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A Simple Orthotropic Finite Elasto-Plasticity Model

Based on Generalized Stress-Strain Measures

Jorg Schrodera, Friedrich Gruttmannb & Joachim Lobleina

aInstitut fur Mechanik, FB 10Universitat Essen, 45117 Essen, Universitatsstr. 15, Germany

bInstitut fur Statik, FB 13Technische Universitat Darmstadt, 64289 Darmstadt, Hochschulstr. 1, Germany

Contents

1. Introduction.2. Kinematics and Generalized Stress-Strain Measures.

2.1. Generalized Stress-Strain Measures.2.2. Matrix Representation.

3. Constitutive Framework.3.1. Invariance Conditions.3.2. Representation of Anisotropic Tensor Functions.3.3. Free Energy Function and Related Polynomial Basis.3.4. Yield Criterion and Related Polynomial Basis.3.5. Solution Algorithm for the Set of Constitutive Equations.

4. Variational Formulation and Finite Element Discretization.5. Numerical Examples.

5.1. Necking of a Circular Bar.5.2. Conical Shell.

5.2.1 Isotropic Material Response with Nonlinear Hardening.5.2.2 Isotropic Material Response with Linear Hardening.5.2.3 Anisotropic Material Response with Nonlinear Hardening.

5.3. Cylindrical Cup Drawing from a Circular Blank.6. Conclusions.

References.Appendix A: General Return Algorithm.Appendix B: Matrix Representation of Tangent Moduli.

published in: Computational Mechanics, 30, 48-64 , 2002

March 2002

A Simple Orthotropic Finite Elasto-Plasticity ModelBased on Generalized Stress-Strain Measures

J. Schrodera, F. Gruttmannb, J. Lobleina

aInstitut fur Mechanik, FB 10Universitat Essen, 45117 Essen, Universitatsstr. 15, Germany

bInstitut fur Statik, FB 13Technische Universitat Darmstadt, 64283 Darmstadt, Alexanderstr. 3, Germany

Abstract

In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based ongeneralized stress-strain measures, which reduces for one special case to the so-called Green-Naghdi theory. The main goal is the representation of the governing constitutive equationswithin the invariant theory. Introducing additional argument tensors, the so-called structuraltensors, the anisotropic constitutive equations, especially the free energy function, the yieldcriterion, the stress-response and the flow rule, are represented by scalar-valued and tensor-valued isotropic tensor functions. The proposed model is formulated in terms of generalizedstress-strain measures in order to maintain the simple additive structure of the infinitesimalelasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detailand some representative numerical examples are discussed.

Keywords: Anisotropy, finite elasto-plasticity, generalized measures

1 . Introduction.

The complex mechanical behaviour of elasto-plastic materials at large strains with an ori-ented internal structure can be described with tensor-valued functions in terms of severaltensor variables, usually deformation tensors and additional structural tensors. Generalinvariant forms of the constitutive equations lead to rational strategies for the modellingof the complex anisotropic response functions. Based on representation theorems for ten-sor functions the general forms can be derived and the type and minimal number of thescalar variables entering the constitutive equations can be given. For an introduction tothe invariant formulation of anisotropic constitutive equations based on the concept ofstructural tensors and their representations as isotropic tensor functions see Spencer [26],Boehler [3], Betten [2] and for some specific model problems see also Schroder [19].These invariant forms of the constitutive equations satisfy automatically the symmetryrelations of the considered body. Thus, they are automatically invariant under coordinatetransformations with elements of the material symmetry group. For the representationof the scalar-valued and tensor-valued functions the set of scalar invariants, the integritybases, and the generating set of tensors are required. For detailed representations of scalar-and tensor-valued functions we refer to Wang [31], [32], Smith [23], [24]. The integritybases for polynomial isotropic scalar-valued functions are given by Smith [23] and thegenerating sets for the tensor functions are derived by Spencer [26]. In this work weformulate a model for anisotropic elasto-plasticity at large strains following the line ofPapadopoulos & Lu [16]. Here we use a representation of the free energy function andthe flow rule which fulfill the material symmetry conditions with respect to the referenceconfiguration a priori.

Papadopoulos & Lu [16] proposed a rate-independent finite elasto-plasticity modelwithin the framework of a Green-Naghdi type theory, see e.g. Green & Naghdi [5], us-

J. Schroder, F. Gruttmann & J. Loblein 3

ing a family of generalized stress-strain measures. An extension of this work to anisotropyeffects is given in Papadopoulos & Lu [17], where the algorithmic treatment deals withnine of the twelve common material symmetry groups. Furthermore, the authors developspecial return algorithms for some anisotropy classes. Generalized measures have beenused e.g. by Doyle & Ericksen [4], Seth [20], Hill [8], Ogden [15] and Miehe &Lambrecht [13] in the case of nonlinear elasticity.For an overview of the developments in the theory and numerics of anisotropic materialsat finite strains we refer to the papers published in the special issue of the InternationalJournal of Solids and Structures Vol. 38 (2001), EUROMECH Colloquium 394, and thereferences therein. In the following we discuss only a few contributions in this field. Ayield criterion which describes the plastic flow of orthotropic metals has been first pro-posed by Hill [7]. A numerical study on integration algorithms for the latter modelat small strains and especially the evaluation of iso-error maps is given in De Borst& Feenstra [14]. A constitutive frame for the formulation of large strain anisotropicelasto-plasticity based on the notion of a plastic metric is proposed by Miehe [11]. Aconsistent Eulerian-type constitutive elasto-plasticity theory with general isotropic andkinematic hardening has been developed by Xiao, Bruhns & Meyers [35], combiningthe additive and multiplicative decomposition of the stretch tensor and the deformationgradient. An anisotropic plasticity model at large strains taking into account the postu-late of Il’iushin is proposed by Tsakmakis [29] and specialized for transverse isotropyin Hausler, Schick & Tsakmakis [6]. The main ingredient is the introduction ofan evolution equation for the rotation of the preferred anisotropy directions. A furtherapproach to describing the anisotropic elasto-plastic material behaviour is based on thehomogenization of polycrystalline meso-structures, in this context see Miehe et al. [12]and references therein.

In this paper a theory of finite elasto-plastic strains using the notion of generalized stressand strain measures is presented. With the essential assumption of the additive decompo-sition of the generalized strains the structure of the constitutive equations corresponds tothe linear theory, in this context see also [16], [17]. The model has been implemented ina brick–type shell element of Klinkel [9], see also Wagner et al. [30] and referencestherein. The element provides an interface to three–dimensional material laws; it is im-plemented in the program FEAP [36]. Due to special interpolation techniques based onmixed variational principles, the element is applicable for the numerical analysis of thinstructures. For a comparison of three dimensional continuum elements with shell elementsfor finite plasticity problems see Wriggers, Eberlein & Reese [33]. Furthermore, theinterface of the program ABAQUS [1] is used for an implementation. We discuss three rep-resentative numerical examples: the necking of a circular bar; the load carrying behaviourof a conical shell; and the deep drawing process of a sheet metal plate.

The new aspects and essential features of the formulation are summarized as follows:(i) In our formulation we exploit the fact that the generalized strains and the Green–Lagrange strains are characterized by the same eigenvectors. Using this feature one canextend results of Ogden [15] which have been derived in the context of isotropic elasticity.In [15] elastic stored energies as functions of the principal stretches are considered andfirst and second derivatives are derived.(ii) Based on these features the components of the projection tensors with respect to theprincipal axes are derived. Using the fourth-order and sixth-order transformation tensorsone can evaluate the Second Piola-Kirchhoff stress tensor and associated linearization.

