a model of crystal plasticity based on the theory of continuously distributed dislocations

Journal of the Mechanics and Physics of Solids49 (2001) 761–784

www.elsevier.com/locate/jmps

A model of crystal plasticity based on the theoryof continuously distributed dislocations

Amit Acharya ∗

Center for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign, 3315 DCL,1304 W. Spring'eld Ave., Urbana, IL-61801, USA

Received 14 June 2000; received in revised form 17 August 2000

Abstract

This work represents an attempt at developing a continuum theory of the elastic–plastic re-sponse of single crystals with structural dimensions of ∼ 100 �m or less, based on ideas rootedin the theory of continuously distributed dislocations. The constitutive inputs of the theory relateexplicitly to dislocation velocity, dislocation generation and crystal elasticity. Constitutive nonlo-cality is a natural consequence of the physical considerations of the model. The theory reduces tothe nonlinear elastic theory of continuously distributed dislocations in the case of a nonevolvingdislocation distribution in the material and the nonlinear theory of elasticity in the absence ofdislocations. A geometrically linear version of the theory is also developed. The work presentedin this paper is intended to be of use in the prediction of time-dependent mechanical response ofbodies containing a single, a few, or a distribution of dislocations. A few examples are solvedto illustrate the recovery of conventional results and physically expected ones within the theory.Based on the theory of exterior di3erential equations, a nonsingular solution for stress=strain4elds of a screw dislocation in an in4nite, isotropic, linear elastic solid is derived. A solutionfor an in4nite, neo-Hookean nonlinear elastic continuum is also derived. Both solutions matchwith existing results outside the core region. Bounded solutions are predicted within the core inboth cases. The edge dislocation in the isotropic, linear theory is also discussed in the context ofthis work. Assuming a constant dislocation velocity for simplifying the analysis, an evolutionarysolution resulting in a slip-step on the boundary of a stress-free crystal produced due to thepassage and exit of an edge dislocation is also described. c© 2001 Elsevier Science Ltd. Allrights reserved.

Keywords: Continuous distribution of dislocations; B. Crystal plasticity; Dislocation mechanics

∗ Correspondence address: Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh15213-3890, USA. Tel.: +1-412-268-4566; Fax: +1-412-268-7813.E-mail address: [email protected] (A. Acharya).

0022-5096/01/$ - see front matter c© 2001 Elsevier Science Ltd. All rights reserved.PII: S0 0 2 2 -5 0 9 6 (00)00060 -0

762 A. Acharya / J. Mech. Phys. Solids 49 (2001) 761–784

1. Introduction

It is well established that permanent deformation in a crystalline solid arises due todislocation motion. It is also believed that the stress 4eld in a body drives the motionof dislocations. Since dislocations themselves cause an internal stress 4eld in the bodywhich depends on their spatial distribution, such stresses, in combination with thosethat arise in accommodating applied boundary conditions, result in a redistribution ofthe dislocation 4eld in a body. Such a spatial redistribution alters the stress state thusinducing further motion, or resistance to motion, of the dislocation 4eld. Conceptu-ally, this cycle can be thought of as a coupled process that may or may not lead toequilibrium states (in the sense of state evolution) depending upon the nature of theapplied boundary and initial conditions, the material constitution of the body whichleads to its elasticity and the generation and motion of dislocations in it. It is theaim of this paper to attempt a mathematical description of the above coupled processthat results in a closed continuum theory (in the sense of eliminating nonuniquenessdue to a lack of suBcient physical statements), whose inputs are constitutive state-ments for crystal elasticity and dislocation generation and motion. It is only reason-able to demand that the theory should reduce to nonlinear elasticity in the absenceof dislocations and to the nonlinear elastic theory of continuously distributed disloca-tions (ECDD) (Willis, 1967) in the absence of evolution of the dislocation density4eld.

The requirement of closure in the theory is an important one when viewed in thecontext of extending ECDD for the calculation of internal stress. Roughly speaking,the 4eld degree of freedom that allows for the solution of problems of internal stresson a known reference con4guration for arbitrary divergence-free dislocation density4elds is the one that has to be eliminated in order to have a theory which admits adisplacement 4eld on the said reference, which varies as the dislocation density 4eldon the body is altered. This observation is discussed in greater detail in the followingsections of the paper.

In considering the evolution of state within the theory, motivation is drawn fromthe work of Mura (1963) and Kosevich (1979). However, the present theory adoptsa di3erent stance with respect to the speci4cation of the plastic deformation and thedislocation Eux in comparison to these earlier works. A statement on dislocation densityevolution appears, and is used, as a partial di3erential equation (PDE) which is coupledto the stress 4eld, if the constitutive equations for the dislocation velocity and sourcesare assumed to depend on the stress. With such a mechanism for generating the currentdislocation density 4eld in hand, evolutionary statements for the dislocation Eux andslip deformation tensors are completely speci4ed within the theory. Roughly speaking,the plastic deformation is developed as an additive sum of geometrically appropriatecompatible and incompatible parts generated from the slip deformation and dislocationdensity 4elds respectively. Mura (1963) and Kosevich (1979), in the context of smalldeformations, develop statements for the dislocation density similar in spirit to the onedeveloped here as part of their kinematical analysis but refrain from using it as a4eld equation for the determination of the dislocation density. In the context of exactkinematics, Fox (1966) derives a relationship for the evolution of his dislocation line

A. Acharya / J. Mech. Phys. Solids 49 (2001) 761–784 763

densities which is conceptually similar to the evolutionary statement derived as part ofthis work if dislocation sources are not included. As will be shown in Section 4, theinclusion of dislocation sources provides the theory with the mechanism of predictingan incrementally inhomogeneous deformation from a homogeneous state, a capabilitythat has signi4cant physical implications (Hutchinson, 2000).

Motivated by the structure of the nonlinear theory, a geometrically linear versionis also developed. It transparently brings to the forefront two major implications ontheories of plasticity based on the postulation of a constitutive statement for plas-tic strain. Firstly, if the dislocation velocity is a function only of stress then theconstitutive statement for the plastic strain rate is necessarily a function of the in-tegrated history of spatial gradients of stress. Secondly, if dislocation sources are in-cluded, then the plastic strain rate is a genuinely nonlocal function of such sourcedistributions.

The theory presented herein is intended to be applicable to dislocation related phe-nomena in crystalline solids that appear well resolved at length scales of 103b− 10b,where b is the lattice spacing in the crystal. If the dislocation density 4eld on theinitial con4guration is provided as an initial condition along with the applied trac-tion 4eld that maintains the body in equilibrium in the initial con4guration, then theequations of the theory for determining the residual stress 4eld on this con4gurationreduce to exactly those for the determination of internal stress in the ECDD (Willis,1967). The deliberations of the paper also provide a conceptual method for generatingsolutions to the above problem on 4nite domains for nonlinear elastic laws of arbitrarysymmetry, a method that can easily form the basis of a numerical algorithm basedon the 4nite-element method. Because the theory is not restricted to linearity in theelastic constitutive law, it appears to be within the bounds of the theory to modelsome aspects of both long and short-range dislocation interactions of distributions ofdislocations and hence the hardening that arises due to back stresses as well as strainhardening. How far such predictions can match up with reality remains to be seen atthis point. The change in stress response from purely elastic behavior associated withthe 4rst generation of dislocations in a virgin crystal appears to be within the pre-dictive range of the theory — interestingly, such a notion is necessarily accompaniedby inhomogeneous deformation. If yielding is associated with the onset of motion ofpreexisting dislocations, the associated stress-response history would also seem to bewithin the predictive scope of this work.

