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the prediction of microstructural evolution in response to thermo - mechanical loading is important for materials design , processing or thermomechanical fatigue phenomena . computational modeling of evolving texture in response to large plastic deformation and recrystallization has been studied extensively  @xcite but less so than that produced by thermally - induced stresses  i.e . stress - induced texture evolution  @xcite . we consider a thermo - mechanical setting in which temperature changes cause stresses to develop due to geometrical constraints . the temperature is sufficiently high to generate grain boundary motion and yet low enough such that recrystallization does not occur . the induced stresses may be associated with both elastic and plastic deformation  @xcite .    in a previous work , a hybrid monte carlo ( hmc ) approach  @xcite was developed by combining a mc algorithm for grain boundary motion  @xcite with the material point method ( mpm )  @xcite for elastic deformation . purely elastic driving forces , originating from the anisotropic mechanical response of individual grains , are treated as a bulk body force in a potts model for grain boundary evolution  @xcite . the approach is time accurate through the use of parametric links  @xcite to sharp - interface ( si ) kinetics  @xcite . it also takes advantage of the fact that mc grain boundary mobility is independent of the driving force  @xcite . the present work extends this paradigm to include the influence of inelastic deformation on texture evolution  @xcite . as in the elastic study  @xcite , texture evolution is assumed to be dominated by grain boundary kinetics  @xcite . furthermore , we consider infinitesimal deformation to distinguish the stress - induced texture from deformation texture . the latter is associated with grain and lattice rotation in response to finite deformations  @xcite . a stochastic , crystal plasticity model , developed from rate - independent crystal plasticity  @xcite , is applied within the mpm framework as the constitutive model to capture the elasto - plastic response of a polycrystalline media  @xcite . as opposed to conventional deterministic algorithms  @xcite , the stochastic algorithm relies on a mc routine to determine the activated slip system which is therefore referred to as the monte carlo plasticity ( mcp ) . when plastic deformation occurs , dislocations are generated , stored and annihilated within the microstructure . the heterogeneous distribution of these dislocations within the polycrystalline medium constitutes a plastic driving force for grain boundary migration . this is treated as a body force within the mc kinetics using parametric links between mc and si models . a red / black ( rb ) updating scheme is used to parallelize the mc algorithm  @xcite , although other methods might also be useful  @xcite . this parallelized hmc approach is used to investigate the microstructural evolution of nickel polycrystals under plastic loading . as expected , the grains with smaller schmid factors gradually dominate the polycrystalline system . the data is subsequently used to construct a macroscopic kinetic equation to predict the evolution of microstructure . plastic response of polycrystalline materials is treated through a classical rate - independent small deformation crystal plasticity formulation  @xcite . the foundations of the constitutive model assume that the elasto - plastic response of single crystals is dominated by slip deformation mechanisms  @xcite . a successful numerical algorithm must carry out three tasks : the determination of activated slip systems ; the calculation of the plastic slip on each activated slip system ; and , the solution of redundant constraints associated with a hardening law  @xcite . various numerical methods have been devised and successfully implemented in deterministic formats  @xcite . as opposed to deterministic algorithms , the current work adopts a probabilistic approach borrowed from concepts in statistical mechanics in which only one slip system is activated during each time step . plastic slip is therefore treated as a series of discrete , probabilistic events that mimic the sequential accumulation of dislocations at the lattice scale . this monte carlo crystal plasticity ( mcp ) is algorithmically simple because plastic slip can be resolved through the solution of one equation with no redundant constraints . on the other hand , the associated computational steps has to be sufficiently small such that a sequence of single slips mimics multiple slip behavior . a probabilistic algorithm , detailed in what follows , is used to determine which slip system is chosen at each step . the constitutive framework and stress updating routine are otherwise standard  @xcite .    