Finite deformation plasticity in principal axes: from a...

Computer methods in applied

moGhaniGs and engineering

ELSEVIER Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

Finite deformation plasticity in principal axes: from a manifold to the euclidean setting

A d n a n I b r a h i m b e g o v i c * , Fadi G h a r z e d d i n e CompiOgne University of Technology, Department GSM, Division MNM, Lab. G2MS, UPRES A 6606 CNRS, BP-20.529,

60205 CompiOgne, France

Received 19 February 1998; revised 1 May 1998

Abstract

This contribution presents a link between early developments in finite deformation plasticity, where the notion of a covariant formulation is introduced and employed, and more recent developments, where rediscovery of a fundamental work of Hill on the method of principal axes led to a very efficient implementation scheme. More precisely, we demonstrate how to develop a covariant theory of finite deformation plasticity in an invariant form, by making use of the elastic principal stretches. We also show how to implement principal axis formulation in the framework of manifold, to carry out all the necessary manipulations by exploiting the Lie derivative formalism and eventually to simplify the final result to the Euclidean setting.

Much of our work on numerical implementation reflects the fruitful cross-fertilization of ideas with those from theoretical formulation. In particular, we show how the operator split method, which is typically used to s~mplify the plastic flow computation, can also be used to reduce the computational cost related to the special finite element interpolation schemes based on incompatible modes. The latter proves to be an indispensable ingredient lbr accommodating the near-incompressibility constraint arising in the finite deformation deviatoric plasticity. An important advantage of the proposed formulation as opposed to alternative remedies (e.g. B-bar method) is that the basic structure of the governing equations need not be modified. © 1999 Elsevier Science S.A. All rights reserved.

I. Introduction

The main objective of this work is to fill the gap between initial works on numerical implementation of finite deformation plasticity on a manifold (see e.g. [1-3] and more recent works using the principal axis representation, which is restricted to the Euclidean setting (see e.g, [4-8], among others).

Initial works on numerical implementation of finite deformation plasticity (see [1-3]) have indicated that the most appropriate setting for exploiting the full generality of multiplicative decomposition of the deformation gradient (see e.g. [9]) and developing a consistent numerical implementation procedure, is a differential manifold ~. The latter offers a clear geometric interpretation of the role of the operator split method in separating the computation of a new deformed configuration (i.e. computation of the total displacements and total deformations) from the computation of a new intermediate configuration (i.e. computation of the current values of the internal variables which control the plastic flow). A differential manifold also provides a general framework in which one can construct a consistent formulation of the finite deformation plasticity which is independent of a particular choice of the reference configuration, yet referred as covariant formulation (see e.g. [10]). Finally, it circumvents (see e.g. [11]) many conflicting issues in finite deformation plasticity (see e.g.

* Corresponding author.

A differential manifold embodies the idea of allowing general coordinate systems and sufficient continuity of the mappings, (see e.g. ll0]).

0045-7825/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 9 8 ) 0 0 2 1 5 - 1

342 A. lbrahimbegovi~, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

[12]), including the most notable one pertinent to the question of the appropriate stress rates, by simply appealing to the notion of the Lie derivative.

In developing an invariant representation of the finite defi~rmation plasticity on a manifold for the case of interest of isotropic elastic response 2, the initial works (see e.g. [1-3]) have invariably resorted to using the invariants of the chosen strain measures. The potential developments in terms of the principal values of the chosen strain measures, or elastic principal stretches were not attempted, judging (see e.g. [10, p. 220]) that they might be too complicated.

Having shown in our previous work (see [11]) that one can indeed provide a very sound formulation of a covariant theory for finite deformation plasticity in the space of principal axes, we hope to show in this work that the same methodology leads to a very efficient numerical implementation. The latter has also been recognized in several recent works on numerical implementation of the finite deformation plasticity (see e.g. [4-8] among others), where it has been observed that a judicious choice of the logarithmic strain measure simplifies drastically the internal variable computation, reducing the latter to the procedure which is fully equivalent to the one used in the case of infinitesimal kinematics. All these recent works follow the fundamental work of Hill 114] and rely crucially on the Euclidean setting.

In this work we show that the same desirable simplification of the internal variable computation can be obtained when departing from a more general finite plasticity tbrmulation set on a manifold. When working with manifolds, one needs to include the metric tensor in the list of state variables, and thus use a general linear eigenvalue problem to define the principle axes (see [I 1]). Whenever possible (e.g. 3d, plane strain and axisymmetric problems), such an eigenvalue problem is simplified to the standard one, by choosing a particular coordinate representation of the metric tensor given as the unit-matrix 3 and recovering some of the recent results on finite defornlation plasticity obtained in the Euclidean setting (see e.g. [4-8]).

However, recovering some of the results recently obtained in the Euclidean setting in an alternative manner, departing from a more general setting of manifolds, is not our only goal. More importantly, we show that the proposed general framework of manifolds offers the only appropriate geometric interpretation which facilitates the development of the method of incompatible modes (see e.g. [16,40]) in finite strain regime. The latter is used as a very effective remedy to eliminate the locking problems, associated with the incompressibility constraint on the plastic deformation field. More precisely, the variational basis of the method of incompatible modes for finite deformation plasticity is developed in a manifold setting, its geometric interpretation is discussed and the simplifications ~tppropriate for the Euclidean setting are then carried out. The feature of a special interest for the proposed implementation of the incompatible mode method relates to exploiting the operator split procedure to simplify the implementation and significantly reduces the secondary storage requirements. The particular implementation of the method of incompatible modes has much in common with the operator split procedure proposed for finite plasticity implementation. This should not come as a surprise, since both problems are of similar natures; Namely, they are both the evolution problems in a constrained configuration space: the constraint to be imposed in plasticity pertains to the plastic admissibility of state variables (in the sense of yield criterion), whereas the constraint to be imposed in the incompatible mode method is pertinent to the satisfaction of the momentum balance equation associated with the variation of the incompatible mode parameters. Similarity of two problems extends further since in both cases imposing the constraint is reduced to a local computation, over a numerical integration point in plasticity, or over a single element in incompatible mode method.

The outline oil" the paper is as follows. In the next section, we briefly recall the fundamental ingredients of the finite deformation plasticity model in principal axes. For a more detailed discussion we refer to Ibrahimbegovic [11]. In Section 3, we discuss the numerical implementation details of the internal variable computations. The method of incompatible modes for finite deformation plasticity is described in Section 4. In Section 5, we present and discuss several illustrative numerical simulations. Some closing remarks are given in Section 6.

2 As noted by Drucker [13], the isotropy of the elastic response is quite a realistic assumption for most polycrystalline metals and alloys. 3 In our previous work (see [15]), we have presented the numerical implementation of finite deformation plasticity for the cases where no

simplification to the Euclidean setting can be obtained (e.g. space-curved membranes or shells).

A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369 343

2. Formulat ion of finite deformation plasticity

2.1. Phenomenological model in terms o f spatial strain measures

Following the fundamental works of Lee [9] and Mandel [18], in the case when both elastic and plastic deformations are of unrestricted size, one can assume that the multiplicative decomposition of the deformation gradient, F.

F = F ~ F p , (1)

gives rise to a stress-free intermediate configuration 4, with F e and F p referred to, respectively, as the elastic and the plastic deformation gradient. See Fig. 1.

Among different possibilities to formulate the theory (see Fig. 1 or Ibrahimbegovic [11 ] for a more detailed discussion), we choose the one employing the spatial strain measure, i.e. the strain measures defined in the current configuration 5. For example, one choice is the left Cauc]hy-Green elastic deformation tensor, b e, which is defined as

b e = F e G JF ~T . (2)

where G is the metric tensor in the intermediate configuration. In the absence of plastic deformation, with F--= F ~, we recover from (2) the left Cauchy-Green total deformation tensor, b = F G - aFT, which represents the corresponding transformation of the metric tensor in the initial configuration, G. Comparing such a result with the metric tensor in the current configuration, g, we can get a usual choice for the spatial strain measure in finite elasticity, the Finger deformation tensor % = ½(g-J - b ) , e.g. see [19]. Hence, the metric tensor g ought to be included among the state variables. Finally, in the case of developed plastic deformation, one should also take into account possible strain-hardening effects. For simplicity, we consider only isotropic hardening, which can be handled by a single, scalar, strain-like internal variable, ~.

In conclusion, b c, g and s c are the state variables of the pre,;ent phenomenological model. We demonstrate

T B , T B ~

cpc \ i / F p F e CI Tt~

Fig. 1. Multiplicative decomposition of deformation gradient. The tensor pair featuring in the general linear eigenvalue problem for elastic principal stretches is identified in each configuration.

