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

Journal of the Mechanics and Physics of Solids 49 (2001) 761 – 784

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A model of crystal plasticity based on the theory of 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 response of single crystals with structural dimensions of ∼ 100
m or less, based on ideas rooted in the theory of continuously distributed dislocations. The constitutive inputs of the theory relate explicitly to dislocation velocity, dislocation generation and crystal elasticity. Constitutive nonlocality is a natural consequence of the physical considerations of the model. The theory reduces to the nonlinear elastic theory of continuously distributed dislocations in the case of a nonevolving dislocation distribution in the material and the nonlinear theory of elasticity in the absence of dislocations. A geometrically linear version of the theory is also developed. The work presented in this paper is intended to be of use in the prediction of time-dependent mechanical response of bodies containing a single, a few, or a distribution of dislocations. A few examples are solved to 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=strain 4elds of a screw dislocation in an in4nite, isotropic, linear elastic solid is derived. A solution for an in4nite, neo-Hookean nonlinear elastic continuum is also derived. Both solutions match with existing results outside the core region. Bounded solutions are predicted within the core in both cases. The edge dislocation in the isotropic, linear theory is also discussed in the context of this work. Assuming a constant dislocation velocity for simplifying the analysis, an evolutionary solution resulting in a slip-step on the boundary of a stress-free crystal produced due to the c 2001 Elsevier Science Ltd. All passage and exit of an edge dislocation is also described.  rights reserved. Keywords: Continuous distribution of dislocations; B. Crystal plasticity; Dislocation mechanics

∗ Correspondence address: Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh 15213-3890, USA. Tel.: +1-412-268-4566; Fax: +1-412-268-7813. E-mail address: [email protected] (A. Acharya).

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

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1. Introduction It is well established that permanent deformation in a crystalline solid arises due to dislocation motion. It is also believed that the stress 4eld in a body drives the motion of dislocations. Since dislocations themselves cause an internal stress 4eld in the body which depends on their spatial distribution, such stresses, in combination with those that arise in accommodating applied boundary conditions, result in a redistribution of the dislocation 4eld in a body. Such a spatial redistribution alters the stress state thus inducing further motion, or resistance to motion, of the dislocation 4eld. Conceptually, this cycle can be thought of as a coupled process that may or may not lead to equilibrium states (in the sense of state evolution) depending upon the nature of the applied boundary and initial conditions, the material constitution of the body which leads to its elasticity and the generation and motion of dislocations in it. It is the aim of this paper to attempt a mathematical description of the above coupled process that results in a closed continuum theory (in the sense of eliminating nonuniqueness due to a lack of suBcient physical statements), whose inputs are constitutive statements for crystal elasticity and dislocation generation and motion. It is only reasonable to demand that the theory should reduce to nonlinear elasticity in the absence of dislocations and to the nonlinear elastic theory of continuously distributed dislocations (ECDD) (Willis, 1967) in the absence of evolution of the dislocation density 4eld. The requirement of closure in the theory is an important one when viewed in the context 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 stress on a known reference con4guration for arbitrary divergence-free dislocation density 4elds is the one that has to be eliminated in order to have a theory which admits a displacement 4eld on the said reference, which varies as the dislocation density 4eld on the body is altered. This observation is discussed in greater detail in the following sections of the paper. In considering the evolution of state within the theory, motivation is drawn from the work of Mura (1963) and Kosevich (1979). However, the present theory adopts a di3erent stance with respect to the speci4cation of the plastic deformation and the dislocation Eux in comparison to these earlier works. A statement on dislocation density evolution appears, and is used, as a partial di3erential equation (PDE) which is coupled to the stress 4eld, if the constitutive equations for the dislocation velocity and sources are assumed to depend on the stress. With such a mechanism for generating the current dislocation density 4eld in hand, evolutionary statements for the dislocation Eux and slip deformation tensors are completely speci4ed within the theory. Roughly speaking, the plastic deformation is developed as an additive sum of geometrically appropriate compatible and incompatible parts generated from the slip deformation and dislocation density 4elds respectively. Mura (1963) and Kosevich (1979), in the context of small deformations, develop statements for the dislocation density similar in spirit to the one developed here as part of their kinematical analysis but refrain from using it as a 4eld equation for the determination of the dislocation density. In the context of exact kinematics, Fox (1966) derives a relationship for the evolution of his dislocation line

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densities which is conceptually similar to the evolutionary statement derived as part of this work if dislocation sources are not included. As will be shown in Section 4, the inclusion of dislocation sources provides the theory with the mechanism of predicting an incrementally inhomogeneous deformation from a homogeneous state, a capability that has signi4cant physical implications (Hutchinson, 2000). Motivated by the structure of the nonlinear theory, a geometrically linear version is also developed. It transparently brings to the forefront two major implications on theories of plasticity based on the postulation of a constitutive statement for plastic strain. Firstly, if the dislocation velocity is a function only of stress then the constitutive statement for the plastic strain rate is necessarily a function of the integrated history of spatial gradients of stress. Secondly, if dislocation sources are included, then the plastic strain rate is a genuinely nonlocal function of such source distributions. The theory presented herein is intended to be applicable to dislocation related phenomena in crystalline solids that appear well resolved at length scales of 103 b − 10b, where b is the lattice spacing in the crystal. If the dislocation density 4eld on the initial con4guration is provided as an initial condition along with the applied traction 4eld that maintains the body in equilibrium in the initial con4guration, then the equations of the theory for determining the residual stress 4eld on this con4guration reduce 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 generating solutions to the above problem on 4nite domains for nonlinear elastic laws of arbitrary symmetry, a method that can easily form the basis of a numerical algorithm based on the 4nite-element method. Because the theory is not restricted to linearity in the elastic constitutive law, it appears to be within the bounds of the theory to model some aspects of both long and short-range dislocation interactions of distributions of dislocations and hence the hardening that arises due to back stresses as well as strain hardening. How far such predictions can match up with reality remains to be seen at this point. The change in stress response from purely elastic behavior associated with the 4rst generation of dislocations in a virgin crystal appears to be within the predictive range of the theory — interestingly, such a notion is necessarily accompanied by inhomogeneous deformation. If yielding is associated with the onset of motion of preexisting dislocations, the associated stress-response history would also seem to be within the predictive scope of this work. The theory appears to be suitable for the response of bodies with structural dimensions in the mesoscale and perhaps even smaller scales, given the occasional success of the linear elastic theory of dislocations for nanomaterials. Rigorous averaging of the theory can lead to a plasticity model with resolution in the meso-microscale which naturally 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

