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Comparison of two models for anisotropic hardening and yield surface

evolution in bcc sheet steels

T. Clausmeyer a ,b, *, B. Svendsen c ,d

a Institute of Mechanics, TU Dortmund University, Leonhard-Euler Strae 5, D-44227 Dortmund, Germanyb Institute of Forming Technology and Lightweight Construction, TU Dortmund University, Baroper Strae 303, D-44227 Dortmund, Germanyc Material Mechanics, RWTH Aachen University, Schinkelstrae 2, D-52062 Aachen, Germanyd Microstructure Physics and Alloy Design, Max-Planck-Institut fr Eisenforschung GmbH, Max-Planck Strae 1, D-40237 Dsseldorf, Germany

a r t i c l e i n f o

Article history:

Received 23 June 2014

Accepted 29 May 2015

Available online 9 June 2015

Keywords:

Material modeling

Cross hardening

Yield surface

a b s t r a c t

The purpose of the current work is the investigation and comparison of aspects of the material behavior

predicted by two models for anisotropic, and in particular cross, hardening in bcc sheet steels subject to

non-proportional loading. The rst model is the modied form (Wang et al., 2008) of that due to Teo-

dosiu and Hu (1995, 1998). In this (modied) Teodosiu-Hu model (THM), cross hardening is assumed to

affect the yield stress and the saturation value of the back stress. The second model is due to Levkovitch

and Svendsen (2007) and Noman et al. (2010). In the Levkovitch-Svendsen model (LSM), cross hardening

is assumed to affect the ow anisotropy. As clearly demonstrated in a number of works applying the THM

(e.g., Boers et al., 2010; Bouvier et al., 2005, 2003; Hiwatashi et al., 1997; Li et al., 2003; Thuillier et al.,

2010; Wang et al., 2008) and the LSM (e.g., Clausmeyer et al., 2014, 2011b; Noman et al., 2010), both of

these are capable of predicting the effect of cross hardening on the stress-deformation behavior observed

experimentally in sheet steels. As shown in the current work, however, these two models differ

signicantly in other aspects, in particular with respect to the development of the yield stress, the back

stress, and the yield surface. For example, the THM predicts no change in the shape of the yield surface

upon change of loading path, in contrast to the LSM and crystal plasticity modeling of bcc sheet steels(Peeters et al., 2002). On the other hand, the LSM predicts no hardening stagnation after cross hardening

as observed in experiments, in contrast to the THM. Examples are given.

2015 Elsevier Masson SAS. All rights reserved.

1. Introduction

Finite-element-based modeling and simulation of the material

and structural behavior of sheet metal parts in various stages of

design and manufacture is today standard. In general, one aim of

this is to benet from the predictive capability of such simulations

(Zienkiewicz et al., 2010). In this regard, Wagoner et al. (2013)emphasize the importance of improving material models to ac-

count for the loading path-dependent behavior of metals during

sheet metal forming. When subject to complex non-proportional

loading processes such as those found in many technological ap-

plications, a number of metals exhibit hardening behavior which is

more complex than isotropic and kinematic hardening alone.

Observed effects in this regard include cross hardening and hard-

ening stagnation during orthogonal loading (e.g., tension to shear).

Cross hardening is observed to occur for example in a number of

steels such as austenitic fcc tube steels (e.g., SUS304: Ishikawa,

1997; Wu, 2003), ferritic bcc tube steels (e.g., S355: Kowalewski

and Sliwowski, 1997), multi-phase tube steels (e.g., X100:

Shinohara et al., 2010), or ferritic bcc sheet steels (e.g., LH800:Ghosh and Backofen, 1973; Noman et al., 2010). Systematic studies

(Bouvier et al., 2005, 2006a, 2003) of interstitial free (IF), high-

strength low-alloyed (HSLA), transformation-induced plasticity

(TRIP), and dual-phase (DP), sheet steels, found signicant kine-

matic hardening, hardening stagnation, as well as cross hardening,

the latter especially in IF sheet steels. In these investigations, the

material was subjected to monotonic shear, reverse shear, as well as

orthogonal tension-shear, loading. Clausmeyer et al. (2012); van

Riel and van den Boogaard (2007); Wang et al. (2008)have docu-

mented these effects in the IF sheet steel DC06 with the help of

monotonic tension, reverse shear, and orthogonal tension-shear,

* Corresponding author. Institute of Forming Technology and Lightweight

Construction, TU Dortmund University, Baroper Strae 303, D-44227 Dortmund,

Germany. Tel.:49 231 755 8429; fax:49 231 755 2489.E-mail address: [email protected](T. Clausmeyer).

Contents lists available at ScienceDirect

European Journal of Mechanics A/Solids

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c om / l o c a t e / e j m s o l

http://dx.doi.org/10.1016/j.euromechsol.2015.05.016

0997-7538/

2015 Elsevier Masson SAS. All rights reserved.

European Journal of Mechanics A/Solids 54 (2015) 120e131


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tests, all under plane-strain conditions. In particular, cross hard-

ening occurs during discontinuous (e.g., tension-shear: Bouvier

et al., 2005, 2006a, 2003) and continuous (e.g., tension-shear:

Noman et al., 2010; van Riel and van den Boogaard, 2007; Wang

et al., 2008) orthogonal tension-shear tests. Similar results were

obtained by Verma et al. (2011) in a series of tension and

compression tests on ultra-low carbon IF sheet steel in which the

tension or compression direction changed from rolling to trans-

verse. In these tests, cross hardening was correlated with a change

of the tension axis. As attested to in particular by the continuous

orthogonal tension-shear test results (Noman et al., 2010; van Riel

and van den Boogaard, 2007; Wang et al., 2008), cross hardening is

transient and strongly depends on the rate of transition. Its

occurrence and strength are strongly inuenced by the particular

path taken in stressspace in changing from one loading direction to

another.