4 Note that F e- ~ is the deformation gradient that releases elastically the stress on the neighborhood ~o(0(x)) in the current configuration. Multiplicative decomposition of the deformation gradient is defined only point-wise, so that the intermediate 'configuration' does not necessarily represent a collection of compatible neighborhoods; However, the latter presents no inconvenience to the finite element solution procedure, due to the use of the numerical integration.

5 However, the path-dependent nature of plasticity still requires that these spatial objects remain functions of the material coordinates, or, in other words be given in material description (see e.g. [19]).

344 A. Ibrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

below how to obtain the corresponding evolution equations for these state variables, and the constitutive equations for the remaining, dependent variables.

We assume e, xistence of the strain energy function defined in terms of state variables

~, = ~(b e, g, so), (3)

The invariance requirements under rigid body motion superposed on the spatial configuration, along with the isotropy requirements in the reference configuration, restrict the acceptable forms of the strain energy to those given in terms ,of invariants of the tensors b e and g. Rather convenient form (see [11]) is obtained with using

elastic principa] stretches A~, i.e.

= ¢,(A, ~, so). (4)

The elastic principal stretches are the solution to the following eigenvalue problem

b e e . 2 - I - ( a i ) g ] ~ : 0 . (5)

The eigenvalue problem in (5) above is set naturally in terms of contravariant tensors, so that the principal vectors ~ are covariant vectors or one-forms (see [10], p. 49). An arbitrary coordinate representation of (5) leads to a general linear eigenvalue problem for the matrix pencil of two symmetric matrices [20]. Principal stretches A~ remain independent of a particular choice of coordinates for the eigenvalue problem in (5) (see e.g. [21, p. 24]). Moreover, the fact that any coordinate representation of g leads to a positive definite matrix guarantees (see e.g. [20, p. 307]) that all A7 are real. We also recall the orthogonality property of the eigenvectors u, as

' := 4 i , ( 6 ) ~ - g ~

where 6ij is the Kronecker symbol. The orthogonality property of eigenvectors in (6) plays an important role in the developments to follow.

Standard thermodynamics considerations directly lead to tile constitutive equations for the Kirchhoff stress tensor, ~.6. Namely, in context of purely mechanical theory the dissipation is defined as

D := 7 . d - - ~ ~b(b e, g, sc)~>0 (7)

where d is the rate-of-deformation tensor. In the present framework of manifold, the latter is computed by the application of the Lie derivative formalism (see e.g. [10]) as follows:

1 d = ~L~,g

1 1 0 = ~ F - ~ [ F T g F I F j

1 - ( g + g l + l X g )

2 (8)

where 1 = F F ~ is the spatial velocity gradient. Although we will eventually target the applications which can be handled within the standard Euclidean

framework (e.g. 3d, plane strain or axisymmetric problems), it proves convenient to retain at this level the general setting of manifolds (with an arbitrary metric tensor g), which allows us to use the Lie derivative formalism. However, we can still take advantage of the forthcoming simplification by assuming at the outset that the time derivative of the metric tensors vanish, i.e. ¢~ = 0 and g = 0. The latter allows that the Lie derivative of these tensors cart be replaced by their autonomous Lie derivative (see e.g. [10]), which leads to the following expression for the rate-of-deformation tensor d

The Kirchhoff stress is defined as via the true or Cauchy stress o', through ~r = Jtr, where J = det F.

A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999)341-369 345

1 d = ~ (gl + 1Tg)

1 -- 2 ~ v g (9)

The material time derivative of the strain energy function in (7), can also be computed by the application of the Lie derivative formalism to get

D--~cb(bC, g , ~ ) = ~ " L v [ b C ] + 2 ~ g ' d + - ~ ( (10)

where the Lie derivative of b e can be written as

0 Zv[b e] = F 07 [F - ' b e F -T]FT

=b e - l b e - b e l T (11)

By making use of the multiplicative decomposition of the deformation gradient in (1) and the definition of the left Cauchy-Green elastic deformation tensor in (2), the last expression can be written in an alternative form as

0 Lo[be]=F ~7[FP-ld-IFP T]FT

= F C P IFT (12)

where C p = F rrt~F p is the right Cauchy-Green plastic deformation tensor. Considering that the elastic process is non-dissipative, with t. ~p= 0, ~ = 0 and ~ = 0, from (7) we get

0g, r = 2 - ~ - g ~,, , (13)

as a generalization of the Doyle-Ericksen formula (see e.g. [22] or [10]) to finite deformation plasticity. Isotropy implies that the Kirchhoff stress tensor r shares the same eigenvectors g found in the eigenvalue

problem in (5), so that we can write

[ r - r , g - 1 ] ~ = 0 ~ r = ~ ~ g - I ~ i @ g - l ~ i , (14) i

where r,. are the principal values of the Kirchhoff stress. One can readily compute an explicit form of the constitutive equation in the principal axis representation. To

that end, we first compute (see Appendix A) the directional derivative of the eigenvalue problem statement in (5) to get

ag - 2 g g ® g g (15)

Next, by using the chain rule and the last result, we can rewrite (13) in the principal axis representation as

~, 0¢, -1 r = a~ -~,~ g ®g- ' v~ (16)

Comparing the last result with the one in (14), we conclude that

0¢, r~ = a~ 0h 7 (17)

Therefore, the principal axis methodology leads to a very efficient computational framework, since the tensor calculus can be reduced to manipulation of scalars (i.e. the con:esponding principal values).

REMARK 1: Key result and an alternative form of constitu,tive equation. In order to obtain the plastic dissipation inequality, one makes use of the following key result (see [11])

346 A. lbrahimbegovic, F. Gharzeddine I Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

a~, a~o e- ab e - g -~g b J

It is interesting to note that by using the key result in (13), we can also obtain an alternative form of the stress constitutive equation as

r = 2 g I 0~e be,

Simplifying the last expression to the Euclidean setting, with unit-matrix coordinate representation of the metric tensor, we can recover the previous result of Simo [8], obtained by using the Euclidean setting at the outset.

Introducing the stress constitutive equations in (13) into the dissipation inequality in (7), we obtain its reduced form in terms of so-called plastic dissipation

1 gLo[be]b e-~) + q ~ > 0 (18) ~ : = r . ( - ~ -

where q is the flux, which is energy-conjugate to the hardening variable (, i.e.

q = - 0~p/0~ c (19)

With the principle of maximum plastic dissipation (see e.g. [23]), we can establish convexity of the yield surface and define all the remaining ingredients of the theory. Namely, among all admissible values of r and q, for which the yield criterion is satisfied, such that

~b := ~b(z, g, q) = &(z~, q) ~< 0. (20)

we select the .ones which maximize the plastic dissipation in (27). The latter leads to the Kuhn-Tucker optimality conditions (see e.g. [24, p. 314] or [25, p. 724]), containing the following evolution equations

0~b.e Lvb ° = - 2 yg -~ -~z t, , (21)

a4, (= Y Oq (22)

and the loading/unloading conditions

"~>0 ; ,~bO',g,q)~<0; 5,~b=0. (23)

REMARK 2: Flow rule principal axis representation. Evolution equation for the elastic deformation Cauchy- Green tensor can be rewritten explicitly in the principal axis representation. To that end, we first note that computing the directional derivative of the eigenvalue problem in (14), we get

0R

By making use of the chain rule and the last result, from (21) we can get

e 2 0~/~ - 1 Lo[b ~ ] = - 2 ( Z (A,) -~r g g ® g - ' g

i

2.2. Model problem: Quadratic strain energy form and isochoric plastic flow

The constitutive theory discussed in the previous section is valid for any form of the strain energy ~h and yield criterion (b, defined in the principal axis space. In this section, we consider a simple model problem where both functions take quadratic forms.

The yield criterion which is used in model problem is the one of J-flow theory, which is valid for majority of metals (e.g. see [13]), This criterion takes a very simple form in the principal axis representation given as


~b(~, q ) : = (r2~ + r22 + r~ - r l r 2 - r2r 3 - r3r ,) ' /2 _ ?( r s - q) = 0 , (24)

It is interesting to note (see [ 11 ]) that any such pressure-insensitive yield criterion for which Z i(0 q~/0 R) = 0), leads to the exact preservation of plastic volume or plastic incompressibility condition, enforcing that

J P : 1 ~ j --- j e , (25)

where J :=det F, jc = det F c and JP : det F p. In the case when the plastic deformation is much larger than the elastic one, the plastic incompressibility condition dictates near-incompressible deformation patterns, which cannot be well represented by the standard displacement-type finite element interpolations. That is the main reason for constructing in this work the special interpolation schemes, based on the incompatible modes.