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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 screw dislocation in an anisotropic material. 1 Given the 4eld-theoretic nature of the above formulation and keeping in mind the current goal of developing a nonequilibrium theory of dislocation distributions which accounts for the stress 4elds of such distributions rigorously, it is perhaps natural to ask if the ECDD could be used with an evolutionary statement for the dislocation density to achieve such a goal. Roughly speaking, the expectation is that as the dislocation density evolves due to the action of stresses in the body and as the applied boundary conditions change, the body should deform in a deterministic way and this should be a natural outcome of the theory. This question is pursued in this section. Consider a star-shaped con4guration 2 R on a part of whose boundary, @Rt ; known Cauchy tractions t1 are applied and the rest of the boundary, @Rx , is held 4xed. Obviously, the boundary constraint on @Rx also provides any traction distribution required to hold the body in overall static equilibrium. Let Tˆ be the elastic constitutive equation and W1 the inverse elastic deformation (or distortion) tensor. 3 A dislocation density 4eld 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, when integrated over such an area. It is further assumed that any dislocation density 4eld is divergence-free on R. Let us assume that there is a dislocation density 4eld 1 speci4ed on the body such that the following equations hold: curl W1 = −1 ;

ejrs (W1 )is; r = (−1 )ij

ˆ −1 ); T1 := T(W 1

div T1 = 0;

on R;

(T1 )ij; j = 0

on R;

(1) (2)

where di3erentiation is understood with respect to coordinates established from a rectangular Cartesian frame with respect to which all tensor indices are also expressed in this paper. Also, the symbol erjk represents a component of the alternating tensor. The stress component Tij represents the component in the xi direction of the force on unit area 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 de4ned to be opposite in direction to the line integral of the inverse elastic deformation along any 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 the stress 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 constraints on @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 of solution. 2

A topological constraint that ensures that there exists a point in the con4guration such that the line joining 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.

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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 loading has changed and the body has not been allowed to deform. However, we 4nd that this physical expectation is not borne out if (1) – (2) form the basic equations of the theory ˜ 2 + ∇w, where ˜ 2 . Then W — for consider a particular solution of (1) given by W2 = W w is any vector 4eld on R and ∇ represents the gradient, is also a solution to (1). In terms of this solution to (1), (2) takes the form ˆ W ˜ 2 + ∇w}−1 ); T2 := T({

div T2 = 0;

(T2 )ij; j = 0

on R;

(3)

where the following boundary conditions can be imposed to determine the function w: ˆ W ˜ 2 + ∇w}−1 ) · n = t2 on @Rt ; T({

w=0

on @Rx ;

(4)

where n is the outward unit normal 4eld on the boundary of R. Eqs. (3) and (4) 4t the speci4cation of a problem in the usual nonlinear theory of elasticity with an unusual ˜ 2 and, consequently, it is constitutive equation for the stress due to the presence of W perhaps reasonable to assume that there exists a solution to the problem (if we consider the linear version of the problem posed by (3) and (4), then it can be shown that a solution de4nitely exists). An alternative way of summarizing the above discussion is to conclude that if a physical displacement 4eld of R were to be included in the problem 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 a 4eld. Based on such motivation, we conclude that the ECDD has to be extended, beyond the speci4cation of an evolution equation for the dislocation density, to enable the prediction of deformation of the body in response to changes in the applied boundary conditions and the internal state of the material characterized by the dislocation density. 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 operator curl on the current con4guration C has a nontrivial null space in that it contains, at least, 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 constitutive assumption, then it can be shown that the solution to such an augmentation of the problem de4ned by (1) is unique in a sense that can be made precise, assuming for the moment that C is known. In reality, C has to be determined and this may be viewed as solving the equilibrium equations (2) and corresponding boundary conditions, assuming W is known from the solution of the augmentation of (1) just described (note that the unknown con4guration enters in the de4nition of the operator div and the boundary conditions). Of course, the combined, nonlinear problem of the determination of C and W has to proceed in tandem. The constitutive equation for the null-space component

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is chosen in such a way that in the absence of dislocations in the body, the problem reduces to solving the problem of nonlinear elasticity; in the presence of a dislocation density distribution 'xed in the material and changing only to the extent of elastic material deformation induced by external loads, the reduction is to the ECDD phrased as a problem on the 4nal deformed con4guration. Henceforth, we choose the con4guration of the as-received body as the reference con4guration 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 as F. It is now our aim to discuss some mathematical machinery by which we can speak about a unique decomposition of tensor 4elds on C into orthogonal components in a suitable sense. These components are such that one part is annihilated by the operator curl and the other part is orthogonal to the part that is annihilated. With this objective in mind, consider the space D of all square-integrable 3×3 matrix 4elds on C (thought of 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 matrix inner product of two matrix 4elds belonging to D, i.e.
Aij Bij d: (A; B)D = C

