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International Journal of Solids and Structures 71 (2015) 277–290

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsols t r

A mesoscopic theory of dislocation and disclination fields for grainboundary-mediated crystal plasticity

http://dx.doi.org/10.1016/j.ijsolstr.2015.06.0310020-7683/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (V. Taupin).

V. Taupin a,⇑, L. Capolungo b, C. Fressengeas a, M. Upadhyay b, B. Beausir a

a Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux, Université de Lorraine/CNRS, Ile du Saulcy, 57045 Metz Cedex, Franceb G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology/CNRS, 57070 Metz Cedex, France

a r t i c l e i n f o

Article history:Received 20 December 2014Received in revised form 5 May 2015Available online 6 July 2015

Keywords:Grain boundaryDisclinationDislocation

a b s t r a c t

A coarse-grained extension of a recent nanoscale elasto-plastic model of polar dislocation and disclina-tion density fields is developed to model grain boundary-mediated plasticity in polycrystals. At a smallresolution length scale, the polar dislocation/disclination densities render continuously the discontinu-ities of the elastic displacements/rotations across grain boundaries. When the resolution length scaleincreases, the net polarities of a crystal defect ensemble decrease, perhaps to the point where nostrain/curvature incompatibility is left in the body. The defect densities are then labeled ‘‘statistical’’.However both polar and statistical dislocation/disclination densities contribute to plastic flow, and acoarse-grained mesoscopic plastic curvature rate needs to be defined. In addition, whereas it is over-looked at nanoscale where grain boundaries are seen as continua, tangential continuity of the elastic/-plastic curvature/strain rates across grain boundaries needs to be considered at mesoscale, because thelatter are seen as singular discontinuity interfaces. It induces long-range, grain-to-grain, elastic/plasticinteractions across interfaces. The mesoscale model allow preserving the essential features of the lowerscale approach. In particular, it is shown that it allows accounting for such plastic deformation mecha-nisms as grain boundary migration and grain boundary misorientation variation by disclination motionand concurrent dislocation nucleation, when plasticity by dislocation glide is unavailable. Accumulationof polar defect densities in the vicinity of the grain boundaries and triple lines, leading to long-rangeinter-granular activation of slip and grain size effects, are also predicted by the model.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

In polycrystals, grain boundaries (GBs) are thin regions wherethe lattice rapidly changes its orientation within nanometers(Sutton and Vitek, 1983; Priester, 2013; Fressengeas et al., 2014;Cordier et al., 2014). GBs play a key role in the mechanical responseof polycrystals. They can strengthen the materials by limiting dis-location glide within grains. Dislocation transmission orre-emission from GBs do also lead to plastic flow, such that the roleof GBs in plasticity is not limited to being obstacles for dislocationglide. In nanocrystalline materials or in slip-deprived materials,GBs can also become important sources of plastic deformation.Well-known examples of GB-mediated plasticity mechanisms areGB migration (Gutkin and Ovid’ko, 2005; Farkas et al., 2006;Cahn et al., 2006; Ovid’ko et al., 2008; Gorkaya et al., 2009;Mompiou et al., 2009; Cordier et al., 2014), nucleation of disloca-tions (Tschopp et al., 2007, 2008) followed by propagation in grains

(Van Swygenhoven et al., 2006, 2008; Bitzek et al., 2008).Importantly, the elasto-plastic response of nanocrystalline materi-als is strongly sensitive to the atomic structure of the GBs(McDowell, 2008). For example, the atomic structural units andassociated excess free volumes (Tucker et al., 2010a) composingthe GBs (Sutton and Vitek, 1983) were shown to correlate withboth nucleation of dislocations at GBs (Tucker et al., 2010a) andshear-coupled boundary migration (Cahn et al., 2006; Taupinet al., 2014).

Several modeling approaches aimed at describing the mechan-ical response of GBs and phase boundaries in relation with theinterface microstructure have been developed in the past (seee.g. Bollmann, 1970; Pond and Hirth, 1994; Hirth and Pond,1996; Hirth et al., 2006; Khater et al., 2012; Vattre et al., 2014).In particular, elasto-static disclination-based approaches havebeen proposed (Romanov and Kolesnikova, 2009; Romanov et al.,2015; Li, 1972; Shih and Li, 1975; Gertsman et al., 1989), whichsuccessfully complemented dislocation-based models and moti-vated further investigation involving dissipative processes.Indeed, in past work by the authors, a nanoscale structure sensitive

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model of GBs was proposed within the framework of anelasto-plastic theory of disclination and dislocation fields(Fressengeas et al., 2011). In this theory, based on seminal worksby Volterra (1907), Nye (1953), Mura (1963), deWit (1970),Kröner (1980) and Acharya (2001), the polar disclination and dislo-cation densities account in a continuous manner for the disconti-nuities of the elastic/plastic rotation and displacement arisingacross bounded surfaces in a crystalline material. These defect den-sities induce incompatible elastic and plastic curvatures andstrains, leading to long-range internal stress and couple stressfields. Their transport, driven by thermodynamically consistentmechanical forces, provides their spatio-temporal dynamics andyields plastic deformation (Fressengeas et al., 2011). This theorywas first used for continuous modeling of the core structure ofsymmetric tilt boundaries in copper (Fressengeas et al., 2014)and more complex boundaries in olivine (Cordier et al., 2014)and fullerene monolayers (Bozhko et al., 2014). In these papers,an atomistic to continuum crossover is realized by substitutingdipoles of wedge disclination densities to the structural units com-posing the core of the GBs (Sutton and Vitek, 1983). A continuousdistribution of elastic energy in the GB core area is obtained fromthe solution of the field equations, in good agreement with atomis-tic simulations and experiments (Hasson et al., 1972; Tschoppet al., 2008). Based on this structure of disclination dipoles,shear-coupled boundary migration of symmetric tilt boundariesin copper is also retrieved (Taupin et al., 2014), in agreement withatomistic and experimental findings (Cahn et al., 2006; Gorkayaet al., 2009). An essential outcome of the model is that, since thediscrete atomic structures are replaced by continuous defect den-sities, the vibrations of atoms do not have to be resolved at thefemto-second time scale. Thus, time-coarsening is built-in, whichallows performing simulations over realistic time scales with strainrate levels amounting to experimental values (Taupin et al., 2014),while keeping essential features of GBs.

The objective of the present study is to provide a coarse-grainedextension of the above Field Dislocation and DisclinationMechanics (FDDM) nanoscale framework that allows retainingsome of its key features at mesoscale, namely, the ability toaccount for the structure of GBs in predicting the mechanicalresponse of polycrystalline media. Such an extension requiresaddressing three fundamental challenges, all arising from the scaledependence of the continuous measures of lattice incompatibilityin the presence of crystal defects. First, although the array of discli-nation dipoles composing the GB structure at nanoscale is over-looked when the resolution length scale is increased in thecoarse-graining process, recent measurements obtained fromEBSD orientation maps reveal the presence of dislocation anddisclination dipoles decorating GBs and triple lines at mesoscale(Beausir and Fressengeas, 2013). Hence, a first challenge will beto account for such mesoscale patterns of crystal defects and theirspatio-temporal evolution. Second, whereas GBs are seen as con-tinua in the nanoscale FDDM framework, the mesoscale approachpromoted here will adopt an interfacial description of the GBs –i.e. GBs of vanishingly small thickness. Hence, the second point tobe addressed in the present paper is the account of continuity con-ditions at interfaces on the strain/curvature tensors, traction vectorand elastic/plastic strain/curvature tensors (Acharya, 2007;Fressengeas et al., 2012), which allow retaining the nonlocal fea-tures of the GBs that are essential to render the grain-to-graininteractions and their consequences on plastic strain and orienta-tion distributions (textures), and mechanical behavior. Third,whereas the coarse-graining process leads to decreasing crystaldefect densities when the resolution length scale is increased, itdoes not impact on the plastic distortion and curvature rates,which do not follow this decreasing trend (Fressengeas et al.,2012). As a result, mesoscale plastic distortion and curvature rates

statistically offsetting for the decreasing polar plastic distortionand curvature rates need to be introduced in the averaging process,in analogy with the work by Acharya and Roy (2006) in the puredislocation case. A finite element implementation of the proposedphenomenological mesoscopic field disclination and dislocationmechanics (PMFDDM) model will be developed, in order to showillustrative examples and to evidence the predictive potentialbrought by the model in mesoscale systems.

