given by

E: = {(a, q) E S x t: f(r, q) < O}. (34.1)

Given a motion t - Vpt consider the rigid body motion (30.1) superposed on thecurrent placement St = pt(•). Since the Cauchy stress transforms objectively asA+ = Q(t)oaQT(t), it necessarily follows that f: x I - IR in (34.1) is objective ifand only if

f(Q(t)aQT (t),q) = f(a,q) VQ(t) G SO(3),

SECTION 34 331

(34.2)

CHAPTER III

F

C Ce

CP-I 1

Reference Configuration Inte

1

be

te Configuration Current Configuration

FIG. 34.1. Illustration of the local multiplicative decomposition of the deformation gradient and the basicdeformation tensors.

which is the statement of isotropy for the function f(., q). In summary, a classicalformulation of the yield criterion in terms of true stresses necessarily restricts theresulting theory to isotropy, as claimed.

In what follows, we shall address the extension to the regime of finite strains ofthe classical model of infinitesimal plasticity based on assumptions (A1)-(A4) and thepreceding observations.

(B) Basic kinematical result for multiplicative plasticity. The generalization of theadditive decomposition in assumption (Al) to finite strains is motivated by the struc-ture of the single crystal model for metal plasticity, and takes the form of a localmultiplicative decomposition of the deformation gradient as

F(X, t) = Fe(X, t)FP(X, t) V(X, t) E Q x , (34.3)

where F(X, t) := D(X, t) denotes the deformation gradient of the motion at X E Q(see Fig. 34.1). From a micromechanical point of view FP is an internal variable re-lated to the amount of dislocation flow through the crystal lattice, whereas Fe modelsthe lattice deformation, see KRONER and TEODOSIU [1972] and HAVNER [1992]. Inthe phenomenological setting considered in LEE and LIU [1967], LEE [1969], KRA-TOCHVIL [1973] and others, Fe-t is viewed as defining a local, stress-free, unloadedconfiguration obtained by releasing the stresses on each neighborhood in the currentplacement.

332 J.C. Simo


Associated with the multiplicative decomposition (34.3) one defines elastic rightand left Cauchy-Green tensors by the conventional relations

Ve = FeTFe and be = FeFeT. (34.4)

Observe that C e is a covariant tensor field defined on the intermediate configurationwhereas be is a contravariant tensor field defined on the current placement St. Localrate of deformations associated with the plastic and the elastic parts of the deformationgradient are defined by the relations

LP = FP - ' and l e = FeF e - l . (34.5)

These definitions are motivated by the expression 1 = Vv o t = FF- 1 found forthe (spatial) velocity gradient. Again, we remark that L P is a generally nonsymmetrictensor field defined on the intermediate configuration while 1e is a nonsymmetric tensorfield defined on the current placement St. The following result provides the relationbetween these two fields.

THEOREM 34.1. The plastic velocity gradient L P and the elastic velocity gradient eare related by the transformation

FeLPFe - 1 = - le where I = FF - 1 = Vv o got. (34.6)

This (push-forward) relation implies that L P and le are contra-covariant tensor fieldson the intermediate configuration and the current placement, respectively.

PROOF. The formula for the derivative of the inverse of a rank two tensor, togetherwith (34.5)1 and the factorization (34.3) give

LP = at [F e-lF]F p -

l = [F e - l

_- F e -

lFeF e -

lF] F p -

l

= F e - 1 [F - 1 - FeFe-] ]Fe. (34.7)

The result then follows from definition (34.5)2.

Expression (34.6) motivates the following definitions in the spatial description forthe plastic velocity gradient, the plastic rate of deformation tensor and the plastic spintensor

IP = 1 - le, dP = sym[lP] and Pi = skew[lP]. (34.8)

Recalling the expressions d = sym[l] and ii = sym[l] for the total rate of deforma-tion tensor and the total spin tensor, respectively, the preceding definitions imply theadditive decomposition

d = de + d with de = sym[le] and w-e = skew[le].

SECTION 34 333

(34.9)

In summary, the multiplicative factorization (34.3) of the deformation gradient alongwith definition (34.8) for the plastic velocity gradient in the spatial description resultin the additive decomposition (34.9) of the rate of deformation tensor d into elasticand plastic parts. The following result characterizes the behavior of these objects undersuperposed rigid body motions on the current placement St.

THEOREM 34.2. In a superposed rigid body motion p+(, t) = r(t) + Q(t)p(o,t)on the current placement St, with Q(t) E SO(3) and r(t) C Rrndi (dim = 3), thefollowing transformations hold:

(i) F+ = QF, Fe + = QFe and FP+ = FP . (34.10a)

(ii) + = QQT + QiQT le+ = QQT + QleQT and /P+ = QlpQ T . (34.10b)

PROOF. (i) Relation F+ = QF was shown to hold in Section 31. Relation FPm = FP

is evident since the intermediate configuration is unaffected by spatial rigid bodymotions. The multiplicative factorization (34.3) then implies the relation F e+ = QFe.

(ii) Relation l+ = QQT + QiQT was also shown to hold in Section 31. Similarly,time differentiation of Fe + = QFe gives relation e+ = QQT + QleQT. The lastrelation IP+ = QIPQT in (ii) follows from the preceding two and the definitionl p = I l e. []

REMARK 34.1. Result (ii) immediately implies the objective transformations de+ =QdeQT and dP+ = QdPQT. All these results are to be expected with the exceptionof the objective transformation IP+ = QlPQT for the full plastic velocity gradient inthe spatial description. This relation implies, in particular, the objective transformationi,-p+ = QipQT for the plastic spin. Obviously, tensor fields defined on the interme-diate configuration, such as Ce and L P, are unaffected by rigid motions superposedon the current placement.

REMARK 34.2. A much debated issue in finite strain plasticity is concerned with theapparent lack of uniqueness inherent to the multiplicative factorization (34.3), whicharises when considering superposed rigid body motions on the intermediate configura-tion. In one such rigid motion, the deformation gradients transform for any Q c SO(3)according to

Fe + = FeQT and FP+ = QFP so that F+ = F. (34.11)

The entire issue depends on an a priori specification of the class of admissible rotationsQ for such transformations. This question is related to a constitutive assumption on thesymmetry group of the material and appears to have little to do with any fundamentalprinciple in continuum physics. In particular, if FP is completely specified by theflow rule, as in the case of the single crystal model (see ASARO [1983]), the subgroupof admissible rotations on the intermediate configuration is the identity. Thus, nouniqueness issues or invariance questions arise.



35. Elastic response and free energy for multiplicative plasticity

As in the infinitesimal theory [assumption (A2)], the free energy function T is as-sumed to depend locally on the deformation through the elastic part of the deformationgradient. From a micromechanical standpoint, T represents the stored energy asso-ciated with the elastic lattice deformation. Assuming that hardening mechanisms areuncoupled from the elastic deformation and enforcing at the outset frame invariancerelative to superposed rigid body motions on the current configuration, one is led tothe functional form

T1(Fe, (se) = W(Ce) + (J ) where Ce = FeTFe. (35.1)

The dependence of T on the deformation via C e is used by a number of authors,MANDEL [1972, 1974] in particular, and occurs in the original work of LEE [1969].The formulation of a model of plasticity incorporating {FP,I } as microstructuralvariables and the internal energy (35.1) then proceeds as follows.

(A) Dissipation function and constitutive relations. Recall that the stress power inthe finite strain theory is given by Pint = P · F = r d. In addition, for the purelymechanical theory, the local dissipation D is the difference between the stress powerand the rate of change in internal energy, i.e.,

D) = *r d - a(Fe, (a) . (35.2)

As in the development of the infinitesimal theory, we accept the inequality D 0,for all possible motions on 2 x II, as a basic principle. The goal is to identify theappropriate constitutive equations implied by this dissipation inequality. Using thekinematic relation

C e = 2FeTsym[FeFe-l]Fe = 2FeTdeFe, (35.3)

which follows from (34.9), the time rate of change of the free energy is evaluated as

atT = aCeW( ¢e) e +a aiy)hint

=2Fe[aeW( Ce)]FeT * de + E 0 (). (35.4)

Inserting this result in (35.2) and making use of the additive decomposition (34.9)1implied by (34.3), results in the inequality

nint

D= (T-2Fe [a, W]FeT ) . de +r . dP + - qa~ > 0, (35.5)CZ=l

SECTION 35 335

where we have set q = -ab7. A standard argument in constitutive theory (seeTRUESDELL and NOLL [1965]) then produces the following constitutive relations

T = 2Fe[ZeW( e)]FeT and q' =-aa(). (35.6)

The dissipation inequality then takes the reduced form

Lint

D = r dP+ q'fl t 0 in x , (35.7)a=1

which is viewed as a restriction to be placed on the evolution equations that describeplastic flow (corresponding to assumptions (A3) and (A4) in the infinitesimal theory).

(B) Description relative to the intermediate configuration. It is a trivial matter toexpress the preceding results entirely in terms of objects defined on the intermediate(local) configuration defined by FP . To do so, let

= Fe- TF e -T (35.8)

denote the symmetric second Piola-Kirchhoff stress tensor relative to the intermediateconfiguration. From (35.8) and (35.6) one obtains the equivalent statement of theconstitutive equations

S = 2a-eW(Ce) and qf = -a(). (35.9)

Similarly, noting that r · dp = -r ·1 (since - is symmetric) and using (35.8) togetherwith result (34.6), the reduced dissipation inequality (35.7) becomes

?int

Dz= [CeS] .LP+ q 0 in x . (35.10)a=1]

This form of the dissipation inequality appears in MANDEL [1972], see also ANAND

([1985], Eq. (23)). The generally nonsymmetric stress tensor H := [C eS] conjugate tothe plastic rate of deformation L P is restricted by the symmetry condition C e- =

T- C e-1 and first appears in MANDEL [1972] who refers to L P as the plastic distortionrate.

36. Plastic flow evolution equations for multiplicative plasticity

To complete the formulation of a model of finite strain plasticity it is necessary tospecify the evolution equations for the internal variables. These are either {IP, , } if aspatial description is adopted, or {L P, , } if a description relative to the intermediateconfiguration is adopted. This is the aspect of the theory where the key differencesbetween existing models arise.

336 J. C. Simo CHAPTER III


(A) Description in the current configuration. Models of plasticity widely used in nu-merical simulations are typically formulated directly in the current configuration. See,e.g., the review articles of TVERGAARD and NEEDLEMAN [1981] and HUGHES [1984].To establish a connection between these models and formulations of multiplicativeplasticity, consider first the case in which the elastic domain is defined in terms of thespatial Kirchhoff stress as

E = {(r, q) E S x Rnin': f(-r,q') < O for = 1,2,..., m}. (36.1)

As pointed out earlier, frame invariance necessarily implies that f, (QrQT,.) =

fil(r, ), thus restricting the yield condition to isotropic functions. The simplest ex-ample is furnished by the von Mises criterion (m = 1) described in Chapter I.

The model of associative plasticity corresponds to the assumption of maximum dis-sipation at a fixed configuration which, in view of (35.7), translates into the inequality

nint

r - ('] .d + E [q - q]~ > (T, qua) E E. (36.2)a=1l

An argument identical to that described in Section 6 then yields the local evolutionequations

m m

dP -zyrf,(r, q), , = q aqvf(r, qf),

/1fb~~~~=l~ m= (36.3)

y >0, f,(r, q) O and Ey"f (, q) =0.t/=1

To complete the theory it is necessary to provide an evolution equation for iwjP so thatthe evolution of P is defined. Two possible options are:

(i) The restriction of the entire formulation to isotropy. In this situation, the ori-entation of the intermediate configuration is irrelevant and the plastic spin remainsarbitrary.

(ii) The specification of a constitutive equation for the plastic spin wiiP, a pointbrought into consideration in KRATOCHVIL [1973]. This is clearly possible since, ac-cording to (34.10), iv-P is objective. A common choice is iiP = 0. Other options areexamined in DAFALIAS [1984] and ANAND [1985].Interestingly, the plastic flow rule (36.3) can be obtained from the corresponding flowrule in the infinitesimal theory merely by replacing EP with dP. This form, however, isnot the most convenient one for the development of numerical algorithms. The follow-ing result, due to SIMO and MIEHE [1992] in the restricted context of isotropy, yieldsthe optimal parametrization of (36.3) from the point of view of numerical analysis. Let

£,b e = F{at [F-lbeF-T] }FT = be - Ibe - belT (36.4)

be the Lie derivative of the elastic left Cauchy-Green tensor.

SECTION 36 337

THEOREM 36.1. The following relation holds: -£fbe = sym[lPbe]. (36.5)

PROOF. By making repeated use of (34.3) and definition (34.5) for L P one obtains theidentity

FeLPFeT = F[(FP-I F PFP- ) FP - T ] FT

= -F[at(FP-) FP -T]FT. (36.6)

Now observe that (34.3) implies be = FFT F[FP-FP - T]FT. Combining thisresult with (36.6) yields, in view of expression (36.4),

2sym[FeLPFeT ] = -F[ t(FP-IFP-T)]FT = -£ b e . (36.7)

The result follows by noting that (34.6) implies FeLpFeT = IPbe .

Consider now the specific constitutive assumption ivP = 0. Using (36.5), the flowrule (36.3) can be rewritten as

vbe = sym [ ( a, ]fp(r, bej, (36.8)

which is the form used in SIMO [1992] in the development of a class of algorithmsthat inherit all the features of the infinitesimal theory. These methods will be describedin the following chapter in a more general setting.

REMARK 36.1. If the additional assumption of isotropy is made, the entire theorycan be described in terms of the variables {be , 5, } and the Kirchhoff stress tensori-; i.e., entirely in the current configuration. Isotropy means full S0(3) invariancewith respect to rigid body motions of the intermediate configuration; hence the storedenergy function depends on be since

W(Q Q T ) = W(Ce) VQ E S0(3) <#= W(C e ) = W(be). (36.9)

The Kirchhoff stress tensor is then computed either by the well-know relation (seeTRUESDELL and NOLL [1965]) r = 2abW(be), or using the representation theoremsfor isotropic tensor functions summarized above. Clearly, r and be are co-axial.

(B) Description in the intermediate configuration. Recall that in a description ofplastic flow relative to the intermediate configuration the basic internal variables are{LP, ,}, along with the second Piola-Kirchhoff stress tensor S defined by (35.8).This is the description adopted in most micromechanical investigations of crystallineplasticity. Here, attention is restricted to the formulation of associative models ofplasticity with elastic domain

(36.10)

338 J.C. Simo CHAPTER II1

= (_, q') E X Rni": f -9, ) < 0 for pI = 1, 2 .... 7mI.


In contrast with the situation found in the spatial description, frame invariance nolonger restricts the convex functions f (-, ,) to isotropy. The first step in the deriva-

tion of the flow rule is to enforce the symmetry condition S = S and cast expression(35.10) for the internal dissipation into the equivalent form

nint

D = S. DP + E qfl > 0 where DP = sym [ eLP]. (36.11)a=l

Definition (36.11)2 appears in SIMO ([1986], Eq. (33.14b)) and gives the symmetriza-tion of LP relative to 0 e, which is consistent with the interpretation of Ce as themetric tensor in the intermediate configuration. The same definition for D P occurs inMORAN, ORTIZ and SHIH [1990] where, consistent with the view of C e as a metric,the plastic spin is also defined as the skew-symmetric part of L P relative to C e. Thisleads to the decomposition

LP = Ve- [D p + WP] where W P = skew[CeLP]. (36.12)

Since in most metals the elastic deformations are typically small in comparison withthe plastic deformations, (36.12) differs only slightly from the usual decompositionof L P into symmetric and skew-symmetric parts. By adopting the inequality

int

[ S-S] DP + E [q -q] O (S, ) E, (36.13)ol=l

the same argument quoted in the derivation of the flow rule (36.3) now yields theevolution equations

DP = CE yasff(s, q'), i = Ypaq f (S, q(),3

(36.14)

¢ly Ž0, fA (Sq) 0 and y/ff(SSqO) = 0./z=1

The formulation is completed by specifying an independent constitutive equation for

the plastic spin W P. For single crystals, this additional constitutive relation arisesas a result of Schmidt's law; see HILL and RICE [1972], ASARO [1983], HILL andHAVNER [1982] and the more recent discussions in BOYCE, WEBER and PARKS [1989],HARREN [1991] and MORAN, ORTIZ and SHIH [1992] among others. For poly-crystals,the widely used assumption [see (36.12)]

WP = LP = e-lDp,

SECTION 36 339

(36.15)

specifies the orientation of the intermediate configuration via (36.14) and (36.15)2.

REMARK 36.2. In the work of MANDEL [1974] the yield surfaces defining the elasticdomain E are assumed to take the functional form f ( , q") < 0, supplemented bythe following dissipation inequality

ninth

[a-i* + E [q - ](a O (+, Y E (36.16)a=1

These hypothesis result in the replacement of the six-dimensional flow rule in (36.14)by the alternative nine-dimensional flow rule

ZL = Ey ,i f q( ). (36.17)/=

1

However, as noted in LUBLINER [1986], Mandel's derivation of (36.17) from theprinciple of maximum dissipation is questionable since the symmetry constraint~e-l = :Te-1 is not accounted for; see LUBLrNER ([1990], p. 460) for

further discussion and an alternative form of (36.17). Observe that the constrainte-'l v= Te-l is a restatement of local balance of angular momentum.

REMARK 36.3. The preceding formalism is consistent with the following geometricinterpretation. Endow the local intermediate configuration with the local Riemannianmetric defined by C. Relation (34.6) then asserts that IP is the push-forward to thecurrent configuration St of a contra-covariant rank-two tensor, L P, in the intermediateconfiguration. Similarly, with the definitions given above, (34.6) implies the relations

dP = Fe-TDPFe- I and jjP = Fe TWPFe-I, (36.18)

which are push-forward transformation rules if D P and W P are viewed as covariantrank-two tensors on the intermediate configuration. This interpretation agrees with theview of (36.11)2 and (36.12)2 as "index lowering transformations" with the metrictensor e. In the same vein, relation (35.8) is the appropriate push-forward transfor-mation rule if the stress field S is viewed as contravariant rank-two tensor field on theintermediate configuration. This interpretation is consistent with the view of the termS DP in the dissipation function (36.11)1 as a contraction between a contravariantand a covariant tensor field. In summary, the description relative to the intermediateconfiguration is the convected description of elastoplasticity with metric tensor C e.This geometric point of view is developed further in SIMO and ORTIZ [1985] and SIMO[1986].


TABLE 36.1

Elastoplastic model of multiplicative plasticity.

(i) Kinematics induced by the multiplicative factorization F= FeFP. Setting

P=l-l wherel=FF-l and le = FeF,

it follows that IP = FeLPFe with LP = ibPFP - 1

.

(ii) Constitutive equations for a free energy function = W(Ce) + 7{()

r= F [2D W(Ce)]FeT and qa =-a-i( a)/ac,

where Ce = Fe

TFe is the elastic right Cauchy-Green tensor.

(iii) Evolution equations describing associative plastic flow

/P = ,ylarf(T,q), la (T,q),

yP f0, f(r, q) <O and y fu(,q) = 0.

These equations imply iiP = skew[lP] = 0.

REMARK 36.4. Using the transformation relations defined by (36.18) and (35.8), theflow rule (36.14)h in the intermediate configuration implies the following flow rule inthe current configuration

m

dP = Y n. where/ = Fe-T [ -Sf . ( q)] F (36.19)

/=1

which furnishes the generalization of the spatial flow rule (36.3)1 to anisotropic re-sponse. Relation (36.19)2 becomes a restatement of the chain rule, with n, givenby n = af,(r, qn) if the yield functions satisfy the relation f (FeTFeT,.) =

f,(S, .). By Noll's rule (TRUESDELL and NOLL [1965]), this covariant property is anequivalent statement of the isotropy condition.

To render matters as concrete as possible, in what follows we shall be concernedalmost exclusively with a formulation of plasticity relative to the current configuration,as explained in detail above. For the convenience of the reader and easy reference,Table 36.1 summarizes the key relations of the elastoplastic model that will be usedthroughout our subsequent developments.

37. Volumetric/deviatoric uncoupled finite plasticity: J2-flow theory

A large body of experimental literature supports the fact that plastic deformationsin metals preserve volume. It is for this reason that the assumption of an isochoric


plastic flow is introduced at the outset in nearly all treatments of metal plasticity,as exemplified by the classical monograph of HILL [1950]. The goal of this sectionis to specialize the theory of finite strain plasticity developed above to this situationrelevant to metal plasticity. The resulting theory is illustrated by constructing an ex-tension of the classical model of J 2-flow in which the elastic response is characterizedby uncoupled volumetric/deviatoric hyperelastic relations. The developments in thissection rely heavily on the results presented in Section 32 for uncoupled models ofelasticity at finite strains.

The multiplicative factorization of the deformation gradient provides a rather con-venient mechanical setting for the enforcement of the volume-preserving constrainton the plastic flow. Within this setting, this condition translates into the kinematicassumption det[FP] = 1 on the determinant of the plastic deformation gradient. Thefollowing purely kinematic results give the rate of change of the total, the elastic andthe plastic volumes.

THEOREM 37.1. The time derivatives of J = det[F] and Je = det[Fe] are given by

J = Jtr[d] and Je = Jetr[de], (37.1)

where d = sym[FF-t], de = sym[FCFe ']. The time derivative of JP = det[FP] isgiven by

JP = JP tr[L] = JPtr[dP], (37.2)

where L P = PPFp - ' and dp = sym[FeL PFe 1] is the plastic rate of deformationtensor in the spatial description.

PROOF. The following standard result gives an explicit formula for the time derivativeof the determinant of an invertible time-dependent matrix:

d det[A(t)] = det[A(t)] tr[A(t)A- (t)] (37.3)

for any A(t) E GL(n). Relations (37.1) follow by setting A(t) = F and A(t) = Fe

in the preceding formula and noting that a skew-symmetric matrix has zero trace.Relation (37.2) also follows from (37.3) by setting A(t) = FP , together with theproperty

tr FeL P Fe- ! ] = tr [P] (37.4)IL~~~~~~~~~~~~~(74

for the trace of a matrix, and the definition of d p in terms of L P.

342 J.C. Simo CHAPTER iI


An immediate consequence of relation (37.2) is that the condition JP = 1 is equiva-lent to requiring that tr[dP] = 0 for all t E If. A formalism based on the use of Lagrangemultipliers provides an obvious way to enforce this constraint. Rather than pursuingthis route, we shall adopt the framework described in Section 32 for the treatment ofmodels of finite elasticity exhibiting uncoupled volumetric/deviatoric response, andexplore the consequences of considering a free energy function of the following form:

= U(J) + W(Ce) + ((~). (37.5)

In this functional form C e denotes the elastic, volume preserving, right Cauchy-Greentensor, defined by the relation

Ce = eTFe where Fe = Je-I/ndimFe (37.6)

which ensures that det[Fe] = det[Ce] = 1. For the moment, no constraints will beplaced on the determinant JP of the plastic deformation gradient.

(A) Hyperelastic stress-strain relations in the spatial description. To derive the elas-tic stress-strain relation associated with the free energy function (37.5) we mimic theprocedure employed in Section 35 and compute first the time rate of change of BT,.Time differentiation of definition (37.6)1 gives

C = 2Je-2/ndimFeT [Fe-T eFe- - dim Jetlel] (37.7)

ndim

By inserting (37.1)2 into this expression and making use of the relation Ce =FeTdeFe derived in Section 35, the time rate of change of Ce can be written usingdefinition (37.6)2 as

C e = eTdev [de]Fe (37.8)

where dev[de] = de - (l/ndim) tr[de]l is the deviator of the elastic spatial rate of de-formation tensor. This relation together with (37.1)1,2 implies the following expressionfor the time rate of change of the free energy function:

at@= [JU'(J)] tr[d] + dev[Fe2ae W(Ce)FeT] de

+ S ~,~1(~). (37.9)a=1

By making use of the additive decomposition d = de + dP and inserting expression(37.9) into the dissipation inequality D = r · d - tg > 0, the same arguments

SECTION 37 343

employed in Section 35 yield, in the present context, the constitutive relations

r = [JU'(J)] 1 + dev [F e28w f(Ce) j eT](37.10)

and q' = -8( ,),

together with the following expression for the reduced dissipation inequality:

D = dev[l] dP + q > O. (37.11)UY=

1

An important feature of this result should be noted. Since dev[T] dP = dev[-r] dev[d] it follows that neither the trace of the Kirchhoff stress tensor nor the traceof the plastic rate of deformation tensor, which describes the evolution of plasticvolume according to the relation tr[dP] = dlog(JP)/dt implied by (37.2), appearin the reduced dissipation inequality. In other words, the Kirchhoff pressure field7t = (1/ndim) tr[-r] does not contribute to the plastic dissipation in the material. Thisresult is possible only if tr[dP] = 0 during plastic flow or, equivalently, if JP = 1throughout the entire deformation process. Therefore, we arrive at the remarkableconclusion that the free energy function (37.5) automatically implies the isochoriccondition on the plastic flow.

(B) Associative evolution equations for isochoric plastic flow. To complete the for-mulation of the theory it only remains to formulate appropriate evolution consistentwith the condition of isochoric plastic flow. We shall do so by adopting the frameworkof associative plasticity, as described in Section 36. Since plastic flow is to be inde-pendent of the pressure field, the yield criterion formulated in the spatial descriptioncan depend only on the deviatoric part dev[-] of the Kirchhoff tensor. This conclusionis one of the key hypotheses made in classical treatments of metal plasticity, see HILL([1950], Chapter I). Accordingly, the elastic domain is assumed to be defined as

E := {(dev[r],q&) E x Rni"': f,(dev[r],q) < 0

for/ = 1, 2,..., m}, (37.12)

and subject to the usual conditions which require the functions f(-, .) to be convexand assume that the m-constraints are qualified. Yield criteria which are independentof the hydrostatic pressure field, of the type involved in this definition of E, arereferred to as pressure insensitive. The evolution equations of associative plasticityemanate from the principle of maximum dissipation which, in the present context,results in the following inequality implied by expression (37.11):

hlint

[dev[r] - dev[;]] . dp + E [q - j] 0 V(dev[r], qc) EI. (37.13)a=1



Once again, a standard argument in convex yields the following associative evolutionequations for isochoric plastic flow:

m m

dP = y 7 af, (dev['r], qa), : = y aq f (dev[,r], q),

(37.14)

' >s 0, fS, (dev[r],q) <0 and Ey7f,(dev[r], q) =0./v=1

Observe that by exploiting the relation dev[r] = - (/ndim) tr[r]l, the evolutionequation for dP can be written in the equivalent form

m

dP = E ,y dev [adev[,] f (dev[r], qa)], (37.15)/L=1

which is consistent with the condition tr[dP] = 0 and implies, therefore, the isochoricconstraint on the plastic flow. As in the general theory, the plastic spin has to be spec-ified via an additional evolution equation, the simplest example of which correspondsto the widely used assumption wP = 0.

EXAMPLE 37.1 (J2-flow theory). The von Mises yield criterion of metal plasticity isthe prototype of a pressure-insensitive yield condition. For isotropic hardening, theelastic domain takes the form

E = {(dev[-r],q) x +: f(dev[T],q)

= dev[r] + [ y - q] < 0}, (37.16)

and the associative evolution equations that describe plastic flow, the so-called Levy-Saint Venant flow rule, become

d = n and = wheren- dev[(37.17)[dev[r]

The plastic multiplier is subject to the standard Kuhn-Tucker conditions y > 0 andyf(dev[r], q) = 0. We provide below two examples of hyperelastic constitutive equa-

tions that complete the extension of this classical model of metal plasticity to the finitestrain regime within the framework of a multiplicative decomposition of the defor-mation gradient. Because the general evolution equations (37.14) ensure that JP = 1throughout the entire range of the deformation, it is permissible to replace J by Jewhenever the former variable appears. In keeping with the preceding framework,however, we have chosen not to perform this replacement.

SECTION 37 345

Model I. The first model is a direct extension of Example 32.1 considered in Sec-tion 32, as defined by the free energy function

T = --_ _ log(J)] + l(tr [Ce] - nfdim) -+ X(), (37.18)

which is clearly of the form (37.5). The function X(() models the specific form ofisotropic hardening in the material via the constitutive equation q = -- '((). Thestress-strain relations defined by (37.10)1 then become

r = ½i(2 _ 1)1 + tdev[b6 ] where be = FeFeT. (37.19)

The free energy function (37.18) was introduced in SIMO [1987].

Model II. The second model is based on a description of elasticity in principalelastic stretches via the following free energy function

ndim

= [log(J)] +/ yMi () + l(), (37.20)A=]

where A' > 0, A = 1,..., ndim, are the eigenvalues of the elastic, volume preserving,deformation gradient Fe. This model clearly falls within the class of free energyfunctions (37.5) since (Ae) 2 are the eigenvalues of C e. The stored energy functionpart in (37.20) is known as the Henky model, see ANAND [1979] for a discussionof the applicability of this model. Using the results in Section 33, the stress-strainrelations (37.10) are easily shown to reduce to

,r = r log(J)1 + log(be). (37.21)

This model is discussed in detail in Chapter IV within the context of a recentlyproposed class of exponential return mapping algorithms.

38. Rate form and variational inequality for multiplicative plasticity

The model of multiplicative plasticity summarized in Table 36.1 provides a hypere-lastic constitutive equation for the Kirchhoff stress field r relative to the intermediate(released) configuration. This model can be cast into a rate constitutive equation forthe Jaumann derivative of the Kirchhoff stress tensor by enforcing the consistencycondition using a technique entirely analogous to that employed in the infinitesimaltheory. The resulting rate constitutive equation arises naturally in the formulation inthe incremental (or rate form) of the initial boundary value problem.

Remarkably, it will be shown below that older models of plasticity, with elastic

response characterized in terms of the hypoelastic constitutive equation -= aid- dPj,which is postulated at the outset, can be exactly recovered from the rate form of themodel of multiplicative plasticity summarized in Table 36.1. These two formulations of



plasticity differ, therefore, exactly in the same manner as the theories of hyperelasticand hypoelastic material response, see TRUESDELL and NOLL [1965]. In short, themultiplicative theory of plasticity provides an explicit expression for the incrementalelastic moduli a, involving derivatives of a stored energy function, while the hypo-elastic theory regards the moduli a as constitutive parameters to be specified a priori.As pointed out in the introduction to this chapter, hypoelastic models of plasticity arewidely used in older algorithmic treatments of the subject, see KEY and KRIEG [1982],TAYLOR and BECKER [1983] and the review articles of NEEDLEMAN and TVERGAARD[1984] and HUGHES [1984].

The first step in the derivation of objective rate forms of the hyperelastic constitutiveequations is the computation of the material time derivative of the Kirchhoff stress.Time differentiation of the hyperelastic constitutive equation for the Kirchhoff stressand use of the kinematic relations in Table 36.1 gives

*= leT q- e T -BFe[c(lVe)]Fe T, (38.1)

where C = 4a2ecW(C e) is the elasticity tensor in the intermediate configuration.Define the elasticity tensor c in the current placement by the push-forward relation

Cijkl = FeAFeBFkeFcFeDCABCD where CABCD = 4 aC Bc (38.2)

Since le = FeF e- and de = sym[le], using the kinematic relation

de = sym[FeF e - l] = Fe-T [FeTFe + FeTFeF e- 1

= Fe-T[½Ce]F e - l, (38.3)

Eq. (38.1) can be written as

= le + leT + cde. (38.4)

Now recall the kinematic relation le = de + ivu e. Denoting by e = .- v e. + rjiethe Jaumann derivative computed with the elastic spin tensor, relation (34.7) takes theform

= a de where aijk = Cijkl + Tik6jl + Tiljk. (38.5)

In the model in Table 36.1 the plastic spin is specified as ii P = 0 which, in viewof the additive decomposition (34.8) implied by the multiplicative factorization of thedeformation gradient, is equivalent to the condition wv = we. By specializing result(38.5) to the current model, the following two alternative forms of the rate constitutiveequations are obtained:

V=ade= a[d- dP] and £r-=cd-adP. (38.6)

SECTION 38 347

The second relation in (38.6) is a direct consequence of definition (30.12), result (38.5)and the additive decomposition l = -IP specific to this model. Expression (38.6)1 interms of the Jaumann derivative arises naturally in the interpretation of multiplicativeplasticity as a variational inequality, while expression (38.6)2 in terms of the Liederivative arises naturally in the incremental formulation of the initial boundary valueproblem. It is clear that any objective rate other than the two appearing in (38.6) canbe used in the formulation of the incremental elastic constitutive equations, althoughlittle is to be gained from such an exercise.

The model summarized in Table 36.1 can be cast in the form of a variational in-equality which arises as a result of the underlying assumption of maximum dissipation.Let hd = D2()/0~ be the generalized plastic moduli. Assuming that boththe tensor a in (38.6)2 and the nint x nint matrix [h'a] are invertible, we have thefollowing result.

THEOREM 38.1. Denoting by (, ) the L2 ( 2)-inner product, the constitutive model inTable 36.1 is equivalent to the variational inequality

nint int

r - 7, (Vv o at)-a-' - (q - 4',h-) 0oa=1 3p=l

V(r,q ) E E, (38.7)

where E = ((,, q) E S x R': f (r, q ) 0 for = 1,2,..., m}.

PROOF. Combining the rate constitutive equation (38.6)1 with the associative flow ruleusing the relation de = d - dP yields, under the assumptions that a and [had] areinvertible, the result

d-a 'v=dP =E 3, "f(r q),/z=1

(38.8)nit m

-- , h:q = CY /Oq = A fit( Tqqu qa)P=1 3y=1

Inequality (38.7) then follows from the convexity property

ninth

[- r] . -f(7rq) + Z [a - qn]qf,(r'qC)

< f (K , q- ) - f (7r, q ), (38.9)

and the Kuhn-Tucker conditions. Since the argument is entirely analogous to that usedin the proof of (6.8), further details are omitted.

All that remains to complete the incremental or rate formulation of multiplicativeplasticity is to eliminate the plastic multipliers y > 0 in the flow rule recorded in



Table 36.1 leading to closed-form expression for the elastoplastic tangent moduli. Todo so, one proceeds exactly as in the infinitesimal theory, see Section 5. Assumingthat the constraints are qualified one defines the trial set of constraints as

Jtr = { E 1,..., m}: f, (r, q) = 0}. (38.10)

The set of active constraints is then J = f{/ E Jtr: f(r, q") = O}. To computef, recall that f(, q ) is isotropic as a result of the restriction imposed by frameinvariance. Consequently a,fu commutes with T and one has the crucial result

arft Te - af, * = [T(a,f,) - (af, )T ] = 0. (38.11)

Using the incremental stress-strain relation = a[d - dP] and the preceding result,it follows that

nint ninth

i, = arf, a[d - dP] - E (a q f)h / ;. (38.12)a=1 p=1

Now define the metric coefficients g = [g] = gT by the relation

nint nint

g= T f,f-a afv+ E (a f)h (aq, fv), /,V E J. (38.13)a=1 O=1

Since the constraints are qualified and the basis vectors ,-f,. are linearly independentat each point (, qu) E aE, it follows that the matrix g = [g,] is invertible. Proceed-ing exactly as in the infinitesimal theory, setting [g4"] = g-1 the plastic multipliersare obtained by enforcing the consistency requirement f, = 0 leading to

y = gv f, .ad for E Jt. (38.14)VEJV

The set J of active constraints then becomes J = {i E J: -yA > 0}. By insertinginto the incremental constitutive equation (38.6)1 the flow rule for dP in Table 36.1,with the multipliers 7y" defined by (38.14), one arrives at the incremental relation

Vr = aP d where aeP = a - g'V[aa,f,] ® [a ,fr]. (38.15a)

ltEJ vEJ

Clearly, the tensor aep of elastoplastic moduli satisfies the symmetry conditions

tr and traijkl alij and aijk = ajlk ajikl.

SECTION 38 349

(38.15b)

TABLE 38.1

Rate model of plasticity and relation with the multiplicative theory.

(i) Elastic rate constitutive equations with r = i - + Tro and d = sym[FF -]

-=a[d-dP] and h =_E h3s,

P=

where aijkl = aklij, aijkl = aijlk = ajikl and ha3= h

3 a .

(ii) Evolution equations describing associative plastic flow

Jp

= E '

al -f1 (r, qa), a = 7 aq f(, q),

y 0, f(r, q) o and f (, q ) = O.i=1

These equations imply ivP = skew[/P] = 0 so that Ip = dp .

(iii) Relation to multiplicative plasticity. Elastic and hardening moduli

aijkl = Ckl l + ik6Sjl + Tijk and ha3 = a2R/acat,

where tf = W(Ce) + 7-(~0) is the free energy and

Tij = FIAFjB2ace , Cijkl = FA FBF kCFID4a e a- eCA B acABaCD

Kinematic relation: It = FeLPFe - l with LP = FPF-I.

The rate equation (38.5) is phrased in terms of the Lie derivative and not the Jaumannrate. By inserting expression (38.14) into the flow rule for dP the rate constitutiveequation (38.6)2 in terms of the Lie derivative reduces to

£7, = cepd where CeP = c - E EgV[a-f ] ® [af,] . (38.16)ktEJ vEJ

Here C is the spatial elasticity tensor defined by (38.2). This expression will be usedin the incremental formulation of the elastoplastic initial value problem.