Generally speaking, anisotropic hardening in sheet steels may

be inuenced by the grain and dislocation (micro)structures. In

particular, the former is related to the grain orientation distribution

(texture). The inuence of texture on the hardening behavior of IF

sheet steel was investigated by Bacroix and Hu (1995) and

Nesterova et al. (2001a,b)using two-stage loading tests (e.g., shear

to reverse shear, tension to shear). In particular, Bacroix and Hu(1995)concluded that, at least up to moderatestrains, the inu-

ence of texture evolution on hardening in the specimens investi-

gated was small compared to that of dislocation structure

evolution. This conclusion was substantiated by later crystal plas-

ticity modeling (e.g.,Peeters et al., 2002). Related to this are more

recent EBSD investigations on DC06 (Boers et al., 2010; Clausmeyer

et al., 2012), which imply that the rolling-induced texture in this

steel does not change considerably for strains lower than 35% in

simple tension. This may be the case in other ferritic steels (e.g.,

LH800:Clausmeyer et al., 2012; Noman et al., 2010) as well. These

results imply that it is sufcient to account for the effect of the

initial (e.g., rolling) texture on the anisotropic hardening and ow

behavior in the material model.

Although texture evolution in this sense may be secondary,grain orientation (i.e., glide-system orientation) in relation to

loading direction certainly inuences dislocation structure devel-

opment. The development of certain characteristic dislocation

structures related to cross hardening have been observed during

quasi-static loading of mild steels such as DC06 at room tempera-

ture (e.g., Rauch and Schmitt, 1989; Rauch and Thuillier, 1993;

Thuillier and Rauch, 1994). These include for example dense

dislocation wall structures. The morphology and orientation of such

walls depends forexample on grain orientation, the type of loading,

and the loading direction in relation to the grain orientation (e.g.,

Clausmeyer et al., 2012; Nesterova et al., 2001a,b; Thuillier and

Rauch, 1994). A change in loading direction or type activates new

glide systems for which existing walls act initially as obstacles,

resulting in cross hardening.One of the rst phenomenological models accounting in

particular for cross hardening is the Teodosiu-Hu model (THM: e.g.,

Hu et al., 1992; Teodosiu and Hu, 1995, 1998). In the THM, cross

hardening is assumed to affect the yield stress sY in the yield

function fY. The THM has been employed in a number of works

(e.g., Bouvier et al., 2005, 2003; Haddadi et al., 2006; Hiwatashi

et al., 1997; Li et al., 2003; Thuillier et al., 2010) to model aniso-

tropic ow and hardening behavior in sheet metals. This has

motivated similar work on models for anisotropic hardening in the

continuum (Barlat et al., 2013; Butuc et al., 2011; Carvalho Resende

et al., 2013; Clausmeyer et al., 2014; Pietryga et al., 2012; Shi and

Mosler, 2012; Tarigopula et al., 2008, 2009) and crystal plastic

(Peeters et al., 2002; Viatkina et al., 2007) contexts. More recently,

the THM has been modi

ed, extended and generalized to deal with

arbitrary changes of loading path by Wang et al. (2008). This

modied version of the THM is that considered in the current work.

A second model for cross hardening was introduced by

Levkovitch and Svendsen (2007) and Noman et al. (2010). This

model has been referred to byShi and Mosler (2012)as the Lev-

kovitcheSvendsen model (LSM), whodiscussed related models for

distortional hardening and the strength differential effect in mag-

nesium alloys. In the LSM, cross hardening is assumed to inuence

the ow anisotropy through the corresponding tensor A deter-

mining fY. Common to both the THM and the LSM is the consti-

tutive form

fYffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM X$AM X

p sY (1)

forfYwith respect to the intermediate (local) conguration. Here,

Mis the Mandel stress (e.g., Mandel, 1971, 1974), and Xis the back

stress. To be more precise, in the THM,sYis assumed to depend on

both isotropic and cross hardening, and A is assumed constant. On

the other hand, in the LSM, cross hardening is assumed to inuence

the evolution ofA, andsYis assumed to depend only on isotropic

hardening. As shown in the previous works discussed above, both

models are capable of quantitatively predicting experimentally

observed cross hardening. The question arises as to how the THMand the LSM compare in other respects. To this end, in the current

work, a direct comparison of these two has been carried out. Tothis

end, both models have been identied from the same data set for

the ferritic sheet steel DC06 (for comparison, the ferritic-pearlitic

steel LH800 is also briey discussed). As the current comparison

of the THM and the LSM shows, the two models are not equivalent

in other respects. Among these, the prediction of yield surface

evolution is perhaps the most prominent.

The current work begins with a brief review of the formulation

of the two models in Section 2. This is carried out within the

framework of the multiplicative decomposition of the deformation

gradient and the assumption of small elastic strain relevant to

metal inelasticity. Again, for a meaningful comparison, the two

models are identied in Section3using the same test data sets forthe ferritic bcc sheet steel DC06. The identied THM and LSM are

then compared on the basis of their respective predictions for yield

and back stress evolution in Section 4 as well as yield surface

development in Section 5. The latter results are also compared

qualitatively with analogous results from the crystal plasticity

model ofPeeters et al. (2002)for IF steel. Lastly, these two models

are compared in the context of their application to the modeling of

sheet metal forming during non-proportional loading in Section6.

The work ends with a summary and discussion in Section 7.

