a comparative study of stress update algorithms for rate-independent and rate-dependent crystal...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2001; 50:273–298

A comparative study of stress update algorithmsfor rate-independent and rate-dependent crystal plasticity

Christian Miehe∗;† and J�org Schr�oder

Institut f�ur Mechanik (Bauwesen) Lehrstuhl I; Universit�at Stuttgart; 70550 Stuttgart; Pfa�enwaldring 7; Germany

SUMMARY

The paper presents a comparative discussion of stress update algorithms for single-crystal plasticity at smallstrains. The key result is a new uni�ed fully implicit multisurface-type return algorithm for both the rate-independent and the rate-dependent setting, endowed with three alternative approaches to the regularizationof possible redundant slip activities. The fundamental problem of the rate-independent theory is the possibleill condition due to linear-dependent active slip systems. We discuss three possible algorithmic approachesto deal with this problem. This includes the use of alternative generalized inverses of the Jacobian of thecurrently active yield criterion functions as well as a new diagonal shift regularization technique, motivated bya limit of the rate-dependent theory. Analytical investigations and numerical experiments show that all threeapproaches result in similar physically acceptable predictions of the active slip of rate-independent single-crystal plasticity, while the new proposed diagonal shift method is the most simple and e�cient concept.Copyright ? 2001 John Wiley & Sons, Ltd.

KEY WORDS: constitutive integration algorithm; crystal plasticity; multisurface plasticity; redundant systems

1. INTRODUCTION

This article presents a comparative discussion of stress update algorithms for single-crystal plasticityat small strains and considers aspects of its �nite-element implementation. The key result is a newuni�ed fully implicit multisurface-type return algorithm for both the rate-independent and the rate-dependent setting, endowed with three alternative approaches to the regularization of possibleredundant slip activities.The description of the phenomenological response of crystalline solids is based on the well-

established so-called continuum slip theory, see e.g. Mandel [1], Havner [2], and references therein.In the small strain format, one assumes locally an additive decomposition of the total strains intoa plastic part solely due to the plastic slip on given crystallographic slip planes and an elastic part

∗Correspondence to: Christian Miehe, Institut fur Mechanik (Bauwesen), Universit�at Stuttgart, Lehrstuhl I,Pfa�enwaldring 7, 70569 Stuttgart, Germany

†E-mail: [email protected]

Contract=grant sponsor: Deutsche Forschungsgemeinschaft; contract=grant number: SFB 404=A8

Received 29 October 1998Copyright ? 2001 John Wiley & Sons, Ltd. Revised 12 November 1999

274 C. MIEHE AND J. SCHR �ODER

which describes the lattice distortion. It is well known that the single-crystal plasticity can be recastinto the mathematical framework of multisurface plasticity as outlined for example in the works ofKoiter [3] and Mandel [1]. In this format, the multiple constraints are the yield criterion functionson the given crystallographic slip planes. In contrast to standard formulations of multisurfaceplasticity, the constraints can be linearly dependent or redundant. This results in a possible non-uniqueness of the set of active systems for a given deformation mode, see also Taylor [4], Kocks [5]and Havner [2] for a discussion of this point. In order to circumvent this problem, many authorshave applied rate-dependent formulations based on power-type creep laws without di�erentiationof slip systems into active and inactive sets via loading functions. These formulations withoutan elastic domain have been applied as a numerical regularization technique even in situationswhere rate dependency is a physically negligible e�ect. We refer in this context to the numericalimplementations of rate-dependent single-crystal plasticity documented in Peirce et al. [6; 7], Asaroand Needleman [8], Mathur and Dawson [9], Becker [10], Steinmann and Stein [11], among others.The recent research on computational single-crystal plasticity focuses on formulations with an

elastic domain and in particular the rate-independent theory. We refer in this context to the refer-ences Cuitino and Ortiz [12], Borja and Wren [13], Anand and Kothari [14] and Miehe [15; 16].These developments di�er in the following format. Cuitino and Ortiz [12] propose an algorithmicsetting for a multisurface-type viscoplastic model with elastic domain. Here, the problem of redun-dant constraints does not occur, due to the viscoplastic regularization e�ect. Borja and Wren [13]propose a so-called ultimate algorithm for the rate-independent theory which follows the succes-sive development of active slip within a typical discrete time interval. Anand and Kothari [14]solve the system of redundant constraints of rate-independent single-crystal plasticity by the use ofa generalized inverse on the basis of the singular-value decomposition of the Jacobian of the activeyield criterion functions. This approach meets least-square-type optimality conditions by minimiz-ing the plastic dissipation due to the slip activities. Motivated by this development, Schr�oder andMiehe [17] have proposed an alternative general inverse where the reduced space is obtained bydropping columns of the local Jacobian associated with zero diagonal elements within a standardfactorization procedure.In this paper we focus on aspects of stress update algorithms for rate-independent single-crystal

plasticity which extend the results of the work cited above. The underlying concept of this paperis a comparison of the rate-independent formulation with the rate-dependent approach on the basisof an elastic domain as used in Cuitino and Ortiz [12]. Clearly, the case of the rate-independenttheory can be motivated by a limit process from the rate-dependent theory, e.g. for very slowprocesses or a vanishing viscosity. In order to achieve this limit process within the algorithmicsetting, we construct a uni�ed implicit stress update algorithm in the multisurface format whichcovers both the rate-independent and the rate-dependent theory. The proposed algorithm providesthe basis for a careful analysis of the above-mentioned limit process which we perform for arepresentative model problem with two redundant slip activities. This consideration motivates afurther new regularization approach of the possibly singular local Jacobian of the rate-independenttheory by means of a simple diagonal shift. It turns out that this new approach is the most e�cienttreatment of algorithmic implementations of rate-independent single-crystal plasticity, comparedwith the approaches proposed by Anand and Kothari [14] and Schr�oder and Miehe [17] basedon the application of general inverses of the local Jacobian. We include all three alternativeregularization techniques as sub-tools in the proposed uni�ed stress update algorithm.The paper is organized as follows. In Section 2 we summarize the constitutive equations

of single-crystal plasticity in the multisurface format for both the rate-independent and the