Let T be a set of continuous test functions with vanishing tangential component on the boundary of C and at least piece-wise continuous 4rst derivatives in C. On the boundary of C with unit normal 4eld n, any such test function satis4es Qri − (Qrj nj )ni = 0. A suitable weak form for curl W = −, arrived at by formally integrating the equation by parts, is

Wrk ekji Qri; j d = − ri Qri d for all Qri in T: C

C

Henceforth, whenever we write an equation of the form curl X = , we always mean it in the sense of such a weak formulation. Similarly, when required to make sense due to smoothness constraints, solutions to the equilibrium equations will also be interpreted as solutions to the corresponding weak problem, i.e. the virtual work equation for statics. The linear subspace N (curl) of D consists of all functions W belonging to D that satisfy
Wrk ekji Qri; j d = 0 for all Qri in T: C

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 that every 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 algorithm to generate the orthogonal projection in the null space.

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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 speci4ed in Section 3.4. Assuming div  = 0 on C, the equations for the determination of the con4guration C and the 4eld W on it are given as curl W = −;

(5)

W = (P · F−1 ) ;

(6)

ˆ −1 ); T = T(W

(7)

div T = 0

(8)

with the usual traction and=or displacement boundary conditions for the force equilibrium part of the problem, where (5) – (8) are meant to hold on C. The constraint div  = 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). This constraint has the status of the node-theorem of dislocation theory, i.e. the continuous analog of the statement that the sum of the Burgers vector of all dislocation lines meeting at a node vanishes, which also implies that a dislocation line cannot end within a body. If 0 , the dislocation density 4eld at the initial instant, is speci4ed on R and the intention is to solve for the state of stress on R then only (5), (7), (8) need be solved (only traction boundary conditions make sense for this problem of internal stress on a known con4guration). The inverse elastic deformation 4eld at the initial instant, W0 , can be determined by following exactly the discussion in Section 2. In essence, (W0 ) becomes the gradient of a vector 4eld that is used to satisfy (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 known to the author (the corresponding situation in the linear theory will be discussed in Section 4). For consistency, the initial condition on P should be chosen as P0 = W0 . 3.2. Dislocation density tensor We de4ne the undeformed dislocation line directions for the th slip system as follows: i1() := n0 ;

i2() := m0 ;

i3() := n0 × m0 ;

m0

(9)

n0

where is the unit slip direction and is the unit slip normal for the system . Corresponding to {im() ; m = 1; 3}, we de4ne the deformed dislocation line directions in the current con4guration given by d1() =

Fe−T · i1() ;  Fe−T · i1() 

d2() =

Fe · i2() ;  Fe · i2() 

d3() =

Fe · i3() ;  Fe · i3() 

(10)

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where Fe := W−1 is introduced to be in accord with familiar notation from crystal plasticity and the symbol  a  represents the magnitude of the vector a. Let {dm() ; ()−1 n() m = 1; 3} be the dual basis corresponding to {dm() ; m = 1; 3}, i.e. dm() = gmn d , () m() · dn() . Then where g represents the matrix whose mth-row, nth-column entry is d () , the th slip system dislocation density tensor, may be expressed as () = mn() im() ⊗ dn() ;

mn() = im() · () · dn() :

(11)

The components (mn() =b) have the natural interpretation of being the number of dislocation lines threading unit area perpendicular to dn() in the deformed lattice whose true Burgers vectors point in the direction im() in the intermediate con4guration. Consequently, these components may be thought of as idealized representations of screw and edge dislocation densities in the body in its current state. In assigning a slip system dislocation density tensor (e.g. for the de4nition of initial conditions), essentially Nye’s (1953) conceptual prescription can be followed. On system , let nl be the number of dislocation lines along the unit direction l per unit of area perpendicular to l. Let these dislocations have Burgers vector in the direction bl . The contribution to mn() from this family of dislocations is (bl · im() )(l · dn() )nl . Similar contributions from dislocation families oriented di3erently may simply be added. In terms of such an initial assignment from observations, it is possible that slip system dislocation density tensors start out with vanishing forest components. Glide dislocations on such systems still experience interactions with forest obstacles provided by glide dislocations on other slip systems since the dislocation velocity typically is a function of stress and the stress is a functional of the total dislocation density in the body 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 dislocation Eux 4eld for the th slip-system which measures the rate of inEow of dislocation lines 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 the rate of generation of dislocations, per unit area, in the region of the deformed lattice traced out by the deforming material surface A. Then it seems reasonable to say that


d () ()  · n da = f · dx + s() · n da (13) dt A(t) C(t) A(t) 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 for the change of dislocation content in the material surface A(t) due to material motion perpendicular to it. Since () is a two-point tensor delivering the true (undeformed)

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Burgers vector of dislocation lines and it is physically reasonable to assume that dislocation lines do not end inside the material, such a contribution to the balance statement is 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 Eow de4ned by the material velocity) of the two-point tensor 4eld . In the above, v represents the material velocity 4eld, ˙ the material time derivative of , and L the velocity gradient. Eq. (15) is a direct consequence of using the identity n da = J F−T · N dA;

