A theoretical investigation of the influence of dislocation sheets on evolution of yield surfaces in...

Journal of the Mechanics and Physics of Solids50 (2002) 783–807

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A theoretical investigation of the in&uence ofdislocation sheets on evolution of yield surfaces

in single-phase B.C.C. polycrystalsBart Peetersa ; ∗, Surya R. Kalidindib, Cristian Teodosiuc,

Paul Van Houttea, Etienne Aernoudta

aDepartment of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, KasteelparkArenberg 44, B-3001 Leuven, Belgium

bDepartment of Materials Engineering, Drexel University, Philadelphia, PA 19104, USAcLPMTM-CNRS, Universit+e Paris Nord F-93430 Villetaneuse, France

Received 18 January 2001; received in revised form 1 June 2001; accepted 1 June 2001

Abstract

Accurate and reliable predictions of yield surfaces and their evolution with deformation re-quire a better physical representation of the important sources of anisotropy in the material.Until recently, the most physical approach employed in the current literature has been theuse of polycrystalline deformation models, where it is assumed that crystallographic textureis the main contributor to the overall anisotropy. However, recent studies have revealed that thegrain-scale mesostructural features (e.g. cell-block boundaries) may have a large impact on theanisotropic stress–strain behaviour, as evidenced during strain-path change tests (e.g. cross e8ect,Bauschinger e8ect).In previous papers, the authors formulated an extension of the Taylor-type crystal plasticity

model by incorporating some details of the grain-scale mesostructural features. The main pur-pose of this paper is to study the evolution of yield surfaces in single-phase b.c.c. polycrystalsduring deformation and strain-path changes using this extended crystal plasticity model. It isdemonstrated that the contribution of the grain-scale substructure in these metals on yield lociis comparable in magnitude to the e8ects caused by the di8erences in texture. Furthermore, itis shown that the shape of yield loci cannot be predicted accurately by the traditional poly-crystalline deformation model with equal slip hardening. The trends predicted by the extendedcrystal plasticity model are in much better agreement with the experimental evidence reported inthe literature than those represented in classical treatments by isotropic and kinematic hardening.? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Plastic anisotropy; A. Microstructure; Texture; Yield surfaces; Changing strain paths

∗ Corresponding author. Tel.: +32-16-32-17-80; fax: +32-16-32-19-90.E-mail address: [email protected] (B. Peeters).

0022-5096/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S0022 -5096(01)00094 -1

784 B. Peeters et al. / J. Mech. Phys. Solids 50 (2002) 783–807

1. Introduction

The traditional approach to modelling plastic deformation in metals has been to for-mulate a phenomenological description of the yield surface, and then to deDne a &owrule based on the geometry of the yield surface (e.g. normality &ow rule). However,it has long been recognized that, under progressing plastic deformation, the yield sur-face of a given material undergoes signiDcant changes. Consequently, there have beenseveral attempts to understand the changes in yield surfaces due to deformation, andto develop models that can capture them accurately.In the simplest phenomenological plasticity models, the evolution of the yield sur-

face of a metal during deformation is often described by a combination of isotropicand kinematic hardening laws. In isotropic hardening laws, it is assumed that the yieldsurface will undergo only a change in size, without any changes in shape or posi-tion. Kinematic hardening laws (see, e.g. Prager, 1949; Ziegler, 1959; Dafalias andPopov, 1975; Cailletaud, 1987; Hu et al., 1992) take the shift in the position of theyield surface in stress space into account. Note that neither of these models accountstrictly for a change in the shape of the yield surface with imposed plastic deforma-tion.The development of crystal plasticity models (see, e.g. Aernoudt et al., 1993), where

the distribution of the crystal orientations in the given material and the geometry of theavailable slip systems in the given crystals are taken into account explicitly, opens thepossibility of characterizing more accurately the anisotropic yield surface of a givenmaterial by measuring its texture (e.g. using di8raction techniques) and employing anadequate polycrystalline model (e.g. a Taylor-like model or a self-consistent scheme).There have been several attempts in the literature (see, e.g. Bunge, 1980; Mols et al.,1984; Van Houtte, 1987; Barlat et al., 1991, 1997; Van Houtte et al., 1995) to developappropriate mathematical descriptions of anisotropic yield surfaces based on the yieldsurface predictions from crystal plasticity models. In prior studies, it was often assumedthat crystallographic texture was the main contributor to the anisotropy in a givenmaterial. Furthermore, all the slip systems in the polycrystal were assigned identicalvalues of the slip resistance in these simulations.It has been demonstrated that the shape of the yield surface predicted by crys-

tal plasticity models is quite sensitive to the hardening laws used, i.e. the nature ofthe evolving slip anisotropy on the di8erent slip systems in the crystals during priorcold work (Asaro and Needleman, 1985; Kalidindi and Schoenfeld, 2000). It is worthnoting that Asaro and Needleman (1985) showed that Kocks-type latent hardeningled to increased Bauschinger e8ect when yield surfaces were calculated for texturesderived from the Taylor averaging criteria, but the work did not separate the rela-tive contributions of texture and deformation-induced slip anisotropy to the overallanisotropy of the aggregate. The results shown by Kalidindi and Schoenfeld (2000) in-dicated that slip anisotropy and latent hardening assumptions are more signiDcant thanpreferred crystallographic orientations (texture) in determining the anisotropic stressresponse.Recently, the present authors have extended the Taylor-type crystal plasticity models

by incorporating more details of the microstructure at the grain scale (Peeters et al.,