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 50:273–298

A COMPARITIVE STUDY OF STRESS UPDATE ALGORITHMS 275

rate-dependent setting. With regard to an application to the analysis of f.c.c. crystals we takeinto account elastically cubic symmetry. Furthermore, we consider a classical constitutive harden-ing model as suggested by Kocks [5], Hutchinson [18], Chang and Asaro [19] and others. Therate-dependent constitutive formulations for the plastic slip are based on formulations used byPerzyna [20] and Cuitino and Ortiz [12].Section 3 is concerned with the construction of the above-mentioned uni�ed stress update algo-

rithm for single-crystal plasticity at small strains. The algorithm is endowed with a straightforwardrobust active set search for the detection of the current active slip systems, allowing the applicationof large time steps. Section 4 investigates in detail the three alternative regularization techniqueswhich become a crucial part of the uni�ed stress update algorithm. These sub-tools provide aphysically acceptable method of handling the possibly ill-conditioned or singular Jacobian of thelocal consistency conditions in the rate-independent limit. A main goal of this part of the paperis an analysis of the physical origins of this behaviour. We therefore investigate in detail an an-alytical model problem with two linearly dependent slip systems and consider the consequencesof the transition from the rate-dependent to the rate-independent setting, as well as the responseof the three regularization techniques considered. For this problem it can be shown that redundantconstraints appear in particular for the case of isotropic Taylor-type hardening.Finally, we demonstrate in Section 5 the performance of the proposed algorithms for two rep-

resentative numerical examples. The �rst example considers a simple shear test with di�erentorientation of the f.c.c. unit cell and records the di�erent current slip activities for the case of therate-independent and the rate-dependent setting. The second example is concerned with the tensionof a strip under plane strain conditions and compares the e�ciency of the proposed alternativealgorithmic approaches. In summary, it turns out that all the three regularization techniques forthe rate-independent setting considered yield the same physical result, which coincides in the limitwith the viscoplastic formulation. Here, the proposed new diagonal shift method appears to be themost e�cient approach.

2. THE CONSTITUTIVE FRAME OF SINGLE-CRYSTAL PLASTICITY

In this section we summarize the constitutive framework of single-crystal plasticity at small strainswithin the continuous setting. Here, we consider successively both the rate-independent elastoplasticsetting and the rate-dependent elastoviscoplastic setting.

2.1. Rate-independent single-crystal plasticity

Let B∈R3 be the body of interest and u :B→R3 a given displacement �eld. The linear strain ten-sor U= sym[∇u] is by de�nition the symmetric part of the displacement gradient and we considerits additive decomposition

U= Ue + Up (1)

into elastic and plastic parts Ue and Up, respectively. The latter remains after stress relaxation.

2.1.1. Free energy function and stress response. The elastic response of a crystalline solid isgoverned by lattice deformations and by local inhomogeneous deformation �elds due to dislocations



and point defects. A general anisotropic linear elastic response of the solid is provided by therepresentation

macro(Ue)= 12U

e :Ce : Ue (2)

of the free energy function governed by a fourth-order tensor Ce of elasticity moduli. The stressesin the crystalline solid are then given by the hyperelastic function

b= @Ue macro(Ue)=Ce : Ue (3)

With regard to an application to f.c.c. crystals, we focus in our treatment on the case of cubicelastic symmetry. Here, the fourth-order tensor of elastic moduli take the form

Ce=Ceijklei ⊗ ej ⊗ ek ⊗ el (4)

in the cartesian base {ei}i=1;3 aligned to the cubic crystal. The matrix representation of thecoe�cients is

Ceijkl=

C11 C12 C12C12 C11 C12C12 C12 C11

C44C44

C44

(5)

see e.g. Hosford [21]. In the case of elastic isotropy we have the identi�cation

C11 = � + 43�; C12 = � − 2

3�; C44 = � (6)

in terms of the bulk modulus � and the shear modulus �.

2.1.2. Yield criterion functions and ow rule. In the framework of rate-independent single-crystalplasticity we consider a non-smooth convex elastic domain in the stress space

E= {(b; g�) | ��(b; g�)60 for �=1; 2; : : : ; m} (7)

based on m-independent ow criterion functions

��(b; g�)= ��(b)− g� for �=1; 2; : : : ; m (8)

These ow criterion functions are formulated in terms of the Schmid resolved shear stresses

�� := b :P� with P� := sym(s� ⊗m�) (9)

on a typical slip system �. The slip system � is de�ned by orthonormal vectors (s�;m�) whichde�ne the slip direction and the slip normal, respectively. g� denotes the current slip resistanceon the slip system �. The evolutions of these resistances within a multislip deformation processof the crystal start from the so-called critical resolved shear stress �0 and are governed by thehardening equations

g�=m∑

�=1h�� with g�(t=0)= �0 (10)



Table I. Multislip single-crystal-plasticity.

Ue := sym[∇u]− Up1: macro stress b=Ce : Ue

2: Schmid stress �� = b :P� with P� := sym(s� ⊗m�)3: ow criteria �� = �� − g� for �=1; : : : ; m4: ow rule Up=

∑m�=1 ��P�

5: evolution A=∑m

�=1 ��

6: hardening g� =∑m

�=1 h(A)[q + (1− q)��

]��

7: loading �� ¿ 0; �� 6 0; �� =0(viscoplastic �� =

1�

[(��+

g�+ 1

)p

− 1])

in terms of the plastic slip rates �� on the slip systems �. Here, h�� are denoted as the hardeningmoduli. A classical assumption is

h��= h(A) [q+ (1− q)��] (11)

as suggested by Hutchinson [18], Peirce et al. [6]. Here, A is a strain-like internal variable for thedescription of the internal hardening state of the crystal on average. It is the sum of the accumulatedplastic slip on all slip systems. The parameter q∈ [1; 1:4] speci�es the type of hardening behaviourand has been speci�ed for f.c.c. crystals by Kocks [5] on the basis of experimental investigations.For q=1 we obtain the so-called isotropic or Taylor-type hardening. A speci�c form of thefunction h(A) in terms of the critical resolved shear stress �0, a saturation strength �s and theinitial hardening modulus h0 has been proposed by Chang and Asaro [19]

h(A)= h0 sech2[

h0A�s − �0

](12)

For an overview of alternative constitutive hardening functions we refer to Cuitino and Ortiz[12]. The evolution equations of the plastic strains Up and the strain-like internal variable A inrate-independent single-crystal plasticity take the typical form

Up=m∑

�=1��P� and A=

m∑�=1

�� (13)

of multisurface plasticity, where the plastic parameters �� for each slip system � are determinedby the Kuhn–Tucker-type loading conditions

��¿0; ��60; ��=0 (14)

The constitutive set of small-strain single-crystal plasticity is summarized in Table I. Note thatwe de�ne slip systems for each possible slip direction, for example 2× 12 systems for a typicalf.c.c. crystal. Insertion of (13)1 into the rate equation for stresses (3) yields with de�nition (9)2the form

b=Ce : U− ∑�∈A

��(Ce :P�) (15)



Here, we denote with A the currently active set of slip systems with ��¿0. The active plasticparameters �� are computed from the consistency conditions

��=P� : b − g�=0 for �∈A (16)

The insertion of (15) and (10)1 then yields the rate equation for the stresses

b=Cep : U (17)

in terms of the continuous elastoplastic tangent moduli

Cep :=Ce − ∑�∈A

∑�∈A

D��−1(Ce :P�)⊗ (P� :Ce) (18)

in terms of the de�nition

D�� :=P� :Ce :P� + h�� (19)

The inversion of D�� needed in (18) is a problem in the case of redundant constraints and com-mented on in Section 4.