(16)

where n and N are the unit normal 4elds on A(t) and A(0), respectively, J is the determinant of F and da and dA are the 4elds representing the areas of the surface elements 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 dislocation density, the Eux and source terms have to be de4ned from physical considerations. s() is undoubtedly related to dislocation generation in the lattice and consequently a function of lattice strain and the inherent instabilities of elastic constitutive equations for crystals which incorporate the notion of lattice periodicity. De4ning such a function is obviously a physically delicate matter which is postponed as a subject for future research — 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 relevant in 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 Vmn() be the dislocation velocity, relative to the material, of dislocation lines in the direction dn() with true Burgers 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, is given by im() ⊗ mn() Vmn() · (dn() × dx) = im() ⊗ mn() (Vmn() × dn() ) · dx (no sum): (17) We now make the assumption that f () is a sum of all possible elementary Euxes of dislocation density arising from the form (17) corresponding to system (), i.e. f () =

3  m; n=1

im() ⊗ mn() (Vmn() × dn() ):

(18)

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

() =

3 

im() ⊗ curl{mn() (Vmn() × dn() )} + s() :

(19)

m; n=1

The kinematical basis of such a balance statement, arising from the choice of Eux given by (18) (which is independent of the orientation of any particular plane), appears to be approximate. A physically reasonable constitutive assumption for the dislocation velocities of glide dislocations with dislocation lines in the directions {dn() ; n = 2; 3} that incorporates the Peach–Koehler idea and experimental observations (Clifton, 1983) is  ()  dn() × d1() % b ; (20) B  dn() × d1()  where %() is the resolved shear stress on system  and B is a dislocation drag coeBcient. Of course, judging whether a particular Vmn() ought to correspond to a velocity of glide dislocations or not also depends on whether the vector Fe · im() is perpendicular to d1() or not. Dislocation climb can be accommodated in the above formalism as well as the motion of forest dislocations (the latter, perhaps, with some degree of non uniqueness due to the possibility of resolving the forest direction as lines on various slip systems). In essence, of course, the speci4cation of dislocation velocities relative to the material has to rest on considerations at an atomic scale or even 4ner scale. There is one important observation that we make in ending this section. If there is no dislocation motion relative to the material and no dislocations are generated, then the dislocation density should evolve up to the extent that
d  · n da = 0 (21) dt A(t) for all material surfaces A(t) in the body. The evolution statement (19) trivially guarantees 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 that are characteristic of plastic response. It is viewed as an indicator of material deformation due to dislocation motion, where such deformation is measured with respect to the con4guration R. It is a two-point tensor 4eld on the con4guration R that maps vectors in R to vectors in the intermediate con4guration. Its evolution is thought of as arising from the homogeneous deformation of neighborhoods of material points. Such an evolution is governed by the dislocation Eux tensors pulled back to the intermediate con4guration as P˙ · P−1 =

3   

m; n=1

im() ⊗ (−mn() )(Vmn() × dn() ) · Fe

(22)

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with due adjustment for the di3erence in sense of the material deformation and the true Burgers vector. If only glide dislocation families with true Burgers vector in the undeformed slip direction are considered and it is assumed that the direction of the dislocation velocity vector for each such family is in the corresponding deformed slip plane, then (22) bears some similarity to the constitutive equation for the plastic deformation in conventional crystal plasticity with the slip rates being constitutively speci4ed through Orowan’s relation for the slip rate. It is possible that when appropriate constitutive equations for the dislocation velocities are incorporated, the slip systems preferred in conventional crystal 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 hence on C. The present theory makes use of only the compatible part (P · F−1 ) on the con4guration C. In the context of a multiplicative decomposition of the deformation gradient, the plastic deformation 4eld Fp in this theory can be expressed as e−1 · F + (P · F−1 ) · F Fp = Fe−1 · F = F⊥

(23)

which amounts to expressing the fact that Fp is an additive sum of geometrically appropriate incompatible and compatible parts arising from the dislocation density dise−1 tribution on the body (F⊥ is a functional of ) and the slip deformation 4eld P respectively. 4. Geometrically linear theory A geometrically linear theory, motivated by the considerations of Section 3, may be posed by considering the initial–boundary-value problem as de4ned on the con4guration R for all times. All di3erential operators are de4ned on function spaces on this con4guration, with the null space of the operator curl being de4ned accordingly. Moreover, the following approximate=exact statements are used: F = I + U; Fe ≈ I + U e ;

F−1 ≈ I − U; W ≈ I − Ue ;

p

P ≈ I + U˜ ; p where U is the displacement gradient, I is the second-order identity and Ue and U˜ are measures of ‘small’ elastic and slip deformation, respectively. The convected derivative is simply replaced by the material time derivative and a linear elastic constitutive law is used, i.e.

ˆ e ) ≡ C : &e ; T = T(F where C is the fourth-order, positive-de4nite tensor of elastic moduli and &e := 1 e eT 2 (U + U ). Additionally, the corresponding deformed and undeformed dislocation line directions are assumed to be identical.

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Under the above assumptions, (5) – (8), (19) and (22) take the form  curl Ue =      p e   U = U − U˜     e   T=C:&      div T = 0 ()

˙

3 

=

i

m()

⊗

curl{mn() (Vmn()

×i

n()

)} + s

()

; =





m; n=1

3   p im() ⊗ (−mn() )(Vmn() × in() ) U˜˙ = 



m; n=1

()

               

on R;

(24)

p p p where the products U˜ 0 · U and U˜˙ · U˜ have been neglected. If the plastic deformation p e is now introduced as U := U − U , then the 4nal set of governing equations for the geometrically linear theory becomes

curl Up = −;

(25)

p Up = U˜  ;

(26)

T = C : (& − &p );

& := 12 (U + UT );

T

&p := 12 (Up + Up );

(27)

div T = 0; ()

˙

=

(28)

3 

im() ⊗ curl{mn() (Vmn() × in() )} + s() ;

m; n=1 3   p im() ⊗ (−mn() )(Vmn() × in() ); U˜˙ = 

=



() ;

(29)