EXAMPLE 38.1. For the convenience of the reader the basic relations that definedthe conventional rate model of classical finite strain plasticity and its relation to themultiplicative theory are summarized in Table 38.1. To illustrate this relation in aconcrete setting, consider a stored energy function introduced in Example 31.1

W(Ce) = ¼A(det[V e ] - 1) - [ (, + ½A)log(det[Ce])+ 2tt[~] 3 (3.2


(38.17)+ 1 /, (tr Ve] 3),

where e = det[ e] and A, p > 0 can be interpreted as Lam6 constants. Using the

relation aZe Je = Jee-l, the constitutive equations in Table 38.1 for the Kirchhoffstress becomes

r = A(J e 2 -1)1 + (b e - 1) where be = FeFeT. (38.18)

Finally, using relations in Table 38.1, the spatial elasticity tensor is easily shown tobe given by the expression

c = Aje2 1 1 + 2 [1 - (Je2 _ 1)A/i] I, (38.19)

where I with components abcd = [6acSbd + 6ad6bc]/2 is the rank-four symmetric unittensor. An explicit expression for the tensor a is easily found by inserting (38.18) and(38.19) into the relation in Table 38.1.

Now suppose that the additional assumption is made that the elastic volumetricstrain is small in the sense that Je = 0(1). This hypothesis is quite reasonable formetal plasticity. In addition, for the von Mises yield criterion one easily concludesthat the magnitude of the deviatoric Kirchhoff stresses are of the order of the flowstress ay. Suppose further that the hyperelastic constitutive equations of multiplicativeplasticity are replaced by the rate constitutive equation in Table 38.1, with the tensora defined by via the ad hoc constitutive assumption

a = A 1 + 2 I, (38.20)

widely used in conventional treatments of plasticity. A formal argument that uses astraightforward asymptotic expansion then reveals that the tensor a defined by (38.20)differs from the tensor a associated with the model of multiplicative plasticity withfree energy function (38.17) by terms of order O(y//). Since the ratio vy/ ofthe flow stress over the shear modulus is of the order of 10- 3 for most metals, thepreceding argument provides a partial justification for the conventional (hypoelastic)rate formulations of finite strain plasticity.

39. Variational formulation: Weak form of the momentum equations

As a first step in the formulation of the initial boundary value problem for elasto-plasticity, we consider the weak formulation of the momentum balance equations forthree-dimensional nonlinear continuum mechanics. Since the interest here is nonlin-ear plasticity, a Lagrangian formulation of the problem is adopted throughout. Forplasticity problems, however, it is conventional and fairly convenient to cast the en-tire variational formulation in terms of fields, such as the Kirchhoff stress, definedon the current placement St of the continuum body. The formulation remains never-theless Lagrangian, since the parametrization of the problem is in terms of materialcoordinates {XA} and the motion itself is the primary variable in the problem.

The weak form of the momentum equations, supplemented by local constitutiveequations for the stress field (say the Kirchhoff stress tensor r), gives rise to a highly


nonlinear initial boundary value problem for the motion t c- got of the continuumbody. The two sources of nonlinearity in this problem are: (i) Geometric, arisingfrom the intrinsic nonlinearity in the kinematic description of the problem and (ii)Material, arising from the nonlinearity in the constitutive relations. These two sourcesof nonlinearity become apparent when the weak form of the momentum equations isformulated in rate or incremental form.

Motivated by the foregoing description of the admissible deformations in a contin-uum body, with reference configuration 2 c Tdi m and prescribed deformations i) onthe part of the boundary F,. C aQ, define the configuration space as the set

C = {yP C WI'P(2)nd im : det[Dp] > 0 in 2 and cplr = c,}. (39.1)

For nonlinear elastostatics, the choice for the exponent p is dictated by growth con-ditions on the stored energy function. As before, St = cPt(•2) denotes the currentplacement of the body under a motion t i got E C. A time-independent displace-ment field superposed on St which does not violate the prescribed Dirichlet boundaryconditions on Fr is called an admissible (spatial) variation of the motion pot at timet C I. Admissible variations span a linear space denoted by Vi and defined as

V = : (Q) Rdim: (,(X)) = O for X E rF}. (39.2)

In short, the space V, is the tangent space to the configuration got C at time t I.Elements in V, are also referred to as spatial test functions. To remove the dependenceof ri E V, on P one defines material test functions, denoted by rlo: Q2 - I d'm, viathe change of variables ro(X) = 7r(cp(X)). Thus, material test functions span thefixed linear space

Vo = {ro: Wl'P(Q)dim: o(X) = 0 for X E j. (39.3)

It is convenient to think of Vo as given and then construct test functions 7 E V bysetting ri = r o c-' for each r0o C Vo. By the chain rule, it follows that

GRAD[7ro] = (Vri o cp)F, i.e., 70a A = 7rabFbA. (39.4)

With this notation in hand the weak form of the momentum balance equations can beformally justified by taking the L2(Q2)-inner product of (29.6) with any 7r0 C Vo andusing the divergence theorem. The result can be written as

Gdyn(P, V; 70) =/ po od dQ + G(P; o) = (39.5)

for any (material) test function i0 E Vo. Here, G(P; r0o) denotes the (static) weakform of the equilibrium equations which is given by

G(P; ro) = J P GRAD[ro1 d2dQ - J B ,od- Jt, r7odP. (39.6)



Statement (39.5) is considerably more general than the local form (29.6), since theconfigurations 'ot are subject to less restrictive continuity requirements. The variationalequation (39.5) is supplemented by the weak form of the kinematic condition att -V =0.

EXAMPLE 39.1. For elasticity the nominal stress P is a local function of the motiondefined by (31.1) in terms of the deformation gradient, i.e., P = aFW(Dwt). If thisconstitutive equation is enforced locally then the static weak form becomes a functionof the motion t - 'Pt C and the dynamic weak form reduces to

Gdyn((Pt, V; r0o) = pov . o0 d2 + G(aFW(Dwpt); o). (39.7)

The dynamic weak form thus becomes a function of (t, V) Z where Z = C x Vo isreferred to as the canonical phase space for elastodynamics. The variational equations

Gdyn(pt,V; 0) =0 and I po(,t - V) 7odQ = 0, (39.8)

for arbitrary r0o c Vo, then furnish the weak form of Hamilton's canonical equationsfor elastodynamics viewed as an infinite-dimensional Hamiltonian system; see SIMo,POSBERGH and MARSDEN [1991] for a detailed account of this formalism. As alludedto above, only short-time existence results are known for this quasilinear hyperbolicsystem (see MARSDEN and HUGHES [1983], Chapter 6).

For the pure traction problem, the weak form of the momentum equations exhibitstwo conservation laws which are a direct consequence of the classical Euler law ofmotion. Recall that the local form of the momentum equations are the direct result ofapplying Euler's law of motion to a neighborhood O(X) of an arbitrary point X CG in the reference configuration, see, e.g., TRUESDELL and NOLL [1965]. The global formof balance of momentum, on the other hand, are obtain when Euler's equations areapplied to the entire reference configuration Q. Explicitly, let Next and Mreac denotethe resultant force of a prescribed system of external loads and the resultant force ofthe reaction forces, respectively defined by

Next =, jBd2+ j T d and Nreac=j Pv d, (39.9)

where 0o: 2 - S2 is the unit outward field normal to the boundary 02. Similarly,let Mext and Mreac denote the resultant torque associated with the prescribed system ofexternal loads and the resultant torque associated with the reaction forces, respectivelygiven by

Next= / t x B d + t x T d,

NFxtfTPtXr d2+J 'Pt x~dP, (39.10)Mreac = j ,t x Pv dr.

SECTION 39 353

A point in the canonical phase space Z = C x Vo of the problem will be denotedby z = (o, p) where p = poV is the generalized momenta. With this convention,the total linear momentum and the total angular momentum about the origin of thesystem are given by the standard expressions

L(z) =Jpd2 and J(z) = t x pd2. (39.11)

The classical Euler equations assert that any admissible motion t C 11 - z(t) C Z inthe canonical phase space is to satisfy the balance laws

dL = Next + Nreac and J = Mext Mrea. (39.12)dt dt

Therefore, if the external forces are equilibrated, in the sense that Next = 0 and Mext =0, then the total linear momentum and the total angular momentum are conserved forthe pure traction initial boundary value problem corresponding to Fr = 0. Theseconservation laws follow immediately from (39.12) merely by noting that NreacTreac = 0 as a result of the condition Fr = 0.

These classical conservation laws can also be derived directly from the weak form ofthe momentum equations. The explicit result is recorded below (for the case ndim = 3)since this is the setting adopted in the numerical treatment of the initial boundary valueproblem. The global conservation laws of momentum will play an important role inthe class of algorithms described in Chapter IV.

THEOREM 39.1. For the pure traction boundary value problem (,, = 0) under equi-librated loads (Next = Met = 0), the weak form of the equations yields the conser-vation laws

d dd L=O and dJ=0 inZxE , (39.13)dt dt

where Z = C x Vo is the canonical phase space and L, J are defined by (39.11).

PROOF. Conservation of linear momentum trivially follows from the dynamic weakform (39.5) by noting that for rp, = 0 arbitrary translations of the current placementSt = ot(Q2), defined by vector field x C St r = C with E Rndim constant,lie in the space V, of admissible test functions. To prove conservation of angularmomentum one observes that arbitrary infinitesimal rotations of the current placementalso lie in V,. Therefore, the vector field

= t(X) E St - r1 o(X) = C x t(X) C V0, (39.14)

is an admissible (material) test function for any E Rt di m, with gradient GRAD[ro0] =

C Dwt where is the skew-symmetric matrix with axial vector C. A straightforward



manipulation that exploits the relation r = P DotT along with the symmetry conditionr = rT then implies

P .GRAD[lo] = P .Dt = PDoT * = r. = 0. (39.15)

Using this result and expression (39.10)1, we conclude that the static part of the weakform of the momentum equations, defined by (39.6), reduces to

G(P, C x t) = - [J t x Bdf2+ I g t x TdFj = C - Mext, (39.16)

which vanishes since Mext = 0 by assumption. Finally, combining (39.16) with ex-pression (39.5) for the dynamic weak form and using definition (39.11)2 yields

Gdyn(P; C X Wt) = ( ot x po dQ = C aJ = 0, (39.17)

which implies the result since ( is arbitrary.

The preceding argument is closely related to techniques used in the derivation ofconservation laws for systems of differential equations which are invariant relative toa one-parameter group of transformations, see, e.g., OLVER [1986].

REMARK 39.1. The variational formulation of the equations of continuum mechanicsdiscussed above is formal in the sense that little is known about the appropriate math-ematical structure for specific models. For instance, in elastostatics the only knownsatisfactory theory is based on minimizers of the potential energy with a poly-convexstored energy function (BALL [1977]). Existence can be proved in a suitable Sobolevspace (typically W 1 ,p with p large enough as dictated by certain growth conditions),but it is not clear in what sense the minimizers satisfy the weak form (39.6). For arecent review of known results we refer to CIARLET [1988]. No rigorous mathemati-cal results are currently known for finite strain plasticity, except in the context of theinfinitesimal theory where a complete existence theory does exist; see the accountsgiven in Chapter I and references cited therein. The so-called deformation theory istreated in TEMAM [1985].

40. The total and incremental weak forms of momentum balance

The numerical solution of the weak form of the momentum equations for nonlinearproblems in solid mechanics is accomplished via an iterative scheme that involvesthe repeated solution of a sequence of linearized problems. These linear problemsarise from the formulation of an incremental or rate version of the weak form abouta sequence of configurations, leading to the so-called incremental method. For non-linear elasticity, the mathematical basis of this widely used technique is examined inBERNARDOU, CIARLET and Hu [1985]. The statement of the weak form of the equa-tions and its associated rate version will be given below in the current configuration,

SECTION 40 355

since this is the description widely adopted in the numerical solution of plasticityproblems.

Although the weak form (39.6) can be phrased entirely in terms of fields definedon the current placement of the body, the formulation remains Lagrangian because,implicitly, the configurations 'pt C are regarded as the primary variables. The keyresult used in this alternative statement of the weak form arises from the followingchain-rule relation

P . GRAD[r70] = F GRAD[ro] = r GRAD[ro]F-

= - (V/o t) (40.1a)

where r = 0 o o - '. In components, (40.1a) is equivalent to the expression

PaA(X)TOa,A(X) = [Tab()Tlab(X)] =ot(X) (40. lb)

Setting G,, (r; ) = G(P; ro), substitution of (40.1a) into (39.6) then producesthe result

Go,(r; ) = r · (V o pt) dQ -J B ( o pt) d2

(- 'i t *(77 o Wt)dF. (40.2)

To simplify the notation we shall often omit explicit indication of the compositionoperation involved in the change of variables from material to spatial coordinates,with the understanding that all the quantities involved are evaluated at appropriatepoints. This will usually be clear from the context.

The rate version of the weak form of momentum balance arises in the followingcontext. Suppose that (pt E C is a specified configuration, with associated Kirchhoffstress field r and spatial velocity field v, in dynamic equilibrium under prescribedloading defined by B in x and t on T x . Suppose further that the loadingis incremented by rates B and leading to an incremental change P in the nominalstress. Assuming sufficient smoothness so that time differentiation under the integralsign is permissible, the incremental version of the dynamic weak form becomes

atGdyn(P, V;o)0) =/ poat (V) o d2 + atG(P, 77) = 0. (40.3)

The key step in the reformulation of this result in terms of objects defined on thecurrent placement St, involves the following identity obtained by time differentiationof the relation P = FS

P = FS + FS = [FSFT (PF - ') (FSFT)]F T.

356 IC. Simo CHAPTER III

(40.4)

Using the relations r = FSFT and Er = [FSFT] along with expression (28.22),identity (40.4) can be written as

P = [£vr + (Vv o t)r] F- T. (40.5)

This result together with the relation GRAD[ro] = (Vr o Wt)F arising from the chainrule gives

a, [P GRAD[1o]] = [£vT- + (Vv o Wt)T] (V7i o 't). (40.6)

Assuming again sufficient smoothness so that time differentiation under the integralsign is permissible, (40.6) and (40.2) yield the following expression for the static term

tG¢p (r; r7) = atG(P; 7o0) in the incremental weak form (40.3) at configuration 'pt

atG, (r; ) = f [£vr + (Vv o pt)r] . (Vq o W't) dQ2

- J B. ( /o ) dQ - t ( o t) dP, (40.7)

for all admissible test functions rq V. Observe that the objective rate that naturallyarises in the incremental form (40.7) is the Lie derivative of the Kirchhoff stress. Itis conventional in the numerical analysis literature to use the following nomenclaturefor the first two terms in (40.7):

(i) Geometric term: [(Vv o 'Pt)r] (Vr o 'Pt), also referred to as the initial stressterm. For fixed stress field at a given configuration this term gives rise to a symmetricbilinear form (see Chapter IV).

(ii) Material term: (7-) · (V o t). For elasticity, the rate constitutive equation£f = (Vv o Pt) gives rise to a symmetric bilinear form at fixed configuration'Pt C (see Chapter IV).

We shall follow this nomenclature here and in subsequent chapters. The most im-portant application of the preceding rate form of the momentum equation arises in thenumerical solution of the initial boundary value problem by the so-called incrementalmethod. For the quasistatic problem, the spatial velocity field v is replaced by a dis-placement increment Au E V,, and weak form is discretized in space via a Galerkinfinite element method. For the dynamic problem it is also necessary to perform a timediscretization. Further details will be given in Section 41 and in Chapter IV.

41. Initial boundary value problem: Dissipativity and a priori estimate

The weak form of the momentum equations discussed in the preceding section, to-gether with the constitutive equations described in Section 34, comprise the initialboundary value problem for elastoplasticity at finite strains. This section providesthe complete statement of this problem in the form used in our subsequent algorith-mic analysis, gives the key (formal) a priori estimate responsible for the dissipativecharacter of the solutions.


To make matters as concrete as possible, attention will be restricted to a conventionalmodel of plasticity in which the elastic domain is specified in terms of true (Kirchhoff)stresses defined on the current placement of the body, as summarized in Table 36.1. Theprimary variables in the initial boundary value problem are the deformation qot C,the velocity field v C V,, and the internal variables (Fe, ~) C , where

= {(Fe,o): Q -* GL+(ndim) X Rnint}. (41.1)

Observe that FP is completely determined in terms of Fe and the deformation viathe local multiplicative factorization as FP = Fe l[Dqp]. Similarly, the Kirchhoffstress r and the hardening stresses q are determined in terms of (Fe, I,) C I viathe constitutive equations in Table 36.1. For convenience, we adopt the followingnotation for the flow associated with the primary variables in the problem

t E II 1- Xt = (t, v, F e , ,) E C x V,, x = Z x . (41.2)

The evolution of this flow is governed by the dynamic weak form of balance ofmomentum, written in the form (see Section 39):

Gdyn(Xt; 7/) fJ pot(v 0 pt) (n 0 cpt) dQ

+ G, (r; 7) =0 Vr C71 Vt. (41.3)

Recall that G, (7r rj) is the static part of the dynamic weak form defined by the(virtual work) expression (40.2), i.e.,

G~,(;r) = 2 r (V o Q.t) d - -(ro (q , ) dQ2

.* (,I Wtp) dFr. (41.4)

It is again emphasized that is viewed here as a dependent function of primaryvariables Xt Z via the local constitutive equations in Table 36.1.

The nonlinear problem for the flow t - Xt defined by (41.3) and (41.4), alongwith the local equations in Table 36.1, is dissipative in the sense that there exists afunction E: Z -* I which decreases along the flow. This function is the total energyof the system and is defined as follows.

Let Vxt: C - IR denote the potential energy of the external loads, i.e., a functionalsatisfying the condition

-dt et('-t) / B VdQ- t Vd, (41.5)dt if'F,



where V = v o t is the material velocity of the motion. For dead loading the functionV,,t(ot) is obviously given by

Vext(q(t) = -J B t d2 - t . t d. (41.6)

The total energy function E: Z -- i R is the sum of the kinetic energy of the system,the total internal free energy and the potential of the external loads, i.e.,

E(Xt) = J [Pjv o Vtl2 + W(FeTFe) + 7i(c)] d + Vext(Wt). (41.7)

The following result gives the rate of change of the functional E(-).

THEOREM 41.1. The time rate of change of the total energy along the flow is

d E(Xt) = -DQ in , where Do = f2dQ (41.8)

is the total instantaneous dissipation at time t i 1[ and

fint

D = r. dP + E q S ,

c=l

is the instantaneous local dissipation.

PROOF. Assume sufficient smoothness so that time differentiation under the integralsign is allowed. Then observe that

atw(e) = 2CeW(Ce) . C e

= Fe [2aeW(Ce)]FeT . Fe-T [Ce]F e - 1. (41.9)

Since 1e = F'eFe- and de = sym[le], by making use of the kinematic relation

de = Fe-T[2e']Fe- derived in (38.3), the constitutive equation for the Kirchhoffstress in Table 36.1 implies the following alternative but completely equivalentexpression for (41.9)

at~w(Ce) = de = r [d-dP]. (41.10)

Using this result in the computation of the time derivative of E(Xt) and recalling thatd = sym[Vv] o Spt gives

dt E(Xt) = Gdy (Xt; v)- g dP d+?. (41.11)dd; / ]

SECTION 41 359

The result then follows by noting that the first term on the right-hand side of (41.11)vanishes along the flow since the spatial velocity field v E V., is an admissible testfunction in the dynamic weak form (41.3) for time-independent essential boundarycondition.

Accepting the dissipation inequality Do > 0 for all t C I as an unquestionedfundamental principle, the preceding result implies that dE(Xt)/ dt < 0 for all t eC ,i.e., the total energy decreases along the flow as claimed. The dissipation inequalitycan often be checked in examples, as illustrated in the following situation alreadyconsidered in the context of the infinitesimal theory.

EXAMPLE 41.1. Suppose that the convex functions f,(r, q) that define the elasticdomain are of the form

f (, q) = ¢ (r, qx) _ ry,, < O where rye > 0, / = ,..., m (41.12)

and the functions 4,(r, q0 ) are convex and homogeneous of degree one. As alreadypointed out in Chapter I, this is the case of interest in applications since all the yieldcriteria used in practice satisfy these conditions. Thus, for the associative flow rule inTable 36.1 and the assumed form of the yield criteria in (41.12), the local dissipationis given by

nint

a=l

E=~i T-l * a (r, qaC) + Eq aq¢i(r, q) . (41.13)

Using Euler's theorem for homogeneous functions, it follows that

nin

· at¢ ~(T, q ) + E q ah (Tr q ) = ¢7 (r, qO). (41.14)o=1

From this relation and (41.13) one concludes that

m r rn

D = ) q) = E yf ( q ) + Eylry,

mr

ince , and as a result o f the Kuhn-Tucker conditions.Thus, the model obeys the local dissipation inequality.

360 J.C. Simo CHAPTER In

CHAPTER IV

The Discrete Initial Boundary

Value Problem:

Exponential Return

Mapping Algorithms

The numerical solution of the initial boundary value problem for dynamic plasticityat finite strains involves the transformation of an infinite-dimensional dynamical sys-tem, governed by a system of quasilinear partial differential equations into a sequenceof discrete algebraic problems by means of the following two steps:

Step 1. The infinite-dimensional phase space Z = C x V is approximated by afinite-dimensional phase space Zh C Z via a Galerkin finite element projection. Thisprojection induces, in turn, a discretization of the space Z h of internal variables. Theprojected (finite-dimensional) dynamics on Zh XZh is governed by a system of couplednonlinear ordinary differential equations arising from the weak form Gdyn: Zh X h X1[ - R of the momentum equations together with the local form of the constitutiveequations for multiplicative elastoplasticity.

Step 2. The coupled system of nonlinear ordinary differential equations describesthe time evolution in the time interval ]I of interest of the nodal degrees of freedomand the internal variables associated with the finite element Galerkin projection ontoZh x Zh . A time discretization of this problem involves a partition

N

1[ U [t. t+l]n=0

of the time interval . Within a typical time subinterval [t, tn+l], a time marchingscheme for the advancement of the configuration and velocity fields in Zh togetherwith a return mapping algorithm for the advancement of the internal variables in 2 h

results in a nonlinear algebraic problem which is solved iteratively.

361

Formally, the two steps outlined above commute in the sense that the same solutionis obtained by reversing the order of application of the preceding two steps, as indicatedin the diagram below.

ELASTOPLASTIC IBVP > i SEMIDISCRETE IVPGalerkin Projection

Time Integration Time Integration

Galerkin Projection|SEMIDISCRETE BVP ALGEBRAIC PROBLEM

The goal in this chapter is to provide a fairly complete account of the time inte-gration schemes required to successfully complete the second step in the precedingnumerical solution scheme. The emphasis is placed on the formulation of a new classof exponential return mapping algorithms for the integration of the plastic flow evolu-tion equations and, to a lesser extent, on the formulation of time marching schemes forthe advancement of the solution in phase space. Issues pertaining to the constructionof the spatial Galerkin finite element projection in the first step are addressed only tothe extent needed to illustrate the main difficulties involved and the structure of theresulting initial value problem. The approach adopted in the design of return mappingalgorithms is motivated by the following considerations.

Conventional generalizations to the finite strain regime of the return mapping al-gorithms described in Chapter II are based on a direct numerical integration of the

Vincremental elastic constitutive equation r = aid - dP] to define an algorithmictrial state. Representative examples of this approach, widely used in commercial fi-nite element software up to the mid-eighties, are found in KEY [1974], KEY, STONEand KRIEG [1981], NAGTEGAAL and VELDPAUS [1984], GOUDREAU and HALLQUIST[1982], ROLPH and BATHE [1984] and HALLQUIST [1984, 1988].

Standard time-stepping algorithms, however, do not guarantee in general that thekey condition of objectivity is inherited by the resulting, numerically integrated, in-cremental constitutive equation. Satisfaction of the fundamental condition of frameinvariance leads to so-called incrementally objective algorithms, a notion formalizedin HUGHES and WINGET [1980], RUBINSTEIN and ATLURI [1983], PINSKY, ORTIZ andPISTER [1983] and others. The condition of incremental objectivity precludes the gen-eration of spurious stresses in rigid body motions. Although it would appear that sucha requirement furnishes a rather obvious criterion, a number of algorithmic treatmentsof plasticity at finite strains exist which violate this condition (e.g., MCMEEKING andRICE [1975]).

The approach described in this chapter, initiated in SIMO and ORTIZ [1985] andSIMO [1986], completely circumvents the need for incrementally objective algorithmsand leads to objective schemes which are exact in the absence of plastic flow. Thekey idea is to exploit the multiplicative decomposition of the deformation gradientand evaluate the trial state via a mere function evaluation of hyperelastic stress-strainrelations formulated relative to a released (moving) configuration. These ideas arefurther extended in SIMO [1988a,b]. Subsequent work within the context of the mul-

362 J.C. Simo CHAPTER IV

The discrete initial boundary value problem

tiplicative decomposition includes WEBER and ANAND [1990] and ETEROVICH andBATHE [1990], where an exponential approximation to the incremental flow rule isemployed, and PERIC, OWEN and HONNOR [1989]. More recently, MORAN, ORTIZ andSHI [1990] addressed a number of computational aspects of multiplicative plasticityand presented explicit/implicit integration algorithms. Related approaches are consid-ered in KIM and ODEN [1990]. Methods of convex analysis, again within the contextof the multiplicative decomposition, are discussed in EVE, REDDY and ROCKEFEL-LAR [1991]. The preceding survey, although by no means comprehensive, conveysthe popularity gained in recent years by computational elastoplasticity based on themultiplicative decomposition.

42. The Galerkin projection: The spatially discrete problem

The goal of this section is to outline the formal steps involved in a spatial discretiza-tion of the initial boundary value problem for finite plasticity (Step 1), leading toa finite-dimensional, second-order initial value problem. The exposition will be re-stricted to the standard Galerkin finite element method since the objective here is toprovide the general algorithmic framework for the generalization to the finite strainregime of the return mapping algorithms described in Chapter II. Details specific tothe formulation of mixed finite element methods, better suited for the problem at hand,are of secondary interest in motivating the main issues arising in the time integrationof the problem of evolution. An account of a rather convenient class of mixed finiteelement methods designed to circumvent the difficulties experienced by low-orderconventional finite element methods will be given in the following section.

Consider a spatial discretization 2 = U"' Qf2 of the reference configurationQ c Rfndim, generically referred to as the triangularization and denoted by Th inwhat follows, into a disjoint collection of nonoverlapping subsets g,. With a slightabuse in notation we shall refer to a typical subset S2? as a finite element and denote byh > 0 the characteristic size of an element in a given triangularization. Precise defini-tions of these notions, along with the technical conditions to be met by a discretizationto qualify as an admissible triangularization, are found in CARLET [1978].

Associated with the triangularization Th one introduces a finite-dimensional ap-proximation Ch C C to the configuration manifold C, defined as

Ch = {(h C: h E [Co(2)] 'd im and oph• 2e . [Pk(Qe)] ndm}, (42.1)

where Pk(S2e) denotes the space of complete polynomials of degree k 1. Thetechnical condition that the approximation be at least Co is dictated by the varia-tional structure of the problem. For second-order elliptic variational problems this is awell-known requirement (see CIARLET [1978]) which, in the present context, is merelyformal and motivated by the presence of generalized derivatives up to order one in the(static) weak form (40.2) of the equilibrium equations. These technical considerations

SECTION 42 363

will be largely ignored in the exposition that follows. The finite-dimensional subspaceoh c Vo of material test functions associated with Ch is then defined as

vh = 71oh Vo: r0lh [cO(Q)] dim and 70hls? e C [Pk(2 e)]ndim}. (42.2)

Recall that maps ph E Ch must satisfy the Dirichlet boundary condition oh|r = h,while material vector fields on the test function space VOh satisfy the homogeneousform of this essential boundary condition, i.e., hlr = 0.

The current configuration associated with a deformation iph Ch is the setSh = h(Q). As in the continuum problem, the tangent space Vhh is spanned by

spatial test functions rh : Sh t- Rndim defined via composition with the deformationas -h = roh o pOh-l. By the chain rule one has the key relation

GRAD [h] = (ioh *)iT h ie, Or _h a,0[77,] D(V o 'Pi) D'Pt, i.e., V0fa = '7a a9b (42.3)

aXA axb XA'

which furnishes the finite element counterpart of (39.4). The discrete canonical phasespace Zh C Z associated with the finite element discretization introduced above isZh = Ch V>h'. The Galerkin projection of the problem is obtained merely by re-

stricting the weak form of the equations to Zh X Ih. Adopting a description relativeto the current placement, the projected motion t I -* iopth C Ch, with associatedspatial velocity field vh and stress field Th, then satisfies

Gdyn (r", h; h) = / PO A(Vh o WIh) (h o io h) d2

+ Gth (T; rh) = 0 (42.4)

for any (spatial) test function rh E VCh,>. As in Section 39, Gh (h ; r7h) denotes the(static) weak form of the equilibrium equations which in a description relative to thecurrent configuration is given by (40.2), namely

G¢, (rh;r h)= T [rh (,V7 *h o 'Ph) - B o ( ° )] dQ

X ' h ( h 0 MOyh) d (42.5)

The dynamic weak form (42.4) together with local constitutive equations that relatethe stress field i-h to the deformation, such as the model of multiplicative finite strainplasticity summarized in Table 36.1, define an initial value problem on the discretephase space Zh for the projected motion and the projected velocity field.

REMARK 42.1. In addition to the spatial approximations for the admissible deforma-tions and the corresponding test functions, for constitutive models of the type givenin Table 36.1 it is also necessary to specify a suitable interpolation for the internal



variables. For C° finite element methods the approach almost universally adopted isbased on the following considerations, made with reference to the specific model inTable 36.1 with internal variables {F e, A,} C I:

(i) A CO-approximation to the admissible deformations via polynomials of orderk > 1 defines an approximation Diph to the deformation gradient of order (k - 1)within each element, discontinuous across elements in the triangularization Th.

(ii) Consistent with this result, the interpolated internal variables {Feh, h} E Zh

are also assumed to be discontinuous across elements and consisting of complete poly-nomials of order (k - 1) within each element. The construction of this local (k - 1)interpolation for the internal variables then involves defining a local triangularizationTTh within each element DI.

The actual implementation of this approach becomes trivial if one observes thatthe integrals appearing in the static weak form are always evaluated via numericalquadrature. By choosing the vertices of the local element triangularization 5Th co-incident with the quadrature points of the element, it follows that the values of theinternal variables at the quadrature points provide all the information needed for theevaluation of the weak form (42.5).

43. The linearized (semidiscrete) initial value problem

The statement given below of the incremental initial boundary value problem is ideallysuited both for nonlinear elasticity and nonlinear elastoplasticity at finite strains, andprovides the basis for the numerical solution strategies discussed in the following chap-ter. For recent accounts of the relative small number of existing mathematical results,mostly restricted to static nonlinear elasticity, see CIARLET [1988] and VALENT [1988].

The solution of the semidiscrete initial value problem, which arises from theGalerkin finite element discretization outlined above, employs an iterative schemebased on the solution of a sequence of incremental problems. These linear problemsare obtained from a Galerkin discretization of the rate version of the weak formof momentum balance (see expression (40.7) for the continuum problem) and arisein the following context. One is given a configuration qoth E Ch with velocity fieldvph e ]Eo, which is not necessarily in dynamic equilibrium for prescribed loading, i.e.,Gdym (-r, Vt; t}) 4 0. The objective, then, is to compute an incremental displacementfield

Auh(, t) :Sth = th(JQ) Indi such that Au(-, t) E 1V,h (43.1)

by solving the linear problem obtained by linearizing the dynamic weak form aboutthe configuration oh G Ch. Typically, this problem is defined for t E [t,, t+l ]. Theupdated configuration Wot + Au o cpt then provides an improved approximation of thesolution of the dynamic weak form for prescribed incremental loading in [t,, t,+l].Our goal is to describe the structure of this linear (incremental) initial value problem.We do so before addressing the design of the time-stepping algorithms to provide amotivation for the structure of the incremental constitutive equations needed in thesolution of this problem.

SECTION 43 365

CHAPTER IV

Current Configurationat Time to

Nearby Configurationat Time t+

FIG. 43.1. Illustration of the configurations involved in the linearization of the semidiscrete, algorithmicweak form. The placement of the body St = () at time t remains fixed. The nearby placement

St+ = t+C(2) is obtained for qot+ = ot + (Au o Wt with > 0.

Consider the situation illustrated in Fig. 43.1 where the spatial velocity field vtand the Kirchhoff stress tensor rth are given fields on the current placement of thebody Sh = th() for a prescribed configuration poth E Ch. A nearby configuration

qoh+ ( Ch associated with the incremental displacement field in (43.1) is defined bythe expression

hP0+h = %olh + AUh for > 0. (43.2)

Associated with this configuration, we have a stress field rth and a spatial velocityvt+¢. The linearized problem about the prescribed configuration oth is then given bythe affine approximation

dn / ,Vt - G t (43.3)~~d aGdcyn d Gdy(t+ 5 t+, T) = G V, (43.3)

where, from expression (42.4) and in view of (43.2), the second term in this affineapproximation is given by

d h h

d Gyn(r , v ;) = j P082(Au 'P th) (n ) d'P

+d- j Gusr:~;1h T+ )- (43.4)

366 J.C. Simo

I


An explicit expression for the second term in (43.4) is derived as follows. Consid-ering for simplicity dead loading, the only contribution to the second term in (43.4)arises from the weak form of the stress-divergence term in (42.5). Assuming sufficientsmoothness so that the interchange of integration at differentiation is permitted, theproblem then reduces to evaluating at C = 0 the derivative with respect to , of theterm *-h+¢ * (Vor/h o Sth+C). In this expression V(() denotes the gradient relative tox = Ih = ptX) If fth+ = Dt+ Dth-1 denotes the relative deformation gradient

between the placements Sh and Sh+(, from (43.2) together with the chain rule wearrive at the relations

f-t =1 + CV(AUh) and V/h = V,7 ft 4 (43-5)

where V(-) denotes the gradient relative to x = ot(X). To recast the result of thecomputation in a form suggestive of the rate constitutive equation (38.16) we definethe Lie derivative of the Kirchhoff stress relative to the vector field Auh(., t) as

£hhh =h L- fhT where t+ i rt+ t+ (43.6)

From a geometric point of view, h+C is a contravariant tensor field defined on Sth

known as the pull-back under the relative deformation gradient of the stress field rt+, ,which is defined on Sh C. Using the chain rule relation (43.5)2 along with expression(43.6)2 for i+th,, a straightforward manipulation yields the identity:

h . , (,q <.; )= (th+ *hth+) V t)_h r i+ ft'+¢'ri+C). (Vr o S ) (43.7)

By observing that the configuration Sth = oth (/2) remains fixed during the linearizationprocess, the preceding identity along with the product rule for differentiation anddefinition (43.6)1 yields

d| r/th+.* (Vc" °o t+c

[V(AU ° Who)rt + £Suhrth] . (Vh o aoh). (43.8)

With this result in hand, the second term in (43.8) can be written as

=d/ ,[V (AUh o t(V ) d. )= j [v(uh~ o wth)tth + AUh-th] t ) d. (43.9)

Now suppose that a time-stepping algorithm is used to numerically integrate the con-stitutive equations in Table 36.1 within a time interval [t,, t] for prescribed initial data

SECTION 43 367

at time t. The linearization of such an algorithm at the configuration 9ot h Ch, notnecessarily in dynamic equilibrium, will be shown below to result in an algorithmicincremental constitutive equation of the form:

£A,,r-h CP[V(Auh) o o], i.e., [CAurh]j = cPk (Au)k,I. (43.10)

The incremental constitutive equation (43.10) is the algorithmic counterpart of the ex-act rate constitutive equation (38.16) derived in Section 38 for the continuum modelin Table 36.1. The key difference between these two incremental constitutive equa-tions lies in the expression taken by the continuum elastoplastic moduli ctP, arisingin (38.16)1 and defined by (38.16)2, and the algorithmic elastoplastic moduli cP ap-pearing in (43.10), which are obtained by linearization of the integration algorithm asdescribed below. As in the infinitesimal theory, these two expressions coincide only inthe limit as the step-size tends to zero. The algorithmic moduli will be shown belowto inherit the symmetry properties (38.15a) and reduce to the spatial elasticity tensorc for nonlinear elasticity.

Collecting the preceding results, the linear initial value problem defined by theaffine approximation (43.4) to the Galerkin projection (42.4) of the weak form of themomentum balance together with the rate form (43.10) of the constitutive equationscan be stated as follows.

THEOREM 43.1. Given a configuration pth Ch, with associated stress field r-t1 andspatial velocity field vth, the linearized problem for an incremental displacement fieldAuh E Vh is given by

(pott (u o tp), (h oth)) + B h (nh, Auh)

= -Gdyn (h, Vt; h), (43.11)

for all rh E VL where (, *) denotes the L2 (f2)-inner product and Bv h(-,o) is thesame bilinear form arising in the incremental variational equation (40.7), defined as

B r, r/2h h) = Bmt (OI, r2h ) + B (I, , r2z) (43.12)

Here B¢,h(-,) is the geometric term defined for fixed configuration Oh c Ch with

given Kirchhoff stress field rth by the bilinear form:

ge (r 2 h)B~,h ( %1: 4 )

=t t *V42 ° ft )dQ \d7 , 42 E Vh: (43.13)

which is clearly symmetric (but need not necessarily be coercive). B,at(,.) is thematerial term defined in view of (43.10) and (40.7) by the bilinear form:



Bth (n , ')

= J (·v ) . V2 o 2 N) d2 VJ71, r2 G Vh (43.14)

which is symmetric (and often coercive in Vh).

PROOF. The result follows by inserting expressions (43.4), (43.9) and (43.10) into theaffine approximation (43.3).

When the initial value problem associated with the Galerkin projection is writtenin matrix form, the bilinear forms Bge°(-, .) and Bmat(., .) give rise to the geometricand material tangent stiffness matrices, respectively, whereas the first term in (43.11)gives rise to the so-called mass matrix of the Galerkin discretization.

REMARK 43.1. Suppose that configuration Woh is in dynamic equilibrium, so that thedynamic weak form satisfies: Gdyn(t h , vth; h) = 0 for all test functions 1rh G Vh.