Following Lubliner [26], we assume an additive decomposition of the strain energy function in (4) into

0(A~, s c) = ~(a~) + ~(~:) (26)

where ~(s c) provides strain-hardening description. It follows from (19) that

d ~ . q = - --~ =. -k(sC). (27)

The admissible forms of ~(A~) should satisfy so-called poly-convexity conditions 7, inherited from the finite elasticity which can be roughly interpreted as physically reasonable requirements that the extreme strains should be accompanied by the extreme stresses. Although one could argue that the extreme stress in plasticity are precluded by the yield criterion, it is worthy to recall that the yon Mises yield criterion places no restrictions on volumetric elastic strain, so that they could be quite large and the poly-convexity conditions could become an issue. Mathematically, this condition can easily be represented in the principal axis space as

44A~)-+ac if {A~a2a3}--+0 + (28)

~(A~)-->~ if { a ~ a 2 a ; } - + ~

Moreover, the strain energy should also be selected in such a way that the standard form of the isotropic elasticity (Hooke's law) is recovered for very small elastic strains, for ,l~ ~ 1.

One simple form of the strain energy, which indeed reduces to the isotropic elasticity for small elastic strains and satisfies the poly-convexity conditions for very large strains, is given as

1 ~(aT) = ~ a(ln A] + In Aez + In /~;)2 q_ /z((In A;) 2 + (In a2) 2 + (In /~ ; )2) , (29)

where A and /z are Lamd's parameters. In the vast majority of the recent works on numerical implementation of finite deformation plasticity this

model is also employed, either to simplify the plastic flow computation (see e.g. [4-8]), or to obtain an explicit two-dimensional reduction of the plane stress plasticity theory for membrane shells, Ibrahimbegovic [15].

3. Numerical implementation

3.1. The role of the operator split method

In summary of the discussion given in the previous section we can state that the state variables in the proposed model of the finite deformation plasticity are: the metric tensor, g, left Cauchy-Green elastic

7 The poly-convexity conditions in finite elasticity require that the strain energy be a convex function of the deformation gradient, its determinant and its cofactor. They guarantee that the solution existence does not preclude the solution uniqueness, i.e. does not preclude the buckling phenomena (see e.g. [27] or [10, p. 20]).

348 A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

deformation tensor, b e and the strain-like hardening variable, (. For any choice of coordinates, the matrix representation of the metric tensor can be computed as

gi) -- ~ i" ~./ (30)

where ~ is the derivative of the point mapping to deformed configuration. Therefore, in the list of state variables the metric tensor can be replaced by the mapping ~p. Dependent variables, such as the Kirchhoff stress tensor and the stress-like hardening variable, can then be computed from the known values of the state variables.

Central problem of computational plasticity (see e.g. [28])is to trace the time histories of the state variables through an inc]remental sequence, 0, t~, t 2 . . . . . t,, t,,+j . . . . . t m a x . When using a single-step time integration procedure to carry out such a computation, it is enough to focus on advancing the solution over a typical time step h = t,+j - t , . The central problem of computational plasticity can then formally be written as follows:

• Given: ~, = ~p(t,~), b~ = b~(t,,), so,, = ~(t,) and h = t,,+ ] - t,, > 0

*Compute: ~,,+~ = ~p(t,,+~), b ~ ~ t = ,,+1 = b ( .+] ) , ( .+l ((t,,+l) The central problem is well posed, with the evolution equations (21) and (22) along with the momentum

balance equations being at our disposal. The latter can be written as

G(q~,be, ~:,w): = r ' ~ L f , , g d ~ - G ~ x , = O (31)

It is only the presence of the constraint on the admissible values of the state variables, given in terms of loading-unloadiing conditions (23), that complicates the solution procedure and calls for the development of dedicated methods.

The special method used to simplify the central problem of computational plasticity belongs to the class of the operator split methods (see e.g. [29]). The crucial simplification is introduced by dividing the state variable computation; First, the admissibility of the internal state variables is enforced for the given value of the deformed configuration, and then the momentum balance equations are solved for the frozen values of the internal variables, providing eventually a new guess for the deformed configuration 8. A compelling physical interpretation can be given to each of these two phases (see [30,2]): the first phase defines the intermediate configuration at time t,+j, whereas the second phase defines a new iterative guess for the deformed configuration. For the present formulation given in terms of principal axes, the operator split method boils down to the computation of the principal directions in the first phase followed by subsequent plastic flow computation which provides the admissible values of the elastic principal stretches in the second phase.

There is a fundamental difference between these two computational phases which concerns the size of the problem to be solved: the new deformed configuration is established by solving the global set of momentum balance equations in (31), whereas the plastic flow computation is performed with a local set of equations concerning only one numerical integration point and one element at the time. In the remainder of this section we discuss the implementation details for both computational procedures.

3.2. Plastic f low computation f o r a general plasticity model

We first elaborate upon the computation of the internal variables corresponding to a given iterative guess for the deformed configuration at time t,,+ ], defined through the deformation gradient -~+JJT"~ ="v~.+~, ") where '(i) ' is the iteration counter of the global solution procedure.

3.2. 1. Elastic trial step

We start the plastic flow computation by assuming the so-called elastic trial state. In other words, assuming ~b'r<0 (where superscript 'tr ' is used to denote the computed trial value), from the loading/unloading conditions in (23) it follows that 3~ tr= 0. The evolution equations (21) and (22) can be written as

Lv b e := b.e _ lb e _ belT = O, b ~(0) = b, ~, (32)

x Following the fundamental ideas behind the product formulae (see [29]), two computation phases are carried out sequentially, using the last computed values for the initial conditions in the subsequent computation phase.

A. Ibrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369 349

= 0 , ~:(0) = ~:n (33)

The last equation directly implies that

(~r+~ = sc ~ (34)

Exploiting the results in (11) and (12), (32) can be rewritten in an equivalent form

be d p 1FT] p jF ~ = ~ [FC - c~ ; b e(0) = b,] := F.C n . (35)

Considering that the plastic deformation tensor C v = FpT(~F p remains fixed during the elastic trial step, we can obtain the exact solutions to the last equation as

'r =F.+,CO leT+,

= f b ~ f T (36)

In (36) above, with f = F~+~F~ ~ as the relative deformation gradient, the trial value of the elastic strain is simply obtained as the convective transport (or push-forward) of the elastic strain computed at time t,,. The corresponding graphical illustration of the trial state computation is given in Fig. 2.

There are two possibilities to proceed from the computed trial state. The simpler one pertains to the case when - - . (i) the yield criterion is indeed satisfied for the computed values of trial variables. Namely, if ~b'.r+j := q)tg,,.-j,

r~r+,, q~+, )<~ 0, with ~'~._, and (i,~+ ~ computed from the constitutive equations (13) and (19), the elastic trial state constitutes indeed an admissible state and the plastic flow computation is completed with a trivial result. In the opposite case, for ,;b~+ ~ > 0 we need to compute the value of ~) > 0, for which the admissibility of the state variables is reestablished, with ~(g~+~, 7.+~, q . + j ) = 0 .

3.2.2. Return mapping algorithm The computation of the admissible values of the internal variables is performed in the fixed current

configuration, starting with their trial values. In other words, the evolution equations in (21) and (22) can be rewritten as

~_~ e tr b ' e = - 2 T g ' b e , b e ( 0 ) = b . + l (37)

~=Y Oq' ~(0)= .+, (38)

An approximate solution to the last equation can readily be obtained by using the first-order accurate, backward Euler method to get

TB @ ~ Fn=-V~% " ~ ~ TB~ f

T/~.+x

Fig. 2. Geometric interpretation of the elastic trial step at time t,+), with the intermediate configuration computed at t kept fixed.

350 A. Ibrahimbegovic, F. Gharzeddine I Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

sc +, = ~:~+, + y,,+, (39) n + l

where y. + l = h ~. + l > 0. As proposed by Weber and Anand [4], an approximate solution of the evolution equation in (37) can be

• • 9 obtained by the first-order accurate, exponential approx~manon

b e = e x - + l g n + l n + l . + l ~ , ( 4 0 )

with exp|A] = Z;= o l / k ! A k.

Remark 3: Plast ic incompressibi l i ty . One advantage of the exponential approximation to the elastic strain rate equations in (40) is that it preserves the plastic incompressibility condition, which, as established in Section 2.2, holds for any pressure insensitive yield criterion. The latter can easily be shown by exploiting the well-known result (see e.g. [19, p. 227]), that det(exp[A]) = exp(trace[A]), which leads to a modified form of (40) given as

de t (b~+~)=exp trace - 2 y g - l - ~ z det(b. '+,) (41)

Furthermore, by taking into account the orthogonality property of eigenvectors in (6) and the result presented in Remark 2. we get

trace - 2 ~ g -~ ~ --- ( - 2 ~ ) ~ = 0 (42~ i = 1

From the last two results it is readily apparent that the plastic flow does not change the elastic volume.