The theory appears to be suitable for the response of bodies with structural dimen-sions in the mesoscale and perhaps even smaller scales, given the occasional successof the linear elastic theory of dislocations for nanomaterials. Rigorous averaging ofthe theory can lead to a plasticity model with resolution in the meso-microscale whichnaturally incorporates length-scale e3ects due to dislocation activity.

2. The nonlinear elastic theory of continuously distributed dislocations

The aim of ECDD, as formulated in terms of the elastic distortions in Willis (1967),is the prediction of the internal stress 4eld in a body in a known con4guration when


the dislocation density 4eld is known on this con4guration and the nonlinear elastic,and possibly anisotropic, constitutive equation of the material is given. Willis (1967)develops an asymptotic solution to these equations for the case of a discrete screwdislocation in an anisotropic material. 1 Given the 4eld-theoretic nature of the aboveformulation and keeping in mind the current goal of developing a nonequilibrium theoryof dislocation distributions which accounts for the stress 4elds of such distributionsrigorously, it is perhaps natural to ask if the ECDD could be used with an evolutionarystatement for the dislocation density to achieve such a goal. Roughly speaking, theexpectation is that as the dislocation density evolves due to the action of stresses inthe body and as the applied boundary conditions change, the body should deform in adeterministic way and this should be a natural outcome of the theory. This question ispursued in this section.

Consider a star-shaped con4guration 2 R on a part of whose boundary, @Rt; knownCauchy tractions t1 are applied and the rest of the boundary, @Rx, is held 4xed. Obvi-ously, the boundary constraint on @Rx also provides any traction distribution requiredto hold the body in overall static equilibrium. Let T be the elastic constitutive equationand W1 the inverse elastic deformation (or distortion) tensor. 3 A dislocation density4eld on R is meant as a two-point tensor 4eld that yields the true Burgers vector(Willis, 1967) of all dislocation lines threading a bounded area in the body, whenintegrated over such an area. It is further assumed that any dislocation density 4eld isdivergence-free on R. Let us assume that there is a dislocation density 4eld �1 speci4edon the body such that the following equations hold:

curlW1 = −�1; ejrs(W1)is; r = (−�1)ij on R; (1)

T1 := T(W−11 ); divT1 = 0; (T1)ij; j = 0 on R; (2)

where di3erentiation is understood with respect to coordinates established from a rect-angular Cartesian frame with respect to which all tensor indices are also expressed inthis paper. Also, the symbol erjk represents a component of the alternating tensor. Thestress component Tij represents the component in the xi direction of the force on unitarea perpendicular to the xj direction. The negative sign on the dislocation density in(1) is due to the fact that the true Burgers vector of a discrete dislocation is de4nedto be opposite in direction to the line integral of the inverse elastic deformation alongany curve encircling the dislocation in the elastically stressed con4guration. Eqs. (1)and (2) above are the usual equations of ECDD. Of course, it is also assumed that thestress 4eld T1 satis4es the traction boundary condition on @Rt .

We now think of keeping the con4guration R 'xed and altering the traction 4eld on@Rt to t2 and the dislocation density 4eld on R to �2. Suitable boundary constraintson @Rx still remain in force such that traction 4elds dictated by equilibrium can be

1 See Teodosiu and Soos (1981a, b, 1982) for related work with a di3erent Eavor for the method ofsolution.

2 A topological constraint that ensures that there exists a point in the con4guration such that the linejoining any point of the con4guration to this point lies completely within the con4guration.

3 Instead of writing the usual Fe−1; the symbol W is chosen to avoid writing the superscripts repeatedly.


provided. It is perhaps reasonable to expect that there cannot exist a solution to system(1) and (2), now posed in terms of the quantities W2; �2; T2; and t2 since the loadinghas changed and the body has not been allowed to deform. However, we 4nd that thisphysical expectation is not borne out if (1)–(2) form the basic equations of the theory— for consider a particular solution of (1) given by W2 = W2. Then W2 +∇w, wherew is any vector 4eld on R and ∇ represents the gradient, is also a solution to (1). Interms of this solution to (1), (2) takes the form

T2 := T({W2 + ∇w}−1); divT2 = 0; (T2)ij; j = 0 on R; (3)

where the following boundary conditions can be imposed to determine the function w:

T({W2 + ∇w}−1) · n = t2 on @Rt; w = 0 on @Rx; (4)

where n is the outward unit normal 4eld on the boundary of R. Eqs. (3) and (4) 4t thespeci4cation of a problem in the usual nonlinear theory of elasticity with an unusualconstitutive equation for the stress due to the presence of W2 and, consequently, it isperhaps reasonable to assume that there exists a solution to the problem (if we considerthe linear version of the problem posed by (3) and (4), then it can be shown that asolution de4nitely exists). An alternative way of summarizing the above discussionis to conclude that if a physical displacement 4eld of R were to be included in theproblem of determination of internal stress in the presence of a dislocation density,then the equations of the ECDD are inadequate for the unique determination of such a4eld.

Based on such motivation, we conclude that the ECDD has to be extended, beyondthe speci4cation of an evolution equation for the dislocation density, to enable theprediction of deformation of the body in response to changes in the applied bound-ary conditions and the internal state of the material characterized by the dislocationdensity.

3. A plasticity theory

3.1. Equations for stress and displacement

The basic idea in the extension of ECDD is as follows: the linear di3erential operatorcurl on the current con4guration C has a nontrivial null space in that it contains, atleast, gradients of vector 4elds on C. If the null-space component of the solution to (1)(taking R as C and dropping all subscripts) is suitably speci4ed through a constitutiveassumption, then it can be shown that the solution to such an augmentation of theproblem de4ned by (1) is unique in a sense that can be made precise, assuming for themoment that C is known. In reality, C has to be determined and this may be viewed assolving the equilibrium equations (2) and corresponding boundary conditions, assumingW is known from the solution of the augmentation of (1) just described (note that theunknown con4guration enters in the de4nition of the operator div and the boundaryconditions). Of course, the combined, nonlinear problem of the determination of C andW has to proceed in tandem. The constitutive equation for the null-space component


is chosen in such a way that in the absence of dislocations in the body, the problemreduces to solving the problem of nonlinear elasticity; in the presence of a dislocationdensity distribution 'xed in the material and changing only to the extent of elasticmaterial deformation induced by external loads, the reduction is to the ECDD phrasedas a problem on the 4nal deformed con4guration.