(30)

m; n=1

where (25) – (30) are meant to hold on R. To solve the problem of internal stress on R given an initial dislocation density 4eld p p p 0 ; U0 is chosen as the gradient of a vector 4eld such that U0p := U0 + Uˆ satis4es p (27) and (28) with U ≡ 0 and appropriate traction boundary conditions, where Uˆ is any particular solution of (25). This procedure for determining the residual stress 4eld on R is essentially the one outlined by Eshelby (1956). The solution can be shown to be unique up to a spatially uniform skew-symmetric second-order tensor 4eld. The problem in this linear setting satis4es the necessary condition for existence of solutions to the Neumann problem that arises when vanishing traction boundary conditions are imposed for the determination of U0p . Material nonlinearity in this geometrically linear theory is expected to arise in (29) from the product of the dislocation density components and the dislocation velocity, the latter generally being a function of stress. For consistency, p the initial value of the slip deformation is chosen as U˜ 0 = U0p .

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The necessity of condition (26) for the unique determination of a displacement 4eld becomes 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 distributed p dislocations, a constitutive equation for U˜ (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 suggestion may be justi4ed within the present work by noting that in the absence of dislocation sources in (29), (25) – (26) and (29) – (30), with suitable initial conditions, imply p Up = U˜ . The current theory has the advantage that such a constitutive description is de4ned by (29) and (30). For situations where a constitutive statement for plastic deformation is to be postulated as an input to the theory, it should be noted that as long as the dislocation velocity is such that its product with the dislocation density components is not independent of the dislocation density, any such statement for the plastic deformation should depend on the history of stress gradients, assuming that the dislocation velocity is a function of stress. Of course, in the presence of dislocation sources such a justi4cation is no longer valid. In the present theory, dislocation sources provide a mechanism for predicting incremental inhomogeneous plastic deformation out of a homogeneous state since a nonvanishing source results in ˙ being nonzero and p consequently U˙ is necessarily inhomogeneous.

5. Some illustrative examples In this section solutions are derived to simple problems corresponding to idealized dislocation distributions belonging to two classes — solutions in which the dislocation distribution does not evolve with time and another in which the dislocation distribution evolves 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 determination of initial conditions on the slip deformation. It is to be carefully noted that these static solutions are not necessarily energy minimizing — instead, as will be shown, for a particular choice of the constitutive equation for the dislocation velocity, they can be shown to be genuine equilibrium solutions of the theory. The problems are solved on in4nite domains. This is merely for simplicity in illustrating some basic features of the work — conventional boundary conditions for the mechanical equilibrium part of the problem, and the interesting interactions that arise due to their presence, are very much 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,

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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 generating particular solutions to exterior di3erential equations on star-shaped domains (Edelen, 1985; Edelen and Lagoudas, 1988). This is a class of di3erential equations to which equations of the form curl X =  can be shown to belong. As pointed out in Edelen and Lagoudas (1988); solutions to such equations for data with compact support have very general spatial decay properties. Such properties have a remarkable resemblance to dislocation 4elds derived from linear elasticity. Let ij be a matrix function satisfying ij; j =0 on a star-shaped domain whose points are generically denoted by x := (xj ; j = 1; 3). Corresponding to ij , the skew-symmetric functions ˆrjk := ejkm rm are de4ned. For an arbitrarily chosen 4xed point x0 , the integral H (to be interpreted as one symbol) is now introduced as
1 0 0 (31) ˆijk (x0 + ((x − x0 )) ( d(: Hik (x; x ) := (xj − xj ) 0

It is now to be demonstrated that )ijk := Hik; j − Hij; k = ˆ ijk

(32)

which implies erjk Hik; j = ir :

curl H = ; Since


Hik; j =

)ijk

0

1

(33)

ˆijk (x0 + ((x − x0 )) ( d(


1 + (xm − xm0 ) ˆimk; j (x0 + ((x − x0 ))(2 d(; 0
1 0 ˆ =2 ijk (x + ((x − x0 )) ( d(

(34)

0

+(xm −

xm0 )




1

0

{ˆimk; j (x0 + ((x − x0 )) − ˆimj; k (x0 + ((x − x0 ))} (2 d(; (35)

where a subscript comma followed by a letter, say j, represents partial di3erentiation with respect to xj . Integrating the 4rst term by parts,
1 ˆ {(xm − xm0 )(ˆimk; j − ˆimj; k ) − (xr − xr0 )ˆijk; r }(2 d(: (36) )ijk = ijk (x) + 0

The constraint ij; j = 0 translates to + ˆ + ˆ = 0: ˆ i23; 1

i31; 2

i12; 3

(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 choice of x0 in (31).

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5.2. Screw dislocation in a isotropic, linear elastic, medium A screw dislocation surfaces of responding

dislocation in a thick in4nite plate in the x1 –x2 plane is considered, with line in the positive x3 direction and Burgers vector of magnitude b. The the plate perpendicular to x3 are assumed to be free of traction. The cordislocation density distribution is assumed to be   33 (x1 ; x2 ; x3 ) = ’ x12 + x22 ; ij = 0 if i = 3 and j = 3; (38)

where ’ is a positive scalar valued function of one variable. The intention is to solve the 4rst equation of (24). Since ˆ312 = −ˆ321 = ’ are the only nonzero components of , ˆ choosing x0 to be the origin, we have
1 2  Hik = xm ˆimk ((x) ( d( (39) 0

m=1

and H32 and H31 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:
2
r0
r0 b b= ’(r) r dr d+ ⇒ ’(r) r dr = : (40) 2 0 0 0

