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Acta Materialia 54 (2006) 2181–2194

On the consideration of interactions between dislocations andgrain boundaries in crystal plasticity finite element modeling –

Theory, experiments, and simulations

A. Ma, F. Roters *, D. Raabe

Max-Planck-Institut fur Eisenforschung, Max-Planck-Str. 1, 40237 Dusseldorf, Germany

Received 26 April 2005; received in revised form 6 January 2006; accepted 6 January 2006Available online 10 March 2006

Abstract

We suggest a dislocation based constitutive model to incorporate the mechanical interaction between mobile dislocations and grainboundaries into a crystal plasticity finite element framework. The approach is based on the introduction of an additional activationenergy into the rate equation for mobile dislocations in the vicinity of grain boundaries. The energy barrier is derived by using a geo-metrical model for thermally activated dislocation penetration events through grain boundaries. The model takes full account of thegeometry of the grain boundaries and of the Schmid factors of the critically stressed incoming and outgoing slip systems and is formu-lated as a vectorial conservation law. The new model is applied to the case of 50% (frictionless) simple shear deformation of Al bicrystalswith either a small, medium, or large angle grain boundary parallel to the shear plane. The simulations are in excellent agreement withthe experiments in terms of the von Mises equivalent strain distributions and textures. The study reveals that the incorporation of themisorientation alone is not sufficient to describe the influence of grain boundaries on polycrystal micro-mechanics. We observe threemechanisms which jointly entail pronounced local hardening in front of grain boundaries (and other interfaces) beyond the classical kine-matic hardening effect which is automatically included in all crystal plasticity finite element models owing to the change in the Schmidfactor across grain boundaries. These are the accumulation of geometrically necessary dislocations (dynamic effect; see [Ma A, Roters F,Raabe D. A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations. ActaMater 2006;58:2169–79]), the resistance against slip penetration (dynamic effect; this paper), and the change in the orientation spread(kinematic effect; this paper) in the vicinity of grain boundaries.� 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Dislocation density; Grain boundary mechanics; Grain boundary element; Slip penetration; Aluminum

1. Introduction

Different types of experiments such as micro-torsion,micro-bending, deformation of particle-reinforced metal–matrix composites, and micro-indentation hardness testshave revealed similar hardening effects when refining thedeformation scale, substantiating a general length scaledependence of the flow stress [2]. The common feature ofthese experiments is that the non-uniform plastic deforma-

1359-6454/$30.00 � 2006 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2006.01.004

* Corresponding author. Tel.: +49 211 6792 393; fax: +49 211 6792 333.E-mail address: [email protected] (F. Roters).

tion imposed in such cases entails the formation of orienta-tion gradients among neighboring material points and,hence, the accumulation of extra dislocations for the pres-ervation of the lattice continuity.

In order to integrate these various interconnectedaspects into one constitutive model, we introduce in Ref.[1] and in this study a joint formulation which includesthe tensorial nature of elastic–plastic deformation in termsof dislocation slip, the evolution of the crystallographictexture at each material point, nonlocal orientation- andstrain-gradient terms occurring between neighboring mate-rial points, and the local mechanical effects associated withinterfaces. The aim is to provide a general framework for

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2182 A. Ma et al. / Acta Materialia 54 (2006) 2181–2194

studying polycrystal micro-mechanics at small spatialscales in a crystal-plasticity finite element framework. Thecentral problem addressed in this paper is the local effectof grain boundaries on the hardening behavior. The gen-eral introduction of geometrically necessary dislocationsinto the crystal plasticity finite element framework has beendiscussed in Ref. [1].

A basic issue associated with the constitutive aspects dis-cussed above is the description of the constraints imposedby the neighborhood on one particular material point.

In finite element simulations which use empirical localconstitutive models the relation between a material pointand its neighborhood is satisfied by accounting for theequilibrium of the forces and the continuity of the totaldeformation, and, of course, the procedure of solving theboundary value problem satisfying these constraints.

For approaches which add nonlocal terms to an other-wise local dislocation model in the bulk crystal, Nye’s dis-location tensor [3] can be used to express the constraintsimposed by the neighborhood on a material point. This isdue to the fact that orientation differences between neigh-boring points which are far away from rigid obstacles areusually small. The accumulation of geometrically necessarydislocations then acts as a penalty term against local reori-entations entailing a resistance against the formation oforientation gradients upon loading. In such cases weencounter three constraint conditions in the boundaryvalue problem, namely, the two constraints mentionedabove plus the accommodation of the lattice curvature interms of Nye’s dislocation tensor.

When additionally considering the slip resistance associ-ated with slip penetration events across grain boundariesthe influence among neighboring material points becomesmore complex. In such a case the total set of conditionsare the three constraints mentioned above plus the trans-mission (penetration), reflection, and absorption of mobiledislocations by grain boundaries including the tensorialand structural nature of that problem.

The latter aspect is central in this study. The approachthat we choose is the introduction of a grain boundarytransmission mechanism, which is added to the nonlocaldislocation model introduced in the preceding study [1].The transmission mechanism enters through an additionalactivation enthalpy term which quantifies the penetrationprobability of mobile dislocations through grain bound-aries. The activation enthalpy is the elastic energy of for-mation of misfit dislocations which remain as debris inthe interface upon slip penetration. This new energy barrieris introduced in the form of a special grain boundary ele-ment which contains the actual interface.