Henceforth, we choose the con4guration of the as-received body as the referencecon4guration R, possibly in a state of stress due to dislocations and=or applied loads.We denote the deformation gradient of the current con4guration C with respect to R asF. It is now our aim to discuss some mathematical machinery by which we can speakabout a unique decomposition of tensor 4elds on C into orthogonal components in asuitable sense. These components are such that one part is annihilated by the operatorcurl and the other part is orthogonal to the part that is annihilated. With this objectivein mind, consider the space D of all square-integrable 3×3 matrix 4elds on C (thoughtof as the components of a tensor 4eld with respect to a rectangular Cartesian frame).An L2 inner product on D is de4ned as the Lebesgue integral over C of the matrixinner product of two matrix 4elds belonging to D, i.e.

(A; B)D =∫CAijBij d�:

Let T be a set of continuous test functions with vanishing tangential component on theboundary of C and at least piece-wise continuous 4rst derivatives in C. On the boundaryof C with unit normal 4eld n, any such test function satis4es Qri − (Qrjnj)ni = 0. Asuitable weak form for curlW=−�, arrived at by formally integrating the equation byparts, is∫

CWrkekjiQri; j d�= −

∫C�riQri d� for all Qri in T:

Henceforth, whenever we write an equation of the form curlX=�, we always mean itin the sense of such a weak formulation. Similarly, when required to make sense due tosmoothness constraints, solutions to the equilibrium equations will also be interpretedas solutions to the corresponding weak problem, i.e. the virtual work equation forstatics.

The linear subspace N (curl) of D consists of all functions W belonging to D thatsatisfy∫

CWrkekjiQri; j d�= 0 for all Qri in T:

Let the set of all functions Z in D satisfying

(Z; Y )D = 0 for all Y ∈ N (curl)

be denoted by N⊥(curl). The fact that N (curl) is a closed subspace of D implies thatevery function X in D admits a unique orthogonal decomposition given by X=X⊥+X‖,with X‖ ∈ N (curl) 4 and X⊥ ∈ N⊥(curl). The unique orthogonal projection of X on

4 When the operator curl is discretized, say by the 4nite element method, there exists a de4nite algorithmto generate the orthogonal projection in the null space.


N (curl) has the property (X −X‖; X −X‖)D=inf Y∈N (curl) (X −Y; X −Y )D. In this paper,we refer to the subspace N (curl) as the null-space of the operator curl.

Let P be the slip deformation 4eld for which a constitutive equation will be speci4edin Section 3.4. Assuming div � = 0 on C, the equations for the determination of thecon4guration C and the 4eld W on it are given as

curlW = −�; (5)

W‖ = (P · F−1)‖; (6)

T = T(W−1); (7)

divT = 0 (8)

with the usual traction and=or displacement boundary conditions for the force equi-librium part of the problem, where (5)–(8) are meant to hold on C. The constraintdiv �=0 (or its corresponding weak form) is suBcient for the existence of solutions to(5) and (6), assuming the con4guration is known (Carlson, 1967; Weyl, 1940). Thisconstraint has the status of the node-theorem of dislocation theory, i.e. the continuousanalog of the statement that the sum of the Burgers vector of all dislocation lines meet-ing at a node vanishes, which also implies that a dislocation line cannot end within abody.

If �0, the dislocation density 4eld at the initial instant, is speci4ed on R and theintention is to solve for the state of stress on R then only (5), (7), (8) need besolved (only traction boundary conditions make sense for this problem of internalstress on a known con4guration). The inverse elastic deformation 4eld at the ini-tial instant, W0, can be determined by following exactly the discussion in Section2. In essence, (W0)‖ becomes the gradient of a vector 4eld that is used to sat-isfy (7) and (8), now driven by boundary conditions and a particular solution to(5) forced by �0. A uniqueness assertion=analysis for this problem is not knownto the author (the corresponding situation in the linear theory will be discussed inSection 4). For consistency, the initial condition on P should be chosen asP0 = W0.

3.2. Dislocation density tensor

We de4ne the undeformed dislocation line directions for the �th slip system asfollows:

i1(�) := n�0 ; i2(�) :=m�0 ; i3(�) := n�0 ×m�

0 ; (9)

where m�0 is the unit slip direction and n�0 is the unit slip normal for the system �.

Corresponding to {im(�); m= 1; 3}, we de4ne the deformed dislocation line directionsin the current con4guration given by

d1(�) =Fe−T · i1(�)

‖ Fe−T · i1(�) ‖ ; d2(�) =Fe · i2(�)

‖ Fe · i2(�) ‖ ; d3(�) =Fe · i3(�)

‖ Fe · i3(�) ‖ ; (10)


where Fe :=W−1 is introduced to be in accord with familiar notation from crystalplasticity and the symbol ‖ a ‖ represents the magnitude of the vector a. Let {d(�)

m ;m = 1; 3} be the dual basis corresponding to {dm(�); m = 1; 3}, i.e. d(�)

m = g(�)−1mn dn(�),

where g(�) represents the matrix whose mth-row, nth-column entry is dm(�) ·dn(�). Then�(�), the �th slip system dislocation density tensor, may be expressed as

�(�) = �n(�)m im(�) ⊗ d(�)

n ; �n(�)m = im(�) · �(�) · dn(�): (11)

The components (�n(�)m =b) have the natural interpretation of being the number of dis-

location lines threading unit area perpendicular to dn(�) in the deformed lattice whosetrue Burgers vectors point in the direction im(�) in the intermediate con4guration. Con-sequently, these components may be thought of as idealized representations of screwand edge dislocation densities in the body in its current state.

In assigning a slip system dislocation density tensor (e.g. for the de4nition of initialconditions), essentially Nye’s (1953) conceptual prescription can be followed. On sys-tem �, let nl be the number of dislocation lines along the unit direction l per unit ofarea perpendicular to l. Let these dislocations have Burgers vector in the direction bl.The contribution to �n(�)

m from this family of dislocations is (bl · im(�))(l · dn(�))nl. Sim-ilar contributions from dislocation families oriented di3erently may simply be added.In terms of such an initial assignment from observations, it is possible that slip systemdislocation density tensors start out with vanishing forest components. Glide disloca-tions on such systems still experience interactions with forest obstacles provided byglide dislocations on other slip systems since the dislocation velocity typically is afunction of stress and the stress is a functional of the total dislocation density in thebody de4ned as

�=∑�

��: (12)

3.3. Evolution equation for slip-system dislocation density

Consider a material surface A bounded by a closed curve C. Let f (�) be the disloca-tion Eux 4eld for the �th slip-system which measures the rate of inEow of dislocationlines through C into A, carrying along with them their corresponding Burgers vectors.Let s(�) be the dislocation source density for the �th slip-system which represents therate of generation of dislocations, per unit area, in the region of the deformed lat-tice traced out by the deforming material surface A. Then it seems reasonable to saythat

ddt

∫A(t)

�(�) · n da=∫C(t)

f (�) · dx +∫A(t)

s(�) · n da (13)

for all material surfaces A(t). Since (13) is a balance statement for an areal density,typically there should be a term on the right-hand side of (13) that accounts forthe change of dislocation content in the material surface A(t) due to material motionperpendicular to it. Since �(�) is a two-point tensor delivering the true (undeformed)


Burgers vector of dislocation lines and it is physically reasonable to assume that dislo-cation lines do not end inside the material, such a contribution to the balance statementis assumed to vanish.