Consequently, with the substitution r ≡ x12 + x22 ,
1 H31 = −x2 ’((r)( d(; (41) 0
1 ’((r)( d(; H32 = x1 0

which further implies   b −x2 H31 = ; 2 r 2 b  x1 ; H32 = 2 r 2 and




−x2 H31 = 2 r x1 H32 = 2 r

r

0


0

r

r0 ¡ r (42)

’(s)s ds; r ≤ r0 :

’(s)s ds;

For the choice    1 1  b ; r ≤ r0 ; − ’(r) = r0 r r0  0; r0 ¡ r;

(43)

(44)

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Eq. (43) evaluates to     r2 b −x2 r− ; H31 = r0 2r0 r2  x  b  r2 1 r− ; H32 = 2 r r0 2r0

r ≤ r0 : (45)

The only nonzero elastic stress components out of the H deformation 4eld are   b −x2 ; T13 = T31 = 2 r 2 r0 ¡ r; b  x1 T23 = T32 = ; 2 r 2     r2 b −x2 ; r − T13 = T31 = r2 r0 2r0 r ≤ r0 ;  x  b  2 r 1 r− ; T23 = T32 = - 2 (46) r r0 2r0 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 are independent of the x3 coordinate, the only nontrivial equilibrium equation that has to be 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 the context of the satisfaction of the virtual work equations with continuous test functions having piecewise continuous derivatives, it is to be noted that the stress tensor is continuous (and consequently the tractions) on r = r0 and on any surface where the test function derivatives are not continuous. The point-wise traction 4eld on the surface of a cylinder V/ of radius r = / vanishes as / → 0, while the virtual work contribution of 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 also satis4es the weak form of the 4rst of (24) if it is assumed that the test functions have vanishing tangential component at r = ∞ . This is so because H and the test functions remain bounded on the surface of V/ as / → 0, the integral
(Hrk ekji Qri; j + ri Qri ) d for all Qri in T V/

vanishes as / → 0; H is continuous on r = r0 and on any surface where the test function derivatives may be discontinuous, and curl H =  is satis4ed in the regions 0 ¡ r ¡ r0 and r0 ¡ r.

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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 causes vanishing strains in the same limit. Willis’ (1967) result for the 4rst-order (linear) problem is consistent with the above conclusion. Indeed, his dislocation distribution is given by a Dirac delta function and he obtains a singular distortion 4eld while solving exactly the same equations as being considered here (when his treatment is restricted to the isotropic case). Interestingly, Willis’ (1967) method of solution is quite di3erent from the one being considered here. Choice (44) was made based on the 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 quite large thus calling into question the use of the linear theory. For instance, for r0 ≈ b it is possible to have shear strains of the order of 30% very close to the axis of the core. The solution presented in this section is formally similar to the method used by Edelen and Lagoudas (1988) which applies to well-de4ned, smooth dislocation density distributions. The di3erence in this work is that a speci4c choice for the dislocation density distribution within the core has been made which is singular, along with an attempt 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 of providing 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 characterized by the strain-energy function E(Fe ) =

(I1 (Fe ) − 3); 2

(48)

where - is the shear modulus for in4nitesimal deformations and I1 is the 4rst invariant of the left Cauchy Green deformation tensor Ge ≡ Fe ·FeT . The material is considered to be elastically incompressible in the sense that det(Fe ) = 1. The Cauchy-stress response is given as T=2

@E e G − pI = -Ge − pI; @I1e

(49)

where I is the second-order identity tensor and p is the constitutively undetermined pressure 4eld to be determined by equilibrium and boundary conditions. If we now consider (5) with  speci4ed exactly as in the previous subsection, then a particular solution is −H where H is speci4ed by (42) and (45). Clearly, W := I − H

(50)

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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  1 0 H31 ; 1 H32 [G e ] =  0 2 2 H32 H32 (1 + H31 + H32 )   -−p 0 T31 ; -−p T32 [T ] =  0 2 2 T31 T32 (- − p + -(H31 + H32 )) 

(52)

where T31 ; T32 are de4ned by (46). If we now take the pressure 4eld to be the constant 4eld given by p(x1 ; x2 ; x3 ) = -, then the only nontrivial equilibrium equation again becomes (47) which is satis4ed as observed in the linear case. Consequently, the internal stress distribution given by (52) with p = - quali4es as a solution to the problem being considered. The only di3erence between the linear and nonlinear solution for this particular material is in the appearance of the stress component T33 given by    2 1 r b2 b2 r0 ¡ r; T33 = - 2 2 1 − ; 0 ≤ r ≤ r0 : (53) T33 = - 2 4 r2 2r0  r0 2 + T2 ¿ T As long as r0 ≥ b, it is interesting to note that in the entire domain T31 33 32 thus lending some justi4cation to the linear result where the normal stress does not appear. 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 presented here and is based on determining strains from vector 4elds with jumps. Their solution 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 region r0 ¡ r. 5.4. Edge dislocation in a isotropic, linear elastic medium An identical geometry and traction boundary condition as in Section 5.2 are considered. In this subsection, we suppose that all quantities are appropriately nondimensionalized so as to make physical sense. The dislocation density distribution is assumed to be 13 (x1 ; x2 ; x3 ) = ’(r);

ij = 0 if i = 1 and j = 3

(54)

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with

779

 2 2 b −r 2  − e−r0 ]; 0 ≤ r ≤ r0 ; C(r0 ) = {1 − e−r0 (1 + r02 )}−1 ;   C(r0 )[e 5 ’(r) = (55)
2
r0    0; r0 ¡ r; ’(r)r dr d+ = b: 0