2. Model formulation

2.1. Notation

In this work, Euclidean vectors (i.e., rst-order Euclidean ten-

sors) are represented by lower-case bold italic characters a,b,; in

particular, let i1,i2,i3represent the Cartesian basis vectors. Likewise,

upper-case bold italic characters A,B, represent second-order

Euclidean tensors; in particular, let I represent the second-order

identity tensor. Such tensors are dened in this work as linear

mappings between (three-dimensional) Euclidean vectors. In other

words, Ab is a vector for allA and all b. Let I,A and devAA(I,A)I/3 represent the trace and deviatoric part, respectively, of any A.

Likewise, let symA:(AAT)/2 and skwA:(AAT)/2 representthe symmetric and skew-symmetric parts, respectively, of any A.

Fourth-order tensors are represented by upper-case calligraphic

characters A; B;, in this work. Interpreting these as linear

mappings between second order tensors,AB

represents a second

T. Clausmeyer, B. Svendsen / European Journal of Mechanics A/Solids 54 (2015) 120e131 121


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order tensor for all A and all B. The dyadic product C A5BofanyA andanyB is a fourth-order tensor with Cartesian components

Cijkl AijBkl. The scalar product of two tensors A and B of any orderis represented by A,BAijBij via the summation convention.The (Euclidean) norm of any A is dened by jAj:(A,A)1/2.

2.2. Relations common to both models

The formulation of both models for cross hardening and yield

surface evolution under consideration in this work is carried out in

the framework of the standard multiplicative decomposition

F FEFP (2)

of the deformation gradient F (e.g., Lee, 1969) into elastic FE and

inelastic FP parts, respectively. Note that such a decomposition

arises naturally in the context of the modeling ofFPas a change of

local reference conguration (Svendsen, 2001). More precisely, the

current formulation is based on evolution relations derived from

(2) and the right polar decompositionFEREUEofFE in the contextof small elastic strain as follows. To begin, note that the time de-

rivative ofF

E FF1

P REUE results in the relation_REUE RE _UE REUELP LREUE (3)

between the velocity gradient L: _FF1 and its inelastic counter-partLP : _FPF1P . Multiplying (3) ontherightby U1E and on the leftby RTE R1E , we obtain

RTE _RE _UEU1E UELPU1E RTELRE: (4)Restrict the formulation now to metals and small elastic strain

as gauged or measured by (the magnitude of) the right logarithmic

stretch lnUE, i.e., jlnUEj1. In this case,

U1E

Iln UE

O

ln UE

2; (5)

and so

UELPU1E LP Ojln UEj; _UEU1E _ln UE Ojln UEj;

(6)

then hold. Here, O(jAjn) represents all terms depending explicitlyon all jAjm form n. Substituting (6) into (4), rearrangement yields

RTE _RE _ln UE LP RTELRE Ojln UEj: (7)Lastly, since RTERE I implies

_R

TERE 2 sym RTE _RE 0, the

symmetric and skew-symmetric parts of (7) then imply the (small-

elastic-strain-based) evolution relations

_ln UE RTEDRE DP; (8a)

_REWRE REWP; (8b)

to O(jlnUEj) for lnUEand the elastic rotation RE, respectively. Here,D symL is the continuum rate of deformation, DP symLP itsinelastic counterpart, W skwL the continuum spin, andWP skwLP the plastic spin. In particular, the usual associated owrule

DP _aPvfY

vM (9)

is assumed in both models for DP in terms of the yield functionfYfrom(1), withaP the equivalent inelastic deformation. Since various

investigations (e.g., Bacroix and Hu,1995; Boers et al., 2010) suggest

that the inuence of grain orientation evolution on hardening in

ferritic bcc sheet steels such as DC06 is not signicant in the range

of deformation relevant to the forming processes considered in this

work, WP is neglected here for simplicity. Consequently, the evo-

lution relation (8b) for REdepends only on the continuum spin W,

and as such is purely kinematic. In addition, attention is restricted

here to forming below the forming limit, and damage or any other

process resulting in inelastic volume changes are assumed negli-

gible. In this case, plastic incompressibility I,DP 0 pertains, andDP devDPis purely deviatoric as usual.

Since the elastic range and elastic strain are small, the effect of

any elastic anisotropy on the material behavior is assumed negli-

gible here. In this case, the isotropic form

M kI$EEI 2m dev EE (10)

holds for the Mandel stress to lowest order in the small elastic

strain measure EE:lnUEfrom (5), withk the bulk modulus, andmthe shear modulus. Likewise in the context of small elastic strain,

K REMRTE (11)

holds between the KirchhoffKand Mandel Mstresses. In this case,

note that the trace I,KI,M is independent of RE. Lastly, bothmodels are based on common saturation-based forms

_r crsr r _aP; (12a)

_X cxsxNP X _aP; (12b)

for the evolution of isotropic and kinematic hardening, respectively.

Here, r represent the contribution of isotropic hardening to sY. In

addition,crand cxrepresent the respective growth rates, srand sx

are the respective saturation values, andN

P:DP/jDPj is the direc-tion of inelastic ow.Since the THM(e.g., Hu et al.,1992; Teodosiu and Hu,1995,1998;

Wang et al., 2008) and the original version (e.g. Levkovitch and

Svendsen, 2007; Noman et al., 2010) of the LSM1 are limited to

rate-independence, the current model comparison is carried out on

this basis. To this end, attention is restricted in this work to quasi-

static loading rates at which the rate-dependent behavior of the

steels in question is small. As is well-known, the rate-independent

formulation is based in particular on the concepts of an elastic

range, yield surface and yield conditions, resulting in the so-called

KaresheKuhneTucker conditions: _fY 0,fY 0, fY _aP 0. In thiscontext, aP is determined as usual by the consistencycondition:_fY 0 whenfY 0. For the case of rate-dependence, of course, theyield condition generalizes to an activation condition, and the

consistency condition no longer applies.