The point-wise statement corresponding to (13) is

◦�(�) =curl f (�) + s(�); (14)

where

◦�(�) =div(v)�+ �− � · LT: (15)

◦�(�) represents the convected derivative (or the Lie derivative with respect to the Eowde4ned by the material velocity) of the two-point tensor 4eld �. In the above, v repre-sents the material velocity 4eld, � the material time derivative of �, and L the velocitygradient. Eq. (15) is a direct consequence of using the identity

n da= JF−T ·N dA; (16)

where n and N are the unit normal 4elds on A(t) and A(0), respectively, J is thedeterminant of F and da and dA are the 4elds representing the areas of the surfaceelements corresponding to any convected coordinate parametrization of A(t)and A(0).

For (14) to be viewed as an evolutionary statement for the slip-system dislocationdensity, the Eux and source terms have to be de4ned from physical considerations.s(�) is undoubtedly related to dislocation generation in the lattice and consequently afunction of lattice strain and the inherent instabilities of elastic constitutive equations forcrystals which incorporate the notion of lattice periodicity. De4ning such a functionis obviously a physically delicate matter which is postponed as a subject for futureresearch — it seems that research in the area of crystal elasticity (e.g. Milstein, 1982)and nanomechanics of defects in solids (e.g. Ortiz and Phillips, 1999) will be relevantin answering this question. As to the de4nition of the local slip-system dislocation Eux,some more progress can be made at this stage.

Consider a line element dx tangent to the curve C(t). Let Vn(�)m be the dislocation

velocity, relative to the material, of dislocation lines in the direction dn(�) with trueBurgers vector in the direction im(�). Then the inward Eux of true Burgers vector,carried by dislocations of the above type crossing C(t) through the element dx, isgiven by

im(�) ⊗ �n(�)m Vn(�)

m · (dn(�) × dx) = im(�) ⊗ �n(�)m (Vn(�)

m × dn(�)) · dx (no sum): (17)

We now make the assumption that f (�) is a sum of all possible elementary Euxes ofdislocation density arising from the form (17) corresponding to system (�), i.e.

f (�) =3∑

m;n=1

im(�) ⊗ �n(�)m (Vn(�)

m × dn(�)): (18)


Consequently, the local statement of balance (14) now becomes

◦�(�) =

3∑m;n=1

im(�) ⊗ curl{�n(�)m (Vn(�)

m × dn(�))} + s(�): (19)

The kinematical basis of such a balance statement, arising from the choice of Eux givenby (18) (which is independent of the orientation of any particular plane), appears tobe approximate.

A physically reasonable constitutive assumption for the dislocation velocities of glidedislocations with dislocation lines in the directions {dn(�); n = 2; 3} that incorporatesthe Peach–Koehler idea and experimental observations (Clifton, 1983) is(

%(�)bB

)dn(�) × d1(�)

‖ dn(�) × d1(�) ‖ ; (20)

where %(�) is the resolved shear stress on system � and B is a dislocation drag coeB-cient. Of course, judging whether a particular Vn(�)

m ought to correspond to a velocityof glide dislocations or not also depends on whether the vector Fe · im(�) is perpendic-ular to d1(�) or not. Dislocation climb can be accommodated in the above formalismas well as the motion of forest dislocations (the latter, perhaps, with some degreeof non uniqueness due to the possibility of resolving the forest direction as lines onvarious slip systems). In essence, of course, the speci4cation of dislocation velocitiesrelative to the material has to rest on considerations at an atomic scale or even 4nerscale.

There is one important observation that we make in ending this section. If there isno dislocation motion relative to the material and no dislocations are generated, thenthe dislocation density should evolve up to the extent that

ddt

∫A(t)

� · n da= 0 (21)

for all material surfaces A(t) in the body. The evolution statement (19) trivially guar-antees this property, as can be observed from its derivation.

3.4. Evolution equation for slip deformation tensor

The slip deformation tensor P is introduced to account for the large deformations thatare characteristic of plastic response. It is viewed as an indicator of material deformationdue to dislocation motion, where such deformation is measured with respect to thecon4guration R. It is a two-point tensor 4eld on the con4guration R that maps vectors inR to vectors in the intermediate con4guration. Its evolution is thought of as arising fromthe homogeneous deformation of neighborhoods of material points. Such an evolutionis governed by the dislocation Eux tensors pulled back to the intermediate con4gurationas

P · P−1 =∑�

3∑m;n=1

im(�) ⊗ (−�n(�)m )(Vn(�)

m × dn(�)) · Fe (22)


with due adjustment for the di3erence in sense of the material deformation and thetrue Burgers vector.

If only glide dislocation families with true Burgers vector in the undeformed slipdirection are considered and it is assumed that the direction of the dislocation velocityvector for each such family is in the corresponding deformed slip plane, then (22) bearssome similarity to the constitutive equation for the plastic deformation in conventionalcrystal plasticity with the slip rates being constitutively speci4ed through Orowan’srelation for the slip rate. It is possible that when appropriate constitutive equations forthe dislocation velocities are incorporated, the slip systems preferred in conventionalcrystal plasticity are the ones that make the most signi4cant contributions in (22).

In general, the tensor 4eld P as de4ned in (22) is incompatible on R and henceon C. The present theory makes use of only the compatible part (P · F−1)‖ on thecon4guration C. In the context of a multiplicative decomposition of the deformationgradient, the plastic deformation 4eld Fp in this theory can be expressed as

Fp = Fe−1 · F = Fe−1⊥ · F + (P · F−1)‖ · F (23)

which amounts to expressing the fact that Fp is an additive sum of geometricallyappropriate incompatible and compatible parts arising from the dislocation density dis-tribution on the body (Fe−1

⊥ is a functional of �) and the slip deformation 4eld Prespectively.

4. Geometrically linear theory

A geometrically linear theory, motivated by the considerations of Section 3, maybe posed by considering the initial–boundary-value problem as de4ned on the con4g-uration R for all times. All di3erential operators are de4ned on function spaces onthis con4guration, with the null space of the operator curl being de4ned accordingly.Moreover, the following approximate=exact statements are used:

F = I + U; F−1 ≈ I −U;

Fe ≈ I + Ue; W ≈ I −Ue;

P ≈ I + Up;

where U is the displacement gradient, I is the second-order identity and Ue and Up

aremeasures of ‘small’ elastic and slip deformation, respectively. The convected derivativeis simply replaced by the material time derivative and a linear elastic constitutive lawis used, i.e.

T = T(Fe) ≡ C : &e;

where C is the fourth-order, positive-de4nite tensor of elastic moduli and &e :=12 (Ue + UeT). Additionally, the corresponding deformed and undeformed dislocationline directions are assumed to be identical.