0

For this distribution of dislocation density, a particular solution of curl Ue =  is given by H + ∇v, where ∇v is the gradient of a vector 4eld to be determined (not to be mistaken for a physical displacement 4eld) and the only nonzero components of H are given by   b −x2 H11 = ; 2 r 2 r0 ¡ r; b  x1 H12 = ; 2 r 2   2 C(r0 ) −x2 H11 = [1 − e−r (1 + r 2 )]; 2 r2 0 ≤ r ≤ r0 C(r0 )  x1 −r 2 2 H12 = [1 − e (1 + r )]; (56) 2 r2 In the following, it is assumed that 3 (x1 ; x2 ; x3 ) ≡ 0 and 1 and 2 are functions of the x1 and x2 coordinates only. Under these conditions, the two non-trivial equilibrium equations 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     x1 −b  x2 b f1 = (( + 2-) +; 2 ;1 2 r 2 r 2 ;2 r0 ¡ r;     x1 b −b  x2 f2 = + ( ; 2 r 2 ;1 25 r 2 ;2   x2 −C(r0 ) −r 2 2 [1 − e (1 + r )] f1 = (( + 2-) ;1 2 r2   2 x1 C(r0 ) +[1 − e−r (1 + r 2 )] ; 2 ;2 2 r 0 ¡ r ¡ r0   2 C(r0 )  x1 [1 − e−r (1 + r 2 )] f2 = ;1 2 r2   x2 −C(r0 ) −r 2 2 +( [1 − e (1 + r )] ; (58) ;2 2 r2

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The functions f1 and f2 are square-integrable and taking the two-dimensional Fourier transform of (57), it can be shown that vˆ1 =

fˆ 1 − fˆ 2 ; !12 − !22

vˆ2 =

{(( + 2-)!12 + -!22 }fˆ 2 − {(( + 2-)!22 + -!12 }fˆ1 ; (( + -)!1 !2 {!12 − !22 }

where


a(! ˆ 1 ; !2 ) =

∞




−∞

∞

−∞

a(x1 ; x2 )e−i(!1 x1 +!2 x2 ) d x1 d x2

(59)

(60)

represents the two-dimensional Fourier transform of the function a. Assuming the expressions in (59) can be inverted, H + ∇v is the elastic deformation solution for the problem in hand and the corresponding linear elastic stress 4eld can be generated from it. Even though an explicit solution has not been generated for this problem, the solution method reveals the main features of the problem of internal stress in the linear setting — the deformation incompatibility translates to apparent body forces which drive a conventional boundary value problem. If boundaries were to be present, then the traction on such boundaries arising from the particular solution satisfying the incompatibility equation becomes an additional agent forcing the conventional boundary value problem. The problem also reveals a systematic method of attack on such problems for the purpose of generating approximate solutions — generate a particular solution to the 4rst of (24) and then generate the complementary 4eld in the null space of the operator curl which satis4es the equilibrium equations driven by the particular solution and 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 expressed () c ⊗ c- . Assuming no dislocation sources are present, (29) in terms of this basis as may be expressed as 3  () ˙c ⊗ c - = (im() ) e-/9 {e9: mn() (Vmn() ) (in() ): }; / c ⊗ c- : (61) m; n=1

Considering only one slip system with (c ;  = 1; 3) coinciding with (im ; m = 1; 3) for that slip system, the component version of (61) is 3  ˙mp = epsq {eqrn mn (Vmn )r }; s (no sum on m) (62) n=1

With the above identi4cation, mn = mn . Additionally, we consider dislocation density distributions of the form ij (x1 ; x2 ; x3 ; t) = bi tj (x1 ; x2 ; x3 ; t);

(63)

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where b (dimension Length) is a constant and spatially uniform vector 4eld and t is another vector 4eld to be determined. We also assume that for all values of m; n the dislocation velocity vectors take the form t × n0 ; (64) Vmn = V t where n0 is the constant and spatially uniform unit normal 4eld of the slip system being considered. Under the above assumptions, (62) reduces to t˙p = epsq { t  (n0 )q }; s :

(65)

Since (n0 )q = <1q , (65) can now be written as the set of equations t˙1 = 0; t˙2 = { t }; 3 ; t˙3 = {−  t }; 2 :

(66)

It is to be noted at this point that if the choice f(x1 ; t) V (x1 ; x2 ; x3 ; t) =  t0 

(67)

is made, where t0 is the initial condition on the 4eld t and f is a function of the arguments displayed, then we have an equilibrium solution of the theory. The problems considered in Sections 5.2 and 5.4 4t into the class of dislocation density distributions being considered here and, as such, those solutions are genuine equilibrium solutions of 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 is the x3 direction, i.e. a screw or edge dislocation, depending upon b, with dislocation line 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 the essential nature of the conclusion to be drawn shortly is not a3ected by taking it to be a 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 t˙3 = −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 the initial conditions (68). Solution (71) corresponds to the initial dislocation travelling along the x2 -axis in the sense of the sign of V with constant speed |V |, as intuitively expected. In an in4nite

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medium, the stress 4elds derived in Sections 5.2 and 5.4 translate with the dislocation along with some accumulation of permanent deformation due to the motion of the dislocation which is dealt with next. Let the dislocation under consideration be an edge with Burgers vector given by b = bm0 , b ¿ 0. Then, with the solution obtained above for t, p p U˜˙ = −bm ⊗ (V m × !i3 ) ⇒ U˜˙ (x ; x ; x ; t) = −b!(x ; x − Vt)V m ⊗ n : 0