The paper is organized as follows. In Section 2, useful mathe-matical notations are settled. In Section 3, the mesoscale extensionof the model is introduced. Essential points in this section are thescale dependence of polar defect densities, the initial representa-tion of GBs at mesoscale, and the implementation of tangentialcontinuity conditions on plastic distortion and curvature rates atsingular GB interfaces. The coarse-grain derivation of a mesoscalestatistical plastic curvature rate, an analysis of scale dependencesof the PMFDDM model as well as the ability of the latter to rendernew GB-driven plastic mechanisms, are discussed in Section 4. InSection 5, illustrative simulations of bicrystals and tricrystals arepresented to show the predictive capability of the model.Conclusions follow.

2. Notations

A bold symbol denotes a tensor. When there may be ambiguity,

an arrow is superposed to represent a vector: ~V. The symmetric

part of tensor A is denoted Asym. Its skew-symmetric part is Askew.The tensor A � B, with rectangular Cartesian components AikBkj,results from the dot product of tensors A and B, and A� B is theirtensorial product, with components AijBkl. A: represents the traceinner product of the two second order tensors A : B ¼ AijBij, in rect-angular Cartesian components, or the product of a higher ordertensor with a second order tensor, e.g., A : B ¼ AijklBkl. The crossproduct of a second-order tensor A and a vector V, the div andcurl operations for second-order tensors are defined row by row,in analogy with the vectorial case. For any base vector ei of the ref-erence frame:

ðA� VÞt � ei ¼ ðAt � eiÞ � V; ð1Þ

ðdivAÞt � ei ¼ divðAt � eiÞ; ð2Þ

ðcurlAÞt � ei ¼ curlðAt � eiÞ: ð3Þ

In rectangular Cartesian components:

ðA� VÞij ¼ ejklAikVl; ð4Þ

ðdivAÞi ¼ Aij;j; ð5Þ

ðcurlAÞij ¼ ejklAil;k: ð6Þ

where ejkl is a component of the third-order alternating Levi–Civita

tensor X. A vector ~A is associated with tensor A by using its traceinner product with tensor X:

ð~AÞk ¼ �12ðA : XÞk ¼ �

12

eijkAij: ð7Þ

In the component representation, the spatial derivative with respectto a Cartesian coordinate is indicated by a comma followed by thecomponent index. A superposed dot represents a material timederivative. Finally, A½ � ¼ Aþ � A� denotes a jump or a discontinuityof tensor A across an oriented interface.

V. Taupin et al. / International Journal of Solids and Structures 71 (2015) 277–290 279

3. Mesoscale model of dislocation and disclination fields

3.1. Polar defect densities

The field theory of dislocations and disclinations is now brieflyrecalled, the reader is referred to the paper by Fressengeas et al.(2011) for a more comprehensive presentation. In a continuummechanics setting in the absence of cracks, the displacement vec-tor field u is defined continuously at any point of anelasto-plastic body. The total distortion tensor, defined as the gra-dient of the displacement U ¼ gradu, is thus curl-free:

curl U ¼ 0: ð8Þ

In the presence of dislocation ensembles (Acharya, 2001), the plas-tic, Up, and elastic, Ue, components of the distortion may containincompatible, non curl-free parts, U?p and U?e . Additional curl-free

compatible components, Uke and Ukp, allow satisfying the balanceof equilibrium and boundary conditions. The following relationsare therefore satisfied:

U ¼ Ue þ Up; ð9Þ

Ue ¼ U?e þ Uke; ð10Þ

Up ¼ U?p þ Ukp; ð11Þ

0 ¼ U?e þ U?p ; ð12Þ

curl U?e ¼ �curl U?p ¼ a – 0: ð13Þ

Eq. (13) are satisfied by the incompatible plastic distortion U?p asso-ciated with the presence of Nye’s polar dislocation density tensor a

(Nye, 1953), and by the incompatible elastic distortion U?e ensuringthe continuity of matter implied by Eq. (8). The polar dislocationdensity can be used to represent the core of a single or partial dis-location as well as an ensemble of dislocations with non-zero netBurgers vector. Since Uke and Ukp are curl-free, Eq. (13) can also bewritten as:

curl Ue ¼ �curl Up ¼ a: ð14Þ

By decomposing the elastic and plastic distortions into their sym-metric and skew-symmetric parts, Eq. (14) can alternatively beexpressed as (deWit, 1970; Fressengeas et al., 2011):

curl �e ¼ þaþ jte � trðjeÞI; ð15Þ

curl �p ¼ �aþ jtp � trðjpÞI: ð16Þ

Eqs. (15) and (16) relate the incompatibilities in elastic strain �e andplastic strain �p arising from the presence of Nye’s dislocation den-sity tensor a, and the elastic, je, and plastic curvature, jp. In anelasto-plastic model considering solely dislocations, the elastic,plastic and total, j, curvature tensors are rotation gradients:

je ¼ grad ~xe; ð17Þ

jp ¼ grad ~xp; ð18Þ

grad ~x ¼ j ¼ je þ jp: ð19Þ

Hence, the elastic and plastic curvatures are curl-free and the asso-ciated rotation vectors ð ~xe; ~xpÞ are integrable single-valued quanti-ties. The sum of the elastic and plastic rotation vectors is thematerial rotation vector ~x ¼ 1

2 curl u. As shown by deWit (1970),(je;jp) may not be curl-free anymore, if the possibility ofmulti-valued elastic and plastic rotations ~xe and ~xp, i:e:, a

discontinuity of the elastic and plastic rotations over some surface,is acknowledged. In such a situation, a non-zero tensor h such that

h ¼ �curl jp ¼ curl je ð20Þ

can be defined. h is the polar disclination density tensor (deWit,1970). Similarly to the case of dislocations, the incompatible partof the plastic curvature tensor due to a polar disclination densityis offset by the incompatible part of the elastic curvature tensorin order to ensure the continuity of matter. Since the rotation vec-tors ð ~xe; ~xpÞ are multi-valued in the presence of disclinations, thecorresponding elastic and plastic rotation and distortion tensorsare undefined. The discontinuity of the elastic rotation is theFrank vector. It is obtained by integrating the incompatible elasticcurvatures along a closed circuit C:

X ¼Z

Cje � dr: ð21Þ

Similarly, the discontinuity of the elastic displacement is theBurgers vector. It contains a possible contribution from thenon-uniformity of elastic curvatures and reads (deWit, 1970):

b ¼Z

Cð�e � ðjt

e � rÞtÞ � dr: ð22Þ

If S is the surface of unit normal n bounded by circuit C, usingStokes’s theorem and Eqs. (15) and (20) allows rewriting Eqs. (21)and (22) as:

X ¼Z

Sh � ndS; ð23Þ

b ¼Z

Sða� ðht � rÞtÞ � ndS: ð24Þ

X and b defined above are point-wise measures of the latticeincompatibility in the presence of disclinations and dislocations.In contrast, the disclination and dislocation densities h and adefined in Eqs. (15) and (20) are continuous tensorial renditionsof this incompatibility. They provide a natural regularization ofthe singular and discontinuous nature of the Frank/Burgers vectorsand they are thus appropriate mathematical tools to model the coreof crystal defects in a fully continuous fashion.

3.2. Scale dependence of polar defect densities

Eqs. (21)–(24) reveal the circuit size/resolution length scaledependence of the Burgers and Frank vectors and of the corre-sponding polar defect densities. Given a prescribed ensemble ofcrystal defects, the Burgers/Frank vectors directly depend on thechoice of surface S and on the characteristic size of circuit C. If thissize is chosen in the nm scale or less, every defect (or part of adefect) can be resolved by a polar defect density, and a completedescription of the incompatible elastic strains and curvatures asso-ciated with this defect configuration is obtained. For example, con-sidering as in Fig. 1 a dipole of dislocations/disclinations ofopposite magnitudes, and adopting a small circuit C allows captur-ing the Burgers/Frank vector of each individual defect and describ-ing the dipoles with defect densities of opposite signs. Hence, theincompatibility induced by the dipoles is fully captured by thepolar defect density field. In contrast, if a mesoscale resolution ischosen, such that the circuit encompasses all defects, then thenet polarity of the defect ensemble may reduce or even cancelout. For instance, if one increases circuit C in Fig. 1 to the pointwhere it fully encompasses the dipole, the net Burgers/Frank vec-tors vanish, as well as the corresponding polar dislocation/disloca-tion densities. In this extreme case, the incompatibility induced bythe dipoles is missed by the representation.