Under the above assumptions, (5)–(8), (19) and (22) take the form

curlUe = �

Ue‖ = U − U

p‖

T = C : &e

divT = 0

�(�) =3∑

m;n=1

im(�) ⊗ curl{�n(�)m (Vn(�)

m × in(�))} + s(�); �=∑�

�(�)

˙Up

=∑�

3∑m;n=1

im(�) ⊗ (−�n(�)m )(Vn(�)

m × in(�))

on R; (24)

where the products Up0 ·U and ˙U

p· Up

have been neglected. If the plastic deformationis now introduced as Up :=U − Ue, then the 4nal set of governing equations for thegeometrically linear theory becomes

curlUp = −�; (25)

Up‖ = U

p‖; (26)

T = C : (&− &p); & := 12 (U + UT); &p := 1

2 (Up + UpT); (27)

divT = 0; (28)

�(�) =3∑

m;n=1

im(�) ⊗ curl{�n(�)m (Vn(�)

m × in(�))} + s(�); �=∑�

�(�); (29)

˙Up

=∑�

3∑m;n=1

im(�) ⊗ (−�n(�)m )(Vn(�)

m × in(�)); (30)

where (25)–(30) are meant to hold on R.To solve the problem of internal stress on R given an initial dislocation density 4eld

�0; Up0‖ is chosen as the gradient of a vector 4eld such that Up

0 :=Up0‖ + U

psatis4es

(27) and (28) with U ≡ 0 and appropriate traction boundary conditions, where Up

isany particular solution of (25). This procedure for determining the residual stress 4eldon R is essentially the one outlined by Eshelby (1956). The solution can be shown to beunique up to a spatially uniform skew-symmetric second-order tensor 4eld. The problemin this linear setting satis4es the necessary condition for existence of solutions to theNeumann problem that arises when vanishing traction boundary conditions are imposedfor the determination of Up

0. Material nonlinearity in this geometrically linear theory isexpected to arise in (29) from the product of the dislocation density components andthe dislocation velocity, the latter generally being a function of stress. For consistency,the initial value of the slip deformation is chosen as U

p0 = Up

0.


The necessity of condition (26) for the unique determination of a displacement 4eldbecomes transparent in this geometrically linear version. For without it, given a 4eld�, a gradient of an arbitrary vector 4eld can be added with impunity to a solution of(25) and for each such combination, there exists a displacement solution that satis4es(27) and (28).

Mura (1963) essentially suggests that for the linear theory of continuously distributeddislocations, a constitutive equation for U

p(nonsymmetric) along with (27) and (28)

suBces to de4ne the theory (up to the symmetry property of the plastic deformation,this is also customary in conventional small deformation plasticity). Such a suggestionmay be justi4ed within the present work by noting that in the absence of disloca-tion sources in (29), (25)–(26) and (29)–(30), with suitable initial conditions, implyUp = U

p. The current theory has the advantage that such a constitutive description

is de4ned by (29) and (30). For situations where a constitutive statement for plasticdeformation is to be postulated as an input to the theory, it should be noted that aslong as the dislocation velocity is such that its product with the dislocation densitycomponents is not independent of the dislocation density, any such statement for theplastic deformation should depend on the history of stress gradients, assuming that thedislocation velocity is a function of stress. Of course, in the presence of dislocationsources such a justi4cation is no longer valid. In the present theory, dislocation sourcesprovide a mechanism for predicting incremental inhomogeneous plastic deformation outof a homogeneous state since a nonvanishing source results in � being nonzero andconsequently U

pis necessarily inhomogeneous.

5. Some illustrative examples

In this section solutions are derived to simple problems corresponding to idealizeddislocation distributions belonging to two classes — solutions in which the dislocationdistribution does not evolve with time and another in which the dislocation distributionevolves with time according to the PDE governing its motion. The problems in the‘static’ category may be viewed as ones that need to be solved for the determinationof initial conditions on the slip deformation. It is to be carefully noted that these staticsolutions are not necessarily energy minimizing — instead, as will be shown, for aparticular choice of the constitutive equation for the dislocation velocity, they can beshown to be genuine equilibrium solutions of the theory. The problems are solved onin4nite domains. This is merely for simplicity in illustrating some basic features ofthe work — conventional boundary conditions for the mechanical equilibrium part ofthe problem, and the interesting interactions that arise due to their presence, are verymuch an integral part of the general theory.

5.1. Static problems

In this class of problems, the dislocation density distribution is assumed to be known.In the linear theory, the aim is to solve the 4rst, third and fourth equations of (24) or,


equivalently, (25), (27) and (28) with U = 0. In the nonlinear case, (5), (7) and (8)will be solved.

To proceed analytically, the Riemann–Graves integral operator is introduced for gen-erating particular solutions to exterior di3erential equations on star-shaped domains(Edelen, 1985; Edelen and Lagoudas, 1988). This is a class of di3erential equationsto which equations of the form curlX = � can be shown to belong. As pointed outin Edelen and Lagoudas (1988); solutions to such equations for data with compactsupport have very general spatial decay properties. Such properties have a remarkableresemblance to dislocation 4elds derived from linear elasticity.

Let �ij be a matrix function satisfying �ij; j=0 on a star-shaped domain whose pointsare generically denoted by x := (xj; j= 1; 3). Corresponding to �ij, the skew-symmetricfunctions �rjk := ejkm�rm are de4ned. For an arbitrarily chosen 4xed point x0, the inte-gral H� (to be interpreted as one symbol) is now introduced as

H�ik(x; x0) := (xj − x0j )∫ 1

0�ijk(x

0 + ((x − x0)) ( d(: (31)

It is now to be demonstrated that

)ijk :=H�ik; j − H�ij;k = �ijk (32)

which implies

curlH� = �; erjkH�ik; j = �ir : (33)

Since

H�ik; j =∫ 1

0�ijk(x

0 + ((x − x0)) ( d(

+ (xm − x0m)

∫ 1

0�imk; j(x

0 + ((x − x0))(2 d(; (34)

)ijk = 2∫ 1

0�ijk(x

0 + ((x − x0)) ( d(

+(xm − x0m)

∫ 1

0{�imk; j(x0 + ((x − x0)) − �imj;k(x0 + ((x − x0))} (2 d(;

(35)

where a subscript comma followed by a letter, say j, represents partial di3erentiationwith respect to xj. Integrating the 4rst term by parts,

)ijk = �ijk(x) +∫ 1

0{(xm − x0

m)(�imk; j − �imj;k) − (xr − x0r )�ijk; r}(2 d(: (36)

The constraint �ij; j = 0 translates to

�i23;1 + �i31;2 + �i12;3 = 0: (37)

A direct expansion of the integrand of the second term on the right-hand side of (36)and use of (37) now yields the desired result (32) which is independent of the choiceof x0 in (31).