Orthotropic Finite Elasto-Plasticity 4

Furthermore explicit matrix representations are given. The expressions are simple andthus allow a very efficient finite element implementation.(iii) The constitutive equations for orthotropy are formulated in an invariant setting. So–called structural tensors describe the privileged directions of the material. For the yieldcondition with isotropic and kinematic hardening we introduce a simple representationin terms of the invariants of the deviatoric part of the relative stress tensor and of thestructural tensors.(iv) The set of constitutive equations is solved applying a so-called general return method,see e.g. Simo & Hughes [22] and Taylor [27]. Additionally, the condition of plastic in-compressibility is fulfilled by a correction of the inelastic part of the generalized strains.(v) For the numerical examples, where we compare different stress-strain measures, anidentification of all models is performed with respect to a given reference uniaxial tensiontest. Thus all formulations in the generalized measures reflect the same (given) macro-scopic stress-strain characteristics, like linear isotropic hardening. This can be seen as a(minimum) essential feature in order to get physically comparable results for the phe-nomenological quantities. In this context see e.g. the discussions in Hill [8].

2 . Kinematics and Generalized Stress-Strain Measures.

The body of interest in the reference configuration is denoted with B ⊂ IR3, parametrizedin X and the current configuration with S ⊂ IR3, parametrized in x. The nonlineardeformation map ϕt : B → S at time t ∈ IR+ maps points X ∈ B onto points x ∈ S. Thedeformation gradient F is defined by

F (X) := Gradϕt(X) (2.1)

with the Jacobian J(X) := det F (X) > 0. The index notation of F is F aA := ∂xa/∂XA.

An important strain measure, the right Cauchy–Green tensor, is defined by

C := F T F with CAB = F aAF b

Bgab , (2.2)

where gab denotes the coefficients of the covariant metric tensor g in the current configu-ration.

2.1. Generalized Stress-Strain Measures.

Following e.g. Doyle & Ericksen [4], Seth [20], Hill [8] and Ogden [15] we definethe generalized strain measures

E(m) :=

1

2m(Cm − 1) for m 6= 0

1

2ln[C] for m = 0

, (2.3)

where 1 denotes the second-order unit tensor. Let λA, A = 1, 2, 3 be the eigenvalues andNA, A = 1, 2, 3 the eigenvectors of C then we arrive at

Cm =∑3

A=1 λmA NA ⊗NA

ln[C] =∑3

A=1 lnλA NA ⊗NA

. (2.4)


In this context we remark that formulations of isotropic metal elasto-plasticity based onlogarithmic strains are successfully used e.g. in Peric et al. [18]. Furthermore, let thegeneralized stress measure S(m) be the work conjugate to E(m). To simplify the notationwe write S := S(1) for the symmetric Second Piola-Kirchhoff stress tensor and E := E(1)

for the Green-Lagrangian strain tensor. The related work conjugate stress measures aredefined by the stress power density

w = S : E = S(m) : E(m)

= S(m) : 2∂C(E(m)) : E = S(m) : IPE : E . (2.5)

This leads to the transformation rule for the Second Piola-Kirchhoff stress tensor

S = S(m) : IPE with IPE = 2∂CE(m) . (2.6)

e1

e2

N 3

N 1

λm2

e3

λ2

N 2

Figure 1: Visualization of the coaxiality of E and E(m).

For the following derivations we exploit the fact that E(m) and E have the same eigen-vectors. An illustration of this is given in Figure 1. Now, following Ogden [15] we arriveat the explicit expression for the fourth-order transformation tensor

IPE =∑3

A=1

∑3B=1 PAABB NA ⊗NA ⊗NB ⊗NB

+∑3

A=1

∑3B 6=A PABAB (NA ⊗NB)⊗ (NA ⊗NB + NB ⊗NA)

. (2.7)

With the eigenvalues EA = 12(λA − 1) of E and the eigenvalues of the generalized strain

measures

E(m)A :=

1

2m(λm

A − 1) for m 6= 0

1

2ln [λA] for m = 0

, (2.8)

we derive the non-zero components of IPE:

PAABB = ∂EBE

(m)A = λm−1

A δAB

PABAB = γ(m)AB =

E(m)A − E

(m)B

2(EA − EB)=

E(m)A − E

(m)B

λA − λB

for λA 6= λB

1

2λm−1

A for λA = λB

. (2.9)

The result for two equal eigenvalues is obtained applying the rule of l’Hospital, in thiscontext see e.g. Miehe & Lambrecht [13].


2.2. Matrix Representation.

Let ¯S = [S11, S22, S33, S12, S13, S23]T and P be the matrix representations of S = Sijei⊗ej

and IPE, respectively. Thus (2.6)1 and (2.7) lead to

¯S = P T S and P = T T L1T , (2.10)

with the Cartesian components of the generalized stress tensor Sij = ei · S(m)ej orga-nized in a vector S = [S11, S22, S33, 2S12, 2S13, 2S23]

T . The ei|i = 1, 2, 3 denote the fixedCartesian basis with respect to the reference configuration. The matrix T contains thecomponents NA

j = NA · ej of the eigenvectors NA

T =

(N11 )2 (N1

2 )2 (N13 )2 N1

1 N12 N1

1 N13 N1

2 N13

(N21 )2 (N2

2 )2 (N23 )2 N2

1 N22 N2

1 N23 N2

2 N23

(N31 )2 (N3

2 )2 (N33 )2 N3

1 N32 N3

1 N33 N3

2 N33

2N11 N2

1 2N12 N2

2 2N13 N2

3 N11 N2

2 + N12 N2

1 N11 N2

3 + N13 N2

1 N12 N2

3 + N13 N2

2

2N11 N3

1 2N12 N3

2 2N13 N3

3 N11 N3

2 + N12 N3

1 N11 N3

3 + N13 N3

1 N12 N3

3 + N13 N3

2

2N21 N3

1 2N22 N3

2 2N23 N3

3 N21 N3

2 + N22 N3

1 N21 N3

3 + N23 N3

1 N22 N3

3 + N23 N3

2

.(2.11)

The matrix L1 is of diagonal form

L1 = diag[

λm−11 , λm−1

2 , λm−13 , γ

(m)12 , γ

(m)13 , γ

(m)23

], (2.12)

where the components γ(m)AB are defined in (2.9). The compact matrix formulation of IPE,

see (2.10)2, (2.11) and (2.12), provides a very efficient finite element implementation.

3 . Constitutive Framework.

In this section, we point out the main components for a simple finite anisotropic plasticitymodel, see also Papadopoulos & Lu [16, 17]. It consists of an additive decompositionof the generalized strain tensor in “elastic” and plastic parts, with Ee(m) := E(m)−Ep(m).For the calculation of the generalized stresses, the back stresses β and the stress-likeisotropic hardening variable ξ we assume the existence of a free energy function ψ, whichis decoupled additively in an elastic part ψe, a plastic part ψp,i due to isotropic hardeningand ψp,k due to kinematic hardening. The yield criterion Φ is formulated in terms of therelative stresses Σ := S(m) − β and the isotropic hardening stress ξ. For the evolutionof the plastic strains Ep(m) and of the internal variables α, ep(m) we use Φ as a plasticpotential. The loading-unloading conditions in Kuhn-Tucker form complete the model.


The constitutive equations are summarized as follows:

additive split E(m) = Ee(m) + Ep(m)

free energy ψ = ψe(J1, ..., J7) + ψp,i(ep(m)) + ψp,k(α)

generalized stresses S(m) = ∂Ee(m)ψe

back stresses β = ∂αψp,k

isotropic hardening ξ = ∂ep(m)ψp,i

relative stresses Σ = S(m) − β

yield criterion Φ = Φ(Σ, iM , ξ) = Φ(I1, ...I6, ξ)

associative flow rule Ep(m)

= λ∂S(m)Φevolution of α α = −λ∂βΦ

evolution of ep(m) ep(m) =

√2

3||Ep(m)||

optimization condition λ ≥ 0, Φ ≤ 0, λΦ = 0

(3.13)

For the explicit formulation of invariant constitutive equations the representation theo-rems of tensor functions are used. The governing constitutive equations have to representthe material symmetries of the body of interest a priori. Furthermore, the minimal numberof independent scalar variables which has to enter the constitutive expression is required.For a detailed discussion of this topic we refer to Boehler [3]. The structural tensorsiM , the invariants Ji|i = 1, ...7 used in free energy and Ii|i = 1, ...6 entering the yieldcriterion are defined in the following sections.