Fig. 1. Scale dependence of polar defect densities and plastic flow. Example of the expansion of an edge dislocation dipole (left) and a wedge disclination dipole (right). At asmall nanoscale resolution sketched by the orange Burgers and Frank circuits, each dislocation/disclination can be rendered by a polar dislocation/disclination density. In thatcase, the plastic strain/curvature rate produced by the expansion of a dislocation/disclination dipole with velocity v can be rendered by the mobility of polar dislocation/disclination densities (terms surrounded by orange lines). At a higher mesoscale resolution sketched by the green circuits, the polarities of the dislocation/disclination dipole‘‘statistically’’ offset, resulting in a statistical dislocation/disclination density. In that case, the plastic strain/curvature rate produced by the expansion of the dislocation/disclination dipole cannot be resolved by the mobility of polar dislocation/disclination densities and must be accounted for by a statistical rate term (terms surrounded bygreen lines). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this article.)


3.3. Scale dependent representation of grain boundaries

When modeling symmetric tilt GBs with nanoscale resolution,dipoles of polar disclination densities are used to continuously rep-resent the atomic structural units composing GBs (Fressengeaset al., 2014). As per discussion in the above, this construct doesnot apply at mesoscale resolution, as the net polar disclinationdensity may vanish. From this standpoint, the need for using polardisclination densities may appear as limited at mesoscale.However, even at such a scale, recent estimates from electronback-scattered diffraction (EBSD) orientation maps withsub-micron resolution evidenced the presence of polar disclinationdipoles along GBs (Beausir and Fressengeas, 2013). Clearly, thesedensity fields do not render the individual crystal defects constitut-ing the grain boundary structure at nanoscale, but rather the netpolarity of ensembles of defects resulting from misorientation gra-dients at mesoscale along the GBs.

Motivated by these observations, we propose to follow themethodology already used in EBSD orientation mapping to buildthe fields of incompatibility that will render initial grain bound-aries at mesoscale. Interestingly, this allows employing experimen-tal orientation maps to build initial conditions for the crystal defectdensity fields. Using the Bunge convention, the local lattice orien-tation is characterized by the set of Euler angles ð/1;/;/2Þ and theassociated sample-to-crystal rotation matrix R. The disorientationbetween two points (in EBSD, two pixels) a and b in the body isgiven by the couple ðr;DwÞ, where r is the axis of rotation andDw the disorientation angle around r. In the Cartesian coordinatesystem with the sample reference frame ðe1; e2; e3Þ, the differenceof elastic rigid body rotation vector Dxe between points a and bsatisfies:

Dwr ¼ Dxei ei: ð25Þ

If Ra and Rb denote the rotation matrices at points a and b, the dis-orientation matrix from point a to b can be calculated asDR ¼ R�1

a :Rb. Then, the variation of the elastic rotation Dxei is

obtained from (Pantleon, 2008):

Dxei ¼ �

eijkDRjkDw2sinðDwÞ : ð26Þ

As a final step, the elastic curvature tensor component jeij is approx-

imated by the variation of the elastic rigid body rotation Dxei over a

distance Dxj:

jeij ¼

Dxei

Dxj: ð27Þ

The initial polar disclination density associated with this elastic cur-vature field is then given by Eq. (20) h ¼ curl je. Overlooking thecurl of elastic strain tensor at the initial time and using Eq. (15),the initial polar dislocation density associated with elastic curva-tures is approximately given by:

a ¼ �jte þ trðjeÞI: ð28Þ

Because the incompatible elastic strain in Eq. (15) is unknown fromorientation mapping, a fraction of the dislocation density is clearlyoverlooked. Fig. 2 depicts the geometry of micron-sized bicrystaland tricrystal configurations, together with their initial dislocationand disclination density fields. Fig. 2(a) shows the initial edge dislo-cation density field associated with a tilt boundary in a bicrystal, asobtained from Eqs. (26)–(28). Because periodic boundary conditionsare used along the directions of the GB plane, the polar disclinationdensity field is uniformly zero. Fig. 2(b) and (c) show the initial fieldfor the norm of the polar dislocation and disclination density ten-sors in a tricrystal with three GBs that have each both tilt and twistcomponents, as obtained from Eqs. (20), (26), (27) and (28). Walls ofedge and screw polar dislocations are found in the GBs, whereaswedge and twist polar disclinations are observed at the triple line.Disclinations are also present at the intersection of the GBs andexternal surfaces, in order to terminate GB dislocation walls. Thepolar defect densities obtained in these simple bicrystal and tricrys-tal configurations are in good agreement with those observed inEBSD orientation maps (Beausir and Fressengeas, 2013).

To verify consistency of the predictions, the polar dislocationdensity obtained for the tilt boundary shown in Fig. 2(a) is nowexplained. The edge polar dislocation density a13 obtained followsfrom Eqs. (27) and (28). The jump of lattice orientation Dxe

3 acrossthe GB of normal e1 results in a bending elastic curvatureje

31 ¼ Dxe3=Dx1, which in turns give rise to an edge dislocation den-

sity field a13. Remembering that aij ¼ bitj, where bi is the areal den-sity of Burgers vector along direction ei and tj is the component ofline vector along ej, the component a13 obtained for this tilt bound-ary renders a wall of edge dislocations with straight lines along thetilt axis direction and with Burgers perpendicular to the grainboundary. This is in agreement with the conventional interpreta-tion of tilt boundaries as arrays of edge dislocations. We note thatthe Burgers vector associated with this mesoscale edge polar

Fig. 2. Initial polar defect densities in mesoscale bicrystals and tricrystals, as estimated from crystal lattice orientations. (a) Edge dislocation density a13 (in rad lm�1) in a20 � 5 � 5 lm3 bicrystal with a tilt misorientation of 25� around e3 axis. Periodic boundary conditions are assumed along the GB plane ðe2; e3Þ. (b) Norm of the dislocationdensity tensor (in rad lm�1) in a 15 � 15 � 10 lm3 tricrystal. The orientation of the three grains are randomly generated such that the grain boundaries have both tilt andtwist components. c: norm of the disclination density tensor (in rad lm�2) in the same tricrystal.


dislocation density field is equivalent to that induced by the wedgedisclination dipole structure of this same tilt boundary at nanos-cale. Hence, the vectorial characteristics of the incompatibility isconserved through the length scales.

3.4. Grain boundaries viewed as interfaces

As already mentionned, the mesocale description of the GBscannot account for their core structure because they are viewedas a singular interfaces where the field variables may encounterdiscontinuities. However the jumps of the elastic/plastic strainand cuvature tensor fields across the interface cannot be arbitraryfor the consistency of the continuum theory. As recalled below,tangential continuity conditions on the elastic/plastic strain andcurvature tensor fields must be satisfied in a framework basedon non-singular disclination and dislocation density fields(Acharya, 2007; Fressengeas et al., 2012). Fig. 3 represents a closedcircuit C crossing two grains, separated by a singular GB interface(I) of unit normal n at point P. l and t ¼ n� l are unit vectorsbelonging to the interface, forming an orthonormal basis with n.

Fig. 3. Mathematical setup used to derive tangential continuity conditions atsingular grain boundary interfaces (Acharya, 2007; Fressengeas et al., 2012). Twograins are delimited by a singular GB interface ðIÞ of unit normal n at point P. Theunit vectors l and n � l belong to the GB plane. A closed circuit C crosses both grainsand the GB.