B. Peeters et al. / J. Mech. Phys. Solids 50 (2002) 783–807 785

2001a). SpeciDc internal state variables to account for the development of dislocationsheets in the form of cell boundaries and cell-block boundaries have been used. SomespeciDc characteristics of these microstructural features, such as latent hardening andpolarity, were built on earlier microstructural models (see, e.g. Teodosiu, 1992; Hu etal., 1992; Teodosiu and Hu, 1995; Hiwatashi et al., 1997; Peeters et al., 2000). Thisnew extended Taylor model was initially formulated with a di8erent rule for decidingthe planes of the dislocation sheets (Peeters et al., 2000), but was modiDed in a laterpaper in light of experimental observations (Peeters et al., 2001a, b).The model has been developed for single-phase b.c.c. polycrystals. In a material con-

taining a high density of precipitates and impurities, one can expect that these woulda8ect the hardening=softening at strain-path changes (Hosford and Zeisloft, 1972).However, the new model does not account for such e8ects and is thus restricted tosingle-phase materials. The model has been validated macroscopically during an elab-orate series of two-stage strain paths for thin rolling plates (Peeters et al., 2001a). Acomplete validation, covering the Dve-dimensional space of the strain modes, is practi-cally impossible for thin metal sheets, since the only available mechanical tests capableof achieving relatively large strains are the tensile test and, more recently, the simpleshear test. Notwithstanding, such tests and the associated TEM evidence are coveringthe main strain paths that are imposed on the material in some technologically im-portant processes, e.g. in sheet metal forming. Also a detailed validation of the modelon the mesoscopic scale has been undertaken: both the spatial orientation of the dis-location sheets and the dislocation densities of the substructural features predicted bythis model were in good qualitative agreement with TEM micrographs for a variety ofcrystal orientations after a series of two-stage strain paths (Peeters et al., 2001a, b).In the present paper, the major features predicted by our new model for the evolution

of yield surfaces during a cross test (i.e. a mechanical test for which the strain ratetensors of the 2 sequential strain paths are orthogonal) are reported. It will be shownthat the predicted trends for the changes in the shape of the yield surface from thismodel are quite reasonable and signiDcantly better than those from the crystal plasticitymodels with equal hardening of all slip systems. The speciDc aims of the present paperare to demonstrate that (i) the grain-scale substructure plays an important role thatcannot be overlooked, i.e. crystallographic texture information alone is often insuIcientto accurately predict the yield surface of a given work-piece material, and (ii) thechanges in the yield surface predicted by the new extended Taylor model during strainpaths are signiDcantly closer to reality than those that can be captured in classicaltreatments by isotropic and kinematic hardening.

2. Extended Taylor model

2.1. Sources of anisotropy

In order to describe accurately the evolution of the yield loci in a given material,it is imperative to take into account the most important sources of anisotropy in thematerial: the slip system activity at the microscopic level (see, e.g. Gil Sevillano,


Fig. 1. Experimental stress–strain behaviour of an IF steel during a monotonic simple shear test, a cross testand a reverse test.

1993), substructure evolution at the mesoscopic level (see, e.g. Teodosiu, 1992) andthe texture development during deformation at the macroscopic level (see, e.g. VanHoutte, 1988).During plastic strain the grains of a polycrystalline material rotate towards stable ori-

entations. This geometrical rearrangement is modelled adequately by current polycrys-talline plasticity models, e.g. the classical Taylor–Bishop–Hill models (Taylor, 1938;Bishop and Hill, 1951a, b) and their extensions (see, e.g. Van Houtte and Aernoudt,1975; Van Houtte, 1988; Aernoudt et al., 1993). In these models, it is assumed thatall the individual grains in the polycrystal experience the same deformation gradienthistory (Taylor, 1938). It has been shown in literature that in spite of this simplifyingassumption, the models produce fairly accurate predictions of the monotonic stress–strain response and the deformation texture of polycrystalline materials. Furthermore, itwas recently shown by Kalidindi and Schoenfeld (2000) that relaxing this assumptionusing Dnite element models did not in&uence signiDcantly the shape of the predictedyield surface. A general discussion of polycrystalline modelling with the emphasis onsingle crystal properties can be found in Kock’s review (Kocks, 1970).Metals are known to develop under plastic deformation organized dislocation struc-

tures at the grain scale. TEM studies and experimental stress–strain curves of single-phase b.c.c. metals during two-stage strain paths have shown that this substructure playsa key role in the anisotropic behaviour of the material (see, e.g. Rauch and Schmitt,1989). The signiDcant in&uence of the substructure on &ow stress anisotropy has alsobeen reported for cold rolled aluminium sheets (Winther et al., 1997; Winther, 1998).Fig. 1 shows the experimental stress–strain curves of a deep-drawing quality IF steelfor (1) a simple shear test with the shear direction (SD) parallel to the rolling direction(RD) and the shear plane normal (SPN) parallel to the transverse direction (TD), (2)a cross test, 10% true tensile strain in RD, followed by simple shear with SD also par-allel to RD and SPN parallel to TD, (3) a reverse test (also called Bauschinger test),30% simple shear in RD, then simple shear in the opposite sense. A short overview


Fig. 2. (a) TEM micrograph in a grain of a specimen after a tensile test, 20% amount of tensiledeformation with the tensile axis parallel to RD, in longitudinal plane view. The grain orientation is(ND)[RD]= (−0:86; 0:23; 0:45)[0:45;−0:08; 0:89]. The tensile axis and the orientation of the CBBs are de-noted in this Dgure. (b) Schematic representation of the microstructure: cell-block boundaries (CBBs) parallelto {1 1 0}-planes, the most active slip systems; cell boundaries (CBs) with a more random character.

of the in&uence of the developed substructure on the speciDc features observed in thestress–strain curves is given here:

(i) In TEM micrographs of single-phase b.c.c. metals a clear distinction can bemade between cell-block boundaries (CBBs), i.e. planar dislocation boundaries, andcell boundaries (CBs), i.e. a more randomized dislocation mosaic in between the CBBs,as shown in Fig. 2a. CBBs are currently generated parallel to the planes of the mostactive slip systems (Fernandes and Schmitt, 1983; Rauch, 1992a) and will cause mostof the latent hardening (Franciosi, 1985). An immediate consequence is that when newslip systems are activated (by rotation of the crystal or a change in strain path), thesedislocation sheets act as e8ective obstacles for the mobile dislocations. It has beensuggested that the CBBs are responsible for the “cross e8ect” observed in cross tests(Sc in Fig. 1; Rauch and Schmitt, 1989). After a small plastic strain along the new de-formation path a transient behaviour in the stress–strain curve is observed (see Fig. 1),which is associated with micro-localizations of plastic &ow. These will lead to the de-velopment of microbands (Thuillier and Rauch, 1994). They are roughly parallel to theplane of the newly most active slip system. These microbands cut through the “old”dislocation sheets and produce channels for easy transport of the mobile dislocations.Progressively with increasing strain along the second path, there is further destructionof the pre-existing structure. In the mean time microbands saturate and are replaced bydislocation sheets (CBBs), i.e. a new dislocation structure is formed corresponding tothe last deformation mode. The macroscopic equivalent of this process can be observedin the stress–strain curve, which eventually tends towards the monotonic stress–straincurve characteriztic of the new strain mode (see Fig. 1).(ii) The polarity of CBBs is caused by the mechanical “charge” (Burgers vector)

of the dislocations, i.e. dislocations of a certain sign are stopped on one side of theboundary and of the opposite sign on the other side of the boundary. This polarity gives


rise to an asymmetry of slip resistance, and has been suggested to be responsible forthe “Bauschinger e8ect” observed in reverse tests (Sb in Fig. 1; Rauch and Schmitt,1989). When the load is reversed, the “polarity” dislocations can escape from the CBBsand will annihilate with new mobile dislocations of opposite sign and with immobiledislocations stored in the cell interiors as well as in the CBs (Hasegawa et al., 1975).The observed transient zone in Fig. 1 during a reverse test is associated with thisprocess. Progressively with increasing strain in the new deformation path, the CBBswill become polarized again but with the opposite sense than during the previous strainmode. The stress–strain curve in Fig. 1 will asymptotically tend towards the monotonicstress–strain curve characteriztic of the new strain mode.

2.2. Microstructural model

A brief, but complete description of the microstructural model is presented in thissection. For more details the readers are referred to previous work of Peeters et al.(2001a).The model starts from a developed substructure consisting of a cell and a cell-block

structure fulDlling Holt-type scaling laws (Holt, 1970). The dislocation patterns are ide-alized in terms of cell-block boundaries (CBBs) and randomly distributed cell bound-aries (CBs) in addition to the statistically stored dislocations. Three internal variablesare introduced to characterize the substructure in a deformed material:

(1) The statistically stored dislocations in the cell interiors as well as the cell boundarydislocations are represented by the local dislocation density �.

(2) The local density of immobile dislocations (�wd) stored in CBBs.(3) The local directionally movable dislocation density (�wp), also called “polarity

dislocation” associated with the CBBs. This density is assumed to have a polarityor sign, i.e. it can also take on negative values.

A comparison between a TEM micrograph after 20% tensile strain and the schematicrepresentation of the internal variables of the model is provided in Fig. 2.

2.2.1. Formation of CBBsDuring the deformation the model constructs at most two families of CBBs with the

following criteria. The primary family of CBBs is generated parallel to the {1 1 0}-planeof the most active slip system. The second family of CBBs is realized in a similarway parallel to the {1 1 0}-plane of the second active slip system. This construction,i.e. the observation of maximum two families of CBBs in a deformed crystal as wellas the {1 1 0}-character of the CBBs, is based on experimental evidence (Fernandesand Schmitt, 1983; Nesterova et al., 2001; Peeters et al., 2001b). Let uw1 and u

w2 denote

the unit vectors perpendicular to these two families of CBBs, respectively. A Kockstype law is used to describe the storage and recovery behaviour of dislocations inthese cellular structures. The double cross-slip mechanism, proposed by Wiedersich(1962), may be appropriate to explain the fact that mobile dislocations will contributeto CBBs parallel to their slip plane. It is assumed that the mean-free path of the mobiledislocations, responsible for the construction of the family of CBBs i, is scaled with


the mean spacing between the dislocation sheets i. Applying Holt’s law (Holt, 1970)separately to each family of CBBs, the dislocation density of each CBB i can becalculated as

�wdi�wdsat

=1b

(Iwd√

�wdi�wdsat

− Rwd�wdi�wdsat

)i (i=1; 2) (1)

with b the magnitude of the Burgers vector, i the total slip rate on the plane of theith most active slip system (e.g. 1 is the total slip rate on the plane of the mostactive slip system) and �wdsat the saturation value of �

wd. The immobilization coeIcientIwd includes the probability of a mobile dislocation being trapped into a CBB. Therecovery coeIcient Rwd scales with the annihilation length.

2.2.2. Polarity of CBBsFor each currently generated CBB i, the net &ux of dislocations from slip systems

non-coplanar with that boundary i is calculated as

�i=n∑

s=1

�sbubs · uwi :

The symbol n denotes the number of slip systems, �s the slip rate on slip system s (canbe positive as well as negative) and ubs the unit vector assigned to the slip directionof the system s. A Kocks type law has been used to describe the storage and recoveryof these directionally movable dislocations

�wpi�wpsat

=

sign(�i)Iwp

√�wdi + |�wpi |

�wpsat− Rwp

�wpi�wpsat

|�i|; (2)

where Iwp and Rwp denote the immobilization coeIcient and the recovery coeIcientof the polarity layers, respectively; �wpsat represents the saturation value of the direc-tionally movable dislocation density. This equation implies that directionally movabledislocations are created at the CBB by interactions of mobile dislocations with eitherCBB dislocations or directionally movable dislocations already stored at the boundary.Recovery takes place only between mobile dislocations and “polarized” dislocations ofopposite sign.