3.1. Invariance Conditions.

Following (3.13)2 we now focus on hyperelastic materials, i.e. we assume the existence ofa so-called Helmholtz free energy function ψe in terms of the generalized strain measures.Assume ψe to be a function solely in the deformation gradient, i.e. ψe = ψe(F , •). Theargument (•) in the free energy function denotes additional tensor arguments. They char-acterize the class of anisotropy of the material; we discuss this topic in the following sec-tions. We consider perfect elastic materials, that means that the internal dissipation Dint iszero for every admissible process. The principle of material frame indifference requires theinvariance of the constitutive equation under superimposed rigid body motions onto thecurrent configuration, i.e. under the mapping x → Qx the condition ψe(F ) = ψe(QF )holds ∀ Q ∈ SO(3). For the stress response this principle leads to the well known reducedconstitutive equations ψe = ψe(C) which fulfill a priori the principle of material objectiv-ity. In the case of anisotropy we introduce a material symmetry group Gk with respect toa local reference configuration, which characterizes the anisotropy class of the material.The elements of Gk are denoted by the unimodular tensors iQ|i = 1, ...n. The concept ofmaterial symmetry requires that the response be invariant under transformations on thereference configuration with elements of the symmetry group

ψe(FQ) = ψe(F ) ∀ Q ∈ Gk,F . (3.14)

We say that the function ψ is a Gk-invariant function. Without any restrictions weset Gk ⊂ SO(3), where SO(3) characterizes the special orthogonal group, becauseonly invariants of (absolute) second order tensors appear in (3.13), see e.g. Spencer[26]. Based on the mapping X → QT X for arbitrary rotation tensors Q ∈ SO(3)


we have, in view of a coordinate free representation to fulfill the transformation ruleQT S(F , •)Q = S(FQ, •) ∀Q ∈ SO(3) . If we assume the free energy function to be afunction of the generalized strain tensor ψe(E(m)) we obtain

S = 2∂Cψe(Ee(m)) = S(m) : 2∂CEe(m) with S(m) := ∂Ee(m)ψ(Ee(m)) . (3.15)

The invariance requirement with respect to the material symmetry group is then given by

ψe(QT Ee(m)Q) = ψe(Ee(m)) ∀ Q ∈ Gk,Ee(m) . (3.16)

Thus it is clear that material symmetries impose several restrictions on the form of theconstitutive functions of the anisotropic material. In order to work out the explicit restric-tions for the individual symmetry groups or more reasonably to point out general forms ofthe functions which fulfill these restrictions it is necessary to use representation theoremsfor anisotropic tensor functions. Similar arguments are used for the construction of thecoordinate invariant yield condition Φ = Φ(Σ, ξ), see below.

3.2. Representation of Anisotropic Tensor Functions.

In order to construct an isotropic tensor function for the anisotropic constitutive behaviourwe have to extend the Gk-invariant functions to functions which are invariant under thespecial orthogonal group. For this purpose we introduce the so-called structural tensors,which reflect the symmetry group of the considered material. The symmetry group of amaterial is defined by (3.14). Here we consider orthotropic material which can be char-acterized by three symmetry planes, where the anisotropy can be described by somesecond-order tensors iM |i = 1, 2, 3 defined with respect to the reference configuration.Let Gk be the invariance group of the structural tensors, i.e

Gk = Q ∈ SO(3), QT iMQ = iM for i = 1, 2, 3 . (3.17)

Thus the invariance group of the structural tensors characterizes the class of anisotropy.Figure 2 illustrates the preferred directions ia and ia := F ia with respect to the referenceand current configuration, respectively.

NX ⊂ B Nx ⊂ S

1a

2a

3a

x

2a F

3a

1a

X

Figure 2: Preferred directions ia and ia in a neighborhood P of the material point X andx defined with respect to the reference B and the actual configuration S, respectively.


With this definition we arrive at a further reduction of the constitutive equation of theform

ψ = ψ(Ee(m), iM |i = 1, 2, 3) = ψ(QT Ee(m)Q, QT iMQ|i = 1, 2, 3) ∀ Q ∈ SO(3) .(3.18)

This is the definition of an isotropic scalar-valued tensor function in the arguments(Ee(m), 1M , 2M , 3M), which fulfills the above postulated transformation rule for thestresses. It should be noted that the function is anisotropic with respect to Ee(m).

Remark: There are further material symmetries which are finite sub–groups of SO(3),for the different crystal classes, see e.g. Smith, Smith & Rivlin[25], Spencer [26] andthe references therein.

3.3. Free Energy Function and Related Polynomial Basis.

The material symmetry group of the considered orthotropic material is defined by

Go := ±I; S1,S2,S3 , (3.19)

where S1,S2,S3 are the reflections with respect to the basis planes (2a, 3a), (3a, 1a) and(1a, 2a), respectively. Here, (1a, 2a, 3a) represents an orthonormal privileged frame. Basedon this, we obtain for this symmetry group the three structural tensors

1M := 1a⊗ 1a, 2M := 2a⊗ 2a and 3M := 3a⊗ 3a , (3.20)

which represent the orthotropic material symmetry. Due to the fact that the sum of thethree structural tensors yields

∑3i=1

iM = 1 we may discard 3M from the set of structuraltensors (3.20). So the integrity basis is given by

P := J1, ...J7 . (3.21)

The invariants (J1, J2, J3) are defined by the traces of powers of E(m), i.e.

J1 := trEe(m) J2 := tr[(Ee(m))2], J3 := tr[(Ee(m))3] . (3.22)

The irreducible mixed invariants are given by

J4 := tr[1MEe(m)], J5 := tr[1M(Ee(m))2]

J6 := tr[2MEe(m)], J7 := tr[2M(Ee(m))2]

(3.23)

see e.g. Spencer [26]. For the free energy function we assume a quadratic form:

ψe = 12λJ2

1 + µJ2 + 12α1J

24 + 1

2α2J

26 + 2α3J5 + 2α4J7 + α5J4J1 + α6J6J1 + α7J4J6 . (3.24)

The generalized stresses appear with (3.15)2 in the form

S(m) = λJ11 + 2µEe(m) + α1J41M + α2J6

2M + 2α3(Ee(m)1M + 1MEe(m))

+2α4(Ee(m)2M + 2MEe(m)) + α5(J1

1M + J41)

+α6(J12M + J61) + α7(J4

2M + J61M)

.