The most important advantage of using the exponential approximation in (40) is in a significant simplification of the plastic flow computation. Namely, the principal axis representation of the exponential approximation for the elastic strain tensor in (40) can be written as

g21+,E.n+, ®g,;,+,E.,,+, ~ . . . . tr . 2 - , tr - , tr = ~Ai.n+,) g~+j Vi,,,+I ®gn+l ~'i,,+1 (43) i..,., 2T.+, o~ .,+,

i = 1 i ~ l

Appealing to the uniqueness of the spectral decomposition of a tensor (see e.g. [19]), from the last expression we can obtain that

(i) the principal axes for the trial and the final state coincidence,

- | t r - 1 l r - ! - 1

gn+llYi,n+l @gn+l]O'im+l ~-gn+ll~i,n+l @gn+ll~i,n+l (44)

(ii) the final values of the elastic principal stretches are obtained from the corresponding trial ones

e [ ~b ] = ( ~ , t ~ ,2 (45) (ai,,,+~)Zexp 27~+1 ~ ,+1 '~--i.n+l.'

Taking the logarithm on both sides of the last equation, the multiplicative update above reduces to the additive update

~c,+l = ei.,,+~ - T~+I (46) n + l

e e where e~.,,~ = log(,~.,+ ~). Hence, with the judicious choice of the logarithmic strain measure, the plastic flow

An approximate solution of the system of the first-order ODEs in the form y' =Ay, y(O)=y,, can be obtained by the exponential approximation (see 131, p. 100l) as y.+~ = exp[AtAlyn; The solution is exact if A is constant.

computation simplifies to a formally same procedure as the one used in the small deformation plasticity. The equation governing the plastic flow computation can be rewriUen as

I O~bn+ll e+, _ + t o+ ,

r,,+l := / ¢"+' = 0 (47) I

Oq /

Solution to the system in (47) above is presented in [32] for a general nonlinear form of the yield function ~b(r,, q) and of the strain energy function ~b(e~, ~:). In this work we discuss only a simplified procedure, which can be used for the chosen model problem.

Once the plastic flow computation in (47) is complete, we can recover the corresponding values of the elastic strain tensor as

3 - e - e /~,,,+, exp(Ei,,,+,) ~ b .+ , ~ ] . ~ e . 2 - , - , = = ~.a.i,.+,) g.+,~i,.+, ®g,,+,~i,.+, (48)

i=1

The graphical illustration of the plastic flow computation is given in Fig. 3.

3.3. Plastic flow computation for the model problem

TB

Due to a particular form of the strain energy function adopted in the model problem in (29), the matrix containing its second derivatives has constant entries, with

0 e O ~ - : C = ,~ a + 2 N . ( 4 9 ) A a ~ + 2 ~

Similarly, for the chosen form of the yield criterion in (24), one can show that

O~)n + 1 • ,II ( 5 o ) 07" - - : n n + l ~ - n n + l ' n n + l = PO'n+l + tr

where o" is a vector containing the principal values of the Kirchhoff stress,

IU Fn- n

A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369 35l

TB~+I

Fig. 3. Plastic flow computation corresponds to the computation of the intermediate configuration with the deformed configuration being held fixed.

3 5 2 A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

i ~ = 7".,

and P is a projection operator onto the deviatoric space

to

(51)

,[2, e : v - 1 2 - .

- 1 - I

(52)

These resul~ts can be exploited to simplify the set of equations in (47), governing the plastic flow computation,

' e e , t r ..}_ t r

~ ' n + l - - ~ ' n + l ~/n+lnn+li

r = 4'.+, t} = 0 (531) !

1 -~°+~ + ~" + ~'÷~ J

The system of nonlinear algebraic equations in (53) can be solved by the Newton iterative procedure. At kth iteration we make use of the following linearized form of the system

I I I • . . , + , . <,, o

= - + l (54)

0: ~ k,r'.c~k~ ) i Aq 0 \ ' a t l - r | n + |

where k' is the isotropic hardening tangent modulus. By taking into account that I / t r . t r ,+~ :C .n ,+~ =2/.t and t r I r

n,+ 1 :n,+j = 1, the linearized system above can be reduced to a single scalar equation

2 t z + ~ - . . . . +~) za~,+~ .~,,+~ (55)

It can easily be verified that (55) above is the linearized form of the yield criterion, when this one is expressed as a function of the consistency parameter, %+~,

y. +, ) ::= Ildev[~'.'+, ]}l- ~ +, 2# - ~ (~, + k(~(~,, +, )) : 0 (56)

Since the plastic flow computation ought to be carried out at each numerical integration point, replacing the set of nonlinear equations in (53) by a single scalar equation in (56) represents a considerable computational saving. It can Ix; further concluded from (56) that, as long as the hardening model is selected so that -k(~:) is a convex function, the local iterative procedure for the plastic flow computation is guaranteed to converge.

3.3.1. Algorithmic tangent modulus Having completed the plastic flow computation resulting with the converged values of the internal state

variables, one proceeds to computing a new iterative guess for the trial elastic strain. The latter is carried out by solving the mon:tentum balance equations, which is described in the next section. An essential ingredient of such a computation i,; the consistent elastoplastic modulus, which, as shown by Ibrahimbegovic et al. [33], can be obtained from a slight modification of the linearized form in (54),

C J+~,,+~ O~r : n .+ j A r t + l A ~ . + 1

O: ~/2/3 / ~ , - ~ / / - -,~ / . + 1 . _ l k q , , 4 - ] - q . + l . )

where the converged values of the internal variables are denoted with a superposed bar. Making use of the static condensation procedure (see e.g. [34]) from the last expression we obtain the elastoplastic compliance, whose

A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999} 341-369 353

inverse provides the consistent tangent modulus, which confers with the previously obtained results (see e.g. [35,36]).

2/. ,r tr (2/z) 'Yn+l ,r ®n,r ® 1 ) (58) V'~P° '=C _, / n, ,+t@n,,+l ifdev[o.,~+,]l I 1-n.+, +,-~(1 ~ . + l 1 + k , + I 3tx

3.4. Weak form o f the balance equations and its consistent linearization

Having established the admissible values of the internal variables at time t.+], 6~+~ and ~,,+~, such that

Y,,,+11>0; ~b(g .+ j ,~ ,+ l ,0 .+ l )~<0 (59)

where ~,,+l and '~.+l are computed from (13) and (19), respectively, we can write the weak form of the momentum balance equation as a function of the position vector ~n + ~ only, i.e.

f G(qg,, + , , /~ ,e + ], ~,, + , , W) " = ~',+1 " 2 " ~ , ~ , [ g n + , l d ~ ' : 3 - G e x t =-0 ( 6 0 )

In the last equation we employ the autonomous Lie derivative of the metric induced by the virtual displacement field, w, which can be computed as

e=o T . -

~ . , [g , ,+ l ] = F-T,,+I [(Fn+l + eVw) R,,+j(Fo+j + eVw ] F . + I ]

= (V*w) T g.+l +g,,+J V*w (61)

where V*w is the spatial gradient of the virtual displacement field, which can be computed from of the material gradient. Vw. as follows:

V~°w = VwF ~,-~ j (62)

If the weak form of the momentum balance equation in (60) is satisfied, within the prescribed tolerance, we proceed to the next time step. In the opposite, the plastic flow computation must be restarted with a new iterative

(i+1) guess for the deformed configuration, q%+] . The latter follows from a linearized form of the momentum balance equations

to~¢,~+ ~. b,,+,. (,,+ ~, w)l := o r e . + ,. b n + 1 , ~n+ 1, w)

d ~G" (i) (i) -e +d--~E , = o [ (q9"+1 + r U n + t ' b n + l ' ~ n + " W ) ] = O ( 6 3 )

where u~'+~ ~ are incremental displacements, at ith iteration. The second term in the last equation gives rise to the tangent stiffness matrix, which can be written as

;, ;Y 1 O~,,,+j l ~ g f . [ g . + l ] d ~ + ~.[~ew[g.+,[l:- d ~ (64) d .=o [G.+,,.] = ~- ~..[g~+,] • 2 Ogn+ l . 2 -~ r,,+,

where

37 [g.+,] = F2+ v .=o FT -~

= ~.V~uO~ )v V ~_ ~,~ . .+J g ~ + l + g . + l u.+]) (65)

and

~---- ( n + l ,E~ 'wgn+ I n+ 1,~) F . + I n + 1 E =0

_ (V,pu o ) - - , ,,+I)V[(V~w)T g~+ i +g.+jV~W]+[(V~w)Tgn+l +g~+lV~wW~ ti) J Un+] (66)

354 A. lbrahimbegovic. F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

Having solved for the incremental displacement u~+~ ~ from the linearized form in (63), we can then provide a new guess for the deformed configuration with

~i+]) _,) ~i~ (67) ~ n + l : ~ T n + I ~-Un+l

3.5. Consistent elastoplastic tangent modulus

In the linearized form of the momentum balance equations in (63), we have to compute the explicit form of the partial derivative of the stress with respect to the metric tensor, which gives rise to the tangent elastoplastic modulus