2.2. Rate-dependent single-crystal plasticity

In the rate-dependent formulation we replace the Kuhn–Tucker-type loading conditions (14) bya constitutive viscose evolution equation for the plastic slip on the slip systems �. A classicalviscoplastic form is provided by the structure

��=1�

[(��+

g� + 1)p

− 1]

(20)

where � represents the viscosity parameter and p a strain-rate-sensitivity exponent, see e.g.Perzyna [20] and references therein. The overstress functions ��+ in (20) are de�ned by

��+ :={�� if ��¿00 otherwise

(21)

Insertion of (20) into (15) then gives the representation of the stress rate

b=Ce : U−m∑

�=1

1�

[(��+

g� + 1)p

− 1](Ce :P�) (22)

for the case of rate-dependent inelastic response.Formulation (20) is identical to the power-type viscosity law used in Cuitino and Ortiz [12]

��=

�0

[(��

g�

)p

− 1]

if ��¿g�

0 otherwise(23)

with �0 := 1=�. An alternative widely used classical creep-type formulation for the evolution of theslip is the power law given by Pierce et al. [7]

�� =

�0

(��

g�

)p

if ��¿0

0 otherwise(24)



where �0 := 1=� denotes a reference slip rate. The ansatz represents a formulation of crystal plastic-ity without an elastic domain. Observe carefully that this formulation in contrast to (20) and (23)does not distinguish between active and inactive slip-systems. Equation (24) only di�erentiates be-tween positive and negative slip directions by forcing the plastic slip to be non-negative. It is there-fore identical to the power law of the form ��= �0 sign(��) (|��|=g�)p used by Hutchinson [18].In this paper we de�ne slip systems for each possible slip direction. As an example, we introduce

2× 12 systems for a typical f.c.c. crystal. Then (24) de�nes an active set of 12 slip systems inthe sence A= {� | sign(��)=+1}.

3. STRESS UPDATE ALGORITHMS FOR SINGLE-CRYSTAL PLASTICITY

We now consider the algorithmic counterpart of the single-crystal plasticity models outlined aboveby constructing multi-surface-type stress update algorithms. Here, we follow conceptually the workof Luenberger [22], Sim�o et al. [23], Cuitino and Ortiz [12], Ortiz and Stainier [24] and Miehe[15; 16]. We �rst consider the rate-independent case and then discuss the modi�cations for therate-dependent case. The key aspect is the treatment of the redundant constraints in the rate-independent theory by means of a generalized-inverse approach and a comparison with viscoplastic-regularization approaches.

3.1. Rate-independent formulation

The basis of the proposed algorithm is an implicit backward Euler update applied to the continuousevolution equations. We consider a typical time interval [tn; tn+1]∈R+ and assume that all variablesat time tn are known. An application of the backward Euler scheme to the evolution equations inTable I yields the system

Up = Upn +m∑

�=1 �P�

A = An +m∑

�=1 �

g� = g�n +

m∑�=1

h(A)[q+ (1− q)��

] �

�¿ 0; ��60; ��=0

(25)

with the incremental plastic parameters � := ��(tn+1− tn) on each slip system �. The initial condi-tions are Up=0, A=0 and g�= �0 at the beginning of the process. In what follows, all variableswithout subscript are assumed to be evaluated at time tn+1. A straightforward algebraic manipula-tion of (25)1 yields the update

Ue= Ue∗ −m∑

�=1 �P� (26)

where Ue∗ := U−Upn denotes a trial value of the elastic strains. Based on these elastic trial strains we

compute the associated trial stresses b∗ :=Ce : Ue∗ and the resolved trial shear stresses ��∗= b∗ :P�



on each slip system �. For

��∗ := b∗ :P� − g�n¡0 ∀�∈S (27)

the step is elastic. If Equation (27) is violated for slip systems �∈S the step is elasto-plastic.In the above expression, S := {1; : : : ; m} denotes the set of possible slip systems. As alreadymentioned above we introduce for a typical f.c.c. crystal m=2× 12=24 possible slip systems bydi�erentiating between positive and negative directions.

3.1.1. Update of plastic slip at given active set. If the ow criterion functions are violated in thesense ��∗¿0 for some �∈S, we have to satisfy the plastic consistency conditions. The mainproblem in this context is that the set of active slip systems

A := {�∈S | �¿0 and ��=0} (28)

at the end of the time interval is not a priori known and not uniquely determined by the trialstate. Thus we have to perform an iterative active set search procedure, which we comment on inthe Section 3.1.2 below.Assume at this stage of the investigation the set A of active slip systems as given. We then com-

pute for all �∈A the actual incremental parameters � from the associated consistency conditions.Based on the representation of the stresses

b=Ce : Ue with Ue= Ue∗ −m∑

�=1 �P� (29)

the consistency conditions can be recast in the form

r� :=��= b :P� − g�=0 for �∈A (30)

The solution of these conditions r �(Ue∗; �)=Ce : Ue∗−∑�∈A �P� :Ce :P�−g�=0 for the plastic

parameters � at frozen trial value Ue∗ of the elastic strains is performed by a local Newton iterationbased on the linearization

r� − D��∗� �=0 for (�; �)∈A (31)

of (30) with the Jacobian matrix

D��∗ :=−@r�

@ �=P� :Ce :P� + h��∗ (32)

Here, we have introduced the hardening moduli

h��∗ := @g�

@ �=

∑�∈A

[q+ (1− q)��

][h(A)�� + h′(A) �

](33)

with the notation h′ :=dh=dA. The resulting update formula for the plastic parameters appears inthe form

� ⇐ � +� � with � �=D��∗−1r� (34)



for the currently active slip systems, i.e. for (�; �)∈A. The computational steps of the localNewton iteration outlined above have to be repeated until convergence is obtained in the sense

|r| :=√ ∑

�∈A

r� 26tol (35)

Alternative approaches to the update (34)2 of the incremental slip in the case where the JacobianD��∗ is ill-conditioned or singular are commented on in the Section 4 below.

3.1.2. Update of the active set of slip systems. The above outlined Newton iteration for the deter-mination of the plastic slip is embedded into the following iterative procedure for the determinationof the active set A of slip systems, which enforces the Kuhn–Tucker-type loading–unloading con-ditions (25)4;5. We start with the �rst estimate

A=An (36)

by assuming that the active set at time tn+1 coincides the one of the previous time step at timetn. If it turns out that this assumption contradicts constraints (25)4;5, we clear the active set andrestart the active set iteration with the condition

Initialization: A= ∅ (37)

Slip systems are then loaded or removed successively one after the other. Here, each single changeof the active set is accompanied by the Newton iteration for the incremental slip outlined in thesubsection above. After consistency has been restored in the sense of the convergence argument(35) for a currently assumed active set, we successively check the conditions (25)4 and (25)5 bymeans of the following update procedure.If some parameters � for �∈A violate the discrete loading conditions (25)4 in the sense �60,

we drop the minimum loaded system(s) from the active working set

Update I: A ⇐ {A=(� := arg[min��] ∈ A)} (38)

and restart the local Newton iteration with the initialization �=0 for all �∈A.Having obtained a converged solution of an active working set with �¿0 for all �∈A, we

check the condition (25)5 by monitoring the yield criteria of the non-active systems. For someviolations in the sense ��¿0 with �∈S=A, we add the maximum loaded system(s) which hasnot been previously in the active set, i.e.