In order to critically evaluate the new model we conductcorresponding simulations of simple shear deformationtests of pure aluminum bicrystals with different grainboundaries (small angle, 7.4�; medium angle, 15.9�; largeangle, 33.2� [4]). The grain boundary plane is in all casesparallel to the shear plane. The predictions are comparedto experiments (50% shear) in terms of the von Mises

equivalent strain distributions, the statistical orientationdistributions, and the local textures on the sample surface.The constitutive parameters required for the constitutivedislocation model are fitted by approximating the averagestress–strain curve of a single crystal as outlined in Ref. [1].

2. Grain boundary micro-mechanics observed in earlier

experiments

Grain boundaries act as obstacles to the motion of dislo-cations. At the onset of plastic deformation of polycrystals,mobile dislocations are first created on the slip system withthe largest local resolved shear stress in the grain with themost favorable orientation. When encountering a grainboundary, such mobile dislocations will accumulate in theform of pile-ups entailing a local stress state which can beapproximated by a super dislocation. At the tip of thatpseudo dislocation stress concentrations arise which addto the external stress field at this material point. Suchmicro-plastic effects, where the local arrangement of dislo-cations plays the dominant role for the local stress, cannotbe mapped one-to-one by crystal plasticity-type continuummechanics which map dislocation mechanics in a phenome-nological statistical or even in an empirical fashion. How-ever, homogenization is admissible at larger plastic strainswhere most of the slip activation processes can be capturedby long-range stresses rather than by local ones [5]. Thismeans that dislocation mechanics can, beyond the micro-plastic regime, be homogenized in the form of statisticaldislocation populations which in turn can be embedded asconstitutive rate equations in a crystal plasticity theory[1,6,7]. The basic applicability of the crystal plasticity finiteelement approach to a large spectrum of intricate micro-mechanical problems has been shown by many studies inwhich both the textures and the strains were properly pre-dicted when compared to corresponding experiments [4,8–13].

At very low plastic strains where isolated dislocationpile-ups can be observed [5], careful studies in the vicinityof grain boundaries by transmission electron microscopyshowed several types of interactions between the mobiledislocations and the interface. These are the direct trans-mission or penetration of dislocations through grainboundaries, the reflection of dislocations at grain bound-aries, and the absorption of dislocations into grain bound-aries [14,15]. Among these interactions, the slip penetration(transmission) especially has been shown to provide a veryimportant contribution to the deformation of polycrystal-line materials. Based on the experimental data obtainedat very small plastic strains, Livingston and Chalmers[16], Clark et al. [5] and Lee et al. [17] have proposed sev-eral criteria for the penetration of slip through a grainboundary. These criteria include the relationship betweenthe Burgers vectors on either side of an interface, the inter-action lines between the grain boundary and the slip planesof the incoming and outgoing dislocations, the resolvedshear stress on the outgoing side related to the stressfield of the pile-up dislocations on the incoming side, and

A. Ma et al. / Acta Materialia 54 (2006) 2181–2194 2183

residual grain boundary dislocations left as debris in theinterface during a transmission event.

Livingston and Chalmers [16], Clark et al. [5] and Leeet al. [17] suggest that, if a slip penetration event occursat a grain boundary, the criteria outlined above can be usedto determine the expected outbound dislocations for agiven set of inbound ones. We apply these criteria as guide-lines for the identification of those mechanisms whichshould enter into an improved crystal plasticity finite ele-ment model for the incorporation of grain boundarymicro-mechanics in a constitutive framework which isbased on the dislocation model developed in Ref. [1].

3. Discussion of grain boundary micro-mechanics inconjunction with crystal plasticity finite element theory

In this section, we discuss various approaches which areconceivable for the introduction of grain boundary micro-mechanics into a crystal plasticity finite element constitu-tive framework.

3.1. Energy equivalence concept

A first possible concept for including grain boundariesin a crystal plasticity finite element (FE) model consistsin using the grain boundary energy as an approach forintroducing an interface resistance against slip penetrationin terms of geometrically necessary dislocations. The ideabehind this approach is to directly translate the grainboundary energy [18] into a generalized Read–Shockley-type formulation and by using this form for extracting anequivalent density of geometrically necessary dislocationswhich replaces the actual grain boundary in a finite elementwhich carries the interface properties. This approach suffersfrom two drawbacks: First, we could not identify a suitedhomogenization model which acts locally. This means thatit is unclear how such equivalent dislocations would act asa barrier and interfere with the motion of the mobile dislo-cations according to the basic framework outlined in Ref.[1]. Second, such an approach does not include the tenso-rial nature of the problem. This means that a robustmicro-mechanical concept of the grain boundary as anobstacle against dislocation motion requires that the slipresistance encountered by mobile dislocations must be dif-ferent for different incoming and outgoing glide systems(which are described by three orthogonal vectors, namely,the glide vector da, the line or tangent vector ta and the slipplane normal vector na). Examples at hand are easy grainboundary transmission events, such as are conceivable forsmall angle grain boundaries or certain coherent twin inter-faces [5,14,15,19], as opposed to difficult penetrationevents, such as in the case for large angle grain boundarieswith large CSL (coincidence site lattice) numbers and alarge misalignment between the slip systems on either side.A locally enhanced equivalent scalar density of geometri-cally necessary dislocations cannot reflect such tensorialphysics of slip penetration.

