gf

m

La

m 7e 22

anrelative critical stresses, and the orientation of the crystals with respect to the loading direction. For a constitutive description of thesematerials to be reliable, it has to account for texture evolution associated with twin reorientation, and for the eect of the twin barriers on

vation stresses and, as a consequence, very anisotropic

Earlier polycrystal models incorporating twinning weremostly concerned with capturing the texture evolution

the predominant twin reorientation (PTR) scheme, consist-

transferring volume fractions from one cell to another inEuler space. A disadvantage of the VFT scheme is thatthe grain identity is lost and does not allow one to imple-ment a realistic hardening scheme. An alternative frame-work was proposed by Kalidindi [4,5] and implementedwithin a Taylor code: this approach keeps track of the

* Corresponding author. Tel.: +1 505 665 0892; fax: +1 505 667 8021.E-mail address: tome@lanl.gov (C.N. Tome).

Acta Materialia 55 (2007) 213721plastic properties. Twinning activity, in particular, playstwo important roles: it has a marked eect on textureevolution and it strongly aects the hardening responseof low-symmetry aggregates. The former eect is due tothe crystallographic reorientation associated with thetwinned portion of the grains, while the latter is associatedwith the barriers that the twin lamellae pose to the propa-gation of dislocations. As a consequence, for a constitutiveplastic description of low-symmetry aggregates to be gen-eral and reliable, it needs to be based on crystallography.

ing of reorienting the grain using the most active twin sys-tem in it. In 1993, Lebensohn and Tome [3] proposed avolume fraction transfer (VFT) scheme and implementedit in a visco-plastic self-consistent (VPSC) code. Withinthe VFT scheme, the Euler space is partitioned in cellsand, instead of keeping the grain volume fraction constantwhile changing its orientation, as done in PTR, the volumefraction assigned to each cell evolves with deformation.The VFT approach permits one to account for the contri-bution to texture of every twin system in every grain bydislocation propagation and on the stressstrain response. In this work, we introduce a model for twinning, which accounts explicitly forthe composite character of the grain, formed by a matrix with embedded twin lamellae which evolve with deformation. Texture evolutiontakes place through reorientation due to slip and twinning. The role of the twins as barriers to dislocations is explicitly incorporated intothe hardening description via geometrically necessary dislocations and a directional HallPetch mechanism. We apply this model to theinterpretation of compression experiments, both monotonic and changing the loading direction, done in rolled Zr at 76 K.Published by Elsevier Ltd on behalf of Acta Materialia Inc.

Keywords: Twinning; Polycrystal model; Hardening; Hexagonal materials

1. Introduction

Low symmetry aggregates exhibit several crystallo-graphic slip and twinning modes with widely dierent acti-

associated with twinning. In 1978, Van Houtte [1] proposeda Monte Carlo approach consisting of randomly sam-pling and reorienting full grains by twinning. In 1991,Tome et al. [2] proposed a variation of such a model, calledModeling texture, twinninduring deformation o

G. Proust, C.N. To

MST-8, MS G755, Los Alamos National

Received 20 June 2006; received in revised forAvailable onlin

Abstract

Hexagonal materials deform plastically by activating diverse slip1359-6454/$30.00 Published by Elsevier Ltd on behalf of Acta Materialia Inc

doi:10.1016/j.actamat.2006.11.017and hardening evolutionhexagonal materials

e *, G.C. Kaschner

boratory, Los Alamos, NM 87545, USA

November 2006; accepted 8 November 2006January 2007

d twinning modes. The activation of such modes depends on their

www.actamat-journals.com

48.

terievolving volume fraction of twins in each grain and of theirorientations with respect to the original grain. The matrixtwin crystallographic relation is assumed to be preserved,and the Cauchy stress in the crystal is obtained as the vol-ume average over matrix and twinned regions. Two draw-backs of Kalidindis model are: (i) grain reorientationfollows only from shear taking place in the matrix, and(ii) no secondary twinning activity is allowed inside the pri-mary twins.

Models for martensitic transformation in TRIP steelsshare many commonalities with the crystallography-basedtwin models described above [68] and have been adapted[9] for modeling twinning. The model by Cherkaoui [9]applies to face-centered cubic (fcc) systems, treats twinsas an increasing number of small at inclusions whichcan be treated via a MoriTanaka approach for compositesystems and does not allow slip or further transformationinside the twins. While these assumptions are relevant tolow-stacking fault energy fcc systems, they are not applica-ble to hexagonal close-packed (hcp) systems, where twinsgrow to sizes comparable to the grains, and where slipand secondary twinning activity inside twins plays a non-negligible role in deformation, as we show here for Zr.

Correctly describing texture evolution is relevant tounderstanding the anisotropy associated with plastic form-ing but accounts only for part of the constitutive descrip-tion. The other part, describing twin-induced hardening,is an even more challenging task because it requires us tounderstand the interaction between the various slip andtwinning modes. Such an eect has been introduced inpolycrystal models, in a rather empirical way, by assumingan evolution with strain of the threshold stress of the slipand twinning modes, and by coupling the two modes usinglatent hardening coecients. In Kalidindis approach [4,5],latent hardening is made a function of the coplanaritybetween twin and slip planes, and predictions of mechani-cal response are made for brass deforming in compressionand shear. A revised version of this model is used by Salemet al. [10,11] to study texture and stress evolution in Tideformed by compression and shear. Tome et al. [12] alsoimplement a latent hardening scheme in a VPSC codewhich used the PTR scheme for handling twin reorienta-tion. They developed constitutive laws for Zr at roomand liquid nitrogen temperature, which reproduce tensileand compressive response of rolled Zr along dierent direc-tions and reproduce the measured texture as well. The con-stitutive law was used by Kaschner et al. [13] to study thefour-point bending of Zr bars. The approach utilized inRef. [12] for Zr was adapted by Brown et al. to describeplastic forming of Mg [14] and Be [15].

