Stress integration schemes for novel homogeneous anisotropic hardening model

Comput. Methods Appl. Mech. Engrg. 247–248 (2012) 73–92

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Comput. Methods Appl. Mech. Engrg.

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Stress integration schemes for novel homogeneous anisotropic hardening model

Jinwoo Lee a, Myoung-Gyu Lee a,⇑, Frédéric Barlat a, Ji Hoon Kim b

a Graduate Institute of Ferrous Technology (GIFT), Pohang University of Science and Technology (POSTECH), 31 Hyoja-dong, Nam-gu, Pohang, Gyeong-buk 790-784, Republic of Koreab Material Deformation Group, Korea Institute of Materials Science, Changwon, Gyeong-nam 642-831, Republic of Korea

a r t i c l e i n f o

Article history:Received 4 March 2012Received in revised form 19 July 2012Accepted 25 July 2012Available online 3 August 2012

Keywords:Elasto-plasticityStress integration algorithmAnisotropic hardeningIso-error mapClosest point projection methodCutting plane algorithm

0045-7825/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.cma.2012.07.013

⇑ Corresponding author. Tel.: +82 54 279 9034; faxE-mail address: [email protected] (M.-G. Lee).

a b s t r a c t

Numerical formulations and implementation of stress integration algorithms in the elasto-plastic finiteelement method are provided for the homogeneous yield function-based anisotropic hardening (HAH)model. This model is able to describe complex material behavior under non-monotonic loading condi-tions. Two numerical algorithms based on the semi-explicit and fully implicit schemes are comparedin terms of accuracy. To efficiently treat the yield locus distortion when the strain path changes, amulti-step Newton–Raphson method is proposed to calculate the first and second derivatives of theHAH yield surface. For the validation of the developed numerical algorithms, the r-value anisotropy iscompared for the conventional yield model with classical isotropic hardening and for the HAH model.Moreover, detailed error analysis is presented using iso-error maps. The results show that the fully impli-cit stress integration algorithm based on the closet point projection method leads to better accuracy ingeneral. However, the semi-explicit algorithm also provides comparable accuracy if an appropriate timeincrement is chosen. Furthermore in spite of the yield surface distortion, the developed numerical algo-rithms can successfully update stress with the equivalent level of the error for the conventional yieldmodel.

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1. Introduction

In sheet metal forming, a numerical approach such as finite ele-ment simulation has been frequently utilized for reducing develop-ment time and cost through optimization of the forming process.For accurate finite element simulations of sheet metal forming pro-cess, constitutive models that account for the material behaviorunder various loading histories are required. Then, significant ef-forts have been devoted to the development of constitutive equa-tions that are able to reproduce the stress–strain behavior underabrupt changes of the loading direction [1–8]. In finite elementanalysis of the elasto-plasticity, the main elements of the constitu-tive descriptions are the yield function and the hardening law. Theconcepts describe the material initial plastic anisotropy and itsevolution during deformation, respectively. The phenomenologicalyield function concept is widely used for its mathematical simplic-ity and low computational effort in the finite element analysis. Forexample, Hill’s quadratic yield function [9] has been widely used inmany industrial applications for its simplicity. However, this yieldfunction was found to lead to limited accuracy for highly aniso-tropic materials like aluminum alloys. Thus, more improved yieldfunctions with non-quadratic property were developed and ap-plied by several researchers [10–13].

ll rights reserved.

: +82 54 279 9299.

Since the usual sheet metal forming processes involve bendingand stretching, the material experiences tension followed by com-pression. From the material modeling aspect, the classical isotropichardening has been successful in providing good predictions offormability or springback for conventional low strength materialsbecause these particular materials do not show significant asym-metry in tension and reversal. However, as materials with light-weight or with increasing strength emerge, in the stress–straincurve under reverse loading, four main characteristics are ob-served: (1) the Bauschinger effect (lower yield stress when the loadis reversed), (2) the transient behavior (rapid change of work hard-ening rate), (3) work-hardening stagnation (low work hardeningrate), and (4) permanent softening (permanent distinction be-tween forward and reverse curve). The four features are schemat-ically shown in Fig. 1. These deformation characteristics are a goodexample that cannot be captured by the isotropic hardening model,and become strong motivation for the development of more ad-vanced constitutive models that can improve the prediction capa-bility in sheet metal forming with advanced sheet materials.

In order to represent the Bauschinger effect, Prager [14] andZiegler [15] proposed linear kinematic hardening. Armstrong andFrederic [16] developed a nonlinear kinematic hardening modelto predict the transient hardening behavior and later Chaboche[17] further improved it to better predict cyclic deformationfeatures. Teodosiu and Hu [18] included the effect of arbitrarystrain-path changes, so the model could capture the Bauschinger

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Tru

e st

ress

Accumulated absolute true strain

Pre-strain Bauschinger effect

Transient behavior

Work-hardening stagnation

Monotonic

Permanentsoftening

Fig. 1. A schematic forward and reverse stress–strain curve after pre-strain.Absolute stress value for the reversed curve (dash) is shown to be compared withforward one (line).

74 J. Lee et al. / Comput. Methods Appl. Mech. Engrg. 247–248 (2012) 73–92

effect, transient behavior, permanent softening and work-hardening stagnation. For sheet metal forming applications, manyresearchers implemented kinematic hardening laws into the finiteelement simulations to predict formability and springback [19–27].

As an alternate approach, distortional plasticity associated withkinematic hardening was also proposed. For example, Ortiz and Po-pov [28] developed a distortional hardening model with an effec-tive stress quantity which affects the yield surface evolution. Thedistortion of the yield surface was described with a fourth ordertensor by Voyiadjis and Foroozesh [29] and Feigenbaum and Daf-alias [30,31]. Wu [32,33] used the convected coordinates to discussthe evolution of the yield surface including proportional expansion,translation, nonlinear distortions and rotation. Francois [34] pro-posed a so-called egg-shape yield surface expressed with a ‘‘dis-torted stress deviator’’.

More recently, Barlat et al. [35] introduced a novel approachcalled homogeneous yield function-based anisotropic hardening(HAH), which can describe the complex anisotropic hardeningbehavior under strain path changes. The model is able to capturethe important anisotropic hardening features such as the Bausch-inger effect, transient behavior and permanent softening. In addi-tion, the HAH model along with a dislocation density-basedhardening law can also describe relevant material phenomena insheet metals undergoing various strain path changes.

In order to use the rate independent elasto-plastic models innonlinear finite element analysis of boundary value problems,the constitutive equations are numerically integrated over a dis-crete sequence of time steps. Therefore, appropriate integrationalgorithms with proper implementations into the finite elementcode are important for overall efficiency and robustness. In otherwords, the accuracy, convergence and stability of the global itera-tive solution are highly dependent on the integration algorithmsused in the simulations. In general, implicit integration approachesfor elasto-plasticity are robust, but they are usually lengthy andcumbersome. On the other hand, explicit or semi-explicit integra-tion approaches have been attractive alternatives to the implicitschemes because of their simplicity and effectiveness. In particular,explicit finite element analyses have been prevalent as sheet metalforming processes become complex.

The stress integration schemes in sheet metal forming simula-tions are often based on the predictor–corrector method. Wilkins[36] proposed a return mapping procedure based on two succes-sive steps. The first step is purely elastic and a trial stress state iscalculated. The second step is a plastic corrector step in whichthe stress is projected onto the yield surface using a flow rule in

order to guarantee the consistency condition. The Closest PointProjection method (denoted as CPPM hereafter) with its fully im-plicit nature has become popular because it is unconditionally sta-ble for various elasto-plasticity models [37–39]. A quadraticconvergence rate with unconditional stability for Newton–Raph-son iterations can be obtained when an algorithmic consistent tan-gent modulus is used. The fully implicit algorithm guaranteed thestability and quadratic convergence for the classical J2 plasticity,but additional efforts should be made for more complex constitu-tive models. In particular, the closed-form second derivatives ofthe yield function often result in lengthy and cumbersome calcula-tions. There have been many approaches to solve the convergenceissue especially when more advanced yield functions are used. Oneexample is the multi-stage return mapping algorithm proposed byYoon et al. [40] in which the deformation path is sub-divided intomany small steps during the time increment.