D i p , = 2,9÷._~,/Og.+, (68)

The tangent modulus in (68) can be obtained by making use the principal axis representation of the stress tensor in (14), as well as the chain rule

0 7 + e0 ,,+ = O m a l n + I , 1 , 1

3 0 3 a~,n+l -~ -] - t -z =~i= 2 ~ g . + , ~ . . + , ® g . + , ~ . . + , + i:1 ~ 2~..+j Og.+ 1 [gn+lPi.n+l@g.+l~i.n+l] (69)

In the last expre.ssion. D~P.~.,,+I represents the 'material ' part of the elastoplastic tangent modulus, which can be computed by exploiting the results in (15) and (58). to get

i= 2 0 ~ . - ~ g . +, ~/..,+ ] @ g . +, ~.n + 1

3 3

EE. e. , , , _, = (g,,+l ~..+l ®g,,+1 ~. .+l) (70) ~ij. .+J.g,,+lE,.+, @gn+lPi,n+l) @ i = 1 j - : l

The second term in (69) represents the 'geometric ' part of the elastoplastic tangent modulus. After some lengthy computation, discussed in Appendix A, we can arrive at the explicit form of this term

e p ~ :--'---2r' - b e ®b,e+l + 13n+l A-2i.n+l[Ig "~- (g,,+l-I --gn+l~i.n+l-I -I Dg = ®g.,+ 1 ~/.. + 1 ...... +1 ai. . . l {l~,~+, .+l . ", ) i = I , n

- I .q_ e 2 - , 1 ® ( g . + J , - g . , + J , ~ , . + , ® g . + , E . . + , ) ] (A~.+,) [(gn+iPi.n+l@gn+ll~i.n+l)

e e 2 - I b ~ - - I ' E,,,+I) + (1, . . - 4 ( h ~ , . + , ) )(gn+l~i,n+, ® ,,+1 + b . + , ® ( g . + l ~ , . + J ® g . + l . J

- - I 1 - - I @ g n + , ~ i n + , i)@(gn+l~i.n+l @g,,+~ ~,,+~)]} (71)

e 2 e 2 / ~ e 2 e 2 where di, , ,+l=((Aj. .+l) -(a~,, ,+z ) ) ( ( k . , , + l ) - ( a i . , , + l ) ) , with i = l , 2, 3, j = l + m o d ( 3 , i) and k = l +

[[be+l]ijkl : l(be,ik te, il q._hejI ]e,jk ,~ ik jl il il jk mod(3, k) j°, whereas , ~. ,,+j~.,+~ _~. ,+,~, ,+j) and [Ig;;2,] = - ~ t g , , + ~ g . + ] + g .÷ , + g . + , g . + ~ ) .

4. Incompatible mode method for finite deformation plasticity

4.1. Variational basis o f the incompatible mode method

As first shown by [43[, the standard isoparametric interpolation is not a suitable choice for discrete approximation of the isochoric plasticity. Namely, this kind of interpolation is not able to accommodate the quasi-incompressible deformation patterns associated with large plastic deformations and leads to overly stiff response, commonly referred to as the locking behavior. Although initially conceived to improve the accuracy of the isoparametric elements in the bending dominated problems (see e.g. [16]), is also proved capable (see e.g.

m We use the FOIq'TRAN-Iike notation with the function mod(i, j )= i - int(i/j)*j.


[37]) of eliminating this kind of locking problem. All the fore-mentioned developments of the incompatible mode method were carried out in the framework of the small deformation problems. In this work, we extend the method of incompatible modes to the finite strain plasticity, foUowing in the footsteps of the previous works on the subject (see e.g. [38] or [39,40]). To that end, the key assumption to be used is the one on the multiplicative decomposition of the deformation gradient in which one abandons the standard deformation gradient, F.+, = V¢.+,, in favor of an enhanced gradient, if,

i f + , : [I +a.+,]F,,+, (72)

~) is the spatial di,;placement gradient, so that I + a. +, represents In (72) above, I is a unit, two-point tensor, a . +, a spatial deformation gradient superposed on the compatible deformation gradient F,,+,. The graphical interpretation of the multiplicative decomposition is given in Fig. 4.

It was shown in [40], that the spatial deformation gradient a can formally be constructed by making use of the material gradient A of the incompatible displacement field a~, i.e.

a.+, = A, ,+ ,F ,,ll

V, -I : ¢e.+lF,,+ j (73)

Hence. in the subsequent consideration we use enhanced gradient A. +, as one of the state variables. The weak form of the balance equations can now be rewritte.n as

f~ 1 ~ . (i) A~i~ -~ w) ~,+1: [(~7'Pw)Tg,,+I + g . + l V ÷ w ] d ~ - G e x t = 0 t.~,,,+~, . + , . b . + , , ~ . ~ , . "= -~ (74)

where the spatial gradient gradient ft. + j as

of the virtual displacement field is recomputed with the enhanced deformation

~ 1

V~w =VwF.+, (75)

However, in contrast with the standard displacement-based interpolation case, we get an additional equation associated with the variation of the incompatible mode parameters ~', which can be written as

(i) A(i) Ce ( 1 G. ~, ,÷ , , n+~, t , .+ , ,~ .÷j ,~ ' ) :=j~ ~,÷,:-~[(V~')Zg,,+~ + g , , + j V 4 ~ ' ] d ~ = 0 (76)

where V ~ " is the spatial form of the variation of the enhanced displacement gradient given as

V ~" = V~'F,~, (77)

f~(~)

T B " TBn + 1

I + an/) 1

TB,~+I

Fig. 4. Muhiplicative decomposition of the deformation gradient into compatible and an enhanced deformation gradient.

356 A. Ibrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999)341-369

4.2. Discrete approximation of the incompatible mode method

The crucial simplification introduced in this section is the choice of Euclidean coordinates, which leads to a simple coordinate representation of the metric tensor in terms of the unit matrix, i.e.

g.+l ~--)I (78)

Moreover, in the Euclidean framework the total displacement vector, d,,+l, is well defined, and the position vectors in the current configuration can be constructed as

~P.+l = x -~d.+ 1 (79)

where x is the corresponding position vector in the reference configuration. In order to alleviate the notation in the subsequent developments, we will drop the subscript n + 1, still

keeping in mind that all the variables are referred to time t. + ~, In each eleme, nt, ~ ~, the displacement vector is interpolated in the standard isoparametric manner (e.g. see

[28] as

hen

al = U . ( x ) d . (80) a - - I

where el. are the nodal displacements and N.(x) are the element shape functions for a particular choice of element with n~. nodes. The displacement gradient can then be constructed as

"'" _ / O N . ON. ON.\ val o = } ] a~, ® V N . VTN. - \ Ox, ' Ox 2 ' Ox3 / (81)

a = I

As indicated before by Ibrahimbegovic and Frey [39,40], the enhanced displacement gradient can formally be constructed as the gradient of an incompatible displacement field, so that we can write

",,,, __4oK,, oM, oK,,; A = ~ % ® v K ~ , V TK,, \Ox, ' Ox 2 ' Ox~/ (82)

b = l

where M~(x) are incompatible mode shape functions jj and nim is the chosen number of incompatible modes. As noted by Ibrahimbegovic and Frey [39,40], the incompatible mode shape functions must be modified to make them orthogonal in the energy norm to a constant stress field, and thus to ensure the patch test satisfaction. This can be carried out according to (see [37])

K b = Mr, - ~ - f~o M h d~3 (83)

Using the isoparametric interpolations for the virtual displacement field and the variation of the enhanced displacement gradient, results with

hen

(84)

him

= ;®V Kb (85) b = l

where w and g . .,h are, respectively, the virtual displacements and virtual enhanced gradient interpolation parameters, and

v~Kb = f v ~TKb (86)

~] In choosing the compatible and incompatible displacement shape functions, one must ensure that N N M h = O; Typically, functions M~ are polynomials of one degree higher that those in N .


The symmetric part of the last two expressions can be written in a matrix notation as

hen

sym(V~w) l~ = ~ B.(d,A)w.

and

llirn

sym(V'~f)l~ = ~ Gb(d,a)f~ b = l

where B and G can be written, respectively, as

e] ( e l

T V~No) e 2 ( e2

T T ~7 ~/V. ) e3(e3 B ~(d,A) = TV%)+ T TV% e~ (e 2 e e (e l

T T V ~ N a ) _~_ T T V ~/Va) e2 (e3 e3 (e2 T T ~ 7 ~ N ) _] - z T ~ 7 ~ N ) . .e3 (el el (%

and

m T T e I (e, V~M~)

T T e2(e 2 V~Mh)

T T e3(e 3 V~Mh) B (d,A)= T T T T

el(e 2 V'~Mt,) +e2 (e , V Mb) T ~VCM,,)+ 1 TV~M,,) e2(e3 e3(e2 T T T T ~b

e 3 ( e , V¢Mb)+ej(e3 V 'Mh)"

(87)

(88)

(89)

(92)

where e i as the unit base vectors which constitute the unit matrix I = [e I ,e2, e3]. The main advantage of choosing the spatial strain measures is resulting sparse structure of the strain-displacement operators B and G, the same as the one from the linear theory.