(3.25)


The second derivative of ψe yields in this special case the constant generalized fourth-orderelasticity tensor

Ce(m) = λ1⊗ 1 + 2µII + α11M ⊗ 1M + α2

2M ⊗ 2M

+2α3IK1 + 2α4IK2 + α5(1M ⊗ 1 + 1⊗ 1M)

+α6(2M ⊗ 1 + 1⊗ 2M) + α7(

2M ⊗ 1M + 1M ⊗ 2M )

(3.26)

with IIIJKL = δIKδJL, IK1IJKL = δIK

1MJL + δJL1MIK and IK2

IJKL = δIK2MJL + δJL

2MIK .The elasticity parameters (λ, µ, αi|i = 1, ..7) can be identified using the matrix notation

S(m)11

S(m)22

S(m)33

S(m)12

S(m)13

S(m)23

=

C11 C12 C13 0 0 0C12 C22 C23 0 0 0C13 C23 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66

Ee(m)11

Ee(m)22

Ee(m)33

2Ee(m)12

2Ee(m)13

2Ee(m)23

, (3.27)

with the elasticity constants Cij. Choosing the preferred directions as 1a = (1, 0, 0)T and2a = (0, 1, 0)T we obtain the material parameters

λ = C33 + 2(C44 − C55 − C66)µ = C55 + C66 − C44

α1 = C11 + C33 − 4C55 − 2C13

α2 = C22 + C33 − 4C66 − 2C23

α3 = C44 − C66

α4 = C44 − C55

α5 = C13 − C33 − 2(C44 − C55 − C66)α6 = C23 − C33 − 2(C44 − C55 − C66)

α7 = C12 − C13 − C23 + C33 + 2(C44 − C55 − C66)

(3.28)

in the invariant setting. In case of isotropy the only remaining constants are λ and µ,which can be directly determined from Young’s modulus E and Poisson’s ratio ν.

3.4 Yield Criterion and Related Polynomial Basis.

In the following, we consider an orthotropic yield condition using isotropic tensor func-tions. We assume that the plastic yield condition should not be influenced by the hydro-static pressure. Thus, the integrity basis for the argument tensor devΣ and the structuraltensors 1M and 2M are given by

I1 := tr [(devΣ)2], I2 := tr [1M(devΣ)2], I3 := tr [2M(devΣ)2]

I4 := tr [1MdevΣ], I5 := tr [2MdevΣ], I6 := tr [(devΣ)3]

. (3.29)

The anisotropic flow criterion is formulated as an isotropic tensor function, i.e.

Φ(devΣ, 1M , 2M ) = Φ(QT devΣQ,QT 1MQ, QT 2MQ) ∀ Q ∈ SO(3) . (3.30)


The quadratic flow criterion function Φ = Φ(I1, I2, I3, I4, I5, ξ) ≤ 0 is given as follows:

Φ = η1 I1 + η2 I2 + η3 I3 + η4 I24 + η5 I2

5 + η6 I4I5 −(

1 +ξ(ep(m))

Y 011

)2

. (3.31)

The material parameters ηi|i = 1, ..6 can be identified by six independent tests. Assumethe tests are relative to the fixed orientation of the specimen 1a = (1, 0, 0)T and 2a =(0, 1, 0)T . Let Y 0

ij be the yield stress in ij-direction, with respect to ia and ja. The linearindependent numerical tests with β = 0 are:

1. uniaxial tension in 1a-direction

S =

Y 011

00

I1 = 2/3(Y 011)

2 I4 = 2/3Y 011

I2 = 4/9(Y 011)

2 I5 = −1/3 Y 011

I3 = 1/9(Y 011)

2 I4 I5 = −2/9 (Y 011)

2

(3.32)


S =

0Y 0

22

0

I1 = 2/3(Y 022)

2 I4 = −1/3 Y 022

I2 = 1/9(Y 022)

2 I5 = 2/3Y 022

I3 = 4/9(Y 022)

2 I4 I5 = −2/9 (Y 022)

2

(3.33)


S =

00

Y 033

I1 = 2/3(Y 033)

2 I4 = −1/3 Y 033

I2 = 1/9(Y 033)

2 I5 = −1/3 Y 033

I3 = 1/9(Y 033)

2 I4 I5 = 1/9 (Y 033)

2

(3.34)

4. shear test in 1a-2a plane

S =

0 Y 012

Y 012 0

0

I1 = 2(Y 012)

2 I4 = 0I2 = (Y 0

12)2 I5 = 0

I3 = (Y 012)

2 I4 I5 = 0

(3.35)


S =

0 Y 013

0Y 0

13 0

I1 = 2(Y 013)

2 I4 = 0I2 = (Y 0

13)2 I5 = 0

I3 = 0 I4 I5 = 0

(3.36)


S =

00 Y 0

23

Y 023 0

I1 = 2(Y 023)

2 I4 = 0I2 = 0 I5 = 0I3 = (Y 0

23)2 I4 I5 = 0

(3.37)


This leads after evaluation of the flow criterion to the parameters

η1 =1

2(−1

(Y 012)

2+

1

(Y 013)

2+

1

(Y 023)

2)

η2 =1

(Y 012)

2− 1

(Y 023)

2

η3 =1

(Y 012)

2− 1

(Y 013)

2

η4 =2

(Y 011)

2− 1

(Y 022)

2+

2

(Y 033)

2− 1

(Y 013)

2

η5 =−1

(Y 011)

2+

2

(Y 022)

2+

2

(Y 033)

2− 1

(Y 023)

2

η6 =−1

(Y 011)

2− 1

(Y 022)

2+

5

(Y 033)

2+

1

(Y 012)

2− 1

(Y 013)

2− 1

(Y 023)

2

. (3.38)

Remark: If we set Y 0ii = Y 0 for i = 1, 2, 3 and Y 0

ij = Y 0/√

3 for i 6= j with i, j = 1, 2, 3

we arrive at the well known von Mises criterion Φ =3||devΣ||2

2(Y 0)2−

(1 +

ξ(ep(m))

Y 0

)2

≤ 0.

3.5 Solution Algorithm for the Set of Constitutive Equations.

To solve the set of equations (3.13) we apply a so–called operator split along with a generalreturn method. The procedure is given here for a typical time step [tn , tn+1] with the time

increment ∆t := tn+1 − tn. We denote the solution at time tn by S(m)n ,αn, E

p(m)n , e

p(m)n .

Furthermore, at time tn+1 the strain tensor E(m)n+1 is given. The procedure is based on the

introduction of the so–called trial state by

Ep(m),trialn+1 = Ep(m)

n , αtrialn+1 = αn, e

p(m),trialn+1 = ep(m)

n , λtrialn+1 = 0 . (3.39)

Hence, the trial stresses are evaluated according to (3.15)

Strialn+1 = ∂Etrialψ(Etrial

n+1 ) with Etrialn+1 := E

(m)n+1 −E

p(m),trialn+1 . (3.40)

An elastic step is present if

Φtrialn+1 = Φ(Strial

n+1 ,βtrialn+1 , e

p(m),trialn+1 ) ≤ 0 (3.41)

and the state variables at time tn+1 are given by the elastic predictor step. In case ofΦtrial

n+1 > 0 one has to perform the corrector step, see Appendix A. This leads to a solution

at time tn+1 which is denoted by S(m)n+1, αn+1,E

p(m)n+1 , e

p(m)n+1 . Furthermore, we obtain the

consistent tangent tensor in the generalized stress-strain space.

Remark: It is well known that the general return algorithm according to Appendix A isnot volume preserving for all generalized measures, except for the case m = 0. In case ofm 6= 0 we perform the following post-processing algorithm in order to fulfill the plasticincompressibility condition det [Cp] = 1 at the beginning of each time interval. Startingfrom (2.3)1 and the additive decomposition (3.13)1 we consider Cp(m), which is implicitlydefined by

Ep(m) =1

2m(Cp(m) − 1) . (3.42)


It should be noted that Ep(m) is a deviatoric tensor. The latter equation yields

Cp(m) = 2mEp(m) + 1 , (3.43)

with det [Cp(m)] 6= 1 in general. The correction is performed as follows:

Cp(m),∗ := Cp(m)/(det Cp(m))1/3 , (3.44)

thus simply constructing the unimodular tensor Cp(m),∗ by scaling the initial tensor. Usingthis procedure, we arrive at the corrected plastic part of the generalized strain tensor

Ep(m),∗ =1

2m(Cp(m),∗ − 1) . (3.45)

In order to set Ep(m) equal to Ep(m),∗ in the next time step the latter quantities arestored in the history array. As mentioned above, for m = 0 the plastic incompressibilitycondition is fulfilled automatically. This statement can be shown using the well knownidentity det [expA] = exp[tr A], where A is a symmetric second-order tensor.