2.3. THM for cross hardening

Besides the common ones just discussed, the THM (Teodosiu

and Hu, 1995, 1998; Wang et al., 2008) is based in particular on

the constitutive relations

sY sY0 rfjSj; (13a)

1

This model was generalized to rate-dependence recently inBarthel et al. (2013).

T. Clausmeyer, B. Svendsen / European Journal of Mechanics A/Solids 54 (2015) 120e131122


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A A 0; (13b)

for the yield stress sYand ow anisotropy tensor A, respectively,

appearing in (1). In (13b), A 0 is the initial value2 ofA determined

by the initial grain structure (i.e., rolling texture). As discussed in

the introduction, for the current case of ferritic bcc sheet metals,

the grain (orientation) structure is not observed to vary signi-

cantly in the loading range of interest, resulting in (13b). In (13a),sY0is the initial (i.e.,aP 0) yield stress, andfis a material constantdetermining the fraction of oriented dislocation structures

contributing to isotropic hardening. In addition, S is a (symmetric

traceless) fourth-order structure tensor, the magnitude of which

represents the effective strength of (persistent) oriented disloca-

tion structures and their effecton the level of yield. The evolution of

S as a standard internal variable3 is modeled by the relation

_S cd

hpssN P

hp hxS d

_aP cljS l=ssjnlS l_aP (14)

in the back-rotated frame with N P NP5NP. Here, S drepresentsthe strength of the dislocation structures (e.g., walls) associated

with the currently active slip systems, and S

lS

S d is thatassociated with the latent (i.e., currently inactive) slip systems.

Note that cd (hp hx) determines the effective saturation rate, andhp ss/(hp hx) the effective saturation magnitude, associated withS d. By analogy, cljS l=ssjnl is the effective saturation rate. Thesequantities contain the material constants cd, cl, ss, and nl. hp rep-

resents a further material function depending on S d, the direction

NP, a further material constant np, and the symmetric traceless

second-order tensor-valued internal variable P with constitutive

relation

_P cpNP P _aP (15)

which accounts for the effect of dislocation polarization on cross

hardening (Teodosiu and Hu, 1998; Wang et al., 2008). cp is thecorresponding saturation rate. The effect of hardening stagnation is

accounted for via material function hx depending in particular on

NP and X (Teodosiu and Hu, 1998; Wang et al., 2008). Lastly, the

saturation magnitudesxofXis given by the constitutive relation

sx x0 1 fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijS dj2 mjS lj2

q (16)

depending on the directional strength of dislocation structures.

Here,x0represents the initial (i.e., jS dj 0 jS lj) value ofsx. Theparameter m determines the inuence of latent as opposed to

current dislocation structures on the saturation magnitude sx of the

back stress X.

This model contains 21 material parameters. These include theelastic bulkk and shearm moduli, the initial yield stresssY0, the six

parameters F, G, H, L, M, and N determining the initial ow

anisotropy tensorA 0, the ve saturation rates cr,cx, cd,cl and cp, the

three saturation valuessr,x0, andss, the two exponentsnlandnp, as

well asfand m.

2.4. LSM for cross hardening

As just discussed, the effect of cross hardening on the material

behavior is accounted for in the THM and (13) by the dependence of

theyield stresssYonS . Instead of this, the effectof cross hardening

on the material behavior is accounted for in the LSM by a depen-

dence of the ow anisotropy tensor A on a structure tensor H

formally analogous to S . Instead of (13), then, the LSM for cross

hardening is based on the constitutive relations

sY sY0 r; (17a)

A A 0 H; (17b)

for the yield stress and ow anisotropy tensor, respectively, in (1).

Since the effect of cross hardening on the material behavior is then

accounted for in the modeling (17b) of A here, that (17a) ofsYaccounts only for dislocation processes resulting in isotropic

hardening. As discussed in detail elsewhere (Levkovitch and

Svendsen, 2007; Noman et al., 2010), the form

_H

cd

hdN P

Hd

_aP

cl

fhl

Idev

N P

Hl

g_aP (18)

of the evolution relation for H (and so for A since A 0 is constant)

in the LSM ismotivatedby that (14) for S in the THM. Here, IdevI 13 I5I is the fourth-order deviatoric identity. Further,Hd N P$HN P is the part ofH parallel, and Hl H Hd thatperpendicular, to N P. The idea here is that currently active dislo-

cation structures oriented parallel to NPpersist after an orthogonal

loading-path in inactive form (e.g., as cell-block boundaries). This

strengthens existing obstacles to glide-system activation in the

new loading direction.

Comparison of (13) and (17) shows clearly the qualitative dif-

ference between the two models in regards to the assumed effect of

cross hardening on the material behavior. In addition, note that

isotropic and kinematic hardening are coupled to cross hardening

in the THM, but basically decoupled from each other in the LSM. Asit turns out, the LSM is an example of the class of models distin-

guished by a yield function of the general form introduced in Baltov

and Sawczuk (1965). As will be shown in detail inwhat follows, one

result of this is that cross hardening results in a change in shape of

the yield surface in the LSM. This is completely analogous to the

association of isotropic hardening with a change in size, and ki-

nematic hardening with a change in center, of this surface. Before

delving into this, however, we rst summarize briey the identi-

cation of the above models upon which their subsequent com-

parison is based.