Update II: A⇐{A∪ (� := arg[max��]∈S=A)} (39)

and restart the local Newton iteration with the initialization �=0 for all �∈A. Otherwise weterminate the local iteration.This update procedure, summarized in Table II, turned out to be a save scheme to handle the

complex structural changes of the slip activity with su�cient accuracy for reasonable time steps,compared with alternative active set searches where the working set is updated during the localNewton iteration. See also Section 5.1, where the accuracy for di�erent choices of the time stepis investigated. The sucessive release of the constaints starting with the initialization (37) is inthe spirit of the algorithm proposed by Cuitino and Ortiz [12], see also the recent work Ortizand Stainier [24]. It avoids stress oszillations which may occur if one starts with an estimate ofthe active set deduced from violations of the yield functions associated with the trial state asconsidered for example in Miehe et al. [25].



Table II. Uni�ed stress update algorithm for crystal plasticity.

(i) Elastic predictor check

Database {U; Upn ; An; g�n;An} and projection tensors P� := sym(s�⊗m�) are given. Compute trial elas-

tic strains Ue∗ := U− Upn and trial stresses b∗ :=Ce : Ue∗. Evaluate for all �∈S trial yield functions��∗ := b∗ :P�−g�

n. If (��∗6tol ∀�∈S) set Up= Upn , A=An, g� = g�

n, b= b∗; Cep=Ce; A= ∅and exit.

(ii) Determination of active slip

1. Initialize active set iteration counter iset=0 and trial set A=An.2. Set iset ⇐ iset + 1. If(iset=2 and An 6= ∅) clear active set A= ∅.3. Set initial values for plastic slip iteration � =0∀�∈S.4. Update elastic strains Ue= Ue∗−∑m

�=1 �P� and A=An +

∑m�=1

�.5. Get current stresses b=Ce : Ue and compute for active slip systems �; �∈A:

r� = b :P� − g� p

√��t

� + 1

D��∗ = P� :Ce :P� + h��∗ ( ��t

� + 1)1=p

+ g�� p�t

( ��t

� + 1)1=p−1

with g� = g�n+

∑�∈A h(A)

[q+(1−q)��

] � and h��∗=∑

�∈A

[q+(1−q)��

][h(A)��+h′(A) �].

6. Try factorization D= ldu. If (|d�|¡� :=minC est �1) for one �∈A compute inverse basedon alternative strategies:

(i) singular value decomposition : �D−1 :=V ��−1UT

(ii) ansatz in reduced space : �D−1 := DT{DTD}−1DT

(iii) diagonal shift method : �D−1 := (D+ �1)−1

Else set �D−1 =D−1 based on standard inversion.7. Update incremental plastic slip � ⇐ � +

∑�∈A

�D��∗−1r�

8. If(√∑

�∈A r� 2¿tol)go to 4.

9. Update I of slip activity: If ( �60 for some � ∈ A) drop minimum loaded system(s) A ⇐{A=(� := arg[min��]∈A)} and go to 2.

10. Update II of slip activity: If (��¿0 for some � ∈ S=A) add maximum loaded system(s)A⇐{A∪ (� := arg[max��]∈S=A)} and go to 2.

(iii) Consistent tangent-moduli

Cep :=Ce − ∑�∈A

∑�∈A

�D��∗−1 (Ce:P�)⊗ (P�:Ce)

3.1.3. Algorithmic elastoplastic moduli. The algorithmic expression for the stresses is obtainedby insertion of (26) into (3) and yields the form

b=Ce : Ue∗ − ∑�∈A

� (Ce : P�) (40)



From the consistency conditions r �(Ue∗; �)= 0 in (30) we derive the relationship@ �

@Ue∗ =D��∗−1(P� :Ce) (41)

Based on this result, the algorithmic elastoplastic moduli which govern the increment of the stresses

�b=Cep∗ : �U with Cep∗ := @Ub= @Ue∗b (42)

are obtained in a straightforward manner and take the form

Cep∗ :=Ce − ∑�∈A

∑�∈A

D��∗−1 (Ce :P�)⊗ (P� :Ce) (43)

For elastically isotropic material response the moduli appear in the simple form

Cep∗ := � 1⊗ 1+ 2�P− 4�2 ∑�∈A

∑�∈A

D��∗−1P� ⊗P� (44)

in terms of the bulk modulus � and the shear modulus �, respectively. P := I − 131 ⊗ 1 is the

fourth-order deviatoric projection tensor. In the case of an ill-conditioned or singular JacobianD��∗ due to redundant constraints, the inverse D��∗−1 is replaced by a generalized inverse �D��−1

which we introduce in Section 4 below. Observe that the algorithmic moduli Cep∗ degenerate tothe continuous moduli Cep de�ned in (18) for tn+1 − tn → 0 due to the limit D��∗→D�� in thecase of plastic loading.

3.2. Modi�cations for rate-dependent viscoplastic response

In this section we point out the modi�cations which have to be taken into account, if the rate-dependent formulations of single-crystal plasticity outlined in Section 2.2 are applied. The timeintegration of formulation (20) in a typical time interval yields the incremental slip

�=�t�

[(��+

g� + 1)p

− 1]

(45)

These equations replace the Kuhn–Tucker-type loading conditions (25)4 of the rate-independentformulation. A reformulation of (45) leads to the set of equations

��+ − g�(

p

√��t

� + 1− 1)=0 for �∈A (46)

Insertion of the active ow criterion functions yields the equation

r� := ��∗ − ∑�∈A

�P� :Ce :P� − g� p

√��t

� + 1=0 (47)

These equations represent a modi�cation of the rate-independent consistency equations (30),denoted here as quasi-consistency conditions. They have to be satis�ed for all active slip sys-tems �∈A and can be solved for the plastic parameters � at frozen elastic trial strains by a localNewton iteration. We obtain the identical update formula as given in (34). The only modi�cationconcerns the Jacobi matrix D��∗ in (32), which now takes the modi�ed form

D��∗vis =P

� :Ce :P� + h��∗ ( ��t

� + 1)1=p

+ g�� p�t

( ��t

� + 1)1=p−1

(48)



in terms of the hardening moduli h��∗ de�ned in (33). Observe carefully that we obtain in thealgorithmic setting for the rate-dependent case a formally identical representation for the stressrate as for rate-independent case (42)1. This is in contrast to the continuous formulations (17)and (22). This fact allows us to analyse within the algorithmic setting the rate-independent caseas a limit case of the rate-dependent case, an observation which is of high importance for thesubsequent treatment in Section 4.1.3. We obtain the limit Cep∗

vis →Cep∗ for D��∗vis →D��∗. This is

obtained for �→ 0, p→∞ or �t→∞, i.e. for vanishing viscosity, high power-law exponents orlarge algorithmic time steps.