2.2.3. Construction of CBsIt is considered that the randomly distributed dislocation mosaic, represented by the

dislocation density �, produces merely an isotropic hardening of the material. A tradi-tional Kocks–Mecking-type law has been adapted for the evolution of this dislocationdensity

��sat

=1b

(I√

��sat

− R��sat

) n∑s=1

|�s|; (3)

where I and R represent the immobilization and the recovery coeIcients of the CBs,respectively; �sat is the saturation value of �.


2.2.4. Behaviour of CBBs and CBs during a change in slip activityA change in deformation path or a rotation of a crystal can lead to the activation of

new slip systems. The mobile dislocations belonging to these systems will graduallyform new CBBs corresponding to the current deformation mode (as explained in Sec-tions 2.2.1, 2.2.2), but will also slowly disintegrate the pre-existing CBBs with furtherstraining, according to the laws:

�wdi�wdsat

=− Rncgb

�wdi�wdsat

new ; (4)

�wpi�wpsat

=− Rncgb

�wpi�wpsat

new ; (5)

where new denotes the total slip rate on the 2 crystallographic planes containing thehighest slip activity and Rncg, the annihilation coeIcient of the latent, i.e. non-currentlygenerated, CBBs.A change in deformation path that leads to the activation of the same slip systems,

but in the opposite sense, as observed during reverse tests, will lead to a reversal of the&uxes contributing to the CBBs. Therefore, the CBBs will not dissolve. However, thedislocations that got stuck at the borders of the CBBs can move very easily away fromthe CBBs. These directionally movable dislocations will be annihilated by dislocationsof opposite sign, e.g. CB dislocations. This will result in an increase of the annihilationrate for the randomly distributed CBs

��sat

=1b

⟨(I√

��sat

− R��sat

) n∑s=1

|�s| − R2��sat

�bausch2�wpsat

n∑s=1

|�s|⟩

(6)

if no &uxes are reversed: =0;if 1 &ux is reversed: =1; �bausch = |�wpi | (i is the family of CBBs whose &ux is

reversed);if 2 &uxes are reversed: =1; �bausch = |�wp1 |+ |�wp2 |.The slip rates �s include the &uxes carried by mobile and remobilized dislocations.

The term �bausch=2�wpsat scales the slip rates, representing the fraction of the &uxes, which

are due to the remobilized dislocations under consideration. Consequently, the last termon the right-hand side of Eq. (6) represents the annihilation of the statistically storeddislocations in cell interiors and CBs, �, due to the remobilization of directionallymovable dislocations. The recovery coeIcient R2 re&ects the in&uence of the interactionof these two dislocation families on the annihilation rate of �. Note that Eq. (6) is ageneralization of Eq. (3).When a &ux �i, associated with a family i of currently generated CBBs, is re-

versed, the directionally movable dislocations stuck at the border of these CBBs willbe annihilated

�wpi�wpsat

=− Rrev�wpi�wpsat

|�i| (7)


with Rrev, the recovery coeIcient of the directionally movable dislocations responsiblefor the polarity of the CBBs due to a reversal of the net &ux.

2.2.5. Critical resolved shear stressThe critical resolved shear stress on slip system s is taken to be composed of several

contributions (1) �0 represents all aspects of the microstructure that are not includedin the internal variables selected here (e.g. solid solution e8ects); (2) the dislocationdensity inbetween the CBBs; (3) the dislocation density of the CBBs, and (4) the di-rectionally movable dislocation density associated with the CBBs. The resultant criticalresolved shear stress is obtained from the rule of mixtures (Mughrabi, 1987) for thestress in the CBBs and in the cell-block boundary interiors, represented by CBs, as

�cs = �0 + (1− f)�CB + f6∑

i=1

�CBBis (8)

with

�CB = �Gb√�; (8a)

�CBBis = �Gb(〈√abs(�wpi )u

bs · uwi sign(�wpi )〉+

√�wdi abs(u

bs · uwi )): (8b)

The Drst term in Eq. (8b) describes the asymmetry of slip resistances due to the polarityof the CBBs, the second term captures latent hardening of the CBBs. The constant �represents the dislocation interaction parameter, f is the volume fraction of CBBs andG the shear modulus.

2.3. Micro–macro transition

To scale up from the single crystal behaviour to the polycrystalline material responsethe microstructural model was implemented in a full-constraints Taylor model (VanHoutte and Rabet, 1997). For b.c.c. metals the systems {1 1 0}〈1 1 1〉 and {1 1 2}〈1 1 1〉were used to calculate the slip activity. Employing the Taylor assumption, the macro-scopic deviatoric Cauchy stress in the polycrystal U�ij can be expressed as

U�ij =1N

N∑k=1

�(k)ij ; (9)

where N is the number of crystals in the aggregate, and �(k)ij is the deviatoric Cauchystress in the kth crystal.It is expected that the implementation of the features described above will strongly

in&uence the shape, size and position of the yield surface. It will be shown in Section 4that the phenomenological descriptions of isotropic and kinematic hardening are fairlylimited in their ability to capture the complex changes in the shape of the yield surface.The model described above is potentially capable of capturing complex changes in theshape of the yield surface, while retaining a more physical description of the actualmicrostructure in the material.


3. Methods

3.1. Materials and experiments

The material studied is a 0:8 mm thick rolled sheet of a deep-drawing Ti-killedIF steel (0:0006 wt% C, 0:16 wt% Mn; 0:038 wt% Ti; 0:024 wt% Al; 0:0015 wt% Si;0:005 wt% P and 0:004 wt% S). The initial texture of the material has been measuredby standard X-ray di8raction techniques, and discretized into 250 orientations using the“statistical method” described by Toth and Van Houtte (1992). The crystal plasticitymodel used in the present study has been calibrated to the measured stress–strainresponses in Dve di8erent mechanical tests:

• a simple shear with SD parallel to RD and SPN parallel to TD;• 2 cross tests: 10% and 20% true tensile strain in RD, followed by simple shear withSD parallel to the tensile axis, and SPN parallel to TD;

• 2 Bauschinger tests: 10% and 30% amount of simple shear in RD and then simpleshear in the opposite sense.