The eorts described above were reasonably successfulin reproducing the measured stressstrain response andtexture evolution associated with monotonic loading intension, compression or torsion. However, the ultimatechallenge of a constitutive description is to predict the plas-

2138 G. Proust et al. / Acta Matic response associated with complex loading histories, suchas changes in loading path. In such a situation, latent hard-ening criteria overlooks the directionality of the grainmicrostructure created during the previous deformationpath, and a model with a realistic description of the twinstructure and the twindislocation interaction inside grainsis required.

This paper addresses the challenge described above. Weuse the well-established VPSC polycrystal approach as aplatform in which we implement a meso-scale compositegrain (CG) model. Such a procedure, which incorporatesmicrostructural information into the polycrystal model, isof general applicability to low-symmetry materials. TheCG enters in the VPSC formulation as an ellipsoidal inclu-sion formed by alternating layers of twin and matrixdomains. The alternating twinmatrix interfaces providea barrier to and dene a directional mean free path forpropagation of slip or twinning. Since strain path changesalter the prevailing sense of shear in the grain associatedwith the previous loading condition, they best reveal therole that microstructure plays on hardening. We addresshere the simulation and interpretation of strain pathchanges in rolled Zr tested in compression at liquid nitro-gen temperature (76 K). Under this condition one caninduce either compressive or tensile twins, depending onthe testing direction.

This work, which focuses on modeling issues, is part of acomprehensive microstructural characterization of themechanical response of Zr: McCabe et al. [16] report atransmission electron microscopy (TEM) and electronbackscattering diraction (EBSD) study of the deforma-tion modes as a function of temperature and test direction;Kaschner et al. [17,18] performed and simulated compres-sion reloads, keeping the test direction constant whilechanging the temperature, to study the role of tensile andcompressive twins as barriers to the propagation of disloca-tions or other twins; and a preliminary version of the CGmodel applied to strain path changes was recently reportedby Tome and Kaschner [19].

In this paper. we discuss the issues that need to beaddressed by the model and we highlight the necessity ofcarrying out a simultaneous experimental and theoreticalstudy at several length scales. In addition to mechanicaltesting and texture measurements, the necessity to charac-terize microstructural mechanisms both qualitatively andquantitatively becomes obvious.

2. Experimental results

Our starting material is a high-purity clock-rolled Zrcrystal bar suciently low in interstitial impurities thattwinning is prevalent under the appropriate conditions.Cubic samples of initial dimensions 7 8 9 mm3 werecut from the rolled plate by electron discharge machiningand annealed at 550 C for 1 h. The resultant microstruc-ture is fully recrystallized with equiaxed grains of1520 lm. The initial texture of the Zr exhibits a strong

alia 55 (2007) 21372148basal component perpendicular to the rolling plane(Fig. 1). The cuboidal shape of the samples permits the

otorfor

terireloading of the same samples along a dierent strain direc-tion without having to remachine them. It also allows aneasy preparation of at specimens for performing EBSDin order to quantify the texture. Neutron diraction is usedto measure intermediate bulk textures on the same samplebetween reloads.

The Zr was tested in compression at 76 K at a con-stant strain rate of 103 s1. This series of tests focuseson the deformation mechanisms prevalent in Zr at76 K. At that temperature, both f1012gh1011i tensileand f1122gh1123i compressive twins are active, whichallows the investigation of their role in plastic deforma-tion. Four dierent tests were performed: monotoniccompression along the through thickness direction(TTC) up to 30% strain, monotonic compression alongthe in plane direction (IPC) up to 30% strain, TTC upto 10% followed by 20% IPC, and 10% IPC followedby 20% TTC. The stress strain response for the four testsis shown in Fig. 1. The basal pole gures, as measuredby neutron diraction at the HIPPO facility (LANSCE-LANL) at the end of each deformation history, areshown in Fig. 3.

In our previous studies of Zr [12,16,17], three hardening

Fig. 1. Initial (0002) poles of rolled Zr. Stress response associated with monby 20% IPC and 10% IPC followed by 20% TTC (solid lines). All tests pe

G. Proust et al. / Acta Maregimes were identied in the monotonic loading curves ofFig. 1: an early slip dominated process (e < 2%) where,most likely, twins nucleate; next there is a twin-dominatedstage ending at 10% or 20% strain, depending on the typeof twinning, and characterized by an increasing hardeningrate and a rapid increase in the twinned volume fraction.Finally, twinning tends to saturate and deformation takesplace via slip. This latter stage is characterized by adecrease in hardening rate, which can be qualitative linkedto slip taking place (either inside the twins or in the matrix)across much reduced grain dimensions. In what concernsthe texture, it is apparent how twin reorientation substan-tially modies the initial texture: in the IPC case the c-axistends to align with the compressive axes as a consequenceof tensile twin activation, in the TTC case compressivetwins are activated which tends to deplete the center andform an equatorial ring as viewed in a 0002 projection ofthe texture [20].3. The CG twin model

In previous simulations of Zr, we used a PTR scheme[12,17] to describe the contribution of twinning to textureevolution. The PTR scheme was incorporated in our VPSCpolycrystal code and consists of twin-reorienting full grainsonce a certain threshold twin fraction (typically 50%) isreached within the grain. A limitation of such an approach,as far as hardening is concerned, is that, by consideringeither the initial orientation or the twin reoriented grain,it becomes dicult to account for the directional barriereect that twins have upon the propagation of dislocations.Such an eect was accounted for in an indirect way byenforcing a strong latent hardening associated with twinshears.