3.2. Misorientation equivalence concept

A second possible concept is to directly use Nye’s dislo-cation tensor [3] for mapping the micro-mechanics of grainboundaries. This approach amounts to using the respectivemisorientation associated with a grain boundary instead ofits energy to derive an equivalent density of geometricaldislocations. The disadvantage of this approach is thatNye’s theory is built on small orientation gradients whichwould exclude the possibility of mapping the nature oflarge angle grain boundaries. Also, grain boundaries repre-sent lattice defects the structure of which may change dis-continuously as a function of the misorientation. Whilegrain boundaries can in the small angle regime be welldescribed by the Read–Shockley concept [20], large anglegrain boundaries do not consist of lattice dislocations. Thismeans that Nye’s tensor does not apply for large anglegrain boundaries since they do not contain, in theunstrained state, any geometrically necessary dislocationswithin their actual grain boundary area but consist ratherof relaxed structural units which do not form lattice dislo-cations [19]. A continuous and dislocation based descrip-tion of grain boundaries with arbitrary misorientation interms of Nye’s dislocation tensor is, therefore, a less perti-nent concept to capture grain boundary micro-mechanicsin a crystal plasticity FE environment. Another drawbackassociated with a continuous Nye-type model of grainboundary mechanics is the tensorial nature of the problem:The translation of a local orientation difference of twoabutting crystals directly into geometrically necessary dis-location populations does not give a unique solution, sincedifferent combinations of slip systems could yield the sameresult. The details associated with these initial values of thelocal densities of geometrically necessary dislocations are,however, important for the very beginning of plasticdeformation.

3.3. Thermally activated slip transmission concept

In the following we give an outline of a third way toapproach the problem.

The transmission probability of incoming mobile dislo-cations which try to penetrate a grain boundary can betreated in terms of an activation concept. The enthalpyfor this activation process stems from the elastic energyof formation of potential misfit dislocations which wouldremain as debris in the interface upon slip penetration. Thisactivation enthalpy enters as an additional contributioninto the activation term for the slip of mobile dislocations(see details of the constitutive equations in Refs. [1] or [21]).It is likely that each transmission event will occur at thesmallest possible energy consumption which provides anatural selection criterion for the outgoing slip system,when the incoming one is known.

If we describe the inbound slip system in the first grainby the shear vector da, the tangent vector ta, and the slipplane normal vector na, the task consists of identifying that


particular abutting (outbound) slip system on the otherside of the boundary (db, tb,nb) which provides the mini-mum energy barrier, i.e., the smallest misalignment relativeto the active inbound slip system. We do not actually checkwhether or not this slip system on the other side of thegrain boundary can be truly activated by the local stressbut assume in a somewhat simplifying fashion, that its acti-vation is possible, as the slip system orientation is close tothe inbound one. For an arbitrary transmission event, it isobvious that some incoming slip system does as a rule notmatch a corresponding one on the outbound side, i.e., ageometrically coherent shear system on the other side ofthe grain boundary. Therefore, in order to meet the conser-vation of the lattice defect vector sum when crossing aninterface, certain misfit dislocations will be left in the grainboundary. The additional energy required to produce suchan extra misfit dislocation acts as an energy barrier mea-sure for this thermally activated slip transmission event.

One should emphasize that this transmission event pro-vides a method to quantify the penalty energy required forsuch an event. However, as we are modeling the materialbehavior on a homogenized continuum scale there neednot be a strict one-to-one correlation between incomingand outgoing dislocations. Moreover, it is conceivable thatthe transmission only rarely takes place owing to the localhardening effect that it introduces and that, instead, theaccumulation of geometrically necessary dislocationsincreases the stiffness locally. These aspects will be dis-cussed in detail in Section 6. A final remark concerns themisfit dislocations. While they serve as a means to quantifythe penalty energy they will very likely not be stored but bedissolved by some relaxation process in the grain bound-ary. This implies that there will be no accumulation of mis-fit dislocations in the boundary, which would alter theprocess for newly incoming dislocations. Moreover, theassumption that the general character of the grain bound-ary is not altered by the penetration process allows us touse the same model of the penetration process for smallstrains and large strains.

4. A grain boundary micro-mechanical mechanism based on

dislocation transmission

In this chapter, we discuss the mathematical treatmentof the third possibility for treating the grain boundaryeffect as discussed in the previous chapter.

We consider two crystals with orientations QI and QII

with slip systems (da,b, ta,b,na,b), a, b = 1,2, . . . , 12,1 and agrain boundary with a normal vector nGB, which separatesthese two crystals, see Fig. 1.

According to Fig. 1(a), we assume that the dislocationline elements, la and lb, will align with the boundary planeduring the transmission event. This means that we use thetwo line elements:

1 The indices a and b always refer to crystals I and II, respectively.

la0 ¼ laðnGB � naÞ ð1Þ

lb0 ¼ lb nGB � nb

� �ð2Þ

instead of la and lb for the conservation law of the latticedefect (expressed by the dislocation tensor) during the pen-etration event

ba � la0 ¼ bb � lb

0 þ baGB � laGB ð3Þ

where b is the respective Burgers vector and the index GBrefers to the grain boundary dislocation, i.e., to the debriswhich remains in or at the grain boundary upon slippenetration.