The planar simple shear test is an eIcient technique to evaluate the mechanical prop-erties of &at samples. The major advantages of this test are the absence of any neckingdevelopment, the large range of achievable homogeneous strains, the simpliDcation ofthe sample geometry and the possibility to perform easily Bauschinger tests (Hu et al.,1992; Bacroix et al., 1994; Rauch, 1998). The conditions concerning the shape of thespecimen to achieve a homogeneous simple shear are reported in Rauch (1992b).Fig. 3 shows the experimental measurements and the corresponding Dtting results

from our model. The material parameters and the corresponding saturation values of thethree dislocation densities for the studied IF steel obtained by this calibration procedureare given in Table 1. Table 2 shows the values of the Burgers vector, the shear modulus,the dislocation interaction parameter, volume fraction of CBBs and the initial criticalresolved shear stress on all slip systems. It is shown elsewhere (Peeters et al., 2001a, b)that there is an excellent agreement between the model predictions and the experimentalresults attesting that the model captures very well the transients in strain-path-changeexperiments. Based on the agreements between experiments and simulation results, theauthors assume that our model is capable of capturing well the changes in the shapeand size of the yield surface for b.c.c. polycrystals even during complex strain-pathchanges.It should be noted that our deDnitions of “cross” and “Bauschinger” tests di8er

slightly from those used customarily in the literature. The authors follow the generaldeDnitions based on the scalar parameter �, proposed by Schmitt (1986), to charac-terize a two-stage strain path. The parameter � is deDned as the double-contractedtensor product between the plastic strain rate mode 1 during the prestrain, Apre, and thesubsequent plastic strain rate mode, A, i.e.

�=cos �=Apre :A: (10)

1 Strain mode denotes the strain rate tensor divided by its Euclidean norm.


Fig. 3. Experimental measurements and the corresponding simulated results for the stress–strain behaviourof an IF steel during di8erent strain-path changes. All tests were performed parallel to the rolling direction.

The Euclidean norm of these tensors is equal to 1. A strain-path change correspondingto �=0, is called a cross test. A strain-path change corresponding to �=−1, is deDnedas a Bauschinger test. Within the frame of this deDnition, there are two possible ways tolook at cross tests and Bauschinger tests. The classical approach (Schmitt et al., 1985;Schmitt, 1986; Raphanel et al., 1987) has been to study the di8erence in mechanicalresponse (e.g. yield strength) immediately before and after the change in strain path.For example, a tensile test followed by a simple shear test with the shear direction alongthe prior tension direction constitutes a cross test. In such a test, the shear yield strengthimmediately after the path change is compared with the tensile yield strength just priorto the path change. However, such comparison is made diIcult by the fact that thestress states in the two paths are distinctly di8erent from each other. In literature,the general approach has been to use a suitable “equivalent stress” in comparing thestrength levels in di8erent stress states, but such a procedure implicitly assumes thatthe material exhibits an isotropic response. To circumvent this diIculty, the authorsdeDne cross tests and Bauschinger tests in this study in a slightly di8erent way. Wesubject two di8erent samples to two di8erent strain modes related to each other bythe desired value of � according to Eq. (10). After a certain amount of prestrain, oneof the samples is subjected to a strain-path change into the strain mode of the second

794B.Peeters

etal.

/J.

Mech.

Phys.

Solids

50(2002)

783–807

Table 1Material parameters of the studied IF steel and the corresponding saturation values of the dislocation densities

I Iwd Iwp R Rwd Rwp Rncg Rrev R2 �wdsat �wpsat �sat(m) (m) (m) (m) (m) (m) (m) (m) (m) (1=m2) (1=m2) (1=m2)

8:5× 10−10 2:6× 10−8 2:1× 10−9 8:5× 10−10 2:6× 10−8 3:8× 10−9 2:3× 10−9 1:0× 10−8 1:0× 10−8 1:3× 1015 5:8× 1014 6:7× 1014

Table 2Values of the Burgers vector b, shear modulus G, dislocation interaction parameter �, volume fraction ofCBBs f and the initial critical resolved shear stress on all slip systems �0

b (m) G (MPa) � f �0 (MPa)

2:48× 10−10 8:16× 104 0.2 0.2 42.0


sample. Now that the strain paths after the path change are the same, we can directlycompare the stress values, without the need for an ad-hoc deDnition of an equivalentstress. For the example mentioned above the response of the monotonic simple sheartest will be compared with the response of the simple shear test after the prestrain intension. These tests use an ad-hoc deDnition of strain instead of an ad-hoc deDnitionof stress. This choice seems more reasonable, since, due to the stress saturation, theerrors introduced by the strain approximation are signiDcantly less than for stress, andthe error decreases at large strains. A further advantage of these tests is, that afterlarge strains the two stress–strain curves will converge to each other, as far as thetwo textures also converge to each other (i.e. the in&uence of the di8erence in startingtexture will disappear eventually). This, however, is not true in the traditional crosstests, where in the example mentioned above the response of the monotonic tensile testis compared with the response of the simple shear test after the prestrain in tension.Note that the tests in Fig. 3 follow the descriptions of cross tests and Bauschinger testsdescribed here. The same method is used throughout this study.

3.2. Computation of yield loci

Since the description of the complete yield surface requires a six-dimensional spaceand advanced visualization tools, the authors focus in this study on speciDc sectionsof this yield surface, e.g. (�11; �22) or the (�11; �12) sections. It is emphasized thatthroughout Section 4 the rolling direction RD corresponds to the 1-axis and the trans-verse direction TD to the 2-axis. For computing the (�11; �22) sections, di8erent velocitygradients in the global reference frame (i.e. RD, TD, ND) are imposed of the form

1 0 0

0 −q 0

0 0 −(1− q)

L11:

The value of q was varied in order to Dnd di8erent (�11; �22) combinations. All otherstress components are zero. The yield loci plotted in Section 4 are calculated at ano8set strain of 0.05% von Mises strain.