The CG model that we present here instead accounts forthe original (parent) and twin reoriented (child) fractions,their interaction and their evolution with deformation.Here we adopt the philosophy of the PTR scheme, in thatwe keep track of the accumulated shear of each twin systemin each grain, and identify as the predominant twin system(PTS) the one that involves the maximum grain volumefraction, given by

nic TTC and IPC (open circles). Stress associated with 10% TTC followedmed at 76 K and 103 s1.

alia 55 (2007) 21372148 2139f PTS DcPTS=S 1Here Dc is the shear contributed by the twin system and Sis the characteristic twin shear. We regard the PTS as thesystem which is going to form the parallel twin lamellaein the grain. The experimental evidence indicates that thethickness of the twin lamellae and their separation are(for a given strain rate and temperature) approximatelyuniform [21,22]. This suggests the introduction of twoparameters in the CG model: the separation d c of the cen-ter planes (typically d c = 0.2d g) and the maximum volumefraction that may twin-transform inside the grain f PTSmax (typ-ically 0.5). We will assume that, following twin nucleation,equally spaced twins of thickness d twin form in a grain ofsize d g and grow under deformation to a possible maxi-mum thickness d twinmax. The latter follows from the previoustwo parameters as: d twinmax f PTSmax dc (typically 0.1d g). Theschematic of such a CG is shown in Fig. 2.

andate etion

teriSimple geometric considerations show that the thicknessof the child and the parent domains evolve with the twinfraction as:

d twin f PTSdcdmatr 1 f PTSdc 2

We dene the PTS by keeping track of the volume fractionassociated with each twin system and assigning the PTS la-bel to the one that rst exceeds a threshold fraction (typi-cally 5% of the grain volume). The idea behind thisprocedure is to allow for the nucleation and initial growthprocess before committing to any particular twin system.

For the situation depicted in Fig. 2, it is reasonable toassume that slip occurring in planes parallel to the twinmatrix interface will not be aected by the twin boundarywhile slip will become more dicult on planes that intersectthe interface. In a way, twinning introduces a directionalgrain size reduction. Once the PTS has been identied inthe grain, its twin plane will have a specic orientation withrespect to the slip and twin planes of the other active sys-tems. As a consequence, a mean free path, dsmfp, is intro-duced (see Fig. 2) for each system s.

dsmfp d twin= sin a in the twin ands matr

3

Fig. 2. EBSD image of Zr deformed 5% in TTC at 76 K. Schematic of CGlengths used in the model in association with twinning. In each case the lamellae are oriented such that the short direction (or the short axis of thdirection (or the long axis of the ellipsoid) is parallel to the twinning direc

2140 G. Proust et al. / Acta Madmfp d = sin a in the matrix

Here a is the angle made by the intersection of the PTSplane and the slip or twin plane of the system s. In addi-tion, dsmfp < dg is enforced. As a consequence of Eq. (3)the mean free path for a given system will gradually de-crease in the matrix and gradually increase in the twin asthe twin lamellae grow. Such evolution is relevant to thedirectional HallPetch hardening of each slip or twin sys-tem. However, it is unlikely that the twins will successfullyimpede the transmission of dislocations until they involve asubstantial volume fraction of the grain. As a consequence,when implementing the idea above in the CG model weintroduce the eect of mean free path upon hardeninggradually and do not allow for a full barrier eect untilthe PTS has accumulated the maximum allowed fractionin the grain f PTSmax .4. The hardening law

Within the CG model, three mechanisms contribute tothe hardening of slip and twinning systems inside thematrix and the twin: the evolution of statistical dislocationswith strain, the evolution of geometrically necessary dislo-cations (GND) and a directional HallPetch eect associ-ated with the presence of twin interfaces

ss ssSTAT ssGND ssHP 4The two mechanisms other than the HallPetch eect

are updated incrementally with deformation. The rst termin Eq. (4) is a classical saturation Voce law associated withstatistical dislocations, plus a latent hardening eect cou-pling shear increments Dcs

0in system s 0 with the increase

in strength in system s:

DssSTAT ds^s

dC

Xs0

hss0Dcs0 5a

where

s^sC s0 s1 1 exp Ch0s1

: 5b

Here C is the accumulated shear in the grain. An eectwhich is central to the CG model comes from considering

schematic of uncoupled twinmatrix approach showing the characteristicdomains (or the ellipsoids) representing the twin lamellae and the matrixllipsoid) is along the normal to the K1 plane of the twin (nk1). The longg1.

alia 55 (2007) 21372148a mean free path for each slip and twin system in relationto the interface posed by the PTS lamella (Eq. (3)). Thedependence of the GND term with the mean free pathhas been derived by Karaman et al. [23] and Kok et al.[24] as:

DssGND hsm

dsmfpssSTAT ssGNDDcs 6

The HallPetch term adopts the familiar form

ssHP hsHPdsmfp

q 7What is dierent with respect to Refs. [23,24] is that in writ-ing Eqs. (6) and (7) we utilize a directional mean free pathand make it specic to each system (Eq. (3)). The parame-ters hsGND and h

sHP are empirical and reect the strength of

The modeling approach described so far is independent

terithe corresponding mechanism. To keep the number ofparameters to a minimum, in this work, we assume thatthe hardening parameters that appear in Eqs. (5)(7) aredierent for each deformation mode but the same for everysystem in the mode.

The characteristics of the CG model require us to makeseveral considerations concerning hardening:

(a) In the past, within the PTR scheme we had to assign ahigh latent hardening coecient to the twin systemsto represent the twin barrier eect [12,17]. Now, suchan eect is specically accounted for by the direc-tional dependence of hardening but it only accountsfor the PTS in the grain. Since the rest of the twin sys-tems are allowed to contribute shear, we still allowfor hardening coecients hss

0> 1 for twinslip and

twintwin interactions, except the PTS, where weuse hs,PTS = 1.

(b) The strength of the PTS is set to a high value whenthe maximum fraction f PTSmax has been reached, inorder to prevent further growth of the twin in thegrain. The rest of the twin systems are allowed toremain active both, in the parent and the child, sub-jected to the hardening constitutive laws describedby Eqs. (4)(7).

(c) Basinski et al. [25] have suggested that a crystallo-graphic transformation of glissile dislocations intosessile should harden the twinned domains.Micro-indentation measurements of the twins, donein Hadeld steel [23] and titanium [11], suggest anincrease of about 20% in the yield stress inside thetwins. This eect, however, is dicult to quantify atthe slip system level and, since its inclusion is not crit-ical for the model, it is obviated in this work. All thesame, twins tend to exhibit a larger strength due tothe HallPetch eect.