given a set of potentially activated slip systems @xmath1 , identified through comparison of resolved shear stress with slip resistance  @xcite , the elastic energy of a crystal , @xmath2 , can be calculated if each slip system of the set @xmath3 is individually activated . this generates @xmath4 possible states for the deformed crystal . the probability , @xmath5 , of a slip system being selected is computed using the partition function  @xcite , @xmath6 : @xmath7{l}$z=\sum\limits_{\alpha=1}^{n}e^{-\beta e_{\alpha}}$\\$p_{\alpha}=\frac{e^{-\beta e_{\alpha}}}{z}$ , \end{tabular } \label{mc_pla}\ ] ] where @xmath8 is the index of a potentially activated slip system , and @xmath9 is the inverse of the fundamental temperature . dislocation energy can be ignored in eqn . ( [ mc_pla ] ) due to the fact that an isotropic hardening model is used  @xcite . the @xmath10 slip system of the set @xmath3 is activated when the following criterion is met : @xmath7{l}$\sum\limits_{\alpha=1}^{i}p_{\alpha}-p_{i}\leq r<\sum\limits_{\alpha=1}^{i}p_{\alpha}$. \end{tabular } \label{random}\ ] ] here @xmath11 is taken as a random number between @xmath12 and @xmath13 . with the activated slip system determined , the deformation can easily be parsed into elastic and plastic components .    as a verification of this mcp method , a uniaxial tension test was carried out on a single crystal of nickel . the material properties  @xcite required for this model are listed in table [ plapara ] . .nickel properties required in mcp model  @xcite . [ plapara ] [ cols="<,<,<",options="header " , ]     a prescribed strain increment , @xmath14 was enforced at each time step : @xmath15 , \ ] ] with @xmath16 fig . [ stress_mcp_svd ] compares the results obtained using the mcp logic with their counterparts obtained with a deterministic model wherein a singular value decomposition ( svd ) algorithm is applied to solve the ill - conditioned constraint equations . the two approaches clearly agree . sharp - interface , grain boundary kinetics  @xcite is the basis for our grain boundary kinetics model . it can be developed by considering a bi - crystal subjected to uniaxial loading that results in plastic deformation as shown in fig . [ si_schem ] . the dislocation energy , @xmath17 , and dislocation density , @xmath18 are related using the relation  @xcite @xmath19 here @xmath20 is the shear modulus , @xmath21 is the burgers vector , and @xmath22 is the flow stress . the thermodynamic driving force for interfacial accretion is given by  @xcite @xmath23\!]+\kappa \gamma^ { * } \ , \label{driv_trac}\ ] ] where @xmath24 is the capillary driving force , @xmath25 is the local mean curvature , and @xmath26\!]$ ] is the jump in a field across the grain boundary . the elastic driving force is temporarily suppressed in order to study the influence of the plastic behavior on texture evolution . the herring relation  @xcite is used to describe the accretive normal speed of the interface , @xmath27 : @xmath28 where @xmath29 is grain boundary mobility . the mc paradigm is intended to implement si kinetics within a computationally efficient setting . continuum , deterministic fields are linked to discrete , probabilistic counterparts , and the physical domain is discretized into a square lattice . a q - state potts model  @xcite is used to represent a crystalline system and grain orientation is described by an integer - valued spin field , @xmath30 , which ranges from @xmath13 to @xmath31 . the system hamiltonian is then written as @xmath32 where @xmath33 is the spatially varying bulk energy associated with the @xmath34 lattice , @xmath35 is the interaction energy between the neighboring spin fields @xmath30 and @xmath36 , @xmath37 is the number of neighbors considered on the selected lattice , and @xmath38 is the kronecker delta . kinetics , within the mc paradigm , is translated to a series of probabilistic trials carried out at all lattice sites . the acceptance of a trial event is determined by a probability function based on the change in the hamiltonian associated with a trial flip . plastic effects can therefore be accounted for if the plastic driving force is treated as a bulk energy term in eqn . ( [ h_1 ] ) .    