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Scripta Materialia 54 (2006) 705–710

Continuum theory and methods for coarse-grained,mesoscopic plasticity

Amit Acharya *, Anish Roy, Aarti Sawant

Civil and Environmental Engineering, Carnegie Mellon University, 119 Porter Hall, Pittsburgh, PA 15213, United States

Received 14 July 2005; received in revised form 13 October 2005; accepted 26 October 2005Available online 1 December 2005

Abstract

Theory and methods involving partial differential equations and their approximate solution are advanced for the study of coarse-grained dislocation behavior.� 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Plasticity; Dislocations; Internal stress; Bauschinger effect; Finite element analysis

1. The modeling challenge posed by mesoscopic plasticity

In our view, the main challenges of a model of plasticityat the micron scale are the accurate representation of

1. the stress field of average, signed dislocation density(geometrically necessary dislocations, GNDs) that doesnot vanish at the scale of resolution of such models,

2. the plastic strain rate arising from the temporal evolu-tion of this density,

3. the plastic strain rate arising from the unresolved dislo-cation density (statistically stored dislocations, SSDs),and

4. the effect of GNDs on the strength of the material.

These features of plastic response at this scale lead to sizeeffects, both at initial yield and during the course of defor-mation, and the development of microstructure undermacroscopically homogeneous conditions; any mechanicalmodel ought to strive to account for these features in asmechanistically rigorous a manner as possible. The scalesof spatial and temporal resolution of such models, how-ever, are large enough that accounting for individual dislo-

1359-6462/$ - see front matter � 2005 Acta Materialia Inc. Published by Else

doi:10.1016/j.scriptamat.2005.10.070

* Corresponding author. Tel.: +1 412 268 4566; fax: +1 412 268 7813.E-mail address: acharyaamit@cmu.edu (A. Acharya).

cations is not possible. A complicating factor is that thespatial averaging cancels pairs of positive and negativedislocations but the plastic strain rate produced from themotion of such pairs (as in loop expansion) does not canceland needs to be accounted for in the model. Other compli-cating factors are the nonlinearity of the underlying dislo-cation dynamics and the large reduction in degrees offreedom implied by the desired coarse-grained model,resulting in memory-dependent response and stochasticbehavior of averaged variables of an autonomous, deter-ministic fine-scale theory.

2. Our perspective

We believe that the aforementioned challenges can beadequately, and in some sense optimally, addressed bycontinuum/field methods of analysis and approximation.Obviously, even the structure of the field theories involvedare not expected to remain the same as the spatial and tem-poral scales at which plasticity is to be represented change;however, the representation of physical processes via partialdifferential equations (PDE) has by now proved its worth asa practically robust and conceptually elegant tool in math-ematical physics, and we would like to capita- lize on thisestablished success in describing the challenging physics ofmesoscopic plasticity.

vier Ltd. All rights reserved.

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3. Rationale behind perspective

Consider, first, the case of developing a dynamicalmodel of dislocation plasticity where the scale of spatialresolution is such as to resolve each and every dislocation(we refer to this case as fine-scale dislocation mechanics).We begin with the realization that all flavors of classicaldislocation theory, whether considering dislocations ascontinuously distributed or as discrete singularities, areactually field theories as they relate to the computation ofinternal stress of a dislocation distribution. Thus, it onlymakes sense to devise field methods for solving such aproblem in finite bodies. When it comes to specifying thedynamics of such a distribution, the discrete-singularitymodel has generally been preferred. For dynamics in thecontinuously distributed case, while the basic kinematicsrelevant to the evolution of a dislocation distribution wasrealized early on, a sufficiently general understanding of awell-posed procedure for linking the dislocation stressproblem to the dynamics of the distribution has onlyrecently emerged. Such a development has importantconsequences: the continuously distributed case can beextended to the case dealing with unrestricted geometricand material nonlinearities as well as elastic anisotropywith ease and, more importantly, to the case of dealingwith material inertia. The discrete-singularity model fallsshort on all of these counts, especially with regard to deal-ing with elastic nonlinearity and inertia: superposition ofsolutions forms the backbone of the approach, and thestress fields of arbitrarily accelerating dislocation segmentsare not known in a practically useful form so that disloca-tion interactions cannot be accounted for correctly. Thusit makes sense to pursue the field approach to fine-scaledislocation mechanics.