5. The self-consistent polycrystal model and CG deformation

We implement the CG model in our VPSC code, and thereader is referred to Ref. [3] and references therein for adetailed description of the VPSC model. During the simu-lation we need to calculate the stress and strain rate in thematrix and twin domains of the CG. The strain rate _e isrelated to the shear rates _cs contributed by slip and twin-ning systems through a rate sensitive law

_eij Xs

msij _cs _c0

Xs

msijms : rss

n M secijklrkl 8

Here r is the stress tensor and ss is the strength associatedwith system s. The strength evolves with deformation andtwinning in the manner discussed above. Within the VPSCapproach we regard each grain as a visco-plastic inclusion

G. Proust et al. / Acta Maembedded in a visco-plastic eective medium characterizedby a constitutive responseof the CG model for the grain. We use in this work two dif-ferent methods to calculate the deformation of matrix andtwin. They are briey described below.

5.1. Coupled deformation of twin and matrix: the CG model

In this case, the twin and the matrix are assumed tointeract across the twin interface that separates them.One may introduce such interaction via a two-site VPSCmodel [27] or via a model for lamellar structures proposedby Lebensohn [28]. Here we implement the latter because itassumes a planar interface separating twin and matrix,which is a closer representation of such conguration inhcp crystals. We dene an orthogonal reference system inwhich axes 1 and 2 are contained in the planar interfaceand axis 3 is perpendicular to it, and we enforce continuityof some stress and strain rate components across the inter-face. Since we further assume that stress and strain rate arehomogeneous in the domain of the twin and the matrix, weenforce the following conditions over twin and matrixmagnitudes:

_eM11 _eT11; _eM22 _eT22; _eM12 _eT12rM13 rT13; rM23 rT23:

13

Note that because ours is an incompressible formulation,only ve of the independent deviatoric components ofstress or strain rate enter in Eq. (13) (the condition_eM33 _eT33 follows from incompressibility). For the purpose_eij M secijklrkl 9If one solves the problem of a visco-plastic inclusionembedded in a visco-plastic medium, one obtains a relationbetween macroscopic stress and strain rate, and the stressand strain rate in the grain [3]:

M sec : rM sec : r _e _e eM : r r 10awhere

eM neffI S1SM sec 10bis the interaction tensor, S is the visco-plastic Eshelby ten-sor, and ne tunes the inclusion-matrix interaction to be inthe range secant (ne = 1) to tangent (ne = n). In the pres-ent application, we use n = 20 and ne = 10 [26]. Combin-ing Eqs. (8)(10) leads to a localization equation for thegrain stress of the form

rij M sec eM 1ijmnM sec eM mnklrkl 11The condition that the average stress over all grains has tobe equal to the macroscopic stress hri r provides arelation for calculating the macroscopic secant complianceM sec iteratively (self-consistently), namely,

hM sec eM 1ijmnM sec eM mnkli Iijkl 12

alia 55 (2007) 21372148 2141of the polycrystal model, we consider a composite grainwith fractions wM and wT assigned to matrix (M) and twin

teri(T), respectively. The stress, the strain rate and the secantcompliance of the composite grain are given by theweighted averages [28]:

_eij wM _eMij wT _eTijrij wMrMij wTrTijM sec wMM sec ;M : A wTM sec ;T : wM : A wTI1:

14Note that the strain rate in matrix and twin is given by thesum of shear rates contributed by slip and twin systems(Eq. (8)). The twinning contribution, of the formmtwij w

M _ctw, may also be written using Eq. (1) as mtwij _wMStw.

The latter form is usually treated as a separate transforma-tion term in martensitic transformation models [68] andsome twinning models [9]. The matrix A in Eq. (14) is afunction of the individual secant moduli for twin and ma-trix (Eq. (8)), and the continuity conditions (Eq. (13)).The interaction equation (Eq. (10)) contains now to theCG magnitudes dened by Eq. (14) and, from the pointof view of the self-consistent procedure, the CG is treatedas an eective grain. Observe that, in the limitswM ! 1 or wT ! 1, the matrix or the twins, respectively,dominate the eective magnitudes and, as a conse-quence, the mediumgrain interaction. When the twin frac-tion is much smaller or much bigger than that of thematrix, this approach is likely to give a good representationof the twinmatrix interaction. However, when twin andmatrix fractions become comparable it is likely that accom-modation will be limited to the interface, without aectingthe bulk of the twin or the matrix. In such a case, the cou-pling assumption could be too restrictive, and the uncou-pled scheme discussed below may be better suited.

5.2. Uncoupled deformation of twin and matrix

In this case, the twin and matrix domains are treated astwo non-interacting orientations (grains) embedded in thehomogeneous eective medium. We still identify the PTSand account for its orientation, its evolving volume frac-tion and the evolving mean free path that the PTS denesfor each system. As soon as the twin is created we give it aat ellipsoid shape. The shortest axis of the ellipsoid is par-allel to the normal to the K1 plane of the twin and anotherof the ellipsoid axes is parallel to the twinning direction g1(see Fig. 2). A similarly oriented ellipsoid is created to rep-resent the untwinned region of the crystal as soon as thetwin volume fraction reaches f PTSmax . The aspect ratios ofboth ellipsoids evolve with deformation. Now twin andmatrix are characterized by independent secant compli-ances Msec and no explicit twinmatrix interaction is con-sidered to solve the self-consistent equations. The relativefraction of each phase is updated incrementally with defor-mation as the grain twins. This approach already repre-

2142 G. Proust et al. / Acta Masents an improvement upon the PTR scheme, in that itkeeps track of both orientations rather than switching fromthe parent to the twin orientation when a threshold isreached. From a numerical point of view, it only requiresus to duplicate the number of initial orientations.

5.3. Reorientation of child and parent

Concerning crystallographic reorientation associatedwith slip and twinning we refer the reader to any of the sev-eral papers where the kinematics associated with plasticspin and the inclusion formalism are described (see, forexample, Refs. [3,8]). The issue that we address in this sec-tion is whether the twinmatrix crystallographic relation ispreserved during deformation. Crystallographic rotation oftwin and matrix follow from the slip and twinning shearstaking place inside them and, in general, their individualrotation rates will not preserve the crystallographic twinmatrix relation across the interface. Here one may specu-late that if the twin lamella growths in the grain, then sucha relation has to be preserved until growth is completed. Asa consequence, parent and child will co-rotate crystallo-graphically, and we have in the past proposed a simpleapproach for enforcing such co-rotation [29]. Past the twingrowth stage, one may assume, based on the fact that twinand matrix are of comparable sizes, that they reorient inde-pendently of each other. While we do not have detailedexperimental evidence about this process, the EBSD char-acterization [30] indicates that the twinmatrix orientationrelation is preserved (within 15) for deformations of 30%or less. However, since the main contribution to reorienta-tion comes from twinning and slip does not induce largechanges in the misorientation between twin and matrix,results are rather insensitive to the assumption adopted.In our calculations, we reorient the matrix and the twinindependently.