We assume the existence of continuous polar defect densitiesðh;aÞ in both grains and, provisionally, possible surface (singular)defect densities (hSðIÞ;aSðIÞ) lying in the interface. The Frank andBurgers vectors associated with circuit C can be obtained fromEqs. (21) and (22). When collapsing circuit C onto point P in theinterface, Eqs. (21) and (22) become (Fressengeas et al., 2012):

je½ � � l ¼ hSðIÞ � t; ð29Þ

�e � je � r0½ � � l ¼ ðaSðIÞ � hSðIÞ � r0Þ � t; ð30Þ

The above equations can be seen as generalized Frank–Bilby inter-facial relations applied to both dislocations and disclinations(Frank, 1950; Bilby, 1955). Eq. (29) means that any tangential dis-continuity of elastic curvature across the interface can be accommo-dated by a singular surface disclination density hSðIÞ lying in theinterface. Similarly, Eq. (30) means that any tangential discontinuityof the elastic strain across the interface, including a possible contri-bution of elastic curvatures and surface disclinations, can be accom-modated by a singular surface dislocation density aSðIÞ lying in theinterface. When disclinations are overlooked, the latter relationreduces to:

Ue½ � � l ¼ aSðIÞ � t; ð31Þ

which means that any tangential discontinuity of the elastic distor-tion across the interface can be accommodated by a singular surfacedislocation density aSðIÞ lying in the interface (Acharya, 2007). Ifmodeling choices are such that surface defect densities are notallowed, for consistency of the continuous framework, Eqs. (29)and (30) reduce to (Fressengeas et al., 2012):

je½ � � l ¼ 0; ð32Þ

�e � je � r0½ � � l ¼ 0: ð33Þ

Eqs. (32) and (33) imply tangential continuity of the elastic curva-ture and strain tensors across the interface. Similar continuity con-ditions also apply to the plastic curvature and strain tensors, andthe plastic curvature rate and strain rate tensors. Through the limitsimposed to the accommodation of discontinuities, non-locality ofelasto-plasticity is induced across the interface. The characteristic


length scale of such non-locality is extremely small. Nevertheless, ithas a strong impact on the plastic strain rate and rotation rate fieldsin the vicinity of the grain boundaries, and it allows retrieving sizeeffects on plasticity, loading path-dependency and directional hard-ening in particle-reinforced alloys, as well as texture peculiaritiesdue to grain interactions (Mach et al., 2010; Richeton et al., 2011;Puri et al., 2011; Taupin et al., 2012). In the forthcoming simula-tions, we thus propose to investigate further the effects of the inter-facial tangential continuity conditions on the predictions of themesoscale model, in terms of polar defect density distributions atgrain boundaries and triple lines, plasticity, material hardeningand size effects.

3.5. Elasticity

In the absence of body forces, the momentum and moment ofmomentum equations reduce to (Mindlin and Tiersten, 1962;Taupin et al., 2014):

div Tsym þ 12

curl div Mdev ¼ 0: ð34Þ

In the above, Tsym and Mdev denote the symmetric Cauchy stress ten-sor and the deviatoric part of the couple stress tensor, respectively.A specific free energy density function w containing contributionsfrom elastic strains and curvatures is now introduced as follows:

w ¼ wð�e;jeÞ: ð35Þ

For a quadratic expansion of the latter, the elastic constitutive rela-

tions for Tsym and Mdev are derived in the linear form:

Tsym ¼ C : �e þ D : je; ð36Þ

Mdev ¼ A : je þ B : �e; ð37Þ

where the B and D tensors of elastic constants are non-zero only indefected areas (Upadhyay et al., 2013). Hence, they will be consid-ered as null at mesoscale. Homogeneous and isotropic elasticity isassumed for C and the couple stress tensor is taken in the form

Mdev ¼ nlb2jdeve , where l and b denote the shear modulus and

the magnitude of the Burgers vector, respectively. n is anon-dimensional parameter whose value and scale dependence willbe discussed in the following.

3.6. Plasticity and scale dependence

To derive a mesoscale measure of plasticity by dislocation glide,we use a space–time running average of the nano-scale plasticstrain rate tensor:

_�p ¼ ða� VaÞsym; ð38Þ

where Va denotes the velocity of an infinitesimal part of a disloca-tion. With provisional overbars for the mesoscopic averages, weobtain:

_�p ¼ ða� Va þ LpÞsym; ð39Þ

where Lp is defined as

Lp ¼ ða� �aÞ � Va ¼ a� Va � �a� �Va: ð40Þ

Hence, Lp represents a part of the average plastic distortion rateproduced by the nanoscale dislocation density. It can benon-vanishing when �a ¼ 0 and, as such, is to be interpreted as theplastic distortion rate produced by the so-called statistical disloca-tions (Acharya and Roy, 2006). Clearly, the velocity �Va is a space–time average of the point-wise, nanoscale dislocation velocitiesVa. �Va and Lp are terms that require constitutive specification, to

be provided below. Dropping the overbars, the polar dislocationdensities are set into motion by the driving force:

Fa ¼ b� b � ajaj

� �ajaj ; ð41Þ

where b ¼ Tsym:a : X and a ¼ 13 Tkka : X. The dislocation velocity is

then related to the driving force Fa by assuming a mobility lawensuring positiveness of the dissipation. The following power lawform:

Val ¼ va

0TVM

T0

!naFa

l

jFajð42Þ

is often used. In this expression, ðva0; T0Þ are reference velocity and

stress values respectively. TVM denotes the Von Mises stress. Eq.(42) reflects a thermally activated mechanism for high values ofthe exponent na and viscous mechanisms when na � 0. The rela-tionship (41) and (42) is sufficient to ensure incompressibility ofplastic straining, by forbidding out of plane motion of edge disloca-tions, such as climb or cross-slip, whereas the motion of screw dis-locations remains unrestricted (Fressengeas et al., 2011). In crystalplasticity, Lp is the conventional plastic slip rate tensor, whichreads:

Lp ¼X

s

_cps ms � ns ¼

Xs

_cps Ps; ð43Þ

if _cps is the plastic slip rate on slip plane s of slip direction ms and

normal ns. Ps is the orientation Schmid tensor. Plastic slip on systems is activated by the resolved shear stress ss ¼ T : Ps through therelation:

_cps ¼ _cp

0ðjss=s0jÞnasignðssÞ; ð44Þ

where _cp0; s0 and n are the reference plastic strain rate, shear stress

and power law exponent, respectively. Well established standardcrystal plasticity representations are therefore included in the pre-sent mesoscale model.

In a similar manner, using space–time running averages of thenanoscale disclination density tensor, disclination velocity Vh andplastic curvature rate tensor _jp, allows writing the mesoscopicplastic curvature rate as:

_jp ¼ h� Vh ¼ h� Vh þ _jp: ð45Þ

The mesoscopic plastic curvature rate _jp may be non-zero when thenet polarity of a disclination ensemble vanishes at mesoscale:ðh ¼ 0Þ, as for instance in the large circuit shown in Fig. 1(b). In suchas case, it becomes:

_jp ¼ h� Vh ¼ _jp: ð46Þ

Dropping the overbars, it will be written as:

_jp ¼ h� Vh þ _jp: ð47Þ

We refer to _jp as the statistical plastic curvature rate. Like Vh, itrequires constitutive specification, to be provided below and inthe next section. Eq. (47) accounts for plasticity processes in rela-tion with the mobility of disclinations, which cannot be describedby a pure dislocation-based model. Such processes include therelaxation of elastic curvatures by grain boundary misorientationvariation following the motion of GB disclination densities andthe concurrent nucleation of GB dislocation densities. Note herethat a plastic curvature rate similar to _jp was proposed byRomanov and Vladimirov (1992) to account for the relaxation ofcouple stresses by GB rotation.

The motion of polar disclination densities is driven by aPeach-Köhler-type force Fh, which writes (Fressengeas et al., 2011):


Fh ¼Mt � h : X; Fhl ¼ ejklMijhik: ð48Þ

We propose that the disclination velocity be related to this drivingforce by the following power law relationship:

Vhl ¼ vh

0MVM

M0

!nhFh

l

jFhj; ð49Þ

in analogy with Eq. (42). In the above expression, (vh0;M0) are refer-

ence velocity and couple stress values, respectively. MVM denotesthe Von Mises couple stress. The exponent nh reflects thermallyactivated mechanisms for high values and viscous mechanisms fornh � 0. It is not necessarily equal to na.