parametric links between si and mc models are used to convert si parameters into mc format  @xcite : @xmath39 where @xmath40 and @xmath41 are reasonable grain boundary stiffness and grain boundary mobility obtained from the literature  @xcite , while @xmath42 is the computed bulk energy density associated with the si model . the counterparts in the mc paradigm , which can be analytically or numerically derived , are @xmath43  @xcite , @xmath44 and @xmath45  @xcite , respectively . a characteristic energy , @xmath46 , length , @xmath47 , and time , @xmath48 along with the mc lattice size @xmath49 , interaction energy @xmath50 , effective temperature @xmath51 , inclination angle @xmath52 and time step @xmath53 , are also required to bridge the si and mc paradigms  @xcite . these links endow the mc simulation with physical time and length scales . in a typical mc implementation , an update of variables immediately follows the acceptance of a trial event as in the metropolis algorithm for example  @xcite . to improve the efficiency of this approach within a parallelized setting , we adopted a red / black ( rb ) updating rule  @xcite wherein the domain is decomposed into a checkerboard , and each mc step is divided into red / black and black / red half steps . this is an alternative to the n - fold way algorithm  @xcite . in contrast to the large number of updates associated with a standard mc time step , states are not updated in the rb format until each half step finishes . when applied within a parallel environment , the domain is divided uniformly among processors with synchronous communication between processors at each half - step . the approach is first implemented within a 2d , bi - crystal setting  @xcite . a circular grain is placed at the center of a square domain and the inner grain shrinks or grows due to the combined effects of capillary and bulk driving forces . the two - state square lattice ising model is then employed to represent the physical system . mc simulations were carried out on a @xmath54 grid , at an effective temperature @xmath55 . the evolution of inner grain was simulated with four combinations of capillary and bulk driving forces . capillary driving force was fixed by giving a constant interaction energy , i.e. , @xmath56 , while bulk driving force , which is half of the bulk energy difference per site between inner and outer grains , varies from @xmath57 to @xmath58 . in particular , the bulk energy term will be used later on to represent mechanical effect ( dislocation energy ) in realistic plastic deformation . corresponding si simulations were also run using the established parametric links between mc and si models . fig . [ 2d_rb ] ( a ) indicates good agreement between the two models in the prediction of inner grain area as a function of time . an additional test case was also performed for a polycrystalline system where the q - state ( @xmath59 ) potts model with isotropic interaction energy ( @xmath56 ) was used . many mc simulations were carried out at effective temperatures @xmath60 and @xmath61 , respectively . [ 2d_rb ] ( b ) shows the measured average grain size with linear fits to the data . the linear grain growth behavior is consistent with that of the _ isotropic grain growth theory _  @xcite . in later applications , a relatively high mc effective temperature is preferred to remove lattice pinning  @xcite . . solid line and discrete points represent mc and si results respectively . grid size is @xmath62 . ( b ) isotropic grain growth at two mc temperatures for 2-d polycrystals . the gray and black dots demonstrate @xmath51=0.5 and @xmath51=1.0 , respectively . grid size is @xmath62 . mc results are averaged over @xmath63 samples.,scaledwidth=60.0% ] the aforementioned parallelized mc algorithm was next applied within the previous polycrystalline setting to study the influence of bulk energy distribution on the evolution of texture . the interaction energy was taken to be isotropic and a gaussian distribution for the bulk energy was assigned to each orientation . fig . [ mc_texture ] ( a ) shows the orientation - dependent gaussian shape bulk energy : @xmath64   , \label{flow}\ ] ] where @xmath4 is orientation , and @xmath44 is the bulk energy associated with this material . the bulk energy is adimensioned within a pure mc setting . mc simulations were carried out at an effective temperature @xmath60 . [ mc_texture ] ( b ) presents the orientation distribution at three time slices . as physically expected , materials points with a lower bulk energy are favored . this indicates that grains which experience the least plastic deformation will be favored as opposed to the ones have larger dislocation content .    , and @xmath60 . simulation results were averaged over @xmath65 runs.,scaledwidth=60.0% ]    the parallelized hmc approach was then used to consider the microstructural evolution of three - dimensional , polycrystalline nickel in response to plastic deformation . attention was restricted to texture development in response to plastic deformation which is dominated by grain boundary kinetics rather than that of triple junction kinetics  @xcite . to meet this requirement , all numerical experiments were performed at an mc effective temperature ( @xmath66 ) , which is a non - physical temperature . note that the corresponding physical temperature ( as opposed to mc temperature ) is still not high enough to initiate recrystallization . in addition , elastic driving forces were suppressed for the sake of clarity . the mcp framework is implemented in mpm as a plastic constitutive model mediating the mechanical response  @xcite . the required fitting parameters are listed in table [ plapara ] . material properties at room temperature were adopted even though they may be temperature dependent  @xcite . in order to illustrate the methodology and facilitate the interpretation of the results , grains were distinguished by a single angle of rotation with respect to the z - direction and allowed it to vary from @xmath67 to @xmath68 in @xmath69 angle steps . the other two euler angles were held fixed . a similar setting has been previously consider with deformation restricted to the elastic regime  @xcite .    as mentioned previously , plastic slip occurs when the resolved shear stress on a slip system exceeds its threshold resistance . as a consequence , orientations with bigger schmid factor will be plastically deformed first [ schmid_orn ] shows the orientation dependent schmid factor of a face centered cubic ( fcc ) single crystal . orientations near @xmath70 and @xmath71 have the largest schmid factor . these orientations slip more easily and thus accumulate dislocations the fastest . therefore , subsequent grain boundary motion will tend to remove such grains . in the current study , an @xmath72 mpa uniaxial loading was applied in the x - direction in @xmath65 loading increments . at each step , the strain increment associated with each mpm particle is computed using the mpm algorithm , and the mcp algorithm is then called to partition elastic and plastic deformations . the quasi - static state of the polycrystalline system is reached after a series of iterations when either the relative stress increment or relative strain increment meets the prescribed convergence criterion for all the particles  @xcite . after each mcp step , dislocation energy is computed and converted into the mc domain using our parametric links . the microstructure is subsequently evolved for one mc step , equal to @xmath63 minutes  @xcite . figs . [ texture_pla ] ( a ) and ( b ) describe the evolution of orientation distribution at five different time slices . negative rotation angles are used to represent orientation from the @xmath73 to @xmath68 . as expected , texture evolution favors grains which have smaller schmid factors with respect to the loading axis . interestingly , orientations between @xmath74 and @xmath75 are not selected with the same frequency as orientations between @xmath67 and @xmath76 even though the two groups have identical schmid factors . this is because the effective young s moduli of the former group elements are larger than that of their counterparts . therefore less elastic , and thus more plastic strain , results from the same loading . to illustrate this principle more clearly , pure mechanical simulations were performed in single crystals of various orientations . [ orn_sgl ] ( a ) shows that more plastic deformation is obtained on the @xmath76 orientation than the other two orientations . fig . [ orn_sgl ] ( b ) highlights the differences in the mechanical response between @xmath67 and @xmath75 orientations which have the same schmid factor . as expected , more plastic deformation is observed in the @xmath75 orientation . independent runs.,scaledwidth=60.0% ]        to quantify the effect of texture evolution with plasticity , the texture histograms were fitted to a time dependent gaussian equation  @xcite : @xmath77 , \label{text_evl}\ ] ] where @xmath78 is the orientation probability density , @xmath79 is the orientation angle , and @xmath80 is a time - dependent gaussian variance . the fitted results are shown as solid curves in fig . [ fitted_texture ] ( a ) along with the gaussian variance described in fig . [ fitted_texture ] ( b ) . the effects of plastic deformation on texture evolution are quantified with a previously developed hybrid algorithm that blends a discrete , probabilistic algorithm for grain boundary motion with a continuum , deterministic model of elastic deformation . plastic deformation is accounted for using a monte carlo plasticity model in which slip occurs through a sequence of probabilistically determined , single slip events . mechanical loading results in an inhomogeneous distribution of dislocation energy which amounts to a plastic driving force for texture evolution . grains with less damage grow at the expense of those which have been more heavily deformed .    