4. Predictions of yield loci

4.1. In<uence of texture

The in&uence of texture on the shape of yield surfaces will be addressed Drst. The(�11; �22) sections of the yield loci are computed from (i) a random texture (willbe called case “rt”); (ii) the initial texture of the deep-drawing IF steel (case “it”)and (iii) the deformation texture after a rolling simulation with a reduction of 70%,neglecting the in&uence of the CBBs (i.e. latent hardening and polarity). Switchingo8 the CBBs in the extended Taylor model transforms it into a full-constraints Taylormodel with equal hardening on all slip systems. The simulation mentioned in (iii)


Fig. 4. The simulated (�11; �22)-yield loci, normalized by the yield strength of a tensile test in the RD (�0),calculated from a random texture, from the initial texture of the IF steel and from the deformation textureafter a rolling simulation with a reduction of 70%, with only isotropic hardening (i.e. only cell boundariesand texture updating).

re&ects the material state due to the deformation texture, assuming isotropic hardening(will be called case “ih”). The (�11; �22) sections of the three yield surfaces are plottedin Fig. 4 after normalizing them by the yield strength �0 obtained by a tensile testparallel to RD (1-axis). Note that there is no grain-scale substructure involved in thesecomputations and thus, the di8erences between the 3 yield loci can be attributed todi8erences in textures alone.Since the normals to the yield surface play an important role in simulating defor-

mation processes the direction of the normals at di8erent points along the predictedyield surfaces is also provided. In fact, a rather small change in the shape of the yieldlocus can drastically in&uence the stress response corresponding to an imposed strainrate mode. To characterize the direction of the normals at certain points of the yieldsurface, two angles � and �, as shown in Fig. 5, are deDned in a reference frame Dxedto the centre of the yield locus (i.e. the point around which the yield surface is assymmetric as possible) and parallel to the global reference frame. In Fig. 6 the angle� is plotted for the cases “rt”, “it” and “ih” as a function of the position on the yieldsurface.In essence, Figs. 4 and 6 depict the e8ect of variations in texture, namely a ran-

dom texture, the initial texture of the IF steel, and the deformation texture using thefull-constraints Taylor model (without substructures). It seems that the texture di8er-


Fig. 5. Schematic representation of a yield surface and the deDnition of surface normals used in this study.

Fig. 6. The angle �, i.e. the angle the normal direction makes with the x-axis, is plotted as a function ofthe position on the yield surface (which itself is expressed as an angle � from the local x-axis).

ences have a substantial e8ect on the shape of the yield surface, which is illustratedboth by the variations in curvature of the yield loci (Fig. 4) and by the slope of the� vs. � diagram (Fig. 6).

4.2. In<uence of the grain-scale substructure

The in&uence of the CBBs, responsible for the latent hardening and the polarity, hasbeen studied in the case of a rolling simulation with a reduction of 70%. The discretizedinitial texture and the material parameters of the IF steel, given in Table 1, were


Fig. 7. (a) The simulated (�11; �22)-yield loci after a rolling simulation with a reduction of 70%, taking onlyisotropic hardening into account (i.e. only cell boundaries and texture updating), and taking all microstruc-tural features into account (i.e. latent hardening, polarity and texture updating). (b) The same simulated(�11; �22)-yield loci normalized by the yield strength of a tensile test in the RD.

provided as input to the extended Taylor model. A rolling simulation with a reductionof 70% was performed taking the CBBs into account. It was found that the deformationtextures for the case “ih” and this new case are identical. This is reasonable in lightof our previous experience that texture does not seem to be signiDcantly in&uencedby hardening laws in cubic materials. The second simulation re&ects the full hardenedstate of the material, which will be referred to as case “fh”. The obtained Dnal textureand the critical resolved shear stresses on each slip system in each grain allow thecomputation of the yield locus. In Fig. 7a the (�11; �22) sections of the yield loci aftera rolling simulation with a reduction of 70% are plotted for the case “ih”, i.e. withoutthe in&uence of the CBBs, and for the case “fh”, i.e. taking the CBBs into account.Since the two textures were identical, the di8erence in shape, position and size of theyield loci shown in Fig. 7a can be totally attributed to the in&uence of the CBBs.To emphasize the importance of the CBBs on the shape of the yield loci, the surface

normal � is also plotted in Fig. 6 for the case “fh” as a function of the position onthe yield surface. The di8erence between the curves for cases “ih” and “fh” can beattributed to the CBBs. It is seen that the e8ects caused by hardening are signiDcant andcomparable in magnitude to the e8ects caused by the di8erences in texture, indicatingthat it is quite important to consider substructural e8ects. Whereas the centre of theyield locus case “ih” is still positioned in (0; 0 MPa), the centre of yield locus case “fh”is shifted to (0:06�0; 0:22�0), where �0 denotes the initial CRSS on each slip system.To explain the in&uence of latent hardening and polarity of the CBBs on the shape of

yield loci, the yield surfaces for the cases “ih” and “fh” (Fig. 7a) were normalized by �0(Fig. 7b). The in&uence of � on the stress response has been studied in the following setof numerical experiments. Two numerical samples of the IF steel sheet with the initialtexture described earlier were subjected to two di8erent strain paths: (1) a simple shear


Fig. 8. The relation between S�, i.e. the di8erence between the initial yield strength of a monotonic simpleshear test (shear(�)) after 10% simple shear strain in RD and the yield strength at a shear strain of 10%obtained during a monotonic simple shear test (shear(�), without prestrain), and the corresponding variable� (Eq. (10)).