3. Parameter identication

For completeness, the results of the model parameter identi-cation are briey summarized and discussed in this section. For this

study, DC06 steel sheets (DIN EN 10130) of 1.0 mm thickness sup-

plied by ThyssenKrupp Steel were tested in uniaxial tension, plane-

strain tension, monotonic shear, shear to reverse-shear, and plane-

strain tension to shear. The advantage of shear tests is that larger

strains can be achieved with common testing facilities (Bouvier

et al., 2006b; Klepaczko et al., 1999; Miyauchi, 1987 ) than gener-

ally possible in uniaxial tension tests. The steel DC06 is often used

in auto-body parts4 due to its high ductility and formability. The

mechanical tests involving shear and plane-strain deformation

2A0 is often assumed to be of the type due to Hill (1948). Other choices (such as

in Cazacu and Barlat, 2004; Shi and Mosler, 2012) for the representation of the

initial ow anisotropy in non-bcc metals can be combined with the presented

approaches for anisotropic hardening.3 In the original form of the THM due to Teodosiu and Hu (1995, 1998), S is not

modeled as such a variable. To this end, and for the purpose of modeling continuous

loading path changes, the model was reformulated byWang et al. (2008), resulting

in ( 14) for _S

.

4 For example, see http://incar.thyssenkrupp.com/download/Broschueren/

Tiefziehstaehle.pdf.



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were carried out using the biaxial testing facility at the Faculty of

Engineering Technology, University Twente, The Netherlands. De-

tails of the mechanical testing procedure are given elsewhere (e.g.,

Clausmeyer et al., 2011b; Noman et al., 2010; van Riel and van den

Boogaard, 2007). The region of the specimen subject to deforma-

tion is initially 45.0 mm wide and 3.0 mm high. The strain eld is

homogeneous in the range of deformation investigated. As an un-

ambiguous and transparent measure of the state of deformation,

the deformation gradient Fis used here; let Fij ii,Fijrepresent itsCartesian components. For the tests, the tension/rolling direction is

oriented in the i2 direction, and the shear/transverse direction in

the i1 direction. Consequently, in the case of plane-strain testing,

F22 is the deformation component for tension, and F12 the defor-

mation component for shear. Given plastic incompressibility and

small elastic strain, the Cauchy stress TK/detF is well-approximated by the Kirchhoff stress K; let Kij ii,Kij representits Cartesian components in what follows.

Among the material parameters, note that k,m,sY0,F,G,H,L,M,

N,cr,sr,cx, andx0are common to both models. The Hill parameters

F,G,H,L,M,Nare computed from average r-values (seeClausmeyer

et al., 2011b). Ther-values are obtained from uniaxial tension tests

performedat 0, 45and 90with respect to the rollingdirection. Inparticular, for DC06 at room temperature, we have k151 GPa,m69.6 GPa, sY0 132 MPa, F 0.252, G 0.302, H 0.698,N 1.36, L 1.5, and M 1.5. The remaining parameters valuesidentied for all models are shown inTable 1. The hardening pa-

rameters cr,sr,cx, andx0are identied from uniaxial tension, plane-

strain tension, shear-reverse shear, and plane-strain tension to

simple shear, tests (Haddadi et al., 2006; Noman et al., 2010; Wang

et al., 2008). Since for example the modeling of kinematic hard-

ening is different in the two models, note that the corresponding

material parameter values in Table 1 are different. In particular,sx is

constant in the LSM and variable in the THM.

The identied models for DC06 are compared with experi-

mental data for the case of plane-strain tension to simple shear

loading in Fig.1 (left). To document the applicabilityof the THMand

the LSM to steels other than DC06, and for comparison, analogousresults for the ferritic-pearlitic steel LH800 (thickness 0.7 mm)

from Noman et al. (2010) and Noman (2010) are also shown in Fig.1

(right). Clearly, for both DC06 and LH800, cross hardening is

captured well by both models; this hardening is particularly pro-

nounced in the former material (i.e., more than 50 MPa;Fig. 1, left).

Also more pronounced in DC06 than in LH800 is hardening stag-

nation after cross hardening and change of loading direction (e.g.,

from tension to shear). Such stagnation is accounted for in the THM

(via (15) and (16)) but not in the LSM. This is the reason why, after

loading path change and cross hardening, the THM-based result

(red dashed curve, in the web version) inFig. 1(left) lies below the

LSM-based one (green dashed curve, in the web version) and closer

to the data (crosses). In any case, this difference between the two

models is hardly surprising and certainly to be expected. As

investigated in more detail in the next section, another difference

between the two models regards the yield surface evolution pre-

dicted by each, to which we turn after a more detailed analysis of

the contribution of isotropic as well as kinematic hardening.

4. Comparison of isotropic and kinematic hardening

modeling

As discussed in Section2, the relations (13a) and (17a) forsYin

the THM and the LSM, respectively, differ in the assumed inuence

of cross hardening via S on the former. Note that sY0 and the

evolution relation (12a) forrare the same in both models. As well,

the evolution relation (12b) forXis of exactly the same form in both

the THM and the LSM. On the other hand, the saturation magnitude

sxin (12b) is treated differently in the two models, i.e., dependent

on S and variable in the THM via (16), and constant in the LSM.

As an example of the consequences of these model differences,

consider the orthogonal loading of DC06 in discontinuous tension-

shear. In particular, this loading path consists of (i) loading in ten-

sion to F22 10.1, (ii) unloading, and (iii) reloading in shear toF22

1

F12

0.5. Consider rst the results for aP and sYfor this

loading case displayed inFig. 2. Although the relation (12a) for the

evolution of r is the same in both models, the coupled nature of

model identication results in different values for the samema-

terial coefcients in Table 1. For example, note thatcris a factor of 3

smaller, andsris an order of magnitude larger, in the LSM than in

the THM. This is the reason why, as shown inFig. 2 (right), isotropic

hardening saturates much more quickly in the THM than in the

LSM. The much lower level ofrin the THM is due of course to the

additional contribution fromS to isotropic hardening not assumed

in the LSM. Given the coupled natureof model identication, from a

quantitative point of view, the contribution ofXto fYin (1) will also

inuence the level of isotropic hardening in both models.