in the current approach , the dislocation density associated with the area swept by a moving interface is replaced with that of the overtaking grain . this implies that , when a grain boundary moves , the dislocation content adjacent to the grain boundary is extended into the material through which the grain boundary migrates . however , this is not always the case in reality . on the other hand , if one assume that no dislocations were transmitted with the boundary motion , then a new dislocation free grain would be introduced and the material would recrystallize  @xcite . though recrystallization undoubtedly occurs , and is certainly important in the materials studied and at these levels of deformation , it is not the only option . a grain boundary can simply migrate in a heavily deformed polycrystal  @xcite . the migration of a grain boundary in a deformed polycrystal involves the propagation of some , but not always all , of the dislocation content that are adjacent to it . the authors are not aware of a meaningful quantitative description of how the dislocation structure propagates with a migrating grain boundary , and thus assume that nearly all the dislocations are carried with a boundary when it moves . this hybrid computational methodology enables the temporal evolution of the microstructure under thermomechanical loading . it offers an alternative to sharp - interface and phase - field modeling . the resulting texture evolution maps are expected to be useful in materials and process design , thermomechanical fatigue as well as in part performance assessment throughout service life . the current work was limited to a single angle of misorientation between grains . an extension of this work to include fully arbitrary misorientation is underway , which will allow us to capture the influence of plastic deformation on the texture evolution of more realistic material systems . the research is funded by sandia national laboratories . sandia national laboratories are operated by the sandia corporation , a lockheed martin company , for the united states department of energy under contract de - ac04 - 94al85000 . we also acknowledge the golden energy computing organization at the colorado school of mines for the use of resources acquired with financial assistance from the national science foundation and the national renewable energy laboratories . 10          f.  roters , p.  eisenlohr , l.  hantcherli , d.d . tjahjanto , t.r . bieler , and d.  raabe . overview of constitutive laws , kinematics , homogenization and multiscale methods in crystal plasticity finite - element modeling : theory , experiments , applications . , 58(4):1152  1211 , 2010 . a.  u. telang , t.  r. bieler , a.  zamiri , and f.  pourboghrat . incremental recrystallization / grain growth driven by elastic strain energy release in a thermomechanically fatigued lead - free solder joint . , 55(7):2265  2277 , 2007 . remi dingreville , corbett  c. battaile , luke  n. brewer , elizabeth  a. holm , and brad  l. boyce . the effect of microstructural representation on simulations of microplastic ratcheting . , 26(5):617  633 , 2010 .
a hybrid monte carlo ( hmc ) approach is employed to quantify the influence of inelastic deformation on the microstructural evolution of polycrystalline materials . 
 this approach couples a time explicit material point method ( mpm ) for deformation with a calibrated monte carlo model for grain boundary motion . 
 a rate - independent crystal plasticity model is implemented to account for localized plastic deformations in polycrystals . 
 the dislocation energy difference between grains provides an additional driving force for texture evolution . 
 this plastic driving force is then brought into a mc paradigm via parametric links between mc and sharp - interface ( si ) kinetic models . 
 the mc algorithm is implemented in a parallelized setting using a checkerboard updating scheme . 
 as expected , plastic loading favors texture evolution for grains which have a bigger schmid factor with respect to the loading direction , and these are the grains most easily removed by grain boundary motion . 
 a macroscopic equation is developed to predict such texture evolution . 
 * @xmath0 * monte carlo , grain boundary , plasticity , anisotropy , texture , driving force