The energy of formation for these misfit dislocations ful-fills the following inequality:

1

2Gb2la0

61

2Gb2lb0 þ 1

2Gba2

GBlaGB ð4Þ

where G is the shear modulus. As both mobile dislocationsin grains I and II are crystal dislocations, we assume thatthey have equal energies. Therefore, the additional energyfor the transmission event is the energy of formation ofthe grain boundary dislocation which has to be formedduring the process. The final task is to identify for everyslip system a of crystal I a slip system b in crystal II withthe boundary condition that the energy of the grain bound-ary dislocation is minimized upon slip transmission

EaGB ¼ min

b

1

2Gba2

GBlaGB ð5Þ

We now have to determine baGB and laGB in such a way that

Eq. (3) is fulfilled. However, as we are interested in the acti-vation energy only, it is sufficient to determine the magni-tude of the vectors, i.e., ba

GB and laGB. This means that we

can rewrite Eq. (3):

kbaGB � laGBk ¼ kba � la

0 � bb � lb0 k ð6Þ

which can be reformulated into

kbaGBkklaGBk ¼ kba � la

0 � bb � lb0 k ð7Þ

This expression does not have a unique solution. Wheneveridentifying a pair of kba

GBk and klaGBk that satisfies Eq. (7)any pair ckba

GBk and 1c kl

aGBk with c being an arbitrary con-

stant would also be a valid solution. However, when usingEq. (5) to calculate Ea

GB the result would change by a factor c.Nevertheless, when fixing, for example, the magnitude

of baGB to cb, where b is the magnitude of the lattice Burgers

vector, one can calculate klaGBk as

laGB ¼

1

cbkba � la

0 � bb � lb0 k ð8Þ

and the energy for the activation event can be calculated as

EaGB ¼ min

bc2 1

2Gb2la

GB ð9Þ

When further writing the length of laGB in multiples of the

magnitude of the lattice Burgers vector, that is

Fig. 2. Normalized activation energy for twist grain boundaries withrotation about the [111] direction using c = 1 in Eq. (11), the normal-ization factor is 1

2Gb3.

Fig. 1. Schematic drawing of penetration events for mobile dislocations through a grain boundary. la0, lb

0, laGB are the line vectors, and ba, bb, ba

GB are theBurgers vectors of the dislocations in the moment of the penetration event. a and b indicate crystals I and II and the index GB the grain boundary dislocation.


laGB ¼ ab ð10Þ

where a is determined by Eq. (8) we finally obtain

EaGB ¼ min

bc2a

1

2Gb3 ð11Þ

It is worthwhile mentioning at this point, that while theabsolute magnitude of Ea

GB can be changed by the choiceof c, the ratio of the activation energies for different bound-aries is not affected by this value.2

The activation energies of an incoming dislocation withlength b are calculated for twist boundaries with rotationsabout the [111] and [110] crystal directions for the face-centred cubic crystal structure under the additional con-straint, that the grain boundary plane is perpendicularto these rotation axes. The results are shown in Figs. 2and 3, where the activation energy has been normalizedby the factor 1

2Gb3 and the constant c was chosen to be

equal to one. Both figures also show the average of theenergy barrier for better comparison. From these curves,it is clear that a grain boundary is a strong obstacle todislocation motion, as the average activation energiesfor the formation of the misfit dislocations reach theorder of magnitude of the activation energy for forestcutting.

It can be seen that the energies for slip penetrationshow a periodic behavior. This periodicity arises fromthe octahedral symmetry of the slip systems in the crystal.Every change in the choice of the outgoing slip system bcauses a discontinuous change in the slope of the energycurve. One can see that the energy barrier stronglydepends on the misorientation of the two crystals, espe-cially when the rotation angle is larger than about 20�.However, the average activation energies show a muchmore constant behavior than the single slip system curvessuggest, which implies that the strong effect for individualslip systems will be averaged out to some extent in mac-roscopic experiments.

2 c, therefore, serves as a fitting parameter in the model.

Fig. 3. Normalized activation energy for twist grain boundaries withrotation about the [110] direction using c = 1 in Eq. (11). The normal-ization factor is 1

2Gb3.

Fig. 4. Schematic drawing of the penetration and cutting mechanism:(a) cutting mechanism; (b) penetration of perfect grain boundary; (c) FEMtreatment of penetration process; qa

m is the density of mobile dislocationsfor slip system a, qa

F the density of forest dislocations, ka the averageobstacle spacing, and LGB the thickness of the grain boundary element.


Finally one has to consider two special situations: first,when the grain boundary plane is parallel to the slip planeof the incoming dislocation the energy barrier is set to zero,because the mobile dislocations do not penetrate the grainboundary, but they move parallel to it; second, when theplane of the outgoing dislocation is parallel to the grainboundary we set lb

0parallel to la

0as there is no intersection

line with the grain boundary.In summary the new model developed in this paragraph

assigns an obstacle strength of the grain boundary whichdoes not only depend on the grain boundary misorienta-tion but also on the grain boundary plane orientation(nGB) and on the slip systems involved on either side ofthe interface.

5. The grain boundary element

In all crystal plasticity FE modeling (FEM) implemen-tations known to the authors the grain boundaries coin-cide with element boundaries. In contrast to thisapproach we use a grain boundary implementation inthe form of a special grain boundary element. In this ele-ment we use a modified version of the flow rule given inRef. [1], namely

_ca ¼_ca

0 exp � Qeff

kBh 1� jsaj�sa

pass

sacut

� �h isignðsaÞ jsaj > sa

pass

0 jsaj 6 sapass

8<:

ð12Þwhere the pre-exponential variable _ca

0 and passing stresssa

pass are the same as equations 15 and 16 in Ref. [1]. Thecutting stress caused jointly by the forest dislocations andthe grain boundary at 0 K now reads

sacut ¼

Qeff

c2c3b2

ffiffiffiffiffiqa

F

pð13Þ

where c2 and c3 are again constants and Qeff is the modifiedeffective activation energy:

Qeff ¼ Qslip þ QaGB ð14Þ

When comparing this flow rule with the one specified inRef. [1], the only difference is the use of Qeff instead of Qslip.The energy Qa

GB is calculated according to Eq. (11) as

QaGB ¼ min

bc9a

1

2Gb3 ð15Þ

with the constant c9 = c2.Modifying the activation energy for an individual dislo-

cation jump in Eq. (12) implies that the boundary is over-come in a single activation event.