of �=0:1, with SD parallel to RD and SPN parallel to TD, (2) a simple shear with SDat an angle � to RD (shear(�)). Each angle � yields a di8erent value of �. After theprestraining, the deformed sample 1 was then subjected to a strain-path change into thestrain mode of the second sample, i.e. shear(�), and the shear yield strength in the newstrain path was probed. The di8erence between the initial shear yield strength of sample1 in the new strain path (shear(�), after prestrain) and the yield strength of sample2 at a shear strain of 10% obtained during a monotonic simple shear test (shear(�),without prestrain) is denoted as S�. Each S� is plotted, after normalization by theaverage yield strength just after the prestrain, against the corresponding � as diamondsin Fig. 8. As expected, for a value � equal to 1, S� was equal to zero (point A inFig. 8). However, when the plastic strain rate modes of the 2 sequential deformationsare exactly perpendicular (i.e. when � equals zero) a “cross e8ect” is obtained (pointB in Fig. 8). The latent hardening due to the CBBs plays an important role whennew slip systems get activated due to the change in the strain path. The CBBs willbe obstacles for the new mobile dislocations and thus, a higher stress response of thematerial is indeed expected. The polarity of the CBBs gives rise to an asymmetry ofslip resistance, namely the resistance of the material is lower on reversal of the strainrate mode (i.e. � close to −1). In that case most of the slip systems that were activeduring the prestrain are activated again, but in the opposite sense. Consequently, thedislocations responsible for the polarity of the CBBs can easily move away from theCBBs giving rise to a decrease in yield strength, i.e. S� becomes negative. When thestrain rate mode of the second deformation is exactly opposite of the strain rate modeof the Drst deformation, a “Bauschinger e8ect” is observed (point C in Fig. 8). Thetrend depicted as diamonds in Fig. 8 has been experimentally conDrmed in severalworks (Schmitt et al., 1985; Schmitt, 1986). The stress–strain curves of the same IFsteel obtained by simple shear=simple shear experiments, reported by Peeters et al.


Fig. 9. Simulation of a cross test, 10% true tensile strain in RD, followed by simple shear with SD parallelto RD and SPN parallel to TD.

(2001a), have been used to validate the simulation results in Fig. 8. The experimentalpoints presented as triangles in Fig. 8 conDrm the trend obtained by the simulations.It should be analysed whether these results are only dependent on the CBBs or

also on the di8erence between the deformation texture after the prestrain and the tex-ture obtained at the same equivalent strain during the last deformation mode withoutprestrain. For this purpose the same numerical experiments have been performed bya full-constraints Taylor code with equal slip resistances on all slip systems (i.e. thework-hardening model without CBBs). In the same Fig. 8 the di8erence between theinitial shear yield strength in the new strain path (shear(�), after prestrain) and theyield strength at the same equivalent strain obtained during a monotonic simple sheartest (shear(�), without prestrain) is plotted against the corresponding � as squares, afternormalisation by the average yield strength just after the prestrain. Note that no CBBsare considered in these simulations. The squares in Fig. 8 denote that texture di8erencesdo not have a signiDcant impact on the characteristic macroscopic features observedin strain-path change tests. The cross e8ect and Bauschinger e8ect are, in our opin-ion, caused by microstructural features related to hardening, and not by texture e8ects.Therefore, the diamonds in Fig. 8 depict the sole in&uence of the CBBs on the stressresponse during changing strain paths. By travelling along the yield locus the valueof � changes (Fig. 7b), providing an important contribution from the microstructure tothe overall plastic response of the material.

4.3. Evolution of yield surface during the cross test

In this section, the evolution of yield loci will be addressed during a cross test (10%true tensile strain in RD, i.e. the 1-axis, followed by simple shear with SD parallelto RD and SPN parallel to TD, i.e. the 2-axis). In Fig. 9 the simulated stress–straincurves of the cross test (thin line) and of a simple shear test without any prestrain(thick line) are depicted. During this simulation the yield loci are calculated at di8erentstrain levels denoted as A, B, C and D in Fig. 9.


Fig. 10. (a–d) Yield loci corresponding to the material states (A–D) deDned in Fig. 9. The arrows denotethe projection in a (�11; �12) plane of the imposed strain rate mode corresponding to the subsequent simpleshear. (e) Yield loci corresponding to the material state (A) deDned in Fig. 9 in the assumption that thesubstructure is not being considered.


The (�11; �12) sections of the yield surfaces plotted in Fig. 10a correspond to thetwo di8erent material states at strain level A denoted in Fig. 9: (i) 10% tensile strainin RD (thin line) and (ii) 10% von Mises equivalent strain by simple shear withSD parallel to RD (thick line). It should be analysed whether the di8erences in theyield surfaces of Fig. 10a are only dependent on the CBBs or also on the di8erencebetween the deformation texture after the tensile strain and the texture after the shearstrain. For this purpose the same numerical experiments have been performed by afull-constraints Taylor code with equal slip resistances on all slip systems (i.e. thework-hardening model without CBBs, assuming isotropic hardening). It seems that theresulting yield loci, plotted in Fig. 10e, are close to each other. Consequently, thedi8erence in shape between the plotted yield surfaces in Fig. 10a re&ects the di8erentanisotropic character of the developed grain-scale substructure during the tensile testand the shear test. For instance, during the tensile test with the tensile axis parallelto RD CBBs are formed making an angle between 35◦ and 65◦ with the tensile axis(Fernandes and Schmitt, 1983). During a subsequent simple shear deformation with SDparallel to the tensile axis (i.e. point Ac in Fig. 10a), most of the mobile dislocationswill move parallel to SD (Rauch and Schmitt, 1989). The CBBs developed during theprestrain, the tensile test, will be the main obstacle for the mobile dislocations, whichresults in a cross e8ect (i.e. the di8erence between Ac and Am in Fig. 10a; !c in Fig.9). A cross e8ect can also be observed when the initial prestrain is performed by thesimple shear test (thick line in Fig. 10a), and the second deformation is either a tensiletest (i.e. the di8erence between Cm and Cc in Fig. 10a) or a compression test (i.e. thedi8erence between Dm and Dc). The yield surfaces plotted in Fig. 10a show that theevolution of yield surfaces of a metal during deformation cannot be described simplyby a combination of isotropic and kinematic hardening laws.Starting from the two material states mentioned above, the samples were deformed