If the bilinear form Bh (, ) is coercive in Vh, then the incremental problem iseasily shown to admit the unique solution Auh = 0. If, on the other hand, fOh is notin dynamic equilibrium, the right-hand side of the incremental problem defines thelinear functional

Lh(rh) = Gdyn(th,vth; h) 2 0 Vn h G Vch, (43.15)

since the configuration Wp, the stress field r n and the spatial velocity field vh areprescribed functions. Problem (43.11) then reduces to a standard second-order linearsystem for the unknown Auh Vh, which again admits a unique solution if thebilinear form Bh (, ) is coercive in V,h.

44. Matrix form of the semidiscrete initial value problem

Let nelem denote the total number of elements, nnode the total number of nodes in atriangularization Th and nnode the number of nodes in a generic element e labeled

as {Xe E I·ndim: e = 1, ... , node}. This local numbering system is related to the

global numbering system via the following fairly standard convention:

XA = X with A = ID(e,a), e = 1,... ,nelem, A = 1,.. , node, (44.1)

where the nelem x node array ID(., ) is defined by the geometry of the triangularization.A rather convenient implementation of the Galerkin finite element method is achievedby writing the local polynomial basis as {Na(C)}, where = ((... , (dim) arenormalized coordinates with domain the unit square i[ in Ii dir , and introducing thefollowing local coordinate change known as the isoparametric map:

node

E -* Xh = ,be() = E Na(() X~ E Q2. (44.2)a=l

SECTION 44 369

The local polynomial basis functions N a : D - I are referred to as the local elementshape functions and satisfy the completeness condition Na((b) = ba, where a =((la, I,(, dim a) are the vertices of the bi-unit square. Piecing together the localshape functions by enforcing the CO-continuity requirement yields the global finiteelement basis functions NA : Q2 -- , A = 1, . . ., node, satisfying

NA o e = N a on Q2e for A = ID(e, a). (44.3)

The key feature that renders this basis useful is small compact support of the functionsNA(X) so defined. The isoparametric mapping can be viewed as providing a collec-tion of charts that results in an extremely convenient parametrization of the referenceconfiguration.

EXAMPLE 44.1. To provide an illustration of these standard ideas in the simplest set-ting, assume ndim = 2, neode = 4 and k = 1 for the local (bilinear) polynomialsPk(f2,). The bi-unit square is [I = [-1, 1] x [-1, 1] and the element shape functionsare

Na ((l , 2) = 'I( - 1 a) (2 - 2 a), a = 1,.. nde (444)

defined so that the completeness condition Na(Cb) = ba is satisfied. The globalfinite element basis functions are the well-known "hat functions", see STRANG andFIx [1973], JOHNSON [1987] or any other textbook on the finite element method.

To enforce the nonhomogeneous Dirichlet boundary condition on rh C aQ2 onetypically introduces the affine decomposition

ho"(X, t) = X +- Uh(X, t) fo gr (X, t) tC Q x I, (44.5)

where U" E Voh is the unknown displacement vector field for t C I, while the functiongh :•2 x - Rndm satisfies the boundary condition

gh(X, t) = (X, t) for (X, t) r, x , (44.6)

and is constructed as follows. Denote by = {I E {l,. .,nnode}: XI E Pn}the set of indices associated with the nodes on the boundary r,, with dimension

dim[g] = nboun Construct modified global shape functions N : 2 -*- R associated

with the global node numbers I C ~ such that N (XJ) = i by suitably piecingtogether local element shape functions. Then define g h as the interpolant

gh (X t) = N' X) (X (X, t). (44.7)leg



Setting = I {l,... ,nnode}: I ¢ 5}, with dimension dim[G] = node, theprojected dynamics defined by t E Uth G Vh, along with the material testfunctions roh E Vh , are written as

Uh(X,t) = NA(X)dA(t) and rh(X) = NA(X)qA, (44.8)AC ACtg

where dA (t) CG Rndim defines the current position at time t C Il of a (material) nodeXA E 2 in the triangularization T h not lying in F. The coupled system of ordinarydifferential equations governing the evolution of the nodal positions, obtained byinserting the interpolations into the weak form (42.4) of the momentum equations,can be written in standard (conservation) form as follows. Define the nnode vectorfields t ~ pA (t) of generalized momenta as

pA(t) = E MAB dB(t) where MAR = / poNANB d2.B=1

The nnode x nnode coefficients MA B define the ndof x ndof positive definite matrix M =[MABI,,dJ] known as the mass matrix of the triangularization. Here nnode = nnode-nboun, where ndof = node x ndim is the total number of degrees of freedom involvedin the discretization (accounting for the prescribed Dirichlet boundary conditions) andIndim is the ndim X ndim identity matrix. Next, define the vector fields t F ofprescribed nodal forces as

Fe = J NABdf + NAT d. (44.9)

Finally, define the internal nodal force vector Fint (dB, rth) by the expression

Fint(dB,r h) = r[ [VNA] df2

where VNA = DOh-TGRAD[NA]. (44.10)

With the preceding definitions in hand, it is easily verified that the projected weakform of the momentum equations (42.4) yields the following problem of evolution:

dpA(t) = F -Fi(d(t),th)

d nThode for A E g. (44.11)dtdA(t)= MApB(t)

B=1

Equations (44.11) define a dissipative dynamical system on the finite-dimensionalphase space P = Rnd

of x ITndof for prescribed initial data, when the stress tensor th

SECTION 44 371

is (implicitly) specified in terms of the dynamics via the local constitutive equationssummarized in Table 36.1. In a subsequent section where global time stepping algo-rithms are described, it will be shown that this finite-dimensional dynamical systemalso inherits the global conservation laws (39.13). Therefore, the Galerkin projec-tion preserves the global conservation laws of momentum as well as the dissipativeproperty of the infinite-dimensional dynamical system.

The matrix form of the linearized problem about a configuration th0 C Ch followsimmediately for the results presented in Section 43 and the matrix expressions givenabove. The incremental displacement field Auh(., t) G Vh is written in the globalfinite element basis as [see (44.8)]

Au th ( h (X), t) = E N A (X)AdA (t). (44.12)Acg

Using the linearity of the Galerkin projection, it is easily seen that the matrix formof the linearized variational problem (43.11) coincides with the linearization of theinitial value problem (44.11) about the finite-dimensional configuration defined by{dA(t),pA(t)} E Zh . The matrix form of (43.11) then yields the linear, finite-dimensional, problem of evolution:

dtAP (t) + E K (t)AdB(t)= RA( d A ( t) , A ( t ) , 'rt)

dt AdA (t) hE B(t)Beg

(44.13)

where KAB (t) RE Indim x I ndim (A, B E 5) is the nodal tangent stiffness matrix, thematrix counterpart of the bilinear form Bh(.,-), and RA e ndim is the residualvector associated with node A E 5 and defined as

RA (dA (t), pA (t), ) = Fext - Fi(dB(t), t) _ dpA(t)/ dt (44.14)

Obviously, RA (dA(t), pA (t), rth) = 0 if the nonlinear system (44.11) is in equilib-rium. The decomposition (43.12) of the bilinear form Bh (, .) into a geometric partand a material part gives, in turn, the decomposition

KAB(t) = KAeo (t) + Kmat (t) (44.15a)

for the tangent stiffness matrix where, in view of (43.13), the component expressionof the geometric part is given by

[K ABEoi= J NN A -JTN d28ij, (44.15b)[geo (] ij kij

k=l 1=1



and component expression of the material part is defined in view of (43.14) as

[Kat (t)] ij I E NAep N d2. (44.15c)I j,1

im i li 1 I

Here Nk = aNA/Oxk denotes the partial derivatives of the shape functions relativeto the current coordinates xk = pk (XA, t).

Before closing this section it is useful to summarize the key implication of thepreceding developments in the design of return mapping algorithms. The numericalsolution of the initial value problem (44.11) for loading prescribed in terms of Fext(t)and specified initial data exploits the following strategy. The stress tensor rh togetherwith the collection {Feh, S} G h of internal variables are viewed as dependentvariables which are solved for in terms of the dynamics t E [t., t.+l] -* dA(t) viathe class of return mapping algorithms described below. Since the integral that definesthe internal nodal force vector FinAt in (44.10) is evaluated by numerical quadrature,knowledge of the stress field r is required only at the discrete points used in thequadrature formula. Moreover, because of the interpolation scheme adopted for theinternal variables described in Remark 42.1, effectively, knowledge of the internalvariables is also required only at the quadrature points.

The specific form taken by the deformation-driven return mapping algorithm isthe subject of the following sections. The last section in this chapter addresses thestructure of the time-stepping algorithm for the advancement within the time interval[t., t+l] of primary variables (dA, PA) in the phase space P.

45. Mixed finite element discretization: An illustration

The key difficulty involved in the construction of the Galerkin projection (Step 1) forclassical plasticity is related to the enforcement of the volume-preserving conditionon the plastic flow. This constraint arises whenever the yield condition is pressureinsensitive, a situation nearly always encountered in metal plasticity and exemplifiedby classical yield criteria such as the Maxwell-Huber-von Mises condition. This prob-lem is currently well understood and, since first pointed out in the pioneering work ofNAGTEGAAL, PARK and RICE [1974], has been extensively addressed in the literature.In particular, use of low-order Galerkin finite element methods is well known to resultin an over-constrained pressure field that may render overly stiff (locking) numeri-cal solutions. In the linear regime, techniques designed to circumvent this difficultyinclude classical LBB-satisfying displacement/pressure mixed finite element meth-ods, see the recent review in BREZZI and FORTIN [1991], higher-order Galerkin finiteelement methods, such as the high-order triangle of SCOTT and VOGELIUS [1985],and special three-field mixed finite element methods as in SIMO and RFAI [1990],among others. Extensions of these methodologies to the finite strain regime have beenconsidered by a number of authors. In particular, generalizations of the approach inNAGTEGAAL, PARK and RICE are presented in HUGHES [1980] and extensions of thedisplacement/pressure mixed methods are considered in GLOWINSKI and LE TALLEC

SECTION 45 373

[1989], SIMO, TAYLOR and PISTER [1985] and SUSSMAN and BATHE [1987] amongothers. Enhanced mixed finite element methods which include nonlinear versions ofthe method of incompatible modes (see TAYLOR, BERESFORD and WILSON [1976] andCIARLET [1978]) are presented in SIMO and ARMERO [1992] and have been recentlyextended in SIMO, ARMERO and TAYLOR [1993].

The goal of this section is to describe a generalization to the finite deformationregime of the assumed strain finite element methods described in Section 26. Theseclass of mixed finite element methods is specifically designed to circumvent the diffi-culties experienced by the conventional Galerkin finite element method in the nearlyincompressible regime. From a computational standpoint, this methodology offers akey advantage over conventional displacement/pressure mixed methods. Specifically,for boundary value problems arising in classical nonlinear elasticity and plasticity, theconvenient strain-driven format of the Galerkin (displacement) finite element methodis preserved. In particular, for classical plasticity, the structure of the local strain-drivenreturn mapping algorithm described in detail in the preceding sections is preserved byassumed strain methods.

In order to present the technique in the simplest possible context, attention will berestricted to quasistatic nonlinear elasticity following the approach described in StMo,TAYLOR and PISTER [1985]. The generalization of the methodology to the elastoplasticproblem is entirely analogous to the situation addressed in Section 26 within thecontext of the infinitesimal theory. As shown in the aforementioned reference, the onlymodification needed to tackle the elastoplastic problem merely involves performingthe local return mapping algorithm with the strain field replaced by the assumed strainfield.

(A) Three-field variational formulation of the problem. The severe locking arisingfrom the point-wise enforcement of a nearly isochoric deformation via a low-orderGalerkin finite element method is circumvented by introducing an independent volumefield, denoted by J, together with its dual variable r that plays the role of a Lagrangemultiplier and is interpreted as the Kirchhoff pressure. The constraint expressing theequality between J and the point-wise value field J(qo) = det[Dqo] is then enforcedweakly via a Lagrange multiplier term. The variational setting for the problem iscompleted by defining the assumed deformation gradient as

F(o, J) J/d... F(o) where F'(q) = [J(o)] D~p (45.1)

is the volume preserving part of the deformation gradient which satisfies the condi-tion det[F(W)] 1. Observe that the assumed volume field J is subjected to theconstraint J > 0. This constraint can be automatically enforced by replacing J withthe unconstrained variable 0 = log(J), so that

J= exp[)] and C(q,v )= [F(=p,0)] F(45,.),

374 J.C. Simo CHAPTER iV

(45.2)


and considering the following three-field Lagrangian functional as a point of departurefor the construction of the assumed strain mixed finite element method:

L(o, , ) ((C, W( ,)) + r [log (J(g)) - ] } d2 - Vext(). (45.3)

Here Vext (g) denotes the potential energy of the external forces which, for simplicity,is assumed to correspond to dead loading. Our first objective is to compute the formalEuler-Lagrange associated with the saddle-point problem defined by (45.3). To do so,use is made of the relations in the result below.

THEOREM 45.1. The directional derivatives of F(o, i9) and C(, '0) are given by thefollowing expressions:

(i) D F(o, 9)- r = (dev[V[r]] o o)F(o, 9), (45.4a)

(ii) DoiF(o, 0)- 0 = (1/ndim), /F(p, '), (45.4b)

(iii) DoC((, 0) . = 2[F(, 0)]T(dev[E[r]] o o)F(, 0), (45.5c)

(iv) DtC(go, 0) * . = (l/ndim)2[F(Wo, 0)]T [/)1]F(o, 29), (45.5d)

for any test spatial function r/ V, and any 0 E L2 (Q), where V[r] is the spatialgradient defined as V[rl] o = GRAD[,7 o o](Do)- ' and e[r/] = [V[r] + (V[q])T].

PROOF. (i) The directional derivative is defined by the formula

DF(, ) r F( C ro V, 0), (45.6)

where, in view of the definition of F(Wo, 0),

F( + ( o O , )

=exp [I 0]J( + 7o o)(Do+ GRAD[r7o ]). (45.7)

From a well-known formula for the derivative of the determinant it follows that

d J(go + r /o g) = det [Do](D) - T · GRAD[rl 0 ]

= J(go) div[r/] o go. (45.8)

Using this result together with the product rule for differentiation and the chain rulerelation GRAD[77o go] = (V[r] o p) D, expression (45.6) becomes

DqF(O, =0) = [J()/J] -/ndim (V[] - l/ndim div[/]) o 4[D5],

SECTION 45 375

(45.9)

which in view of the definition of F(qo, '0) yields result (i). Expressions (ii), (iii) and(iv) are proved in a similar fashion. [

Making use of the preceding results, the Euler-Lagrange equations associated with(45.3) take the following form strongly reminiscent of the results found in Section 26for the linear theory. Upon defining the Kirchhoff stress and pressure fields

r(o, 9, r) = rl + dev[F(o, 0)2acW(C(, 9))F (,p 0)], (45.10a)

p(Wo, V) =-tr[ F(o, 0)2acW(C(po, 0)) F (o, 0)], (45.10b)ndim

a straightforward manipulation yields the following three variational equations thatprovide the stationarity conditions for (45.3):

J T(Wp,, , 7r) . (V[ri o o) dQ - Gext() = (45.1 la)

( log (J( )) - 0) dQ = 0, (45.1 b)

|f (p( ) -o , t) - 7(45. 11c)

which are to hold for any spatial test function GC V, and any q, b E L2 (Q2). HereGext(,q) denotes the virtual work of the external (dead) loads given by the standardexpression

Gext() = B. (o o )dQ + T ( ) d. (45.12)

The weak forms (45.11a,b,c) furnish the point of departure for the construction of amixed finite element approximation to the problem at hand.

(B) Discontinuous mixed finite element discretization. Consider a finite element tri-angulation Th of the reference placement 2 l2h = Ue=1 Q•e, a conforming finiteelement approximation Ch C C to the configuration space and the correspondingfinite-dimensional approximating subspace Vh' C Vv of test functions, as describedin Section 42. To recover the structure of the assumed strain method presented inSection 26 within the scope of the infinitesimal theory, we introduce discontinuousapproximations both for pressure and the volume field via the finite element subspace

ph = A{sh L2(?): h In2 = FT(X) ,b for ibe C I }. (45.13)

As in Section 26, r(X) = [ (X) ... r,(X)]T is a vector of prescribed elementlocal element shape functions the specific form of which is left open for the moment.The lack of interelement continuity requirements on functions in the finite element



space ph is the key design condition placed on this approximation. It is precisely thiscondition that enables us to obtain an explicit expression for the pressure at the volumefield at the element level in terms of the deformations eph E Ch, thus recovering thestructure of an assumed strain method. We do so by substituting the approximationsin (45.13) into the variational equations (45.1 lb,c) and solving at the element level toobtain:

7e (qe) = FT(X)H

1 J rF(X) log(J ( )) df, (45.14a)

e ( re) = (X)p(pe , e (p)) dQ, (45.14b)

where ()h = (.)hlne denotes the restriction of the finite element approximation ()hto a typical element / 2e and H is the m x m element matrix defined as

He= F(X) ) F(X) dQ. (45.15)

The collection of element functions {0 9(po)}=L and {r7r(qOh)}c=l defines a discon-tinuous global approximation to the (logarithmic) volume field and pressure field interms of the deformation Woh E Ch, which we shall denote in what follows by 19h(Wph)and 7rh(Wph), respectively. By substituting these approximations into the remainingvariational equation (45.1 la) we obtain the following variational problem:

Problem Gh: Find the deformation qoh C Ch such that

J (W (Wh), th ( Ph), Th ( Wh)) (V [h] o qoh ) d2

- Gext(77h) = 0, (45.16)

for all finite element (spatial) test functions r/h C Vhh, with the "equivalent" Kirchhoffstress field defined by expressions (45.10a,b).

The preceding variational problem merely amounts to a generalized displacementGalerkin finite element method with "modified" constitutive equations defined in termsof the deformation field via expressions (45.10a,b).

REMARK 45.1. Given a finite element conforming subspace, the possible choices forthe functions r(X) that define the finite element pressure/volume subspace ph areseverely restricted by the well-known inf-sup condition. While a rather completetheory exists for linear problems, see, e.g., the account in BREZZI and FORTIN [1991],very limited results currently exist for quasilinear problems exemplified by quasistaticnonlinear elasticity. The conventional approach in the design of the pair of finiteelement spaces Vph /ph is to adopt pairs that satisfy the inf-sup condition in the lineartheory. A well-known example is furnished by the choice Q2/P1 which renders an

SECTION 45 377

optimal element in the linear regime. We refer to the aforementioned reference andthe brief discussion given in Section 26 for additional details.

From an algorithmic standpoint, the solution of the "modified" variational prob-lem Gh is accomplished via an iterative technique, say Newton's method, in whichthe deformation p h plays the role of the primary "driving" variable. For a givennumerical quadrature formula is given, the specific structure of which being ir-relevant for present purposes, the evaluation of the assumed stress field h =S(qth(cht), h(,), 7rh(ph)) in (45.16) over a typical element Q, involves the fol-lowing steps to be performed at each quadrature point:

Step (i). Let Nf: Q2 - 1R denote the local element shape functions associatedwith the conforming approximation to the deformation field. Compute the elementdeformation field via the standard interpolation formula

node

(X) = ZNa(X)r' where r = X' + d' (45.17)a=1

and de , a = 1,. ., nen, denote the current displacement vectors of the nodenodal points of element 2?,, which are assumed to be prescribed in a conventional"displacement-driven" iterative solution format.

Step (ii). Evaluate the conforming approximation to the displacement gradient andthe volume field via the standard expressions

Tnode

Dp = r e0GRAD[_Na ] and J(oh) =det [DWh]. (45.18)ala=1

The preceding two steps are exactly the same as in the conventional Galerkin finiteelement method.

Step (iii). Evaluate the element volume field Jh = exp[nh(qehn)] by making use ofexpression (45.14a), namely

e = exp [FTrx) f F(X) log(J(e)) dQ]. (45.19)

Observe that this interpolation ensures that condition Jn > 0 holds for any X Ce. Then evaluate the assume displacement gradient Fe and the corresponding rightCauchy-Green tensor Ch by setting

-h (j( h)/h)-l/hnd h d = -hT

h (45.20)ThF = a /e; D and =_ _e (45.20)

This step represents an additional computation not present in the conventional Galerkinfinite element method.



Step (iv). Perform a constitutive evaluation (for each quadrature point of the ele-ment) and compute stress field Fph [2acW(Ch)]FbT. This step is exactly the same asin the Galerkin finite element method, the only difference being that the conformingright deformation gradient Fh = Dh is replaced by the "assumed" deformationgradient Fe defined in Step (iii). The same observation holds for models other thanelasticity. In particular, for plasticity all that is needed is to "drive" the local returnmapping algorithms with the assumed deformation gradient Fh.

Step (v). Evaluate the assumed pressure field 7rh over the element via expressions(45.14b) and (45.10b), i.e.,

= r T (X)H J F(X) 1 tr[F e 2OcW( h)FhT] df?. (45.21)efo ndim

Then compute the assumed stress field rh over the element from expression (45.10a)as

Teh = rh 1 + dev[Fh2acW( Ce)FhT ]. (45.22)

The pressure evaluation via (45.21) is an additional computation over the Galerkinfinite element method while the stress computation in (45.22) is formally identical.

Once the element stress field reh is computed at each quadrature point of an ele-ment S29 according to the preceding steps, the evaluation of the weak form (45.16)proceeds exactly in the same fashion as in the conventional (displacement) Galerkinfinite element method, see Section 26. In particular, the matrix formalism described inSection 44 carries over to the present context without essential modifications. Hence,further details will be omitted. Observe that the generalization of (45.16) to the dy-namic problem is straightforward and merely involves appending the weak form ofthe inertial term poo h.

From a computational standpoint, an important property of the Galerkin finite ele-ment method is the sparse structure of the Wl~-matrix that arises in the conventionalmatrix expression

,node

E[r7eh = sym{V[17Jh]] = ):'eaa, where a' e Rndinm. (45.23)a aa

a=l

for the symmetric gradient of the spatial test functions nh C Vhh. A noteworthyfeature of the mixed finite element method outline above, which becomes apparent byinspection of the reduced weak form (45.16), is the preservation of this sparse structure.This property holds independently of any possible coupling between volumetric anddeviatoric response induced by the specific form of the constitutive model.

REMARK 45.2. The preceding mixed finite element method can be cast into an al-ternative format suggested in HUGHES [1980] as an extension for linear problems ofthe methodology introduced in NAGTEGAAL, PARKS and RICE [1974], without making

SECTION 45 379

any reference to the underlying variational structure. As noted in SIMO, TAYLOR andPISTER [1985], such a reformulation relies on the following identity for the term in(45.16) involving the element pressure field 7rh:

7h (div [ ] ) d= PJ(div [te] ° , P) dh , (45.24)¢ e e

where CP) = eSic 9()) is defined by (45.10b), with ( e) given by (45.14a),and div[r/h] is defined by the expression

dv[] ° W = FT(X)H- J rF(X)(div[r] ) dQ. (45.25)

Identity (45.24) is an immediate consequence of expression (45.14b) for the elementpressure field 7rh and is verified by following the same steps used in the proof of thetheorem in Section 26. Expression (45.25) arises naturally within the present varia-tional framework as the linearization of the element volume field Je = exp[0h( o)]defined by formula (45.19) in Step (iii), since

DJho r 7eh = Je (djiv[reh] ph). (45.26)

This relation is easily verified and furnishes the counterpart for the assumed volumeelement field of the standard result DJ(Wh) * r7h = (Wh) (div[r7h] 0o Wh). Substitutionof (45.24) into (45.16) then gives the following equivalent weak formulation:

J [Fh23cW(lh)F hT] ( 7 h /h] 0a h) d2 - Gext( 7 h) = 0, (45.27)

where Fh is the assumed deformation gradient defined by formula (45.20) in Step(iii) and Vh [.] is a modified spatial gradient defined on each element 2e by

1Vh[re] = dev[V[r1*]] + di iv[r/]1. (45.28)

The implementation of the weak form (45.27) involves the same steps described above,with the exception that the element pressure field 7rh defined by formula (45.21) inStep (v) need not be explicitly evaluated. The drawback of this implementation overthe one outlined above lies in the structure of the matrix expression for the modifiedgradient Vh[-], as defined by (45.28), which no longer retains the sparse structurepresent in expression (45.23) for V[-].



(C) Linearization: The incremental problem. Under the assumption of dead loads,the linearization of the weak form (45.16) yields the following bilinear form, thestructure of which was derived in Section 43:

B (rh, Auh)

= J [h] . (v[Auh]fhl) e+hE [ l] . (aulh h) d2, (45.29)

Geometric term Material term

where explicit indication of the composition with the deformation is omitted to sim-plify the notation. We shall follow this convention throughout this section. Observethat the stress field appearing in the geometric term is the assumed stress field de-fined by (45.10a,b). What remains to be done is to compute the material term £aUht h

by linearizing expressions (45.10a,b). We shall do so below for the particular casein which the volumetric and deviatoric response are uncoupled via a stored energyfunction of the form

W(C) = U(J) + W(C) with C = J-'l/ndC. (45.30)

This situation is of special interest in metal plasticity (see Section 37) and also arises inthe numerical treatment of incompressible elastic materials via a penalty regularizationthat employs elastic models of the type describe in Section 32, see, e.g., SIMO andTAYLOR [1991]. In contrast with the general case, in this setting the deviatoric partdev[(eh] of the element stress field is independent of the volume element Jh whilethe pressure field Trh depends only on Jh. This simplification results in the followingremarkably simple result.

THEOREM 45.2. For the uncoupled stored energy function (45.30), the material termarising in the linearization (45.29) of the reduced residual (45.16) at a configurationW h e Ch is given by the bilinear form

Bhat (h, uh) [ h]. [* 27rhI + 6] [Au ] dQ

+ f [JhU'(Jh)]'d v[h]div[Au ] dQ, (45.31)

where Ch are the elastic moduli associated with the volume-preserving part W(Ch)of the stored energy function, as defined by the closed-form expression (32.20), anddiv[-] is the modified divergence operator defined by (45.26).

PROOF. In view of the result obtained in Section 43, expression (45.10a) for the elementstress field th implies

LtuhTh = jAUh [ 1] + CE[AUh].

SECTION 45 381

(45.32)

A literal application of the definition for the Lie derivative together with the rule forproduct differentiation then yields

£Auh = -27rE [Auh] + CE [AUh] + 1 Drh .Au. (4533)

The first two terms in this result give rise to the first integral in expression (45.31).All that remains to show is that the last term in (45.33) involving the linearizationof the element pressure field rh produces the second integral in (45.31). In view ofresult (45.26), for the uncoupled stored energy function (45.30) the linearization ofexpression (45.10b) reduces to

D7rh . Au = FT(X)H j (X) [JhU (Jh)] d- v[Auh ] dQ. (45.34)

The final step in the proof makes use of the following identity similar to that leadingto (45.24):

| (D7n * Au'I) V [] dS

= [JhU'(Jh)]' div['h] div[Auh] dQ. (45.35)

The result then follows from (45.35) and (45.33). El

To complete the implementation of the method outlined in Step (i) to Step (v)it remains to specify the iterative solution strategy for the solution of the nonlinearproblem (45.16). Considering Newton's method as a representative algorithm, theresults (45.29) and (45.31) show that the evaluation of the tangent stiffness matrixassociated with the bilinear form B(., ) requires the additional computation of themodified divergence operator div[.] via expression (45.25). We remark that expression(45.31) retains the sparse structure of the Galerkin finite element method. From acomputational standpoint, the additional term involving the modified divergence div[.]translates into a rank-one update.

The key conclusion to be drawn from the foregoing developments is that the formu-lation of integration algorithms for plasticity, the objective of the following sectionsin this chapter, can be addressed within the context of the conventional Galerkin finiteelement method. The extension of methodology to be developed below to the mixedfinite element methods described in this section merely involves the replacement of thestandard deformation gradient Fh = Dqph with the assumed deformation gradient Fh.

REMARK 45.3. The preceding class of methods is design to circumvent the difficultiesexperienced by low-order Galerkin finite element methods but is not suitable forplasticity problems involving highly localized plastic deformations. This situation is ofinterest in a number of applications and always arises in materials exhibiting softeningresponse, such as metals subject to very high strain rates where thermoplastic softening

382 J.C Simo CHAPTER IV


results in adiabatic shear banding. A class of recently proposed enhanced-strain mixedmethods appear to be ideally suited for this type of problems while retaining the goodperformance of mixed methods in the nearly incompressible problem. We refer to SIMOand RIFAI [1990] and SIMO, ARMERO and TAYLOR [1993] for further background onthis subject and additional information.

46. Exponential return mapping algorithms for multiplicative plasticity

Consistent with the solution strategy outlined above, the goal of this section is toprovide a local constitutive equation for the stress field within a typical time in-terval [tn, t+l] in terms of the motion pt : - ]Rd'm, which is regarded as aprescribed field in [t, tq+l]. This algorithmic constitutive equation is derived by in-tegration of the local constitutive equations in Table 36.1 and has the remarkableproperties of being exact for incrementally elastic processes, independent of the spe-cific form adopted for the stored energy function and exact plastic volume preservingfor pressure insensitive plasticity models. Furthermore, it will be shown that the im-plementation of the resulting scheme, referred to as a local exponential return map-ping algorithm in what follows, takes a form essentially identical to the standardreturn maps of the infinitesimal theory for the important case of isotropic elasticresponse.

In an attempt to simplify the notation involved in the exposition of the algorithm,the following conventions are employed:

(i) The superscript h that refers to the finite element discretization is droppedthroughout. All the fields are assumed to emanate from a Galerkin projection andinterpolated as discussed in the preceding section. Accordingly, qo stands for oh E Ch,

F stands for Fh = Dph and so on.(ii) The internal variables in h, written as Fe , ~} according to the preceding

convention, are assumed to be evaluated at a specific point X EG identified with anarbitrary quadrature point in the triangularization Th. Thus, {F e , GL} E GL(ndim) RiRn t since the target space for fields in I is GL(ndim) x R]nRn.

(iii) A generic variable () is always assumed to be evaluated at (X,r) EQ2 x [ta, t,+l]. The subscript r attached to that variable, as in (-),, is always un-derstood to denote the algorithmic approximation at an arbitrary time t E [t., t,+l].The expressions (.),, ()n+o and (),+1 denote the algorithmic approximations to thevariable () evaluated at X E 2 at times t, t,+, = 9tn + (1 - t)t,+l and t+l,respectively, for specified E (0, 1].

Unless explicitly indicated otherwise, these notational conventions are implicitlyassumed throughout this and the subsequent sections.

The local problem of evolution to be integrated numerically is formulated as fol-lows. Suppose that the primary variables X, = (n~, Vn, F, nn) E Zh x 1 h thatapproximate the exact solution are given at time t so that, for each X E Q2, one hasthe initial data:

F n = Dpo(X) E GL(ndim) and (F,5 ,,) E GL(ndim) X (4int.

SECTION 46 383

(46.1)

The problem to be addressed below is the construction of an algorithmic approximationto the plastic flow evolution equations in Table 36.1 for the prescribed deformationgradient

F = Dp,(X) at X CG for r E [t1 , tn+l]. (46.2)

Leaving [t,, t,+l] as an arbitrary point within the given time interval allows aneasy generalization of the two-stage projected Runge-Kutta methods introduced inChapter II by specialization of the results described below. The differential algebraicsystem equations to be integrated numerically is defined by the associative flow rulein Table 36.1 within the interval [t. tn+l], namely,

= and alP:= Doy ? Tf.('r,q' ) and ( Y = ~ Oq-f,(r,q'),ii=1 11=1

(46.3a)

with 'y" > 0, fm(Tr, q) 0 and ylyf(r,q )= 0.z=1

As it stands, however, this problem is not in standard form. The time derivative on theleft-hand side of (46.3a)i is hidden in the definition of the spatial plastic rate IP, whichis related to the time derivative PP of the plastic part of the deformation gradient bythe crucial relation (34.6) derived in Chapter III, namely,

P = Fe[LP]F e- 1 with FP = LPF P . (46.3b)

By appending to the system defined by (46.3a,b) the elastic constitutive equations forthe Kirchhoff stress r and the stress-like variables q° (see Table 36.1), one arrives at aproblem in which the strain-like variables {Fe, ( } become the independent variablesin the problem, while the deformation gradient (46.2) plays the role of a prescribedforcing function. The feature to keep in mind in the design of any suitable algorithmicapproximation to (46.3a,b) is the restriction emanating from the requirement that Feand FP are to remain in GL(ndim) for any X E 2. Such a restriction translates intothe following constraints:

Je = det[Fe] >0 and JP det[FP] >0 in 2 x[t,, t+l]. (46.4)

These two constraints are of fundamental physical significance and must, therefore,be preserved by the algorithmic flow rule. In particular, plastic flow is often incom-pressible for metal plasticity, a feature that translates into the constraint JP = 1 inQ2 x . The algorithmic procedure derived by means of the two steps described belowautomatically guarantees the satisfaction of these conditions.

Step 1. Consider the following exponential approximation of the evolution of theplastic deformation gradient defined by (46.3b)2 :

FP = exp[(- t)L P] Fp for T E [t, t(+l] -


(46.5)

The discrete initial boundary value problems

By well-known properties of the exponential map this approximation implies

JP = det[exp[( - tn)LP]] det[FP] = exp[(r- t)tr[L P ] J > 0, (46.6)

for any T E [t,, t+l], provided that the initial data satisfies J = det[F P] > 0.

Step 2. Use the multiplicative factorization F = FeFP of the deformation gradientalong with standard properties of the exponential map to conclude from (46.6) that

F, = Ffe exp [( - t-)L P] F 1e-FFP

= exp[(T - t,)F L PFe-'] FrePn (46.7)

Now use definition (46.3b)1 for the plastic rate of deformation tensor IP in the spatialdescription to rewrite (46.7) as

F = exp[-(r - t)lP]Fef' where Fe = FF-. (46.8)

The justification for the notation Fe' in (46.8)2 should be clear. According to thisdefinition, Fetr is the elastic deformation gradient computed via the multiplicativefactorization under the assumption that FP = FnP; i.e., by freezing plastic flow.

Step 3. Finally, by inserting the evolution equation (46.3a)1 into (46.8), subsumingthe time step factor ( - t") into the definition of the algorithmic plastic multipliersA-y > 0 and using a conventional backward Euler approximation for (46.3a)2 onearrives at the following result:

Fr = exp_ ttyM 8 r f ) Fre tr,ll=

m

a r = ( n+ A- y aq f (r, q), (46.9a)/=1

m

Ayl >0, flu (r, qr ) < 0 and Ay f, (r,q)=O.'U=J

In this algorithmic version of the flow rule, the generalized stresses (-r, qT) are definedvia the elastic stress-strain relations evaluated at (F, ( T), namely,

r- = Fe [2ae(FeTFe)]FeT and q7 = -aH(T). (46.9b)

A geometric illustration of the algorithm defined by (46.9a,b) is given in Fig. 46.1.The derivation of the algorithmic flow rule (46.9) is a generalization of a scheme

proposed in SIMO [1992] for the case of isotropic elasticity, improving on earlierschemes introduced in SIMO [1988b] and SIMO and MIEHE [1992] which is based

SECTION 46 385

Fixed Reference Configuration

Fixed Intermediate Configuration Updated Current Configuration

FIG. 46.1. Interpretation of the update procedure defined by the trial elastic state. The (local) referenceand intermediate configurations remain fixed, while the current configuration is updated by the prescribed

relative deformation gradient.

on conventional backward Euler approximations. A noteworthy feature of the presentapproach is provided by the following result.

THEOREM 46.1. For models of plasticity possessing a pressure insensitive yield cri-terion, defined by the condition tr[0,-fp(r, q`)] = 0, the algorithm (46.9) preservesplastic volume in the sense that JP = 1.

PROOF. Set Jr = det[FT]. Taking the determinant on both sides of (46.9a)1 and usinga well-known property of the exponential map gives

det F] = exp [ Ay4 tr(a,-fT (r,, q-)) det[Fe tr] (46.10)

The condition that the yield criterion is pressure insensitive then implies det[Fe] =det[Fetr]. Moreover, from the definition of Fetr in (46.8)2 it follows that det[F e tr ] =

det[Fr] det[F P - l] = J/J p. Consequently,

JP = det[Fee-lFr] = J/det[F e tr] = JT(JP/JT) = JP. (46.11)

By induction it follows that JP = 1 since J = JP = Je = 1 at the reference state.

REMARK 46.1. In the present context, the generalization of the classical backwardEuler return mapping algorithms of the infinitesimal theory is obtained by settingT = t+l so that t,+l - t = At is the full step size. This scheme is first-orderaccurate. The generalization within the present approach of the second-order accu-rate return mapping algorithms based on the use of either backward differentiationformulae or projected Runge-Kutta schemes is described below.


I

""\ fl

be n+1 5


47. Exponential return mappings for isotropic elastic response

The actual implementation of the general algorithm described above involves the so-lution of the highly nonlinear system of equations (46.9a,b) by an iterative solutionscheme, typically Newton's method. Remarkably, for the important case of isotropicelastic response a closed-form solution is possible, as described below. This situationcovers most of the plasticity models currently used in large-scale metal forming appli-cations including the classical J2 -flow theory at finite strains. The results that renderthe implementation of the scheme closed-form are summarized below.