4 . Variational Formulation and Finite Element Discretization.

In the following we give a brief summary of the corresponding boundary value problemand finite element formulation in the material description. Let B be the reference body ofinterest which is bounded by the surface ∂B. The surface is partitioned into two disjoinedparts ∂B = ∂Bu

⋃∂Bt with ∂Bu

⋂∂Bt = ∅. The equation of balance of linear momentum

for the static case is governed by the First Piola-Kirchhoff stresses P = FS and the bodyforce f in the reference configuration

Div[FS] + f = 0 . (4.46)

The Dirichlet boundary conditions and the Neumann boundary conditions are given by

u = u on ∂Bu and t = t = PN on ∂Bt , (4.47)

respectively. Here N represents the unit exterior normal to the boundary surface ∂Bt.With standard arguments of variational calculus we arrive at the variational problem

G(u, δu) =

∫

B

S : δE dV + Gext with Gext(δu) := −∫

B

fδu dV −∫

∂Bt

tδu dA , (4.48)

where δE := 12(δF T F + F T δF ) characterizes the virtual Green-Lagrangian strain tensor

in terms of the virtual deformation gradient δF := Gradδu. The principle of virtual work(4.48) for a static equilibrium state of the considered body requires G = 0. For the solutionof this nonlinear equation we apply a standard Newton iteration scheme which requiresthe consistent linearization of (4.48) in order to guarantee the quadratic convergence ratenear the solution. Since the stress tensor S is symmetric, the linear increment of G denotedby ∆G is given by

∆G(u, δu, ∆u) :=

∫

B

(δE : ∆S + ∆δE : S) dV , (4.49)


where ∆δE := 12(∆F T δF +δF T ∆F ) denotes the linearized virtual Green-Lagrange strain

tensor as a function of the incremental deformation gradient ∆F := Grad∆u. The incre-mental Second Piola-Kirchhoff stress tensor ∆S can be derived using (2.6) as

∆S = ∆S(m) : IPE + S(m) : ∆IPE = C : ∆E , (4.50)

with ∆E := 12(∆F T F + F T ∆F ). The moduli appear in the form

C = IPE : C(m)ep : IPE + S(m) : IK , (4.51)

with the consistent tangent tensor C(m)ep = ∂E(m)S(m), in this context see Appendix A,

Equation (A.71).

The sixth-order tensor IK = 2∂CIP is derived following Ogden [15]

IK =∑3

A=1

∑3B=1

∑3C=1[KAABBCC NA ⊗NA ⊗NB ⊗NB ⊗NC ⊗NC

+ KABABCC NA ⊗NB ⊗ (NA ⊗NB + NB ⊗NA)⊗NC ⊗NC

+ KABCCAB NA ⊗NB ⊗NC ⊗NC ⊗ (NA ⊗NB + NB ⊗NA)

+ KAABCBC NA ⊗NA ⊗NB ⊗NC ⊗ (NB ⊗NC + NC ⊗NB)

+ KABBCCA NA ⊗NB ⊗ (NB ⊗NC + NC ⊗NB)⊗ (NC ⊗NA + NA ⊗NC)

+ KABCABC NA ⊗NB ⊗ (NC ⊗NA + NA ⊗NC)⊗ (NB ⊗NC + NC ⊗NB)

.

(4.52)


The non-zero components of IK are found to be

KAABBCC = ∂ECPAABB = 2 (m− 1)λm−2

A δABδAC

KABABCC = KABCCAB = ∂ECPABAB

where A 6= B

=

γ(m)AAB δAC + γ

(m)BBA δBC for λA 6= λB

1

2(m− 1) λm−2

A for λA = λB

0 for A 6= B 6= C 6= A

KAABCBC =1

2

PAABB − PAACC

EB − EC

− PBCBCδAB − δAC

EB − EC

where B 6= C

=

γ(m)AAC δAB + γ

(m)AAB δAC for λB 6= λC

1

2(m− 1) λm−2

B for λB = λC

0 for A 6= B 6= C 6= A

KABBCCA = KABCABC =1

4

PABAB − PCACA

EB − EC

+1

4

PBCBC − PABAB

EC − EA

where A 6= B 6= C 6= A

=

γ(m) for λA 6= λB 6= λC

1

2γ

(m)AAC for λA = λB 6= λC

1

4(m− 1) λm−2

A for λA = λB = λC

. (4.53)

Here, we use the abbreviations

γ(m)AAB =

λm−1A (λA − λB)− 2(E

(m)A − E

(m)B )

(λA − λB)2,

γ(m) =λ1(E

(m)2 − E

(m)3 ) + λ2(E

(m)3 − E

(m)1 ) + λ3(E

(m)1 − E

(m)2 )

(λ1 − λ2)(λ2 − λ3)(λ3 − λ1).

(4.54)

The contraction S(m) with IK in Equation (4.51) yields

S(m) : IK =∑3

A=1

∑3B=1

∑3C=1 K

(AA)BBCC NB ⊗NB ⊗NC ⊗NC

+ K(AB)ABCC (NA ⊗NB + NB ⊗NA)⊗NC ⊗NC

+ K(AB)CCAB NC ⊗NC ⊗ (NA ⊗NB + NB ⊗NA)

+ K(AA)BCBC NB ⊗NC ⊗ (NB ⊗NC + NC ⊗NB)

+ K(AB)BCCA (NB ⊗NC + NC ⊗NB)⊗ (NC ⊗NA + NA ⊗NC)

+ K(AB)CABC (NC ⊗NA + NA ⊗NC)⊗ (NB ⊗NC + NC ⊗NB)

,

(4.55)


with the components

K(AB)CDEF = S

(m)(AB)K(AB)CDEF . (4.56)

In (4.56) no summation over the indices (AB) takes place. Furthermore, the projectionof the generalized stress tensors into the space of principal directions yields

S(m)AB = S(m) : NA ⊗NB . (4.57)

The matrix representation of (4.51) considering (4.55) is given in Appendix B. The expres-sions are very simple and allow an efficient computer implementation. It should be noted,that in case of isotropic material behaviour the eigenvectors of S(m) and E(m) coincide,thus S

(m)AB = 0 holds for A 6= B.

The spatial discretization of the considered body B ≈ ⋃nele

e=1 Be with nele finite elements Be

leads within a standard displacement approximation u =∑nel

I=1 N IdI , δu =∑nel

I=1 N IδdI ,

and ∆u =∑nel

I=1 N I∆dI , of the actual-, virtual-, and incremental-displacement fields,respectively, to a set of algebraic equations of the form which can be solved for thesolution point d. For a detailed discussion of this point we refer to the standard text bookZienkiewicz & Taylor [36] or others.

5 . Numerical Examples.

The constitutive model described in the previous sections has been implemented in thefinite element programs FEAP [28] and ABAQUS [1]. In FEAP we use an 8–noded brick–type shell element as is documented in Klinkel [9] or Wagner et al. [30] and referencestherein. The basis of the element formulation is given with a standard isoparametric dis-placement approach. Based on so–called ANS–methods the transverse shear strains andthe thickness normal strains are independently interpolated using special shape functions.Furthermore the membrane behaviour of the element is essentially improved applying theenhanced strain method with 5 additional parameters, [30]. Due to the different interpola-tion techniques the element orientation has to be considered when discretizing the mesh.An interface to arbitrary three–dimensional material laws is available. In ABAQUS thematerial model is programmed via the user interface umat.