further in simple shear with SD parallel to RD. The stress–strain curves of thesesimulations are plotted in Fig. 9. The arrows depicted in Fig. 10a–d show the projectionof the imposed strain rate mode in a (�11; �12) plane. During the subsequent simpleshear the cross e8ect (di8erence between Ac and Am) gets smaller and will disappear(Figs. 9 and 10a–d). This is due to the annihilation of the CBBs of the previousdeformation mode (tensile test) and the creation of new ones corresponding to thenew mode (Rauch and Thuillier, 1993). After a certain deformation the yield loci arealmost identical indicating that the history of the tensile test is almost completely wipedout and replaced by a new grain-scale substructure corresponding to the simple sheardeformation.In Fig. 11a–d the yield surface normals (�) are plotted for each yield surface of Fig.

10a–d as a function of the position on the yield surface. These curves show again thatthe yield surface undergoes signiDcant changes in shape during the strain-path change.An evolution parameter #, deDned by

#=∫ 2$

0(�complex test(�)− �monotonic test(�))2 d�;

is introduced to evaluate the di8erence between these curves. The values of #, indicatedin Fig. 11a–d, conDrm that the curves tend to each other during further straining in


Fig. 11. (a–d) The angle �, i.e. the angle the normal direction makes with the x-axis, is plotted for the yieldloci in Fig. 10a–d as a function of the position on the yield surface (which itself is expressed as an angle� from the local x-axis). (e) The evolution of the centre of the yield loci plotted in Fig. 10a–d.

the second deformation path. Eventually, after a certain transition zone, the yield lociwill become almost identical.Fig. 11e shows the evolution of the centre of the yield locus at various strain levels.

The movement of the yield surface during plastic deformation is mainly determinedby the polarity associated to the dislocation sheets and the changes in crystallographictexture. Since the directionally movable dislocations cause an asymmetry of slip re-sistance, they produce a Bauschinger e8ect during a reverse test. Therefore, the yieldsurface is expected to shift in the six-dimensional stress space parallel to the stressmode. In fact, phenomenological theories (e.g. Dafalias, 1983) often assume that the


yield surface translates in the stress space parallel to the stress mode, which is describedin a material frame. This is a frame in which are written the evolution equations ofthe Cauchy stress and of the internal variables, the latter being the macroscopic coun-terpart of the material microstructure. For calculating the rotation of this frame severalsolutions have been proposed in the literature (see, e.g. Mandel, 1982; Dafalias, 1985).The simplest methods are either based on the skew-symmetric part of the velocity gra-dient or based on the rigid body rotation rate. More elaborate formulations considerthe rotation rate of the frame as an average of the rotation rates of the crystal orien-tations (Mandel, 1982). During a tensile test no large rotations are involved, and theyield locus is expected to translate along the axis �11. This is in agreement with theshift predicted by our crystal plasticity model (point A in Fig. 11e). Taking the largerotations that accompany a simple shear test into account (Dafalias, 1983), the yieldsurface during such a test is indeed translating along the �12-axis in a rotating materialframe (diamonds in Fig. 11e). The sense of rotation agrees well with the sense of thesimple shear test. After a change in strain path the yield locus translates in the stressspace according to the last deformation mode. Indeed, the squares in Fig. 11e depict theevolution of the centre of the yield surface during a simple shear test after a prestrainin tension. Since the substructure built up during the prestrain will be gradually re-placed by the induced anisotropy of the subsequent simple shear test, the yield surfacewill shift simultaneously in the stress space along the negative �11-axis, re&ecting thedestruction of the old dislocation sheets, and along the �12-axis in a rotating materialframe, re&ecting the microstructural changes during the new deformation mode.

5. Conclusions

Most phenomenological plasticity models describe the evolution of yield surfaces bya combination of isotropic and kinematic hardening. This implies that the shape of theyield surface stays constant during the deformation. However, the induced anisotropyassociated with the development of preferred grain orientations and of the grain-scalemicrostructure during deformation is expected to a8ect signiDcantly the shape of theyield surface. This is an important feature in plasticity mechanics because even a smallchange in the shape of the yield locus can drastically in&uence the stress responsecorresponding to an imposed strain rate mode. The authors developed a crystal plasticitybased work-hardening=softening model (Peeters et al., 2001a) that attempts to take intoaccount the microstructural details at the grain scale, consisting of dislocation sheetsin each crystal of the polycrystalline aggregate. This model allowed us to study thee8ects of texture and of microstructure on the shape, size and centre of the yield lociduring deformation, in an IF steel. The following conclusions can be drawn:

(1) The development of preferred grain orientations during deformation has a largeimpact on the shape of the yield locus.(2) Also the grain-scale substructure plays an important role on the shape, size and

centre of the yield surfaces. Consequently, the crystallographic texture information aloneis inadequate to accurately predict the yield surface of a given material under complex


strain paths. It has been shown that the predicted trends for the changes in the shape ofthe yield surfaces due to the dislocation sheets are quite reasonable. The e8ects causedby these sheets are signiDcant and comparable in magnitude to the e8ects caused bydi8erences in texture. These results emphasize the importance of the contribution fromthe grain-scale substructure to the overall anisotropy in the material.(3) The changes in the yield surface during strain-path change experiments that are

predicted by the model, are signiDcantly closer to reality than those that can be capturedin classical treatments by isotropic and kinematic hardening.

Acknowledgements

BP acknowledges the Dnancial support through a scholarship of the Fonds voorWetenschappelijk Onderzoek—Vlaanderen (FWO). Financial support was also providedby the Federal Government of Belgium (DWTC) through the contract IUAP P4=33.SK acknowledges support from NSF-DMR9612343 and from the University ResearchCouncil of Katholieke Universiteit, Belgium.

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