Consider next the development of the components of the back

stress

bX :

REXR

TE in the current conguration during tension-

shear loading displayed in Fig. 3. During the tension stage, theevolution of the normal componentsbX11,bX22 andbX33 is qualita-tively similar for both models. The reduction in these components

predicted by both models during shear (re)loading after tension

(pre)loading evident in Fig. 3 is due to the fact that the corre-

sponding components ofNPvanish in the shear stage, resulting in a

zero saturation level for the evolution of these components as

determined by (12b). As is the case for DC06 in Fig. 3, due to the

increase ofjSj (i.e., jS dj; jS lj) and its contribution to sx via (16), thelevel of kinematic hardening predicted by the THM will generally

be higher than that of the LSM, in which sx is assumed constant.

This together with the effect of hardening stagnation on S is also

the reason for qualitative differences in the results forbX12 in Fig. 3in the shear stage of discontinuous tension-shear loading. Indeed,

Table 1

Material parameter values for DC06 determined for the THM (above) and the LSM (below) from room-temperature uniaxial tension, monotonic shear, cyclic shear, and

orthogonal tension-shear, test data (Clausmeyer et al., 2011b). See text for details.

THM

parameter cr sr cx x0 cd ss cl nl cp np f m

units MPa MPa MPa

value 20.1 20.8 499.0 8.1 8.4 221.0 4.1 1.0 4.0 90.0 0.2 1.6

LSM

parameter cr sr cx sx cd hd cl hl

units MPa MPa

value 6.64 192.0 33.1 56.0 23.9 0.0 87.3 -0.447



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in the LSM (Fig. 3, right), sx is constant (e.g., 56 MPa for DC06 in

Table 1). Consequently, the result forbX12 from the LSM has thetypical Voce-based growth-saturation form as expected from (12b).

In contrast, as shown inFig. 3 (left), the coupling of the evolution of

X to S via sx in the THM results in qualitative deviations of the

result forbX12 from ideal Voce form. In particular, the effect ofhardening stagnation and the increase of sx are evident in Fig. 3

(left). As well, in the case of DC06 at least, the much larger value

ofcxfor the THM in Table 1results in much faster growth ofbX12inthe THM (Fig. 3, left) than in the LSM (Fig. 3, right). Note the sim-

ilarity ofbX12 in Fig. 3(left) andK12in Fig. 1(left).5. Comparison of yield surface modeling

In this section, the THM and the LSM identied for DC06 via the

model parameter values inTable 1are used to model yield surface

Fig. 1. Comparison of model and experimental results for shear stress as a function of deformation in DC06 (left) and LH800 (right: Noman et al., 2010; Noman, 2010). Both

materials were subject to plane-strain tension in the rolling direction (up to 10% in DC06 and 13% in LH800) followed by an orthogonal loading path change to simple shear in the

transverse direction. See text for discussion and web version for color.

Fig. 2. Development of aP (left) and isotropic hardening (right) predicted by the THM and the LSM during discontinuous tension-shear loading. This loading path consists of (i)

loading in tension to F2210.1, (ii) unloading, and (iii) reloading in shear to F221 F12 0.5. See text for details.

Fig. 3. Back stress evolution predicted by the THM (left) and the LSM (right) during discontinuous tension-shear loading. Here, the material is (i) loaded in tension to F221 0.1,

(ii) unloaded, and (iii) reloaded in shear toF221 F12 0.5. See text for details and web version for color.



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development in DC06. In the current model context and relation (1)

for the yield functionfY, the yield surface represents the boundary

fY 0 in stress space of the so-called elastic range of the material,i.e., those stress states for which fY


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proles (consisting of mild steel and DP steel: Haddag et al., 2007),split rings from a drawncup (consisting of DC06: Wang et al., 2008),

or even more complex parts (e.g., channel die and S-rail consisting

of DC06:Clausmeyer et al., 2014). Depending on factors such as the

geometry of the workpiece, different types of loading path changes

occur locally in the structure.

For example, in the case of deep drawing with a cross-shaped

die, nite-element (FE) simulations (e.g., Clausmeyer et al., 2011a)

show that, during this process, certain regions in the sides of the

workpiece structure experience a loading path change from plane-

strain tension to uniaxial tension. Stress-deformation results for

DC06 subject to such loading are shown in Fig. 6. As shown, both

models predict a moderate increase of cross hardening with

increasing (pre)tension. In particular, after 10% (pre)tension (Fig. 6,

right), the level of cross hardening predicted by the THM is largerthan predicted by the LSM. This is in contrast to the tension-shear

case inFig. 1. Although less pronounced than in the tension-shear

case, cross hardening in the tensionetension case is also transient

and tends to disappear with increasing loading after the loading

path change. Consequently, for this case as well, dislocation struc-

tures responsible for cross hardening appear to break down and

disappear after change of loading path, i.e., as along as the loading

path direction remains constant. The results in Fig. 6imply that the

level of (pre)deformation during the rst stage of a multi-stage

loading path inuences the amount of resulting cross hardening.

As shown in Fig. 7, such a dependence is also reected in theconcomitant yield surface development. As before, the cross

hardening evident in stress-deformation results like those in Fig. 6

results in the LSM (Fig. 7, right), but not in the THM (Fig. 7, left), in a

change in shape (and orientation) of the yield surface. In particular,

as evident inFig. 7(right), the LSM predicts a rotation of the major

axes of the (in the K11; K22 plane) elliptic yield surface away from

the equibiaxial tension orientation towards K22 with increasing

deformation.

Compared to the case of plane-strain tension followed by simple

shear (Fig. 1), the amount of cross hardening in the case of plane-

strain tension in the rolling direction followed by uniaxial tension

(Fig. 6) is generally much smaller (e.g., Clausmeyer et al., 2011a).