In this section three representative numerical examples with finite plastic deformationsare presented. The computations of the first two examples are performed with FEAP.First we investigate necking of a circular bar subjected to uniaxial tension. Our solutionfor m = 0 is compared with a reference model. In the second example we simulate themechanical behaviour of a conical shell considering different material parameters. For agiven linear isotropic hardening law we adjust the hardening functions of the generalizedstress–strain models with different parameters m for a simple tension test. The resultsfor nonlinear isotropic hardening are compared with a reference solution. Furthermorewe consider orthotropic elastic and plastic material properties. Finally the deep drawingprocess of a circular blank is simulated using the 3-D hybrid solid element C3D8H ofthe programm ABAQUS/Standard. The material is assumed to be isotropic in the elasticrange and orthotropic within plastic deformations. With the simulation the earing effectcaused by the anisotropy of the blank is investigated.


5.1 Necking of a Circular Bar.

The three dimensional necking of a circular bar is an example widely investigated inthe literature, see e.g. Simo & Armero [21] or Klinkel [9]. The geometrical data areR = 6.413 mm, R0 = 0.982R and L = 26.667 mm.

x

y

R0

z

R

L

Figure 3: Geometry of circular bar.

To initialize the necking process we introduce an imperfection with the reduced radiusR0 at z = L. The material data for isotropy are given as follows. The hardening functionconsists of a linear part and an exponential part. It is approximated by piecewise linearfunctions.

Elasticity constants: E = 206.9 GPa ν = 0.29 m = 0

Yield parameter: Y 0 = 0.45 GPa Y ∞ = 0.715 GPa

h = 0.12924 GPa δ = 16.93

η1 =3

2

1

(Y 0)2ηi = 0|i=2,...,6

Hardening function: ξ(ep(m)) = hep(m) + (Y ∞ − Y 0)(1− exp(−δep(m)))

(5.58)

Figure 3 shows a finite element discretization of half the bar. At z = L we impose thesymmetry boundary conditions, whereas in a displacement-controlled computation theaxial displacements u(z = 0) are given. Furthermore, we consider symmetry conditions inthe cross–section of the plane. Thus, one quarter is discretized with 960 elements, wherethe thickness direction of the shell elements corresponds to the global z-axis. We onlyconsider the parameter m = 0. Figure 4b displays the deformed structure at u = 7 andthe equivalent plastic strains. As can be seen large plastic strains occur in the neckingrange. The results are in very good agreement with the reference solution of Klinkel [9].

5.2 Conical Shell.

The second example is a conical shell subjected to a constant ring load λp with p =1 GN/m, see Wagner et al. [30] and references therein. The problem and the finiteelement discretization with 8-node shell elements are depicted in Figure 5. The geometrical


0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7

m = 0KlinkelSimo & Armero

8.924E-02

2.083E-01

3.274E-01

4.465E-01

5.656E-01

6.847E-01

8.038E-01

9.229E-01

1.042E+00

1.161E+00

1.280E+00

1.399E+00

1.518E+00

1.637E+00

Eps plast

a) b)

F

u

Figure 4: Necking of circular bar: a) Load-displacement curve. b) Deformed structure foru = 7 and equivalent plastic strains.

data are r = 1 m, R = 2 m, L = 1 m, t = 0.1 m. For isotropic material behaviour onequarter is discretized with 8 × 8 × 1 elements, whereas for orthotropy the whole conehas to be considered. In our computations the vertical displacement w is controlled. Atw = 2.25 the structure is completely unloaded. It should be remarked that we have useda nine-point Gaussian quadrature.

w

rλpλp t

R

L

p

a) b)

Figure 5: Conical shell: a) Geometry and boundary conditions, b) Finite Element Dis-cretization.

5.2.1 Isotropic Material Response with Nonlinear Hardening. Here, the pa-rameter m of the generalized stress strain model is set to m = 0. The material data aregiven in (5.58). The nonlinear hardening function is approximated by piecewise linearfunctions. The results are depicted in Figure 6a. The load factor λ is plotted versus thevertical deflection w. There is exact agreement with the reference solution of Klinkel [9].The equivalent plastic strains for the deformed structure (w = 2.25) are shown in Figure6b).

5.2.2 Isotropic Material Response with Linear Hardening. Next we comparethe structural response of the conical shell considering the parameters m = 0, m = 0.5,m = 1, m = −1. The material response is restricted to isotropy with material data


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

Referencem=0

1.070E-01

1.380E-01

1.689E-01

1.999E-01

2.309E-01

2.618E-01

2.928E-01

3.237E-01

3.547E-01

3.856E-01

4.166E-01

4.475E-01

4.785E-01

5.095E-01

Eps plast

a) b)

λ

w

Figure 6: Conical shell, isotropy and nonlinear hardening: a) Load displacement curvesusing the reference model (Klinkel) [9] and for the present model with m = 0, b) Equivalentplastic strains at w = 2.25.

according to (5.58). Now the hardening function for the model with m = 0 contains onlythe linear part with h = 0.125 GPa. For each parameter m except m = 0 the functionξ(ep(m)) is adjusted in such a way that all five models lead to the same results for a simpletension test. The test is performed with one element subjected to a homogeneous stressstate.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6

Referencem=0m=0.5m=1m=-1analytical

-0.5

0

0.5

1

1.5

2

2.5

0 0.25 0.5 0.75 1

m=-1m=0m=0.5m=1

a) b)

τ11

ln F11

ξ

ep(m)

Figure 7: Tension test with linear hardening. a) τ11-ln F11 curve for reference model andfor m = 1, m = 0.5, m = 0, m = −1. b) ξ-ep(m) curve for the different models.

The solutions are depicted in Figure 7a, where the uniaxial Kirchhoff stress τ11 is plottedversus lnF11. As a result of the fitting process one can see nearly the same response forthe parameters m = 0, m = 0.5, m = 1, m = −1 and the reference model of Klinkel[9] in comparison to the given linear hardening law. The corresponding hardening curvesξ − ep(m) are shown in Figure 7b).

Using the adjusted functions ξ(ep(m)) we now calculate the load displacement behaviourof the conical shell for each parameter m. Figure 8 shows that the results obtained withthe model in [9] are identical to m = 0. For a negative parameter m we obtain a stifferbehaviour, whereas for positive m the computed load factors are below the referencesolution.

Plots of the equivalent plastic strains at a displacement w = 2.25 are depicted in Figure


0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2 2.5

Referencem = 1m=0.5m = 0m=-1

λ

w

Figure 8: Load displacement curves for the reference model and for the generalized modelsm = 1, m = 0.5, m = 0 and −1.

9.049E-02

1.241E-01

1.578E-01

1.915E-01

2.251E-01

2.588E-01

2.924E-01

3.261E-01

3.597E-01

3.934E-01

4.271E-01

4.607E-01

4.944E-01

5.280E-01

Eps plast

9.044E-02

1.241E-01

1.578E-01

1.915E-01

2.251E-01

2.588E-01

2.925E-01

3.262E-01

3.599E-01

3.935E-01

4.272E-01

4.609E-01

4.946E-01

5.282E-01

Eps plast

1.778E-01

2.095E-01

2.413E-01

2.730E-01

3.048E-01

3.366E-01

3.683E-01

4.001E-01

4.319E-01

4.636E-01

4.954E-01

5.272E-01

5.589E-01

5.907E-01

Eps plast 5.338E-02

9.270E-02

1.320E-01

1.713E-01

2.107E-01

2.500E-01

2.893E-01

3.286E-01

3.680E-01

4.073E-01

4.466E-01

4.859E-01

5.252E-01

5.646E-01

Eps plast

Reference m = 0

m = −1 m = 0.5

Figure 9: Equivalent plastic strains for the reference model (Klinkel) and the generalizedmodels m = 0, m = −1 and m = 0.5.