6. Application of the CG model to Zr

The model described above in abstracto is applied hereto describe the constitutive response of pure Zr at 76 K.We adjust the parameters of the model to a variety ofexperimental data: stressstrain curves, textures, twin frac-tions, observed active modes. The reader should be awarethat the requirement of consistency with such an abun-dance of experimental data not only reduces the arbitrari-ness in the optimization of the model parameters but alsosupports the validity of the physical mechanism assumedby the model. TEM and EBSD microscopy [16] indicatethat our Zr deforms at 76 K by f1010gh1120i prismaticslip, f1012gh1011i tensile twins and f1122gh1123i com-pressive twins. Prismatic slip and tensile twinning areeasy to activate but the hard compressive twinningis required to accommodate compression along the c-axisof the grains.

Table 1 shows the hardening parameters adjusted usingthe CG coupled approach described in Section 5.1. The

alia 55 (2007) 21372148various parameters are dened in Section 4. For both slipand twinning, we use a saturation-type Voce law to

HP

550

s de o

teridescribe the hardening until stage III. For slip, we use theconventional response that follows from dislocation multi-plication and recombination: the initial critical resolvedshear stress starts at s0 and increases until reaching the sat-uration value s0 + s1 (Eq. 5b). For the twinning modes,however, we use an unorthodox softening response,obtained by assigning to s1 and h0 negative values (Table 1).The initial value of the critical resolved shear stress fortwinning is s0, but then it decreases rapidly and saturatesat s0 + s1. Such a response is meant to empirically accountfor the twin nucleation stage, typically taking place withinthe initial 2% strain. Microscopic observation indicatesthat twins always initiate at grain boundaries, which sug-gests that nucleation is mediated by stress concentrationsat grain boundaries in the initial stages of loading. Furtherstraining leads to propagation of nucleated twins, which isthe mechanism that we address in our model. Making thetwinning modes harder at the beginning of the simulationmakes it possible to reduce the twinning activity at the startof the deformation and empirically account for a twinnucleation phase that reduces the presence of twins atlow strains.

The latent hardening parameters coupling prism slipwith twinning are negative (see Table 1) because the deriv-ative of the Voce law is also negative for both twinningmodes and we want to enforce a positive hardeningthrough Eq. (5a). The other latent hardening parametersare equal to 1 to enforce an asymptotic decrease in the twinstrength, as discussed above. This result diers from thelatent hardening coecients reported by Tome et al. [12]in connection with the PTR scheme, where the couplingbetween twinning and the other slip or twinning systemswas mediated by hardening parameters ranging from 2 to20. In the CG model, such coupling is introduced by explic-itly accounting for the presence of the twin interface barrier

Table 1Single crystal hardening parameters for zirconium at 76 K

Modes s0 (MPa) s1 (MPa) h0 (MPa) H

Prismatic 45 275 300 6Tensile twin 105 20 60 6Compressive twin 330 75 20 20The parameter hs,prism represents the latent hardening of deformation modemaximum twin fraction in a grain. See Eqs. (5)(7) for the denition of th

G. Proust et al. / Acta Marather than by means of ad hoc latent hardeningcoecients.

Also, in the previous PTR-based simulations [12,17], thestage IV of hardening for the deformation systems wasaccounted for by a non-saturation Voce law. In the CGmodel, it is represented by the GNDs, required to accom-modate the lattice curvature caused by non-uniform plasticdeformation [24]. These GNDs act as obstacles to themotion of other dislocations, hardening the various slipmodes [31,32] and twinning modes [23].

Concerning the HallPetch eect, twin boundaries areconsidered in this model as having the same properties asgrain boundaries as far as providing obstacles to the prop-agation of dislocations is concerned. Therefore, the variousdeformation modes become harder as the value of themean free path in the matrix decreases due to the presenceof these twin boundaries. The situation reverses inside thetwin domains, where deformation modes become easieras twins grow. There is little experimental informationavailable for Zr in what concerns grain size eects uponthe yield stress to justify using separate HallPetch coe-cients for twins and grain boundaries. The experiments at76 K done by Song and Gray [33] on Zr with 25 and75 lm grain sizes, and oriented to activate compressivetwinning, set an upper bound for hHP of about1000 MPa lm1/2 for the compressive twins. Work realizedby Armstrong et al. [34] on various cubic and hexagonalmetals seems to indicate that the HallPetch coecient isalways greater for twinning than for slip and that the val-ues are independent of temperature and strain rate for allthe deformation modes. Obviously, substantial experimen-tal characterization will be required for elucidating theHallPetch parameters associated with the other deforma-tion modes and to verify their independence on tempera-ture and strain rate for Zr. The parameters listed inTable 1 provide a reasonable t to the monotonic loadingtests. We will see that the HallPetch eect plays an impor-tant role when strain path changes take place.

For our calculations, we represent the initial texture(Fig. 1) using 1944 orientations with properly assignedweights and an initial grain size of 30 lm. We will use asa benchmark the case in which there is coupling betweenthe twin and matrix phases (parameters listed in Table 1),and will discuss the role and inuence of the various modelmechanisms by comparison to this one case.

The predicted stressstrain curves for the monotonicand strain path change deformations are compared with

(MPa lm1/2) HGND (MPa2 lm) hs,prism dc fmax

1e5 1 N/A N/A3e5 1 0.2 0.54e5 5 0.2 0.5

ue to prismatic slip activity. dc represents the twin separation and fmax thether parameters.

alia 55 (2007) 21372148 2143the experimental measurements in Fig. 4. The calculatedmode activities in the matrix and twin phases are shownin Fig. 4 with the evolution of their relative fraction.Fig. 3 shows the measured and predicted textures at 30%strain for the four deformation paths studied. Table 2 givesa comparison of the volume fractions of primary and sec-ondary twins obtained experimentally [30] and from ourbenchmark simulation (the twin volume fractions were cal-culated from the model using only the PTS in the matrixand in the primary twins).