THEOREM 47.1. For isotropic elastic response the following properties hold:(I1) The stored energy function takes the functionalform

w(ce) = W(be ) (,...,e ) (47.1)

where be = FeFeT, eeA = log(AA) are the elastic logarithmic stretches, and (Ae) 2

are the eigenvalues of be (A = 1,..., ndim).(12) The Kirchhoff stress tensor r and the elastic left Cauchy-Green tensor be are

coaxial, i.e., their principal directions coincide. In addition, we have

ndim

r=2be[abeW(be)] orequivalently r =E rAn(A) ®n(A) (47.2)A=l

where TA = aw/laeA and be = Edml( eA) 2 n(A) ® n (A) (spectral decomposition).(13) Denoting by e = log(be) the elastic logarithmic strain tensor, one has the

constitutive equations

r = a,e(E e ) where w (Ee) = w(Er,..., Edi) (47.3)

and Ee is defined via the spectral decomposition Ee = EAdiml Een (A) n(A).

PROOF. (11) Elastic isotropy is equivalent to the assertion: W( e) = W(QieQT)for any Q G S0(3). Setting Q = Re , where F e = ReUe is the polar decompositionof Fe , and noting that be = ReCeReT gives the first part of (I1). The second partof (I1) follows from the representation theorem for isotropic functions [see (31.23)],since the principal invariants are a function of the principal stretches and hence oftheir logarithms.

(12) That r and be are coaxial follows immediately from the representation theoremfor isotropic tensor functions [simply replace b by be and II, I2, 13 } by {If, I2, I3e}in expression (31.27)]. For a direct proof of (47.2)1 observe that

be = a, (FeFeT) = lebe + beleT where l e = FeFe-1 (47.4)

Time differentiation of the stored energy function and use of this relation gives

W(be) = [2abeW(be)be] le = [2dabW(be) be ]

* de,

SECTION 47 387

(47.5)

since abeW(be) commutes with be by isotropy. The result then follows by noting that

W(be) = r . de [see (41.10)]. Expression (47.2)2 is a rewrite of result (33.7) since,by the chain rule, A'aiv/8aA = aw/a log(A).

(13) Result (47.3) is merely a restatement of expression (47.2)2.

Since the elastic response is completely determined from the elastic left Cauchy-Green tensor, consistent with definition (46.8)2 for the trial elastic deformation gradi-ent, we define the trial elastic left Cauchy-Green tensor as

ndim

bet= Fetr (Fetr)T = (Ae 2nr() n( ). (47.6)A=]

With this definition, the algorithmic flow rule (46.9) can be written as

bet = exp [Z>AY"a.f/ b exp [-Eaiaf 1 1 j (47.7)

By result (12) in the preceding theorem both be and e = 1 log(b) commute with rand, therefore, they also commute with a,f, since fS(, qu) is an isotropic function.In particular, we have the spectral decomposition

ndim

Tr = E T An (A ) ® n(A) (47.8)A=l

By observing that the two sides of (47.8) must have the same spectral decomposition,the preceding considerations yield the following remarkable results:

(i) The principal directions nrA) of the (unknown) stress Tr and the (unknown)left Cauchy-Green tensor coincide with the principal directions nr(A) of the trial left

elr t,(A ) (A)Cauchy-Green tensor b., i.e., nr = n,t) forA 1 dim

(ii) By taking the logarithm on both sides of (47.7), using results (47.3) in item(13) and noting that all the tensors on the right-hand side of (47.7) commute, thealgorithmic flow rule (46.9) reduces to

e = etr -_ an ftb(T~,Xq )v

/~=1

m

5A 'y ' S= ) + fAT,)q-f (r, a), (47.9a)



(iii) The stresses r and qa are defined from (47.3) via the hyperelastic constitutiveequations

r = eew(e. ) and q' = -O~ 7-(). (47.9b)

The algorithmic equations (47.9a,b) define a return mapping algorithm in logarithmicstrain space with functional form identical to that arising in the infinitesimal theory.Observe that result (i) implies that the return mapping takes place at fixed principalaxes defined by the trial state b. tr

The advantage of the preceding scheme is clear: It allows the extension to the finitestrain regime of the classical return mapping algorithms of the infinitesimal theorywithout modification but with the following additional simplification. The return mapis now formulated in the principal (Eulerian) axes defined by the trial state, which canbe computed in closed form directly from ber, as described below.

REMARK 47.1. From the point of view of implementation, the optimal coordinateexpression of (47.9a) is in the principal axes n() = ntr ( A). Setting f, (ri, T2, r3, qO) =fz(r, q) one arrives at the reduced system:

mA= t'- jAY]aTA f ( A,qc),

gerA = ErA -7;tar~fil~rA~/r

/b=1

m

,a r = ~n + ATlaq l (rA, qT ), (47.10)P/=1

Ay 0, f(TTA, q) <0, YAy" (TrTA,qT) 0./1=1

It should be noted that no added computations are needed since the principal directionsn() (which coincide with nA) must be computed in order to evaluate the triallogarithmic strain tensor e'trial - lg(b).

REMARK 47.2. Observe that in the isotropic case only the stretching part Ve in thepolar decomposition Fe = VeRe is defined. The orientation of the intermediate con-figuration remains completely arbitrary. The preceding algorithm is consistent withthis observation. In fact, by making use of the kinematic relation

be = FeFe T = FCP-FT where CP = FpTFp , (47.11)

the trial elastic left Cauchy-Green tensor defined by (47.6) can be written as

b = FC- 1FT with CP = F'PTFP. (47.12)

Therefore, for given F,, the algorithm can be completed without any knowledgeof the plastic rotation tensor R which defines the orientation of the intermediateconfiguration. Thus, only CP is required.

SECTION 48 389

48. Implementation of exponential return mapping algorithms

This section gives a detailed implementation of the exponential return mapping al-gorithms described above. Throughout this presentation, attention is restricted to thecase of isotropic elastic response. Consistent with the observation made in the lastremark, in the implementation of the algorithm described below the set of internalvariables {be, , n} is adopted in the data structure. Accordingly, denoting by f therelative deformation gradient defined by

fr O (,, = FFn -1 ' f(Xn) = I + V7uT(r), xn S, (48.1)

the trial elastic left Cauchy-Green tensor is computed from (47.12) via the identity

be= LFF-]Fh IFnTF ] = f,be f. (48.2)

Obviously, the choice {C P, a n} as a set of internal variables is equally admissible.The step-by-step procedure outlined below summarizes the actual implementation of

the updating scheme defined by the preceding algorithm within the interval [tn, t,+l ].In a finite element context, this update procedure takes place at each quadrature pointof a typical element.

Step 1. Trial state. Given {b, ,,n} [at a specific quadrature point x, E (p,,)]compute the relative deformation gradient fT and the trial elastic left Cauchy-Greentensor be tr for prescribed deformation ur: , () - IRndim as

f-(x,): = I + V(x)) be tr = fbef.t (48.3)

Step 2. Spectral decomposition ofb tr. Find the principal stretches {)etA`} by solvingthe characteristic (cubic) equation in closed-form via Cardano's formulae. Computethe principal directions {n (A)} via the closed-form formula in (33.2). In the case ofthree different roots:

/ _ ,etrB) I I - betr etr 2.r= _ ( e tB )2_ ( XetA)21 2 (2 C trI ) (48.4)

for A = 1,2, 3, with B = 1 + mod(3, A) and C = 1 + mod(3, B).

Step 3. Return mapping in principal (elastic trial state) axes. Compute the loga-rithmic stretches and the principal trial stresses:

etA = log (eT A) A 1,.. dim,(48.5)

TrA = aEA ( qT a

If the trial stress (in principal axes) lies outside of the elastic domain expressed interm of principal stresses, then perform the return mapping algorithm (see remark

390 J.C. Simo CHAPTER V


below) to compute the principal logarithmic strains {ET A, : T }. Then reconstruct theKirchhoff stress tensor via the explicit formula

n'dim

T =Cr~nrL~(")e: n~'"': (48.6)?~ = Z a; A nfl (A) ® tr (A) (48.6)A=l

where {r, A, q' } are the principal stresses, also computed as part of the return mappingalgorithm, defined by (48.5).

Step 4. Update of the intermediate configuration. Reconstruct the updated leftCauchy-Green tensor via the spectral decomposition

7ndim

b_ = E exp [2 ETA] n ) ®( ) (48.7)A=i

where ntr (A) and e_ A are defined in Step 2 and Step 3, respectively.

For an arbitrary stored energy function (Ae, A', AS), the return mapping algorithmrequired in Step 3 of the preceding implementation, must be formulated in strain space.The classical return mapping algorithms in stress space of the infinitesimal theory de-scribed in Chapter I are exactly recovered in the present finite strain context, for a freeenergy function quadratic in the principal logarithmic stretches, of the form (ndim = 3)

(E A) 2 A [El + £2 + _,] + / [(le)2 (Ee)2 + (e)e] (48.8)

where A > 0 and pt > 0 are the Lamd constants, and E := log(AeA), for A = 1,2,3.This stored energy function (48.8) is often referred to as the Henky model. From thedefinition of the elastic logarithmic stretches, for the case ndim = 3 it follows that

3 3

4e= lIog(e) 10g(2A3) = log(J ) (48.9)A=l A=l

Now let A = Je-11/3e denote the volume-preserving principal stretches, introducedin FLORY [1961], which satisfy the condition 1/2)3 = 1. From (48.9) one concludesthat

3 3

E A = E~ - 3 E iE = log(i;) = E £A = ° (48.10)B=I A=1

By analogy with the infinitesimal theory, one refers to e as the deviatoric (logarith-mic) stretches. The Henky stored energy function (48.8) can then be written as

w(E ) = ½ [log(J)]J+ " [(,) 2+ (2)2+ (?e)], (48.11)

where = A + 2 is the bulk modulus.3~ i I UI IVUL~

SECTION 48 391

REMARK 48.1. From expression (48.11) one concludes that wr has the correct behav-ior for large strains in the sense that - oo as Je 0 and, likewise, tD ooas Je - o. Unfortunately, Cw is not a convex function of Je and hence fw cannotbe a poly-convex function of the deformation gradient; see, e.g., CIARLET [1988] foran explanation of this terminology. Therefore, the stored energy function defined by(48.11) cannot be accepted as a correct model of elasticity for extreme elastic strains.Despite this shortcoming, the model provides an excellent approximation for moder-ately large elastic strains, vastly superior to the usual Saint Venant-Kirchhoff modelof finite elasticity. Furthermore, this limitation has negligible practical implications inrealistic models of classical plasticity, which are typically restricted to small elasticstrains, and is more than offset by the simplicity of the return mapping algorithm instress space described below.

Below, a brief discussion is given of the actual implementation of the return mappingalgorithm in principal stress space, with attention restricted to an elastic responsegoverned by the Henky stored energy function defined in (48.11). The generalizationto arbitrary elastic models is straightforward, see SIMO [1992]. The goal here is todemonstrate that the algorithms described in Chapter II within the context of theinfinitesimal theory apply literally to this model without any modification. To doso, we introduce the following vector notation for vectors and matrices containingcomponents relative to the principal axes:

ndim

ndim

= and = . (48.12)

Enint qnint

The ndim x ndim matrix of elastic moduli in the principal axes n(A) = ntr(A)

associated with the Henky model (48.11) is given by

ndim

and :=,, N DIAG1,..., 1] is the ndim x ndi identity matrix. Here,= r: ( -A ]is the bulk-modulus and f > 0 is the shear modulus. We shall use the notationh to designate the nint x nint matrix of generalized plastic moduli, with entriesh = 2( )/aa assumed to be constant for simplicity. Following the sameconvention used in Chapter II, we define the following (ndim + nint) x 1 vectors of



generalized stresses and generalized strains along with the corresponding matrix ofgeneralized moduli:

Ee= { e } {~} and G=DIAG[C,h]. (48.14)

We use the convention E to label the elastic domain in stress space, defined as:

E {Z E ]Indim+ni-t' f () = f (, q) 0, t = 1,2,..., m}. (48.15)

The gradient Vf,,(L) is interpreted throughout as an (ndim + nit ) x 1 vector. Withthis notation in hand the return mapping algorithm (46.9) in principal axes takes theform

m

= tr _ A-yVf, (), with r = GEtr,

(48.16)

fv(-r) < 0, Ay> O and Af'f ,(E ) = 0.

which is identical to that discussed in detail in Chapter II. The analysis presentedthere shows that this constrained algebraic problem arises as the optimality conditionsof the dissipation inequality

[T - X] G - [ -X,], VTE . (48.17)

Accordingly, the final state E, in principal axes is the closest-point projection ontothe elastic domain E (specified in terms of principal stresses) in the metric definedby the generalized elasticity matrix G (see Fig. 48.1). The two general strategiesdescribed in Chapter II, i.e., the general closest-point projection algorithm and thecutting plane algorithm, can be used to compute £-. No modifications in these twoalgorithms are required.

To summarize, from a practical standpoint, the preceding developments translateinto the following prescription for the implementation in the finite strain regime ofany infinitesimal model of classical plasticity.

(i) Express the model in the principal axes defined by the trial state. The classicallinear infinitesimal elastic stress-strain relations then translate into a linear relationbetween principal Kirchhoff stresses and principal logarithmic strains.

(ii) Construct the algorithmic counterpart of the model (in principal axes) by ap-plying a standard return mapping algorithm in the principal axes defined by the trialelastic state.

This defines Step 3 in the step-by-step outline of the algorithm given above. Theother steps in this algorithm are independent of the specific elastoplastic constitutivemodel and remain, therefore, unchanged. To complete the formulation of the algorithm

SECTION 48 393

CHAPTER IV

tr1

tn+ 1

T3=U

FIG. 48.1. Geometric interpretation of the return mapping algorithm for multisurface plasticity in principalstress space for two possible trial elastic states.

it only remains to compute the algorithmic elastoplastic moduli in the incrementalconstitutive equation (43.10). Below it is shown that results derived in the context ofthe infinitesimal theory remain essentially unchanged.

49. Linearization: The exact algorithmic tangent moduli

The step-by-step procedure outlined in the preceding section defines the Kirchhoffstress tensor r7 at time r C [t, t+l], along with the updated internal variables, interms of the motion pT. By inserting this algorithmic expression into the Galerkinprojection (44.2) of the dynamic weak form, with matrix expression defined by (44.11),one obtains a nonlinear dynamical system in P = lRdo x Ridof for the motion ~coand the velocity field. The iterative solution of this nonlinear initial value problem isaccomplished by solving a sequence of linearized problems, each of them defined by(43.4), with matrix expression given by (44.13). The linearized problem is completelyspecified once an explicit expression is given for the consistent elastoplastic moduliceP in the algorithmic incremental constitutive equation (43.10), i.e.,

£AUTr-: = CrP[V(Au) o r], L equivalently, [AT],,ij = jPkl(AU)k,1. (49.1)

Recall that Au E V, is the incremental displacement, a vector field on S, = qp(D)which is the primary unknown in the linearized problem, and £ar,- is the Lie deriva-tive defined by (43.6). The goal of this section is to provide an explicit expressionfor the algorithmic moduli CeP for the case of elastic isotropy formulated in principalstretches.

394 J.C. Simo


\ f =FxFnl

current Configurationat Time tn

FLIT1]Fixed Inten

Ic C S.I X / = 1 + V(A)

Nearby Configurationat Time tn+5

Fi-, -X<

FIG. 49.1. Local configurations involved in the linearization of the algorithmic constitutive equations.

Consider a nearby configuration ,+ E C and let f+C denote the relative defor-mation gradient between the placements S, and S,+, respectively defined by

(,+ = T + (Au and f-+C = 1 + (V(Au), (49.2)

where V(-) denotes the spatial gradient relative to the coordinates xk = 'pk(XA, r).The strategy used below in the derivation of the algorithmic moduli mimics the pro-cedure used in Chapter III to obtain the elastic moduli for models of elasticity for-mulated in principal stretches. Briefly, the algorithmic constitutive equations (49.1)are first reformulated in a fixed reference configuration, chosen to be the intermediateconfiguration at time t. The algorithmic moduli are then computed in this fixed con-figuration using ordinary differentiation and the result is finally push-forward to thecurrent configuration. Recall that the trial elastic deformation gradient Fet,: a linearmap between the local intermediate configuration at time t and the current (local)configuration, is defined as (see Fig. 49.1)

F'e, = F+(Fjp-t = fT+FF - = f+Fe t r. (49.3)

To implement this strategy we introduce the second Piola-Kirchhoff and the rightCauchy-Green tensors, relative to the intermediate configuration at time t, respec-tively defined as the pull-backs of the Kirchhoff stress Tr,+¢ and the unit tensors onthe placement S,+(, i.e.,

S =and r = (Fe()T (FIr (and ,+¢ (F+) lt+(]

SECTION 49 395

-

/

on. a nd11 : 1 ll L,,-

s,,+ = F'e,) 1Tr,+(F'er( - (49.4)

Using (49.3), (49.4)1 and definition (43.6) for the Lie derivative, it is straightforwardto verify the following push-forward relation:

£ar = F tr [dS,+t/ dl=o] (Fr )T. (49.5a)

An identical manipulation using (49.4)2 and (49.3) yields the analogous result for theright Cauchy-Green tensor, namely,

2sym[V(Au)] = (Fetr)-T[djC/etd ( =£ (49.5b)

Relations (49.5a,b) are the key results used in the derivation of the consistentelastoplastic moduli Cep appearing in the incremental form (49.1) of the algorithmicconstitutive equations. The closed-form expression is given in the following.

THEOREM 49.1. Let [cerPAB] denote the ndim x ndim matrix obtained by linearizationof the algorithm (47.9a,b) formulated in principal axes, i.e.,

CrA B := a-rA/e',tB, for A, B = 1,..., ndim. (49.6)

The tensor of algorithmic elastoplastic moduli in the constitutive equation (49.1) isgiven by the closed-form expression

ndim dim

ep = E E [CtpABn ) nt(A) (A) tr(B) 3 ntr(B)] + gt, (49.7a)A=1 B=I

where the nonzero components gtIJKL of the tensor gt relative to the basis {nt(A) are defined by the following expressions (I, J = 1,..., ndim):

gHlIII =-2%,

(49.7b)-tr -tr T, I (re tr )2 T (,\e 2tr -t( )r J T I for I J.

(Ae tr

2(Ae

tr

2

The tensor gtr thus depends only on the form of the stored energy function in principalstretches; only the matrix [CeAB] depends on the specific model of plasticity underconsideration via the algorithm (47.9a,b).

PROOF. Since the principal directions of both r, and b coincide with those of betr

defined in the trial state phase, the spectral decomposition of the Kirchhoff stress r,+and the second Piola-Kirchhoff stress tensor Sr+( take the form

ndim

Antr(A) ntr(A)~l'r = .Tr+~ATr+ 0 '~*+~

A=I(49.8a)

ndim

S =ES+T·tr(A ) O N-r(A)

A=I



where

nt(A)Aetr Ar= Fe Ntr(A) and S,+CA = TT+(A/( +4CA) . (49.8b)

Obviously, the unit vectors {N±tr(A) } are the eigenvectors of the right Cauchy-Green

tensor The unit norm condition INtrA)I = 1 then implies the relation

di m

d Ntr(A) = W x N7t(A )= W AN(9)= AfB=l

_ L A

where WT AB = -W BA. As in the derivation of the tangent elastic moduli given inSection 33, the argument now proceeds in two steps.

Step (i). Computation of the Lie derivative £fAr, using expression (49.5a). Thespectral decomposition (49.8a)2 for S,+ together with (49.9) imply

d( dSZZ d((=O A=l B=1

ndim ndim

+ 5 I [S B - ST-A]W ABNrt(A) ® Nr( B) . (49.10)A=1 BOA=I

Using the chain rule, from expression (49.8b)2 one easily concludes that

,s + A = 1 )2=ld [je}r 26ABTB -e d.Er+CB (49.11)

C =0 - 7A I r B I t;

Inserting relations (49.11) and (49.10) into expression (49.5a) for the Lie derivativeand using (49.8a,b) yields the result

ndim dim T

ahT, fdE FEt 2 B Ar 2S nl d e r | tr(A) tr(A)

A=I/3=1 I '" B I d =O

ndim ndim e tr 2_

q- T- (T A) -rA()X (49.12)±e t etr IWABtrA) B) (49.12)A=I B3A= I A rB

Step (ii). Computation of the Lie derivative £Aul using expression (49.5b). Fromthe spectral decomposition of the right Cauchy-Green tensor C er, relation (49.10),r+(,

SECTION 49 397

and the chain rule, it follows that

d fl1rtdim dim de etr N(A)

I = E E (Aet ) 4•A2 dC ( r A d_ A = (=O A=l B=I

nZdi. ndim

+ - E lf[(Aetr) 22r 21(A)2 E B[(,\ 2 _ (AectA) JU2 WrABNT ) -Nr ' (49.13)

A=1 B#A=I

Inserting this result into (49.5b) and noting that [Fetr]-TNtr(A) = tr(A) /.e t, yieldsthe result

ndim n dim i

I E E5 (etr A) 2 _E r B tr(A) ntr(A)A=l B=

ndim fndim (etr )2

- (etr )2E " T -/A WT_ ABtr) One(B).E E 2 ,etr e tr Wrgnr(A) 3 nr) (49.14)

A= B7A=~1i A

A direct comparison of formulae (49.12) and (49.14) yields the closed-form result(49.7a,b) thus completing the proof of the theorem.

The ndim x ndim matrix [CTPAB] of algorithmic moduli appearing in expression(49.7) contains the derivatives of the principal stresses Tr A relative to the elastic triallogarithmic strains e trB. This matrix has an expression identical to the algorithmicelastoplastic moduli of the infinitesimal theory derived in Chapter II. In terms of thenotation introduced in Section 3, the general result (17.6) now yields

-eBp -Gp-PG Vf(Xr) ® PGTVf(ZT)IcAB] =PG Vf(-) . PGTPVf( E) (49.15)

where G, = [G- l + AyV 2f(,)] - l and P denotes the orthogonal projection fromR(ndi+nLin t) onto R"ndim, thus defined by P[Z] = .

In summary, the computation of the algorithmic moduli reduces to evaluating thendim x ndim matrix [ce AB] using expression (49.15) since the components in principalaxes of the tensor t, defined by (49.7b), can be computed once and for all with totalindependence of the specific model of plasticity.

50. A generalization of J2 -flow theory to finite strains

The preceding theory and numerical implementation will be illustrated within thespecific setting of J2-flow theory. The generalization of this classical model of metalplasticity considered below follows the lines outlined in Example 37.1 and incorporatesfinite elastic strains within the framework of a multiplicative decomposition of thedeformation gradient. A main goal is to demonstrate that the radial return algorithmis exactly recovered within the algorithmic framework described in the preceding



sections merely by setting = t,+l so that At = t, - t+l, i.e., for the backwardEuler form of the preceding class of exponential return mapping algorithms.

(A) The case of nonlinear isotropic hardening. Recall that for J2 -flow theory theelastic domain E in principal (true) Kirchhoff stresses is the cylinder defined by thevon Mises yield criterion

f(, ) := [dev[t] | - [y + K'()] < 0, (50.1)

where oy > 0 is the flow stress, is the equivalent plastic strain and dev[r] is thevector containing the deviatoric principal Kirchhoff stresses, i.e.,

dev[t] = - tr[]i and Idev[r][ = v/dev[e] dev[], (50.2)ndim

where tr[e] = i1 is the sum of the principal Kirchhoff stresses, i.e., the truehydrostatic pressure. The unit normal to the von Mises cylinder is v = dev[]/Idev[r] d

and we have v * 1 = 0. By setting ;r,+ = de"t,, the specialization of the generalscheme (48.16) to the present example yields the radial return algorithm described inChapter II, now formulated in principal axes as

Tn+l = 2+l -2LA-yn+l and ,+i = + Ay, (50.3)

supplemented by the standard Kuhn-Tucker loading/unloading conditions. The con-straint i'+l *i = 0 yields n+ 1 1= T+ 1, which implies J++ = Jtr = J+As in the infinitesimal theory, Eq. (50.3)1 determines Pv+l solely in terms of the trialstate. The resulting expression along with the yield condition evaluated at the trialstate take the form

dev[- I and f t+ = dev[[ml] - V/[y + K(n)].

(50.4)

Finally, again as in the infinitesimal theory, one observes that the algorithmic loadingcondition f +l > 0 implies that Ay > 0. The consistency condition fn+l = 0 thenyields the following scalar equation which defines A-y during plastic loading:

fn-+I: = 3-2/iA-' [K- ( + ) - ()] = . (50.5)

Due to the convexity of K(.), Eq. (50.5) is ideally suited for an iterative solution byNewton's method for fixed f4+4 . Of course, the resulting algorithm is identical to the

SECTION 50 399

radial return method with nonlinear isotropic hardening, now formulated in principalstresses.

REMARK 50.1. An extension of the preceding algorithm to accommodate viscoplasticresponse is constructed exactly in the same fashion as described in Chapter II. Oneconsiders the standard viscoplastic regularization in which the Kuhn-Tucker relationsare replaced by the constitutive equation y =]g(f(r, q)[ / )> 0, where ]x[:= (x +H(x))/2 with H(.) denoting the Heaviside function. To circumvent the characteristicill-conditioning exhibited by standard viscoplastic algorithms in the rate-independentlimit as the viscosity q1 - 0, the viscoplastic regularization is rewritten in terms ofthe inverse function g-l () as

f+ = g-' A) for ft+l > 0 A: Ay > 0. (50.6)

Observe that g' l(.) is given in closed-form for practically all rate-dependent modelsof interest, which typically characterize g(-) via exponential functions or power laws.Combining (50.6) with the consistency condition (50.5) results in the modified scalarequation

-9 (t A-y) + 1 nl-2/jAy-[k (tin + J23K'(n)] = 0, (50.7)

which defines Ay > 0 and can be easily solved by Newton's method. Observe that(50.7) is well-conditioned for any value of the viscosity coefficient [0, oo). Inparticular, it is possible to set = 0 in (50.7) to recover exactly the inviscid limitdefined by Eq. (50.5).

(B) A unified scheme for kinematic hardening and viscoplasticity. The extensionof the preceding model of J2-flow theory at finite strains to incorporate kinematichardening relies on the following observation. By frame invariance, a general yieldcriterion of the form f (r, q ) < 0 must be an isotropic function of its arguments andcan only depend on the principal values of r and q. Accordingly, in the presence ofkinematic hardening, the von Mises yield criterion (50.1) in principal axes takes theform

f (dev[I]- , ) = dev[t] - - [yu + K'()] , (50.8)

where E R'nd m denotes the vector of principal values of the back-stress tensor inprincipal deviatoric (Kirchhoff) stress space; i.e., · I = 0. Equations (50.3) definingthe return mapping algorithm for J2-flow theory in principal stresses are then replaced



by the following relations

+1 =+ - 2pAyvi',

~n+l = Cn + >A2HV+l, (50.9)

n+ = + Ay,

/ 7 = ]g(f(dev[] - L,))[ 3 0,

where H is the kinematic hardening modulus, r+ = a e and v'+l is the normalto von Mises yield surface now defined by

,+ = n + 1/13/n+l i, with n+: = dev[in+] - n+,. (50.10)

Relations (50.9) define a return mapping algorithm in principal stress space whichincorporates a form of kinematic hardening consistent with the Prager-Ziegler evolu-tion law of the infinitesimal theory. Proceeding exactly as in the derivation leading to(50.7), one concludes that the trial state completely determines the unit normal to thevon Mises yield surface by expression (50.10). Similarly, one sees that plastic loadingoccurs whenever fr+l > 0. The multiplier A-y > 0 is then determined by solving thescalar equation

- (+ -9 1 (A ) - A-y[2y + 2H]

- [K '( + -A)-k/(( )) = 0. (50.11)

Since g(x) = 0 X x < 0, Eqs. (50.4)-(50.11) collapse to the standard radial returnmethod described in Chapter II for combined linear isotropic/kinematic hardening bysetting q = 0 and assuming that K'(.) is linear (see Fig. 50.1).

Observe that the form of (50.11) precludes ill-conditioning for any value of theviscosity parameter Trl [0, oo). To complete the algorithmic treatment of J2-flowtheory it only remains to compute the algorithmic elastoplastic moduli P+, associatedwith the return map (50.4)-(50.11), and defined by the general expression (49.14) inthe rate-independent case. A direct calculation identical to that described in Chapter IIyields the result

[e'np+IAB] = rd ; 1i + 2[sn,+(I - 311 1) - +,(Fn+, (n+l)],\ (50.12)(50.12)

SECTION 50 401

A- FWi tr I

U L nJ

FIG. 50.1. Geometric illustration of the standard radial return method for combined isotropic-kinematichardening in principal stretches obtained here for zero viscosity ( = 0). In the present approach, the

algorithm takes place in principal Kirchhoff stress space and is the same as in the infinitesimal theory.

where D~+l is defined by (50.4), while the scaling factor s,l and the coefficient6,+, are given by the explicit formulae

2pAy 1 2/A-ysn+ := and A+1:= (50.13)

1)3 tr l K"+H q_+ t '3t3Observe that the (n x ) atrx [f algorithmic elastoplastic moduli in

Observe that the (dim X fldim) matrix [c + IAB] of algorithmic elastoplastic moduli inprincipal axes defined by (50.12) is symmetric. For r7 = 0 one recovers from (50.12)the algorithmic moduli of the rate-independent infinitesimal theory.

51. Two-stage projected exponential return mapping algorithms

The goal of this section is to design the counterpart of the two-stage projected im-plicit Runge-Kutta scheme described in Section 15 of Chapter II and summarized inTable 15.1 within the framework of the exponential return mapping algorithms intro-duced in this chapter. The main difference between this scheme and its generalizationpresented below lies in the form taken by the linear extrapolation step that definesthe trial stress field in the second stage of the product algorithm. For the finite straintheory, a literal transcription of Step 3 in Table 15.1, for the purpose of defining thetrial stress rt+l in Stage II of the algorithm, would result in open violation of thefundamental principle of frame invariance, thus leading to completely meaninglessnumerical results. The underlying reason is that the stress fields 7-,+, and r,+l in-volved in the extrapolation update are defined in different configurations and cannot

402 JC. Simo CHAPTER IV

i

��12


be naively added. Substantiation of this claim requires a detailed examination of thebasic kinematics involved in the generalization of the two-stage algorithm to the fi-nite strain regime. Before doing so, and to motivate the subsequent developments, werecord below the form of the linear extrapolation update in Stage II which is amenableto generalization when finite deformations are involved.

First, one operates with elastic strains in place of stresses. This distinction is imma-terial if the constitutive equations are linear but results in considerable simplificationin the large deformation regime. Second, the linear extrapolation update is carried outin two steps. Recall that Au is the total incremental displacement field in the timestep [t,, t,+l] and set

En+g = En + Ae,+ where AE,+ti = 9 sym[V(Au)]. (51.1)

The first step in the extrapolation update produces an intermediate strain field E *

defined by the linear extrapolation formula, while the second step defines the finaltrial state t, in Stage II of the algorithm according to

1 e 1I - e an etr e* e+ n +1 and zn+ = n+ + - 9- AEn+,d (51.2)

By combining the two relations in (51.2), using (51.1) and multiplying the resultby the elasticity matrix C, one recovers the linear extrapolation formula given interms of stresses in Table 36.1. The key observation to keep in mind, however, isthat the extrapolation formula (51.2)1 remains meaningful in the finite deformationtheory since the finite strains involved are all defined on the same configuration. It willbe shown below that the extrapolation formula (51.2)1 together with an appropriategeneralization of the update formulae (51.1) and (51.2)2 arise naturally by applyingthe methodology presented in Section 46.

To give a concise formulation of the algorithm it proves convenient to rewrite theevolution equations that define plastic flow as follows. Define the forcing functions

ndm fndim

/P = ylaTf,(r,qa) and ha = yaqf,(r,q), (51.3)JL=1 I=1

where the Lagrange multipliers y > 0 either obey the usual Kuhn-Tucker conditionsfor the rate-independent problem or, for single surface rate-independent models, aredefined via the viscoplastic constitutive equation

a'y = ]g(f, (r,qA))[ with = 1. (51.4)

Here ]x[= (x + H(x))/2 is the ramp function, 7r E [0+, oo) is a viscosity parameterand g(-) is a function satisfying g(x) = 0 X x = 0. For multisurface plasticity(/ > 1) models of this type are generally not well-defined and must be replaced by

SECTION 51 403

the Duvaut-Lions viscoplastic regularization described in Chapter II. The problem ofevolution is then defined by the rate equations

FP = LPFP and ds = ha in [tn,tI+l], (51.5a)

where LP = F e- PFe, together with the initial conditions

FPl t=t, = F P and (alt=t, = Sn. (51.5b)

The forcing functions IP and ha in this problem, defined by relations (51.3), dependon the generalized stresses (r, qa) which, in turn, are functions of the internal variables{Fe , Iai} via the constitutive equations

' = Fe[CueW(FeTFe)]FeT and qa = -Oa7(). (51.6)

The primary variables in the problem are, therefore, the internal variables {Fe, a},although the evolution equations (51.5) are cast in terms of FP to facilitate the formu-lation of the algorithm described below. Before doing so, we introduce some kinematicpreliminaries.

(A) Kinematic preliminaries: The generalized mid-point configuration. Recall thatin the infinitesimal theory the displacement field in the first stage of the algorithm,corresponding to time t,+ = t,+l + (1 - )t,, is a convex combination of thedisplacement fields at time t, and time t+, respectively. To generalize this set-upto the finite deformation theory, consider the one-parameter family of generalizedmid-point configurations po+ C defined by the convex combination

Wn+f: = qOn+l + (1 - dE)(,, 0 e (0 1], (51.7)

and illustrated in Fig. 51.1. Setting F,: = Dp, and F,+l := DP,,+l the deforma-tion gradient associated with the configuration p,+o is also given by the convexcombination

F,,+,: = F,,+ + (1 - 2O)F.. (51.8)

In the finite element implementation of the algorithm it is convenient to work with therelative displacement field between the configuration S,+., = (p+ (Q) and S, =p, (), parameterized by points in S,+; i.e., the vector field u,+ :'Sn+ ,+ Rndim

defined as

un+9(xn+z9) = n+a O - W.n ° qn+,(a(X+z) for Xz+, E Sn,+. (51.9)

As before, let f,+o = Fn+,gF - 1 denote the relative deformation gradients betweenthe configurations S,+s and S,. Similarly, we shall denote by f,+ = Fn,+1F-'the relative deformation gradient between the configurations S,+ and S+,y (see



ReferencConfigu:

iguratione tn

FIG. 51.1. Illustration of the three configurations involved in the formulation of the modified return mappingalgorithm and the corresponding relative deformation gradients. Attached to each configuration are shown

the corresponding deformation tensors.

Fig. 51.1). The following expressions in terms of u,,+ are an immediate consequenceof the chain rule and (51.8):

fn+,7 = [1 - Vu+] - and f+l = 1 + 0 - Vun+J. (51.10)

Observe that (51.10)1 differs from (48.1) since the vector field u is assumed param-eterized in this latter expression by points x, in the configuration S,. Finally, let

{n,, be}, ~{,+o, b±n+,} and {T+l , ben+ } denote the (true) Kirchhoff stress field andthe left Cauchy-Green tensor at the configurations Sn, Sn+ and S,,+, respectively.These tensor fields are also illustrated pictorially in Fig. 51.1. Consistent with theremarks made at the beginning of this section, two crucial facts should be kept inmind when a Lagrangian description of the motion is employed:

(i) Fields defined on a given configuration, say for example Tr+d defined on Sn+ ,obey the usual frame invariance requirement only relative to isometrics superposedon that particular configuration; for the example at hand this configuration is S,+e.Superposed isometries on any other configuration leave these fields unchanged. Forthe example at hand, isometries superposed on either S, or S,+1 do not affect r,+g.

(ii) A linear combination of frame invariant tensor fields defined on different con-figurations renders tensor fields which are no longer frame invariant. For instance,ar,+o + 3r,+1 is not frame invariant for a, 3 E IR.

It is precisely because of these two facts that a literal transcription of the linearextrapolation formula in Step 3 of the two-stage algorithm summarized in Table 51.1

SECTION 51 405

TABLE 51.1Return mapping algorithm in principal axes.

For prescribed input data {ber, , } perform the following steps:

(i) Solve the eigenvalue problem: betrn (A) = etrnttr(A), for all the eigen-pairs {Ae, nt()},

A = ,..., ndim.

REMARK. Although closed-form solutions are possible the most robust technique is Jacobi iterationgiven the small problem size (ndim < 3).

(ii) Compute the trial logarithmic strain components setr = log(AAt), for A = 1,..., ndim to evaluatethe trial stresses. For the Henky model we have

rAt = Alog(J.") + 21EAr t while qtr = -a(- ) /a..

REMARK. Note that log(Je tr) = Adm EeAr(iii) Compute the principal stresses {TA ,, q } and the principal logarithmic strains {EeA r, T } from

the trial state by performing a closest-point projection return mapping algorithm in principal axes.REMARK. For the Henky model the return map is the classical stress-space return mapping algo-

rithm.(iv) Reconstruct the Kirchhoff stress and the elastic left Cauchy-Green tensor via the spectral decom-

positions as

ndim ndim

VT- E tr T (A) tr(A) b= (Ae )2t() ntr(A)

A=L A=i

where XeA 7 = exp[EA T] for A = 1,. , ndim.

is completely meaningless in the finite deformation regime. The two-stage algorithmdescribed below carefully avoids updates of this type.

REMARK 51.1. The practical implications of an integration algorithm which involvesobjects lacking frame invariance are hardly ignorable. For instance, numerical resultscomputed with such an algorithm can be altered merely by reorienting the coordinateaxes for the initial finite element mesh. It is clear that a method which produces resultsthat depend on the mesh orientation can hardly be regarded as useful.