4. TREATMENT OF ILL-CONDITIONED AND SINGULAR LOCAL JACOBIANS

In the case of multislip the constraints of the multisurface framework outlined above are possiblyredundant. This e�ect occurs in the case of the rate-independent theory under particular hardeningsituations. As pointed out in the model problem below, the e�ect occurs in particular in the caseof ideal plastic and Taylor-type isotropic hardening. As a consequence, the continuous Jacobian(19) as well as the algorithmic Jacobian (32) becomes ill-conditioned or singular. Our subsequentconsiderations focus on the Newton equation (31) which we here represent in the matrix form

DS= r with D∈Rn×n; S∈Rn; r∈Rn (49)

in terms of the Jacobian D and the incremental plastic slip vector S. Here, n6m=2 is the number ofcurrently active slip systems. In Section 4.1 we discuss physically motivated special solutions forthe incremental slip S for the case where D is ill-conditioned or singular. The �rst two approachesare based on the introduction of general inverse matrices which meet optimality conditions. Thethird approach is a simple perturbation technique motivated by the rate-dependent setting of crystalplasticity. Section 4.2 analyses a simple model problem and discusses the physical consequencesof the three regularization methods mentioned above.

4.1. Regularization techniques

4.1.1. Generalized inverse based on singular-value decomposition. The application of singular-value decomposition to the problem of redundant constraints in rate-independent single-crystalplasticity has been proposed by Anand and Kothari [14]. A singular-value decomposition of thematrix D in (49) has the form

D=U�VT (50)

where U∈Rn×n and V∈Rn×n are orthogonal matrices and �∈Rn×n is a diagonal matrix

�=diag[�1; �2; : : : ; �n] with �1¿�2¿ · · ·¿�n¿0 (51)

The decomposition is unique, see for example Golub and Van Loan [26] or Press et al. [27].Formulation (50) is consistent with the two spectral decompositions

DDT=U�2UT and DTD=V�2VT (52)

As a consequence, the columns {u�}�=1; :::; n of U and {v�}�=1; :::; n of V can be computed from thespecial eigenvalue problems

DDTu�= ��2u� and DTDv�= ��2v� (53)



associated with the symmetric matrices DDT and DTD, respectively. We denote the eigenvalues{��}�=1; :::; n as the singular values of D. The �th columns u� and v� of U and V are denoted asthe �th left singular vector and the �th right singular vector of D. For the case where D is asymmetric matrix, we have the situation U=V.Firstly, the singular value decomposition (50) can be used for an accompanying check of the

condition of the matrix D. The inverse condition number C is computed from the singular-valuesde�ned in (51) by

C := �n=�1 for �1¿0 (54)

and we have the situation C→ 0 for the case of a singular matrix D. Secondly, a generalizedinverse can be constructed as follows. The insertion of the decomposition (50) into (49) yieldsthe spectral form of the Newton equation

�� s = r�s for �=1; : : : ; n (55)

based on the vector transformations Ss=VTS=∑n

�=1 �sv

� and rs=UTr=∑n

�=1 �su

�. We con-sider Equation (49) with the spectral counterpart (55) as ill-conditioned or singular for

��¡� with � :=minC�1 (56)

where minC is a machine-dependent admissible minimum inverse condition number which isassumed to be given. Restriction (56) de�nes the numerically admissible range r6n of the matrixD in the sense

�1¿�2¿ · · ·¿�r¿� and �r+1¿�r+2¿ · · ·¿�n¿0 (57)

A special solution of an ill-conditioned or singular system (49) is then obtained in the spectralform

� �s = ��−1r�s for �=1; : : : ; n (58)

based on the de�nition

��−1 :={1=�� for ��¿�0 otherwise (59)

Thus the incremental slip { � �s}�=r+1; :::; n associated with the null space of D have been simply setto zero. Based on (59) one computes the generalized inverse

�D−1 :=V ��−1UT (60)

where the diagonal matrix ��−1 contains the singular values de�ned in (59). The special solutionof (49) is then

�S= �D−1r (61)

As pointed out in Press et al. [27], the approach (61) is a unique solution of system (49) forthe case where the vector r lies in the range of D. Then the special solution �S in (58) and(61) minimizes the norm of all possible solutions in the sense | �S|=min| �S + � �S| where � �S is aperturbation of the solution which lies in the null space of D. For positive slip � �

s ¿0 we realize



at once that this condition is identical to a minimization of the plastic power, which in the caseof ideal plasticity takes the form

D :=∑�∈A

�� =�t¿0 with ��= constant (62)

Taking this interpretation into account, the special solution (61) can be considered as a physicallywell-motivated result. In the case where the right-hand vector r is not in the range of the matrixD, system (49) and (55) has no solution. Then the special solution (61) minimizes the residualexpression |D �S− r|=min|D(�S+��S)− r| in the least-square sense, where ��S is a perturbation ofthe solution which lies in the range of D. Obviously, under the conditions discussed above thisresults again in a minimization of expression (62) for the plastic power.

4.1.2. Generalized inverse based on an ansatz in a reduced space. An alternative approach forthe setup of a general inverse for the solution of the ill-conditioned or singular system (49)can be obtained as follows. The range of the matrix D∈Rn×n is checked out during a standardfactorization process of the form

DT= ldu with d=diag[d1; d2; : : : ; dn] (63)

where l∈Rn×n and u∈Rn×n are lower and upper tridiagonal matrices. During the factorizationprocess, we drop columns of DT with diagonal elements

|d�|¡� with �=minC est �1 (64)

and introduce the rectangular matrix DT ∈Rn×r where r6n is the numerically admissible rangeof the matrix D. The machine-dependent tolerance value �, computed similar to (56), is basedon the given admissible inverse condition number minC and the estmation est �1 for the largesteigenvalue. This estimation can be obtained by the maximum norm

est �1 = ‖D‖∞= max16�6n

{n∑

�=1|D��|

}(65)

The key idea then is that an unique solution for the incremental slips S can only be obtained inthe reduced space Rr with r6n. We therefore introduce the ansatz

�S= DTS (66)

for the special solution of the redundant plastic slips. The insertion of this ansatz into a least-square-type minimization problem yields

min|DS − r| with D :=DDT (67)

This problem has the solution

S=(DTD)−1DTr (68)

in the reduced space, where the matrix is usually denoted as the Moore–Penrose pseudo-inverseof D. For known S, we then obtain a special solution �S from the ansatz (66). This induces therepresentation

�D−1 = DT(DTD)−1D

T(69)



of the generalized inverse alternatively to (60), which avoids the spectral decompositions associatedwith the singular-value-decomposition. Clearly, in the case where the range r of the matrix D isidentical to the dimension n, the generalized inverse reduces to the inverse of D.