5.2. Screw dislocation in a isotropic, linear elastic, medium

A screw dislocation in a thick in4nite plate in the x1–x2 plane is considered, withdislocation line in the positive x3 direction and Burgers vector of magnitude b. Thesurfaces of the plate perpendicular to x3 are assumed to be free of traction. The cor-responding dislocation density distribution is assumed to be

�33(x1; x2; x3) = ’(√

x21 + x2

2

); �ij = 0 if i �= 3 and j �= 3; (38)

where ’ is a positive scalar valued function of one variable. The intention is to solvethe 4rst equation of (24). Since �312 = −�321 =’ are the only nonzero components of�, choosing x0 to be the origin, we have

H�ik =2∑

m=1

xm

∫ 1

0�imk((x) ( d( (39)

and H�32 and H�31 are the only nonzero components of H�.The function ’ is chosen to vanish on and outside a cylinder of radius r0 centered

on the x3 axis and the following relation is assumed to hold:

b=∫ 2�

0

∫ r0

0’(r) r dr d+⇒

∫ r0

0’(r) r dr =

b2� : (40)

Consequently, with the substitution r ≡√x2

1 + x22,

H�31 = −x2

∫ 1

0’((r)( d(; (41)

H�32 = x1

∫ 1

0’((r)( d(;

which further implies

H�31 =b

2�

(−x2

r 2

);

r0¡r

H�32 =b

2�

( x1

r 2

); (42)

and

H�31 =−x2

r 2

∫ r

0’(s)s ds;

r ≤ r0:

H�32 =x1

r 2

∫ r

0’(s)s ds; (43)

For the choice

’(r) =

b�r0

(1r− 1r0

); r ≤ r0;

0; r0¡r;(44)


Eq. (43) evaluates to

H�31 =(−x2

r 2

)[b�r0

(r − r 2

2r0

)];

r ≤ r0:

H�32 =( x1

r 2

)[b�r0

(r − r 2

2r0

)]; (45)

The only nonzero elastic stress components out of the H� deformation 4eld are

T13 = T31 = -b

2�

(−x2

r 2

);

r0¡r;

T23 = T32 = -b

2�

( x1

r 2

);

T13 = T31 = -(−x2

r 2

)[b�r0

(r − r 2

2r0

)];

r ≤ r0;

T23 = T32 = -( x1

r 2

)[b�r0

(r − r 2

2r0

)]; (46)

where - is the shear modulus.It is now to be checked if the stress 4eld given by (46) itself satis4es equilibrium.

If so, then we are done with deriving a solution to the screw dislocation problem,assuming the tractions vanish at in4nity. Since the nonzero stress components areindependent of the x3 coordinate, the only nontrivial equilibrium equation that has tobe satis4ed is

T31;1 + T32;2 = 0: (47)

It can be checked that (47) is indeed satis4ed by (46) on 0¡r¡r0 and r0¡r. In thecontext of the satisfaction of the virtual work equations with continuous test functionshaving piecewise continuous derivatives, it is to be noted that the stress tensor iscontinuous (and consequently the tractions) on r = r0 and on any surface where thetest function derivatives are not continuous. The point-wise traction 4eld on the surfaceof a cylinder V/ of radius r=/ vanishes as /→ 0, while the virtual work contributionof the stress tensor from within the cylinder vanishes as / → 0. Consequently, (46)satis4es the virtual work equations. The elastic deformation given by the H� 4eld alsosatis4es the weak form of the 4rst of (24) if it is assumed that the test functions havevanishing tangential component at r=∞ . This is so because H� and the test functionsremain bounded on the surface of V/ as /→ 0, the integral∫

V/(H�rkekjiQri; j + �riQri) d� for all Qri in T

vanishes as / → 0; H� is continuous on r = r0 and on any surface where the testfunction derivatives may be discontinuous, and curlH� = � is satis4ed in the regions0¡r¡r0 and r0¡r.


The shear strains=stresses remain 4nite and bounded as r → 0. A singularity in ’stronger than in (44) causes unbounded strains as r → 0 and anything weaker causesvanishing strains in the same limit. Willis’ (1967) result for the 4rst-order (linear)problem is consistent with the above conclusion. Indeed, his dislocation distributionis given by a Dirac delta function and he obtains a singular distortion 4eld whilesolving exactly the same equations as being considered here (when his treatment isrestricted to the isotropic case). Interestingly, Willis’ (1967) method of solution isquite di3erent from the one being considered here. Choice (44) was made based onthe requirement of bounded but 4nite strains as r → 0 and the satisfaction of (40).It should also be noted that even though the strains are bounded they can be quitelarge thus calling into question the use of the linear theory. For instance, for r0 ≈ bit is possible to have shear strains of the order of 30% very close to the axis of thecore.

The solution presented in this section is formally similar to the method used byEdelen and Lagoudas (1988) which applies to well-de4ned, smooth dislocation densitydistributions. The di3erence in this work is that a speci4c choice for the dislocationdensity distribution within the core has been made which is singular, along with anattempt to justify the satisfaction of the 4eld equations in the weak sense.

5.3. Screw dislocation in a neo-Hookean elastic solid

The geometric description of the problem is exactly as in the previous subsection.The plate-ends perpendicular to the x3 direction have boundary constraints capable ofproviding arbitrary traction distributions on these ends. The aim is to solve (5), (7) and(8) on the con4guration R just described for a neo-Hookean elastic solid characterizedby the strain-energy function

E(Fe) =-2

(I1(Fe) − 3); (48)

where - is the shear modulus for in4nitesimal deformations and I1 is the 4rst invariantof the left Cauchy Green deformation tensor Ge ≡ Fe ·FeT. The material is considered tobe elastically incompressible in the sense that det(Fe) = 1. The Cauchy-stress responseis given as

T = 2@E@Ie1

Ge − pI = -Ge − pI; (49)

where I is the second-order identity tensor and p is the constitutively undeterminedpressure 4eld to be determined by equilibrium and boundary conditions.

If we now consider (5) with � speci4ed exactly as in the previous subsection, thena particular solution is −H� where H� is speci4ed by (42) and (45). Clearly,

W := I −H� (50)


is also a particular solution of (5). The Fe ≡ W−1 corresponding to (50) is given by

Fe := I + H�: (51)

Clearly, (51) satis4es the incompressibility constraint. The corresponding tensors Ge

and T are given in matrix form as

[Ge] =

1 0 H31

0 1 H32

H32 H32 (1 + H 231 + H 2

32)

;

[T ] =

- − p 0 T31

0 - − p T32

T31 T32 (- − p+ -(H 231 + H 2

32))

; (52)

where T31; T32 are de4ned by (46). If we now take the pressure 4eld to be the constant4eld given by p(x1; x2; x3) = -, then the only nontrivial equilibrium equation againbecomes (47) which is satis4ed as observed in the linear case. Consequently, theinternal stress distribution given by (52) with p = - quali4es as a solution to theproblem being considered. The only di3erence between the linear and nonlinear solutionfor this particular material is in the appearance of the stress component T33 given by

T33 = -b2

4�2

(1r 2

)r0¡r; T33 = -

b2

�2r20

(1 − r

2r0

)2

; 0 ≤ r ≤ r0: (53)

As long as r0 ≥ b, it is interesting to note that in the entire domain√T 2

31 + T 232¿T33

thus lending some justi4cation to the linear result where the normal stress does notappear.