9. A comparison of the contour values is not possible, since ep(m) is defined differently foreach parameter m.

For m = 1 a solution can not be obtained, because numerical instabilities due to themixed element formulation occur, see the load deflection curve for m = 1 in Figure 8.In this context, we refer to investigations of the numerical stability of enhanced strainformulations in Wriggers & Reese [34]. Due to these difficulties we finally analyzethe load deflection behaviour for the cases m = 1 and m = 0 without activating the five


0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

m=1 without EASm=0 without EAS

λ

w

Figure 10: Load displacement curves for the generalized models m = 0 and m = 1 withoutconsidering the EAS-parameters.

enhanced assumed strain parameters. The corresponding results are depicted in Figure10. It can be seen that the structure shows now a relatively stiff behaviour.

5.2.3 Orthotropic Material Response with Nonlinear Hardening. Next weassume orthotropic elastic and plastic material behaviour. Here we only consider thecase m = 0. The material parameters for elastic and plastic orthotropy are given below.According to Equation (3.20) the privileged directions are denoted by 1a and 2a.

Elasticity constants: C11 = 240.71 C12 = 62.68 C13 = 69.25

C22 = 211.05 C23 = 59.70 C33 = 229.25

C44 = 66.00 C55 = 66.00 C66 = 81.00

Yield parameter: Y11 = 0.585 Y22 = 0.81 Y33 = 0.36

Y12 = 0.286 Y13 = 0.234 Y23 = 0.260

Orientation: 1a = [1 , 0 , 0]T 2a = [0 , 1 , 0]T

The yield parameters and the elasticity constants are given in GPa. The nonlinear hard-ening curve is shown in Figure 11a). Figure 11b) depicts the computed load deflectioncurve.


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

m = 0

a) b)

ξ

ep(m)

λ

w

Figure 11: a)Nonlinear hardening function b) Load displacement curve.

Finally, Figure 12 shows the equivalent plastic strains for a sequence of deformed config-urations. The orthotropy of the material is reflected by the distribution of the equivalentplastic strains. It can be seen that four regions with higher values of ep(m) evolve withrespect to the orientation of the preferred directions.

2a1a

3a

1.047E-01

1.449E-01

1.851E-01

2.253E-01

2.654E-01

3.056E-01

3.458E-01

3.860E-01

4.262E-01

4.664E-01

5.066E-01

5.468E-01

5.870E-01

6.271E-01

Eps plast

w = 0.02 w = 0.50

w = 1.75 w = 2.25

Figure 12: Equivalent plastic strains for anisotropic material behaviour.


5.3 Cylindrical Cup Drawing from a Circular Blank.

In this section, the implemented material model is applied for the simulation of a cylin-drical cup drawing. Deep drawing is an important process in the domain of the formingtechnique. As an example we mention sheet metal forming in the automobile industry. Foran overview we refer to e.g. Lange [10]. The tools of the deep drawing process includedie, holder and punch, see Figure 13. The geometrical data for punch, holder and die aregiven in cm in Figure 14a). The blank radius is 3.95 cm and the thickness is 0.081 cm.

c)b)a)

Figure 13: Tools for deep drawing process: a) die, b) holder, c) punch.

In our simulation the blank is assumed to behave isotropic in the elastic range and or-thotropic in the plastic range. Again, we choose m = 0 in the material model.

0.65

0.650.460.081

holder

3.95

sym.

2.4

2.26

die

punch

0

5

10

15

20

25

30

35

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

a) b)

ξ

ep(m)

Figure 14: Deep drawing process: a) geometry, and b) nonlinear hardening curve.

The nonlinear isotropic hardening law is depicted in Figure 14. It is again approximatedby piecewise linear functions. The privileged directions of the material are described by1a and 2a. The constitutive parameters are summarized as follows:


E = 19600 kN/cm2 ν = 0, 3 m = 0

Y11 = 20.00 kN/cm2 Y22 = 25.00 kN/cm2 Y33 = 18.00 kN/cm2

Y12 = 6.61 kN/cm2 Y13 = 6.61 kN/cm2 Y23 = 6.61 kN/cm2

1a = [1 , 0 , 0]T 2a = [0 , 1 , 0]T .

The blank is discretized with 936 solid elements C3D8H using ABAQUS, where 2 ele-ments are positioned in thickness direction. Die, punch and holder are modelled as rigidbodies. Frictionless contact is considered along the interfaces. The present calculation iscarried out to investigate the effect of the planar anisotropy of the blank and the earingphenomenon due to the anisotropic material behaviour.

(Ave. Crit.: 75%)SDV7

-5.121e-03+9.398e-02+1.931e-01+2.922e-01+3.913e-01+4.904e-01+5.895e-01+6.886e-01+7.877e-01+8.868e-01+9.859e-01+1.085e+00+1.184e+00

u = 0.000 u = 1.479

u = 2.370 u = 3.070

Figure 15: Equivalent plastic strains at different punch displacements.

Figure 15 shows the equivalent plastic strains of the blank for different punch displace-ments u. The earing phenomenon is visible. In total four ears are formed. The maximumequivalent tensile plastic strains are located at the ears.

6 . Conclusions.

In this paper a finite element model for orthotropic elasto–plastic material behaviour atfinite strains has been presented. The theory is based on so–called generalized stress andstrain measures, which allow an adaptation of constitutive models of the infinitesimal the-ory. The governing constitutive equations have been written in an invariant setting, wherethe privileged directions of the material are described by so–called structural tensors. Theadditive decomposition of the generalized strains is the essential kinematic assumption.As a consequence standard return algorithms can be applied to solve the set of material


equations. The condition of plastic incompressibility has been fulfilled by a correction ofthe inelastic strains for the case m 6= 0. As an essential contribution we have derivedexplicit matrix representations for the transformation relations of the generalized stresstensors and the associated linearized expressions. This allows a simple and effective finiteelement implementation. The examples show the robustness of the developed formula-tion. The computed results show for the parameter m = 0 good agreement with availablesolutions from the literature. With the simulation of a deep drawing process the earingphenomenon of an anisotropic metal sheet has been demonstrated. To sum up, the gener-alized model with m = 0 seems to be the most suitable one within this additive frameworkfor the analysis of anisotropic finite plasticity, in this context see also Xiao, Bruhns &Meyers [35].

References.[1] “ABAQUS/Standard User’s Manual 6.2“[2001], Hibbitt, Karlson & Sorensen, Inc.[2] Betten, J. [1987], “Formulation of Anisotropic Constitutive Equations”, in J.P.

Boehler (ed.): “Applications of Tensor Functions in Solid Mechanics” CISM CourseNo. 292, Springer-Verlag

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A General Return Algorithm.