Two dierent techniques have been used to evaluateexperimentally the twin volume fractions: neutron

r 30he

TT

and

teriMea

sure

dPr

edict

ed

Fig. 3. Comparison of measured and predicted basal pole distributions afte(c) 10% TTC followed by 20% IPC; (d) 10% IPC followed by 20% TTC. T

Table 2Measured and predicted primary and secondary twin volume fractions for

Strains Volume fraction primary twins

Measured Predicted

14% TTC 0.40 0.3230% TTC 0.35 0.4017% IPC 0.41 0.3030% IPC 0.48 0.41

The measured values were obtained using neutron diraction at strain 0.3

2144 G. Proust et al. / Acta Madiraction after 30% deformation and EBSD at lowerstrains. Each method has advantages and drawbacks. Forexample, during TTC, secondary tensile twins form insidethe primary compressive twins. As a consequence, the c-axis ips back from the equator to the center of the polegure (Fig. 3a), and the twin fraction will be underevalu-ated when using the pole gures obtained by neutron dif-fraction. On the other hand, EBSD allows theidentication and quantication of twins within twins,but EBSD scans are less precise at large deformationbecause dislocation structure deteriorates diraction pat-tern quality. In addition, since EBSD scans are done on asmall window of a cross-section of the deformed sample,they are not necessarily representative of the bulk texture.This is especially true for our TTC samples where heteroge-neity of the microstructure has been observed. The EBSDresults presented in Table 2 were obtained from the middlecross-section of the deformed sample, where the presenceof twins is greater. Therefore, the EBSD reported twin frac-tions are probably an upper bound. Work is in progress toderive a more precise quantication of twin fraction [30].

The comparison of the measured and calculated resultsshows that the CG model predicts accurately the hardeningrate associated with the twinning deformation for bothmonotonic deformation processes and predicts satisfactorythe hardening rates for the reloading cases (see Fig. 4). Thepredicted yield strengths at reloading are in agreement withthe experimental values, meaning that the eect of micro-structure upon shear of new systems is correctly accounted 0.7 1.0 1.4 2.0 2.8 4.0

% accumulated deformation for: (a) monotonic TTC; (b) monotonic IPC;measured pole gures were obtained by neutron diraction.

C and IPC

Volume fraction secondary twins

Measured Predicted

0.14 0.02N/A 0.100.02 0.001N/A 0.04

EBSD at strain 0.14 for TTC and 0.17 for IPC.

alia 55 (2007) 21372148for by the CG model. As for the evolution of the yieldstress upon reloading, the increase in hardening rate forboth TTC! IPC and IPC! TTC is evidence of twinningbeing activated. The CG model does predict compressivetwinning for the IPC! TTC case, but not enough tensiletwinning for the TTC! IPC case. In the latter case, adecreasing hardening rate is predicted.

The stressstrain response is only one piece of experi-mental information. In addition, the predictions of themodel need to be consistent with measured textures andobserved system activity. The deformation modes activeat 76 K in the matrix and in the twins have been experimen-tally identied and discussed in Ref. [16]. For the TTCcase, the model predicts that deformation is accommodatedby equal amounts of prismatic slip and compressive twin-ning in the matrix plus an important contribution of pris-matic slip and secondary tensile twinning inside theprimary compressive twins (Fig. 4a). Such activity is con-sistent with the equator that forms in the (0002) pole g-ure, induced by the 64 reorientation by compressivetwinning of the initial orientations (Fig. 3a). Subsequently,secondary f1012gtensile twinning reorients the c-axis by85 and realigns it again with the compression direction(Fig. 3b). We would like to highlight the non-negligible roleplayed by secondary twinning in strain accommodation: wepredict 4% of secondary twinning after 14% TTC (experi-mentally we observed 14% of secondary twinning in themiddle cross-section of the sample [30] which, we know,is an upper bound) and 10% after 30% TTC. For

ck-rh st

teriFig. 4. Experimental (s) and predicted () stressstrain response for cloactivity: relative shear contribution by each mode in matrix and twins at eac

G. Proust et al. / Acta Mamonotonic IPC, deformation is accommodated by pris-matic slip and tensile twinning in the matrix and, past astrain of 15%, by compressive twinning inside the primarytensile twins (Fig. 4b). These predicted mode activities arein agreement with the experimental ndings obtained byEBSD and TEM [16]. The texture (Fig. 3b) is consistentwith an alignment of the c-axis with the compression direc-tion. The measured and predicted primary twin fractionsare consistent, while the secondary twin fractions are lowin both cases at 17% deformation (Table 2). Note that sec-ondary twinning does not play an important role in IPC bycomparison with TTC for the same percentage of deforma-tion. As discussed above, quantitative measurements oftwin volume fractions are still preliminary due to speciclimitations of the neutron diraction and EBSD tech-niques, which, respectively, tend to under- and overpredictthe volume fraction of twins. This experimental biasexplains the inconsistency of the primary twin volume frac-tions observed for TTC at 14% and 30% deformation. ForIPC, the model seems to predict lower volume fractions oftensile twinning than the measurements by eithertechnique.

For the reloads, we predict that the microstructureinduced during preload aects substantially the modeactivities by comparison with the monotonic deformation.For the TTC! IPC reload, deformation is carried by thematrix, where prism and tensile twin activities are similarwith the ones observed and predicted during monotonic

IPC; (d) 10% IPC followed by 20% TTC. Also shown as a solid line is thecomplementary).olled Zr deformed in compression at 76 K and 103 s1. Predicted modeep. (a) Monotonic TTC; (b) monotonic IPC; (c) 10% TTC followed by 20%

alia 55 (2007) 21372148 2145IPC. The initial yield upon reload is aected by the HallPetch mechanism but the hardening rate is indicative oftwin dominated deformation in the matrix (Fig. 4c),although the model downplays the latter eect. No furtherbuild-up or shear activity is predicted inside the (compres-sive) twin phase. The pole gure (Fig. 3c) is consistent withsuch an analysis: while the pole gure is similar to the oneobtained after 30% IPC (compare Fig. 3b and c), it showsthat there is less tensile twinning activity than in the mono-tonic case. It is likely that the microstructure introducedduring TTC pre-loading acts as a barrier for the tensiletwins. The predicted pole gure shows an extra ring atthe equator due to an overprediction of compressive twinsduring pre-loading.