When considering statistical plastic strain and curvature ratesat mesoscale, the transport equations for polar defect densities(Fressengeas et al., 2011) read:

_h ¼ �curlðh� Vh þ _jpÞ; ð50Þ

_a ¼ �curlða� Va þ LpÞsym þ _jtp � trð _jpÞI: ð51Þ

The above equations are of the form:

_h ¼ �curlðh� VhÞ þ sjh ; ð52Þ

_a ¼ �curlða� VaÞsym þ s�a þ sja : ð53Þ

where the disclination and dislocation sink/source terms sjh and

(sja ; s

�a) are evidenced. Incompatibility of the plastic curvature rate

at grain boundaries and triple nodes is expected to generate polardisclination densities h through the term sj

h in Eq. (52).Interestingly, this disclination sink/source term is not present inthe nanoscale version of the model (FDDM), where only a disloca-tion sink/source term arises from the motion of the existing discli-nations. Concurrently, plastic curvature rates and incompatibility inthe plastic slip rate tensor generate polar dislocation densities athrough the terms sj

a and s�a in Eq. (53), respectively. The terms sja

and sjh provide new disclination-mediated plastic relaxation mech-

anisms for grain boundaries, such as dislocation nucleation orabsorption (Tschopp et al., 2007; Van Swygenhoven et al., 2006),emergence of disclination lines at triple/quadruple junctions(Rösner et al., 2011), that the pure dislocation version of the modelcan not render. The predictive capability brought by such sink/-source terms will be investigated in the next section.

4. Mesoscale plastic curvature rate, scale dependence anddisclination-mediated GB mechanisms

4.1. Expression for the statistical plastic curvature rate

In this subsection, we aim at deriving an expression of the sta-tistical plastic curvature rate _jp as the average over a mesoscopicvolume element of the nanoscale polarized plastic curvature rate_jp ¼ h� Vh (see Eq. (46)). In component form, the latter writes:

_jpij ¼

1Bh ejklepqlMrphrqhik; ð54Þ

where, using Eq. (49), Bh denotes the drag coefficient:

Bh ¼ jFhj

vh0

M0

MVM

� �nh

: ð55Þ

Some manipulations allow rewriting Eq. (54) as:

_jpij ¼

1Bh hikðht

klMlj �MtklhljÞ; _jp ¼ 1

Bh h � ðht �M� h �MtÞ: ð56Þ

Now, breaking hij ¼ Xitj into the Frank vectors around axis ei perunit surface, Xi, and the component of disclination line along direc-tion ej; tj, Eq. (56) can be written in the form:

_jp ¼ Ph : M; Phijkl ¼

1Bh XiXkðdjl � tjtlÞ: ð57Þ

In Eq. (57), Ph contains information on the crystallography of grainboundaries in terms of disclinations, as well as on mobility of thelatter. Such information is important as it directly controls the capa-bility of grain boundaries to relax elastic curvatures. Future work onthis model includes determination of Ph as a function of grainboundary nature, material, temperature etc. Using Eq. (55) for thedrag coefficient Bh, we obtain the statistical plastic curvature rateas:

_jpij ¼

1V

ZV

vh0

jFhjMVM

M0

!nh

XiXkðdjl � tjtlÞMkldV : ð58Þ

Assuming V to be large enough to contain a statistical distributionof the Frank vector and disclination line orientations, we reduce thisexpression to:

_jpij ¼

1V

ZV

vh0

jFhjMVM

M0

!nh

g2X2dikdjlMkldV ; ð59Þ

where X is a characteristic Frank vector and g a non-dimensionalparameter, and finally obtain:

_jpij ¼

1V

ZV

_jp0ðM

VM=M0Þnh

ðMdevij =MVMÞdV : ð60Þ

For the sake of simplicity, _jp0 ¼ g2X2vh

0MVM=jFhj ¼ qXvh0 is taken as

a constant. qX represents a statistical disclination density in unitsof a characteristic Frank vector times a length of disclination lineper unit volume (rad m�2). Note the similarity of this relationshipwith the conventional Orowan’s law for the plastic strain rateresulting from a statistically distributed ensemble of dislocations.Invoking the mean value theorem, the mesoscale plastic curvaturerate is finally approximated as follows:

_jp ¼ _jp0ðM

VM=M0Þnh

ðMdev=MVMÞ: ð61Þ

Identification of constitutive parameters and scale dependences inthe statistical plastic curvature rate (61) is performed in the nextsubsection.

4.2. Scale dependence of curvatures and constitutive parameters

In the presence of a discontinuity of the elastic/plastic rotationat a GB interface, the elastic/plastic curvatures across the interfaceare scale dependent, meaning that their values depend on thechoice of the resolution length scale. Clearly, they are inverselyproportional to the resolution length scale. Hence, the questionarises as to what should be the scale dependence of the constitu-tive parameters to ensure the conservation through the resolutionlength scales of the GB energy and dissipated energy in aGB-mediated plasticity process. Put differently, we look for scalinglaws such that the GB elastic energy and GB-mediated plasticitymechanisms modeled smoothly at nanoscale via fields of polardisclination densities, be rendered consistently at mesoscale whenusing sharp interfaces and curvatures defined in an averaged sense.To illustrate this point, we simulate the bending of a bicrystal con-taining a symmetric tilt boundary with various resolution lengthscales, as shown in Fig. 4. The configuration of the bicrystal issketched at the top of the figure. The tilt axis is perpendicular tothe figure and the tilt angle is indicated by two dashed lines. A typ-ical high angle h001iR5ð310Þ tilt boundary of misorientation 36:9�

is considered. The mechanical behavior of each crystal is taken aspurely elastic. In addition the crystals are dislocation free, so thatany plastic distortion of the bicrystal is produced via GB mechan-ims. A constant and uniform bending curvature is then applied to

Fig. 4. Bending of a bicrystal containing a symmetric tilt GB. The configuration of the bicrystal is shown on top. The two crystals are supposed purely elastic and do notcontain any defect. The tilt boundary rotation axis is perpendicular to the figure and the GB is represented by its atomic structure featuring structural units. The GBmisorientation is indicated by the two dashed lines. A constant and uniform bending curvature is applied to the bicrystal, such that the left and right external surfaces are at a10� angle. Results from simulations of this configuration at nanoscale with FDDM (Case 1a) and FDM (Case 2a) and at mesoscopic scale with PMFDDM (Case 1b) and PMFDM(Case 2b) are sketched below. See the main text for explanations.


the bicrystal, such that the left and right external surfaces are at anangle of 10�.

We first perform a reference simulation at nanoscale (Case 1a),as sketched in Fig. 4(1a). In this simulation, FDDM is used. The sim-ulation parameters are listed in Table 1. The resolution lengthscale, i.e. the mesh size in the finite element simulation, isl0 ¼ 0:1 nm, a value taken as reference. The structural units com-posing the tilt boundary at nanoscale are modeled smoothly viadipoles of wedge disclination densities, as done in previous work(Fressengeas et al., 2014). The corresponding initial energy of theGB (excess energy per unit length of GB) is Egbðl0Þ ¼ 1 J=m2. Thisvalue will be used as a reference in the following. The followingmechanism, referred to as grain boundary misorientation variationthroughout the present paper, is predicted by the model. Due tothe internal elastic fields of the GB disclinations and the imposedbending, strong elastic curvatures exist in thedisclination-populated areas along the GB. Their conjugate couplestresses drive the motion of disclinations, as implied by Eqs. (48)and (49). Negative disclinations move up along the boundaryplane. From Eqs. (47) and (53), such disclination motion generatespositive plastic curvatures and leaves positive polar GB edge dislo-cations in their wake. The Burgers vector of the generated edge dis-location densities is perpendicular to the GB plane and their line is

Table 1Numerical constants used and measured in the reference bending simulations withFDDM.

nh;a _jp0ðl0Þ je

0ðl0Þ T0 va0

10 109 rad m�1 s�1 109 rad m�1 1 GPa 0:1 nm=s

vh0 E m nðl0Þ l0

0:1 nm=s 62:780 GPa 0.3647 10�10 0:1 nm

normal to the figure. As a consequence of the generated plastic cur-vatures and associated dislocation densities, the strong elastic cur-vatures in the GB area are relaxed and the GB misorientationdecreases by a 10� angle, as suggested by the dashed lines in thefigure. Hence, this process can be described as a GB misorientationvariation mediated by GB disclination motion and accompanied byGB dislocation nucleation. Thus, in contrast with previous genericconsiderations on disclinations (Kleman and Friedel, 2008), thecrystal is not seen as ‘‘suffering’’ from the motion of disclinations,but rather as being relaxed in this dissipative process. Moreover,the so-called relaxation dislocations (Kleman and Friedel, 2008)produced by the motion of disclinations do not need to move awayfrom the GB. The energy dissipated per unit GB length in reducingthe misorientation is calculated as:

Ed ¼1

LGB

Zt

ZV

T : _�p þM : _jp� �

dV� �

dt: ð62Þ

It is found to be Edðl0Þ � 0:2 J=m2, which clearly amounts to 20% ofthe initial GB energy. This value will also be used as a reference forthe energy dissipated by the change of misorientation of the tiltboundary in what follows. Other reference parameters to be identi-fied from this simulation are M0ðl0Þ and _jp

0ðl0Þ. _jp0ðl0Þ is estimated

from the plastic curvature variation due to disclination mobility,divided by the time during which bending is applied. It is foundto be _jp

0ðl0Þ ¼ 109 rad m�1 s�1. The reference couple stress is of the

form M0ðl0Þ ¼ nðl0Þlb2je0ðl0Þ, where nðl0Þ is the value of the couple

stress coefficient in the elastic law Mdev ¼ nlb2jdeve at resolution

length scale l0. As suggested above, this coefficient must be scaledependent. Its reference value will be nðl0Þ ¼ 10�10 as furtherdetailed below. je

0ðl0Þ reflects a characteristic elastic misorientationover the reference resolution length scale, estimated from the initial

Fig. 5. Log–log plot as a function of resolution length scale l of couple stresselasticity coefficient n, reference plastic curvature rate _jp

0, energy dissipated duringGB misorientation change Ed , reference elastic curvature je

0 and initial GB energyEgb . Results are normalized by values at reference resolution length scale l0.


values of the elastic curvatures within the boundary area. The valueje

0ðl0Þ ¼ 109 rad=m is retained.We now proceed with simulations at larger resolution length

scales l, from the reference scale l0 up to micron size (Case 1bshown in Fig. 4(1b)). In these simulations, the mesoscopic exten-sion of the FDDM model (to be referred to as PMFDDM:Phenomenological Mesoscale FDDM) is used to model the samebending experiment. As already indicated, we aim at producingresults that are consistent with those obtained in Case 1a, by usingthe mesoscopic plastic curvature rate _jp. The objective is to cali-brate the constitutive parameters such that the mesoscopic modelquantitatively retrieves the reference GB energy, as well as themisorientation change and the reference energy dissipated in theprocess. The characteristic resolution length scale (i.e. the meshsize) used in the finite element simulations is denoted by l. Itwas varied from 0.1 nm up to 1 lm. As aforementioned, the initialGB is not rendered by smooth wedge disclinations in these mesos-cale simulations, but by a polar edge dislocation density, as impliedby Eq. (31). As shown in Fig. 4, the initial tilt boundary supports auniform wall of negative edge dislocation density with Burgersvector normal to the boundary plane. At nanoscale, where the corestructure of GBs is fully captured by wedge disclination dipoles, theinitial GB energy is due almost exclusively to elastic strains(Fressengeas et al., 2014). At mesoscale however, where the corestructure is overlooked, the initial energy Egb of the GB arises solelyfrom the elastic curvatures and their couple stress conjugates. Forthe smallest resolution length scale tested ðl ¼ 0:1nm ¼ l0Þ, we cal-ibrate the coefficient n ¼ nðl0Þ entering the couple stress elasticitylaw to ensure that the GB energy equals the reference valueobtained with FDDM in Case 1a. The value nðl0Þ ¼ 10�10 is found.Now, we increase l to find the scale dependence of nðlÞ such thatthe GB energy remains scale independent. We find that nðlÞ andthus the couple stresses should follow the quadratic dependence:

nðlÞ ¼ nðl0Þll0

� �2

; MdevðlÞ ¼ nðl0Þll0

� �2

lb2jdeve : ð63Þ

As suggested above, this result derives from the scale dependence ofthe elastic curvatures. Since the jump in elastic/plastic rotationacross the interface is invariant when the resolution length scale lis changed, je

0ðlÞ and _jp0ðlÞ follow a 1=l scaling law:

je0ðlÞ ¼ je

0ðl0Þl0

l; _jp

0ðlÞ ¼ _jp0ðl0Þ

l0

l: ð64Þ

With the identified constitutive parameters and scaling laws, bend-ing simulations are now performed by using the mesoscale modelPMFDDM. Due to the applied couple stresses, the statistical plasticcurvature rate _jp generates uniform positive plastic curvatures andpositive edge dislocation densities in the interface. Therefore theinitial negative dislocation density decreases, as sketched in Fig. 4.Hence, similarly to Case 1a, the GB misorientation decreases, actu-ally by an amount of 6�, in rather good agreement with the refer-ence simulation using nanoscale smooth FDDM in Case 1a. Inaddition, the energy dissipated during this plastic relaxation pro-cess is found to be Ed ¼ 0:125 J=m2, also close to the reference valueof Case 1a. Simulations at different scale resolutions l show that thedissipated energy is scale independent. All the above results onscale-dependence in the mesoscale bending simulations withPMFDDM are summarized in Fig. 5.

4.3. Disclination vs. dislocation-mediated plasticity at grainboundaries

In this Section, the intent is to show that the pure dislocationversion of our model (no disclinations) cannot render mechanisms

such as the GB misorientation variation presented in the abovesubsection, in order to make clear that the mesoscale term _jpeffectively adds new plasticity mechanisms to more conventionalmodels relying only on dislocation glide. To this end, two differentsimulations were performed, which are sketched in Fig. 4(2a) and(2b). In the Case 2a, FDM (Field Dislocation Mechanics) was used tomodel the tilt boundary at nanoscale with resolution lengthl ¼ l0 ¼ 0:1 nm. In this case, each atomic structural unit is modeledby a negative edge polar dislocation density spot, with Burgers vec-tor perpendicular to the GB plane and line vector parallel to the tiltaxis. If climb is allowed, the edge dislocation can move up or downalong the GB, depending on whether they are located in a region oftension or compression. However, such motion does not relax thebending curvatures and the GB misorientation is quasi-invariant.In Case 2b, the tilt boundary is modeled at micron scaleðl ¼ 1 lmÞ using PMFDM. Here, as in PMFDDM simulations, the ini-tial GB is composed of a uniform wall of negative edge polar dislo-cation density. Transport of the polar dislocation density by climbis also possible, as in Case 1a, but no plastic curvature is producedand the GB misorientation does not change. To summarize, oursimulations clearly show that disclination motion provides uncon-ventional plastic relaxation mechanisms at GBs, such as GB misori-entation variation accompanied by GB dislocation nucleation, thatcannot be rendered by dislocation models. Further, these mecha-nisms can be described by both nanoscale and mesoscaledisclination-based models.

5. Mesoscale simulations of bicrystals and tricrystals

5.1. Simulation details

In this section, illustrative applications of the PMFDDMapproach are presented. Aluminum bi- and tri-crystals are usedas supporting configurations. The intent is to probe the impact ofthe tangential continuity conditions on elastic/plastic strain/curva-ture and strain/curvature rates at the interfaces on the mechanicalresponse of these structures. A particular focus will be on the influ-ence of these tangential continuity conditions on the sample sizedependence of the mechanical response. Further, the coupled con-tributions of polar and statistical defect densities to bulk vs. GBmediated plasticity will be examined, together with the role of tri-ple lines and temperature.

To this end, the field equations of the mesoscale model pre-sented in Sections 3 and 4 are solved approximately by using the


finite element method. The open source software FreeFem++ isused (Hecht et al., 2014). The tangential continuity conditions(32) and (33) are imposed by spreading the GB interface over afinite element (with small but non-zero width), which regularizesthe discontinuity. Adding a penalty term to the transport Eqs. (50)and (51) allows ensuring that the nucleation of polar defect densi-ties corresponding to tangential discontinuity does not occur.Then, by correcting the incompatible elastic–plastic strains andcurvatures from Eqs. (15) and (20), tangential discontinuity iseffectively removed and relations (32) and (33) are satisfied. Thematerial constants representative of pure aluminum are used. Forthe sake of simplicity, no work hardening is implemented.Parameters used for the dislocation crystal plasticity formulationare _cp

0 ¼ 10�7 s�1 and s0 ¼ 10 MPa. Other parameters are alreadylisted in Table 1.