In contrast to CPPM, since explicit integration does not need thesecond derivatives of the yield function, the overall implementationof the stress update algorithms is generally simple and straightfor-ward. However the major drawback of this integration algorithm isthat it provides conditionally stable solution. Therefore, carefultime step control is essential to obtain numerical stability. Due tothis limitation, fully explicit schemes are usually less efficient thanimplicit schemes. A new class of integration algorithm wasproposed by Simo and Ortiz [41] called General convex CuttingPlane Method (GCPM). The GCPM has a semi-explicit characteristicand has been attractive due to its simplicity and efficiency. Thealgorithm takes advantage of the known reference stress state todetermine the direction of plastic flow. However, the major defi-ciency of GCPM lies in the weak enforcement of the consistencycondition, which might produce drift-off errors if the proper timestepping is not provided [42].

In this paper, two stress integration algorithms based on CPPMand GCPM are considered for the integration of the new distor-tional hardening model recently proposed by Barlat et al. [35],which is suitable to the sheet metal forming simulations. In partic-ular, the model does not rely on the conventional kinematic hard-ening to represent the Bauschinger effect but severe distortion andrecovery repeatedly. Therefore, the availability and effectiveness ofthe present stress integration algorithms should be studied in de-tail. In Section 2, the HAH model is highlighted with new featuresthat reproduce the complex material behavior especially whenloading direction changes. In Section 3, the two algorithms areimplemented into the general-purpose finite element program,ABAQUS via the user-material subroutine. The finite element for-mulations are presented with special attention to the fluctuatingterms. In Section 4, the performance of the considered integrationalgorithms is evaluated and the iso-error maps test the accuracy ofboth algorithms.

2. Homogeneous yield function-based anisotropic hardening(HAH) model

Barlat et al. [35] proposed the homogeneous yield function-based anisotropic hardening model, called hereafter HAH model,which accounts for asymmetric yielding after pre-strain. This crite-rion is defined as

UðsÞ ¼ ð/q þ /qhÞ

1q ¼ ð/q þ f q

1 jhs : s� jhs : sjjq þ f q2 jhs : sþ jhs : sjjqÞ

1q

¼ �r; ð1Þ

where stable component /, fluctuating component /h, two statevariables f1, f2 and the microstructure deviator hs. The stablecomponent can be any regular yield function describing the materialanisotropy, while the fluctuating component introduces a yieldsurface distortion resulting from the loading history. The

J. Lee et al. / Comput. Methods Appl. Mech. Engrg. 247–248 (2012) 73–92 75

microstructure deviator evolves when the strain path changes. Thenormalized quantity hs is defined as

hsij ¼

hsijffiffiffiffiffiffiffiffiffiffiffiffiffiffi

83 hs

klhskl

q ð2Þ

using Einstein summation convention. The tensor hs is initialized tothe stress deviator s corresponding to the first plastic strainhardening.

In Eq. (1), q is a constant exponent while f1 and f2 are expressedwith two state variables g1 and g2 as follows

f1 ¼ ðg�q1 � 1Þ

1q and f 2 ¼ ðg

�q2 � 1Þ

1q: ð3Þ

The two state variables g1 and g2 correspond to the ratio of the flowstress to that for isotropic hardening law, i.e., r = gkriso, where k = 1or 2 depending on the sign of hs : s, and riso is the stress for the iso-tropic hardening material.

Fig. 2 shows an example of yield locus when the material issubjected to uniaxial tension. The parameters in Eq. (3) are theexponent q = 2 and the coefficient g1 = 0.2, g2 = 1.0.

When the material is plastically deformed, the proposed evolu-tion laws for these state variables are following.

dg1

d�e¼

k2 k3Hð0ÞHð�eÞ � g1

� �ðhs : s P 0Þ;

k1g4�g1

g1ðhs : s < 0Þ;

8<: ð4Þ

dg2

d�e¼

k1g3�g2

g2ðhs : s P 0Þ;


� �ðhs : s < 0Þ;

8<: ð5Þ

dg3

d�e¼ 0 ðhs : s P 0Þ;

k5ðk4 � g3Þ ðhs : s < 0Þ;

(ð6Þ

dg4

d�e¼ k5ðk4 � g4Þ ðhs : s P 0Þ;

0 ðhs : s < 0Þ;

(ð7Þ

dhs

d�e¼

k s� 83 hsðhs : sÞ

� �ðhs : s P 0Þ;

k �sþ 83 hsðhs : sÞ

� �ðhs : s < 0Þ:

8><>: ð8Þ

Fig. 2. Distortion of the yield locus predicted by the HAH model.

In the above equations, k and k1�5 are constant coefficients. Hð�eÞ isthe classical hardening curve. The two state variables g1 and g2 con-trol the Bauschinger effect (early re-yielding) and the transient hard-ening behavior depending on the sign of hs : s. The other two statevariables g3 and g4 describe permanent softening, that is, whenthe flow stress after load reversal is permanently lower than themonotonic curve even after large plastic strains. Note that these sys-tems reduce to the classical isotropic hardening law when k1�5 = 0.

The anisotropy using coefficients of the stable yield functionand those of the isotropic hardening curve can be determinedusing uniaxial and balanced biaxial tension test results. The param-eters k1�5 are obtained from forward-reverse loading tests such asuniaxial tension–compression or shear tests with various pre-strains. The parameter k, related to the rotation of the microstruc-ture deviator can be determined from non-proportional loadingexperiments such as uniaxial tension followed by tension in theorthogonal direction.

3. Stress update algorithms of HAH approach in elasto-plasticFEM

The HAH model for elasto-plasticity has been implemented inthe implicit finite element code using the material subroutine ofthe commercial software ABAQUS/Standard. At the equilibriumstate, a strain increment is calculated at each node of the finite ele-ment. The strain increment can be expressed by the discrete truestrain increment based on the incremental deformation theory[43]. For a given strain increment De, the numerical formulationsupdate the stress tensor. The total strain increment is assumed todecompose into elastic Dee and plastic Dep parts and the linearelastic relationships are used as follows:

De ¼ Dee þ Dep ð9Þ

Dr ¼ C � Dee; ð10Þ

where C is stiffness tensor. The equivalent plastic strain incrementD�� can be calculated using the nature of a first degree homogeneousfunction �rðrÞ ¼ r : @�r

@r and the associated flow rule

D�� ¼ r : Dep

�rðrÞ ¼r : Dc @�r

@r�rðrÞ ¼ Dc

�rðrÞ�rðrÞ ¼ Dc ð11Þ

and

Dep ¼ Dc@�r@r¼ D��

@�r@r

: ð12Þ

The state variables in the HAH model are updated when the equiv-alent plastic strain increment is obtained. In general, D�e is the onlyunknown to be determined during the stress integration procedureand other state variables can be updated once this variable is ob-tained. Numerous numerical schemes are available to solve the un-known parameter D�e in the elasto-plastic formulations [44–48]. Inthe present study, two stress update algorithms namely, the generalconvex cutting-plane method (GCPM) and the closest point projec-tion method (CPPM), are investigated considering the special natureof the novel HAH approach.

3.1. General convex cutting-plane algorithm

In the general convex cutting plane method (GCPM), the advan-tage is that the unknown parameter D�e is obtained only from theconsistency condition [49,50] expressed as

�rðr0 þ Dr; f1;0 þ Df1;0; f2;0 þ Df2;0; hs0 þ DhsÞ ¼ Hð�e0 þ D�eÞ; ð13Þ

where the subscript ‘0’ means the (known) initial value from theprevious time step.