For the IPC! TTC reload, prism slip has hardened thematrix during IPC preload and compressive twinningmakes a larger contribution to deformation than in the caseof monotonic TTC. Considerable prism activity and sec-ondary tensile twin activity take place in the twin phaseafter preload. It is likely that the secondary twinning ismainly de-twinning of the tensile twins formed during theIPC preload. The fact that hardening rate is low (Fig. 4d)and that the nal pole gure (Fig. 3d) shows a reconsti-tuted center component seem to support the idea of de-twinning. Experimental and modeling work is underwayto clarify this issue. The remainder of this section focuseson the eect of the hardening parameters on the simulatedstressstrain curves.

volume fraction of the matrix phase in the aggregate (the twin phase is

6.1. Sensitivity of the model to the parameters

In order to characterize the sensitivity of the model tothe various parameters we modify one parameter at a timewhile keeping the rest unchanged. The resulting stressstrain curves are presented in Fig. 5 and compared withthe benchmark case in Fig. 4 for the two monotonic andthe two strain path change deformations. The solid linesin Fig. 5 represent the simulation done using the uncoupledscheme of Section 5.2, that is, without imposing stress orstrain continuity (Eq. (13)) at the interface between matrixand twin and by giving an ellipsoidal shape to bothtwinned and untwinned regions of the crystal. Theobtained stressstrain responses are the results of the com-petition between the three following eects: (i) removingthe continuity condition decreases the stress ow; (ii) givinga shape to the twin and matrix grains increases the stressow; and (iii) the deformation mode activity inside thematrix and twins can vary from the benchmark case. Therole of these eects on the material response varies fromone case to another. The removal of the continuity condi-tion inuences more the monotonic IPC simulation thanthe others; the reason for this is the dierence in the num-ber of grains being treated as CGs: 1112 grains out of 1944for IPC, compared with 764 for TTC. Also, we observe thegreatest variation in the deformation activities for the

monotonic IPC case: the tensile twinning activity decreasesas the prismatic slip activity increases, which lowers thestress ow as the critical resolved shear stress for prismaticslip is smaller than for tensile twinning. Giving an ellipsoi-dal shape to the twin and matrix grains cannot compensatethe two other eects in the case of monotonic IPC; there-fore, we observe a lower stress ow during the uncoupledsimulation. For the three other cases, the deformationmode activities do not vary much and fewer grains are con-sidered as CGs. Therefore, the shape given to twins andmatrix grains has a predominant eect on the materialresponse, and we observe an increase of the stress ow.The uncoupled approach reproduces the rapid increase inthe ow stress experimentally observed during the TTCreload following IPC. This phenomenon can be attributedto a rapid increase in the compressive twinning activity inthe matrix and a rapid decrease of the tensile twin activityin the tensile twins created during the preloading phase.The ellipsoid shape given to the primary twins facilitates,at the beginning of the reloading, secondary twinning activ-ity (this can be due to an increase of stress ow in the pri-mary twins); however, shortly after this it becomes easierfor matrix regions to deform by compressive twinning.

The separation of the center planes of the lamellae (d c inFig. 2) and the maximum twin volume fraction per grain,f PTSmax , determine the value of the mean free path for all

sssenc

2146 G. Proust et al. / Acta Materialia 55 (2007) 213721480.0 0.1 0.2 0.3300

400

500

600

700

800

Benchmark no coupling dc=0.5 Hgnd=0 Hhp=0

Stre

ss (M

Pa)

Strain

0.0 0.1 0.2 0.3300

400

500

600

700

800

Stre

ss (M

Pa)

Strain

a

c

Fig. 5. Eect of the various hardening parameters on the simulated stre(c) preload in TTC, reload in IPC; (d) preload in IPC, reload in TTC. (- - -) B

matrixtwin interface; (*) increasing the separation d

c between the mid-plane ofand (h) removing the HallPetch hardening.0.0 0.1 0.2 0.30

200

400

600

800

Stre

ss (M

Pa)

Straind

b

0.0 0.1 0.2 0.30

200

400

600

800

Stre

ss (M

Pa)

Strain

train response of Zr at 76 K. (a) Monotonic TTC; (b) monotonic IPC;hmark simulations using the full CG model; () eliminating coupling at the

lamellae; (m) removing the geometrically necessary dislocation hardening;

terithe deformation systems in the matrix and in the twins. Themean free path controls the magnitude of the GNDs andHallPetch terms (Eqs. (6) and (7)). The smaller the meanfree path for a specic deformation system, the harder thissystem becomes. During the optimization of the hardeningparameters, we observed that the nal twin volume fractionis mostly controlled by f PTSmax and much less by the d

c

parameter. Thus, we kept f PTSmax 0.5 in order to retain areasonable prediction of twin volume fraction at 30% strainand varied d c. In Fig. 5, we compare the benchmark case(dotted line), for which d c = 0.2, and simulations (repre-sented by the star symbols) run with d c = 0.5. For all thedeformation processes the ow stress decreases and the rel-ative mode activities vary in the matrix and in the twins asd c changes. For example, in the IPC case the prismaticactivity increases while the tensile twin activity decreasesas d c increases. As a consequence, fewer primary twinsare created by comparison with the benchmark case. Theincrease in prismatic activity over twinning is the reasonfor the reduction of the ow stress during IPC.