(B) The general two-stage projected return mapping algorithm. The general set-upfor the construction of the two stages involved in the single step projected returnmapping algorithm is the same as presented in Section 46. To provide a conciseformulation of the algorithm that accommodates both the rate-independent and rate-dependent problems it proves convenient to introduce the following notation:

m

Alp = E AY ,i fi (r, E),=1

(51.11)



As in the continuum problem, the Lagrange multipliers Ayl > 0 obey either thediscrete version of the Kuhn-Tucker conditions for the rate-independent problem or,for single surface rate-dependent models, are defined via the algorithmic constitutiveequation

A7y = -]g(fp( r,T)) [ with /t = 1. (51.12)

With this notation in hand, the update of the given initial data (Fen, c, n) G I attime t, for a prescribed deformation gradient obeying relation (51.8) proceeds in twostages as follows.

STAGE I. The algorithmic solution {F+o(X), cu, n+(X)} C ]GL(rndim) x ] nint thatapproximates the actual solution at time t,+g is computed by the same algorithmdescribed in Section 46. Specializing the results given there to t = tn+s gives

FP+, = exp [ -tL 1] F nP and ( n+o = n + t9 Aha+,9. (51.13)

Using the multiplicative factorization F,+, = F +,FP+,9, the same arguments pre-sented in Section 46 now yield the following update formula for the elastic part ofthe deformation gradient:

F,+, = exp[-6i9Pn+,9F,,% where Fn+4" = F,f+FP-. (51.14)

Equations (51.13)2 and (51.14), along with (51.11) and the elastic constitutive equa-tions (46.9b) define Stage I.

STAGE II. This stage constructs an algorithmic approximation {Fe+lE, n+l} EGL(ndim) x RniR to the exact solution for the internal variables at time t+l via aproduct formula algorithm. Consistent with a mid-point rule approximation, a trialstate is defined in Stage II by the forward Euler step

FP,= exp [(1- )AtL P+,] F-+ P(51.15)

a n+ = o + + (1- r)Ah n+ .

This explicit update defines internal variables for which the yield condition will, ingeneral, be violated. Accordingly, they are regarded as trial values which are subse-quently corrected via a return mapping algorithm. To recast the preceding formulae interms of the elastic part of the deformation gradient, which is regarded as the primaryinternal variable, the trial elastic deformation gradient is defined via the multiplicativefactorization as Fn+l = F FP Using this definition, the update formula (51.15)is expressed in terms of the elastic deformation gradient as

r= Fn exp[(1-F+=F." exp [(I - )Af Pn+l,] FP+,+

Fe' IF,,-' exp [(I - e)Fee+29 Alp+19F p61- ]e -1 p,)+oA F+O + o

= F, t r exp[( - )AlrP,,+]F+. (51.16)

SECTION 51 407

Using expression f,+l = F,,+lF-,+9 for the relative deformation gradient, a rear-rangement of (51.16) gives the following explicit formulae

F.+ = fn+l exp[-(1 - )Al1P+]Fn+,9,(51.17)

i n+I = ao+9 + (1 - ) Sho n+9

which define the trial state for Stage II of the algorithm. The final state {Fn+ l, n.}

is then computed via an exponential return mapping algorithm essentially identicalto that used in Stage I. It will be shown below that the resulting two-stage schemebecomes closed-form under the assumption of elastic isotropy.

52. Closed-form of the two-stage projected IRK for elastic isotropy

Under the assumption of elastic isotropy, the constitutive response is completely char-acterized in terms of the internal variables be, }. The results in Section 48 thenapply literally to Stage I, without any modification, leading to the algorithmic flowrule

n En+ - 9AIP and I, +r = {o,+, + 0Ah,+oa, (52.1)

where E' +19 log~b' + and e tr

where En+o9 = 2 log(bn+og) and %En = log(bt +) are the final and trial logarithmicelastic strain tensors at the configurations Sn+o, respectively. Using these results,Stage II can be reformulated in a form ideally suited for implementation as follows.Upon defining the strial elastic left Cauchy-Green tensor via the standard expression

b+ = F (Fe )T, the update formula (51.17)1 implies the sequential update

b; = exp[-(l - )AtlP+ ] be+l exp[-(l - 9)AIP +(52.2)

bt + = f+bn+ tfT+1 .

Now observe that the right-hand side of (52.2)1 involves symmetric tensors definedon the same configuration Sr,+, all of which commute as a result of the restrictionto isotropy. Consequently, taking logarithms on both sides of this expression andeliminating both AlP,+d and Ah, +,o0 using (52.1) gives

ns = +1O9 - n+ and

wC*7+9 = t n+9 L+ (52.3)

where Ee,+ = 2 log(ben+ ). These formulae involve tensor fields defined in the sameconfiguration Sn,+g, thus furnishing the correct generalization of the linear extrapola-tion update (51.2)1. In addition, from (51.17b)l we conclude that

bn+l = fn,+l exp[2e+ ] fT+ and trn+1 =52n+n~~~~~~~~~~+ l =+ a cn+0'


(52.4)


We show below that the actual implementation of this two stage scheme can beperformed in a rather efficient and cost-effective fashion.

A concise implementation is achieved by grouping together the steps common toeach of the two stages of the algorithm, as summarized in Table 51.1, and making asystematic use of the spectral decomposition in principal Eulerian axes. For prescribedtrial values {bt, i}, the steps in Table 51.1 define a exponential return mappingalgorithm in principal axes, thus producing the following results: (1) The spectraldecomposition of b, (2) The principal logarithmic strains, { eA, T}, (3) Theprincipal Kirchhoff stresses {rA,,q' }, and (4) The reconstructed Kirchhoff stresstensor Tr and elastic left Cauchy-Green tensor be.

From the point of view of implementation, it is convenient to view Table 51.1 as asubroutine that is invoked with different values of the input parameters {b etr, tr} atappropriate points during the computation, according to the following strategy.

STAGE I. The first stage of the algorithm is identical to the scheme already describedin Section 48. Explicitly, the following steps are performed for prescribed initial data{b, e n} and a given displacement increment un+o:

Step 1. Define the trial state according to

be+tr =f.+ b + and 'n+ = n, (52.5)

where fn+g is the relative deformation gradient computed via (51.10)1.

Step 2. Perform the return mapping algorithm in Table 51.1, with input parametersbe tr = b and tr= +

At this point, the following converged results are available: the principal logarithmicstretches eA n+, , nh} and the principal stresses {rA , qT} together with the Kirch-hoff stress tensor r,+, and trial principal directions n+ ), which coincide with finalprincipal directions at the configuration Sn+og. Observe that the left Cauchy-Greentensor b+g, also computed in Table 51.1, is not necessary in Stage I.

STAGE IIA. Since the trial values Ee tr and the converged principal values A

are available from Stage I, the computation of the tensor fields e= 1 and e enter-ing the linear extrapolation formula (52.3) is trivially accomplished via the spectraldecompositions

ndim

etr _E etr tr(A) tr(A)n+d-d HA .+,9 n + n+9X

A=1(52.6)

ndim

ce ~ Fe n ntr (A) tr (A): An+ ¢ n+¢ ( n n + O9 '

A=1

The computation of the linear extrapolation formula (52.3) and the push-forwardrelation (52.4) is then carried out according to the following steps:

SECTION 52 409

Step 1. Compute the linearly extrapolated components etr* and n as-An+ C,

1e* e - 0 etrAn+O - A An+O , no

19 1 -W A = I,.. , ndim- (52.7)1 1- tr(oe n+,9 , - nd - oE~ n+9

This defines

ndim

e = log(be*) = Z e* +,r(Atr (A) on Sn+79 2i 'Y~iiAn n+ An n+oo n+7.A=l

Step 2. Compute the tensor field ben*, by exponentiation as

ndi

be*+ exp [2Ee * +o] tr(A) (52.8)n+ 9 - L~"~ A n n+ 0T12 9 n+9

A=I

Note that ben~_ = exp[2en+*] is a (contravariant) tensor field on Sn -.

Stage IIA can be viewed as defining the initial data {ben*+9 o (n+o} for the secondstage in a product formula algorithm. Observe that these formulae have the remarkableproperty of preserving the constraint det[be * 9] > 0. In other words, the initial databe'_*o lies in S+ C GL(ndim) for all x.,+ E Sn+ , as required.

STAGE IIB. The second part of Stage II is identical to Stage I. The prescribed initialdata now becomes be * ,*+,} and the steps involved in the update are:

Step 1. Define the trial state according to

betr = f+IbefT+1 and Ean+l = (52.9)

where f+l is the relative deformation gradient computed via (51.10)2.

Step 2. Perform the return mapping algorithm in Table 51.1, with input parametersbetr = be t and tr = ,1

Stage IIB thus defines the final values {be+, > n+l} which become the initial data ina subsequent time step. These updated variables satisfy the constraint det[bet+ ] > 0 inS+ I thus remaining within the admissible set 2i. Observe that the full Kirchhoff r+ Ialso computed in Table 51.1 is admissible (lies within the elastic domain) but is notnecessary in this stage and its computation can be omitted. Furthermore, because ofthe structure of the conserving time stepping algorithms described in the next section,it will be shown that Stage II of the algorithm need be performed only once within atypical time step.



53. Global time-stepping algorithms for dynamic plasticity

This section is concerned with the last step in the time-discretization of the initialboundary values problem for dynamic plasticity at finite strains: The formulation ofglobal time stepping algorithms that relate the velocity field to the motion in configu-ration space. This global time stepping algorithm, together with the local exponentialreturn mapping algorithm, produces a nonlinear algebraic system after the Galerkindiscretization in space described in Section 42 is introduced (see the commutativediagram in the introduction of this chapter). Before describing alternative global timestepping algorithms we prove an important property of the Galerkin projection alreadymentioned in Section 42: The finite dynamical system (44.11) inherits the global con-servation of the infinite-dimensional system. One of the goals in this section is to de-sign global time-stepping algorithms that preserve this unique feature of the Galerkinprojection.

Let rA = XA + dA be the position vector of a node XA in the triangularization.Define the Galerkin approximation to the resultant force and the resultant torque ofthe prescribed system of loads by

Nxt = Fx and M ,,xt = rA Fxt (53.1)AEg AE9

From (44.9) it is easily concluded that if body force B and prescribed traction T areconstant, expression (53.1) reproduces the exact results (39.9)1 and (39.10)1 for theinfinite-dimensional dynamical system. Similarly, the Galerkin approximation to thetotal linear momentum and the total angular momentum are defined by

Lh = EpA and jh = E rA x pA. (53.2)AEg Ac9

The following result provides the counterpart for the finite-dimensional dynamicalsystem (44.11) of Theorem 34.1.

THEOREM 53.1. The Galerkin projection preserves the conservation laws for the puretraction initial boundary value problem (r, = 0) under equilibrated loads (Nhxt =Mehx = 0), in the sense that

dLh=O and dJh=0 inlIxP, (53.3)dt dt

where P = Rnd°f x ]tRdof is the phase space for the system (44.11).

PROOF. The result follows from the proof of Theorem 39.1 since Voh C Vo. A directverification of (53.3)2 starts by noting that

(C ErA x Fin (A)t = oN (53.4)AEg Ac9

SECTION 53 411

for any constant E I di., as a result of (44.10). Using expression (44.10)2, astraightforward manipulation yields

( x r4A) ThVNA = Th (rA X GRAD[NA]) Dh-i (535)AEg Ag O

Upon noting that the expression DSDh = ZAEg rA ® GRAD[NA] is the Galerkin ap-

proximation to the deformation gradient, relation (53.5) collapses to h .· whichvanishes since rh is symmetric and C is skew-symmetric. Consequently, (53.4) van-ishes for any C RInd

im which implies the result ZAEg rA x Fi = 0. This result

together with the identity

d jh d A

dt dt rA x pAeQ

= [ MAIpB X PA] - rA X Ft (53.6)ACg BEg AE9

which follows from (44.11) and the fact that MA is symmetric while PB x PA isskew-symmetric, completes the proof of relation (53.3)2.

To describe a number of widely used global time-stepping algorithms it provesconvenient to adopt a conventional matrix notation and rewrite the finite-dimensionaldynamical system (44.11) in the standard form

Pt = Ftext - Fint(dt, rt)ddt for t EG , (53.7)dt = M-pt = vt

dt

where the global vector dt of nodal displacements is subject to the initial condition(dt,pt)lt=o = (do,po). Here M is the mass matrix and t G 1I H- (dt,pt) E Pis the flow in the finite-dimensional phase space P = IRdo° x RLdof, with ndof =3 x n.ode. In addition, t E I Fext denotes the vector containing the prescribed nodalforces F At(t) and Fint(dt, t) is the vector containing the internal nodal forces Fjnat,respectively defined by (44.9) and (44.10) for A c g.

The following time-stepping algorithms provide a representative sample of alter-native global discretization schemes for the initial value problem (53.7) which arewidely used in large-scale simulations. Only one of this schemes exactly preservesthe conservation laws of total linear and total angular momentum inherited by theGalerkin discretization. For a proof of the results quoted below see SIMO, TARNOWand WONG [1992] and SIMO and TARNOW [1992].



(A) The classical Newmark family of algorithms. This time stepping algorithm isdefined by the semidiscrete momentum equation (53.7)1 enforced at t+l along withthe standard Newmark formulae; i.e.,

lMa+la = 'next - Fmnt(Tn+l, dn+ ),

dn+l = dn + Atvn + At 2 [(2 -3)an + 3a+], (53.8)

v,+ = v + At [(1 - y)an + yan+l].

Two noteworthy properties of this class of schemes are(i) The only member of the classical Newmark family that preserves exactly con-

servation of total angular momentum (for equilibrated loading) is the explicit centraldifference method obtained for /3 = 0 and y =

(ii) The Newmark-trapezoidal rule, corresponding to the values -y = and / = 1,

does not inherit the conservation property of total angular momentum.Observe that the Newmark-trapezoidal rule defines an acceleration-dependent time

stepping algorithm. Furthermore, in the nonlinear regime, the trapezoidal rule doesnot define an A-contractive algorithm (see the counterexample of WANNER [1976]).Since the implementation of the preceding class of algorithms is fairly standard furtherdetailswill be omitted.

(B T e-stepping algorithms in conservation form. As an illustration of this classof methods, consider the algorithm defined by the semidiscrete momentum equation(53.7) enforced in conservation form at t+,g along with the standard Newmark for-mulae; i.e.,

M[vn+l- v.] = FX, - Fint(rn+,9 d.+ ),

dn+ = ds + At[(1 -/3/y)v + /3yvn+l] + At2 (' - /3/')an, (53.9)

a.+l = [v.+ - vn]/'At + (1 - 1/?)an,

where d,+a: = dn,+l + (1 -)d, and Fext. = F'+ + ( - )Fxt. In the precedingexpressions r+o denotes the Kirchhoff stress field evaluated at the generalized mid-point configuration p,+ : B -- IR3, with nodal position vector dn,+. Noteworthyproperties of this algorithm are:

(i) Exact conservation of the total angular momentum (for equilibrated loading)is achieved for the values = l3/-y = 2 corresponding to the conservation formof the mid-point rule. These values define an acceleration-independent time-steppingalgorithm.

(ii) In general, formulae (53.9) define a one-leg multistep method. Second-orderaccuracy is obtained if and only if = . For the parameter values = /-y = alinear analysis shows that the spurious root at zero sampling frequency vanishes if andonly if y = 1. This value yields the post-processing formula a,+l (v,+l - v,)/At.

In sharp contrast with the trapezoidal rule, the implicit mid-point rule does definean A-contractive algorithm in the nonlinear regime. In addition to the property of

SECTION 53 413

exact momentum conservation, both the explicit central difference method and mid-point rule possess the strong property of defining symplectic schemes for nonlinearelastodynamics.

(C) Algorithms incorporating high frequency dissipation. Extensions to the nonlin-ear regime of a number of algorithms exhibiting numerical dissipation in the highfrequency range often require the evaluation of the stress field at a generalized mid-point configuration. As an example, consider the algorithm defined by a modifiedmomentum equation and the standard Newmark formulae as

Man+ = Fext - Fint(Tn+, d+ ),

dn+l = dn + Atvn + At2 [( - 3)a n + aa,+I], (53.10)

Vn+ l = Vn + At [(1 - y)an + 'yan+l1.

For linear elastodynamics, this scheme reduces to the 9i-method of HILBER, HUGHESand TAYLOR [1977]. However, in the nonlinear regime the preceding scheme uses ageneralized mid-point rule evaluation of the stress divergence term in place of thecommonly used trapezoidal rule evaluation. The former preserves A-contractivity andleads to nonlinearly stable schemes for infinitesimal plasticity (SIMO [1991]) whereasthe latter does not. Noteworthy properties of this algorithm are

(i) The values 9 = , = , y = 1 reduce the algorithm to (53.9) and yields theacceleration-independent, exact momentum-preserving, mid-point rule in conservationform. As pointed out above, this algorithm possesses zero spurious roots (comparewith HILBER [1976]).

(ii) The values 9 = 1, = 4 and y = give the acceleration-dependent Newmark-trapezoidal rule in (53.8). The acceleration field predicted by this algorithm is noto-riously noisy.

For 0 E (, 1) the algorithm exhibits numerical dissipation in the high frequencyrange but the property of exact conservation of total angular momentum no longerholds.

54. Remarks on the implementation of return mapping algorithms

In the classical Newmark family of algorithms, the discrete version of the momentumbalance equation is enforced at the end of each time step. However, in the last twoglobal time-stepping algorithms outlined above, enforcement of the discrete version ofthe momentum balance involves a single evaluation of the external force vector at anintermediate time t+, a task easily accomplished via interpolation, as well as a singleevaluation of the internal nodal force vector Fint (rn+O, dn+o) also at time t,,+. Thisevaluation requires knowledge of the stress field Tr,+ at the generalized mid-pointconfiguration. It is precisely for this class of conserving time-stepping algorithms thatthe two-stage projected return mapping algorithms described in Section 51 are ideallysuited.



The reason lies in the observation that the stress field ,n+o is computed in Stage Iwhich is uncoupled from Stage II. The only purpose of the second stage is to computethe stress field r,+1 at time t,+l once r,+o is determined. Consequently, the entireiterative solution procedure of the discretized weak form of momentum balance canbe performed by repeated evaluation of Stage I. Stage II of the algorithm need beperformed only once per time step, once convergence of the iterative solution schemeis attained, and can be viewed merely as a means of updating the initial data fromtime t, to time t+l from the solution at time t,+,g. The nodal configuration andvelocity updates are trivially accomplished either via formulae (53-9)2,3 or throughformulae (53.10)2,3. Stage II thus defines the mapping

(b+,c+o} e S+ X IR+ x -{be+,,~n+l } E S+ x 1R+ (54.1)

which provides the initial conditions for the internal variables in the return mappingalgorithm at the subsequent time step.

From the preceding observation it also follows that only the algorithmic moduliassociated with Stage I, with a closed form expression given by (49.7) for t, = t+o,enter into the iterative solution of the algorithmic version of the projected momentumequation (44.10). In fact, the entire iterative solution of the nonlinear algebraic sys-tem arising in each time step is independent of Stage II and remain unaffected evenif a scheme other than the one described above is adopted. From a computationalstandpoint the preceding two-stage scheme is particularly attractive since the imple-mentation of Stages I and II is essentially identical, the added computational cost ofStage II is negligible since the update is purely local, while the overall accuracy ofthe resulting scheme is increased to order two.

As in the infinitesimal theory, Stage II of the algorithm furnishes one among themany possibilities for specifying the map (54.1). Two alternative definitions of thismap considered in SIMO [1992] are constructed as follows. Define the trial elastic leftCauchy-Green tensor by the following formulae:

fn+l (+~) :: I + I Vun+9(x+)],(

(54.2)

It follows that b+ 1 defined in this fashion is the push-forward of the convergedsolution b+ to the current configuration S,+ 1 with the relative deformation gradientf,+l between the configurations S,+0 and Sn+1. The final values {b',+, n+l } canbe defined by the following two alternative schemes that replace Stage II:

(i) Shifted backward Euler scheme (SBE). This update is defined merely by setting{b+l), Gn+l} = {btr+l, n+q} It can be shown that the resulting scheme yieldsa backward Euler return mapping algorithm but within time steps shifted by 9; i.e.,[tn+,9 tn+l +], for n = 0, 1,....

(ii) Product formula algorithm (PFA). This update procedure defines the internalvariables {be,+l, n+l} via a return mapping algorithm identical to the one em-

SECTION 54 415

ployed in Stage I1 of the projected IRK method, but with the trial state now given by{bet , >+o}, as defined by (54.2).

In the first of these two alternative strategies, labeled as (SBE), only the stressfield at the generalized mid-point configuration Sn,,+ remains in the elastic domain,i.e, (+,g q,, ) E E. In general, however, the stresses (n+l, qn+l) will not lie inthe elastic domain E. This feature is often viewed as a disadvantage from a practicalstandpoint. In the second of these two strategies, labeled as (PFA), consistency isenforced at the end of the time step via an additional return mapping algorithm exactlyas in Stage II of the projected IRK method. The only difference lies in the alternativedefinition used for the trial state. For the linear theory, the accuracy analysis describedin Section 20 shows that definition (52.7) results in a second-order accurate scheme.The same analysis also shows that the alternative definition (54.2) yields a schemewhich is only second-order accurate.

REMARK 54.1. At the time of writing this article, we have had no occasion to con-duct extensive numerical experiments with the projected return mapping algorithmdescribed in Section 53. On the other hand, extensive numerical experiments con-ducted with the two alternative strategies outlined above, some of which are reportedbelow, support their excellent performance in spite of being only first-order accurate.Preliminary numerical experiments also suggest that the second-order accurate pro-jected scheme described in Section 53 does not appear to exhibit a performance nearlyas robust as these two, nominally less accurate, update schemes. For this reason, thenumerical simulations described below will be conducted only with the well-testedalternative schemes (SBE) and (PFA).

55. Representative numerical simulations

The formulation presented in the preceding sections is illustrated below in a number offull three-dimensional numerical simulations taken from SIMo [1992]. The goals areto provide a practical accuracy assessment of the different return mapping algorithmsin actual calculations and to demonstrate the robustness of the overall finite elementformulation in both static and dynamic analyses. The calculations are performed withan enhanced version of the finite element program FEAP developed by R.L. Taylorand the author from the version documented in ZiENKIEWICZ and TAYLOR [1989].

(A) Three-dimensional necking f a circular bar. This simulation is the three-dimensional version presented in SIMO and ARMERO [1992] of a well-known two-dimensional problem considered by a number of authors. A bar possessing a circularcross section with radius Ro = 6.413 and total length L = 53.334 is subjected to adisplacement-controlled pure tension test with simply supported boundary conditionscorresponding to perfectly lubricated end grips. For a perfect specimen these bound-ary conditions lead to a bifurcation problem from an initially homogeneous uniaxialstate of stress. The bifurcation problem is transformed into a limit point problem via ageometric imperfection induced by a linear reduction in the radius along the length ofthe bar to a maximum value Rsym = 0.982 x Ro at the cross section in the symmetry



plane. By obvious symmetry considerations only one eighth of the specimen is dis-cretized in the analysis. The constitutive response of the material is characterized bythe model of J2 -flow theory described above, together with the Henky stored energyfunction (48.11) and the von Mises yield criterion (50.1). The material properties are

= 164.21, j/ = 80.1938 and 0y = 0.45. (55.1)

Hardening in the material is characterized by an isotropic hardening function of thesaturation type, with functional form

K(,) := HJ + [y - ay] (1 - exp[-6]), (55.2a)

where a-y > 0y > 0, H > 0 and 6 > 0 are material constants chosen here as

ar = 0.715, 6 = 16.93, H = 0.12924. (55.2b)

The specimen is subjected to a total increase in length of AL/L x 100 = 26.25%.The finite element discretization used consists of three-dimensional Qi/PO mixed finiteelements, as described in SMO, TAYLOR and PISTER [1985]. This simulation is usedin SIMO [1992] as a practical accuracy assessment of a number of alternative returnmapping algorithms in different finite element meshes. The results presented there(see Fig. 55.1 for a representative sample) support the remarkably good performanceexhibited by the exponential return mapping algorithms described in the precedingsections. In particular, the lack of significant degradation in accuracy exhibited by thesealgorithms suggests that the choice of time step should be dictated by the convergenceof the global solution scheme.

The simulations described above are performed with a global Newton iterative solu-tion procedure together with a linear line-search algorithm which renders the schemeglobally convergent. Remarkably, successful computations are completed in only tenload increments; a feature which demonstrates the robustness of the formulation. Inspite of the increase in the condition number of the Hessian with finer meshes, asuccessful solution for the 960 finite element mesh can also be accomplished in 10load steps scheme employing 6 BFGS updates, a line search with factor 0.6 and pe-riodic refactorizations. Simulations with 20, 50 and 100 load steps were successfullycompleted with only two refactorizations per load step.

(B) Three-dimensional dynamic impact of a circular bar The goal of this simulationis to illustrate the performance of the time-stepping algorithms for dynamic plasticitydescribed in the preceding section; in particular, the conservation form of the mid-point rule. These results are compared with with those obtained via the standard formof the Newmark time-stepping algorithm. The model problem selected is the dynamicimpact on a rigid frictionless wall of a three-dimensional bar, with length L = 32.4 mmand circular cross section with radius Ro = 3.2mm, presented in HALLQUIST andBENSON [1987]. The initial velocity is v0o = 0.2 2 7 mm//t s. The material response is

SECTION 55 417

CHAPTER IV

Q)

M

g2

0

-0.5

-1.0

-1.5

-2.0

-2.5

-3.0

-3.5

-4.00 1 2 3 4 5 6 7

Total Elongation (7 = nsteps A· l)

FIG. 55.1. Displacement at the section of extreme necking versus total elongation. Exponential returnmapping algorithm. 960 finite element mesh.

S TRESS 15< 2.133E-01

sTRESS 2< -3,845E-01

STRESS 3< 3775E-03

FI-. 55.2. Fine finite element mesh consisting of 960 QI/PO elements. Contours of the equivalent plasticstrain and stress distribution for the three-dimensional bar. Final solution obtained in 20 time steps with

the product formula (PF) algorithm.

418 J.C. Simo

p�o$P


STRESS 3< -6.659E-01

I HL 0< -4.238E-01

STRESS 3< -3.21 E-01

w

FIG. 55.3. Three-dimensional impact of a circular bar. Small mesh consisting of 144 QI/P0 mixed finiteelements. Sequence of deformed shapes corresponding to times t = 20, t = 40 and t = 80, obtained with

the conservation form of the mid-point rule in 16 time steps.

again characterized by the same model of J2 -flow theory as in the preceding example,with the following values of the material constants

K = 0.4444 GPa, wy = ao = 0.40 GPa, H = 0.1 GPa. (55.3)

The density in the reference configuration is taken as p = 8930.00kg/m3 .The sequence of deformed configurations corresponding to times t = 20 s, t = 40 s

and t = 80 s, computed with the conservation form of the mid-point rule, is shown inFig. 55.3 (for the 144-finite element mesh) and Fig. 55.4 (for the 972-element mesh).These simulations are successfully completed in 16 and 32 time steps, respectively.Despite the extremely large time steps used in the simulation, the final shape of thespecimen at time t = 80 s agrees well with the results reported in HALLQUIST andBENSON [1987]. The time histories of the maximum lateral displacement at the impactsection, computed with different time steps, are shown in Fig. 55.5 (144-elementmesh) and in the following figure for the 972-element mesh. The same simulationswere attempted for the same two meshes with the standard Newmark algorithm andthe one-step exponential backward Euler return map. Successful computations werecompleted only when the time steps where reduced to 2.5 and 1.25, respectively. In

SECTION 55 419

Adinhhh,

t

I

CHAPTER IV

STRESS 3< -7.830E-01

FIG. 55.4. Three-dimensional impact of a circular bar. Large mesh consisting of 972 Ql/PO mixed finiteelements. Sequence of deformed shapes and axial stress distribution corresponding to times t = 20, t = 40

and t = 80, obtained with the conservation form of the mid-point rule in 32 time steps.

all these simulations a quadratic rate of asymptotic convergence was attained in aNewton iterative solution strategy.

(C) Simple shear of a block: Pure kinematic hardening law. This final simulation isconcerned with the simple shear of a rectangular three-dimensional block, with con-stitutive response characterized by J2 -flow theory, in the ideal case of pure kinematichardening. As first noted in NAGTEGAAL and DE JONG [1981], this situation gives riseto spurious oscillations in the stress-strain response when the kinematic hardening lawis described by the Jaumann derivative. The goal of this simulation is to assess theresponse exhibited by the extension to the finite strain regime of the classical Prager-Ziegler kinematic hardening law described above. The finite element mesh consistsof 27 three-dimensional Ql/PO elements as shown in Fig. 55.8. The elastic propertiesare the same as in Example 47.1 while the model of J2-flow is now characterizedby a flow stress y = 0.45 and a kinematic hardening modulus H = 0.1. The finalconfiguration is also shown in Fig. 55.8 and the plot of the Kirchhoff shear stress rl2versus the amount of shear is recorded in the last figure. It is apparent from theseresults that the stress response is monotonically increasing; in fact, essentially linear,and exhibits no spurious oscillations in the entire range of deformations.

420 J.C. Simo

I


5

4

3

2

1

0

q2

a)

0 20 40 60 80

Total Simulation Time t

FIG. 55.5. Impact of a circular bar: 144-element mesh (Ql/PO elements). Time history of the maximumlateral displacement at the impact section for different time steps, computed with two schemes: Newmark-trapezoidal rule with exponential backward Euler (NT/BE) and mid-point with exponential product formula

(MP/PF).

Pi

a)

._la

Cd

X

,--q

.2,--C~

-4

5

4

3

2

1

00 20 40 60 80

Total Simulation Time t

FIG. 55.6. Impact of a circular bar: 972-element mesh (Ql/PO elements). Time history of the maximumlateral displacement at the impact section for different time steps, computed with two schemes: Newmark-trapezoidal rule with exponential backward Euler (NT/BE) and mid-point with exponential product formula

(MP/PF).

SECTION 55 421

CHAPTER IV

- K1

DISPLACEMENT 2< 4.286E+00

FIG. 55.7. Simple shear of a three-dimensional block. J2-flow theory with pure kinematic hardening. Initialmesh and final meshes consisting of 27 Q1/PO mixed finite elements.

a~ An _U.4U

. 0.38

U) 0.36

X 0.34

M 0.32

0.30

- 0.28

a 0.26

0 0.5 1.0 1.5 2.0 2.5 3.0

Amount of Shear Strain

FIG. 55.8. Simple shear of a three-dimensional block. J2-flow theory with pure kinematic hardening. Plotof the Kirchhoff shear stress versus the amount of shear deformation.

(D) Plane strain localization problem. This example is concerned with the planestrain tensile test of a rectangular bar, and has been studied by a number of authors;in particular, TVERGAARD, NEEDLEMAN and Lo [1981]. The goal is to demonstratethe inability of conventional mixed methods to accurately resolve the sharp gradientsarising in localization problems. This unsatisfactory performance is contrasted with arecently proposed technique (SIMO and ARMERO [1992]) ideally suited for this classof problems. A strong localization in the specimen is induced by considering a steep

422 J.C. Simo


U.Y

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

n

Previous law (satur.) ----. -

, I I I , I I

0 0.5 1.0 1.5 2.0

Equivalent plastic strain a

FIG. 55.9. Plane strain localization problem. Steep softening law, defined by a double-parabola, used in thesimulation.

softening law, the double-parabola depicted in Fig. 55.9. Steep softening laws of thistype are often used in the numerical simulation of localization problems.

The specimen considered in this study possesses a width of 12.826 and a lengthof 53.334 and is subject to ideal plain strain loading conditions; hence, the thirddimension is assumed to be infinite. To trigger the localization, an initial geometricimperfection is considered in the form of a linear reduction of the width from its initialvalue at the top to 0.982% of this value at the center of the specimen. A quarter of thespecimen is discretized imposing symmetry boundary conditions along the boundaries.The finite element mesh consists of 200 quadrilateral elements and the simulation isperformed under displacement control of the top surface.

Figure 55.10 shows the load/displacement curves obtained for the double-parabolahardening/softening law with a conventional mixed method (Q1/PO) and the recentlyproposed enhanced-assumed-strain method (SIMO and ARMERO [1992]) labeled asQ1/E4. The conventional Q1/P0 mixed method leads to an overly diffuse responsethat provides a very poor resolution of the sharp strain localization in the specimen. Insharp contrast with these results, the load displacement curve computed with Q1/E4exhibits the sudden drop which is typically found in localization problems of thistype. These results are confirmed by the deformation patterns shown in Fig. 55.11(a).The Q1/E4 element method exhibits a very well defined line of discontinuity in thespecimen at 450 with the load direction while the Q1/P0 element shows a very diffuselocalization pattern. Figure 55.11(b) depicts the distribution of the equivalent plasticstrain for both formulations at the same imposed top displacement. Once more, the

SECTION 55 423

,,

A

1u

8

m0 6

4

2

0 1 2 3 4 5

Top displacement

FIG. 55.10. Plane strain localization problem with double-parabola hardening/softening law. Computedload/displacement curve.

clear shear band exhibited by the Q1/E4 formulation should be contrasted with thediffuse shear pattern exhibited by the mixed QI/PO element.

(E) Sheet metal forming. This three-dimensional example, very similar to one con-sidered in REBELO, NAGTEGAAL and HIBBITT [1990], is taken from LAURSEN andSIMO [1993] and proved an illustration of an industrial application of the precedingtheory. A flat elastoplastic sheet is formed into a pan, by forcing it to conform to theshape of a rigid die pressed against it (see Fig. 55.12). A J 2-model of multiplicativeplasticity is assumed, with material properties E = 70 GPa, v = 0.3, ay = 140 MPaand isotropic hardening with hardening modulus H = 100 MPa. A Coulomb frictionmodel for the contact interface between the die and the sheet is assumed with co-efficient of friction of 0.25. A detailed account of the algorithm issues involved inthe treatment of the contact interface is given in LAURSEN and SIMO [19931. Thisreference should be consulted for additional information on this aspect of the simu-lation. The sheet initially measured 600 mm long, 560 mm wide, and 5 mm thick. Asmay be noted in Fig. 55.12, the die consists of a lower flat region with an inclinedsection leading to it; the simulation was continued until the lower region had beenmoved through a distance of 100 mm. For symmetry reasons only half the geometrywas modeled, with 800 continuum elements being utilized for the discretization of thesheet. The loading was achieved in 100 load steps, through displacement control ofthe die (the table was fixed).

The mixed finite element method described in Section 45 is employed in this simu-lation, with pice-wise constant interpolations for both pressure and volume along with


SECTION 55 The discrete initial boundary value problem 425

Q1/E4 Q1/PO

a

I

0.15

0.60

b

IIF

FF

FFI

F

I

I

I4:

FIG. 55.11. Plane strain localization problem (double-parabola softening law). Results at an imposed topdisplacement of ui = 4.24: (a) Deformed configuration. (b) Distribution of the equivalent plastic strain.

r

l III HHH

H

I

I]H tIIH

I

� Hill

I: -11F i

-

I

Ii4

.:.F-h4

M; :; i i [ 5

CHAPTER IV

FIG. 55.12. Initial geometry for the pan forming problem.

FIG. 55.13. Deformed configurations for the pan forming problem.

a bilinear interpolation for the displacement field. This choice of continuum elementsis not optimal for the problem at hand, particularly with regard to capturing bend-ing behavior. It nonetheless demonstrates the capability of the model of plasticity to

handle large deformations of the type typically encountered in metal forming appli-cations. The outer edges of the plate were considered to be fixed, but even in lightof this constraint some redistribution of material occurs between plate and table and

(obviously) between plate and die. Figure 55.13 shows deformed configurations forthe plate at four equally spaced intervals during the calculation.

One notices immediately that the most severe straining of the material occurs at thecorners of the deepest part of the forming. This assertion is borne out by Fig. 55.14,which depicts a contour of the effective plastic strains at the final state. The computedstrain in this corner (over 100 percent) is partly an artifact of the coarse zoning inthis region, but also indicates the likelihood of tearing of the pan in this corner for

the geometry considered.

(F) Post buckling of a cylinder This two-dimensional problem, taken from LAURSENand SIMO [1993], combines the phenomena of inelastic buckling, frictional contact,and post buckling response. This reference should be consulted for a detailed ac-count of the algorithmic treatment of the contact problem not discussed here. We

426 J.C. Simo


Plastic Strain

< 2.304E-01

> 1.382E+00

FIG. 55.14. Contour of effective plastic strains for the pan forming problem, at the final state.

remark that problems of this type are encountered in crash-worthiness research, wherestructural elements may buckle and subsequently crumple, creating regions of selfcontact. Figure 55.15 depicts the physical situation considered. In this axisymmet-ric problem, a steel cylinder (inner diameter = 27 mm, outer diameter = 31.75 mm,length = 180 mm) is forced downward (via displacement control of its top surface) intoa rigid fixture. The J2 -flow theory described in the preceding sections is employed,with material properties defined as K = 175 GPa, G = 80.8 GPa, y = 700 MPa, andH (isotropic hardening modulus) = G/100. The spatial discretization of the cylinderemploys the same mixed finite element methods used in the preceding example, withfinite element mesh consisting of a total of 177 elements. As the calculation pro-ceeds, the cylinder initially moves down into the die, but when a critical axial load isreached, it begins a series of buckles, as is shown in Fig. 55.15. Importantly, this buck-ling occurs without any initial geometric imperfections. As indicated in Fig. 55.15,calculations are performed both with and without friction between the cylinder anddie, with frictionless response assumed in the self-contacting buckle regions.

Initial examination of Fig. 55.16 may not reveal significant differences between thefrictionless and frictional responses. Careful review of the figure will show, however,that in the frictionless case the cylinder moves a little farther into the fixture beforebuckling is induced. This result is as one would expect, since in the frictional casethe axial force builds more rapidly and buckling is induced sooner. This assertion issubstantiated by Fig. 55.17, which shows the total axial load induced in the cylinderas a function of the top displacement. As expected, the frictional fixture induces thebuckling (indicated by the drop in load) sooner than does the frictionless fixture.Interestingly, this offset in the curves due to the friction only persists through the firstbuckle cycle, after which time the curves rejoin each other for the last two bucklecycles. This suggests that once the initial buckling has been induced and completedit is the conditions existing above the first buckle which dominate the response (andthus the presence of friction below the first buckle becomes of secondary importance).