This is correlated with the degree of orthogonality of the loading-

path change. According to Schmitt et al. (1994), a two-stageloading path is characterized in this regard by the angle

q arccos(NP1,NP2) determined by the direction of the rate of in-elastic ow NPiduring the rst (i 1) and second (i 2) stages ofthe two-stage path. In terms of this angle, plane-strain tension to

simple shear is characterized by q 90, and plane-strain tension touniaxial tension byq 30. As has been discussed elsewhere (e.g.,Wang et al., 2008), this angle can be used to characterize the ex-

pected degree of orthogonal loading-path change and similarly

represent the strength of possible cross hardening as a function of

position in sheet metal parts subject to forming operations like

Fig. 5. Comparison of yield surface development inK12; K11space predicted by the THM and the LSM for DC06, and by the PM for IF steel, after uniaxial tension in the rolling

direction (left) and simple shear in the rolling direction (right), up to 10% von Mises equivalent strain. As done for the results from the THM and the LSM via ( 19), those for Kfrom

the PM were normalized using the initial yield stress of the material as determined by Peeters et al. (2002). See text for details and web version for color.

Fig. 6. Development ofK11in DC06 during loading in plane-strain tension (in the rolling direction) to 5% (left) and 10% (right) deformation followed by unloading and reloading in

uniaxial tension (again in the rolling direction) predicted by the THM and the LSM. For comparison, experimental and model results are also shown for monotonic uniaxial tension

(Clausmeyer et al., 2011a) and web version for color.



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deep drawing. An extension to this concept is the recently proposed

cross hardening indicator (Clausmeyer et al., 2014) which considersthe degree of loading-path change and the material specic ten-

dency to exhibit cross hardening.

As mentioned above and shown in Fig. 6 (right), more cross

hardening is predicted by the THM than the LSM after 10% pre-

tension. To examine this in more detail, different contributions to

the general hardening level are displayed inFig. 8. As evident, the

THM predicts a transient increase inX11(Fig. 8, left) not predicted

by the LSM (Fig. 8, right) resulting from the coupling ofsxin (16) to

S. For this loading path change, note that the change in the shape

of the yield surface predicted by the LSM in the tensionetension

case (Fig. 7, right) is notas pronounced as for the tension-shear case

(Fig. 4, right).

Lastly, consider the FE-based simulation of the tension-bending

of a DC06 sheet metal workpiece. For this purpose, the THM and theLSM were implemented into LS-DYNA (LSTC, 2012) via the user

material interface. Implicit global time integration was used for the

solution of the initial boundary value problem. The specimen of

dimension 10 mm 10 mm 1 mm was discretized with a regularmesh of 2000 (20 20 5) tri-linear hexahedral nite elements(seeFig. 9) with reduced integration and hourglass control (LSTC,

2012). Two tension-bending cases are considered: (i) plane strain

tension in e1 (rolling direction), and (ii) equibiaxial tension in e1and e2, both followed by bending in the e2 direction (seeFig. 11

below). After predeformation in tension, a bending moment is

applied on the lower right hand edge of the specimen, and the left

edge is xed. The sheet is bent until a bending angle of 90 is ob-tained. Bending moment-angle results are shown in Fig. 10; the

deformed sheet and nal spatial stress distribution are displayed in

Fig. 11.

As evident in Fig.10, bending-moment results based on the THM

and the LSM are qualitatively similar for both loading cases.

Consideration of hardening stagnation in the THM results in faster

saturation of the bending moment after about 60 in particular forthe equibiaxial (pre)tension. The expected trend in stress state and

Fig. 7. Yield surface development in theK11; K22plane from the THM (left) and the LSM (right) due to plane-strain tension in the rolling direction to different levels of deformation.

The symbols x, and *mark stress states on the yield surface at F1110.05,F1110.10, and F111 0.15, respectively.

Fig. 8. Development ofrandX11predicted by the THM (left) and the LSM (right) during loading in plane-strain tension (in the rolling direction to 10%) to uniaxial tension ( Fig. 6).

Fig. 9. Initial dimensions and FE discretization of sheet for tension-bending

simulations.



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stress distribution from tension (above) to compression (below) the

midplane of the workpiece is clearly predicted by both models. In-

plane inhomogeneity of the stress eld perpendicular to this is

related to thenite dimensions of the specimen. As indicated by the

bending moment results inFig. 10, the THM predicts more hard-ening in the specimen than the LSM for both plane-strain tension

andequibiaxial tension followed by bending. This is consistent with

the results forKIshown inFig. 11, i.e., the stress level predicted by

the THM is higher than that predicted by the LSM. Since bending

results in a transition from tension to compression, the above ob-

servations for plane strain tension to uniaxial tension loading can

be transferred to the bending case. Up to a bending angle of about

8 bending of the sheet is mainly governed by elastic deformationbecause the release of the displacement constraints in the e1- and

e2 directions is only partly compensated by the incipient straining

in the e1-direction due to bending. Similar to the plane strain

tension to uniaxial tension case, a transient hardening regime fol-

lows. Due to the inhomogeneous nature of the bending deforma-

tion, the duration of this regime (in terms of the bending angle) islarger compared to the homogeneous plane strain tension to uni-

axial tension case. However, the same tendencies are observed. The

level of stress predicted by the THM is larger compared to the LSM.

Consequently, the resulting bending moment is also larger. For

larger bending angles, the differences between the two models

become smaller. Note, that the transition between the elastic

regimeand the plastic transient hardening regime is sharper for the

LSMcompared tothe THM. This is in agreement with the results for

plane strain tension to uniaxial tension.