5.2. Non-local impact of tangential continuity on plasticity

Fig. 6(a) and (b) displays the polar dislocation density and ten-sile plastic strain generated during tensile deformation of thetricrystal shown in Fig. 2 when the tangential continuity conditionsare not imposed. A 1% tensile strain �33 is applied along the direc-tion of the triple line, with constant quasi-static applied strain rate.The tensile plastic strain exhibits discontinuities across the GBs. Itis quasi-uniform within the grains, and there does not seem to beany grain-to-grain elasto-plastic interaction across the GBs. As aconsequence of tangential discontinuity of the plastic straining,the polar dislocation density that is generated is localized in theGB interfaces, as can be seen in Fig. 6(a). Conversely, Fig. 6(c) and(d) shows the results predicted for the same tricrystal when thetangential continuity conditions are imposed. Continuity of thetensile plastic strain, grain-to-grain elasto-plastic interactionsand a re-distribution of the plastic strain rate between grains arenow seen to occur across the GBs. The grain where the plasticstrain was the lowest in the absence of the continuity conditions,now has the largest plastic strain. As shown in Fig. 6(c), this isdue to polar dislocation densities piling up at GBs in the neighbor-ing grains, which produces internal stresses leading to intergranu-lar transfer of plastic deformation. Note in Fig. 6(c) theconcomitant absence of interface dislocations. Fig. 7 shows themacroscopic tension–compression curves corresponding to thedeformation of the tricrystal, with and without tangential

Fig. 6. Tensile deformation of the 15 � 15 � 10 lm3 tricrystal shown in Fig. 2, withoutalong the triple line direction and the applied strain rate is 10�5 s�1. (a), (c) Slice showindeformation. (b), (d) Slice showing the tensile plastic strain at the end of loading.

continuity conditions. The accumulation of polar dislocations atgrain boundaries and the grain-to-grain interactions predictedwhen the tangential continuity conditions are activated lead tomore pronounced Bauschinger effect, and particularly to transientelasto-plastic behavior during load reversal.

Additionnally, the polar disclination densities generated duringplastic deformation of the tricrystal, with and without tangentialcontinuity conditions, are shown in Fig. 8. Without tangential con-tinuity, most of the disclination density is localized within the tri-ple line. When continuity conditions are imposed, the disclinationdensity is not restricted anymore to the triple line, but rather accu-mulated along the triple line and the GBs. As for dislocation densi-ties, tangential continuity of the elastic curvatures tends to removeinterface disclinations, in favor of more heterogeneous distribu-tions within grains, near triple line and GBs. It also leads tograin-to-grain interactions of elastic curvatures. Such non-localbehavior is expected to generate considerable kinematic hardeningin situations involving strong elastic curvatures, such as high accu-mulation of dislocations at grain boundaries at large strains, andbending or torsion solicitations. One should again note that thenucleation of polar disclination densities within grains and alonginterfaces stems from the presence of the statistical plastic curva-ture rate introduced in the above section. The latter is also a sourceor sink of dislocations in Eq. (51). As aforementioned, a careful val-idation of this statistical term is needed in future work to quantifyits impact on the deformation of polycrystals.

5.3. Long-range and size effects induced by tangential continuity

Fig. 9 now illustrates the effects of sample and grain sizeinduced by the tangential continuity conditions on the distributionof plastic strain in a bicrystal and the mechanical behavior of thelatter. In a first step, a reference bicrystal composed of two grainsof width H ¼ 5 lm and length L ¼ 5:5 lm with a symetric tiltboundary of misorientation 10�, is deformed in simple tension toa strain of 10% along the e3 axis. The prefactor _cp

0 in the plastic sliprate is divided by a factor 10 in the right hand side grain in order tomimic lower dislocation activity in that grain, and to promote dif-ferences in the plastic strain across the GB. Fig. 9(a) depicts the ten-sile plastic strain field �p

33 by the end of the tension process iftangential continuity of the plastic strain and curvature tensors isnot required. �p

33 is seen to be uniform within the grains, while a

(a), (b) and with (c), (d) tangential continuity conditions. The applied strain is 0.01g the norm of the dislocation density tensor (in rad lm�1) generated during plastic

Fig. 7. Tension–compression macroscopic stress–strain curves for the tricrystal shown in Fig. 6. Tension and compression are applied along the triple line direction. (a) Curvesobtained with na ¼ nh ¼ 10, with (red line) and without (blue line) interface tangential continuity conditions. (b) Curves obtained with na ¼ nh ¼ 2, with (purple line) andwithout (green line) interface tangential continuity conditions. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version ofthis article.)

Fig. 8. Compression of a tricrystal, without (a) and with (b) interface tangential continuity conditions. The applied strain is �10% along the triple line direction and theapplied strain rate is �10�5 s�1. (a), (b) Color contours showing the norm of the disclination density tensor (in rad lm�2) generated during plastic deformation. The norm ofthe dislocation density tensor generated along the process is shown in gray scale (in rad lm�1).


discontinuity in �p33 shows up across the GB. The configuration of

the bicrystal is actually such that tangential continuity of the plas-tic strain at the interface requires that �p

33 be continuous across theinterface. This is indeed verified in Fig. 9(b), obtained from calcula-tions where tangential continuity is implemented. In addition,polar dislocation pile-ups build-up at the GB, and slip transferacross GB takes place, similar to that previously shown in Fig. 6.The width of the bicrystal is now doubled from H ¼ 5 lm to2H ¼ 10 lm, while its length L ¼ 5:5 lm is kept constant. Theresulting tensile plastic strain field is shown in Fig. 9(c), with tan-gential continuity still implemented. As compared to Fig. 9(b), it isseen that the heterogeneity of the plastic strain is enhanced,because the amount of GB area is increased. The role of the dislo-cation pile-ups in slip transfer is also promoted. As a result, moreplastic strain is produced in the hard grain on the right. Finally,while maintaining the width of the grains equal to 2H ¼ 10 lmand keeping tangential continuity in place, their length is nowincreased to 2L ¼ 11 lm. The resulting plastic strain field is shownin Fig. 9(d). As compared to Fig. 9(c), it is seen that the effect of sliptransfer progressively decreases as one moves away from the GB,while the influence of the boundary conditions on this regiondecreases. This clearly shows the non-local and long-range charac-ter of the effects of tangential continuity on the plastic strain field.The impact of the above changes in size on the stress–strain curvesof the bicrystal are shown in Fig. 9(e). For the grains with dimen-sion 2H ¼ 10 lm and L ¼ 5:5 lm, slip transfer is effective in almostall parts of the hard grain, including the vicinity of the externalboundaries. Hence, the Bauschinger effect and transient elastoplas-tic transition during strain reversal are increased. For the grains

with dimension 2H ¼ 10 lm and 2L ¼ 11 lm, slip transfer is lesseffective away from the GB. Hence, the Bauschinger effect andtransient elastoplastic transition during strain reversal are less pro-nounced. Importantly, it is to be noted that changing the dimen-sions of the bicrystal does not have any effect on the plasticstrain field and the overall mechanical response when the tangen-tial continuity conditions are not imposed.

5.4. GB vs. dislocation mediated plasticity

We now look for the predictive capabilities of the model interms of dislocation vs. GB mediated plasticity. Fig. 10 shows theshear deformation of a 20 � 1 � 1 lm3 bicrystal with a 25� h001isymmetric tilt boundary supporting the appropriate edge disloca-tion density field. Dislocation glide within grains is possibly pre-vented by arbitrarily setting _cp

0 ¼ 0 in the plastic slip rate tensor.Periodic boundary conditions are assumed along the two directionsdefining the GB plane. Simple shear with constant quasi-staticapplied strain rate is imposed on the external surfaces parallel tothe GB plane. Unsurprisingly, the imposed shear is accommodatedby intragranular plastic shear when _cp

0 – 0, as can be seen inFig. 10(b), where the residual strain after loading removal isshown. If instead intragranular dislocation glide is not permitted:_cp

0 ¼ 0, the bicrystal behaves elastically until the applied shearstress reaches a critical value where transport of the edge disloca-tion density decorating the tilt boundary takes place. Accordingly,Fig. 10(c) depicts the occurrence of shear-coupled boundary migra-tion. Further, Fig. 11 shows the predicted shear-coupling factor

Fig. 9. Grain size effects during tension–compression of a bicrystal. (a) Referencebicrystal of dimension H ¼ 5 lm and L ¼ 5:5 lm, deformed to a 10% strain in simpletension along e3 axis (green z axis in the figure), without tangential continuityconditions. The applied strain rate is 10�5 s�1. The misorientation of the symetrictilt GB is 10� around the e3 axis. The prefactor _cp