SECTION 55 427

I I

CHAPTER TV

DIltFIG. 55.15. Initial and final configurations for the post buckling cylinder problem, shown for the case of

frictionless response between cylinder and fixture.

Also apparent in Fig. 55.17 is the contribution of the self-contact occurring betweenbuckles. It is this contact, manifested in the figure as a reversal of load at the bottomof each buckle cycle, which provides the stiffening mechanism necessary to triggerthe next buckle. The regions in which self-contact occurs are also readily apparent inFigs. 55.15 and 55.16. Finally, it is important to emphasize that the deformation in-duces large plastic strains in the cylinder, rendering the problem extremely nonlinear.Figure 55.18 shows the plastic strains at the final state for the frictionless case. As canbe seen, virtually the entire specimen is yielded, with particularly large strains (over

428 J.C. Simo

7 C-1


ImL m(a)

L liiii LJ JL1 L ALLJT I LLj L LL

r(b)

FIG. 55.16. Buckling sequence in the steel cylinder problem, shown for both the frictionless (a) and frictional

(b) cases.

100%) occurring in the buckles. The fact that this problem was executed in a moder-ate number of load steps (approximately 260 for the frictionless case and 350 for thefrictional case) even in the presence of the complicated phenomena present demon-strates the efficiency and utility of the algorithmic treatment of both the multiplicativeplasticity and the contact techniques.

(G) Cylinder impact on deformable rails. In this final example, also taken fromLAURSEN and SIMo [1993], a steel cylinder (ID = 9cm, OD = 10cm, length =80 cm) is dropped, with initial velocity 80 m/s, onto a pair of steel rails (5 cm deep,4.67 cm wide, and 80 cm long), as shown in Fig. 55.19 The rails are parallel to eachother and 58.67 cm apart. Because of the symmetry of the problem only one-quarter ofthe geometry was modeled. Both the cylinder and the rails were given the propertiesK = 175 GPa, G = 80.77 GPa, ay = 700 MPa, H = 400 MPa, and p = 7850 kg/m 3.Roller boundary conditions were assumed on the bottom of the rails, and the ends

SECTION 55 429

F � F :Ulil~~~~~~~~~~~ci~~1

VLI


250

200

zZ 150

' 100

50

fi

0 20 40 60 80 100 120

Top Surface Displacement (m)

FIG. 55.17. Computed load-displacement curve for the steel cylinder post buckling problem.

Plastic Strain

< 1747E-01

_I_ ..........

i -

> 1.048E-00

FIG. 55.18. Computed plastic strains at the final state for the steel cylinder post buckling problem (frictionlesscase).


FIG. 55.19. Initial geometry for the cylinder-rail impact problem.

Eff. Plas. Strain

< 2.259E-02

1_

> 1.356E-01

FIG. 55.20. Contour of plasticity in the cylinder at time t = 0.625 ms.

of each were fixed. Coulomb friction, with = 0.1, was assumed to prevail in thecontact between cylinder and rails. A conventional Newmark time integration (withy = 0.9 and = 0.49) was used to integrate the global equations.

Figure 55.21 shows deformed configurations at various stages of the calculation,which was achieved in 100 time steps. By the final state shown (2.5 ms), the cylinderhas obviously rebounded. The contact region, which is quite localized in this problem,is manifested by large dents in the cylinder, as well as small yielded regions near

SECTION 55 431

CHAPTER IV

(a)

(b) (c)

(d) (e)

FIG. 55.21. Deformed configurations for the cylinder-rail impact problem at (a) t = O.Oms, (b) t =0.625 ms, (c) t = 1.25 ms, (d) t = 1.875 ms, and (e) t = 2.5 ms.

the contact points on the rails. Figure 55.20 shows the contours of plastic strainin the cylinder at the time t = 0.625 ms. As the figure indicates, this is yet anotherexample where significant plasticity occurs. Since this problem also involves dynamicsand the frictional contact between deformable bodies, it provides a particularly goodillustration of the applicability of the proposed techniques to a wide range of problems.

432 J.C. Simo

CHAPTER V

The Coupled

Thermomechanical Problem:

Product Formula Algorithms

A significant portion of the energy dissipated during plastic deformation in met-als is transformed into heat. Heat conduction in the solid affects, in turn, charac-teristic properties in the solid that govern plastic flow, such as the flow stress. Themodeling and simulation of this combined process, which accounts for the couplingeffects between mechanical deformation and heat conduction in a solid undergoinglarge plastic deformations in the cold-working regime, leads to a coupled thermo-mechanical problem. The numerical analysis of this problem constitutes the centraltopic of this chapter. The exposition below emphasizes the formulation of the coupledinitial boundary value problem and, in particular, the description of numerical tech-niques currently used in scientific computing and large-scale simulation of coupledproblems. No attempt is made to provide a comprehensive treatment of the physicalmechanisms involved in the deformation of metals in the cold work range. Back-ground material on the physical foundations of the subject, initiated in the fundamen-tal work of G.I. Taylor and his associates (see, e.g., TAYLOR and QUINNEY [1933]),can be found in the books of REED [1964], COTTRELL [1967] and the monographof BEVER, HOLT and TITCHENER [1973]. Expositions of thermoplasticity that empha-size the connection between continuum theories, underlying physical mechanisms andavailable experimental results are given in DILLON [1963], PHILLIPS [1974], ZDEBELand LEHMANN [1987] and references therein. An outline of the topics covered belowis as follows.

In the first part of this chapter, a brief account is given of the basic balance lawsgoverning the coupled thermomechanical problem, together with an extension of themultiplicative model of plasticity described in Chapter III that incorporates thermaleffects. This completes the formulation of the coupled initial boundary value prob-lem. Next, a formal a priori stability estimate for this initial boundary value problemintroduced in ARMERO and SIMO [1992] is described which generalizes to couplednonlinear thermoplasticity the canonical free energy of DUHEM [1911]. This func-tional was shown in ERICKSEN [1966] to define a Lyapunov function for coupled

433

thermoelasticity and plays a central role in the mathematical treatment of the qua-sistatic thermoelastic problem (BALL and KNOWLES [1986]). For thermoplasticity, thecanonical free energy defines a functional of the dynamics which does not increasealong the flow. This property provides a natural notion of nonlinear numerical stabilitywhich does not preclude interesting physical phenomena such as formation of shearbands in the presence of thermoplastic softening. Alternative notions of nonlinear sta-bility, B-stability in particular, are tailored to contractive problems of evolution and,therefore, not applicable to the problem at hand.

The second part of this chapter is devoted to numerical analysis aspects involved inthe simulation of the thermoplastic, coupled, initial boundary value problem. Coupledthermomechanical problems typically involve different time scales associated withthe thermal and mechanical fields. It is widely accepted that an effective numericalintegration scheme for the full coupled thermomechanical problem should take ad-vantage of these different time scales. It is considerations of this type that motivatethe so-called staggered algorithms, whereby the problem is partitioned into severalsmaller sub-problems which are solved sequentially. This technique is specially at-tractive from a computational standpoint since the large and generally nonsymmetricsystem that results from a simultaneous solution scheme is replaced by much smaller,typically symmetric, sub-systems. A sound numerical analysis is possible by inter-preting staggered schemes as product formula algorithms arising from an operatorsplit of the coupled problem, in the spirit of classical fractional step methods (see,e.g., YANENKO [1971]). Within this framework, the following two specific staggeredschemes are examined.

First, the conventional approach which, as noted in SIMO and MIEHE [1992], arisesfrom a formal split of the problem into a mechanical phase with the temperature heldconstant, followed by a thermal phase at fixed configuration. Staggered schemes of thistype are widely used in practice and go back to the work of ARGYRIS and DOLTSlNIS[1981] and others, see, e.g., LEMONDS and NEEDLEMAN [1986] and WRIGGERS, MIEHE,KLEIBER and SIMO [1992] along with the review articles of PARK and FELIPPA [1983]and DOLTSNIS [1990] for an overview. The well-known restriction to conditionalstability is the crucial limitation of this approach, which often becomes critical forstrongly coupled problems. Stabilization techniques designed to remove the restrictionto conditional stability have been devised by a number of authors with mixed successand, typically, within the context of the linearized problem. The early augmentationschemes of PARK, FELIPPA and DERUNTZ [1977], the iterative scheme of ARGYRISand DOLTSINIS [1980] and the more recent augmentation strategy of FARHAT, PARKand DUBOIS-PELERIN [1991] furnish sample representative approaches.

Second, a staggered scheme was recently proposed in ARMERO and SIMO [1992,1993], which completely circumvents the stability restrictions inherent to the conven-tional approach while retaining its computational approaches. The idea is to par-tition the coupled problem into an adiabatic mechanical phase, in which the to-tal entropy of the system is held constant, followed by a thermal phase at fixedconfiguration. The replacement of the isothermal phase in the conventional stag-gered scheme with an adiabatic phase results in an operator split which, remark-ably, preserves the a priori stability estimate for the coupled problem. An uncon-


The coupled thermomechanical problem

ditionally stable staggered algorithm is then constructed as the product formula oftwo unconditionally stable fractional steps. The chapter concludes with an accountof several representatives of numerical simulations of thermoplasticity. The numeri-cal analysis results described in this chapter rely on joint work of this author withF. Armero.

56. Integral, local and weak forms of the general conservation laws

The first step towards the statement of the initial boundary value problem for cou-pled thermoplasticity is the formulation of the balance laws of momentum, energyand entropy for a continuum body undergoing finite strains, in the presence of heatconduction. The statement of these basic laws of mechanics is standard, see, e.g.,TRUESDELL and TOUPIN [1960] for a complete account including the effects of jumpdiscontinuities. This section provides a summary of these classical results starting withthe integral form of the balance laws and concluding with their weak formulation,which is the form ideally suited for numerical approximations employing the finiteelement method. The background material and the notation on nonlinear continuummechanics introduced in Chapter IV is assumed throughout this exposition.

The simplest statement of the conservation laws is obtained by adopting a La-grangian description of the motion t E I -- cpt C of the continuum body in termsof the nominal stress tensor P, the material velocity field V = Cbt and the referencedensity po. Recall that B denotes the body force per unit of reference volume andT = Pv is the nominal traction vector field specified on the part of the boundaryFT C a.

(A) Integral and local form of the balance laws. Consider an arbitrary open set13 c Q in the reference placement of a continuum body that undergoes a motion pot.In terms of the preceding notations, the integral form of the balance law of linearmomentum takes the form

dt /poVd J= BdQ+ Pm dr, (56.1a)

while the integral form of the law of balance of angular momentum requires

-t 1Jt x poVd = gJ t x Bd2+ +J t Pvod. (56.lb)

Here vo: - S2 is the outward unit normal to the boundary aB of the openset B, which is assumed to be smooth. Equations (56.1a,b) furnish the statement fora continuum body of Euler's laws of motion. In a thermodynamical context, theseclassical laws are supplement by the balance of energy equation and the entropyproduction inequality.

Let E denote the internal energy in the solid per unit of reference volume, Q thenominal heat flux and R the heat source per unit of reference volume. The integral

SECTION 56 435

form of the balance of energy in an arbitrary open set B C 2Q is the statement that

d/ [ppoIV 2 + E] d Qdt

=Ij[B. V + R]d+ [V Pvo-Q-vo]dF, (56.2)

while the integral form of the Clausius-Duhem inequality requires that the net entropyproduction in B be nonnegative in time. The entropy production per unit of volume,denoted by a, is an extensive quantity defined in terms of the time rate of change ofthe (nonequilibrium) entropy in the solid, denoted by H, and the absolute temperatureO > 0 by the following expression:

d =- H d - R/ - /O d Q v/ d > 0. (56.3)fs dt B B los

Within the scope of rational thermodynamics, it is assumed that the internal energy E,the entropy function H, the nominal stress P and the nominal heat flux Q are consti-tutive functions defined in terms of the deformation o and the absolute temperatureO. Inequality (56.3) is then viewed as a restriction to be satisfied by these constitutivefunctions for prescribed body force B and specified heat source R in 2. This pointof view is advocated in COLEMAN and NOLL [1963], COLEMAN and GURTIN [1967]and is also adopted in this exposition.

Assume next that the Lagrangian fields involved in the statement of the precedingbalance laws are smooth. A standard result known as the localization theorem, see,e.g., TRUESDELL and TouPIN [1960] or GURTIN [1981], then yields the following localform of the balance of linear momentum equation (56.1a) introduced in Chapter III:

po V = DIV[P] + B in x . (56.4)

The local form of the balance of angular momentum equation (56.1b) results in thesymmetry condition PFT = FPT on the nominal stress where, as usual, F = Dtdenotes the deformation gradient. The local balance law (56.4) together with thelocalization theorem leads to the following local form of the energy balance equation(56.2):

E = P GRAD[V] +- R - DIV[Q] in 2 x 11, (56.5)

which is known as the reduced energy equation. Finally, the localization theoremapplied to the entropy production inequality (56.3) yields the result

eO = OH - R + DIV[Q]h- 5 GRAD[9] . Q > in x , (56.6)

known as the Clausius-Duhem form of the second law. It is conventional to recastinequality (56.6) in the following alternative form which can be found in TRUESDELL



and NOLL [1965]. Define the total dissipation D, the dissipation arising from heatconduction D,,, and the internal dissipation Dint in the solid by the expressions

D = OE, Dcon =-- GRAD[O] Q and D = Dint + Dcon. (56.7)

Introducing the additional hypothesis that the constitutive equation for the nominalheat flux Q is such that D,,,on 0 in F2 x , the Clausius-Duhem inequality D > 0 in$2 x implies (but is not implied by) the reduced inequality

Dint = (9/H- R + DIV[Q] > 0 in Q x II, (56.8)

often referred to as the Clausius-Plank form of the second law.

REMARK 56.1. The assumption that D,,on > 0 leading to inequality (56.8) is quitereasonable and always satisfied for models of heat conduction such as Fourier's law.According to this classical model, the Kirchhoff heat flux q = F-TQ is proportionalto minus the spatial gradient V[9] = F-TGRAD[19] of the temperature field. In otherwords, the following constitutive equation is assumed to hold:

q = -k(9)V[] Q = -k(O)C-GRAD[9], (56.9)

where k(O) is the conductivity and C = FTF is the right Cauchy-Green tensor.Assuming that the conductivity k(°) > 0, this model of isotropic heat conductionclearly satisfies the condition q V[(9] 0 or, since C is positive definite, theequivalent condition Q GRAD[9] < 0 which implies Dcon > 0. The conductivityk(9) is replaced by a positive definite, symmetric, conductivity tensor k(O) in theconventional extension of the classical Fourier's law to anisotropic heat conduction.

(B) Local governing equations for the initial boundary value problem. To summa-rize the preceding discussion, the local momentum equation (56.4) together with thereduced dissipation inequality (56.8) provide the two basic balance laws used in thestatement of the initial boundary value problem, i.e.,

poV = DIV[P] + Bin Q x ]l. (56.10)

/H = -DIV[Q] + R + Dint

In this equations, the motion pt and the absolute temperature field 9 are regarded asthe primary variables in the problem while the body force B and the heat source Rare prescribed data and the nominal heat flux Q is defined via constitutive equations,say Fourier's law, subject to the restriction Dcon > O. In addition, both the entropy Has well as the nominal stress P are also defined via constitutive relations, typicallyformulated in terms of the internal energy E, as described below. These constitutive

SECTION 56 437

equations are subject to the following restriction on the internal dissipation Dint in thesolid:

Dint = O9H + P GRAD[V] -E > 0 in x 1[, (56.11)

which is obtained by combining the energy equation (56.5) with inequality (56.8).Equation (56.11) asserts that the internal dissipation Dint appearing in (56.10)2 equalsthe thermal power 9H plus the mechanical power P GRAD[V] minus the time rate ofchange of the internal energy E, and must be nonnegative. Effectively, this equationprovides a constitutive relation for D)int > 0 in terms of the constitutive equationsfor P, H and E. Local balance of angular momentum is automatically enforced byconsidering constitutive equations that satisfy the symmetry condition PFT = FPT.The basic governing equations (56.10) and the constitutive restriction (56.11) aresupplemented by the standard boundary conditions for the mechanical field

Sot = t on , x[ and Pv 0 = T on T x , (56.12a)

together with the analogous essential and natural boundary conditions for the thermalfield, namely,

6=0 onFe x I and Qvo=-Q on Q x [. (56.12b)

Here (6 and Q are specified functions on Fr C as2 and Q C aQ, respectively. Inaddition to the usual restriction placed on Fr, and and £T, described in Chapter III,we also require that Fo n r = 0 and Fe U rQ = Oa2. Finally, we assume that thefollowing initial data is specified for the mechanical and thermal fields

(Pt It=o = so0, ott=o = Vo and Olt=o = 9o in Q. (56.13)

The preceding relations, together with the constitutive equations introduced in thenext section, comprise the local formulation of the initial boundary value problem forcoupled thermoplasticity.

(C) Weak formulation of the governing equations. As repeatedly pointed out in thismonograph, the weak form of the governing equations provides the most conve-nient setting for the design of numerical approximations to the initial boundary valueproblem. For the mechanical field, the weak form of Eq. (56.10)1 has already beendescribed in Chapter III and is summarized below for the convenience of the reader.Assuming for simplicity time-independent essential boundary conditions, the admis-sible deformations comprise the infinite-dimensional configuration manifold denotedhere by

Cmech = J{g E W'P()ndim: det[DSo] > 0 in 2 and lu = 6c.}.


(56.14)


Associated with the configuration manifold, we have the tangent space of material testfunctions defined by

V = {r0 C WI'P(Q)nd i : r0o = 0 on rF (56.15)

Recall that for fixed time t E Il the material velocity field V(., t) is a vector field onQ that lies in Vo. Similarly, the admissible temperature fields comprise the infinite-dimensional manifold

Cther = {O G Wlq(SQ)ndi: > 0 in •2 and 9lr, = 9}, (56.16)

while the admissible temperature test functions are contained in the associated linearspace

0 = {o0 e Wl'q() ind:im Go = O on ro}. (56.17)

As pointed out in Chapter III, the choice of exponents p and q in the precedingdefinitions are dictated by growth conditions on the response functions. These technicalconsiderations, however, will play no role in our subsequent developments. With thesedefinitions in hand, the weak form of the local governing equations (56.10) take thefollowing form:

(poV, 'r0) + (P, GRAD[ro]) = (B, 70 ) + ( T, 76.)r1-,(56.18)

(O, ) - (Q, ] RAD[]) = ( ) + (R, Go ) + ( , (o,o)pr,

for all material test functions r0o E Vo and 0 E To. In the preceding statement,the symbol (., ) denotes the standard of the L2(Q2)-inner product, whereas (., )rand (., *)r¥ stand for the inner product of vector fields and functions defined on FTand FQ, respectively. The weak form of the equations can be derived directly fromthe integral form of the balance laws (see ANTMAN and OSBORNE [1979]) and, asthis latter statement, is considerably more general than the local form of the balancelaws. We remark that in the weak formulation of the problem defined by (56.18), therestriction on the constitutive equations placed by the second law is still enforced viathe local inequality (56.11). The role played by this local condition becomes apparentin the developments described in the next section.

57. Constitutive equations for multiplicative thermoplasticity

The goal of this section is to generalize the model of multiplicative plasticity describedin Chapter III to incorporate thermomechanical (coupling) effects. The methodologyemployed in this extension is patterned after the construction presented in Section 35and Section 36 within the context of the purely mechanical theory. In short, motivatedby underlying physics of the problem, we consider a specific functional form for the

SECTION 57 439

internal energy function E and exploit the dissipation inequality to arrive at constitu-tive equations for the fields involved in the coupled problem. The model is completedby introducing evolution equations for the internal variables that describe plastic flowin the solid.

In classical thermodynamics, it is conventional to regard the internal energy E as afunction of the entropy H in the system. For crystalline plasticity, two basic physicalmechanisms contribute to the total entropy in the solid, see, e.g., COTTRELL [1967]and BEVER, HOLT and TITCHENER [1973]. The entropy associated with the lattice isherein denoted by He, and the configurational entropy associated with the motion ofdislocations and defects through the crystal lattice is denoted by HP . Noting that theentropy of a system is an extensive quantity, the total entropy in the solid is writtenas H = He + HP. This decomposition, along with the multiplicative factorization ofthe deformation gradient motivated by the micromechanics of single-crystal plasticity(see Section 34), yield the two basic relations

F= FeFP and H = He + HP (57.1)

Consistent with the physical interpretation assigned to the part He of the total entropy,the internal energy associated with the lattice is assumed to depend both on the latticedistortion, characterized by Fe , and the lattice entropy He. The frame-invarianceargument used in Section 35 then yields an internal energy function of the form

E(Ce, He,f ) = E(Ce, He) + ( ), (57.2)

where we have used the notation C e = FeTFe introduced in Section 34. This func-tional form for the internal energy in the elastoplastic solid is motivated by microme-chanical considerations (justifiable only for crystalline plasticity) and will be regardedas the central constitutive assumption on the basis of which phenomenological con-stitutive relations are developed. We remark that, in a phenomenological context,the assumption implicit in expression (57.2) that thermal effects are uncoupled fromhardening mechanisms is usually made in formulations of metal plasticity, see, e.g.,GREEN and NAGHDI [1966], GERMAIN, NGUYEN and SUQUET [1983] and LEHMANN[1982]. From a micromechanical point of view, such an assumption is motivated bythe observation that the lattice structure, which determines the thermoelastic response,remains essentially unaffected by the plastic deformation.

(A) Constitutive equations for stress, temperature and internal variables. To identifythe restrictions placed by the dissipation inequality (56.11) on the assumed form (57.2)for the internal energy in the solid, we compute the time rate of change of E usingthe identity

Ce = sym[FTGRAD[V]FP-I - FeTIPFe],

440 J. C. Simo CHAPTER V

(57.3)


also derived in Section 34, with 1p = FeLPFe-l and LP = FPFp - 1. The time rate

of change of the internal energy then becomes

nint

E [Fe(2ace,)F P- T] . GRAD[V] + [agHE]H e + [a,,,. (57.4)

Inserting this expression into the constitutive restriction (56.11), a standard argumentin irreversible thermodynamics yields the following potential relations:

P = Fe(2aceE)FP-T , 9 = aHeE and qa = aW. (57.5)

Since the second Piola-Kirchhoff stress tensor S relative to the intermediate config-uration, the nominal stress tensor P and the Kirchhoff stress tensor r are related viathe transformation rules

P = FeSFP T F= FT (57.6a)

the constitutive equation (57.5)1 for the nominal stress tensor P is equivalent to thestress-strain relations

S= 2 cE and r = Fe(2aceE)FT . (57.6b)

Expression (57.4), together with the preceding constitutive equations and the decom-position (57.1)2, also yields the following reduced form for the dissipation inequality(56.11):

nint

Dint = T · dP + 9HP + q 5q > 0 in x [, (57.7)a=1

where dP = sym[lP]. Using the kinematic relations derived in Section 34, the plasticwork term in (57.7) can be written in a number of equivalent forms depending ofwhich stress measured is preferred, in particular, r dP = S · DP. This equationdefines the internal dissipation Dint in the solid once the rate equations that specifythe evolution of dP and ai in terms of r and qa are given.

REMARK 57.1. As indicated in the preceding section, the absolute temperature field(9 along with the motion pt are the primary variables in the formulation of the initialboundary value problem for thermoplasticity. Accordingly, it is customary to castthe constitutive equations in terms of an alternative thermodynamical potential, thefree energy function T(F, 9, ~), obtained by performing the change of variablesHe (9 in the internal energy function for the solid via the Legendre transformation

(57.8)

SECTION 57 441

T1(Co, e, ~,, = E (Fe, He, ) - HOO.

Setting f(Ce, O) = E(C e H e) - He H, the uncoupled form (57.2) for the internalenergy along with (57.8) results in the following functional form for the free energyfunction in the solid:

T( eI, ,) ( c9 , o) +T( z, 0) (57.9)

It is readily seen that the preceding constitutive equations in terms of the internalenergy can be written in the equivalent form

7 = Fe[2IC,(Ce,)]FeT and He = -a( (e,). (57.10)

As before, the corresponding stress-strain relations for either the nominal stress P orthe Piola-Kirchhoff stress tensor S follow immediately from the transformation rules(57.6a). In the subsequent developments, the free energy function will be regarded asthe basic thermodynamic potential.

(B) The evolution of the total entropy. For the uncoupled free energy function (57.9),the local evolution equation for the total entropy entropy H in the solid is computedas follows. Time differentiation of the constitutive relation He = _-ao (C e, 9) anduse of the additive decomposition (57.1)2 yields, after changing the order of partialdifferentiation,

OH = OH + co q- 7- heat, (57. 11)

where c is the specific heat at constant deformation (and internal variables) per unit ofthe reference volume and h eat is the structural elastic heating, respectively, definedby the relations

c= -O2e0 2o9(Oe,e) and

'Heat = -oa [2a=fi(Oe,H)] ce. (57.12)

Because of the form (57.9) adopted for the free energy function in the solid, thethermoelastic heating -heat contains no latent plastic terms. Therefore, as in elasticsolids, the thermoelastic heating is associated in this model with nondissipative (la-tent) elastic structural changes and leads to the so-called Gough-Joule effect, see,e.g., CARLSON [1972]. Using the stress relations along with the kinematic relation

= FeTdeFe, which is implied by the multiplicative decomposition (57.1)1 andwas derived in Section 34, the structural elastic heating can be expressed in terms ofthe temperature derivative of the stress field as

heat = -ea 9 s. -e D= -Oar - de.


(57.13)


The additive decomposition H = He + HP implies that the internal dissipation in thesolid can be written as the additive contribution

Dint = Dmech + Dther in Q2 x ]I, (57.14a)

where Dmech is the mechanical part of the internal dissipation and Dther is the partof the internal dissipation induced by the temperature field in the solid, respectivelydefined by

hint

Dther = OHP and Dmech = r d + E q' . (57.14b)a=1

The local evolution equation for the absolute temperature field in the solid is ob-tained by inserting into the local balance law (56.10)2 the constitutive equation forthe nominal heat flux and Eq. (57.11) for the rate of entropy in the solid. In view ofthe preceding relations we have

c/ = -DIv[Q] + R + [Dmecb - YhFeat] in Q x . (57.15)

It should be noted that the plastic entropy HP does not appear explicitly in the temper-ature evolution equation. The second bracket in (57.15) is the result of heat conductionin the solid and, therefore, vanishes in an adiabatic process.

EXAMPLE 57.1. To provide a specific illustration of the preceding ideas, consider onepossible thermomechanical extension of a purely mechanical model characterized bya stored energy function W(C e) and a hardening potential 7-I( ). Such an extensionrelies on the rather strong assumption that the heat capacity c of the solid is a constant.Integration of the defining relation (57.12)1 then yields the following expression forthe free energy function

kr(Ce, , ~,) = T(O)- ( - Oo)M(C e) + W(Ue) + ±t(), (57.16)

where 90 is interpreted as a reference temperature and the function T(O) is given bythe explicit expression

T(O) = co[(9 - 0o) - log(/0o)] with c = co (constant). (57.17)

The term -(6 - Oo)M(C e) describes the thermomechanical coupling due to thermalexpansion and furnishes the potential for the associated elastic structural entropy,whereas the function T(O) is the potential for the purely thermal entropy. The partHe of the total entropy then becomes

He = co log(9/90) + V(ee). (57.18)

SECTION 57 443

This relation can be solved in closed form for the absolute temperature field 6 toobtain the inverse constitutive relation

9 = (ce, He) = 90 exp{ [He - M(e)]/co} (57.19)

Using this result, the closed-form expression for the internal energy in the solid followsfrom (57.8) as

E(Ce,He, ) = (C e, (Ce,He), ) + He9(Ce, He). (57.20)

It is immediate to verify that constitutive equation (57.19) satisfies the potential rela-tion O(C e, He) = aHE(C e, He, ).

Consider next the specialization of this result to the two specific models of interest inmetal plasticity examined in Example 37.1, labeled as Model I and Model II, respec-tively. These two models are isotropic and exhibit uncoupled volumetric/deviatoricresponse. Consistent with these two restrictions, it is reasonable to assume that thefunction M(Ce) specifying the coupling term in (57.16) depends only on the volumet-ric part Je = J of Ce. Setting M(Ce) = G(J) and using the notation in Section 37,the free energy function becomes

(Ce , , a) = T(O)- ( - Oo)G(J) + U(J) iF(6e) + 7(), (57.21)

where Ce = FeTFe and Fe is the volume preserving part of the elastic deformationgradient Fe . The functions W(Ce) and U(J) are those defined in Example 37.1.Specifically, setting be = FeF eT we have

Model I. 1[J 2 1 - log(J)] and W(e) - (tr [be] _ dim)

Model II. U(J) = 1j[log(J)]2 and (C e) = 4 log (e)

The constitutive equation (57.18) for He, together with the inverse relation (57.19)for the absolute temperature field 9, then take the following forms:

He = colog(9/0o) + G(J) and O = 90 exp{ [He - G(J)]/co}. (57.22)

A possible choice for the function G(J) inspired by the linear theory is G(J) =ndimoU(J) where a > 0 is the linear coefficient of thermal expansion.

(C) Evolution equations for thermoplastic flow. To generalize the results describedin Section 36 to a temperature-dependent yield criterion, it suffices to consider thecase in which the elastic domain E is specified in the current configuration in termsof the Kirchhoff stress tensor. An identical extension holds if E is specified in terms



of the second Piola-Kirchhoff tensor S. To account for a temperature-dependent yieldcriterion it is assumed that E is specified as

E = (r, q{ O) S x Rni t X IR+:

fu (r, q, 69) < O for = 1,2, ... , m. (57.23)

As usual, it is further assumed that the m-constraints, now depending also on the ab-solute temperature, are qualified and defined via convex functions. To gain insight intothe effect of the temperature field, consider once more the case of associative plas-ticity. Under this hypothesis, the evolution equations that define the rate-independentmodel are those which render a maximum for the internal dissipation Dint in the solid,subject to the constraints specified by (57.23). In view of expression (56.11) for Dint,this assumption translates into the inequality

ninth

IT- ] dp + [- 6] H p + E [qa - 0] ooa1

V(t, E, (57.24)

which, given the functional form (57.23) adopted for the elastic domain E, impliesthe following evolution equations:

m.

= h a, f ( q X

z=l

Eo = A yl q fp (r, q , ), (57.25)

m=1

iHP = y aeOof4(r, q=, O).

For the rate-independent theory, the multipliers in these evolution equations obey thestandard Kuhn-Tucker conditions

m

,yl > 0 and 75f,(r,q, ) = 0, (57.26)=1

which enforce the constraints f(-r, q, 6) < 0, for t = 1,2,..., m. Interestingly,the evolution equation (57.25)3 arising from the principle of maximum dissipationprovides a phenomenological interpretation for the evolution of the configurationalentropy HP in terms of the temperature change of the yield criterion. This interpre-tation first appears in StMO and MIEHE [1992]. The extension of the associative flow

SECTION 57 445

rule (57.25) to the rate-dependent theory is accomplished via the same regularizationtechnique described in Chapter II within the context of the infinitesimal theory.

EXAMPLE 57.2. To illustrate the simplest form of the rate-dependent model, considerfor simplicity single-surface plasticity obtained by setting m = 1 in (57.23). The inter-nal dissipation Dint defined by (56.11) is then replaced by the regularized dissipationfunction

IDi = int = Di -j+(f, ) > 0 in Q2 x , (57.27)

where the regularization parameter 7r C (0, oo) has the dimensions of a viscosity coef-ficient. In this expression j+: RI x JR+ R is a differentiable function which satisfies,for fixed but arbitrary temperature 9 E 1t+, the two conditions: (i) j+ (x, 6) > 0, forall x E IR, and (ii) j+(x, 6) = 0 if and only if x < 0. Setting g(f, (9) = afj+(f, 6),the rate-dependent flow rule for single-surface associative thermoelasticity is obtainedby replacing the Kuhn-Tucker conditions (57.26) with the constitutive equation

= g(f(, q, )), 6) if f(r, q, 9) > 0, (57.28)0 otherwise.

Recall that the simplest choice of the regularization function in the purely mechani-cal theory is obtained by setting g(x) = 2[H(x) + x], where H(-) is the Heavisidefunction, and corresponds to the linear viscous model (see Chapter II). For the ther-momechanical theory, the temperature-dependent part g(-, (9) of the regularizationfunction is often assumed to correspond to an Arrhenius term with functional formexp[-A/k6)], where A is the activation energy and k is the Boltzmann constant, seeMCCLINTOCK and ARGON [1968] for additional details.

It remains to show that the evolution equations (57.25) are consistent with theconstitutive restriction (56.11) implied by the Clausius-Plank form of the second law.It is shown below that this restriction implies thermoplastic softening in the solid fora wide class of material models arising in metal plasticity. In other words, the flowstress must decrease with increasing temperature field. This result is consistent withexperimental observations on plastic flow in metals within temperature ranges belowphase transitions, see PHILLIPS [1974] and ZDEBEL and LEHMANN [1987], and will bedemonstrated within the setting afforded by a representative model problem.

EXAMPLE 57.3. Assume that the yield criteria in definition (57.23) for the thermo-elastic domain IE have the functional form

f (r, a,(9) = p (,q ) - y(9) < 0 for i = 1,2,..., m, (57.29)

where the functions 4, (-, -) are convex and homogeneous of degree one anduy,((9) > 0 are interpreted as suitable flow stresses. As pointed out in ChapterII, this is the conventional setting adopted in the description of plastic flow in metal



plasticity. Implicit in (57.29) is the assumption that the effect of the temperature fieldon the yield criterion arises only via the temperature dependence on the flow stress,since the hardening potential 7-( ) is assumed temperature-independent. This simpli-fying hypothesis agrees, at least to first-order effects, with a number of experimentalobservations, see the references quoted above and LEHMANN and BLIX [1985].

The issue to be addressed here is the sign of the derivatives ayL, (9), the caseyus, () < 0 corresponding to thermoplastic softening. By recalling that Euler's the-

orem for homogeneous functions implies the property

q') q ")a~r(~ q) .r±qaqp, rIL'7Q ) qY, (57.30)

the associative flow rule (57.25) together with (57.29) yield the following expressionfor the internal dissipation Dint in inequality (56.11):

m m

Dint= y ylf, (r, q, O) + E -' [aiy(e) -ar (y,)] ) 0. (57.31)/z=l =1

To fix ideas, consider the rate-independent theory. The first term in (57.31) thenvanishes as a result of (57.26)2 while the second term is positive since yM" > 0 andory,() > 0. The last term in (57.31) is the dissipation Dther in the solid arising fromthe dependence of the flow condition on the temperature field. In view of (57.14b)and (57.31), it follows that

m

Diher =- 9P = -E Zyr y((9) in /2 x 1[. (57.32)

From (57.31) and (57.32) one concludes that the requirement Dther > 0, which ensuressatisfaction of the inequality Dint 0, automatically holds under the assumptiond (9) < 0, i.e., for thermoplastic softening. The same conclusion is easily shownto hold for the rate-dependent theory.

58. Formal a priori stability estimate and conservation laws

The central issue in a nonlinear stability analysis of coupled problems concerns theappropriate notion of nonlinear stability. In the linear regime, there is a well notion ofstability going back to the fundamental work of Lax, see RICHTMYER and MORTON([1967], Chapter 2), which exploits the underlying semigroup structure of the prob-lem of evolution. A brief exposition of this notion is given in the following section.For nonlinear dissipative problems of evolution, nonlinear stability is often phrasedin terms of an a priori estimate on the dynamics. Typical examples include the in-compressible Navier-Stokes equations in fluid mechanics, see TEMAM [1984], and thesystem of coupled nonlinear thermoelasticity in solid mechanics, where the a priori

SECTION 58 447

estimate arises from the second law, see ERICKSEN [1966], GURTIN [1975] and BALLand KNOWLES [1986].

The goal of this section is to present a (formal) stability result (ARMERO andSIMO [1993]) for the nonlinear initial boundary value problem of thermoplasticity.Existing stability results are either restricted to the mechanical theory, as described inChapter II and Chapter III, or limited to a linear perturbation analysis for the rate-dependent coupled problem of the type initiated in CLIFTON [1980], see MOLINARIand CLIFTON [1983], ANAND, KIM and SHAWKI [1987] and the review in SHAWKI andCLIFTON [1989]. The nonlinear stability estimate described below will be shown toarise as a direct consequence of the Clausius-Duhem form of the second law.

To state the result, recall that the deformation po E Cmech, the material velocity V =q of the motion, and the absolute temperature field (9 E Cther comprise the primaryvariables of the initial boundary value problem. The plastic part FP of the deformationgradient and the hardening variables A, comprise the internal variables, with evolutiondefined in terms of the primary variables via the flow rule. For convenience, thefollowing compact notation will be adopted

Z = {., V, } and F = {FP,, HP}. (58.1)

We shall further assume that the mechanical loads derive from a potential Vext(pO)according to the standard potential relation

DVext(p) -*0 = -Gext(n) = -(B,,) - (T, no)r Vn0o Vo. (58.2)

The form Gext(r0o) defined by the preceding expression is linear in r70 E Vo and givesthe virtual work of the external loads. For a given system of conservative loads withpotential Vet(qo) and a general free energy function !(Ce, O,(), not necessarilyof the uncoupled form (57.9), in the absence of heat sources (R O0) we define theLyapunov-like function

r(Z, r)

= J [ POlVI2 + (C'e . ,) + ( - O)He] dQ2 + Vext(q), (58.3)

where He = -a,9fo(Ce, 0, a). We emphasize that in this expression Ce = FeTFeis regarded as a dependent variable defined in terms of (Z, F) via the multiplicativefactorization Fe = DqoF p- 1. With these conventions we have the following result.