4.1.3. Inverse based on a perturbation technique. A third possible approach to overcoming theproblem of the ill-conditioned or singular matrix in rate-independent single-crystal plasticity ismotivated by the limit of viscoplasticity. The local Jacobian (48) of the viscoplastic formulationassumes the form

�D=D+ �1 with �=g��p�t

¿0 (70)

for the case of ideal viscoplasticity at �=0. Here, 1∈Rn×n is the identity matrix. Clearly, thesecond term in (70) regularizes the matrix by shifting the singular values of the rate-independentformulation in (57) in the sense

��= �� + � for �=1; : : : ; n (71)

The basic idea is now to apply a constant shift of the form (70) to an ill-conditioned or singularmatrix D as a purely numerical perturbation. We therefore �rst check in a standard factorizationprocedure the diagonal elements by analogy with the preceding subsection. If we have

|d�|¡� with �=minC est �1 (72)

for only one �, we apply the shift in (70) by using the above expression � as a perturbation. Hereest �1 is an estimation of the largest singular value which can be obtained based on the estimation(65). As a consequence of the shift, we then invert the now well-conditioned matrix Jacobian �Din the sense

�D−1 := (D+ �1)−1 (73)

based on a standard procedure. This is by far the simplest method of overcoming the problem ofan ill-conditioned matrix D.

4.2. Physical motivation. Analysis of a model problem

For an analysis of the performance of the local Newton iteration we consider the problem depictedin Figure 1. It is a strip in tension with two perpendicular slip systems with an angle of 45◦ tothe tensile axis.We assume a typical incremental step with a tensile stress 2�∗ associated with a given trial state,

yielding the resolved shear stresses �1 = �2 = �∗ on the two slip systems indicated in Figure 1. Weassume that the Schmid stresses exceed within the time step [tn; tn+1] under consideration the givencritical values g1n and g2n. Clearly, we then expect that the incremental slip 1 and 2 of the timeinterval assume the identical values provided that the resistance to slip on each systems is the same.These incremental slip are computed on the basis of the local Newton update Equation (34).

Taking into account the geometry depicted in Figure 1, we obtain the representation of residual (47)

r�=

�∗ − �

( 1 + 2

)− g1( ��t

1 + 1)1=p

�∗ − �( 1 + 2

)− g2( ��t

2 + 1)1=p

(74)



Figure 1. Tension of a strip with perpendicular slip systems. The applied tensile stress 2�∗ yieldsthe identical Schmid stress �1 = �2 = �∗ on both systems. In the rate–independent case, where a singu-lar Jacobian D��∗ appears, one expects for physical reasons an identical plastic slip on both systems.

This is provided by the three alternative regularizations of D��∗ discussed in Section 4.

and the local Jacobian matrix (48)

D��∗ =[� ��

]+

[h11∗( �

�t 1 + 1

)1=ph12∗( �

�t 1 + 1

)1=ph21∗( �

�t 2 + 1

)1=ph22∗( �

�t 2 + 1

)1=p]

+

[g1

�p�t

( ��t

1 + 1)1=p−1

0

0 g2�

p�t

( ��t

2 + 1)1=p−1

] (75)

for the model problem under consideration. Here, we have assumed elastically isotropic stressresponse. The current critical shear stresses and hardening moduli take the form

g�=

[g1ng2n

]+ h(A)

[ 1 + q 2

q 1 + 2

](76)

and

h��∗= h(A)

[1 q

q 1

]+ h′(A)

[ 1 + q 2 1 + q 2

q 1 + 2 q 1 + 2

](77)

respectively, with A=An+ 1+ 2. Formulations (74) and (75) represent the viscoplastic formulationgoverned by structure (20). Observe, that the rate-independent limit is obtained for the cases�t→∞, �→ 0 or p→∞, i.e. for very slow processes, for vanishing viscosity or for very largeexponents in the power law (20). All of these limit processes yield the representations

r�=

[�∗ − �( 1 + q 2)

�∗ − �( 1 + q 2)

]− g� and D��∗=

[� �

� �

]+ h��∗ (78)

which are formulations (30) and (32) of the rate-independent theory.



We now start with the analysis of the rate-independent case governed by the two equationsoutlined above. From Equations (78) it can be easily seen that the Jacobian D��∗ associated withthe rate-independent case becomes singular in the case of ideal plasticity with h(A)= 0 as wellas in the case of Taylor-type isotropic hardening associated with the cross-hardening value q=1.In this case the residual and Jacobian in (78) take the form

r�=

[�∗ − ��

( 1 + 2

)− g1n�∗ − ��

( 1 + 2

)− g2n

]and D��∗= �

[1 1

1 1

](79)

with the current sti�ness values ��= �+ h(A) and �= ��+ h′(A)( 1+ 2). Clearly, (79) is a singularmatrix.We now apply the three possible regularization techniques proposed above to the solution of

(49) for the model problem (78). The underlying physically expected result requires that the slipactivity on both systems has to be identical. This is always achieved when the residual remainsunchanged. Recall in this context that the exact tangent matrix in (49) ensures the quadraticconvergence of the Newton iteration. A perturbation of the matrix does not change the physicalresult. The singular-value decomposition (50) is based on the matrices

U=V=1√2

[1 −11 1

]and �=

[4�2 00 0

](80)

The evaluation of (60) then yields the generalized inverse

�D−1∗= 14�

[1 11 1

](81)

and the special solution

�S= 12�

[�∗ − ��

( 1 + 2

)− g1n�∗ − ��

( 1 + 2

)− g2n

](82)

for the plastic slip based on (61). This is the physically expected result where both slip systemsexhibit the identical activity. The ansatz in the reduced space (66) is governed by the non-squarematrix

D= �[11

](83)

The straightforward exploitation of (69) then yields again the generalized inverse (81) and thusto the identical special solution (82) as the above-described approach based on the singular-valuedecomposition. Finally, the perturbation technique (73) results in the inverse

�D−1 =1

�(2� + �)

[� + � −�−� � + �

](84)



Table III. Labels of the slip directions s� and planes m�.

� s� m� � s� m� � s� m�

1 [0 1 �1] (1 1 1) 5 [1 0 1] (�1 1 1) 9 [�1 �1 0] (�1 1 �1)2 [�1 0 1] (1 1 1) 6 [�1 �1 0] (�1 1 1) 10 [0 1 1] (1 1 �1)3 [1 �1 0] (1 1 1) 7 [0 1 1] (�1 1 �1) 11 [1 �1 0] (1 1 �1)4 [0 1 �1] (�1 1 1) 8 [1 0 �1] (�1 1 �1) 12 [1 0 1] (1 1 �1)

of the local Jacobian and thus the special solution

�S= 1(2� + �)

[�∗ − ��

( 1 + 2

)− g1n�∗ − ��

( 1 + 2

)− g2n

](85)

for the plastic slip based on (61).All three regularization approaches considered here provide the same physical reasonable result

for the double slip in Figure 1. The uni�ed stress update algorithm for rate-independent and rate-dependent single-crystal plasticity is summarized in Table II.