With this matrix notation in hand, the weak form of the governing equations in (74) can be restated as

"°' T[f G(d,a, bc,~,w): = A w. e = l L ° = I e

:= A w.(r - = 0 (91) e = l I_a= l

wherefo is the external load vector, ~.T = (Tlj ' r22, r~3, ~'J 2, r23, T~l ) is the vector containing all the components of the Kirchhoff stress tensor in (16), and Ae"~ j is the finite element assembly operator (see e.g. [41]) over all nej elements in the mesh.

Similarly, incompatible mode based residual in (76) can be rewritten as

him F - e T / T - - e - G(d,A,b ,~, ~ ' ) := ~ ~"b Gb(d,A)~d,A,b ,~)d~3

b = l J3~ e

him

:~---" 2 T ~bhh =0 , Ve@[1, nez] (92) b = l

The crucial difference between the last two momentum balance equations concerns the continuity requirements and the resulting finite element discretization. The momentum balance equation in (91) is associated with the variation of displacement field with the nodal values of the interpolation parameters, w , which are shared by all the elements in the neighborhood of node: a. On the other hand, the momentum balance equation in (92) is associated with the variation of the incompatible mode field, constructed for each element


independently. ']'he former momentum balance equations lead to a set of global, whereas the latter to a set of local nonlinear algebraic equations. As described in the following section, a special solution procedure is proposed in order to take advantage of this special structure of the nonlinear equations on hand.

4.3. Operator split in the method of incompatible modes

Exploiting the idea of the operator split (see e.g. [29]), we divide the computation of the discrete momentum balance equations into finding admissible values of the incompatible mode parameters which satisfy the second group of discrete momentum balance equation in (92) for fixed values of the nodal displacements (the best iterative guess),, followed by the subsequent computation of a new iterative value of the nodal displacement vector from satisfying the momentum balance equations in (91). In the subsequent discussion we provide further details of the proposed procedure.

To that end, we assume that d (i) is the best (iterative) value provided for the nodal displacement vector. Keeping this value fixed, we proceed to compute the corresponding incompatible mode parameters which will satisfy the momentum balance equation in (92). Since the latter is a set of nonlinear equations, we need to use an iterative procedure; At a typical iterative sweep, (j), of such a procedure, we make use of the linearized form of the momentum balance equation in (92)

f 1 0i" 1 Lin[G(d": ,a ' i ) , l~C,~, ( ) l :=G(d( ' ) ,AU) , l~e ,~ , ( )]+ ~ ~ ¢ g : 2 ~ g : - ~ ~ng

f l + ~ e 5 5 f n [ S f c g ] : ' ? d N = O " VeE[1,ne~] (93)

where

,J2ng = (V~q(J~)l"g ~/)+ g~i)V~'lq(J~ (94)

and

V~7/= 7n'P - ' (95)

Introducing the same isoparametric interpolations for the incremental values of incompatible modes as those already used for their variations, we get

¢lil n

~/l:~ = ~] 7/,,®Ve/~/h (96) b = l

The discrete approximation of the linearized form of the element-based balance equation can be rewritten in matrix notation as

~ ¢ g : 2 _ ~ _ g : ~ O T g d ~ = ~ ~ ~.v GV(d,a)D~VGh(d,A)d~ql, e a = l b = l

him him

= ~, J,~b r/b (97) a = l h = l

and

f ",m 'qr, f~ 1 3

~ e 2 "Lg,[~; g]: "? d ~ = ~'. ~] ~.v Sab] 3 d~3~,k, ; s.b ~, v ~^ v ¢^ = r,,(e i V M,)(ej V Mh) a = l b = l e i , j=l

nirn nim

Z 2 " T H ~ = b~ 2,~, ~ (98) a = ] h = l

A typical iterative step of this element-based procedure reduces to

A. lbrahirnbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 3 4 1 - 3 6 9 359

)1 (j~ = H(J~,-lh(J) (99)

~]l((j+l) : ~ ( J ) .~- ~ ( J )

The converged values of incompatible mode parameters, satisfying the momentum balance equation in (92), are used to construct an element-wise approximation of the incompatible mode field, dr. Having obtained the incompatible mode values, we proceed to the next sweep of the global iterative procedure in order to provide an improved value of the total displacement field. In that respect we make use of the following linearized forms of the discrete momentum balance equations

Lin[G(d(i), A,/)~, ~, w)] := G(d('),A, b~+,, ~, w)

f~, o41 f l + -~ ~L~Wwg : 2 -~g: -~ ~Wg d ~ + ~ ~LP,,[~..gl : 4 d ~

f , O4l f l + -~ ~wg : 2-~-g : ~- Af, Tg dS~ + ~- ~ ,T[~g] : d ~ = 0 (t00)

and

• (i) - - e (i) - - e Lln[G(d ,A,b ,~ ,~ ' ) :=G(d ,A,b ,~,~)

f 1 0 4 1 f~ 1 + e y 5C¢g : 2 ~g : -~ ~L,(~g d ~ + "~ ~. [~q~¢g] : 4 d ~

f,~ 1 0F + e 2 5(:g: 2-~-g : 5( g,+, d ~

L' + o ~ , [ ~ g l 4 d N = 0 ; g e E [ 1 , n ~ , l (101)

By definition of incompatible mode approximation .,{ it follows that all element-based residuals vanish, i.e.

(i) - - e G(d ,A,b ,~,;~)=O; Ve@[1 ,n~] (102)

Hence, (101) can be used to compute the increments in incompatible mode parameters in terms of incremental displacement. The latter can be rewritten in the discrete approximation, by introducing the isoparametric interpolation of the iterative displacement field

hen

ul:~ = ~ N.(x)u. (103) a = l

which gives

hen

$1a E - I = - [H F]~bU b (104)

and

b = l

The material and geometric parts of matrix H are given, respectively, in (97) and (98). Similarly, the explicit form of matrix F in (104) follows from

a~-I T T . ep e2~Lt~¢g:2~g:2 5 ~ . g d ~ = = = ~'. eGa(d,A)O Bb(d,A)d~:3u b

him tlen

E E "r = ( a l,abUb ( 1 0 5 ) a = l b = l

3 6 0 A. Ibrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

f~ t~im hen f ~ 3 1 T ÷d~= ~ ~ ¢ ~r bl 3 d ~ l u l , r h ~e~-~u[~'G~,r,~ ~ : ' = Z - T ~b ^ T ~b . r,i(e i V M.) (e j V Nh) a = l b = l i , j=l

him hen

Z Z ' T F e = ~ a 2,abUb

a = l b = l (106)

The result in (104) can then be replaced into the linearized form of the global balance equations in (100). in order to eliminate the incompatible mode parameters and to obtain the system of reduced size

~o, I ~ d = r - f ; g - F T H - ' F me=l (107)

where K is defined from the linearized form in (100) as

2 ~ ' , ~ g : 2 ~ g : ~ ~ . g d ~ = ~ Z wV. B T ~ ( d , A ) D e P B b ( d , A ) d ~ u h e a = l b = l e

hen hen

Z 2 T = W a K l , a b l l b a = l h = l

(108)

and

L 1 tlel hen f ~ Z Z T d ~ = w ~ P.bl3 d ~ u b .

a = l b = l e

hen hen £ Z T

= W a g 2 , a b i l b a ~ l b = l

3

Pab Z - T ~p T ~p = ~j(e i V N. l (e j V N~) i , j= 1

(109)

For clarity, 1the complete computation procedure is summarized in Table 1.

T a b l e 1

G l o b a l i te ra t ive p r o c e d u r e f o r f ini te d e f o r m a t i o n p la s t i c i ty

fo r n = 0 , 1 , 2 . . . .

fo r i = 1 , 2 . . . .

' f o r j = 1, 2 . . . .

fo r k = l , 2 . . . .