For a given total strain increment Den+1 at the current time step,the trial stress is assumed to be pure elastic state and calculatedusing Eq. (10):

rTnþ1 ¼ rn þ CDenþ1; ð14Þ

where ‘T’ denotes the trial state and subscripts n and n + 1 are theprevious and current time steps, respectively. If the followingcondition

H ¼ ð/ðrTÞq þ /hðrTÞqÞ1q � Hð�enÞ ¼ �rðrTÞ � Hð�enÞ < 0 ð15Þ

is satisfied, the stress state is pure elastic with the same value of thetrial stress and the algorithm goes to the next time step by updatingthe stress only (other state variables are preserved). If the conditionin Eq. (15) is not met or if H > 0, the current time step is consideredas plastic and the predictor–corrector iterative scheme begins withthe trial elastic stress as an initial estimate. Then, the variation ofthe equivalent plastic strain increment for the kth iteration is ob-tained by the linearization of Eq. (15) as follows:

dðD�enþ1Þðkþ1Þ ¼ � HðkÞ

@H@D�enþ1

� �ðkÞ ; ð16Þ

where

@H@D�enþ1

¼ @H@rnþ1

@rnþ1

@D�enþ1þ @H@Hnþ1

@Hnþ1

@D�enþ1ð17Þ

with@H@rnþ1

¼ @�rnþ1

@rnþ1;

@H@Hnþ1

¼ �1 and@rnþ1

@D�enþ1¼ �C

@�rnþ1

@rnþ1:

ð18Þ

In the above relationships, the higher order terms caused by thevariation of the stress components with respect to the variation ofthe effective strain increment are neglected. Finally, using Eqs.(16)–(18), the following estimate is obtained

dðD�enþ1Þðkþ1Þ ¼ HðkÞ

@�rnþ1@rnþ1

C @�rnþ1@rnþ1

þ H0ð�enþ1Þ� �ðkÞ ð19Þ

where H0ð�enþ1Þ ¼ @Hð�enþ1Þ@�e

� �is the slope of the monotonic flow curve.

Note on the implementation of HAH (1): Since the HAH modeldoes not involve additional terms such as the back stress tensor,

Fig. 3. Geometric interpretation: (a) the semi-e

the variation of the equivalent plastic strain increment in Eq.(19) preserves the same simple form as the one in the classical iso-tropic hardening formulation.

During the iterations the stress, plastic strain and other statevariables in the fluctuating components uh should be updated withEqs. (4)–(8).

dðDg1Þðkþ1Þ ¼


� �� dðD�enþ1Þðkþ1Þ ðhs : s P 0Þ;

k1g4�g1

g1� dðD�enþ1Þðkþ1Þ ðhs : s < 0Þ;

8<: ð20Þ

dðDg2Þðkþ1Þ ¼

k1g3�g2

g2� dðD�enþ1Þðkþ1Þ ðhs : s P 0Þ;


� �� dðD�enþ1Þðkþ1Þ ðhs : s < 0Þ;

8<: ð21Þ

dðDg3Þðkþ1Þ ¼

0 ðhs : s P 0Þ;

k5ðk4 � g3Þ � dðD�enþ1Þðkþ1Þ ðhs : s < 0Þ;

(ð22Þ

dðDg4Þðkþ1Þ ¼ k5ðk4 � g4Þ � dðD�enþ1Þðkþ1Þ ðhs : s P 0Þ;

0 ðhs : s < 0Þ;

(ð23Þ

dðDhsÞðkþ1Þ ¼k s� 8

3 hsðhs : sÞ� �

� dðD�enþ1Þðkþ1Þ ðhs : s P 0Þ;

k �sþ 83 hsðhs : sÞ

� �� dðD�enþ1Þðkþ1Þ ðhs : s < 0Þ;

8><>:

ð24Þ

epðkþ1Þ

nþ1 ¼ epðkÞ

nþ1 þ d D�enþ1@�rnþ1

@rnþ1

� �ðkþ1Þ

; ð25Þ

D�eðkþ1Þnþ1 ¼ D�eðkÞnþ1 þ dðD�enþ1Þðkþ1Þ

; ð26Þ

gðkþ1Þi;nþ1 ¼ gðkÞi;nþ1 þ dðDgi;nþ1Þ

ðkþ1Þ for i ¼ 1 � 4; ð27Þ

hsðkþ1Þnþ1 ¼ hsðkÞ

nþ1 þ dðDhsnþ1Þ

ðkþ1Þ: ð28Þ

The iterative scheme continues until the consistency condition issatisfied within a prescribed numerical tolerance

Hðkþ1Þnþ1 ¼ / rðkþ1Þ

nþ1

� �qþ /h rðkþ1Þ

nþ1

� �q� �1q

� H �eðkþ1Þnþ1

� �< Tol: ð29Þ

xplicit and (b) the fully implicit algorithm.


The geometrical interpretation of the GCPM is illustrated inFig. 3(a).

Note on the implementation of HAH (2): The gradient of the yieldfunction @ �rnþ1

@rnþ1in Eq. (19) should be carefully determined consider-

ing the changes of the microstructure deviator, loading directionand the evolution of plastic parameters because the yield surfacein the HAH model is changing with respect to the evolution ofthe state variables. This can be handled by a multi-step Newton–Raphson algorithm proposed by Lee et al. [51]. A brief illustrationof the developed multi-step algorithm will be given in the nextsection.

The consistent tangent modulus Cep, which preserves thequadratic rate of convergence, is required for the implicit finiteelement method. It is obtained by enforcing the consistencycondition in Eq. (15)

dHnþ1 ¼ d½ð/nþ1Þq þ ð/h;nþ1Þ

q�1q � dHnþ1ð�enþ1Þ

¼ @�rðrnþ1Þ@rnþ1

drnþ1 � H0 � d�enþ1 ¼ 0; ð30Þ

where H0 � @�rð�enþ1Þ@�enþ1

.

The stress increment drn+1 in the above equation is defined as

drnþ1 ¼ C denþ1 � d�enþ1@�r

@rnþ1

� �: ð31Þ

Then, the stress components are obtained by substituting Eq.(30) into Eq. (31)

drnþ1 ¼ C�C @�r@rnþ1

� C @�r@rnþ1

@�r@rnþ1

C @�r@rnþ1

þ H0ð�enþ1Þ

!denþ1 ¼ Cepdenþ1; ð32Þ

where � stands for the open product; i.e., (a � b)ij = aibj.Recently, Cardoso and Yoon [42] reported that small time-steps

can provide valid results when the semi-explicit algorithm is usedwith the implicit finite element approach. More in-depth investi-gation on the accuracy and efficiency of the algorithm will be pre-sented in the following section.

3.2. Closest point projection method (CPPM)

In the closest point projection method (CPPM), the gradient ofthe yield surface is evaluated at the current step in a fully implicitmanner. Consequently, the variation of the normal in the yield sur-face should be taken into account during the iteration process in Eq.(29). Compared to the previous GCPM based on the semi-explicitalgorithm, the formulation for the solution of the nonlinear equa-tions become more cumbersome but it provides more accuratesolutions. With the fully implicit scheme, the exact linearizationcan be achieved together with the Hessian matrix or the secondderivative of the yield function to determine the consistent tangentmodulus.

For the condition H > 0 in Eq. (15), the following seven residualsRi (i = 1�7) can be defined from the previous Eqs. (9)–(13) for asub-step k,

R1ðD�eðkÞnþ1Þ ¼ �DepðkÞ

nþ1 þ D�eðkÞnþ1

@�rðrðkÞnþ1Þ@rðkÞnþ1

; ð33Þ

R2ðD�eðkÞnþ1Þ ¼ �rðrðkÞnþ1Þ � Hð�en þ D�eðkÞnþ1Þ; ð34Þ

R3ðD�eðkÞnþ1Þ ¼ gðkÞ1;nþ1 � g1;n � Dg1ðD�eðkÞnþ1Þ; ð35Þ




R7ðD�eðkÞnþ1Þ ¼ hsðkÞnþ1 � hs

n � DhsðD�eðkÞnþ1Þ: ð39Þ

By applying the Talyor’s expansion at the current configuration, thefollowing linearized residuals in Eqs. (33) and (34) for the kth iter-ation at the current time are

R1ðD�eðkÞnþ1Þ þ N�1ðD�eðkÞnþ1Þ : dðDrnþ1Þðkþ1Þ þ@�rðrðkÞnþ1Þ@rðkÞnþ1

dðD�enþ1Þðkþ1Þ ¼ 0;

ð40Þ

R2ðD�eðkÞnþ1Þ þ@�rðrðkÞnþ1Þ

@rdðDrnþ1Þðkþ1Þ � H0ð�eðkÞnþ1ÞdðD�enþ1Þðkþ1Þ ¼ 0:

ð41ÞIn Eqs. (40) and (41), the quantities were introduced,

NðxÞ ¼ Sþ D�eðkÞnþ1

@2 �rðrðkÞnþ1Þ@rðkÞnþ1@r

ðkÞnþ1

!�1

; �eðkÞnþ1 ¼ �en þ D�eðkÞnþ1; ð42Þ

with S = C�1 denoting the compliance tensor.Solving Eq. (40) for d(Drn+1)(k+1) leads to

dðDrnþ1Þðkþ1Þ ¼ NðD�eðkÞnþ1Þ : �R1ðD�eðkÞnþ1Þ �@�rðrðkÞnþ1Þ@rðkÞnþ1

dðD�enþ1Þðkþ1Þ

!:

ð43Þ

Finally by substituting Eq. (43) into Eq. (41) the increment of theequivalent plastic strain dD�e becomes

dðD�enþ1Þðkþ1Þ ¼R2ðD�eðkÞnþ1Þ � R1ðD�eðkÞnþ1Þ : NðD�eðkÞnþ1Þ :

@�rðrðkÞnþ1Þ

@r@�rðrðkÞ

nþ1Þ

@r : NðD�eðkÞnþ1Þ :@�rðrðkÞ

nþ1Þ

@rðkÞnþ1

þ H0ð�eðkÞnþ1Þð44Þ

Then the plastic multiplier and other plastic state variables areupdated for step (k + 1) with Eqs. (20)–(28). This iterative cyclecontinues until the size of the residuals Ri (i = 1–7) are smaller thanthe prescribed tolerances (e.g., 10�6). The schematic illustration ofthe CPPM is shown in Fig. 3(b).

To preserve the quadratic rate of convergence in the global fi-nite element equilibrium, the consistent elasto-plastic tangentmodulus Cep is necessary. By differentiating Eqs. (10) and (12),

drnþ1 ¼ Cnþ1 : ðdenþ1 � depnþ1Þ; ð45Þ

depnþ1 ¼ Dcnþ1

@2 �r@rnþ1@rnþ1

: drnþ1 þ dDcnþ1@�r

@rnþ1ð46Þ

Thus, inserting Eq. (46) into Eq. (45) gives

drnþ1 ¼ Nnþ1 : denþ1 � dDcnþ1@�r@rnþ1

� �: ð47Þ

In addition, differentiating the consistency condition and Eq. (47)yields

dDcnþ1 ¼@�r

@rnþ1: Nnþ1 : denþ1

@�r@rnþ1

: Nnþ1 : @�r@rnþ1

þ H0nþ1

: ð48Þ

Finally, substituting Eq. (48) into Eq. (47) provides

drnþ1 ¼ Nnþ1 �Nnþ1 : @�r

@rnþ1� Nnþ1 : @�r

@rnþ1

@�r@rnþ1

: Nnþ1 : @�r@rnþ1

þ H0nþ1

!denþ1 ¼ Cepdenþ1: ð49Þ

3.3. Special algorithmic issues in the HAH model

Contrary to the classical hardening models including isotropicand kinematic hardening models, the HAH yield surface is


continuously changing during deformation. Therefore, the gradi-ents of the evolving yield surface in the HAH model should be care-fully determined.

In a mathematical viewpoint, Eq. (1) indicates that the gradientof the HAH yield function depends on both stable and fluctuatingterms. Without the latter term, it would simply be the gradientof the regular yield surface, which can be obtained by the normal-ity rule. However, the difficulty occurs when the gradient of thefluctuating term with respect to the current stress is calculated be-cause the parameters f1 and f2 (or equivalently g1 and g2) are func-tions of stress or equivalent plastic strain.

The first derivative of the HAH model was derived in the previ-ous work (refer to Lee et al. [51]). Thus, only the result is shown inEq. (50) for the case of hs : s < 0.

@�r@r¼ 1

q�r1�q q/q�1 @/

@rþ qð�g�ðqþ1Þ

1 Þk1g4 � g1

g1

� �j2hs : sjq

� ��

� �C :@�r@r

� ��1

þ ðg�q1 � 1Þð�2qqÞjhs : sjq�1

� s : k �sþ 83

hsðs : hsÞ� �

� �C :@�r@r

� ��1 !

þ hs :@s@r

!!

ð50Þ

where C ¼ kI� Iþ 2lJ with the second-order identity tensor I andthe symmetric part of the fourth-order identity tensor J.

Alternatively, in a component form, Eq. (50) becomes

@�r@rij

� �¼ �r1�q /q�1 @/

@rij

� �þ A � Dij þ B1 � Dij þ B2 � hs

mn �@smn

@rij

� �� ;

ð51Þ

where A¼ k1ð�g�ðqþ1Þ1 Þ g4�g1

g1

� �j2hs

klskljq; B2 ¼ ½ð�2Þðg�q1 �1Þj2hs

klskljq�1�;

B3 ¼ k �smn þ83ðhs

mnÞðhsklsklÞ

� �� smn

� �; B1 ¼ B2 � B3 and

Dij ¼ � Cijlk :@�r@rkl

� ��1

¼inv 0ijdet

:

@�r@r is given as an implicit function of r, g1 and hs in Eq. (51). Thesolution of Eq. (51) should be obtained by an iterative method. Inthis study, a new multi-step Newton–Raphson method is employed

Fig. 4. Schematic representation of a multi-step method to calculate the normalgradient.

as illustrated in Fig. 4. First, intermediate locals corresponding todifferent values gðkÞi are considered. If hs : s is positive, then the va-lue of g2 is subdivided. And the solution of the first step m(1) is cal-culated from the initial guess m(0), which is the normal gradient ofstable yield function. m(k) is the following definition,

mðkÞ ¼@�rðr; gðkÞi Þ

@r: ð52Þ

The next iterative solutions are obtained from the previous solutionand it continues until subdivided g1 or g2 reach the destination. Thismethod is similar with a sub-stepping procedure which is usefulwhen the plastic potential exhibits strong variations of curvature[52].

Since the solution procedure of the proposed method should becarried out at each integration point during the iterations, it mightbe computationally inefficient. As an alternate way to reduce thecomputation cost, a simplification of the formulation is suggested.In this simplified strategy, the parameters related to the fluctuatingterms are frozen without updating the equivalent plastic strainincrement during the calculation. However other variables, includ-ing stress and microstructure deviator, are updated as in the regu-lar numerical treatment. Then, the simplified approach for the HAHgradient gives

@�r@r¼ 1

q�r1�q q/q�1 @/

@r� ð2qÞf q

1 j2hs : sjq�1 � @ðhs : sÞ@r

!

¼ �r1�q /q�1 @/@r� 2f q

1 j2hs : sjq�1 � hs :@s@r

� �: ð53Þ

The Hessian matrix for the HAH model is derived from Eq. (50)using the chain rule.

Case : hs : s < 0 (similar for hs : s P 0)

@2 �r@r@r

" #¼ @

@r�r1�q /q�1 @/

@r

� �þ A � Dþ B1 � Dþ B2 � hs :

@s@r

� ��

¼ @�r1�q

@r�Wþ �r1�q � @W

@r

¼ ð1� qÞ�r�q @�r@r�Wþ �r1�q � @W

@r;

ð54Þ

where W is /q�1 @/@r

þ A � Dþ B1 � Dþ B2 � ðhs � @s

@rÞ and

@W@r

� �¼ @

@r/q�1 @/

@r

� �þA � inv0

det

� �þ B1 �

inv0

det

� �þ B2 � hs :

@s@r

� ��

¼ @

@r/q�1 @/

@r

� �� þ @

@rA � inv0

det

� ��

þ @

@rB1 �

inv0

det

� �� þ @

@rB2 � hs :

@s@r

� �� :

ð55Þ

First, in the calculation of the above equation, the second derivative

of the stable part @2/@r@r should be obtained. For example, the second

derivative for the Yld2000-2d yield function is given in theAppendix. Then, by the chain rule the other terms in Eq. (55)becomes

@

@rA � inv0

det

� �� ¼ @A@r� inv0

det

� �þ A � @

@rinv0

det

� �; ð56Þ

@

@rB1 �

inv0

det

� �� ¼ @B1

@rinv0

det

� �þ B1 �

@

@rinv0

det

� �; ð57Þ

@

@rB2 � hs :