The activation area for this event is of the order kab. Inthis expression ka is the trapping length of the mobiledislocation (see Fig. 4(a)) and b is the magnitude of thelattice Burgers vector, which is used as the obstacle widthfor the piercing dislocation density. However, when sim-ply adding the activation energies, we use the same activa-tion area for the transmission event. Therefore, when

adopting a in Eq. (11) in such a way that the length ofthe transmitted dislocations equals ka, the element thick-ness of the boundary element should amount to a valueclose to b (see Fig. 4(b)). If we further assume that westart with a mesh of brick elements the element volume(bkaka) would be of the order of 1 lm3 or less. Even ifwe use small samples in the mm size range, this wouldmean that we need about one billion elements to meshour sample. This number is too high for practicalapplications.

The only way to circumvent this problem is to increasethe element thickness to LGB (Fig. 4(c)). However, by doingso we assign an overly stiff material behavior to a volumemuch bigger than the actual grain boundary volume. Inorder to compensate for this large volume we can artifi-cially decrease the grain boundary strength by the choiceof the constant c9 / b

LGB. It has to be stressed though, that

this is a purely empirical adjustment due to the use of largerfinite elements and does not describe the scaling of theunderlying physics one-to-one.

6. Simple shear tests

The grain boundary model introduced in this paper isimplemented in the commercial finite element codeMSC.Marc200x in terms of the user defined material sub-routine HYPELA2 [22]. The experimental setup and theboundary conditions are described in Ref. [1]. The only dif-ference is that for the bicrystal simulations quoted in thispaper, each set of four integration points on either sideof the interface in the newly introduced grain boundary ele-ments shown in Fig. 5 carries the initial orientations of the

Fig. 5. Experimental setup and finite element mesh used for the simulations indicating also the boundary conditions applied. The grain boundary elementsare highlighted.


abutting crystals. In such elements, which initially containa change in crystal orientation within their element bordersaccording to the real bicrystal, the new constitutive grain

Fig. 6. Simple shear test of a bicrystal (3.1 mm long, 2.0 mm thick and 2.2 mmstrain patterns of the experiment obtained by digital image correlation (Dconstitutive law (middle column), and from the simulation which uses the new

boundary law is applied resulting in a locally increasedstiffness in terms of an additional energy barrier for eachincoming slip system as outlined above.

high) with a low angle grain boundary (7.4�). Comparison of the von MisesIC, left column), from the simulation with a conventional viscoplastic

set of constitutive laws (right column).

Fig. 7. Simple shear test of a bicrystal (3.1 mm long, 2.0 mm thick and 2.2 mm high) with a medium angle grain boundary (15.9�). Comparison of the vonMises strain patterns of the experiment obtained by digital image correlation (DIC, left column), from the simulation with a conventional viscoplasticconstitutive law (middle column), and from the simulation which uses the new set of constitutive laws (right column).


6.1. Simple shear tests for aluminum bicrystals

The same constitutive parameters which were fitted withthe help of the single crystal simple shear tests in Ref. [1]were used for simulations of three bicrystal shear tests.The first bicrystal had a 7.4� small angle grain boundarywith the initial orientations (u1 = 277.0�, U = 32.3�,u2 = 37.4�)I and (u1 = 264.7�, U = 32.3�, u2 = 44.3�)II.The second bicrystal had a 15.9� grain boundary with theinitial orientations (u1 = 74.3�, U = 37.1�, u2 = 50.1�)I

and (u1 = 87.9�, U = 36.7�, u2 = 52.9�)II. The third bicrys-tal had a 33.2� grain boundary with the initial orientations(u1 = 105.9�, U = 34.1�, u2 = 43.4�)I and (u1 = 64.1�,U = 34.8�, u2 = 54.6�)II [23]. All three bicrystals have a pla-nar symmetric tilt grain boundary with the [112] crystaldirection as common tilt axis which is parallel to the Y axis(TD direction of the finite element mesh, see Fig. 5).

6.1.1. Comparison of the accumulated plastic strain

Figs. 6–8 show the comparison of the von Mises strainpatterns obtained from the experiment (left column), from

the simulation with a conventional viscoplastic constitutivelaw (middle column), and from the simulation series whichis based on the new set of constitutive laws including dislo-cation rate formulations on each slip system, geometricallynecessary dislocations on each slip system, and a geometri-cal model for the grain boundary resistance against sliptransmission events as introduced in this work (right col-umn). The figures show the von Mises equivalent strain dis-tributions for five subsequent stages of shear with aconstant increase of 10% per load step.

The experimental data clearly reveal in all three cases thestrong micro-mechanical effect imposed by the presence ofthe respective grain boundaries.

Even for the small angle grain boundary (7.4�) the shearexperiment shows an effect of strain separation among thetwo abutting crystals. With increasing grain boundary mis-orientation the heterogeneous distribution of the von Misesstrain becomes more pronounced.