The simulations excluding the contribution of the GNDterm (Eq. (6)) are represented by black triangles in Fig. 5.Excluding this term lowers the ow stress for all the defor-mation processes, as was to be expected. The monotonicIPC case is strongly aected, with the hardening rate shift-ing from increasing to decreasing at high strains (Fig. 5b).Omitting the GND contribution has the same eect asincreasing the mean free path: prismatic slip activityincreases while tensile twinning activity decreases.Although the suppression of the GND term does notgreatly aect the stressstrain response for TTC, it doesinuence the nal texture at 30% strain because the tensiletwinning activity inside the primary compressive twinsincreases. And, although the model does not account forsecondary twin reorientation, the shear associated with sec-ondary twins reorients the primary twins. Since this phe-nomenon is not observed experimentally, this exampleproves the necessity of preventing excessive secondarytwinning in the model, which can be accomplished throughthe GND hardening.

The square symbols in Fig. 5 represent the stressstrainresponses obtained omitting the HallPetch term (Eq. (7))in the hardening law. To start from the same initial yieldstress, we modify the value of s0. Therefore, only theHallPetch eect due to the change of the mean free pathcaused by twins has been suppressed in the simulations.The elimination of the HallPetch term from the hardeninglaw lowers the eective ow stress in all four stressstraincurves. This procedure aects mostly compressive twinningin the TTC cases, as it is the deformation mode with thehighest HallPetch coecient. For the IPC cases, thereduction of the ow stress has two origins. First, prismaticslip takes the advantage over tensile twinning in the matrix,inducing a further decrease in ow stress (same phenome-non described for the increase of d c). In addition, compres-

G. Proust et al. / Acta Masive twinning is easier than for the benchmark simulationinside the primary tensile twins, which also causes adecrease of the ow stress. Moreover, the omission of theHallPetch eect lowers the values of the reload yieldstrength. In the benchmark case, the twins created duringpreloading make it harder for the deformation modes tobe activated at reload due to a decrease of the mean freepath. This eect is greater for the TTC! IPC than forthe IPC! TTC because the TTC preload induces moretwins than the IPC preload.

7. Discussion

The plastic response of hcp aggregates is complex andanisotropic due to the variety of crystallographic deforma-tion modes and, in particular, the presence of twinning.Twinning reorients portions of the grain, poses barriersto the propagation of slip and other twin systems, and pro-vides domains which may be either hard or softdepending on the stress state acting on the grain. As a con-sequence, in hcp aggregates, twinning and hardening evolu-tion are intimately coupled. For a constitutive descriptionto be reliable such coupling needs to be accounted forexplicitly. In this work, we propose a structural model ofthe grain which incorporates the above-mentioned mecha-nisms and implement such a model in a self-consistentpolycrystal scheme. While the model is general, we denespecic parameters for pure Zr deforming at 76 K, a tem-perature at which tensile and compressive twins are presentin addition to prism slip. We choose to perform and simu-late strain path changes as a paradigm for revealing therole of twinning in deformation and as a stricter test ofour constitutive law. Crystallography and relative direc-tionality of slip and twinning are explicitly incorporatedinto this CG model.

The advantage and the challenge of our approach isthat the predictions of the model can be compared with avariety of experimental information, such as system activ-ity, texture, stressstrain response and twin fractions. Thiscreates, in turn, a synergy between experiments and model:modeling considerations suggest the experiments, and theexperimental results provide evidence to conrm or revisemodeling mechanisms. We have tried to minimize the num-ber of parameters and the complexity of the mechanismsassociated with the CG model, especially when the experi-mental information available is not conclusive.

Implicit in our model is the assumption that twins nucle-ate at the beginning of deformation, most likely due tostress build-up at grain boundaries. We only claim todescribe the twin propagation and growth that takes placeafter nucleation. In addition, we assume uniform twin sep-aration and uniform twin thickness in the grains. These arereasonable assumptions if one envisages twinning as astress relaxation mechanism which, once activated, lowersthe shear stress in the vicinity of the twinmatrix interfaceand favors new twinning activity away from the recentlyformed twin region. We also consider at most one predom-

alia 55 (2007) 21372148 2147inant twin system per grain: while we know from EBSDthat this is not always the case; it is, however, a reasonable

assumption. In addition, we allow for secondary twins tocontribute shear but we do not consider a layered structureinside the primary twins because this would complicate theCG model. Further, it is to be expected that twinning willnot only depend on the relative orientation of the grainwith respect to the external load, but also on the orienta-

are several related issues. Theoretical calculations justifyusing directional hardening for GNDs [24,35]. Concerning

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2148 G. Proust et al. / Acta Materialia 55 (2007) 21372148the HallPetch eect for twins, although it has been docu-mented for grain boundaries [33,34], using a directionalHallPetch for describing the eect of the twin interfaceupon dislocations is to be regarded as speculative. The sim-ulation results presented here indirectly support the validityof such a concept, but careful experiments will be requiredto characterize the mechanism. A related issue has to dowith transmissibility and reaction of dislocations at twininterfaces, which will modify the twin-barrier hardeningmechanisms. Transmissibility mechanisms have been char-acterized or proposed for cubic [22,36] and hexagonal[37,38] materials. Until we have better experimental charac-terization, we assume that twin interfaces pose perfect bar-riers to dislocations. However, since twins do not form aperfectly layered structure across the whole grain fromthe beginning of deformation, we introduce a smooth tran-sition to the full barrier eect, at which point we start treat-ing the grain according to the CG scheme.

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Acknowledgements

This work was supported by the Oce of Basic EnergySciences, Project FWP 06SCPE401. The authors are grate-ful to Rodney McCabe for sharing unpublished results oftwin area fraction characterized using EBSD.tion of the neighbors, which will introduce local uctua-tions. Some of the considerations above may beincorporated in the model but will require a detailed micro-scopic and statistical characterization of spatial correla-tions and observed twin structures.

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Modeling texture, twinning and hardening evolution during deformation of hexagonal materialsIntroductionExperimental resultsThe CG twin modelThe hardening lawThe self-consistent polycrystal model and CG deformationCoupled deformation of twin and matrix: the CG modelUncoupled deformation of twin and matrixReorientation of child and parent

Application of the CG model to ZrSensitivity of the model to the parameters

DiscussionAcknowledgementsReferences