L [ ~ * ~ , ] = OhiO+ ) , + DO~I~+ ~ , = 0 nex t k---~b~.+~, (,,+l, K,,+]

a "+~'=a ~'' -(IH'./~ ' 'j' ,,+, .+, ,1 h.,+.[a~,#,a~:),

nex t j ---~P~ ,+ ~

f o r k - 1. 2 . . . .

t ^(k, (k, = [6.+,1= ~°+, +D~L', 0 nex t k---~b~.+l, ~ .+ t , ~ .+ l

- d,,+,) - r ib .... ( [ x ]I.I:+,.A.,+,(d . . . . ' ") = L . . . . d ...... ,,+,)

nex t i--+d.+j

nex t n

A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl• Mech. Engrg. 171 (1999) 341-369 361

5. Numerica l s imulat ions

Several numerical simulations are presented in this section in order to illustrate the performance of the proposed solution procedure. The finite element interpolations are based on bilinear shape functions of the standard 4-node isoparametric element (see e.g. [28] or [4l]), enriched by two quadratic, Wilson-type incompatible modes (see e.g. [16] or [37]).

All the numerical simulations are performed by a research ver,;ion of the computer program FEAP, developed by R.L. Taylor at UC Berkeley (see e.g. [41]).

5.1. Sphere under internal pressure

The first test problem consider an elastoplastic sphere under iinternal pressure. This problem is of particular interest for evaluating the performance of the developed element, for it is one of the very few which lends itself to analytic solution (see e.g. [42]). The chosen geometric characteristics of the sphere are: the inner radius a = 5, the outer radius b = 10. The material model is considered as elastic-perfectly plastic, with Young's modulus E = 1000 and the yield stress cr, = 100. Moreover, the material is assumed to be plastically incompressible, with the chosen von Mises form of the plasticity criterion, and elastically nearly-incompressible, with Poisson's ratio p = 0.4999•

Choosing the spherical coordinates, the problem is reduced to one-dimensional and one can construct an analytic solution, providing the limit value of the internal pressure at which the sphere becomes completely plastified (see e.g. [42]). The corresponding numerical results for the spread of plastic zone are obtained by an axisymmetric finite element model for a quarter of the sphere, using the mesh of 10 × 20 finite elements with incompatible modes (see Fig. 5). The analysis is performed first by imposing the pressure and then the displacement. Both sets of results are plotted in Fig. 6. We can see that the numerical solution converges to the analytically computed value of the limit load, which is also indicated in Fig. 6.

5.2. Cylinder under internal pressure

This example is yet another standard test problem (see [ 1 ]) of a n elastoplastic cylinder under internal pressure. Internal and external radii of the cylinder are given, respectively, as a = 10 and b = 20. The material model is considered to be elastic-perfectly plastic, with Young's modulus E = 11 050 and the yield stress o- = 5000.

Fig. 5. FE mesh used in analysis of elastoplastic sphere.

,-n

0~

Ct.

E e -

105 ~ , i i

100

95 . . . . . . . . " : . i ' " ( , . . . .

9 0 " '

85 ' . . .

80 ......... * . . . . . . . . . . . . . . . . . . . .

7 5 - .,

7 0 - ~ : , - Limit load

65 i " Imposed pressure

60 * . - Imposed .displacement

55 . . . . . .

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 c/b, c is the plastic radius

Fig. 6. Spreading of the plastic zone through thickness.


Moreover, the chosen von Mises form of the plasticity criterion implies that the material is plastically incompressible, and the choice of Poisson's ratio v = 0.454 implies that the material is elastically nearly- incompressible. The near-incompressibility constraint leads to rather poor performance of the standard isoparametric tinite element interpolations even in the elastic regime, and the addition of incompatible modes proves to be very helpful. This is confirmed by the results in Fig. 8, presenting for radial displacement distributions obtained for the standard displacement interpolation and the incompatible mode interpolation.

The numerical results presented in Fig. 8 are obtained by an axisymmetric model composed of 32 elements (see Fig. 7). The same model is also used for computing the elastoplastic deformation of the cylinder. The analysis is also carried out with the corresponding plane strain finite element model, constructed for a segment of cylinder. The analysis is carried out for increasing values of the internal pressure, leading, eventually, to the corresponding value of internal radius, a = 85, which is 8.5 times larger than the one in the initial configuration, inducing very large plastic deformations. The distributions of the radial stress component through the thickness, obtained for several different values of internal radius are plotted in Fig. 9, for both axisymmetric and plane strain models. 'We can see a very good agreement for these two sets of results.

5.3. Cook's membrane problem

This example presents an extension to nonlinear regime of a standard test problem for establishing the accuracy of a finite element approximations in capturing the bending dominated response. The test considers the bending of a built-in tapered panel under shear force applied at the free end. We first consider the finite

Fig. 7. FE meshes used in analysis of elastoplastic cylinder-axisymmetric and plane strain cases.

l . . . . . .

z,.. v

5

4

3

2

1 y

0 10

-*- Incompatible Mode Solut ion

" : + : D i s p l a c e m e n t Mbdel Solul ioh

, i , ;

12 14 16 18 20 r

Fig. 8. Radial displacement-elastic nearly-incompressible case.

0

-200 /

~- - 4 0 0 O

/ -600 ,

10 o - 8 0 0 a-~20 . . . . . . . .

- 1 0 0 0

r r -1200 . . . . -*, Axisymmetr ic A n a l y s i s

-1400 , -+:- Plane ~ r a i n An~ys is

-1600 0 2 4 6 8 10

T h i c k n e s s in deformed configurat ion

Fig. 9. Radial stress distribution through thickness-elastoplstic case.


deformation elasticity model with quasi-incompressible behavior, obtained by choosing Young's modulus E = 240.5 and Poisson's ratio ~, = 0.4999. The total value of the applied force is equal to 100. The finite element solution, obtained for uniform mesh grading with different number of elements, is presented in Fig. 10 along with the corresponding results obtained by the same number of standard displacement-based elements. We can see rather a superior performance of the incompatible mode method with respect to standard displacement approximation in handling the near-incompressible and bending dominated deformation patterns.

The same test is then carried out for an elastoplastic material model with isotropic hardening. The chosen material characteristics are : Young's modulus E = 206.9, Poisson's ratio ~, = 0.29, the yield stress o~ = 0.45 and q = (0.715 - o')(1 - e - 1 6 ' 9 3 ' f ) q- 0.12924~:. The total value of the applied force is equal to 5.

The finite element computation is carried out with the same mesh layouts as those used in the elastic case. The resulting force-displacement diagram is plotted in Fig. 11.

5.4. Double edge notched specimen

This test problem was first introduced by [43] in order to demonstrate the presence of near-incompressibility constraint in deviatoric small strain perfect plasticity, which has an important influence on result accuracy in the limit load computations. In this test, we consider a geometrically nonlinear version of this problem, using the perfect plasticity model with yield stress ~ = 0.45, Young's modulus E = 206.9 and Poisson's ratio p = 0.29. The dimensions of the specimen are kept the same as in the original example of [43] with total width w = 10, height h = 30 and ligament thickness b = 2. The plane strain conditions are assumed. The finite element model uses a rather coarse mesh with 5 × 15 elements for a quarter of the specimen.

The finite element analysis is carried out by imposing the displacement increments of 0.01 at the end of the specimen, until the total displacement value of 0.3 is reached. For such a value of displacements, the numerical results obtained by the incompatible mode based approximation indicate that the specimen has reached the limit load, see Fig. 13. As opposed to the geometrically linear case, where the limit load can clearly be observed, the geometrically nonlinear case exhibits a slight amount of softening, which is probably related to geometrically nonlinear effects. The standard displacement-based model is not capable of predicting this limit load, yielding a monotonically increasing curve.

E 6 E u

m

" O

E O O

O I..--

- Icompatible Mode Solution

* QI/E4 [Simo & Armero 1992]

--Displacement Model Solu!ion

Fig. 10. Convergence of nearly-incompressible case.

i i i i i J

5 10 15 20 25 30 35 Elements per side

the finite element solution-elastic

"E 6

E O

" 5

E O o 4 EX O

F.-

/ / - Ic0mpatible Mode So Ut on t

/ * Q1/E4 [Simo & Armero 1992]

2 I i i ~ I I

0 5 10 15 20 25 30 35 Elements per side

Fig. 11 Convergence of the finite element solution-elastoplastic case.


b = 2

- - ' ) c

W = I O

I l l i i i i i J i i i i i i i I I I I I I I l l l l i J i l l

J l l l l l l l l l J I J l l l l l I J I I I J l l l

L=:30

Fig. 12. Double edge notched specimen and FE mesh used.

5.5. Plane strain necking problem

In this test we consider the necking problem of a plane strain rectangular specimen with the length of 53.334 and width of 12.826, subjected to tensile forces at both ends. In a first stage of deformation process the specimen is maintained in an essentially homogeneous deformation state. Subsequently, when the load reaches the maximum value, a diffuse necking starts to develop. For an ideal specimen this is a bifurcation problem, where necking can dew,lop in any cross-section. In order to transfer this bifurcation problem into a limit load problem, we have introduced a small perturbation by means of reducing the specimen width from its initial values at both ends to 98.2% of this value in the middle. The material properties are Young's modulus E = 206.9 and Poisson ratio u = 0.29. We take into account isotropic hardening rule of exponential type, defined according to the expression below,

k ( s c) = (o:~ - ~ ) ( 1 - e - '8~) + Ks c

"O

O

"6 I--

4

3.5

3

2.5

2

1.5

1

0.5

0

......... i i y y J

/

- - I ncompat ib le M o d e Solution

-- Displacement Model SOlution

• i

~ i i i i

0 0.05 0.1 0.15 0.2 0.25 0.3 Top displacement

Fig. 13. Total extensional load vs. deflection.

A. lbrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 17t (1999) 341-369 365

FEA Fig. 14. FE mesh used for localization problem.

" 0

t~5 o

0 I---

10 i

. . . . . . i

i i

. . . . . . . . . . i . . . . . . i ....

i

0 0.5 1 1.5 2 2.5 3 3.5 4 Top displacement

Fig. 15. Load-displacement curve for the enhanced formulation.

The values of the parameters in the equation above are taken as

o-~ = 0 . 7 1 5 , o~ = 0 . 4 5 , K - - 0.12924, fl = 16.93

The finite element model, prepared for a quarter of the specimen only, consists of 200 elements. In the inner fourth of the specimen the mesh is refined, using altogether 100 elements. This mesh refinement helps capture the shear band formation at 45 ° with the axial direction of loading, see Fig. 14.

Fig. 15 shows the load-displacement diagram computed with the present formulation. While the present model captures both the peak and the softening post-peak response, the isoparametric elements produce overly stiff response, which is not shown in the figure.

6. C l o s i n g r e m a r k s

In this work we have addressed both the theoretical basis and the numerical implementation of finite deformation plasticity. The contribution provides a missing link between early developments on the subject, where the importance of a general setting of the manifolds was recognized for constructing an invariant (or covariant) formulation |2, and more recent works restricted to the Euclidean setting, where the principal axis representation is recognized to lead to a very efficient numerical implementation. The main contributions, which deserve a special attention, can be stated as follows:

(i) The key idea facilitating the development of a covariant formulation of finite deformation plasticity on a manifold pertains to the general linear eigenvalue problem, making use of a pair of deformation tensors. In the deformed configuration, the latter consists of the left Cauchy-Green elastic deformation tensor and the metric tensor, which define the corresponding principal axes.

(ii) The general framework of manifold allows us to exploit the Lie derivative formalism when developing all the governing equations in the most direct way, circumventing the issue of the stress and strain rates. These equations are then simplified in the implementation phase by choosing a particular coordinate representation of the metric tensor corresponding to a unit matrix.

~2 Following the terminology of Marsden and Hughes [10], a formulation is covariant if it is not only invariant with respect to a superposed rigid body motion, but also with respect to a superposed diffeomorphism, provided that the metric tensor is accounted for and properly transformed.

3 6 6 A.. Ibrahimbegovic, F. Gharzeddine I Comput. Methods Appl. Mech. Engrg. 171 (1999).341-369

(iii) It is shown that the plastic-incompressibility-induced locking phenomena can be handled by the method of incompatible modes. Cross-fertilization of the ideas from plastic flow and incompatible mode computations led us to develop a special implementation scheme reducing the secondary storage requirements for the incompatible mode method.

(iv) The computational efficiency of the numerical implementation is further reinforced by selecting the spatial strain measure, which leads to a very sparse structure of the strain-displacement arrays, and thus reduces the cost of computing the tangent operator.

Acknowledgments

This work wa;s supported by the Lab G2MS UPRES A 6066 CNRS. FG was supported by a Grant from the French Ministry of Education, Research and Technology, MENRT.

Appendix A. Geometric part of the tangent modulus

As indicated m Section 3, the main advantage of the principal axis formulation of the finite deformation plasticity is in a simple form of the material part of the elastoplastic tangent modulus, which is formally the same as the one obtained in small deformation case. However, the principal axis formulation also requires the geometric part of the tangent modulus, defined as

3 0 - 1 - 1

Dgeo mep =~ ' ,2 r~ ~ [g g ® g ~1 (A.1) i = l

Generalizing the result of [44], one can obtain a principal vector representation for a general linear eigenvalue problem,

1 - I 2 1 2 - 1 = - _ akg ) a j g )g(b g g ® g - ' ~ ~ ( b (g.2)

where d i = ( A ~ -- 3. 2 2 2 j ) ( a i - ak), with i = 1, 2, 3, j = 1 + mod(3, i) and k = 1 + mod(3, j). A couple of anxiliary results are first derived. In particular, the directional derivative of the inverse of metric

tensor can be computed from

- l O g -_____~_l. - l l g g = l ~ 0g d g = - g d g g (A.3)

Making use of tlhis result and the eigenvalue problem in (5), we can obtain the directional derivative of the elastic principal stretch with respect to the metric tensor as

0 O=-~g[~ - a ~ g - ~ ) ~ l . d g

/ /Oai ~ 1 g-t 2 --1 = - 2 a i ~ , - ~ g ' d g ) g ~i+~.~g-ldg v i + ( b - A i g )dr/ (A.4)

Scalar multiplying the last equation by principal vector ~, and exploiting orthogonality of principal vectors in (6), we can obtain

Oai Ai - l Og . d g = - ~ - ( g ' v , ® g a , , ) .dg (A.5)

A. Ibrahimbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369 367

The directional derivative of the principal vectors with respect to the metric tensor can then be obtained from (A.2), by systematic application of the chain rule and the auxiliary results in (A.3) and (A.5) to get

0 _ l ( O d i . ) 1 0 _ _ , . = - -J -1 _ akg )] dg og(g- 'v~®g v~)dg 7 \ O g g_d,(g v~(~g v , ) + ~ 7 ~ g [ ( b a2g-~)g((b - 2 - ' '

(A.6)

which after somewhat lengthy, but straightforward manipulation results with

0 7g g - 1 1 - 1 v~ ®g v~) =-dT Ib dg b - (b . dg)b

2 2 - I - - I - 1 - ' - 1 + a j h k ( ( g - g E ® g l{i)'dg)(g-l-g v~®g g)

+ a~((g-' - g - I e (~g-' v~) • dg)b + (b. dg)(g -l - g - ' e @ g - ' e )

2 2 - g - l v i @ g v , ) - d g ) ( g - ' - g v / ® g - g)] + a~(aj + a k - 3a~)((g -1 - ' , 1 (A.7)

In a special case when two of the principal stretches coincide, say aj = a 2 # A3, the development presented in the foregoing no longer applies, since d, = d 2 = 0. In this case any vector v~, for which v ~ - g - I v 3 = 0 can be considered as the principal vector. Therefore, the geometric part of the tangent modulus can be written as

ep 0 [ - 1 - 1 0 I - 1 D g ~ o m = 2 r l ~ g g - - g v , ® g ' v 3 ] + 2 r 3 ~ g ( g - v3(g)g v3) (A.8)

using the result in (A.3), we can further obtain

ev 0 0 Dg~om'dg=-2r, g - l dg g l + 2(r3-r,)-@g ~g (g ' v a ( ~ g - ' v 3 ) ' d g (A.9)

where the last term can be obtained explicitly from (A.6). Finally, if all three principal stretches happen to be the same, i.e. aj

vector. The Kirchhoff stress tensor can be written as = /~2 = /~3' any vector is a principal

- 1 7 = ~'lg (A.12)

so that the geometric part of the tangent modulus can simply be written as

ep . - 1 D g e o m d e : --2"rig i dg g (A.13)

Appendix B. Glossary of notation

Objects defined on the reference configuration ~3 G covariant metric tensor in the reference (undeformed) configuration C covariant right Cauchy-Green deformation tensor C p covariant right Cauchy-Green plastic deformation ten,;or

Objects defined on the current (deformed) configuration g covariant metric tensor in the current configuration r contravariant Kirchhoff stress tensor b e contravariant left Cauchy-Green elastic deformation tensor v principal directions in the current configuration

368 .4. lbrahirnbegovic, F. Gharzeddine / Comput. Methods Appl. Mech. Engrg. 171 (1999) 341-369

Two point tensors F d e f o r m a t i o n g r a d i e n t ( m i x e d t e n s o r )

FeFp m u l t i p l i c a t i v e d e c o m p o s i t i o n o f F

Scalars e

Ai ri

ei

q

t

31

principal elastic stretches principal stresses logarithmic strains internal variable which controls hardening thermodynamically conjugate variable to s x strain energy function yield function pseudo-time parameter plastic multiplier

Matrix notation

x

~o d w

u

D = Vd

A =Va~

position vector in initial configuration position vector in deformed configuration total displacement vector virtual displacement vector iterative displacement increment displacement gradient enhanced displacement gradient incompatible displacement field incompatible virtual displacement iterative increment in incompatible displacement

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Finite deformation plasticity in principal axes: from a...

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