5. NUMERICAL EXAMPLES

The formulations described above have been implemented in the program CMP which is basedon of the general-purpose �nite element program FEAP, originally developed by R. L. Taylorand partly documented in Chapter 24 of Zienkiewicz and Taylor [28]. The subsequent numericalexamples are based on a formulation for f.c.c. crystals with 2 × 12=24 possible slip systems.The structure of f.c.c. single crystals is characterized by eight {1 1 1} slip planes and three 〈1 1 0〉directions in each plane. The �rst 12 slip systems are listed in Table III with respect to an or-thogonal frame. The further crystallographically similar slip systems are generated with the coaxialnormal vectors on the opposite facing octahedral planes.The orientation of the f.c.c. unit cell is described by three angles of rotation with respect to

a �xed orthogonal frame. Figure 2 illustrates the relation of the �xed coordinate-system to therotated unit cell. The local axis of the crystals {ei}i=1;3 in (4) are related to the �xed orthogonalframe { �ei}i= 1;3 by the rotation

ei=R �ei with R=�3�2�1 ∈SO(3) (86)

The matrices �1;�2 and �3 represent rotations about the x3-, the x2- and the x3-axes, respectively.They are determined by the explicit expressions

�1 =

cos#1 sin#1 0−sin#1 cos#1 00 0 1

; �2 =

cos#2 0 −sin#2

0 1 0sin#2 0 cos#2

; �3 =

cos#3 sin#3 0−sin#3 cos#3 00 0 1

In the �rst example we investigate a strain-driven simple shear test for a set of 36 di�erentcrystal orientations. A comparison of the rate-independent formulation with the rate-dependentviscoplastic formulation documents the physical acceptance of the proposed approaches to rate-independent crystal plasticity. The second example is concerned with the localization of a tensile



Figure 2. Orientation of the f.c.c. unit cell. The standard cartesian base {�ei}i=1; 3 is rotated to the base {ei}i=1; 3

aligned to the f.c.c. crystal. The (1 1 1) slip plane of the f.c.c. crystal is marked by the shading.

specimen of mono-crystalline material. Here, the in uence of di�erent orientations of the f.c.c.single-crystal cell on the shear band development are investigated. For this case we assume anelastically anisotropic crystal with cubic symmetry.

5.1. Simple shear problem

In this example we assume elastically isotropic crystals. For this purpose we choose the values�=1500:0N=mm2, �=562:5N=mm2 for the bulk modulus and the shear modulus, respectively,and assume a critical Schmid stress �0 = 1:0N=mm2 on all slip systems. In the rate-dependentviscoplastic formulations considered, the strain-rate-sensitivity exponent p has been set to 200 andthe viscosity parameter to �=5× 10−4 s. The crystals are assumed to be stress free in the initialstate.We now de�ne a set of 36 di�erent but equally spaced crystal orientations identical to a problem

considered in Borja and Wren [13]. This set is generated by the angles of rotation #1 and #2. Thethird angle has been set to #3 = 0◦. The angles considered are listed in Table IV.The simple shear problem in the �e1– �e2 plane has been discretized with four bilinear displace-

ment-type elements. In a deformation-driven process we deform the �nite element mesh in 100equal increments up to the �nal value of the shear strain �12 = �21 = 0:01. All other componentsof the strain tensor are zero. Thus, the crystal deforms without volume change. The subsequentnumerical study compares solutions of the rate-independent theory and the rate-dependent theoryfor the 36 crystal orientations listed in Table IV.Table V depicts the active slip systems and the maximum shear stress �xy at the �nal shear strain

�xy =0:01. The linearly dependent slip systems in the generalized inverse approach are denotedby a minus sign. For the visco-plastic and the rate-independent formulation regularized with theperturbation technique we do not get linearly dependent slip systems; thus the minus sign is notvalid in these cases.



Table IV. Orientations of the f.c.c.-unit cell.

No. −#2 −#1 No. −#2 −#1 No. −#2 −#1

1 0 0 13 36 0 25 72 02 0 18 14 36 18 26 72 183 0 36 15 36 36 27 72 364 0 54 16 36 54 28 72 545 0 72 17 36 72 29 72 726 0 90 18 36 90 30 72 907 18 0 19 54 0 31 90 08 18 18 20 54 18 32 90 189 18 36 21 54 36 33 90 3610 18 54 22 54 54 34 90 5411 18 72 23 54 72 35 90 7212 18 90 24 54 90 36 90 90

Table V. Simple shear test. Data at �xy = 0:01.

No. Active slip systems �xy No. Active slip systems �xy

1 1,2,4,5,−7,8,−10,−12 2.44949 19 2,3,6,8,9,−11 1.981682 1,2,4,5,−7,8,−10,−12 1.98168 20 2,8,10,11 1.741293 1,3,5,6,8,−9,−10,−11 1.54327 21 4,10,11 1.243114 2,3,4,6,−7,9,−11,−12 1.54327 22 4,6,10 1.243115 1,2,4,5,−7,8,−10,−12 1.98168 23 2,4,6,8 1.741296 1,2,4,5,−7,8,−10,−12 2.44949 24 2,3,6,8,9,−11 1.981687 1,2,4,5,−7,8,−10,−12 2.32960 25 2,3,5,6,8,−9,−11,−12 2.329608 1,5,8,10 1.87521 26 2,5,9,11 1.875219 1,3,5,6,8,−9,−10,−11 1.46913 27 1,2,4,5,−7,9,−10,−11 1.4691310 2,3,4,6,−7,9,−11,−12 1.46913 28 1,3,4,6,−7,8,−10,−12 1.4691311 2,4,7,12 1.87521 29 3,6,8,12 1.8752112 1,2,4,5,−7,8,−10,−12 2.32960 30 2,3,5,6,8,−9,−11,−12 2.3296013 1,2,4,7,8,−10 1.98168 31 2,3,5,6,8,−9,−11,−12 2.4494914 2,8,10,11 1.74129 32 2,3,5,6,8,−9,−11,−12 1.9816815 6,10,11 1.24311 33 1,2,4,5,−7,9,−10,−11 1.5432716 4,6,11 1.24311 34 1,3,4,6,−7,8,−10,−12 1.5432717 2,4,6,8 1.74129 35 2,3,5,6,8,−9,−11,−12 1.9816818 1,2,4,7,8,−10 1.98168 36 2,3,5,6,8,−9,−11,−12 2.44949

The �nal stresses �xy obtained by the generalized inverse based on the singular-value decom-position, the generalized inverse based on the ansatz in the reduced space and inverse based onthe perturbation technique are identical with the results obtained by the rate-dependent formula-tion. Furthermore, the active slip systems obtained in the rate-independent formulation and therate-dependent formulation are also identical.In the following �gures the results obtained from the viscoplastic formulation are plotted with

dashed and the results from the rate-independent formulation with solid lines. These results areobtained by increasing the �nal boundary displacements in 20 equal steps. Figure 3(a) depictsthe evolution of the equivalent plastic strains of the �rst orientation of the f.c.c. unit cell for therate-independent and rate-dependent formulation versus the applied shear strain �xy. Due to the



Figure 3. Orientation 1: (a) Equivalent plastic strain versus shear strains �xy per cent; (b) Accmulated plasticslips {A� | � = 1; 2; 4; 5; 7; 8; 10; 12} versus shear strains �xy (per cent).