The experimental results are different from thosereported earlier in Ref. [4,23] where channel-die testswere used for deforming a set of similar bicrystals. These

Fig. 8. Simple shear test of a bicrystal (3.1 mm long, 2.0 mm thick and 2.2 mm high) with a large angle grain boundary (33.2�). Comparison of the vonMises strain patterns of the experiment obtained by digital image correlation (DIC, left column), from the simulation with a conventional viscoplasticconstitutive law (middle column), and from the simulation which uses the new set of constitutive laws (right column).


earlier experiments revealed a disturbing influence offriction and of the length to height ratio of the specimensso that the mechanical effect arising from the presence ofinterfaces was overlapped by these macro-mechanicaleffects.

Also, in the channel-die experiments the load wasimposed in a symmetrical fashion and the symmetry ofthe plane strain state relative to the initial orientations ofthe two abutting crystals produced a weak mirror symme-try during plastic deformation in the RD–ND plane. Thiseffect is avoided in our current shear experiments.

A further reason for the absence of symmetry effects inthe shear experiments is the non-linear material behaviorresulting from the highly non-linear flow and hardeningrules which may yield very different results even for tinydeviations in the initial state or in the local boundaryconditions.

The complicated cross-hardening effect may also con-tribute to the absence of symmetry. Probably the mostimportant influence for the absence of strain symmetry,however, are micro-mechanical effects arising from thepresence of the grain boundaries. The experimental data

reveal that in all three bicrystals one of the two abuttingcrystals accumulates more plastic strain when comparedto the other.

Concerning the corresponding simulation results it isessential to note that the use of the empirical viscoplastic(local) law (second rows of Figs. 6–8) does not adequatelyreproduce the influence of the grain boundaries on the lat-eral distribution of the accumulated von Mises strain. Thisapplies in particular to the two bicrystals which have asmall and medium angle grain boundary, respectively.For the bicrystal with the large angle grain boundary,Fig. 8, the empirical (local) model is capable to predictsome although not all characteristics of the strain separa-tion between the two crystals. This partial success of thesimulation with the empirical viscoplastic hardening lawin case of the large angle grain boundary is attributed tothe strong effect of the change in the Schmid factor acrossthe interface. This means that the kinematic effect whicharises from distinct differences in the slip system selectionon either side of a grain boundary plays an essential rolein this case. We refer to this effect as kinematic hardeningimposed by grain boundaries.


The simulation results obtained by using the new modelmatch the experimental data much better. For the bicrys-tals with the small angle grain boundary and that withthe large angle grain boundary, one grain is deformed moresignificantly than the other which reproduces the experi-mental observation very well. The predictions for thebicrystal with the medium angle grain boundary, Fig. 7,show some deviations to the experimental findings. Inorder to analyse possible reasons for this deviation we pres-ent a section through the predicted equivalent total strainmap in the ND–TD plane of the mesh near the corner ofthe sample in Fig. 9.

The section reveals that the cross-section of the specimennear the corner point shrinks considerably compared to theundeformed sample shape. In the experiment the externalload is imposed via friction between the sample surfaceand the tool under strong external pressure. During thedeformation experiment the sample detaches partially fromthe clamping tool, Fig. 9(d), due to the shrinking process.

Although the specimen experiences the same tendency tochange its shape locally in the finite element simulations theboundary conditions imposed (Fig. 5) do not allow fordetachment. This gives rise to artificially large stretches inthis area, Fig. 9(c). We, therefore, attribute the small devi-ations between the experiment and the simulation with thenew nonlocal model observed for the medium angle grainboundary to local deviations in the boundary conditions.

The photogrammetric method used in the experiment tocharacterize the strain can only provide information aboutthe deformation at the surface of the bicrystal samples. Inorder to better understand the micro-mechanical behavior

Fig. 9. Deviation of the boundary conditions of the finite element calculationcut; (b) shape change in the plane shown in (a); (c) modified boundary conditi(schematic).

of the entire bicrystal, Fig. 10 shows for the three bicrystalsseries of seven subsequent through-thickness sections of theequivalent total strain along the Z- or, respectively, NDdirection.

An interesting result from these inner sections is theobservation that in the small angle bicrystal the grainboundary has been penetrated by a deformation band.Although no specific symmetries can be observed in the firstsection (neither in the simulation nor in the experiment),localized deformation zones proceed from the lower crystalinto the upper one along the ND direction. The middle sec-tion reveals an anti-symmetric deformation pattern.

The comparison between the three bicrystals shown inFig. 10 reveals that the specimen with the small angle grainboundary (7.4�) does not show a substantial grain bound-ary resistance against the penetration of plastic slip in theform of deformation bands from one crystal into the other.The other two bicrystal samples with larger misorientationangles (15.9�, 33.2�) between the abutting crystals showstronger resistance against deformation penetration. Thebicrystal with the large angle grain boundary (33.2�)reveals a strongly deformed zone in the upper grain inthe simulations. In the lower grain a less heavy deformedzone begins from the back layer and decays towards thefront layer. The bicrystal with the medium angle grainboundary (15.9�) reveals a similar distribution of the strain.