@s@r

� �� ¼ @B2

@rhs :

@s@r

� �þ B2 �

@

@rhs :

@s@r

� �: ð58Þ


Using Eq. (4), the derivative of A in Eq. (56) is calculated as

@A@r¼ � @g�q�1

1

@r� A1 þ

@

@rg4 � g1

g1

� �� A2 þ

@

@rðj2hs : sjqÞ � A3;

A1 ¼A

�g�ðqþ1Þ1

; A2 ¼A

g4�g1g1

� � ; A3 ¼A

j2hs : sjq; ð59Þ

@g�q�11

@r¼ ð�q� 1Þg�q�2

1@g1

@r;

@

@rg4 � g1

g1

� �¼ �g4

@g1

@r

� ��g2

1:

ð60Þ

Similarly, B1 in Eq. (57) is defined with two terms B2 and B3 in Eq.(51) which are differentiated

@B1

@r¼ @B2

@r� B3 þ B2 �

@B3

@r; ð61Þ

@B2

@r¼ @ð�2f q

1 j2hs : sjq�1Þ@r

¼ @f q1

@r� B21 þ

@j2hs : sjq�1

@r� B22;

B21 ¼B2

f q1

; B22 ¼B2

j2hs : sjq�1; ð62Þ

@B3

@r¼@ðkð�sþ 8

3 ðhsÞðhs : sÞÞ : sÞ@r

¼ k@ð�sþ 8

3 ðhsÞðhs : sÞÞ@r

: sþ @s@r

: �sþ 83ðhsÞðhs : sÞ

� � !:

ð63Þ

Then differentiation of hs : @s@r with respect to r in Eq. (58) is calcu-

lated as follows:

@ðhs : @s@rÞ

@r¼ @hs

@r:@s@rþ hs � @

@r@s@r

� �¼ @hs

@r:@s@r

: ð64Þ

Again, similarly to the solution for the first order gradient of theyield surface Eq. (54) is nonlinear. Therefore, the multistep New-ton–Raphson method, which was applied to solve Eq. (50), is ap-plied for the second derivative of the HAH model.For thecalculation of the second derivative of the yield function in the

HAH approach, the Hessian matrix of the stable yield function @2/@r@r

is assumed as an initial guess. As for the gradient of the HAH model,the second derivative is also sequentially calculated between thestable yield function and the target HAH yield function with the gi-ven fluctuating terms. Then, g1 for each subdivided yield function

becomes gðkÞ1 ¼ g01 �

g01�g1

N

� �� k where g0

1 ¼ 1:0 and N is the total

Fig. 5. Schematic view of the iterative solv

number of multiple steps. The solution of the first sub-step H(1) iscalculated from the initial guess H(0), which is the Hessian matrixof the initial stable yield function with g0

1 ¼ 1:0 . After finding thesolution of the first sub-step, the solution of the second sub-step

is evaluated from H(1), which is the second derivative of gð1Þ1 . This

procedure continues until gðkþ1Þ1 ¼ g1 as shown in Fig. 5.

Similar to the simplified approach for the gradient of the HAHmodel, this simplified strategy is also applied to the second deriv-ative. By freezing the state variables related to the fluctuatingterms during iteration, Eq. (54) becomes

@2 �r@r@r

¼ @

@r�r1�q /q�1 @/

@r� 2f q

1 j2hs : sjq�1 � hs :@s@r

� ��

¼ @

@rð�r1�q �WÞ ¼ @

�r1�q

@r�Wþ �r1�q � @W

@r

¼ ð1� qÞ�r�q @�r@r�Wþ �r1�q � @W

@rð65Þ

where W is /q�1 @/@r� 2f q

1 j2hs : sjq�1ðhs : @s@rÞ.

Using the chain rule, @W@r becomes

@W@r¼ @

@r/q�1 @/

@r� 2f q

1 j2hs : sjq�1 hs :@s@r

� ��

¼ ðq� 1Þ/q�2 @/@r

@/@rþ /q�1 @2/

@r@rþ 2 � 2ðq� 1Þf q

1 j2hs : sjq�2

� hs :@s@r

� �hs :

@s@r

� �� 2f q

1 j2hs : sjq�1 hs :@2s@r@r

!

¼ ðq� 1Þ/q�2 @/@r

@/@rþ /q�1 @2/

@r@r� 2 � 2ðq� 1Þf q

1 j2hs : sjq�2

� hs :@s@r

� �hs :

@s@r

� �: ð66Þ

3.4. Overall time integration algorithm

The numerical algorithms for the HAH hardening model aresummarized as shown in Fig. 6.

4. Numerical verification

The accuracy of each of the proposed elasto-plastic GCPM(semi-explicit algorithm) and CPPM (fully implicit1) algorithmswere evaluated. Uniaxial tensile tests in different direction are sim-ulated with single element FE analysis. The stress state and r-valueare important values that have been frequently used to evaluate

ing procedure for the hessian matrix.


the accuracy of the elasto-plastic constitutive models and theirstress integration algorithms in the finite element simulations.Especially, the r-value describes the deformation anisotropy, whichhighly depends on the choice of anisotropic model. In this section,the calculated r-values which are equivalent to the gradient of theyield surface at given stress states will be validated by comparingwith analytically derived values. The r-values are calculated for se-ven different orientations in 15� steps starting from the rollingdirection.

The formulations described previously were implemented inthe commercial FE software ABAQUS through the user-definedmaterial subroutine. A shell element with reduced integration(S4R) and size of 1 mm � 1 mm � 0.1 mm (length �width � thick-ness) was used for all simulations.

Fig. 6. Flow charts of the stress update s

4.1. Isotropic hardening with initial anisotropy

As a first example, the simplest isotropic hardening is assumed.The element is elongated up to 5% strain and unloaded to removean elastic effect in each test. The schematics of the boundary con-ditions are shown in Fig. 7.

Then, the r-values for tension at angle h from the rolling direc-tion were calculated as

rh ¼ep

w

ept¼ � ep

22

ep11 þ ep

22

; ð67Þ

where the superscript ‘p’ denotes the plastic component and eachplastic strain component is calculated by the logarithmic relation;

chemes for (a) GCPM and (b) CPPM.

Fig. 6 (continued)

Fig. 7. Schematics of boundary condition for r-value prediction.


i.e., ep11 ¼ lnð1þ Dup

1Þ; ep22 ¼ lnð1þ Dup

2Þ. Note that these two straincomponents are principal values according to the boundary condi-tions shown in Fig. 7.

Two different materials are considered; i.e., a planar anisotropicmaterial aluminum alloy, AA5042, and planar isotropic materialAKDQ steel. The yield stresses and r-values of the two materialsin three different directions are listed in Table 1 (from the bench-mark of NUMISHEET 2011). The Young’s modulus for AA5042 andAKDQ steel are 70 GPa and 210 GPa, respectively. A Poisson’s ratioof 0.33 is used for both materials.

For an initial anisotropy, Yld2000-2d non-quadratic anisotropicyield function [11] is also implemented in the FE analysis. A briefsummary of the model is shown in the Appendix.

Table 1Mechanical properties of the material samples.

Direction AA5042 AKDQ steel

Yield stress(MPa)

r- Value Yield stress(MPa)

r- Value

Rolling direction (0�) 277.17 0.354 315.68 1.093Diagonal direction (45�) 281.31 1.069 309.82 1.170Transverse direction (90�) 289.58 1.396 306.96 1.180

Table 2Yld2000-2d yield function anisotropy coefficients for AA5042 and AKDQ steel.

a1 a2 a3 a4 a5 a6 a7 a8

AA5042 0.600 1.213 1.047 0.916 0.988 0.719 0.980 1.041AKDQ steel 0.903 1.082 0.685 0.947 0.946 0.691 1.000 1.214


The parameters of Yld2000-2d anisotropic yield function foreach material are calculated from eight experimental data andare shown in Table 2.