Figure 4. Orientation 10: (a) Accumulated plastic slips {A� | �=6; 11; 4; 2} versus shear strains �xy (per cent);(b) Accumulated plastic slips {A� | �=3; 12; 7; 9} versus shear strains �xy (per cent), rate-independent model

(solid line); viscosity model (dashed line).

symmetry of the problem and the crystallographic structure (characterized by the orientation of thef.c.c. unit cell) it is expected that plastic slip will occur simultaneously in the eight slip directions{s� | �=1; 2; 4; 5; 7; 8; 10; 12} with the same amount. The evolution of the individually accumulatedplastic slip A� on the active slip systems is presented in Figure 3(b). For the crystal orientationsconsidered, the accumulated plastic slip on the active slip systems obtained by the rate-independentand the rate-dependent formulation are identical.Figure 4 presents the numerical results of the rate-independent model and the viscoplastic for-

mulation of the 10th orientation of the unit cell. This orientation is characterized by a high ac-tivity of slip systems which start to develop more or less in a sequence. The plastic parameters{ � | �=6; 11; 4; 2} and { � | �=3; 12; 7; 9} are plotted in Figure 4(a) and 4(b), respectively. Thegraphs coincide quite well for all slip systems.In order to investigate the accuracy of the proposed algorithm for the rate-independent formu-

lation in Table II, di�erent time discretizations of the shear problem have been considered. The�nal value �xy =0:01 for the 10th orientation of the f.c.c. unit cell has been obtained within 100(solid line), 50 (solid line with vertical marks) and 10 (solid line with quadrilateral marks) equaltime steps. The results are documented and compared in Figure 5(a) and 5(b). It can be seen



Figure 5. Orientation 10: (a) Accumulated plastic slips {A� | �=6; 11; 4; 2}, versus shear strains�xy (per cent); (b) Accumulated plastic slips {A� | �=3; 12; 7; 9}, versus shear strains �xy

(per cent) for 100; 50 and 10 equal time steps.

Figure 6. Tension of a strip. Geometry, boundary conditions and discretization with 10×40 elements. Materialimperfection at the shaded element.

that in the deformation-controlled test under consideration the accumulated plastic slip are almostindependent of the step size.

5.2. Tension of a strip

We now consider the localization of a rectangular strip under plane strain conditions, where thez-direction is constrained. Here we treat the problem within the framework of rate-independentsingle-crystal plasticity under quasi-static conditions. The geometry of the strip is characterizedby the relation width=length=6=15:4mm. The system and the discretization with 400 Q1=E5enhanced strain elements are depicted in Figure 6. In this example we assume anisotropic elasticmaterial response with cubic symmetry. For this calculation we set the hardening parameter q=1:4and use the scalar-valued hardening function proposed by Chang and Asaro [20] which has been



Table VI. Material parameters.

Moduli C11 107:300GPaModuli C22 60:900GPaModuli C44 28:300GPaFlow stress �0 0:060GPaSaturation stress �s 0:108GPaInitial hardening h0 0:534GPaLinear hardening hl 0:001GPaHardening parameter q 1:4—

Figure 7. Tension of a strip with the orientation of the fcc-unit cell #1 =#2 =#3 = 0◦. Equivalent plasticstrains at the load parameters: (a) �=101; (b) �=102; and (c) �=110.

extended by an additional linear hardening parameter hl. The material parameters are summarizedin Table VI.In order to trigger the localization of the geometrically perfect specimen we assume a material

imperfection on the left-hand side of the specimen as indicated by the shaded element in Figure 6.Here the ow-stress values in Table VI are reduced by the factor 0:9. The prescribed mechanicalboundary conditions at both ends of the strip allow free contraction of the specimen.As in the previous example we consider a f.c.c. unit cell with di�erent orientations. In a

displacement-controlled numerical test we deform the specimen by a prescribed vertical elongationu= �15:4× 10−5 mm at both ends.Figure 7(a)–7(c) presents the equivalent plastic strains for the orientation #1 =#2 =#3 = 0◦

for the sequence of load parameters � = 101; 102 and 110, respectively. The elongation of thespecimen was obtained in nine equal time steps up to �=90 followed by 20 equal time steps upto the �nal value �=110.



Figure 8. Equivalent plastic strains at load parameter �=120 for the orientations of the fcc-unit cell#2 =#3 = 0◦ and (a) #1 = 0◦; (b) #1 = 15◦; and (c) #1 =− 15◦.

Table VII. Comparison of CPU time.

#1(◦) (i)(s) (ii)(s) (iii)(s)

0 407 423 26015 396 413 24730 315 303 260

Figure 8 depicts the equivalent plastic strains at the load parameter �=120 for three di�erentorientations of the f.c.c. unit cell. The elongation of the specimen was obtained in 120 equal timesteps up to the �nal value �=120. Figure 8(a)–8(c) depict the distribution of the equivalentplastic strains for the �rst angle of rotation #1 = 0◦; #1 = 15◦ and #1 =−15◦, respectively. Theother angles are set constant to #2 =#3 = 0◦.The orientations of the localized bands of equivalent plastic strains depend on the arrangement

of the internal structure, characterized by the orientation of the f.c.c. unit cell. In all examples twoslip bands develop. The �rst example leads to two slip bands which are approximately orientatedunder ±45◦ with respect to the horizontal axis. The second and third orientation lead to moredistinct bands under approximately −30 and 30◦, respectively.For three di�erent orientations of the f.c.c. unit cell, characterized by #1 = 0; 15; 30◦ and

#2 =#3 = 0◦, the computing times are compared for the di�erent regularization techniques. Allcomputations were performed on an IBM RISC 6000-43P-140 workstation under the UNIX op-erating system. The strip is therefore deformed by the above described vertical elongation up toa load parameter of �=200 in 200 equal steps. For these calculations the hardening parameter qis set to 1. Table VII compares the computing times for the local solution procedure based on



(i) the generalized inverse based on the singular-value decomposition, (ii) the generalized inversebased on ansatz in the reduced space and (iii) the inverse based on the perturbation technique.Observe that the solution of the boundary-value problem under consideration based on the

perturbation technique needs less CPU time then the other regularization techniques.

6. CONCLUSION

A new uni�ed fully implicit multisurface-type return algorithm for both the rate-independent andthe rate-dependent setting of singly crystal plasticity has been proposed. The algorithm is endowedwith three alternative approaches to the regularization of the possible redundant slip activities ofthe rate-independent theory. This includes the use of alternative generalized inverses of the Jaco-bian of the currently active yield criterion functions as well as a new diagonal shift regularizationtechnique, motivated by a limit of the rate-dependent theory. Analytical investigations and numeri-cal experiments showed that all three approaches result in similar physically acceptable predictionsof the active slips of rate-independent single-crystal plasticity, while the new proposed diagonalshift method is the most simple and e�cient concept.

ACKNOWLEDGEMENTS

Partial support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG) under grantSFB 404=A8.

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