THEOREM 58.1. Assume that the following conditions on the data for the initial bound-ary value problem of dynamic thermoplasticity hold:

(i) No heat sources, i.e., R = 0 in Q x I.(ii) Conservative mechanical loading with potential Vext(cp).(iii) Boundary condition for the temperature field of the Dirichlet type, with pre-

scribed constant temperature 9o > O, i.e., 0 = 0o on a2Q x I and FQ = 0.



Assume further that the constitutive equation for the heat flux Q and the evolutionequations for the internal variables F satisfy the dissipation inequalities

'int

Dcon = -- Q GRAD[] O0 and )mech = S DP + Ž qa 0,c=l

(58.4)

where DP = sym[CeLP] is the plastic rate of deformation tensor; with LP =FPFp -1, while the generalized stresses (, qu) satisfy the constitutive equations

S= 20ueT(fe , , 5,) and q' = -al P(ze,o, ). (58.5)

Under these hypotheses, the weak form of the balance laws (56.10) in the statementof the initial boundary value problem imply the a priori estimate

d (Z, ) = 0 [D con + Dmech] d2 < 0 in Il, (58.6)

where £(Z, F) is defined by (58.3) with He = -oeP(Ce, O, ).

PROOF. Since the domain of integration 2 in the integrals involved in definition (58.3)is fixed, assuming sufficient smoothness it is permissible to interchange integrationand time differentiation. The key identity used in the computation below is

nint

l(Ve 6?, ) =P GRAD[V] - S P - S qc ',Ca=I

= P GRAD[V] - Dmech, (58.7)

which follows from (57.3), the relation P = FeSFP-T, constitutive equations (58.5)and definition (58.4)2 by exactly the same argument as leading to (57.4). Using thisresult, the time derivative of (58.3) reduces to

d-L(Z, F)= [(poV, V) + (P, GRAD[V]) - Get(V)]

+ X2[He + a ( e 0, ),] d2

+ J [-- mech + (9 - o0)fe] d. (58.8)

The first term within brackets in (58.8) is precisely the weak form (56.10)1 of thebalance of momentum evaluated with o = V chosen as test function which, therefore,vanishes. The second term within brackets in (58.8) also vanishes as a result of theconstitutive equation for the elastic part He of the total entropy in the solid. Finally,

SECTION 58 449

since Dint = D1)mech + Dther with Dther = OHtP, using the additive decompositionH = H + HP the remaining term in (58.8) can be written as

dt (Z Lr) = Dint + ,,Dther + 90°) Ot dQ. (58.9)

The proof is concluded by noting, in view of definition (56.17) for the space Toof admissible temperature test functions, that assumption (iii) on the boundary datarenders the function ( - Oo)/e an admissible test function, i.e.,

~o = (O - o)/O E Vo since 60 = 0 on a2 x E1. (58.10)

For this specific choice of test function, the weak form (56.10)2 reduces to

etH ( 0) eo)

= J Q- GRD [] + Din.) 'Din d. (58.11)

Combining (58.11) with (58.9) and using definition (58.4)1 yields the result.

It is again emphasized that the formal a priori estimate (58.6) does not precludephysical phenomena to be expected in the presence of thermoplastic strain soften-ing, such as the possible formation of shear bands and strongly localized and plasticdeformations. Therefore, estimate (58.6) provides a meaningful notion of nonlinearstability suitable for general models of multiplicative plasticity. This estimate plays acentral role in the analysis of nonlinear numerical stability described below.

REMARK 58.1. The case of coupled nonlinear thermoelasticity is recovered within thepresent context merely by setting FP = 1 and (>. = 0 in 2 x . In this situation, Fe =Dcp and the preceding a priori estimate reduces to the classical result in DUHEM [1911].For thermoelasticity the functional L(Z, F) is a Lyapunov function, a fact first provedin ERICKSEN [1966] and subsequently exploited in the analysis in BALL and KNOWLES[1986] of the quasistatic problem. For linear thermoelasticity in the absence of externalloads, the linearization of (Z, F) about the reference configuration reduces to aquadratic functional shown in DAFERMOS [1976] to define a norm in the Hilbert spaceZ = H(f)nd im x [L2 (f] n dim x L2 (Q) of admissible displacements, velocities and(linearized) temperature fields. The a priori estimate (58.6) then renders linearizedthermoelasticity as a semigroup of contractions in Z.

In addition to the preceding a priori stability estimate, for traction boundary condi-tions the initial boundary value problem retains the two global conservation laws ofmomentum described in Section 39 and summarized in the following.

450 JC. Simo CHAPTER V


THEOREM 58.2. For the pure traction boundary value problem ( = 0) under equi-librated loads (Next = Mext = 0), the weak form (56.10) of the governing equationsyields the conservation laws

d d-L(Z,r)=O0 and J(Z,r) = 0 in , (58.12)dt dt

where L(Z,r) = f 2 poV d2 and J(Z,r) = f cp x poVdf2 are the total linearmomentum and the total angular momentum, respectively. These definitions agree withthose given in (39.11).

PROOF. The proof of these results are identical to that given in Theorem 39.1. D

The remaining of this chapter is concerned with the design of numerical schemesfor the solution of the initial boundary value problem of dynamic thermoplasticitywhich preserve the a priori estimate (58.6) and inherit exactly the conservation lawsof momentum (58.12).

59. Time integration of the coupled problem: General considerations

Given a partition II = UNO=[t,, t,+I] of the time interval II of interest, algorithms forthe numerical integration in time of the initial boundary value problem of dynamicthermoplasticity are typically designed by rewriting this system as an abstract first-order problem of evolution for the primary variables Z and r defined by (58.1). Themain numerical analysis issues involved can be addressed with reference to a generalfirst-order system with the following form:

dtZ(.,'t)=A[z(-,t)]+ f(.) in x ,(59.1)

z(, to) = zo(.) in 2,

where z(o, t) lies in a suitable function space, typically a Banach space of the Sobolevclass with norm denoted by lI zI for z e Z, A[.] is a nonlinear elliptic operator withdomain dense in Z, f is a prescribed forcing term and zo C Z is some specifiedinitial data. Under suitable technical assumptions; see, e.g., MARSDEN [1973, 1974],the homogeneous version (f = 0) of the abstract problem (59.1) defines a localsemiflow, denoted by

Ft: Z x [t t+l] -- Z, (59.2)

which advances the initial data z(., to) E Z to the solution of problem (59.1) at timet according to z(., t) = Ft[z(-, to)], and satisfies Ft+, = FPto F for t > s. In thejargon of dynamical systems, one refers to A[-] as the vector field associated with theflow Ft. In what follows, we shall assume that the technical conditions which ensurethe existence (at least locally in time) of the flow hold, and concern ourselves solelywith its algorithmic approximation.

SECTION 59 451

Let Kat: Z x R - Z be a one-parameter family of maps, referred to as the algorithmin what follows, which depends continuously on the parameter At > 0 herein referredto as the time step. Recall that the algorithm IMAt is said to be consistent with theflow (59.2) if the following two conditions hold for any z Z (see RICHTMYER andMORTON [1967]):

im IAt[z] = z and lim [KAt[z] - = A[z]. (59.3)At-,0 At- At

Stability considerations on the algorithmic approximation are typically made for thehomogeneous problem obtained from (59.1) by setting f = 0. For the linear problem,A[] is the infinitesimal generator of a semigroup for which a well-defined notion ofstability due to Lax exists. This concept of stability essentially amounts to requiringthat the iterated algorithm KI = KAt o... o Kat, with At = (t - t,)/k, be uniformlybounded in bounded t-intervals.

For quasicontractive semigroups, the appropriate notion of stability is furnished bya stronger condition known as B-stability. Recall that a semigroup IFt is said to bequasicontractive if the condition lFt[z] - Ft[] liz < liz - llz holds for all t E E1and any z,z E Z, relative to some norm I I lz, equivalent to the standard II onZ and called the natural norm for the problem. An algorithm IKAt consistent withthe semigroup, in the sense of (59.3), is called B-stable if it inherits the contractivityproperty relative to the natural norm II · z, i.e.,

|IKAt[Zn] - I;At[n] 11 Z lizn - Zni z for At > 0. (59.4)

and any two initial conditions z,, ,, E Z. This definition was exploited in ChapterIII in the stability analysis of the initial boundary value problem for infinitesimalelastoplasticity formulated in stress space. Unfortunately, contractivity of the flowis a very restrictive assumption which does not hold for many dissipative dynamicalsystems of interest and, in particular, for thermoplasticity at finite strains. The situationfor general nonlinear problems of the type (59.1) is, therefore, significantly morecomplex and several alternative notions of nonlinear stability are possible, see theintroductory exposition given in Section 13 and references therein. One possibility isto assume that the algorithm IKAt is consistent, in the sense of (59.3) and linearizedstable (or locally Z-stable, see CHORIN, HUGHES, MARSDEN and MCCRACKEN [1978],pp. 237, 239). These two assumptions plus technical hypotheses on the flow yieldconvergence of the algorithm in the sense that

lim AKt[z] = Ft[Zn], for At = (t- t)/k and t G [tn,t+], (59.5)k-oo

thus providing a nonlinear version of the classical Lax equivalence theorem.Alternatively, for dissipative problems of evolution, nonlinear stability can be for-

mally phrased in terms of a (formal) a priori estimate on the dynamics of the form

[Ft(zo)] < 0 for t e I9,

452 JC. Simo CHAPTER V

(59.6)

The coupled thermnnomechanical problem

where £: Z -- IR is a nonnegative function. Under suitable technical assumptions, itis the existence of such a (Lyapunov-like) function that provides the natural notionof nonlinear stability for a dissipative problem of evolution of the form (59.1). Inparticular, for nonlinear thermoplasticity, the function £ is the canonical free energyfunctional defined by (58.3) and the a priori estimate is precisely inequality (58.6)with zero forcing. An algorithm KAt for problem (59.1) will be called dissipativestable (or simply stable for short) if it inherits the estimate (59.6), in the sense that

£(Kt[z]) - £(z) ~ 0 for z C Z and some At < At,,i,, (59.7)

for zero forcing, i.e., f = 0 in Q x I and T = 0 on FT x I. If no restrictions are placedon the critical time step Atcit > 0 or, equivalently, if Atcit = o00, the algorithm is saidto be unconditionally stable; otherwise, the algorithm is said to be only conditionallystable. This is the notion of stability adopted in the remainder of this chapter.

EXAMPLE 59.1. The simplest example of a linear dissipative problem of evolution isprovided by heat conduction in a rigid isotropic solid with linearized heat capacity coand constant conductivity k > 0. The standard governing equations are

azcat = kA[z] + f in x [,

z=0 on a x , (59.8)

zlt=o = z0 in Q,

which is obtained from (59.1) by specifying A[.] as the constant k/co times theLaplacian with zero boundary conditions, and letting Z = L 2 (Q). In this situationA[.] is an unbounded closed operator with domain the Sobolev space Ho' () densein the space Z = L 2 (9). Thus 11-11 is the standard L 2 (2)-norm and the associatedinner-product (-, ) is the standard L2 (9)-pairing of square integrable functions. Thenatural inner product (, )z and the associated natural norm 1I · z for the problemat hand are respectively defined by

(z, Z2)z = (Po zi, z2) and lizliz = Vz,/)z. (59.9)

Obviously, the standard L2-norm 11 11 and natural norm 11- 1iz are equivalent forco > 0. Under the assumption of no forcing term (i.e., f = 0 in Q x II), the a prioriestimate (59.6) for the flow map follows via integration by parts as

dt [lZ = (z, kA[z])- = -klVzlj 2 • -Ckll Il 2 < 0, (59.10)

where C > 0 is the positive constant arising in Poincare's inequality. Thus, the positivefunction £(z) = I Z11 satisfies (59.6) and, therefore, decreases along the flow. It iswell known that problem (59.8) is also contractive relative to the natural (and hencerelative to the standard L2-norm). As an example of a (generally implicit) definition of

SECTION 59 453

the algorithm z, + l = KAt [zn], consider the approximation of (59.8) by the generalizedmid-point rule

co(Zn+ - z,) = AtkA[zn+,] where z,n+ = Oz,+X + (1 - )zn,. (59.11)

Multiplying this expression by (zn + z,+ ) and integrating over Q, a straightforwardmanipulation that exploits the identity z,+o = 2(Zn + zn+L) + ( - 2[Z+- yields

21z+Ilz 11 - -( -- l -tktIlVznol z. (59.12)

Since L(z) = [z112, this result implies that (z,,+) - (zn) < 0 for any At > 0 if0 > and k > 0. Therefore, the concept of stability embodied by inequality (59.7)reproduces the well-known unconditional stability result for the generalized mid-pointrule when applied to the simple linear problem (59.8).

Considerations on the stability of algorithms for the first-order system (59.1) aregenerally made without reference to the forcing term f (). In the linear regime,for instance, the classical concept of A-stability for an algorithm is a test on thehomogeneous problem z' = z, with A e C, known as Dahlquist's model equation(see Chapter II). For dissipative systems, considerations on the effect of the forcingterm lead to questions concerning the long-term behavior of both the dynamical systemand the algorithm, as the following representative example illustrates.

EXAMPLE 59.2. Consider again the classical problem of heat conduction, now withoutintroducing the assumption of a vanishing forcing term, i.e., for f f 0. The objectiveis to examine the effect of the forcing term on the dynamical system under the simplesthypothesis that f is time-independent. Following the same steps that were leading to(59.10) one arrives at

d( z)= (z, ) z -k1 IVz2 + (f,z)z -CkIlzl + f,z)zl. (59.13)dt Z Z

The term in (59.13) involving the forcing term f can be estimated using Young'sinequality as

| Z) Z I< IfIIZ IzI 2£<, 2 (59.14)

for any > 0. Setting = Ck and inserting the result into (59.13) yields

d L(z) + Ck L(z) < fllk (59.15)dt 2Ck


SECTION 59 The coupled thermomechanical problem 455

By application of the classical Gronwall lemma, see, e.g., TEMAM [1979], one arrivesthe result

L(z) < exp[-Ckt]£(Zo) + 2 (1 - exp[-Ckt]). (59.16)

This inequality implies, in particular, the following estimate governing the long-termbehavior of the dynamical system for a given (time-independent) forcing function:

lim sup[C(z)] < [fllz/(Ck)]2. (59.17)

A similar estimate holds for nonlinear dissipative dynamical systems far more gen-eral than the classical model of heat conduction considered here, the incompressibleNavier-Stokes equations being an specific example, see TEMAM [1979] and referencestherein. A natural question to be asked is whether a given algorithm IKt [-] inherits thelong-term dissipative property embodied in the long-term estimate (59.17). To answerthis question consider again as a model algorithm the generalized mid-point rule, withstability properties derived in the preceding example. Following the same steps asleading to (59.12) one arrives at the identity

( +,) - r(z) -- (oL -½)lz,+ - ll -AtklJVz+o1

+ At(f, zn+)z. (59.18)

Assume that the algorithm is unconditionally stable, so that 9 I Then the firstterm in (59.18) is negative while the second term in (59.18) can be estimated usingPoincare's inequality as -lVzn+ll < -C llz+2Z. The last term in (59.18) isestimated using Young's inequality as

(f Zn+9)3 IZ((fzn+)zI < + 2 2 (59.19)

for any £ > 0. Setting e = Ck and inserting the result into (59.18) yields the followingalgorithmic counterpart of inequality (59.15):

(Zn+l) - L(zn) + AtCkL(zn+o) t AtlIlk (59.20)2Ck

To proceed further, one uses an analog of the classical Gronwall's lemma, which isobtained by estimating the term £(zn+,) using Young's inequality (with = 2) asfollows. First observe that

(Z,+9) ) 092C(Z+, 1) + (1 - ) 2L(Zn) -0(1 - O)(Z Z,n+l)zl

> 2£ (Zn+l) + (1 - -)2(z,) - (1 - ) [(Zn+1) + ,C(zn)]

= 0(20 - l)L(zn+l) - (1 - )(29 - 1)Z(zn). (59.21)

Next, by inserting this result into (59.20) one obtains the recurrence relation

At IfI [L(zn+,) 4 A(At)L(z.) + 1 + AXt 2Ck (59.22a)

where A(At) is a is a norm-amplification factor defined as

1 + 2At = 1 + AtCk(2 - 1)0,A(At) = with (59.22b)

1 + (1At w2 = 1 + AtCk(21 - 1)(1 - ).

Recursive application of (59.22a) and use of the well-known formula for the sumof the terms in an arithmetic progression then yields the result

At IlI 1 - [A(At)](9(z) < [A(At) (z) + 1 + At 2Ck 1 - A(At) (59.23)

which furnishes the algorithmic counterpart of relation (59.16) implied by Gronwall'slemma. If (59.23) is to have a limit for fixed At as n -- o, the norm amplificationfactor A(At) must satisfy the condition IA(At) I < 1 which implies that [A(At)]" - 0as n o. A straightforward computation gives the At-independent estimate

limn sup[L(zn)] 2C2k2(20 if (t) < 1, (59.24)2C2 2 (29 - 1) if

which completely characterizes the long-term behavior of the algorithm for a fixed(time-independent) forcing function f and all possible initial data z0o. The conditionil(At)l < 1 is easily seen to imply the strict inequality > , which is more re-

strictive than the condition 9 > 2 arising solely from stability considerations. In fact,for the value = 2 corresponding to the second-order accurate Crank-Nicholsonscheme, the long-term estimate (59.24) blows up. Remarkably, by comparing (59.24)with (59.10) we conclude that the algorithm reproduces the long-term a priori esti-mate of the dynamics for the value L9 = 1 which corresponds to the backward Eulermethod. The preceding considerations can be shown to generalize to arbitrary dissi-pative dynamical systems and are motivated by the analysis for the incompressibleNavier-Stokes equations presented in SIMO and ARMERO [1993]. Issues related tothe long-term behavior of the algorithmic dynamics will not be pursued further hereand the analysis of coupled problems described below will be restricted to standardstability and accuracy considerations.

60. Monolithic and staggered schemes: Product formula algorithms

With the preceding background in hand we are in the position of addressing the mainissues arising in time integration of the initial boundary-value problem for dynamicthermoplasticity. The first step is to cast the coupled problem into a first-order systemof the form (59.1). Once this is accomplished, the time integration of the coupled



problem is typically performed via two alternative class of algorithms referred toas monolithic and staggered schemes, respectively. The main characteristic of thesetwo methods are summarized below. The reader is referred to LEWIS and SCHREFLER[1987] and WOOD [1990] for an overview of some representative schemes currentlyused in coupled problems arising in a wide range of engineering applications.

(A) Monolithic schemes. These methods are obtained by applying a suitable time in-tegrator to the full first-order system (59.1), such as the the generalized mid-point rulealgorithm considered in the preceding example. For coupled thermoplasticity, the treat-ment of algebraic constraints introduced by the Kuhn-Tucker conditions necessitatesan additional projection scheme leading to the return mapping algorithms described inChapter III. This approach lends itself to a relatively easy development of algorithmspossessing unconditional stability, by adaptation of classical schemes for systems of or-dinary differential equations. However, for coupled problems such as thermoplasticity,monolithic schemes do not take advantage of the different time scales often involvedin this class of problems. Moreover, the linearization of the algebraic problem resultingfrom a combined spatial and temporal discretizations typically leads to large nonsym-metric matrices that add considerably to the total computational cost. It is for thisreason that monolithic schemes are not widely used in the numerical solution of cou-pled problems, in spite of their good stability properties. A fairly complete treatment ofmonolithic schemes for coupled thermoplasticity, including the design of mixed finiteelement methods and the closed-form expressions arising from the exact linearizationof the nonlinear algebraic problem, can be found in SIMO and MIEHE [1992].

(B) Staggered schemes. This class of algorithms is designed to exploit the differenttime scales often involved in a coupled problem via partitioning of the associatedfirst-order system into two or more sub-problems. For coupled thermoplasticity thispartitioning of the problem is accomplished by considering a mechanical phase, typ-ically at fixed temperature, and taking the result as initial conditions of a subsequentthermal phase, performed at a fixed configuration, that defines the temperature field.Strategies of this type will be examined in detail in the next section and are stronglyreminiscent of fractional step methods designed on the basis of a global operator splitof the first-order coupled problem of evolution. To set the stage for the discussion ofthese methods, we sketch below the basic strategy for the abstract problem of evolu-tion (59.1). For a detailed exposition of fractional step methods the reader is referredto the review article of MARCHUK [1990] and references therein.

The key idea is to introduce the additive operator split A[-] = A(')[.] + A( 2) [.]of the vector field A[-] that defines the first-order problem of evolution (59.1), whereA(') [] and A(2) [] are two (hopefully simpler) vector fields, and consider followingtwo sub-problems:

Problem 1: -z(., t) =A() [z(, t)].(60.1)

Problem 2: dz(., t) = A(2) [z(, t).dt

SECTION 60 457

The critical restriction on the design of the operator split is that each of the subprob-lems must preserve the underlying dissipative structure of the original problem. Inother words, if IFta) [] denotes the semiflow associated with the vector field A(a) [.],(c = 1,2), then the following estimates must hold:

dL(IFt )[z]) < 0 for t C I[ and a = 1,2. (60.2)

Now consider algorithms IK() [] consistent with the flows F? ) [-], a = 1,2, and stablein the sense that each algorithm obeys the a priori stability estimate (59.7). Then, thealgorithm KAt[.] defined by the product formula

Kat[-] = (2) t ,)[A], in Z x [tn, t+l], (60.3)

is also consistent and dissipative stable. Formally, the stability of the product formula(60.3) in the sense of the a priori estimate (59.7) is easily concluded from the stabilityof each algorithm, merely by noting that

(KAIt[Z]) - L(z) {(KAt [At [z]] ) (KAt [])

+- 2{ (K(it)[z]) - (Z)} < 0, (60.4)

since neither of the two terms within braces in this identity is nonnegative as a resultof the assumed stability on each of the algorithms. Under appropriate technical condi-tions, consistency and stability of the algorithm defined by (60.3) implies convergenceof the iterated algorithm as the time step At -- O0; i.e.,

lim [K(2) o K[]k k- - ltZtoo A t A [Zn]

= Ft[z,], for At = (t - t,,)/k, with t e [ta, t,+l]. (60.5)

Expression (60.5) is a classical result going back to Lie, see KATO [1976]. Nonlinearversions of this formula have been given by a number of authors, in particular BREZIS[1973] and MARSDEN [1974] within the context of a product formula for the numericalsolution of the incompressible Navier-Stokes equations due to CHORIN [1969].

REMARK 60.1. For quasicontractive semigroups, if each of the algorithms is uncon-ditionally B-stable in the sense of (59.4), then the product algorithm (60.3) is alsounconditionally B-stable in the sense of (59.4). This property follows immediatelyfrom the inequalities

n [n [] | | -KAt at n At 6At

<_ ||K(1 [z,,] Bat1 [.]lZ 1Zn -lz il (60.6)

which are implied by the assumed B-stability of the algorithms.



It should be noted that the product formula algorithm defined by (60.3) is onlyfirst-order accurate, even if each of the underlying algorithms is second- or higher-order accurate. A second-order accurate product formula of two (at least) second-orderaccurate algorithms is given by the modified expression

ast = (t) 2 K(2) o K( 1) (60.7)At/2 At At/2

which goes back to STRANG [1969] and is known as the double pass technique (or"Strang's trick"). A simple example illustrates this result.

EXAMPLE 60.1. Let A E S+ (n) be a symmetric, positive definite matrix and considerthe standard initial value problem

dz(t) = Az(t) with z(0) = zo In. (60.8)dt

For this elementary example it follows that Z = REn. The exact solution to problem(60.8) is given in closed-form via the the matrix exponential as

z(t) = Ft[zo] = exp[-At]zo. (60.9)

The problem is obviously dissipative, with (z) = zTAz and natural norm thematrix norm IlzI2I = 2£(z). Since Z is finite-dimensional, the equivalence betweenthis norm and any other matrix norm follows by standard results in linear algebra. Let

A = Al + A2, with A 1,A 2 cE $+(n), (60.10)

be any decomposition of the matrix A into two positive definite, symmetric matrices.Such a decomposition defines an operator split of problem (60.8) since the dissipativeproperty relative to the natural norm is preserved by the additive partition (60.10). Theexact solution of each sub-problem is again given by the matrix exponential formula(60.9). Accordingly, the algorithms defined by

K(?)[z] = exp[-A~At]z, for z E Rn and c = 1,2, (60.11)

furnish the exact solution to each of the sub-problems in (60.1). Let Tat denote thelocal truncation error associated with a specified product formula algorithm IKat[[],defined by

Tat [z,] = exp[-(Al + A 2 )At] zn - KAt [z,(],

SECTION 60 459

(60.12)

where z, E R is the prescribed initial data in a typical time step [t, tn + At].Assume that the matrices Al and A2 do not commute, so that the matrix commutator[Al, A2] = A 1A 2 - A 2A1 7 0, and consider the following two product algorithms:

(i) One-pass product formula. This algorithm is defined by the conventional ex-

pression KAt[zn] =K K(I o IK( ) [zn]. Making use of the standard series expansionfor the matrix exponential, an elementary computation yields

Tt[Zn] = At[A 1, A2][z,] + O(At3 ). (60.13)

This result demonstrates that the standard product formula is only first-order accurate,even if each of the sub-algorithms is exact, as in the present elementary example.

(ii) Two-pass product formula. This algorithm is defined by expression (60.7). Us-ing again the series expansion for the matrix exponential, a similar computation revealsthat the modified formula (60.7) is designed precisely as to cancel the second-orderterm involving the matrix commutator in the conventional formula (60.13). Sincethe calculation involves only terms up to order O(At2 ) in each of the algorithms

K(t) [ ], a = 1, 2, second-order accuracy of both algorithms suffices for this conclu-sion to remain valid. For instance, the product formula (60.7) with algorithms nowdefined by the Crank-Nicholson approximation

K(t)[Z] = (I - tA) - (I + AtAa)z, ac = 1,2, (60.14)

is also second-order accurate, while the conventional formula is not.

The preceding considerations generalize immediately to multiple operator splits ofthe form A[.] = rN=I A() [.]. From a computational standpoint, the success of thetechnique relies critically on the choice of the specific operator split on the basis ofwhich the fractional step method is constructed. This choice is problem-dependentand nearly always motivated by physical considerations, as illustrated below withinthe context of a classical model problem.

61. Model problem: The coupled system of linearized thermoelasticity

This section provides an illustration of the ideas introduced above within the con-text of a concrete model problem, the coupled system of linearized thermoelasticity,for which a complete mathematical theory currently exists going back to DAFERMOS[1976]. Specifically, the goals are the construction of an operator split and the de-sign of a product formula algorithm which render an unconditionally stable staggeredalgorithm for this coupled problem of evolution. Conceptually, the results describedbelow can be formally generalized to the significantly more complex system of cou-pled thermoplasticity at finite strains. We shall do so in the next section by exploitingthe insight gained in the analysis of the present model problem.

The linearized system of thermoelasticity is obtained by coupling the hyperbolicsystem of linear elastodynamics with the parabolic system of transient heat conduction,via two terms: The linearized structural elastic heating and the thermally induced



stress field arising from thermal expansion. The resulting first-order system can beobtained by linearization of the nonlinear theory described in the preceding sections,with internal variables set to FP 1 and , _ 0, a = 1,..., nint, i.e., from thequasilinear system of nonlinear thermoelasticity. The linearized conservation laws(56.10), together with the elastic constitutive equations and Fourier's law of heatconduction, yield the following first-order system (see, e.g., CARLSON [1972]):

i =v,

poi = div[CVu - AOm] + b in 2 x l, (61.1a)

co = div[kV] -m Vv + r,

where u(., t) E (Ho (2))ndm is the linearized displacement field and 9(., t) L2 (Q)is the relative temperature field. In this statement of the problem po and co are thereference density and the linearized heat capacity of the solid, respectively, C denotesthe linearized elasticity tensor and k is the conductivity tensor, which is assumed tobe symmetric and positive definite. In addition, m is a rank-two symmetric tensorthat couples the displacement and temperature fields via two terms: The linearizedstructural elastic heating 'heat = -m · Vv and the thermal stress field Other = -m.For the isotropic theory, the coupling tensor m is given by the familiar expressionm = ndimal, where c > 0 is the coefficient of thermal expansion. Finally, b andr denote the body force and the heat source term, respectively, both assumed to beconstant in Q x I. For simplicity the following Dirichlet boundary conditions will beassumed throughout the analysis:

'0=0 onOa2x:1 and u=O onaf2x:. (61.lb)

The formulation of the coupled initial boundary value problem is completed by spec-ifying the initial data

91t=o = 90, ult= = u0 and vlt=o = v0o in 2. (61.1c)

In keeping with the notation introduced in the preceding section, we let z ={u, v, V} denote the primary variables and, according to the well-established func-tional setting for this problem (see DAFERMOS [1976]), adopt the function space

Z = Hl(Q)ndi m x L2 (Q)ndim x L2(Q2). (61.2)

The standard Sobolev norm of any z Z will be denoted by lzilz. This normis induced by an inner product denoted by (., .). Recall that Ilzll is defined by thestandard expression

Ilzll2 = J [vl 2 + Vu 2 + Pl2] df?, (61.3)

SECTION 61 461

46 2 J. Simo CHPTERimwhere Vu 2 = E1i"'I i,jui,j. It can be easily shown that for the homogeneousproblem, obtained from (61.1a) by setting b = 0 and r = 0, a quadratic approximationto the canonical free energy function introduced in Section 58 gives the functionalL: Z R defined as

(z) = 1 J [POIV12 + Vu. CVu + co02] dQ. (61.4)

The first two terms in (61.4) give the kinetic energy and the elastic stored energy inthe solid, respectively, while the last term in (61.4) agrees with the function £ usedin the analysis of the heat conduction problem in Example 59.1. The natural norm forthe problem is defined as lztlIz = 2(z, for any z E Z. As before, the associatednatural inner product on Z is denoted by (, *)z. Using Korn's inequality, see, e.g.,CIARLET [1988], it is easily shown that 11 · Hlz does in fact define a norm equivalentto the standard Sobolev norm 11- II on Z. This well-known fact implies, in particular,that £(.) > 0 in Z x 11. The analysis below rests on the following result.

THEOREM 61.1. The canonical free energy function defined by (61.4) satisfies thefollowing properties:

(i) For the homogeneous problem, L(.) satisfies the estimate

dt (z) = (z, A[z])z = - VV * kV df < in . (61.5)

It follows that £(.) is a positive function that decreases along the dynamics, thusqualifying as a Lyapunov function for the problem.

(ii) Let t e l - z() C Z, a = 1,2, denote the solutions of (61.1a) correspondingto two initial data z~t ) Z. Then

Z(2) z < (1) z(2 in 1, (61.6)1z( ) -Z(2)I1z < 14ZO>- z

so that the problem is contractive relative to the norm z 2/I.)=(iii) The continuous problem has a unique solution t - z for prescribed initial

data zo e Z.

PROOF. (i) Time differentiation of (61.4) and use of integration by parts together withEq. (61.1a)l and boundary condition (61.1b)1 yields

dt £() : [ v * div[mO9] + 9cor] dQ. (61.7)

Inserting Eq. (61.1a) 2 into this expression gives

dt(z) = f (-v. div[m] - im Vv + 9 div[kVt9]) dQ. (61.8)



Using integration by parts along with boundary condition (61.1b)l results in the can-cellation of the first two terms in (61.8) and produces the a priori estimate (61.5).

(ii) The difference z = z() _ Z(2 ) of the two solutions satisfies the homogeneousversion of the initial boundary value problem (61.1a), with initial data to = z ) -o2)It follows that the flow t E II - z E Z satisfies the a priori estimate (61.5). Integrationin time yields the contractivity result (61.6).

(iii) Suppose that C - z('), ca = 1,2, are two solutions associated with the initialdata z0 G Z. The contractivity property (61.6) then yields 0 < Ilz() - z(2 ) lIz < 0,which implies that z(1) = z(2 ) thus proving uniqueness. El

Consider next the numerical approximation in time of the thermoelastic problem(61.1 a) via algorithms which inherit the contractivity property proved in the precedingtheorem. It is a fairly simple matter to construct a monolithic scheme that preservesthe contractivity property. In fact, in view of the results summarized in Chapter II, anymember of the sub-class of the Runge-Kutta methods characterized by the conditionthat the matrix M be positive semidefinite (see Section 13) is a B-stable time-steppingalgorithm which, therefore, inherits the contractivity property. A typical example isthe generalized mid-point rule considered in Example 59.1, with 9 > . Our objectiveis to construct a staggered scheme, to be interpreted as a product formula algorithm,designed so that the contractivity property (61.6) is preserved.

(A) The conventional isothermal split. The obvious strategy in the design of a stag-gered scheme for the solution of (61.1 la) is to partition the problem into a mechanicalphase, solved at constant temperature, followed by a purely thermal phase solved forfixed displacement and velocity fields. As noted in SIMO and MIEHE [1992], this ap-proach can be formally interpreted as a product formula algorithm based on the splitA[-] = A(s [.] + A2 [], where

V

As)O [z] = -div [CVu - m ]

(61.9)

A[Z] div[kV] - m * Vv

A first observation is that neither Al) [] nor As2o)[] generates a quasicontractivesemigroup in Z. For instance, if F( )[.] is the semigroup associated with AsW) [], acalculation analogous to that used in the proof of Theorem 61.1 reveals that

dt ( [z] ) =(A ([z]' z)dt ISO

(61.10)

SECTION 61 463

=-div[md],v) ~CIll112 VZ(EZ 2

for some constant C, since the inequality 11V9 11L2 • Ml|10L2 does not hold for someconstant M > 0 and V0 C H l ( 2) . The Lumer-Phillips theorem (see, e.g., PAZY[1983]) then implies that Ai(s [] cannot generate a quasicontractive semigroup, thusbreaking the contractive structure of the original problem. In fact, further analysisreveals that each operator does not even generate a semigroup in Z implying thateach problem, when considered separately, is improperly posed.

In spite of this negative result, for the discrete problem a mesh-dependent constantMh = M(h-l) does exist, such that llV9hllL 2 < Mhll 09hlL2 as a consequence of astandard inverse estimate for finite element spaces. This last result implies, therefore,that time-stepping algorithms based on the isothermal split will work in practice fora given mesh-size h (maximum element size), although the nonuniformity of theestimate in h renders the method conditionally stable at best.

The practical implications of this stability restriction will be illustrated within thecontext of a specific model problem. Consider a fractional step method in which eachphase in the product formula is approximated by the (unconditionally stable) Crank-Nicholson scheme, together with a conforming finite element approximation in spaceZ h C Z defined by piece-wise linear polynomials. The resulting product formulaalgorithm is clearly consistent and first-order accurate in time. Sufficient conditionsfor stability are provided by the following result (ARMERO and SIMO [1992]).

THEOREM 61.2. Consider a generalized Crank-Nicholson algorithm associated with(61.1 a), defined by formulae

n+l Un = AVn+l/27

po (Vn+ -Vnh) = At div [CVU+ 1 /2 -_19m], (61.11)

CO(On+ - Vn) = At(div[kVOn+l/2] -m . Vvh+,),

where f E [0, 1] is an algorithmic parameter and the superscript h refers to thediscretized field via a conforming Co-finite element approximation defined by piece-wise linear polynomials. The scheme is stable provided that

(i) The algorithmic parameter is restricted to the value = 1.(ii) The following restrictions on the time step hold

At < 2¥-pc0 CFL = aisohAt 2 (61.12)h ljmllM h -- m

where ai,, is the largest wave speed for the isothermal problem, CFL denotes theCourant number 6 > 0 is a normalized parameter that measures the strengthof the coupling and M is the constant in the inverse estimate 11V9h L2 (Q) <

(M/h) ll hllL2().For the isotropic case, aiso = /(A + 2 l)/po and = m2 /(2/ co), where A, /• > 0are the Lame constants and m = ml with lImllo, = I ml.



PROOF. Sufficient conditions for stability can be derived via the energy method,whereby an algorithmic norm of the numerical solution to the problem is constructedwhich does not increase along the algorithmic flow. Multiplying Eqs. (61.11)2,3 byn+l/2 and 9n+l/2, respectively, and integrating by parts yields the identity

Ch (znh+l ) h (zh)

= -(1 - 3)At[(div [d+1 ,] v n)- 2(div [9h+ / 2], Vh+l)]

- (kV n+l/2, V6n+/ 2), (61.13)

with the function £h: Zh - iR defined by

Lh(zh) = (zh) - ½At(div[mth],vh) Vzh E Zh. (61.14)

For l = 1, the first term on the right-hand side of identity (61.13) vanishes leadingto the estimate h(zh+l) - (zn) < 0 implied by the assumption of positive defi-niteness on k. The property that Lh (.) decreases along the algorithm flow for /3 = 1implies stability of the algorithm, provided that the function h(-) defines a norm,see RICHTMYER and MORTON [1967]. The stability analysis then reduces to estimatinga lower-bound that ensures positive definiteness of the function £h(-). To do so, wefirst estimate an upper bound for the additional term in (61.13) using integration partsand standard inequalities as

(div [hm], vh) <j (m. V0h, vh) < mlVIh{L2lvhL2. (61.15)

Inserting this estimate into (61.14) and making use of Young's inequality yields

Ch(zh) > £(zh) -Atlmll. [(, pllv L) (C<oI I hL)]2 cop 0 0 1

£(zh) _ At llll [PIIv [hI + o0h 2 /], (61.16)

for any constant y > 0. The crucial result used in the proof is the standard inverseestimate IlV9h llL2 < (M/h)[dhllL2 , for some constant M > 0, valid for conforminglinear finite element spaces see CIARLET [1978]. Substituting this inverse estimate intoinequality (61.16) gives

h (zh)> I hVU . CVuh + [1 I2mlI[MAth

x ( PoIvh12 + M Coh12) dQ.