0 ¼ 0 in the plastic slip rate tensor isdivided by a factor 10 in the right grain in order to mimic lower dislocation activityin that grain. (b) Same bicrystal than that shown in a, but tangential continuityconditions are imposed. (c) Same bicrystal than that shown in a, but its width isincreased from H ¼ 5 lm to 2H ¼ 10 lm and tangential continuity conditions areimposed. (d) Same bicrystal than that shown in a, but its width is increased fromH ¼ 5 lm to 2H ¼ 10 lm, the length of grains are increased from L ¼ 5:5 lm to2L ¼ 11 lm and tangential continuity conditions are imposed. In (a,b,c,d), thetensile plastic strain is color-coded. (e) Corresponding stress/strain curves of thebicrystal. Without continuity conditions, changing H and/or L yields the samemacroscopic curve without any size effect. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Shear deformation of a 20 � 1 � 1 lm3 bicrystal with tilt misorientation of25� , the tilt axis e3 is normal to the figure. Periodic boundary conditions areassumed along the GB plane ðe1; e3Þ. (a) Initial edge dislocation density a13 (inrad lm�1) composing the tilt GB. (b) Edge dislocation density a13 (in rad lm�1) inthe same bicrystal deformed by a 15% shear strain. (c) The two crystals are madepurely elastic by setting _cp

0 ¼ 0 in the plastic slip rate. In that case, the 15% appliedshear strain is accommodated by the motion of the edge dislocation density a13 (inrad lm�1), leading to shear-coupled boundary migration. In (b), (c), the deformedshape after removal of the applied shear strain is shown to evidence the residualshear strain in the bicrystal.


when the tilt angle is varied by appropriately choosing the initialGB dislocation density. The experimentally observed positive andnegative branches (Gorkaya et al., 2009) are retrieved. The switchbetween the branches is predicted at a 45� tilt, due to cubic sym-metry reasons, whereas the actual value is 37�, as predicted bythe nanoscale FDDM model (Taupin et al., 2014) and observed inexperiments (Gorkaya et al., 2009). Despite their mesoscale charac-ter, which prevents from recovering fine scale features, theseresults compare favorably with the experimental values. The pre-dicted critical shear stress for boundary migration is also shownin Fig. 11 as a function of the misorientation. It is seen to be highat low angles and conversely low at high angles, consistent withthe elastic energy and dislocation content of these boundaries.Indeed, if a constant critical driving force is assumed at all misori-entations, Eqs. (41) and (42) show that the higher the GB disloca-tion density is taken, the lower the shear stress becomes.

5.5. Effect of triple lines and temperature on GB-mediated plasticity

Finally, Fig. 12 shows GB-mediated plasticity in a tricrystal con-figuration. Again, intragranular slip is prevented by setting _cp

0 ¼ 0in the plastic slip rate. Periodic boundary conditions are imposedalong the triple line direction. The GBs are of tilt/twist characterand simple shear is applied on the external surfaces, both alongand normal to the triple line direction, in order to activate glideof both the edge and screw polar dislocation densities present inthe GBs. The applied strain rates are constant and quasi-static.An initial edge dislocation component of the boundaries is shownin Fig. 12(a). Because of the mixed character of the GBs and thepresence of a triple line and free surfaces close to the GBs, thedefect dynamics is much more complex than in the bicrystalshown in Fig. 10(c). In particular, it can be seen in Fig. 12(b) thatboundary migration is hindered by the presence of the triple line.As a result the applied shear is accommodated by local variationsin GB misorientation associated with the nucleation of GB edgeand screw dislocations and with their eventual emission in graininteriors. For instance, one of the GBs in Fig. 12(b), essentially of tiltcharacter, encounters a 60% decrease of its tilt angle (a reduction ofabout 15�), while dislocation emission into a neighbor grain inte-rior is clearly in evidence. The same tricrystal configuration is

Fig. 11. Shear-coupling factor of h001i tilt boundaries, as predicted by themesoscale model. Diamonds show the predicted shear-coupling factors, formisorientations ranging from 0 up to 90� . Two positive and negative branches areobtained with a switch at 45� due to cubic symmetry. The two dotted linesrepresent the two branches obtained experimentally with a switch at 37� (Gorkayaet al., 2009). Squares show the predicted critical shear stress (MPa) for the onset ofboundary migration, as a function of misorientation.

Fig. 12. Deformation of a 15 � 15 � 1 lm3 tricrystal with three grain boundaries that have each both tilt and twist components. The three grains are made purely elastic bysetting _cp

0 ¼ 0 in the plastic slip rate tensor. Periodic boundary conditions are assumed along the triple line direction. (a) Initial edge dislocation density a13 (in rad lm�1)decorating part of the grain boundaries. (b) Evolution of the dislocation density a13 (in rad lm�1) when a 15% shear strain is applied normal and along the triple line direction,with na ¼ nh ¼ 10. The triple line and grain boundaries are almost motionless. The upper GB has accommodated deformation by emitting edge dislocations to its right and byconsequently decreasing its tilt misorientation by 60% locally. (c) Evolution of the dislocation density a13 (in rad lm�1) when a 15% shear strain is applied normal and alongthe triple line direction, with na ¼ nh ¼ 1. A 15% tensile strain is also applied normal to the triple line to promote out-of-plane motion of polar edge dislocations. Climb andglide of edge dislocations are treated on an equal footing. In that case, the triple line and grain boundaries migrate to accommodate deformation.


looked at in Fig. 12(c) when additional tension is applied along adirection normal to the triple line, in order to favor climb of edgedislocations. In addition, the power law exponents are set tona ¼ nh ¼ 1 in the mobility laws (42) and (49), in order to mimicviscous motion of defects at high temperature. Climb of edge dislo-cations is also allowed. As can be seen in Fig. 12(c), GB migration issubstituted to the grain boundary misorientation variation mecha-nism and to dislocation nucleation and emission. To summarize,these simulations suggest that viscosity and dislocation climb tendto favor GB migration, whereas GB misorientation variations anddislocation nucleation and emission tend to prevail when thestrength of the obstacles increases.

6. Conclusions

In the present paper, a mesoscale extension of a previouslydevelopped elasto-plastic theory of dislocation and disclinationfields has been introduced. The approach is useful in polycrys-talline materials where plasticity by dislocation glide is limited,either because the average grain size becomes too small, as inultrafine-grained materials, or because the material does not haveenough independent slip systems, as in orthorhombic olivine.Motivated by recent experimental results (Beausir andFressengeas, 2013), where mesoscale disclination dipoles werefound to decorate GBs and triple lines, the tensorial densities ofcrystal defects (dislocations and disclinations) are used to rendercontinuously the initial structure of the GBs and triple lines.Their transport provides the framework for the dynamics of theGBs and triple lines. Consistency of the latter in thecoarse-graining process requires introducing a statistical plasticcurvature rate tensor, whose role is to reproduce GB-mediatedplasticity at mesoscale. Appended with tangential continuity con-ditions on the plastic strain and curvature rate tensors at GBs,the theory offers a nonlocal description of the plasticity of poly-crystals. Nonlocality arises at GBs and triple lines, and haslong-range character. Predicted consequences of non-localityinclude the occurrence of pile-ups of dislocation densities at GBsand activation of slip across GBs, interactions between neighborgrains, sample and grain size effects on mechanical behavior, ani-sotropy of hardening induced by the loading process (‘‘kinematic’’

or directional hardening), GB migration and misorientation varia-tion accompanied with dislocation nucleation and emission,etc.Standard crystal plasticity is included in the theory as a limitingcase, when the crystal defect density fields and the statistical plas-tic curvature rate vanish and when the tangential continuity con-ditions on the plastic strain and curvature rate tensors areoverlooked at GBs. The above nonlocal features are naturallyobtained by the theory, without introducing any adhoc phe-nomenological term. As a challenge for future work, the approachhas in particular the capacity for predicting the influence of grainsize on the work hardening of polycrystals, as a function of temper-ature and strain rate.

Acknowledgements

The authors benefited of financial support from the ANR(Agence Nationale de la Recherche) under GrantANR-11-JS09-007-01, NanoMec. The authors would like to thankA. J. Beaudoin, S. Forest and R. Lebensohn for helpful discussions.

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