For the isotropic hardening, the Voce type hardening law is em-ployed for both materials. For AA5042,

�r½MPa� ¼ 375:08� 107:28 � expð�17:859 � �epÞ: ð68Þ

And for AKDQ steel

�r½MPa� ¼ 471:76� 173:97 � expð�15:886 � �epÞ: ð69Þ

First, the r-values calculated using two different strain incrementsDe are compared; i.e., 5 � 10�3 and 5 � 10�4, which correspond totwice and one-fifth of the yield strain ey (strain at which plasticdeformation initiates), respectively.

The relative error between analytical and algorithmic r-values(ranalytical and ralgorithmic, respectively) defined as

(a) (b

Fig. 9. r-Value predictions for AKDQ

(a) (b

Fig. 8. r-Value predictions for AA50

eRA ¼jranalytical � ralgorithmicj

ranalytical: ð70Þ

The GCPM with the strain increment of 5 � 10�3 results in maxi-mum absolute relative errors of 19% and 29% for AA5042 and AKDQsteel, respectively. For AA5042, the relative error eRA with GCPM be-comes more pronounced when the angle a between tension and RDis larger than 32.5�. However, Figs. 8 and 9 show that the differencebetween the two algorithms is almost negligible when the strainincrement is small enough, i.e., De = 5 � 10�4 in this particularexample. In order to obtain the relative error of r-value betweentwo algorithms, ranalytical and ralgorithmic are replaced by rCPPM andrGCPM, respectively in Eq. (70). The relative error for the two materi-als and two r-values are listed in Table3. These two angles are cho-sen because they correspond to maximum r-value differencesbetween them.

Table 3 indicates that the error becomes negligible only if thestrain increment is significantly reduced. However, if the strainincrement is of the order of the yield strain, the solution by thesemi-explicit method might not be guaranteed. Therefore, theanalysis shows that a careful time step control is necessary whenthe semi-explicit algorithm is used for the finite element analysis.Recent trends in sheet metal forming simulations using finite ele-ment methods is that the explicit FE program usually carries outthe forming simulation, while the implicit FE program calculatesspringback. Since for the explicit FE analysis, a small enough timestep is usually secured, the semi-explicit stress integration algo-rithm seems to be a reasonable approach for both steps.

4.2. HAH anisotropic hardening

In this section, the two stress integration algorithms are appliedto the HAH model. The main objective for this analysis is to studythe influence of the numerical algorithm on the accuracy of the

)

steel : (a) De = 2ey , (b) De = 2ey.

)

42 : (a) De = 2ey, (b) De = 2ey.

Table 3The absolute relative error for r-value prediction with two methods.

AA5042 AKDQ steel

45� 60� 45� 60�

De = 2ey 0.16 0.27 0.18 0.25De = 0.2ey 0.014 0.024 0.017 0.023De = 0.02ey 0.0014 0.0024 0.0017 0.0023

Table 4Yld2000-2d yield function anisotropy coefficients and evolutionary coefficients (HAH)for DP590.

HAH model a q k k1 k2 k3 k4 k5

6 2 30 80 8 0.2 0.9 6Yld2000-2d a1 a2 a3 a4 a5 a6 a7 a8

0.91 1.10 0.99 0.98 1.00 0.93 0.98 1.08


stress calculation for the distorting yield surface. Because of thisdistortion, the r-value is also evolving when the material is sub-jected to a loading path change. A same single element approachis used. The prestrain is applied before the tensile tests for differentspecimen angles from the rolling direction.

A 1.2 mm thick DP590 steel sheet with was considered becausethe coefficients of Yld2000-2d and HAH models are known from aprevious investigation [51]. Table 4 summarizes those parameters.The Young’s modulus and Poisson’s ratio are 194 GPa and 0.3,respectively. The isotropic hardening law is based on Swift model

Fig. 10. (a) Yield loci for the isotropic hardening model and the HAH model, r-value pdescription of relative error between two stress integration methods.

�r½MPa� ¼ 942:92 � ð0:0035þ �epÞ0:149: ð71Þ

Fig. 10(a) shows the normalized yield surface based on Yld2000-2dfor the HAH approach. The yield surfaces represented the two load-ing paths. First, the material is compressed up to 5% so that the yieldsurface is distorted near uniaxial tension. After compression, thematerial is subjected to tension up to 7%, which results in the dis-tortion in the compressive region, while fast recovery occurs inthe tensile direction. For comparison purpose, the normalized yieldsurface based on isotropic hardening is also presented in Fig. 10(a).The evaluation of the r-value by the two stress integration algo-rithms is shown in Fig. 10(b). The strain increment used for GCPMis De = (5 � 10�3) � ey, while that for CPPM is De = (5 � 10�2) � ey. Asexpected, the classical isotropic hardening results in a constant r-value as shown in Fig. 10(b). However, the r-value predicted bythe HAH model changes as the plastic deformation in the secondstage proceeds. This is due to the distortion of the yield surface dur-ing the compressive loading. Fig. 10(c) shows the evolution of r-val-ues with different strain increments for the semi-explicit algorithmduring the tensile stage after 5% compressive strain and with theimplicit algorithm for De = (5 � 10�2) � ey. The relative error of r-va-lue between CPPM and GCPM after 7% tension, rCPPM and rGCPM takeplace of ranalytical and ralgorithmic, respectively in Eq. (70), are illus-trated in Fig. 10(d). Note that the solution by the CPPM is assumedas an exact solution. Like the previous example for the isotropichardening case, the predicted r-values by the semi-explicit algo-rithm converges towards the fully implicit one when the strainincrement is De = 0.05 � ey.

rediction (b) for CPPM and GCPM (c) for the different strain increments, and (d)

Fig. 12. Plane stress yield surface represented in the principal stress plane. Pointsfor iso-error maps plots (A, B, C).

Table 5Strain conditions for plastic deformation in Fig. 10.

ei11 ei

12

State A ey �m � ey

State B ð1� mÞ � ey ð1� mÞ � ey

State C 1þmffiffi3p� �

� ey1þmffiffi

3p� �

� ey


To check the efficiency of the proposed HAH model, the CPUtime was compared with the classical isotropic hardening model.The CPU time for the GCPM with the HAH model was approxi-mately 1.5 times larger than that of the GCPM with the isotropichardening model when the same time increment was used. Con-sidering the complexity in the constitutive model that is able to de-scribe more complex material behavior, it can be said that theefficiency of the HAH model is reasonably good to be used for fu-ture industrial applications.

Finally, the r-value predicted with the two different gradients ofthe HAH yield function calculated in Eqs. (50) and (53) are com-pared in Fig. 11. The material properties of DP590 are used. The er-ror from the simplified gradient Eq. (53) becomes virtuallynegligible for the given increments. The r-values calculated withthe simple and multistep N-R converge as strain increments aresmall.

4.3. Analysis of the iso-error map

In the elasto-plasticity, the iso-error map has been used as asystematic tool to evaluate the accuracy of the stress update algo-rithm in the space of applied strain increments [53,54].

Three points, which represent different possible stress states onthe plane stress yield surface, are selected as shown in Fig. 12.Points A, B and C correspond to uniaxial, balanced biaxial and pureshear stress states, respectively. Because only principal values ofstress and strain are considered, the shear strain component is as-sumed to vanish, i.e., e12 = 0. In order to construct the iso-errormap, two steps are required. First, a pure elastic path is requiredto proceed from the zero stress state to one of the three points inFig. 12. In Table 5, the required strain components for this stepare summarized for the three corresponding points. ey in Table 5denotes the strain component when plastic yielding is initiatedin the monotonic stress–strain curve. Second, a deformation pathgiven in terms of the strain increments De11 and De22 leads to a fi-nal state.

e11 ¼ ei11 þ De11: ð72Þ

e22 ¼ ei22 þ De22; ð73Þ

where superscript ‘i’ stands for yield strain to reach the points A, Bor C. In this study, the normalized strain increments are applied upto 6.0 with an interval of 0.05. Then,

Fig. 11. r-Value prediction for different strain increments between Multi N-R andsimplified normal gradient.