SECTION 61 465

Setting y = M/h > 0, a sufficient condition for positive definiteness of £h(-) isobtained by requiring that the bracket in this expression be positive, which givescondition (ii) in the theorem. D

Observe that the algorithmic norm £h () is not uniformly equivalent to the naturalnorm in Z, even for a fixed value of the Courant number satisfying the stability con-dition (61.12). Thus, strictly speaking, we cannot bound (uniformly) the algorithmicnorm of the solution at some t in terms of the natural norm of the initial conditions.As a result, it is not possible to infer conditional B-stability of the algorithm. Theconstant M can be estimated first for a single element and then maximizing the resultover all the elements. This technique is also used in the derivation of the inverseestimates (JOHNSON [1987]). For the 1D case, the optimal value is /12 (> 1).

REMARK 61.1. Necessary conditions for stability are provided by a classical von Neu-mann analysis of the finite difference stencil associated with the one-dimensional prob-lem, see RICHTMYER and MORTON [1967]. Denoting by E the Young modulus andconsidering, for simplicity, lumped approximations to the mass and capacity matrices,a von Neumann stability analysis yields the following stability condition (ARMEROand SIMO [1992]):

At 2 /poo At 2A 2 P CFL = aiso <- ' (61.17)h jml h e

where ais = V/Epo is the uncoupled or isothermal wave speed and 6 = m 2/(Eco)is the nondimensional parameter that measures the strength of the coupling. In themultidimensional isotropic case, ai,, is to be understood as the isothermal longitudinalwave speed, i.e., ai,, = /(A + 2 p)/po. Note that the adiabatic wave speed aad is givenby aad = a 0(1 + 3). The necessary condition (61.17) should be compared with thesufficient condition (61.12).

REMARK 61.2. Similar stability analyses can be performed for the quasistatic problem,formally obtained by setting po = 0 in (61.1) while keeping co > 0. We assumethat k 0 since, otherwise, the problem becomes rate-independent and reduces toelastostatics with adiabatic elasticities. To give an indication of the restrictions imposedby stability considerations, consider the simplest situation afforded by a single one-dimensional finite element discretization with piece-wise linear polynomials. In thissituation, the stability requirements are easily shown to imply

At m2 - 2Eco 2k Atj 2Ek (S - 2), (61.18)

hZ ~> 2Ek C (2

a condition which is automatically satisfied if 6 2. Thus, for strongly coupledproblems (corresponding to 6 > 2), condition (61.18) is seen to place a restriction ofAt/h 2 reminiscent of the conditional consistency condition arising in certain explicitunconditionally stable methods. An alternative interpretation of this stability conditionarises when the product algorithm in the quasistatic regime is viewed, following the



terminology of ORTEGA and RHEINBOLDT [1970], as a one-pass block Gauss-Seidelmethod for the solution of a (coupled) linear system of equations. It is easily shownthat the condition for convergence of this scheme is precisely given by the result in(61.18). We refer to ARMERO and SIMO [1992] for further details.

The preceding analysis demonstrates that algorithms based on the isothermal split(61.9) are not suitable for strongly coupled problems, since the stability restrictionphrased in terms of the Courant number becomes increasingly restrictive the higherthe strength of the coupling, as measured by the nondimensional parameter . On theother hand, these algorithms become rather useful in the solution of weakly coupledproblems for which the stability conditions derived above do not place severe restric-tions on the time step. Nevertheless, an implicit algorithm possessing only conditionalstability, as in the present case, is always difficult to justify from a computationalstandpoint. We show below that these difficulties can be completely circumvented at,essentially, no additional cost.

(B) The adiabatic operator split. The proof of a priori stability estimate (61.5) inTheorem 61.1 relies on the cancellation of the coupling terms t9m. Vv and v. div[9m]arising as a result of integration by parts, see Eq. (61.8). Preservation of this property,suggests considering an operator split of problem (61.1) in which the coupling terms-div[nm] and -m * Vv appear in the same phase. These considerations lead to thealternative split A[.] = A()[.] + Ad)[], where

* Vv

A( [z]: o div [Q [u] - n],

(61.19)

Ad [z] : 0

a 1div[k V]

Thus, the treatment of the structural elastic heating term Heeat = -m Vv is thekey difference between the conventional split (61.9) and the operator split definedby (61.19). As a result, the temperature field in the mechanical phase of the split(61.19) now changes in such a way that the (linearized) entropy remains constant inthis phase, hence the denomination adiabatic split. From a mathematical standpoint,this seemingly unimportant modification has profound implications. First, A 2) [-] nowgenerates a semigroup (in fact, the analytic semigroup of classical heat conduction)and, as a consequence, A( )[] also generates a semigroup. Second, the semigroups

generated by Al )[-] and Ad2)[.] are contractive relative to the same natural norminduced by £(.). In fact, denoting by t - F(a) [-], a = 1,2, the flows generated in

SECION 61 467

each of the phases, the same calculation used in the proof of Theorem 61.1 now givesthe two estimates

dtlF()[z]) =0 and +d£(F(2)[z]) = - 9 V0 kV,9dQ2 < 0. (61.20)dt d

From result (61.20)1 it follows that the infinitesimal generator A(J) [-] defining themechanical phase produces a semigroup of isometries relative to the natural norminduced by C(). Consequently, no dissipation takes place in the mechanical phase,an obvious conclusion since the total entropy is held constant, while the dissipationin the subsequent thermal phase is precisely the total dissipation Doon generated byheat conduction.

From the results described in the preceding section one concludes that a stablealgorithm KI[at for problem (61.1) can be obtained merely as the product formula 1Kat =KI(2) o K( of two stable algorithms consistent with each of the phases in (61.19). TheAt Atproduct algorithm is B-stable if each of the algorithms is B-stable. Convergence ofthe product formula (in the linear case) is an immediate result of the Lax equivalencetheorem, see RICHTMYER and MORTON [1967].

REMARK 61.3. In principle, the quasistatic case is not encompassed by the estimatesin (61.20). A separate stability analysis confirms the unconditional stability of theproduct formula for the quasistatic case. The reader is referred to ARMERO and SIMO[1992] for the details involved in the proof of this result.

Although both phases in the operator split (61.19) involve the evolution of the tem-perature field, it will be shown below that the actual implementation of the product for-mula reduces the first phase to a purely mechanical problem with modified (adiabatic)elasticities, thus retaining the computational advantages inherent to the isothermal split.For a typical time step [t,, t+l] with prescribed initial data z,, = {Ur, vn, n} e Z,the key idea is to enforce the strong form of the temperature evolution equation inthe mechanical phase, even though the evolution equation for the mechanical field istypically enforced weakly in a standard finite element discretization of the problem.Doing so, the temperature field in the mechanical phase can be solved for explicitlyin terms of the displacement field as

19 = hn - -m V[u - u,,], for (x, t) 2 Q x [tn, t + At], (61.21)

CO

since it = v. Inserting this result into the second evolution equation in the mechan-ical phase, which defines the evolution of the mechanical field, yields the reducedmomentum equation

pot = div (C +-m m VU

+ b- div9nm+ (m m)Vu] , (61.22)1 CO Y Iv 11



which involves only the displacement field. Equations (61.21) and (61.22) comprisethe reduced mechanical phase in the operator split (61.19). The solution of this prob-lem is then taken as initial conditions for the thermal phase in (61.19), which remainsunchanged. Observe that the reduced momentum equation (61.22) is essentially iden-tical to that arising in the mechanical phase of the conventional isothermal split (61.9),with the elasticity tensor C replaced by the adiabatic elasticity tensor C+(1/co)m mand the body force b replaced by the effective body force in (61.22). The same elimi-nation process can be performed if the momentum equation in the mechanical phase iswritten in weak form. A scheme directly formulated on the discretized system arisingin soil consolidations problems, related to the product formula algorithm based on thesplit (61.19), is suggested in ZIENKIEWICZ, PAUL and CHAN [1988].

62. Generalization: A staggered scheme for nonlinear thermoplasticity

The goal of this section is to generalize the product formula algorithm based on anadiabatic operator split of the linearized thermoelastic problem, as described in thepreceding section, to the full nonlinear system of multiplicative plasticity. An entirelyanalogous extension of the product algorithm based on a formal isothermal operatorsplit of the problem of evolution is also possible. However, in view of the inherent sta-bility restrictions alluded to above, and since no additional computational advantagesare to be gained from this class methods, further details on the construction of suchan extension are omitted. The interested reader is referred to SIMO and MIEHE [1992]for a full account of staggered schemes for thermoplasticity based on an isothermalsplit of the problem.

(A) Multiplicative plasticity as a first-order system. The first step in the constructionof the staggered scheme is to cast the initial boundary value problem for multiplicativeplasticity as a first-order problem of evolution. Since the developments in Section 61show that the constant entropy condition plays a key role in the design of the operatorsplit, it is convenient to replace the absolute temperature field 09 by the entropyfunction H of the solid as the primary variable. In view of the decomposition H =He + HP, we replace the set Z of primary variables defined by (58.1) with the set ofconservation/entropy variables Z defined as

Z = {o, p, He, HP} along with r = {FP,,}, (62.1)

where p = poV denotes the material momentum. All the remaining variables in theproblem are defined in terms of Z and r by the kinematic and constitutive relationsderived in Section 57. In particular,

(i) The elastic kinematic variables Fe = DqpFP- I and Ce = FeTFe.(ii) The stress fields P = Fe[23ceE(Ce, He)]F p- T and r = PDp-T.

(iii) The temperature = aHeE(Ce,He) and the nominal heat flux Q =

Q(Ce, o).Whenever any of the variables listed above appears, it will be implicitly understood

that the appropriate relation linking the specific variable to the primary variables

SECTION 62 469

(Z, F) in the problem is employed. With this convention, the evolution equationsfor the internal variables r and the conservation/entropy variables Z can be writtenas follows. Define the nonlinear operators A[.] and G,[.], t = 1,..., m, via theexpressions

DIV[P]-A[ZDIV[Q] + Dmech

A1 Dther (62.2)

G [2, r Fe-I [arf,] D

a qlit A,

Using the kinematic relation FP = Fe-dD Do, which follows from the result inTheorem 45.1 under the assumption of zero plastic spin, the associative flow rule(57.25) gives the local evolution equations

m

r= yGZ,F] in x , (62.3)

with the multiplier 7y > 0 defined either by the Kuhn-Tucker conditions (57.26) inthe rate-independent theory, or by constitutive equations analogous to (57.28) in therate-dependent theory. The mechanical and thermal dissipations in definition (62.2)1are understood to be given by expressions (57.14b) or, equivalently, as

m m

Dmech = Z G,[Z,r] T 0 and Dthle = 9eyZ"af a 0,h=Ll j=1

(62.4)

where we have set T = [pTqi .. qfn,]. With this notation in hand, the first-orderproblem of evolution for multiplicative plasticity takes the form

d=A[Z,r] ins2xa,jjZA[dt AZ[FZ. r ] in x(62.5)

Zt=0 = ZO in Q,

with the internal variables r defined via the local evolution equations (62.3) and sub-ject to the initial conditions Flt=o = {1, O} in Q. In the formulation of the staggeredmethod described below, it is essential to regard the internal variables r as implicitlydefined in terms of the conservation/entropy variables Z via the rate equations (62.3).



Recall that an identical point of view is adopted in the formulation of the return map-ping algorithms described in Chapter IV. Effectively, therefore, the only independentvariables in the problem are the conservation/entropy variables Z.

The stability considerations in the preceding section can be formally extended tothe problem at hand by considering the canonical free energy function for the homo-geneous problem (i.e., B = 0 and R = 0 in S2 x 1[ together with T = 0 on ET x IIand Q on EQ x Il), driven only by initial data now expressed in terms of the conserva-tion/entropy variables Z. This change in variables is accomplished via the Legendretransformation (57.8) and yields the following expression for C£() defined by (58.3):

£(Z )= J [I p2/Po + E(CeHe) + 9l(o,) - OHe] dO. (62.6)

A straightforward computation again verifies that time rate of change of £(o) obeysthe a priori estimate (58.6) along the flow generated by the homogeneous problem.

(B) The adiabatic operator split. Motivated by the strategy used in the design of theadiabatic split for linear thermoelasticity, we consider an operator split with mechanicalphase at fixed total entropy and a thermal phase at fixed configuration. To implementthis idea, we define the vector fields

A(d[,r = - DIV[P]

(62.7)0

AdI)[[ Z.T- { -DIV[Q] + Dmech

D )ther

and consider the following two problems of evolution:

Problem 1: d A= )[ :G], El/1=1

(62.8)

Problem 2: -Z=Ad) [Z, ] =Ea G,[ ,Y]dt

j/=1

Observe that local evolution equations governing plastic flow are appended to bothproblems as a means of defining the internal variables F in terms of the primaryvariable Z in each phase of the split. The mechanical phase defined by Problem 1takes place at constant total entropy since H = He + HP = 0, hence the nameadiabatic split attached to the partition of the problem specified by (62.8). The design

SECTION 62 471

condition Hp = 0 placed on the mechanical phase requires further elaboration. As inthe thermoelastic problem, a change in the temperature field occurs in the mechanicalphase in order to enforce the condition He = 0 of constant structural entropy. Ifplastic flow also takes place, then there exists at least one /lo e {1,.. ,m} suchthat y"0 > 0. As a result, we would have y1 0aef 0 > 0 and, therefore, a nonzeroevolution fHP 0 of the configurational entropy, unless the yield criterion is held atconstant temperature. It follows that the design condition

ftP = 0 A= aDf, = ( = 1,..., m) for Problem 1. (62.9)

In other words, the plastic flow takes place at fixed temperature in the mechanicalphase. This condition becomes critical for preservation of symmetry in the linearized(incremental) problem.

By construction, it is clear the A[] = Al )[ A []). Therefore, according to thegeneral results in Section 59, all that remains is to show that each of the two problemsdefined by (62.7) generate flows that obey the a priori estimate L[Z, F] < 0 for thesplit to be (formally) well-posed. The verification of this property is given in thefollowing result.

THEOREM 62.1. Let t - (Z(a),F(r)), a = 1,2, denote the flows generated by eachof the problems in (62.9). Then

dt )dt 12 r ] =-j smechdQ < 0,

(62.10)

d L [ (2 ), (2 )] = el (2) [D)ch + D(] dQ h 0,

where Dnmech 0 (a = 1, 2) is the mechanical dissipation in Problem a, computed

using expression (62.4)1, and Dn = _Q(2). GRAD[0(2 )]/9(2) > 0 is the dissipationdue to the heat flux Q(2) in the second problem.

PROOF. Since the structural part He is held constant in the mechanical phase, thecanonical free energy £(.) coincides, modulo the constant term O90 He, with the totalenergy in the functional defined by (41.7) for the purely mechanical problem. Result(62.10)1 is therefore a restatement of the energy decay estimate in Theorem 41.1 forthe mechanical problem, since the yield criterion is held at constant temperature as aresult of condition (62.9). Result (62.10)2 is proved by making use of the relation

d [(Ce ,He) + -i() - oH e] V=constant

= -Dmech + ( - 9 0 )He, (62.11)

together with the evolution equation for He in Problem 2 and integration by parts.The computations involved are essentially identical to those in the last part of theproof of Theorem 58.1 and hence are omitted.



An unconditionally stable algorithm KAt for coupled thermoplasticity at finite strainsis therefore formally constructed as the product formula IKAt = (2) o K(1) of two

unconditionally stable algorithms IK(), a = 1,2, consistent with the two problems in(62.7). A brief indication on how this product algorithm is implemented in practice isgiven below.

(C) Implementation of the product formula as a staggered scheme. Concerning al-gorithm KI1) for the mechanical phase (Problem 1), the implementation relies on theobservation that the resulting algorithmic problem is essentially identical to that arisingin the treatment of the purely mechanical theory, as described in Chapter IV. The onlydifference results from the replacement of the stored energy function W(Ce) in thepurely mechanical problem with the internal energy function E(C e, He ) at constantstructural entropy He for the problem at hand. Assume that an explicit expression forE(Ce, He ) is available. The temperature change in the mechanical phase within atypical time step [t,, t, + At] is then computed via the local constitutive equation

O = aHeE(Ce He) He=He, (62.12)

where Hn is the structural entropy at the beginning of the time step. Similarly, thenominal stress P in the mechanical phase is evaluated via the local constitutive equa-tion

P = FSFP -T where S = [aeE(Ce, He)] He=H * (62.13)

The internal variable FP in this expression is expressed in terms of the deformation inthe mechanical phase via the return mapping algorithms described in Chapter IV. TheKirchhoff stress field r appearing in the yield criterion is computed via the standardformula r = FeSFeT. Inserting the resulting expression for the stress field into thealgorithmic version of the weak form (56.18)1 of the momentum equation producesa nonlinear algebraic problem for the deformation field. The iterative solution of thisproblem involves the computation of the linearized weak form of the momentumequations. All the expressions given in Chapter IV remain valid, with the elasticitytensor C = 4C2 )W( C e) now replaced by the rank-four tensor

Cad = [4ae eE(Ce, He) He:He, (62.14)

which defines the adiabatic elasticities of the solid relative to the intermediate con-figuration. The corresponding expression for the adiabatic elasticity tensor cad in thecurrent configuration is obtained via the standard push-forward transformation withthe elastic deformation gradient Fe.

REMARK 62.1. In practice, one is often given the free energy function (C e , 6) inplace of the internal energy function E(C , He). In such a situation, application of

SECTION 62 473

the preceding strategy requires the local solution for the absolute temperature field ofthe (generally nonlinear) constitutive equation

9 (Cle) - H, = 0 to obtain = O(Ce,He)IH=He. (62.15)

With this expression in hand, defined implicitly for the general case via a local it-erative solution scheme, the internal energy function is computed via the Legendretransformation as

E(e, He) = (l e, (e He)) += a(ee He) + e (62.16)

Denoting by Ciso = 4 a2_c(ce l , ) the rank-four tensor of isothermal elasticities,a straightforward application of the chain rule gives the expression

Cad = Ciso + - (2) ( (2e9)l c H (62.17)

for the adiabatic elasticity tensor, where c = -ae9o is the heat capacity of thesolid. As noted in Example 57.1, all the preceding expressions become closed-formunder the assumption of constant heat capacity. In particular, the solution of (62.15)for the temperature field is given by (57.19), while expression (59.15) for the internalenergy is given by (57.20). Finally, expression (62.17) for the adiabatic elasticities istrivially evaluated by noting that 2ae,,t = -2u,,M(Ce).

The thermal phase of the product formula algorithm (Problem 2) is performed byapplying the algorithm KI(2) with initial data defined by the solution of Problem 1. Inview of (62.7) both the deformation and the velocity field remain fixed at the solutioncomputed in the mechanical phase, while the temperature field is updated by solvingthe heat conduction problem in conservation form. To do so, it is necessary to evaluateboth the mechanical dissipation Dmech and the thermal dissipation t)int induced by theevolution of plastic flow. This results in a further update of the internal variables r inthe second phase, performed by applying the return mapping algorithms described inChapter IV. Observe that plastic flow will, in general, take place during the thermalphase of the product formula as a result of thermoplastic softening. A detailed step-by-step description of the implementation of the product formula algorithm is givenin the appendix in ARMERO and SIMO [1993]. The reader is directed to this referencefor additional information.

REMARK 62.2. A noteworthy feature of the product formula algorithm outlined aboveis the possibility of exact preservation of the global conservation laws of momentumfor the pure traction problem (see Theorem 58.2). This is accomplished merely byemploying in the mechanical phase a momentum conserving algorithm of the typedescribed in Chapter IV. Concerning the design of spatial finite element discretiza-tions, all the results described in Chapter IV carry over to the present setting withoutmodification.

474 .C. Simo CHAPTER V


63. Representative numerical simulations

The performance of staggered algorithms based on either the classical isothermal splitor on the proposed adiabatic split is assessed below in three representative simulationswhich closely replicate the stability estimates derived in the preceding sections whileconfirming the excellent performance anticipated for the adiabatic split. The three-dimensional simulations described below are performed with a slightly more generalversion of the model of coupled thermoplasticity labeled as Model II in Section 57.Specifically, the internal energy function is assumed to take the following form:

E( be , 6, He) = K[log J]2 + 4 'l| log be + ndimt9eo0 log J

- co ( - exp [(He - ncdimna log J + Ke ()) /co])

+ [K(, 0o) - OoKe(()], (63.la)

with the temperature hardening potential K((, 09) assumed to be given by

K(I, ) = h(e) 2 + [yo(0) - y ()] H(), (63.1b)

where H(~) = - (1 - exp[-6])/8, and the function Ko(() defined by the closed-from expression:

Ke(6) = h(Oo)Wh~2 + [YO(90)wo - Y (O0)Wh] H(). (63.lc)

Observe that this expression for the free energy incorporates the more generalsituation in which the hardening potential is also temperature-dependent. The modelis completed by considering the standard von Mises yield criterion with isotropichardening, written as

f(-ir, q, 9) = 3ldev[r] + q- (), (63.2a)

with the stress-like hardening variable q defined by the potential relation

q = -{h(O)J + [yo(O) - y.()] (1 - exp[-])}. (63.2b)

For simplicity, attention is restricted to material properties that experience a linearthermal softening, as defined by the relations

yo(O) = yo(Oo) [1 - wo(¢ - o)],

h(O) = h(6o) [1 - wh(O - 0)], (63.2c)

yoo () = y. (Oo) [ - Wh(o - 0)],

where wo and Wh are material constants. In addition, to make the model agree withwidely used formulations of thermoplasticity, the term Dint in the heat conduction

SECTION 63 475

equation has been replaced by a fraction of the total plastic power, as defined by theterm X-r . dP. Here X is a numerical factor estimated in TAYLOR and QUINNEY [1933]as X = 0.9.

(A) D linear thermoelastic vibration problem. The objective of this simulation,taken from ARMERO and SMO [1992], is to demonstrate the unconditional stabilityproperty of the adiabatic split and the conditional stability estimates for the isother-mal split by means of a D linear thermoelastic vibration problem. Consider a one-dimensional problem governed by the equations of linear thermoelasticity written inthe following nondimensional form:

a2u a a2f a a a2w a2uaT2and a- at2- a (63.3a)

02 -- 2 f ar

The dimensionless variables are defined as

cais ca 2 cEai, -=-x, T = t = -- u and =-. (63.3b)k k a cmdo 90

Recall that the dimensionless parameter 6 measures the strength of coupling presentin the problem, and that the restriction on the Courant number are defined in termsof this parameter. The problem is posed in the interval [O, L], with the boundaryconditions:

tU(0, T) = 'u(L, ) = 0 and 09(0, T) = (L, ) = 0, (63.4)

supplemented by the initial data

7t6u((, 0) = 0, v(f, 0) = sin and (f,0) =0. (63.5)

L

The uncoupled or isothermal solution to this problem corresponds to free vibrationin the first mode, with a phase speed of di, = 1 or, equivalently, with a periodTis = 2L. By contrast, the coupled solution exhibits amplitude damping as well asdispersion. It is easily concluded that low wave numbers propagate adiabatically, i.e.,at a phase speed near d2 = 1 + 6, while high wave numbers propagate effectivelyin an isothermal manner. This result is well known (see, e.g., ACHENBACH ([1987],p. 394) or PARKUS ([1976], p. 100)), and implies that the damping induced by thermaleffects has a significant effect only on the high wave numbers. Since the problemunder consideration involves low wave numbers, one expects a decrease of the periodof the order T Tis//I1 + , with a moderate amplitude decay.

The results reported below correspond to simulations performed with the values L =100 and 6 = 1. This value of 6, although leading to a relatively strong coupling for thisclass of problems, is often considered as a suitable choice in the standard literature onthe subject assessing the performance of algorithms. The numerical solution involves

476 J. C. Simo CHAPTER V


CFL = 1

60

40

20

0

-20

-40

-600 50 100 150 200 250 300

Time, r

1.0

0.5

0

-0.5

-1.00 50 100 150 200 250 300

Time, r

0 50 100 150 200 250 300

Time,

0 50 100 '1'"00 250 300

Time, 7

FIG. 63.1. Coupled D vibration problem. Evolution of the displacement at L/2 and the temperature atL/4 obtained with different algorithms for CFL = 1 and CFL = 3 (CFLcrit = 2 for the isothermal split).

Coupling parameter 6 = 1.

a spatial discretization consisting of 100 C° linear finite elements (i.e., h = 1.0) andthree time-stepping strategies:

(i) The adiabatic split, with a Crank-Nicholson scheme for the adiabatic mechan-ical phase and a backward Euler algorithm for the heat conduction phase. Recall thatthe resulting single pass staggered scheme is unconditionally stable.

(ii) The isothermal split, with the algorithm defined by Eqs. (61.11). Recall thatthe resulting staggered algorithm is only conditionally stable.

(iii) A simultaneous solution (or monolithic) scheme obtained by applying a Crank-Nicholson scheme to the full coupled system written in conservation form. Thisscheme is unconditionally stable.

SECTION 63 477

CFL = 3

60

40

20

0

-20

-40

-60

AD ---

SIM - --uncoupl. ----

I, 1 , 1, I , I .

1.0

0.5

0

-0.5

II

-1.0

111

I I Ili lIlllllll ll '

SAD -

... uncoup]. _ iiiiiIII IuncoupI.I

1 111111111111I

Id

Z2

CHAPTER V

20

10

Ar O

1' -10

-20

0 50 100 150 200 250 300

Time, T

FIG. 63.2. Coupled D vibration problem. Accuracy of the adiabatic split (AD) with respect to the monolithicscheme based on a simultaneous solution (SIM). Coupling parameter 6 = 1.

Lumped mass and capacity matrices are used throughout the simulation. Two nu-merical simulations are considered, corresponding to the values CFL = 1 and CFL = 3respectively, which are below and above the critical value CFLcrit = 2 for stabilityof the isothermal split. Figure 63.1 shows the displacement i at the center of thebar = L/2 and the temperature at = L/4 computed in these two cases. Weobserve a good agreement of the three solutions in both displacements and tempera-ture for CFL = 1. On the other hand, while the adiabatic and simultaneous solutionsagree perfectly for CFL = 3, the isothermal split shows disastrous oscillations leadingeventually to overflow. This unstable behavior confirms the conditional stability ofthis split and replicates closely the estimate (61.12). It is interesting to observe thatthe temperature appears to start oscillating first in this unstable computation, thenfollowed by oscillations in the displacements. Note also that the damping and perioddecrease of the stable solutions agree with the comments above.

A more detailed comparison between the solutions obtained with the adiabatic stag-gered algorithm and the monolithic scheme is contained in Fig. 63.2. These resultsdemonstrate the good accuracy exhibited by the adiabatic split. Figure 63.3 shows thespatial distribution of the displacements and temperature at T = Tis/4 = 50 and r =3Ti,/2 = 300 for the three strategies with CFL = 1. The physical dispersion present inthe problem is apparent. Again, a good agreement is found between the three solutions.We remark that results not reported here obtained in a series of numerical tests involv-ing the high wave number range (i.e., wave propagation problems) confirmed also thesame good accuracy and unconditional stability properties of the adiabatic split.

478 J.C. Simo


r = Tji/4 = 50

co-

+=

11

60 80 100

0.6

0.4

0.2

0

-0.2

-0.4

-0.60 20 40 60 80 100

r = 3Tis/2 = 300

0 20 40 60 80 100

0.2

'1 0.1

0

[ -0.1

-0.20 20 40 60 80 100

FIG. 63.3. Coupled ID vibration problem: Spatial distribution of the displacement U and temperature i atT = 50 and T = 300 (CFL = 1). Coupling parameter 6 = 1.

The conditional stability property of the isothermal split is of special concern insituations for which the response is dominated by low wave numbers, as in the presentexample. The reason is that, in the uncoupled case, problems of this type (i.e., struc-tural dynamics) are often solved in practice with implicit algorithms and large valuesof CFL. If the same approach is used in the coupled problem, condition (61.12) maybe violated leading to unstable computations.

(B) Dynamic impact of a thermoplastic cylindrical rod. This example correspondsto the dynamic impact of a three-dimensional bar on a rigid frictionless hot wall.The goal of this problem is to check the performance of the algorithms based on theadiabatic split in this dynamic setting.

40

'; 30

20

i 10

00 20 40

AD-

IS --SIM - - -

uncoupl. ---.

[ , I , I , I . I

2

'- 1

8 O

co

i -1

-2

l l ·l l '

SECTION 63 479

CHAPTER V

FIG. 63.4. Dynamic impact of a cylindrical rod. Discretization of the reference configuration (972 3D mixedfinite element brick).

TABLE 63.1

Dynamic impact of a cylindrical bar: Material properties.

Bulk modulus K 130. GPa

Shear modulus /p 43.3333 GPa

Flow stress Yo 0.40 GPa

Linear hardening h 0.10 GPa

Density Po 8930. kg/m3

Expansion coefficient c 1.0 x 10 5 K-

Conductivity k 45. N/sK

Specific capacity c, 460. m2/s2K

Flow stress softening wo 2.0 x 10 3 K -

Hardening softening wh 0. K-'

The bar considered in the simulation has a length of lo = 32.4 mm and a circularcross section of ro = 3.2 mm. Figure 63.4 shows the reference mesh. A quarter ofthe bar is discretized, with 972 isoparametric 8-node tri-linear bricks with piece-wiseconstant pressure and volume along with a tri-linear interpolation for the displacementfield, as described in Section 45. The model of J2 -flow theory summarized at thebeginning of this section is used in the simulation, with material properties recordedin Table 63.1. The initial velocity of the bar is vo = 0.227 mm//t s along the axis ofthe cylinder. The temperature at the free face is assumed fixed at the reference value090 = 293.15 K, while the wall temperature is i = 00 + 100 K. The lateral face of thebar is assumed to be thermally insulated.

480 J.C. Simo


t = 20 s t = 40 is

TEMPERATURE [K]

1 50E*Cl

2.90E-C14.33E01

7. IE-0 1

8 5E-01

t = 60 us t = 80 s

FIG. 63.5. Dynamic impact of a cylindrical rod. Relative temperature distribution at t = 20, 40, 60, 80 / s.All the deformed configurations are at the same scale. Observe that the wall is hot, at a temperature

100 + o0 K.

Simulations are performed with both the adiabatic and isothermal splits. In bothcases, the dynamical mechanical phase is integrated with the standard Newmark

= , Y = - algorithm, i.e., trapezoidal rule (see SIMO [1991] for a detailed discus-sion of other alternatives) with a lumped mass matrix. The thermal phase is integratedby a backward Euler scheme in both simulations, together with a lumped capacitymatrix. The simulation is carried out for t [0, 80] u s, with equal time incrementsof At = 1.25 ts.

SECTION 63 481


t = 20 I/s t = 40 s

EQ P STN

z_ -__ 1 OE-01

4.8OE-01

1 24Et001.62E00200E+00

t = 60 s t = 80 s

FIG. 63.6. Dynamic impact of a cylindrical rod. Equivalent plastic strain distribution at t =20, 40, 60, 80 p s. All the deformed configurations are at the same scale.

Figure 63.6 shows the temperature distribution as well as the deformed configura-tions at t = 40/ s and t = 80 s, obtained with the adiabatic split. Figure 63.7 showsthe maximum radial displacement versus time for both simulations. We observe aperfect agreement between both simulations, which permits us to conclude that thepresented adiabatic split results in a good numerical accuracy in this dynamic contextas well. We note that the same remarks pointed out in the previous section hold in this

I I

I

I

I


Hot wall

5

*r:L

Cd6r

4

3

2

1

00

Isothermal

20 40 60 80

Time [ps]

FIG. 63.7. Dynamic impact of a cylindrical rod. Equivalent plastic strain distribution at t = 80 s obtainedfor a hot wall and an isothermal simulation (same scale of temperatures as in Fig. 63.6). The lower plot

depicts the maximal radial displacement versus time for these two solutions.

SECTION 63 483


= 11~~~~~ f U~~~~i

R

R

R

fr

K

R

q,=0

sSE

frSr

L

K-F-

7

r

q.= 0

qFIG.= 0--shear banding. Reference configuration and boundary conditions.

FIG. 63.8. Plane strain nearly adiabatic shear banding. Reference configuration and boundary conditions.

case. The isothermal split presents conditional stability in these dynamic simulations;but, as shown in ARMERO and SIMO [1992], the stability condition is inversely propor-tional to the strength of coupling. For these weakly coupled problems, the isothermaland adiabatic splits will then perform very similarly. The superior stability propertiesof the adiabatic split become critical, however, in the presence of a stronger coupling.

(C) Plane strain, nearly adiabatic shear banding. This final example, taken alsofrom ARMERO and SIMO [1993], furnishes the extension to the full thermomechanicalregime of the localization in plane strain of a rectangular bar under uniaxial tensionconsidered in Section 55. In this more general setting, thermoplastic softening is thephysical mechanism that induces response in the material leading to the formation ofshear bands. For high strain rates, this process is nearly adiabatic since heat conductionin the material is precluded by the extremely small characteristic time involved in theloading. A localized generation of heat then takes place at the center of the bar asa consequence of plastic dissipation, resulting in a high temperature rise and a sharpdecrease in the effective value of the yield stress due to the thermal softening. It isthis softening response that triggers the localization of the deformation leading to theformation of shear bands which, for the problem at hand, are oriented at 450 with theaxial direction of loading. This phenomenon is commonly referred to as "adiabatic"shear banding.

E

7

7

I-

�t�U,P I I L

SECTION 63 The coupled thermomechanical problem

TABLE 63.2

Nearly adiabatic shear banding: Material properties.

Bulk modulus Kt 164.206 GPa

Shear modulus A 80.1938 GPa

Flow stress Yo 0.450 GPa

Linear hardening h 0.12924 GPa

Saturation hardening yo 0.715 GPa

Hardening exponent 6 16.93

Density Po 7800. kg/m 3

Expansion coefficient c 1 x 10-

5 K- 1

Conductivity k 45. N/sK

Specific capacity c, 460. m2/s2 K

Flow stress softening wo 2.0 x 103 K-1

Hardening softening wh 2.0 x 10 - 3 K- 1

485

The specimen considered in the numerical simulations has a width of wo =12.826mm, a length of lo = 53.334mm and is subjected to uniaxial loading un-der plane strain conditions. Figure 63.8 shows the mesh of the initial configurationwith the assumed boundary conditions. The bar is assumed insulated along its lateralface, while the temperature is kept constant to the reference value (90 = 293.15Kon the upper and lower faces. Because of the symmetry in the problem, a quarterof the specimen is discretized by imposing the corresponding symmetry boundaryconditions. The finite element mesh consists of 200 4-node isoparametric quadrilat-erals based on the enhanced formulation presented in SIMo and ARMERO [1992].The enhanced strain interpolations are chosen so that, in the linear regime, the one-point quadrature element is recovered. The additional (nonlinear) terms arising in theenhanced formulation provide a stabilization mechanism for the spurious modes asso-ciated with the one-point quadrature element. The model of J2 -flow theory outlinedat the beginning of this section is employed, with material properties summarized inTable 63.2 and chosen to replicate those associated with steel. It should be noted thatthe effect of work hardening is eventually overcome at high strain rates by the (small)thermal softening. As in the preceding simulation, the source term Dint in the energyequation is taken to be a fraction of the total plastic work. Neither geometric normaterial imperfections are introduced in the simulation; the final localized pattern ofthe deformation is triggered solely by the thermal field.

Figure 63.9 shows the final configuration at an imposed top displacement i =5.0 mm. The simulations are performed under quasistatic conditions with a staggeredalgorithm based on the adiabatic split. Two different nominal strain rates are consid-ered:

(a) i/lo = 4 x 102 s-l and (b) I/lo = 4 x 10 4 s.

The first strain rate leads to a diffuse necking mode while the second strain rateproduces sharp shear bands which appear to be well-resolved in the simulation. Fig-

CHAPTER V

TEMPERATURE [K

< 1.000E+01

_ m I

, 6.200E+01

a

TEMPERATURE K

6 OOOE+01

=-~~

> 1.500E+02

b

FIG. 63.9. Plane strain nearly adiabatic shear banding. Distribution of the relative temperature ((9 - (90) andthe equivalent plastic strain at u = 5.0 mm for two different nominal strain rates: (a) /lo = 4 x 10- 2 s - 1,

(b) i/lo = 4 x 104 - 1 . (Notice the different scales between (a) and (b).)

EC, PL ST

< 2.30DE-1

EQ. PL ST.

< 6000E-9

, '.200E+30

486 J.C. Simo

I'

I

i

1. i l I

- ------ 11

-11- ----- 11


lU

8

6

o 4

2

n

I rs

0 1 2 3 4 5

Top displacement [mm]

FIG. 63.10. Plane strain nearly adiabatic shear banding. Plots of the load/displacement curve for two differentnominal strain rates.

TABLE 63.3

Residual norm for a typical time increment(i/lo = 4 x 104 s-l).

Mechanical phase Thermal phase

1.96287 x 10+ 01 2.41033 x 10+°5

1.13417 x 10-02 2.00488 x 10+02

1.08594 x 10- 05 1.48953 x 10- 04

5.09266 x 10-10 6.13528 x 10- 09

ure 63.10 shows the load/displacement curves obtained for these two strain rates.Table 63.3 summarizes the values of the Euclidean norm of the residual, obtainedwithin typical time increment, in an iterative solution procedure employing Newton'smethod. The quadratic rate of convergence exhibited by the iteration is the result of anexact linearization of the two symmetric sub-problems leading to an exact expressionfor the algorithmic tangent moduli.

SECTION 63 487

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