The availability of a completely defined fine-scale theoryof dislocation mechanics allows at least two possibilities forestablishing a theory of plasticity at the mesoscopicscale. First, techniques from mathematical homogenizationtheory for PDE may be utilized [1,2]; unfortunately, thistheory is not at the stage where it can be applied to theproblem at hand but future progress is to be expected. Asa second alternative, a theory posed in terms of PDEnaturally yields important information on the form of theequations that running space–time averages of its basicfields must satisfy; when the underlying fine-scale theoryis nonlinear, there are terms in these averaged equationsthat require closure as they are defined only in terms offine-scale fields and not their averages, e.g. the terma� V :¼ �a� V þ Lp that appears on the right-hand sideof (4)6. Here one may resort to phenomenology in such pre-scription or to some form of coarse-graining since the latterquestion is (almost) well-posed; an underlying fine dynam-ics is available, desired coarse quantities are well-defined interms of ingredients of the fine theory or an appropriateaugmentation of it and the target is to derive an evolutionlaw for the coarse variables from these available ingre-dients. When converted into a system of ODE by some

spatial discretization, the above coarse-graining problemcan be addressed partially by a method that we have calledthe method of parametrized locally invariant manifolds(PLIM) that extends, in essential ways, prior work byMuncaster [3] which was a practically important abstrac-tion of the essential concepts and methods of the kinetictheory [4]. Interestingly, our approach is again a methodbased on solving a system of PDE, thus connecting nicelyto our overall theme. The qualification of PLIM as a con-ceptual tool that can only partially address the problem ofclosure for averaged quantities at the present time has todo with the incomplete knowledge and the possible non-autonomous time-dependence of fine-scale boundaryconditions for the space–time averaging domains. Also,our method transparently illustrates the fact that fine-scaleinitial conditions may not be so easily ignored for consis-tent coarse-graining, even if the chosen coarse variablesare physically reasonable slow variables. In this sense italso underscores the tremendous utility of the developmentof robust continuum theories for macroscopic responsethat utilize physically faithful constitutive rules.

4. Theory and some methods to carry out our program

A theory of fine-scale dislocation mechanics, namedfield dislocation mechanics (FDM), has been proposedrecently [5–7]. Its equations may be written in the form

curl v ¼ a

div v ¼ 0

divðgrad _zÞ ¼ divða� VÞdiv½C : fgradðu� zÞ þ vg� ¼ 0

_a ¼ �curlða� VÞ þ s.

ð1Þ

Here, v is the incompatible part of the elastic distortiontensor U e, u is the total displacement field, u � z is a vectorfield whose gradient is the compatible part of the elasticdistortion tensor, C is the fourth-order, possibly aniso-tropic, tensor of linear elastic moduli, a is the dislocationdensity tensor, V is the dislocation velocity vector, and sis a dislocation nucleation rate tensor (not related to dislo-cation line length increase from existing dislocations). Theargument of the div operator in (1)4 is the stress tensor, andthe functions V and s are constitutively specified. Theseequations admit well-defined initial and boundary condi-tions that have also been worked out [6,8]. In this modelof dislocation mechanics, the total displacement does notrepresent the actual physical motion of atoms involvingtopological changes but only a consistent shape changeand hence is not required to be discontinuous. However,the stress produced by these topological changes in thelattice is adequately reflected in the theory through the uti-lization of incompatible elastic/plastic distortions. Indeed,the compatible part (i.e. a part that can be represented asa gradient of a vector field) of the plastic distortion is givenby grad z and the total displacement gradient is simply the

A. Acharya et al. / Scripta Materialia 54 (2006) 705–710 707

sum of the compatible parts of the elastic and plasticdistortions

grad u ¼ ðUe � vÞ þ grad z. ð2ÞTo derive an averaged theory corresponding to (1), we

adapt a commonly used averaging procedure utilized inthe study of multiphase flows (see e.g. Ref. [9]) for our pur-poses. For a microscopic field f given as a function of spaceand time, we define the mesoscopic space–time-averagedfield �f as follows:

�f ðx; tÞ :¼ 1RIðtÞ

RXðxÞ wðx� x0; t � t0Þdx0 dt0

�ZI

ZBwðx� x0; t � t0Þ f ðx0; t0Þdx0 dt0; ð3Þ

where B is the body and I a sufficiently large interval oftime. In the above, X(x) is a bounded region within thebody around the point x with linear dimension of the orderof the spatial resolution of the macroscopic model we seek,and I(t) is a bounded interval in I containing t. The aver-aged field �f is simply a weighted, space–time, running aver-age of the microscopic field f. The weighting function w isnon-dimensional, assumed to be smooth in the variablesx,x 0,t,t 0 and, for fixed x and t, have support (i.e. to benon-zero) only in X(x) · I(t) when viewed as a functionof (x 0, t 0). Applying this operator to the equations in (1),we obtain [10] an exact set of equations for the averagesgiven as

curl �v ¼ �a

div �v ¼ 0

divðgrad _�zÞ ¼ divð�a� V þ LpÞ

Ue ¼ gradð�u� �zÞ þ �v

div T ¼ 0

_�a ¼ �curlð�a� V þ LpÞ

ð4Þ

Fig. 1. (a) Development of inhomogeneous edge dislocation density distributiosimple shearing for plastically constrained grains; Bauschinger effect in unload

where Lp, defined as

Lpðx; tÞ :¼ ða� �aÞ � Vðx; tÞ¼ a� Vðx; tÞ � �aðx; tÞ � Vðx; tÞ; ð5Þ

and V are the terms that require closure. Physically, Lp isrepresentative of a portion of the average slip strain rateproduced by the �microscopic� dislocation density; inparticular, it can be non-vanishing even when �a ¼ 0 and,as such, it is to be physically interpreted as the strain rateproduced by so-called SSD, as is also indicated by theextreme right-hand side of (5). The variable V has the obviousphysical meaning of being a space–time average of thepointwise, microscopic dislocation velocity. Initial andboundary conditions for (4) are important from the physicalmodeling point of view, particularly in the context of trig-gering inhomogeneity under boundary conditions corre-sponding to homogeneous deformation in conventionalplasticity theory. These have also been worked out [10].

It should be noted here that as the terms Lp and V areboth explicitly defined in terms of microscopic ingredients,this model sets a natural question for discrete dislocationmethodology to provide constitutive equations for thesecomponents of the mesoscopic model.

4.1. Closure via phenomenological approach

Physically reasonable choices for V and Lp can be madebased on the requirement of non-negativity of plasticworking and ingredients of conventional plasticity theoryresulting in a model that requires only one extra materialparameter over size-independent classical plasticity. Theresulting model can be approximated computationallywithin the finite element method [11]. It is quite likely thata rigorous examination of solutions to this phenomenolog-ical model involving nonlinear transport will indicate thatsolutions may objectively be interpreted only as probabilis-tic measure-valued solutions [12]. We also have reason to

n in a 1 lm thick film under simple shearing. (b) Stress–strain size effect ining for the (1 lm)3 grain.

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hope that the finite element method based approximationswill aid in the precise determination of such probability dis-tributions. This raises the interesting prospect of a deter-ministic theory yielding results of a probabilistic nature.In any case, Fig. 1 shows the capability of our approxima-tion scheme in producing patterned microstructure, sizeeffects, and a strong Bauschinger effect in simple shearingthat would otherwise result in homogeneous deformationin conventional plasticity theory.

4.2. Possible closure via the method of parametrized

locally invariant manifolds

In this approach to semi-rigorous coarse-graining, theidea is to spatially discretize (1) in an averaging cell X oflinear dimension corresponding to the resolution of thedesired coarse theory to obtain a nonlinear system ofODE. An ansatz for a best possible set of boundary condi-tions that may be stated, at most, as an autonomous time-dependent system is also required in the above step. Thenext step is to augment the obtained ODE system withdegrees of freedom (dof) representing forward-in-time-shifts of the discretized dofs by a fixed time interval. Thattime interval represents the desired scale for temporal aver-aging, e.g. if f is a collection of dofs, fs are the correspond-ing s time-shifted variables defined by

fsðtÞ :¼ f ðt þ sÞ. ð6ÞOne now chooses coarse variables as the running time aver-age of some (generally nonlinear) function/map K of thediscretized fine dofs:

cðtÞ :¼ 1

s

Z tþs

tKðf ðsÞÞds. ð7Þ

A term like Lp may be so represented. Then

_cðtÞ ¼ 1

s½KðfsðtÞÞ � Kðf ðtÞÞ�. ð8Þ

Fig. 2. Hysteresis curve for time-averaged

The idea now is to represent the fine dofs f and time-shifted ones fs as mappings of the coarse variables c andthen try to seek these mappings as solutions to a time-inde-pendent system of PDE defined from the definition of thecoarse variable and the fine system of equations. Onceobtained, these mappings are substituted in place of fsand f in (8) to define a closed coarse dynamics, completelyin terms of c. The details of the procedure, and the issuesinvolved in doing so, are provided in Refs. [13–15]. In gen-eral, the approximation of solutions of the time-indepen-dent PDE system is computationally intense but needsto be performed only once to set up the coarse theory.We mention here that interesting features like memorydependence in the coarse response arise naturally whenapproaching the problem in this fashion. Our approach isadapted to deal with a physically-motivated choice ofcoarse variables without restrictions on the number of dofsin the coarse system, and this is an all-important differenceof principle, with practical implications, between ourapproach and other approaches like inertial manifoldtheory [16].

Here, two results are provided that illustrate temporalcoarse-graining and reduction of degrees of freedom inlow-dimensional systems displaying complicated dynamics.The first example involves the evolution of a variant ofmartensite during cyclic biaxial loading of a plate [17]. Agradient flow dynamics based on a �wiggly� energy functionis introduced for the fine evolution. The time step requiredto simulate the process according to this set of equations isvery small. We introduce a time-averaged volume fraction,�k, as the coarse variable as in (7) and define a closed, coarsemodel of evolution, as described in Ref. [15], for this vari-able. Fig. 2 shows hysteresis curves for the average volumefraction evaluated from the solution of the fine model com-pared to the solution of the coarse theory. The latter wasobtained from simulations using time-steps much largerthan those for the fine evolution, as indicated by thecoarse-to-fine (c/f) ratio.

variable �k vs. loading variable r1 � r2.

Fig. 3. Trajectories in coarse phase space (a) fine, (b) coarse.

A. Acharya et al. / Scripta Materialia 54 (2006) 705–710 709

A physically interesting feature of this model is that�yield� type behavior in the hysteresis curve is solely a con-sequence of the wiggles in the microscopic energy withoutthe adoption of any explicit yield criterion [17]. Clearly,the coarse theory we develop inherits this feature as canbe seen from its solutions. This raises the interesting pros-pect of depicting macroscopic yield behavior in plasticitythrough methods like ours, perhaps even by using a quali-tative model of microscopic free-flight-and-wait processesfor dislocation motion.

The second example illustrates reduction of degrees offreedom in the Lorenz system [18] that displays chaoticbehavior. Details are to be found in Ref. [14]. The Lorenzsystem is an autonomous system having three dofs: x, y, z.We would like to retain only x, z as the dofs of the coarsetheory. Fig. 3 illustrates the comparison of results for a sin-gle representative trajectory. The three dof system isevolved numerically from the indicated initial conditionand the (x(t),z(t)) trajectory is plotted in Fig. 3a. Notethe self-intersection of the trajectory; the three-dof systemis autonomous and hence its trajectories cannot intersecttransversally in phase space. But the �reduced� trajectory,i.e. (x(t),z(t)), is actually a projection of the actual trajec-tory (x(t),y(t),z(t)) and it is natural to expect the reducedtrajectory to self-intersect. We interpret this feature as ageometric explanation of the emergence of history depen-dence in coarse response due to reduction in dofs. InFig. 3b, the closed coarse theory is evolves and the corre-sponding (x(t),z(t)) trajectory is plotted. Given the chaoticnature of the underlying system, the correspondence ofresults may be considered satisfactory; especially notewor-thy is the recovery of the self-intersection of the trajectoryin the coarse theory solution.

5. Concluding remarks

We have outlined a point of view for the practicalcoarse-graining of dislocation behavior in plasticity. Wehave also tried to substantiate this point of view with

concrete theoretical ideas and representative results fromsimulations that implement these ideas within robust algo-rithms for the solution of PDE and ODE. It is our hopethat an appropriate grafting of ideas from mathematical[1] and stochastic (see, e.g., Ref. [19]) homogenization the-ory to our approach of continuum dislocation mechanicsand coarse graining will lead to significant advances inplasticity within the next 5–10 years.

Acknowledgements

We thank Dennis Dimiduk for helpful comments relatedto the presentation of this paper. Support from the Programin Computational Mechanics of the US ONR, the Programin Physical Mathematics and Applied Analysis of the USAFOSR, and the AFRL at WPAFB is gratefully acknowl-edged.

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