De11

ei11

¼ 0:0;0:05;0:1;0:15;0:2; . . . ;6:0; ð74Þ

De22

ei22

¼ 0:0;0:05;0:1;0:15;0:2; . . . ;6:0: ð75Þ

The stress corresponding to the prescribed strain is calculated byapplying the stress integration algorithm. The error is defined asthe relative root mean square of the difference between the exactand computed stress value by the specific stress integrationalgorithm.

eRB ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr� rexactÞ : ðr� rexactÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirexact : rexactp � 100 ð%Þ; ð76Þ

where r is the calculated stress tensor and rexact is the exact solu-tion. Because it is difficult to get the exact solution, it is usually as-sumed that the exact solution is equivalent to the stress tensorcalculated by repeatedly applying the algorithm with many sub-increments. In the following calculation, 1000 sub-increments areused.

DP590 was used as a model material and the coefficients forYld2000-2d are those as listed in Table 4. Note that perfect plastic-ity is generally assumed for the iso-error map calculations. Twocases are considered in this study; iso-error map for the conven-tional non-quadratic anisotropic yield function, Yld2000-2d withisotropic hardening and distortional hardening (HAH). For the lat-ter case, the initial yield surface defined by Yld2000-2d is distortedby applying deformation histories such as uniaxial tension, bal-anced biaxial tension and pure shear. Then, the two stress integra-tion algorithms are evaluated.

4.3.1. Iso-error map for Yld2000-2dThe calculated iso-error maps for Yld2000-2d and isotropic

hardening are shown in Fig. 13. Fig. 13(a)–(c) represents the iso-er-ror maps calculated by the fully implicit method based on CPPM,while Fig. 13(d)–(f) are those by GCPM. In general the fully implicitalgorithm provides less error than the semi-explicit algorithm.

Fig. 13. Iso-error map corresponding to point A, B, C: the fully implicit (a–c) and the semi-explicit (d–f) algorithm for Yld2000-2d.


Although not shown here, as the strain increment is reduced, thedifference in error converges.

This result is in agreement with the evaluation of the r-valueconsidered in the previous section. Among the three evaluationpoints, the maximum error from the semi-explicit method islower at point C compared to the other points. To conveniently

compare the accuracy of the two algorithms, the relative errorsbetween the two algorithms are plotted at the same principalstrain space.

eRC ¼jeCPPM � eGCPM j

eCPPM; ð77Þ


where eCPPM and eGCPM are the error from the fully implicit algo-rithm and the semi explicit algorithm, respectively.

Fig. 14(a) and (b) show that the fully implicit algorithm leads tobetter accuracy than the semi explicit algorithm. When the stresspoint is on the pure shear, i.e., point C in Fig. 12, the two algorithmsproduce similar iso-error results as shown in Fig. 14(c).

4.3.2. Iso-error map for HAHFor the HAH model, the shape of the yield surface is distorted,

depending on the deformation history. Therefore, in this section,the iso-error map taking the yield surface distortion into accountis analyzed. Fig. 15 shows distorted yield surfaces based on theHAH approach with respect to three different deformation paths.To simplify the problem, two plastic internal variables controllingthe level of distortion are set to g1 = 0.6 and g2 = 0.8. Three casescorresponding to (a) uniaxial tension, (b) balanced biaxial tensionand (c) pure shear are considered by taking three microstructuredeviators hs.

hs ¼1=2 0 0

0 �1=4 0

0 0 �1=4

264

375 for uniaxial tension ; ð78Þ

Fig. 14. Relative error maps with two m

hs ¼�1=4 0 0

0 �1=4 00 0 1=2

264

375 for balanced biaxial tension; ð79Þ

hs ¼

ffiffiffi3p

=4 0 00 �

ffiffiffi3p

=4 00 0 0

264

375 for pure shear: ð80Þ

For comparison purpose, the initial yield surface based on Yld2000–2d before and after deformation is also drawn in Fig. 15. For the dis-torted HAH yield surface with prescribed state variables, the analy-sis of the iso-error maps is carried out for the three different stressstates A, B and C shown in Fig. 12.

Figs. 16–18 show the HAH iso-error maps for the differentdeformation paths. The iso-error maps calculated by the fully im-plicit algorithm show that the pattern of the map is similar to thatof the conventional anisotropic yield function without distortion.The effect of the distortion on the accuracy of the stress integrationscheme is observed but a consistent trend is not clearly seen. Forexample, for the HAH yield surface distorted by the microstructuredeviator in uniaxial direction, the error at point A decreases com-pared to that of Yld2000-2d surface with isotropic hardening but

ethods corresponding point A, B, C.

Fig. 15. Plane stress yield surface for the HAH models: hs is (a) uniaxial tension direction, (b) biaxial tension direction and (c) pure shear direction.


the error at point B increases. For all cases, the errors at the equi-biaxial stress state (point B) are relatively smaller than those at theuniaxial and pure shear stress states. The pattern of the iso-errormaps for the fully implicit algorithm is the same, regardless ofthe deformation history (or microstructure deviator). When theshape of yield locus is rounded by the distortion in the directionof equi-biaxial tension, the patterns of the iso-error maps calcu-lated by the semi-explicit algorithm shows some similarities tothose by the fully implicit algorithm (see Fig. 17). In general, theproposed algorithms shows stable convergences in the entire testregions for the HAH model. Moreover, compared with previouslyreported works for other conventional yield functions, the errormagnitudes are within an acceptable range when proper strainincrements are selected.

5. Summary and conclusion

In this paper, numerical formulations and implementation ofthe recently proposed homogeneous yield function-based aniso-tropic hardening (HAH) model were presented. Two frequentlyused stress integration algorithms were considered: general cut-ting plane method based on a semi-explicit algorithm and closetpoint projection method based on a fully implicit algorithm. Sincethe HAH model is accompanied by a continuous distortion of theyield locus during plastic deformation when the strain path

changes, a new multi-step Newton–Raphson method was pro-posed under the elasto-plasticity framework to calculate the firstand second derivatives of the evolving yield surface.

The numerical validations on the prediction the r-value ofanisotropic materials were performed to assess the accuracy ofthe algorithms. Moreover, analyses for numerical details were alsocarried out by the iso-error maps. When the non-quadratic aniso-tropic yield function Yld2000-2d was used with classical isotropichardening, the r-values calculated by the fully implicit methodwere shown more accurate than those by the semi-explicit algo-rithm for large time increments. However, as the time step de-creases, the r-values calculated by the semi-explicit algorithmconverged toward the analytical solutions.

A similar analysis was carried out for the HAH model, which in-volves a distortion of the yield surface. The material is subject tocompression followed by tension so that the yield surface distorts.In contrast to classical isotropic hardening, the r-value predictedby the HAH model changes as a second plastic deformation stepproceeds. In terms of the accuracy of the algorithm, like the isotro-pic hardening case, the r-values predicted by the semi-explicitalgorithm converged to the solutions of the fully implicit methodwhen the strain increment becomes smaller. Since the HAH modeldoes not include the back stress tensor, the CPU time for the FEcomputation is comparable to that of the classical isotropic hard-ening model.

Fig. 16. Iso-error map corresponding to point A, B, C when hs is uniaxial tension direction: the fully implicit (a–c) and the semi-explicit (d–f) algorithm.


The analyses of the iso-error maps were performed for threeloading conditions; uniaxial tension, biaxial tension and pureshear. A special procedure was employed to conduct the accuracyassessment for the HAH model; i.e., Pre-plastic deformations in

directions opposite to the three loading conditions were first ap-plied. In general, the iso-error maps leaded to already known re-sults for the conventional elasto-plastic algorithms employedwith anisotropic yield functions. An effect of the yield surface

Fig. 17. Iso-error map corresponding to point A, B, C when hs is biaxial tension direction: the fully implicit (a–c) and the semi-explicit (d–f) algorithm.


distortion on the accuracy of the stress integration scheme was ob-served, although no consistent trends did clearly emerge. For in-stance, the errors at the equi-biaxial stress state were smallerthan those at the uniaxial and pure shear stress states. Also, when

the shape of the yield surface becomes rounded by the distortion,the pattern of the iso-error maps calculated with the semi-explicitalgorithm became similar to those with the fully implicitalgorithm.

Fig. 18. Iso-error map corresponding to point A, B, C when hs is pure shear direction: the fully implicit (a–c) and the semi-explicit (d–f) algorithm.