The evolution laws for the plastic strains and the internal variables according to (3.13) areintegrated approximately in time using an implicit backward Euler integration procedure

Epn+1 = Ep

n + ∆Epn+1

αn+1 = αn + ∆αn+1

epn+1 = ep

n + γn+1

√23∂S(m)Φ : ∂S(m)Φ

∣∣∣n+1

, (A.59)

with γn+1 = ∆tλn+1. The incremental plastic parameters are given with ∂S(m)Φ = ∂ΣΦ :∂S(m)Σ = ∂ΣΦ and ∂βΦ = ∂ΣΦ : ∂βΣ = −∂ΣΦ by

∆Epn+1 = γn+1∂S(m)Φn+1 = γn+1∂ΣΦn+1

∆αn+1 = −γn+1∂βΦn+1 = γn+1∂ΣΦn+1

. (A.60)

We introduce the residual vectors

Rpn+1 = −Ep

n+1 + Epn + ∆Ep

n+1 = 0Rα

n+1 = αn+1 − (αn + ∆αn+1) = 0

RΦn+1 = Φn+1(S

(m)n+1, βn+1, e

pn+1) = 0

, (A.61)

which are solved with respect to the variables

S(m)n+1,βn+1, γn+1 . (A.62)

For this purpose a Newton iteration scheme is applied. Linearization of (A.61) yields asystem of linear equations

Rpn+1 + ∂S(m)R

pn+1 : ∆S(m) + ∂βRp

n+1 : ∆β + ∂γRpn+1∆γn+1 = 0

Rαn+1 + ∂S(m)Rα

n+1 : ∆S(m) + ∂βRαn+1 : ∆β + ∂γR

αn+1∆γn+1 = 0

RΦn+1 + ∂S(m)RΦ

n+1 : ∆S(m) + ∂βRΦn+1 : ∆β + ∂γR

Φn+1∆γn+1 = 0

(A.63)

for the stress and strain increments. Equation (A.63) can be specified more precisely as

Rp + Ξ−1 : ∆S(m) + γ∂2βS(m)Φ : ∆β + ∂S(m)Φ∆γ = 0

Rα + γ∂2S(m)β

Φ : ∆S(m) + Ω−1 : ∆β + ∂βΦ∆γ = 0

RΦ + ∂S(m)Φ : ∆S(m) + ∂βΦ : ∆β + ∂γΦ∆γ = 0

(A.64)

with

Ξ−1 := (Ce(m))−1 + γn+1∂2S(m)S(m)Φn+1 and Ω−1 := H−1 + γn+1∂

2ββΦn+1 (A.65)

and the abbreviation H := ∂2ααψp,k. Formally the solution of (A.64) can be written as

∆S(m)n+1

∆βn+1

∆γn+1

=

Ξ−1 γn+1∂2βS(m)Φn+1 ∂S(m)Φn+1

γn+1∂2S(m)β

Φn+1 Ω−1 ∂βΦn+1

∂S(m)Φn+1 ∂βΦn+1 ∂γΦn+1

−1

−Rp

−Rα

−RΦ

.

(A.66)


The derivative ∂γΦn+1 is given by

∂γΦ = ∂ξΦ ∂2epepψp,i

√2

3(∂S(m)Φ : ∂S(m)Φ)

∣∣∣∣∣n+1

. (A.67)

Using submatrices we can write (A.66) as follows

∆S(m)n+1

∆βn+1

∆γn+1

:=

A11 A12 A13

A21 A22 A23

A31 A32 A33

−Rp

−Rα

−RΦ

.

(A.68)

Considering ∆Ep(l)n+1 = −(Ce(m))−1 ∆S

(m)n+1, ∆α

(l)n+1 = H−1 ∆βn+1 and ∆γ

(l)n+1 = ∆γn+1 the

update of the internal variables is performed as follows:

Ep(l+1)n+1 = E

p(l)n+1 + ∆E

p(l)n+1

α(l+1)n+1 = α

(l)n+1 + ∆α

(l)n+1

γ(l+1)n+1 = γ

(l)n+1 + ∆γ

(l)n+1

(A.69)

where l denotes the local iteration index and Ep(0)n+1 = Ep

n , α(0)n+1 = αn and γ

(0)n+1 = γn. The

next iteration step starts with the evaluation of the generalized stresses (3.13)3 considering(3.13)1 and the back stresses with (3.13)4. The iteration is aborted if the components ofthe residual vectors

||RΦn+1|| < tol , ||Rp

n+1|| < tol , ||Rαn+1|| < tol (A.70)

vanish at the solution point.

Linearization of (A.68) with respect to ∆E(m)n+1 yields the increment of the generalized

stresses as∆S

(m)n+1 = C(m)

ep ∆E(m)n+1 with C(m)

ep := A11 (A.71)

where C(m)ep denotes the so–called consistent tangent tensor.


B Matrix Representation of Tangent Moduli.

The matrix representation of (4.51) considering (4.55) yields

C = P T C(m)ep P + T T L2T (B.72)

with the consistent tangent matrix according to (A.71)

C(m)ep =

C11ep C12

ep C13ep 2C14

ep 2C15ep 2C16

ep

C22ep C23

ep 2C24ep 2C25

ep 2C26ep

C33ep 2C34

ep 2C35ep 2C36

ep

4C44ep 4C45

ep 4C46ep

sym. 4C55ep 4C56

ep

4C66ep

(B.73)

and the symmetric matrix

L2 =

L1111 0 0 L1112 L1113 0L2222 0 L2212 0 L2223

L3333 0 L3313 L3323

L1212 L1213 L1223

sym. L1313 L1323

L2323

. (B.74)

In case of λ1 6= λ2 6= λ3 the components read with γ(m)AAB and γ(m) according to (4.54)

L1111 = 2 S(m)11 (m− 1) λm−2

1

L2222 = 2 S(m)22 (m− 1) λm−2

2

L3333 = 2 S(m)33 (m− 1) λm−2

3

L1212 = S(m)11 γ

(m)112 + S

(m)22 γ

(m)221

L1313 = S(m)11 γ

(m)113 + S

(m)33 γ

(m)331

L2323 = S(m)22 γ

(m)223 + S

(m)33 γ

(m)332

L1112 = 2 S(m)12 γ

(m)112

L2212 = 2 S(m)12 γ

(m)221

L1113 = 2 S(m)13 γ

(m)113

L3313 = 2 S(m)13 γ

(m)331

L2223 = 2 S(m)23 γ

(m)223

L3323 = 2 S(m)23 γ

(m)332

L1223 = 2 S(m)13 γ(m)

L1323 = 2 S(m)12 γ(m)

L1213 = 2 S(m)23 γ(m)

. (B.75)


In the case of two equal eigenvalues we only specify the components which distinguish itfrom (B.75):

λ1 = λ2 6= λ3 L1212 = 12(S

(m)11 + S

(m)22 ) (m− 1) λm−2

1

L1112 = S(m)12 (m− 1) λm−2

1

L2212 = S(m)12 (m− 1) λm−2

1

L1223 = S(m)13 γ

(m)113

L1323 = S(m)12 γ

(m)113

L1213 = S(m)23 γ

(m)113

(B.76)

λ2 = λ3 6= λ1 L2323 = 12(S

(m)22 + S

(m)33 ) (m− 1) λm−2

2

L2223 = S(m)23 (m− 1) λm−2

2

L3323 = S(m)23 (m− 1) λm−2

2

L1223 = S(m)13 γ

(m)221

L1323 = S(m)12 γ

(m)221

L1213 = S(m)23 γ

(m)221

(B.77)

λ3 = λ1 6= λ2 L1313 = 12(S

(m)11 + S

(m)33 ) (m− 1) λm−2

3

L1113 = S(m)13 (m− 1) λm−2

3

L3313 = S(m)13 (m− 1) λm−2

3

L1223 = S(m)13 γ

(m)332

L1323 = S(m)12 γ

(m)332

L1213 = S(m)23 γ

(m)332

. (B.78)

In the case of three equal eigenvalues we only specify the components which distinguishit from (B.76) - (B.78):

λ1 = λ2 = λ3 L1223 = 12S

(m)13 (m− 1) λm−2

1

L1323 = 12S

(m)12 (m− 1) λm−2

1

L1213 = 12S

(m)23 (m− 1) λm−2

1

. (B.79)

The components S(m)AB of the generalized stress tensor with respect to the eigenvector basis

NA considering (4.57) are evaluated as

S = T S

S = [S(m)11 , S

(m)22 , S

(m)33 , 2S

(m)12 , 2S

(m)13 , 2S

(m)23 ]T

, (B.80)

where T is given in (2.11). The Cartesian components of the generalized stress tensorSij = ei · S(m)ej are organized in a vector S = [S11, S22, S33, 2S12, 2S13, 2S23]

T .