6.1.2. Comparison of the experimental and simulated

textures

Fig. 11 shows the {111} pole figures for the three bicrys-tals as obtained from the experiments and simulations with

with respect to the boundary conditions in the experiment: (a) position ofon used in the simulation; (d) boundary conditions in the real experiment

Fig. 10. Von Mises strain distribution in subsequent sections along the Z-axis (ND) as obtained from the simulations. The first row shows the results forthe simulations for the bicrystal with the 7.4� grain boundary; the middle row shows the results for the simulations for the bicrystal with the 15.9� grainboundary; the bottom row shows the results for the simulations for the bicrystal with the 33.2� grain boundary. In the front layer the von Mises strain wasalso measured in experiments as shown in the previous diagrams.



a conventional local model and with the new nonlocalmodel (50% shear deformation).

The experimental pole figures show reorientation zoneswith a strong Z (ND) rotation axis which is common toboth abutting crystals for all three bicrystal specimens.The scatter of the texture is rather weak in all samples,i.e., the reorientation took place rather homogeneouslythroughout the crystal when considering that the two abut-ting orientations were not stable in all three cases under theload imposed. The scatter is due to minor differences in thereorientation rates (similar rotation direction) which origi-nate from local differences in the accumulated strain as isevident from von Mises strain diagrams shown before.

The texture predictions which were obtained by usingthe local viscoplastic constitutive model (Fig. 11(d)–(f))show a rather large orientation scatter in all three cases.

Fig. 11. Texture comparison for the simple shear tests conducted with three bmodel; (g, h, i) dislocation and grain boundary nonlocal model; (a, d, g) bicrysangle grain boundary (15.9�); (c, f, i) bicrystal with large angle grain boundar

The texture spread substantially exceeds the scatterobserved in the experiments. A distinction between the tex-ture evolution in the two abutting crystals is not possible,i.e., the character of a bicrystal pole figure is lost. Thisobservation matches the strain distributions which weremade when using the viscoplastic constitutive law. Thesedata also did not show any influence of the grain boundaryon the micro-mechanical behavior except for the case of thelarge angle grain boundary with 33.2� misorientation wherethe kinematics of the interface prevailed in terms of thechange in the Schmid factor across the boundary.

The texture predictions obtained by the use of the jointnonlocal dislocation and grain boundary constitutivemodel (Fig. 11(g)–(i)) reveal much smaller orientation scat-ter and smaller reorientation rates when compared to thesimulations obtained by the local phenomenological model

icrystals: (a, b, c) experimental textures; (d, e, f) phenomenological localtal with small angle grain boundary (7.4�); (b, e, h) bicrystal with mediumy (33.2�).

Fig. 12. Positions of the path plot for the local orientation comparison between experiment and numerical simulation. Bicrystal with large angle grainboundary (33.2�) after 50% shear deformation. Red areas in the electron backscatter diffraction map indicate that no orientation could be determinedowing to the large deformation. The FEM figure shows the calculated misorientation.

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

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isor

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atio

ns fr

om th

e in

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orie

ntat

ion

[˚]

Distance from the upper shear boundary line [μm]

experimental results simulation results by

the nonlocal model

Fig. 13. Comparison of the local misorientation showing experimentaland numerical results for the bicrystal with the large angle grain boundary(33.2�) after 50% shear deformation.


(Fig. 11(d)–(f)). The simulated textures are in excellentaccord with the experimental pole figures.

We explain this smoothing effect on the texture evolutionmainly in terms of the influence of the geometrically neces-sary dislocations which act twofold: First, their necessaryaccumulation in conjunction with the generation of orienta-tion gradients introduces a direct mechanical resistance tothe further deformation of the material points affected bysuch gradients in terms of the increase in the overall localdislocation density. The second aspect (coupled to the firstone) is the resulting constitutive tendency of the nonlocalmodel to reduce the difference in lattice rotation betweenneighboring material points. This means that the implicitintroduction of the geometrically necessary dislocationsimposes a strong penalty or, respectively, drag force againstlateral gradients in the reorientation rates.

The {111} pole figures shown in Fig. 11(c) and (i) revealthat the projected orientation points cluster in the form oftwo groups, while in (f) this effect is less pronounced. Thisis attributed to the influence of the grain boundary on the tex-ture evolution, in particular to the anisotropy of the modeledresistance of the grain boundaries to different slip systems.

The influence of the nonlocal and of the grain boundarymodel on the overall reduction in the rate at which thedeformation textures evolve is a long standing problem intexture research since many classical texture models(including the crystal plasticity finite element approach)suffer from the drawback that the predicted textures aretwo sharp and the predicted evolution rates are often toohigh when compared to experiments.

6.1.3. Comparison of the local orientation

In the preceding section we showed how the new modelimproves the orientation prediction. However, pole figuresonly give a statistical impression of the texture evolution.The ultimate goal of any micro-structural model has to

be the correct prediction of the local mechanics andmicro-structures. We, therefore, compare the local misori-entation with respect to the initial orientation along oneexemplary line crossing the sample from top to bottom asindicated in Fig. 12, see Fig. 13. Again we observe an excel-lent agreement between the experimental and the simulatedvalues. The strongest deviations in the misorientation arefound near the sample edges, which can be attributed tothe non-perfect boundary conditions of the FE model dis-cussed above.

7. Conclusions

In an extension of the nonlocal dislocation model sug-gested in Ref. [1], the current paper introduced a novelconstitutive law which accounts for the interaction between


mobile dislocations and grain boundaries in a crystal plas-ticity finite element framework. The theoretical approach isbased on corresponding experimental findings which havebeen reported in the literature.

The grain boundary mechanism is introduced via anadditional activation energy barrier into the rate equationfor the slip of mobile dislocations in the vicinity of grainboundaries. The energy barrier is derived by using a geo-metrical model for thermally activated dislocation penetra-tion events through grain boundaries. The model takes fullaccount of the geometry of the grain boundaries and of theSchmid factors of the critically stressed incoming and out-going slip systems.