Acknowledgments

This research was supported by POSCO and by WCU (WorldClass University) program through the National Research

Foundation of Korea funded by MEST (R32-10147). M.G.L. appreci-ates the support by the grants from the Industrial Source Technol-ogy Development Program (#10040078) of MKE and from the BasicScience Research Program (#2011-0009801).


Appendix A. Yld2000-2d

In plane stress, a non-quadratic anisotropic yield function has aform [11]

/ ¼ /0 þ /00

2

� �1a

¼ �r; ðA:1Þ

where

/0 ¼ jXð1Þ1 � Xð1Þ2 ja and /00 ¼ j2Xð2Þ2 þ Xð2Þ1 j

a þ j2Xð2Þ1 þ Xð2Þ2 ja: ðA:2Þ

X(j) (j = 1, 2) are linearly transformed stress tensors and the sub-scripts 1 and 2 indicate their principal values. The transformationsof the Cauchy stress r into X(j) (j = 1, 2) involve two lineartransformations

XðjÞ ¼ CðjÞ � s ¼ CðjÞ � T � r ¼ LðjÞ � r with j ¼ 1;2: ðA:3Þ

@2/@rk@rl

¼X2

r

X3

s

X2

c

X3

b

@2W

@gð1Þr @gð1Þc

@gð1Þr

@Xð1Þs

@Xð1Þs

@rk

!@gð1Þc

@Xð1Þb

@Xð1Þb

@rl

!þ @2W

@gð2Þr @gð2Þc

@gð2Þr

@Xð2Þs

@Xð2Þs

@rk

!@gð2Þc

@Xð2Þb

@Xð2Þb

@rl

! !

þX2

c

X3

r

X3

b

@W

@gð1Þc

@2gð1Þc

@Xð1Þr @Xð1Þb

@Xð1Þr

@rk

!@Xð1Þb

@rl

!þ @W

@gð2Þc

@2gð2Þc

@Xð2Þr @Xð2Þb

@Xð2Þr

@rk

!@Xð2Þb

@rl

! !

þX2

c

X3

b

@W

@gð1Þc

@gð1Þc

@Xð1Þb

@2Xð1Þb

@rk@rl

!þ @W

@gð2Þc

@gð2Þc

@Xð2Þb

@2Xð2Þb

@rk@rl

! !with k; l ¼ 1 � 3: ðA:11Þ

The transformation T calculates the deviatoric stress s and othertransformations C(1) and C(2) induce anisotropy. The products oftwo transformations are represented as L(1) and L(2) which can beexpressed in matrix form as below

XðjÞxx

XðjÞyy

XðjÞxy

2664

3775 ¼

LðjÞ11 LðjÞ12 0

LðjÞ21 LðjÞ22 0

0 0 LðjÞ66

2664

3775

rxx

ryy

rxy

264

375 with j ¼ 1;2; ðA:4Þ

Lð1Þ11

Lð1Þ12

Lð1Þ21

Lð1Þ22

Lð1Þ66

266666664

377777775¼

2=3 0 0�1=3 0 0

0 �1=3 00 2=3 00 0 1

26666664

37777775

a1

a2

a7

264

375 and

Lð2Þ11

Lð2Þ12

Lð2Þ21

Lð2Þ22

Lð2Þ66

266666664

377777775

¼ 19

�2 2 8 �2 01 �4 �4 4 04 �4 �4 1 0�2 8 2 �2 00 0 0 0 9

26666664

37777775

a3

a4

a5

a6

a8

26666664

37777775: ðA:5Þ

The anisotropy coefficients a1–a8 can be determined from uniaxialtension tests in various loading directions and balanced biaxial ten-sion tests.

The first derivative @/@rk

is obtained by the chain rule,

@/@rk¼ 1

2a/1�a

X2

i

X3

j

@W

@gð1Þi

@gð1Þi

@Xð1Þj

@Xð1Þj

@rkþ @W

@gð2Þi

@gð2Þi

@Xð2Þj

@Xð2Þj

@rk

with k ¼ 1 � 3: ðA:6Þ

Now,

@W

@gð1Þi

" #¼

aðXð1Þ1 � Xð1Þ2 ÞjXð1Þ1 � Xð1Þ2 2j

a�2

�aðXð1Þ1 � Xð1Þ2 ÞjXð1Þ1 � Xð1Þ2 j

a�2

" #ðA:7Þ

@W

@gð2Þi

" #¼ a

fðXð2Þ1 þ2Xð2Þ2 ÞjXð2Þ1 þ2Xð2Þ2 j

a�2þ2ðXð2Þ2 þ2Xð2Þ1 ÞjXð2Þ2 þ2Xð2Þ1 j

a�2gf2ðXð2Þ1 þ2Xð2Þ2 ÞjX

ð2Þ1 þ2Xð2Þ2 j

a�2þðXð2Þ2 þ2Xð2Þ1 ÞjXð2Þ2 þ2Xð2Þ1 j

a�2g

" #

ðA:8Þand

@gðlÞi

@XðlÞj

" #¼

12þ

ðXðlÞ1 �XðlÞ2 Þ4rðlÞ

12�


XðlÞ3rðlÞ

12�


12þ


� XðlÞ3rðlÞ

24

35; ðA:9Þ

where rðlÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXðlÞ1 �XðlÞ2

2

� �2

þ ðXðlÞ3 Þ2

sand

@XðlÞj

@rk

� �¼

LðjÞ11 LðjÞ12 0LðjÞ21 LðjÞ22 00 0 LðjÞ66

264

375 with

l = 1, 2.The second derivative @2/

@rk@rlis obtained by following chain rule.

@2/@rk@rl

¼ 12a

/1�a @2W@rk@rl

� a� 1/

@/@rk

@/@rl

with k; l ¼ 1 � 3:

ðA:10Þ

From Eq. (A.11),@2Xð1Þ

b@rk@rl

and@2Xð2Þ

b@rk@rl

are zero.

@2W

@gð1Þi @gð1Þi

¼ aða� 1ÞjXð1Þ1 � Xð1Þ2 j

a�2 �jXð1Þ1 � Xð1Þ2 ja�2

�jXð1Þ1 � Xð1Þ2 ja�2 jXð1Þ1 � Xð1Þ2 j

a�2

" #ðA:12Þ

@2W

@gð2Þi @gð2Þi

" #

¼aða�1Þ

�jXð2Þ1 þ2Xð2Þ1 j

a�2þ4jXð2Þ2 þ2Xð2Þ1 ja�2 2jXð2Þ1 þ2Xð2Þ1 j

a�2þ2jXð2Þ2 þ2Xð2Þ1 ja�2

2jXð2Þ1 þ2Xð2Þ1 ja�2þ2jXð2Þ2 þ2Xð2Þ1 j

a�2 4jXð2Þ1 þ2Xð2Þ1 ja�2þjXð2Þ2 þ2Xð2Þ1 j

a�2

" #

ðA:13Þ

and

@2gðlÞ1

@XðlÞi @XðlÞj

" #¼

14rðlÞ� ðX

ðlÞ1�XðlÞ

2Þ2

16ðrðlÞÞ3� 1

4rðlÞþ ðX

ðlÞ1�XðlÞ

2Þ2

16ðrðlÞÞ3� ðX

ðlÞ1�XðlÞ

2ÞXðlÞ

3

4ðrðlÞÞ3

� l4rðlÞþ ðX

ðlÞ1�XðlÞ

2Þ2

16ðrðlÞÞ31

4rðlÞ� ðX

ðlÞ1�XðlÞ

2Þ2

16r03ðXðlÞ

1�XðlÞ

2ÞXðlÞ

3

4ðrðlÞÞ3

� ðXðlÞ1 �XðlÞ2 ÞX

ðlÞ3

4ðrðlÞÞ3ðXðlÞ1 �XðlÞ2 ÞX

ðlÞ3

4ðrðlÞÞ31

rðlÞ� ðX

ðlÞ3 Þ

2

ðrðlÞÞ3

26666664

37777775

with l ¼ 1;2: ðA:14Þ

@2gðlÞ1

@XðlÞi @XðlÞj

" #¼ � @2gðlÞ2

@XðlÞi @XðlÞj

" #ðl ¼ 1;2Þ: ðA:15Þ

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Stress integration schemes for novel homogeneous anisotropic hardening model