Essentially, the same physical problem has been dealt with in Rosakis and Rosakis(1988) as part of a more general study of the screw dislocation in 4nite elastostatics.Their formulation of the problem and solution method is di3erent from the one pre-sented here and is based on determining strains from vector 4elds with jumps. Theirsolution for the neo-Hookean solid displays unbounded strains and stresses as r → 0.However, the solution presented in this work agrees with their solution in the regionr0¡r.

5.4. Edge dislocation in a isotropic, linear elastic medium

An identical geometry and traction boundary condition as in Section 5.2 are consid-ered. In this subsection, we suppose that all quantities are appropriately nondimension-alized so as to make physical sense. The dislocation density distribution is assumed tobe

�13(x1; x2; x3) = ’(r); �ij = 0 if i �= 1 and j �= 3 (54)


with

’(r) =

C(r0)[e−r

2 − e−r20 ]; 0 ≤ r ≤ r0; C(r0) =

b5{1 − e−r

20 (1 + r2

0)}−1;

0; r0¡r;∫ 2�

0

∫ r0

0’(r)r dr d+= b:

(55)

For this distribution of dislocation density, a particular solution of curlUe = � is givenby H� + ∇v, where ∇v is the gradient of a vector 4eld to be determined (not to bemistaken for a physical displacement 4eld) and the only nonzero components of H�are given by

H�11 =b

2�

(−x2

r 2

);

H�12 =b

2�

( x1

r 2

);

r0¡r;

H�11 =C(r0)

2

(−x2

r 2

)[1 − e−r

2(1 + r 2)];

H�12 =C(r0)

2

( x1

r 2

)[1 − e−r

2(1 + r 2)];

0 ≤ r ≤ r0

(56)

In the following, it is assumed that �3(x1; x2; x3) ≡ 0 and �1 and �2 are functions ofthe x1 and x2 coordinates only. Under these conditions, the two non-trivial equilibriumequations that have to be satis4ed by the functions �1 and �2 are

((+ 2-)�1;11 + -�1;22 + ((+ -)�2;12 = −f1;

((+ 2-)�2;22 + -�2;11 + ((+ -)�1;12 = −f2; (57)

where ( and - are the Lame parameters and the functions f1 and f2 are given by

f1 = ((+ 2-)(−b

2�

)( x2

r 2

);1

+ -(b

2�

)( x1

r 2

);2;

f2 = -(b

2�

)( x1

r 2

);1

+ ((−b

25

)( x2

r 2

);2;

r0¡r;

f1 = ((+ 2-)(−C(r0)

2

)( x2

r 2 [1 − e−r2(1 + r 2)]

);1

+-(C(r0)

2

)( x1

r 2 [1 − e−r2(1 + r 2)]

);2;

0¡r¡r0

f2 = -(C(r0)

2

)( x1

r 2 [1 − e−r2(1 + r 2)]

);1

+ ((−C(r0)

2

)( x2

r 2 [1 − e−r2(1 + r 2)]

);2; (58)


The functions f1 and f2 are square-integrable and taking the two-dimensional Fouriertransform of (57), it can be shown that

v1 =f 1 − f 2

!21 − !2

2;

v2 ={((+ 2-)!2

1 + -!22}f 2 − {((+ 2-)!2

2 + -!21}f1

((+ -)!1!2{!21 − !2

2}; (59)

where

a(!1; !2) =∫ ∞

−∞

∫ ∞

−∞a(x1; x2)e−i(!1x1+!2x2) dx1 dx2 (60)

represents the two-dimensional Fourier transform of the function a. Assuming the ex-pressions in (59) can be inverted, H�+ ∇v is the elastic deformation solution for theproblem in hand and the corresponding linear elastic stress 4eld can be generated fromit.

Even though an explicit solution has not been generated for this problem, the solutionmethod reveals the main features of the problem of internal stress in the linear set-ting — the deformation incompatibility translates to apparent body forces which drivea conventional boundary value problem. If boundaries were to be present, then thetraction on such boundaries arising from the particular solution satisfying the incom-patibility equation becomes an additional agent forcing the conventional boundary valueproblem. The problem also reveals a systematic method of attack on such problemsfor the purpose of generating approximate solutions — generate a particular solutionto the 4rst of (24) and then generate the complementary 4eld in the null space of theoperator curl which satis4es the equilibrium equations driven by the particular solutionand the boundary conditions.

5.5. Solution for the motion of a dislocation and the associated deformation

Let (c�; �= 1; 3) be a background rectangular Cartesian frame. Let �(�) be expressedin terms of this basis as �(�)

�- c�⊗ c-. Assuming no dislocation sources are present, (29)may be expressed as

�(�)�- c� ⊗ c- =

3∑m;n=1

(im(�))�e-/9{e9�:�n(�)m (Vn(�)

m )�(in(�)):};/c� ⊗ c-: (61)

Considering only one slip system with (c�; � = 1; 3) coinciding with (im; m = 1; 3) forthat slip system, the component version of (61) is

�mp =3∑n=1

epsq{eqrn�nm(Vnm)r}; s (no sum on m) (62)

With the above identi4cation, �nm = �mn. Additionally, we consider dislocation densitydistributions of the form

�ij(x1; x2; x3; t) = bitj(x1; x2; x3; t); (63)


where b (dimension Length) is a constant and spatially uniform vector 4eld and t isanother vector 4eld to be determined. We also assume that for all values of m; n thedislocation velocity vectors take the form

Vnm = Vt × n0

‖ t ‖ ; (64)

where n0 is the constant and spatially uniform unit normal 4eld of the slip systembeing considered. Under the above assumptions, (62) reduces to

tp = epsq{‖ t ‖ (n0)q}; s: (65)

Since (n0)q = <1q, (65) can now be written as the set of equations

t1 = 0;

t2 = {‖ t ‖};3;t3 = {− ‖ t ‖};2: (66)

It is to be noted at this point that if the choice

V (x1; x2; x3; t) =f(x1; t)‖ t0 ‖ (67)

is made, where t0 is the initial condition on the 4eld t and f is a function of thearguments displayed, then we have an equilibrium solution of the theory. The problemsconsidered in Sections 5.2 and 5.4 4t into the class of dislocation density distributionsbeing considered here and, as such, those solutions are genuine equilibrium solutionsof the nonequilibrium theory for the choice of dislocation velocity given in (67).

It is now assumed for initial conditions that

t1(x1; x2; x3; 0) = t2(x1; x2; x3; 0) ≡ 0; t3(x1; x2; x3; 0) = !(x1; x2) ≥ 0; (68)

where ! vanishes on the surface and outside a cylinder of radius r0 whose axis isthe x3 direction, i.e. a screw or edge dislocation, depending upon b, with dislocationline in the slip plane in the direction n0 × m0. ! is assumed to have dimensions of(1=Length2). For simplicity it is further assumed that V is a constant, although theessential nature of the conclusion to be drawn shortly is not a3ected by taking it to bea known function of x1 and x2. It is assumed for the moment that

t1(x1; x2; x3; t) = t2(x1; x2; x3; t) ≡ 0; (69)

in which case the only equation that remains to be solved is

t3 = −V |t3|;2: (70)

The solution to (70) and the second of (68) is given by

t3(x1; x2; x3; t) = !(x1; x2 − Vt): (71)

It can now be seen that (69) and (71) indeed constitute a solution of (66) with theinitial conditions (68).