A special grain boundary element was introduced topractically implement the new approach into numericalsimulations of three bicrystals with a small, medium andlarge angle flat grain boundary, respectively. The predic-tions were compared to experiments (von Mises equivalenttotal strain, textures) for the case of simple shear.

The result show three main effects which arise from thenew model and provide excellent agreement between theoryand experiment. These are the accumulation of geometri-cally necessary dislocations and the resulting pronouncedlocal hardening in front of grain boundaries, the resistanceagainst slip penetration across the grain boundaries, andthe reduction of the orientation spread and of the rotationrates in the vicinity of grain boundaries (owing to the effectof the interfaces) and in the crystal interiors (owing to theeffect of the geometrically necessary dislocations).

The simulations and experiments clearly show that theclassical kinematic treatment of grain boundaries, whichis automatically included in all crystal plasticity FE modelsowing to the change in the Schmid factor across the inter-faces is not sufficient to adequately reproduce the micro-mechanics associated with the presence of grain boundaries.

Acknowledgment

The authors thank Professor G. Gottstein for valuablediscussions.

References

[1] Ma A, Roters F, Raabe D. A dislocation density based constitutivemodel for crystal plasticity FEM including geometrically necessarydislocations. Acta Mater 2006;58:2169–79.

[2] Gao H, Huang Y. Geometrically necessary dislocation and size-dependent plasticity. Scripta Mater 2003;48(2):113–8.

[3] Nye JF. Some geometrical relations in dislocated crystals. ActaMetall 1953;1:153–62.

[4] Zaefferer S, Kuo J-C, Zhao Z, Winning M, Raabe D. On the influenceof the grain boundary misorientation on the plastic deformation ofaluminum bicrystals. Acta Mater 2003;51(16):4719–35.

[5] Clark WAT, Wagoner RH, Shen ZY, Lee TC, Robertson IM,Birnbaum HK. On the criteria for slip transmission acrossinterfaces in polycrystals. Scripta Metallurgica et Materialia1992;26(2):203–6.

[6] Evers LP, Parks DM, Brekelmans WAM, Geers MGD. Crystalplasticity model with enhanced hardening by geometrically neces-sary dislocation accumulation. J Mech Phys Solids 2002;50(11):2403–24.

[7] Arsenlis A, Parks DM. Modeling the evolution of crystallographicdislocation density in crystal plasticity. J Mech Phys Solids2002;50(9):1979–2009.

[8] Kalidindi SR, Bronkhorst CA, Anand L. Crystallographic textureevolution in bulk deformation processing of fcc metals. J Mech PhysSolids 1992;40:537–69.

[9] Beaudoin AJ, Dawson PR, Mathur KK, Kocks UF. A hybrid finiteelement formulation for polycrystal plasticity with consideration ofmacrostructural and microstructural linking. Int J Plasticity 1995;11:501–21.

[10] Raabe D, Sachtleber M, Zhao Z, Roters F, Zaefferer S. Microme-chanical and macromechanical effects in grain scale polycrystalplasticity experimentation and simulation. Acta Mater 2001;49(17):3433–41.

[11] Sachtleber M, Zhao Z, Raabe D. Experimental investigation ofplastic grain interaction. Mater Sci Eng A 2002;336(1–2):81–7.

[12] Roters F, Wang Y, Kuo J-C, Raabe D. Comparison of single crystalsimple shear deformation experiments with crystal plasticity finiteelement simulations. Adv Eng Mater 2004;6(8):653–6.

[13] Zhao Z, Radovitzky R, Cuitio A. A study of surface roughening infcc metals using direct numerical simulation. Acta Mater2004;52(20):5791–804.

[14] Shen Z, Wagoner RH, Clark WAT. Dislocation pile-up and grainboundary interactions in 304 stainless steel. Scripta Metall 1986;20(6):921–6.

[15] Kehagias T, Komninou P, Dimitrakopulos GP, Antonopoulos JG,Karakostas T. Slip transfer across low-angle grain boundaries ofdeformed titanium. Scripta Metallurgica et Materialia 1995;33(12):1883–8.

[16] Livingston JD, Chalmers B. Multiple slip in bicrystal deformation.Acta Metall 1957;5:322–7.

[17] Lee TC, Robertson IM, Birnbaum HK. An in situ transmissionelectron microscope deformation study of the slip transfer mecha-nisms in metals. Metall Trans 1990;21A:2437–47.

[18] Gottstein G, Shvindlerman LS. Grain boundary migration in metals –thermodynamics, kinetics, applications. Boca Raton, FL: CRCPress; 1999.

[19] Sutton AP, Baluffi RW. Interfaces in crystalline materi-als. Oxford: Clarendon Press; 1995.

[20] Read WT, Shockley W. Dislocation pile-up and grain boundaryinteractions in 304 stainless steel. Phys Rev B 1950;78:275.

[21] Ma A, Roters F. A constitutive model for fcc single crystals based ondislocation densities and its application to uniaxial compression ofaluminium single crystals. Acta Mater 2004;52(12):3603–12.

[22] MSC. Marc user’s manual 2003, User Subroutines and SpecialRoutines, vol. D; 2003.

[23] Kuo J-C. Mikrostrukturmechanik von Bikristallen mit Kippkorn-grenzen. PhD thesis, RWTH Aachen; 2004.

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