7. Summary and discussion

In the current work, two models for anisotropic, and in partic-

ular cross, hardening in bccsheet steels subject to non-proportional

loading have been compared with each other in detail. The rst

model is the modied form (Wang et al., 2008) of that due to

Teodosiu and Hu (1995, 1998). In this (modied) Teodosiu-Hu

model (THM), cross hardening is assumed to affect (i) the yield

stress and (ii) the saturation value of the back stress. The secondmodel is due toLevkovitch and Svendsen (2007)andNoman et al.

(2010). In this Levkovitch-Svendsen model (LSM), cross hardening

is assumed to affect theow anisotropy. As attestedto by numerous

applications of the THM (e.g., Boers et al., 2010; Bouvier et al., 2005,

2003; Hiwatashi et al., 1997; Li et al., 2003; Thuillier et al., 2010;

Wang et al., 2008) and the LSM (e.g., Barthel et al., 2013;

Clausmeyer et al., 2014, 2011b; Noman et al., 2010), both models

are able to account for the observed effect of cross hardening on

experimental stress-deformation data. As investigated and docu-

mented in the current work, there are otherwise a number of dif-

ferences between the two. For example, in contrast to the THM, the

LSM predicts no hardening stagnation after cross hardening as

observed in experiments. On the other hand, in contrast to the LSM,

the THM predicts no change in yield surface shape during non-proportional loading. As documented in the current work, this is

in contrast to experimental results (for ferritic tube steel:

Kowalewski and Sliwowski, 1997) as well as to predictions of more

sophisticated micromechanical models (e.g.,Holmedal et al., 2008;

Peeters et al., 2002).

As discussed in many previous works as well as here, in the

context of a yield functionfYof the form (1), the (ow) anisotropy

tensor A determines the shape of the yield surface. In the THM, this

shape is attributed to the effect of an (initial) rolling texture, i.e., the

grain (orientation) structure, on the ow behavior. On the other

hand, in the LSM, A is associated with both the grain orientation

and dislocation structures, i.e., AA gra H . In the yield surfacecontext and model form (1) for fY, then, (13) and (17) imply the

correspondence

Fig. 10. Bending moment as a function of bending angle during tension-bending of DC06 predicted by FE simulations based on the THM and the L SM. Left: plane-strain (pre)tension

toF111.2. Right: equibiaxial (pre)tension to F11 F221.1. See text for details.

Fig. 11. Deformed sheet geometry (bending angle 90) after equibiaxial-tension-bending and spatial distribution of the rst principle componentKIof the Kirchhoff stress predicted

by the THM (left) and the LSM (right). Refer to the web version for color.



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ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS$A gra H

S

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS$A gra S

p fS

(20)between H and S, where SMX. Consequently, in the THM, thecoupling between the state of stress Mon the yield surface and the

cross hardening tensor S is linear and scalar. As in the case of

standard isotropic hardening itself, then, it is not surprising that

this model decouples any change in form of the yield surface from

cross hardening effects. This is in contrast to the LSM, which cou-

ples M and the contribution Hfrom cross hardening to the ow

anisotropy tensor non-linearly and tensorially. In this case, a

directional dependence is generic and inherent. As implied by the

comparisons in Section 5 and Fig. 5, micromechanically based

models like the PM (Peeters et al., 2001b,a; 2002) appear to support

the idea of a change inyield surface shape due tocross hardeningas

assumed in the LSM. Being physically based, the PM provides at

least indirect physical evidence for the LSM, since it directly in-

corporates the physical knowledge of the contribution of disloca-

tion interaction to the evolution of the yield surface.

With respect to the hardening modeling in general, note that

isotropic, kinematic, and cross, hardening are decoupled in the

LSM, but coupled in the THM. From the point of view of the ge-

ometry and morphology of the yield surface, then, only in the LSMare isotropic, kinematic, and cross, hardening uniquely related to

the size, center, and shape, respectively, of the yield surface. As

implied by the current comparison, however, to account for pro-

cesses such as hardening stagnation, coupling of hardening types

such as that assumed in the THM may be necessary. Indeed, as

implied by the results of the current workas well,except in the case

of the coupling of yield surface morphology and development to

anisotropic hardening in the sense of a change of yield surface

shape, however, the THM is more sophisticated and micro-

mechanical in nature.

The authors are not aware of comprehensive experimental

evidence which shows the evolution of the yield surface of DC06

for larger (pre)strains (z10%). Experiments on tubular specimens

have shown that there is an evolution of the yield surface (Phillipset al., 1974). In the case of other steels (e.g., SUS304: Ishikawa,

1997), experimentally observed changes in the yield surface

shape during tension-torsion loading are accounted for by the

modeling of cross hardening in the LSM as based on Hand related

to dislocation structure evolution. Kuwabara et al. (2000) and

Kuwabara (2007)investigated the effect of uniaxial, plane strain

and biaxial tension on the yield surface of IF steel up to 1.0% strain.

This work focused on the initial yield surface alone. In any case,

more experimental work on sheet steels along these lines is

clearly needed and required, and will hopefully be available in the

future.

Acknowledgments

The authors thank the reviewers of the rst version of this work

for their constructive criticism and comments which have lead to

major improvement. The authors would like to thank Ton van den

Boogaard from the Faculty of Engineering Technology, University

Twente, The Netherlands, for providing the opportunity to use the

biaxial tester, and Alper Gner from the Institute of Forming Tech-

nology and Lightweight Construction, TU Dortmund University,

Dortmund, Germany, for providing the uniaxial tension test data.

Partial nancial support from the German Research Foundation

(DFG) under contract PAK 250 Identikation und Modellierung der

Werkstoffcharakteristik fr die Finite-Element-Analyse von Ble-

chumformprozessen - TP4 is gratefullyacknowledged. The material

investigated was provided by ThyssenKrupp Steel Europe AG.

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