Solution (71) corresponds to the initial dislocation travelling along the x2-axis in thesense of the sign of V with constant speed |V |, as intuitively expected. In an in4nite


medium, the stress 4elds derived in Sections 5.2 and 5.4 translate with the dislocationalong with some accumulation of permanent deformation due to the motion of thedislocation which is dealt with next.

Let the dislocation under consideration be an edge with Burgers vector given byb = bm0, b¿ 0. Then, with the solution obtained above for t,

˙Up

= −bm0 ⊗ (Vm0 × !i3) ⇒ ˙Up(x1; x2; x3; t) = −b!(x1; x2 − Vt)Vm0 ⊗ n0:

(72)

Consequently,

Up(x1; x2; x3; t) = U

p(x1; x2; x3; 0) − b

{∫ x2

x2−Vt!(x1; s) ds

}m0 ⊗ n0: (73)

Let x2 = L be a free edge of the crystal and let the dislocation axis be situated atx1 = x2 = 0 at time t = 0. It should be noted that in the presence of the boundary, thestress 4eld in the body at times when the dislocation is within it is not the same aswould be derived by translating the 4eld for an in4nite body along with the dislocation,even if the problem is treated as a quasi-static one from the point of view of mechanicalforce balance.

It is now of interest to know the state of the body at times t ¿ (L+ r0=V ) when allexternal loads are released. Since the entire dislocation exits the body for such times,the � distribution in the body is given by � ≡ 0. Moreover, (73) implies that the onlypossible nonzero component of curl U

pis given by

(curl Up)23(x1; x2; x3; t) = (curl U

p)23(x1; x2; x3; 0) + b

{∫ x2

x2−Vt!(x1; s) ds

};2

= −b!(x1; x2) + b{!(x1; x2) − !(x1; x2 − Vt)}= −b!(x1; x2 − Vt) (74)

and for the times of interest, (74) implies that the Up

distribution on the body iscompatible. Consequently, utilizing the unique solution to (25) and (26) given by

Up = Up

(75)

(for all times in the absence of sources) and the fact that there are no loads on thebody, (25)–(28) are all satis4ed by (75) and U = U

pfor t ¿L + r0=V . Hence, the

only long-term e3ect of the passage and exit of the edge dislocation from the bodyis a ‘stress-free’ compatible shear strain distribution along the path of the dislocation,as intuitively expected. The strain distribution is directly related to the nature of thedislocation distribution de4ned by !(x1; x2). In particular, if it is a bell-shaped pro4leon the x1 − x2 plane as in Section 5.4, then (73) indicates (assuming r0=L�1 so thatU

p0(x1; L; x3; 0) ≈ 0) that the shape of the deformed boundary at x2 = L is given by a

smoothly varying displacement u2 in the x2 direction which decreases monotonicallyfrom a constant maximum with vanishing slope (u2;1) for all x1 ≤ −r0 to a minimumat x1 = r0 where the slope again vanishes and remains 0 for x1 ≥ r0. This pro4le maybe interpreted as a slip-step over a distance 2r0.


6. Conclusion

A theory of crystal plasticity has been proposed based on dislocation density asthe primary internal variable. The instantaneous spatial distribution of this internalvariable, along with the applied loads, determines the current state of stress in thebody. The permanent deformation produced is a function of the history of this internalvariable. The internal variable itself evolves according to an equation based on anidealization of the motion of dislocation lines on slip planes. The theory appears toproduce physically reasonable results but much work remains to be done in evaluatingit further and exploring its physical implications.

Future work will involve exploring the nature of boundary conditions that are re-quired, if any, for the evolution equations for the slip system dislocation densities. Inan incremental setting this is not an issue — however, it should be kept in mind thatthe knowledge of the time rate of a function at the beginning of a time interval isnot enough to generate the function exactly in that interval, no matter how small. Thisis the consideration that changes an algebraic problem for the rates in an incrementalversion of the present theory to a PDE. It is not clear to this author that such a tran-sition necessarily needs extra boundary conditions for the evolutionary equations. Thestress and displacement boundary conditions could conceivably pose implicit boundaryconditions. For example, in the problem considered in Section 5.5, if the boundary atx2 = L were not allowed to deform then it is clear that the dislocation velocity cannotbe a constant since that would produce an inconsistency — the dislocation would haveto exit the material producing a slip-step which the boundary condition disallows. Ifthe dislocation velocity is a function of the stress, however, the dislocation would beobstructed before reaching the boundary by a suitable stress 4eld being set up so thatthe rigid boundary condition as well as the constitutive equation can be respected. Bythe same token, however, it is not clear to this author what the situation would be ifthere was a 4nite boundary at some value of negative x2. In Section 5.5 treating thegeometry as in4nite was necessary for the choice of constant dislocation velocity, forotherwise the solution would not be de4ned for all times without boundary conditions.

However, with the existing relationship between U, ˙Up, and � and the introduction of

another one through the constitutive equation for the dislocation density, it is hard tojudge the adequacy or inadequacy of the existing boundary conditions for the problemwithout further mathematical analysis of, what appears to be, a rather diBcult nature. Ifsuch conditions are indeed necessary, the physical nature and geometric interpretationof the dislocation Eux tensors in the theory can aid in their speci4cation.

It would also be desirable to explore the relationship between the incompatibilityof the slip deformation tensor 4eld and the dislocation density in the context of thenonlinear theory in the absence of sources. In the proposed geometrically linear theorywith no source distributions, these quantities are identical up to a sign.

Thermodynamic restrictions on the theory also need to be explored as an elementaryguide to avoiding physically unrealistic predictions. Such restrictions can also provideuseful insights into some necessary conditions that the constitutive equations for thevarious dislocation velocities have to obey. Given the nonlocal nature of the theory,deriving such restrictions does not appear to be a simple matter. Moreover, what the


appropriate statement of the second law of thermodynamics ought to be for genuinelynonlocal materials is also a matter that is not completely obvious or settled. Someadvances along these lines, as they pertain to the work presented in this paper, areclearly desirable.

Finally, a numerical implementation of the theory needs to be pursued. The basicingredient here will be an eBcient procedure to generate a basis for the null space ofthe discretized operator corresponding to the curl, and a method to generate projectionsof discretized functions on this null space. This paper provides a general method fordetermining stress 4elds of 3-D dislocation distributions that can be translated intoa numerical method in a conceptually straightforward manner. It is hoped that thenumerical implementation of such a method, when combined with an implementationof the non(thermodynamic) equilibrium aspects of the theory, will lead to a capabilityof solving problems in dislocation mechanics routinely to enable a better understandingof the fundamentals of plastic deformation.

Acknowledgements

This work was supported by the Center for Simulation of Advanced Rockets at theUniversity of Illinois at Urbana-Champaign under U.S. Department of Energy subcon-tract B341494.

References

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a model of crystal plasticity based on the theory of continuously distributed dislocations

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