0

1

2

3

1

2

0

0

(72) Consequently, p p U˜ (x1 ; x2 ; x3 ; t) = U˜ (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 at x1 = x2 = 0 at time t = 0. It should be noted that in the presence of the boundary, the stress 4eld in the body at times when the dislocation is within it is not the same as would 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 mechanical force balance. It is now of interest to know the state of the body at times t ¿ (L + r0 =V ) when all external 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 only p possible nonzero component of curl U˜ is given by 
x2  p p ˜ ˜ !(x1 ; s) ds (curl U )23 (x1 ; x2 ; x3 ; t) = (curl U )23 (x1 ; x2 ; x3 ; 0) + b x2 −Vt

;2

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

(74) p

and for the times of interest, (74) implies that the U˜ distribution on the body is compatible. Consequently, utilizing the unique solution to (25) and (26) given by p Up = U˜

(75)

(for all times in the absence of sources) and the fact that there are no loads on the p body, (25) – (28) are all satis4ed by (75) and U = U˜ for t ¿ L + r0 =V . Hence, the only long-term e3ect of the passage and exit of the edge dislocation from the body is 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 the dislocation distribution de4ned by !(x1 ; x2 ). In particular, if it is a bell-shaped pro4le on the x1 − x2 plane as in Section 5.4, then (73) indicates (assuming r0 =L1 so that p U˜ 0 (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 monotonically from a constant maximum with vanishing slope (u2; 1 ) for all x1 ≤ −r0 to a minimum at x1 = r0 where the slope again vanishes and remains 0 for x1 ≥ r0 . This pro4le may be interpreted as a slip-step over a distance 2r0 .

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783

6. Conclusion A theory of crystal plasticity has been proposed based on dislocation density as the primary internal variable. The instantaneous spatial distribution of this internal variable, along with the applied loads, determines the current state of stress in the body. The permanent deformation produced is a function of the history of this internal variable. The internal variable itself evolves according to an equation based on an idealization of the motion of dislocation lines on slip planes. The theory appears to produce physically reasonable results but much work remains to be done in evaluating it further and exploring its physical implications. Future work will involve exploring the nature of boundary conditions that are required, if any, for the evolution equations for the slip system dislocation densities. In an incremental setting this is not an issue — however, it should be kept in mind that the knowledge of the time rate of a function at the beginning of a time interval is not enough to generate the function exactly in that interval, no matter how small. This is the consideration that changes an algebraic problem for the rates in an incremental version of the present theory to a PDE. It is not clear to this author that such a transition necessarily needs extra boundary conditions for the evolutionary equations. The stress and displacement boundary conditions could conceivably pose implicit boundary conditions. For example, in the problem considered in Section 5.5, if the boundary at x2 = L were not allowed to deform then it is clear that the dislocation velocity cannot be a constant since that would produce an inconsistency — the dislocation would have to exit the material producing a slip-step which the boundary condition disallows. If the dislocation velocity is a function of the stress, however, the dislocation would be obstructed before reaching the boundary by a suitable stress 4eld being set up so that the rigid boundary condition as well as the constitutive equation can be respected. By the same token, however, it is not clear to this author what the situation would be if there was a 4nite boundary at some value of negative x2 . In Section 5.5 treating the geometry as in4nite was necessary for the choice of constant dislocation velocity, for otherwise the solution would not be de4ned for all times without boundary conditions. p ˙ U˜˙ , and  and the introduction of However, with the existing relationship between U, another one through the constitutive equation for the dislocation density, it is hard to judge the adequacy or inadequacy of the existing boundary conditions for the problem without further mathematical analysis of, what appears to be, a rather diBcult nature. If such conditions are indeed necessary, the physical nature and geometric interpretation of the dislocation Eux tensors in the theory can aid in their speci4cation. It would also be desirable to explore the relationship between the incompatibility of the slip deformation tensor 4eld and the dislocation density in the context of the nonlinear theory in the absence of sources. In the proposed geometrically linear theory with no source distributions, these quantities are identical up to a sign. Thermodynamic restrictions on the theory also need to be explored as an elementary guide to avoiding physically unrealistic predictions. Such restrictions can also provide useful insights into some necessary conditions that the constitutive equations for the various 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

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appropriate statement of the second law of thermodynamics ought to be for genuinely nonlocal materials is also a matter that is not completely obvious or settled. Some advances along these lines, as they pertain to the work presented in this paper, are clearly desirable. Finally, a numerical implementation of the theory needs to be pursued. The basic ingredient here will be an eBcient procedure to generate a basis for the null space of the discretized operator corresponding to the curl, and a method to generate projections of discretized functions on this null space. This paper provides a general method for determining stress 4elds of 3-D dislocation distributions that can be translated into a numerical method in a conceptually straightforward manner. It is hoped that the numerical implementation of such a method, when combined with an implementation of the non(thermodynamic) equilibrium aspects of the theory, will lead to a capability of solving problems in dislocation mechanics routinely to enable a better understanding of the fundamentals of plastic deformation. Acknowledgements This work was supported by the Center for Simulation of Advanced Rockets at the University of Illinois at Urbana-Champaign under U.S. Department of Energy subcontract B341494. References Carlson, D.E., 1967. On Gunther’s stress functions for couple stresses. Quart. Appl. Math. XXV, 139–146. Clifton, R.J., 1983. Dynamic plasticity. J. Appl. Mech. 105, 941–952. Edelen, D.G.B., 1985. Applied Exterior Calculus. Wiley-Interscience, Wiley, New York. Edelen, D.G.B., Lagoudas, D.C., 1988. Gauge theory and defects in solids. In: Sih, G.C. (Ed.), Mechanics and Physics of Discrete Systems, Vol. 1. North-Holland, Elsevier Science Publishers, Amsterdam. Eshelby, J.D., 1956. The continuum theory of lattice defects. In: Seitz, Turnbull (Eds.), Progress in Solid State Physics, Vol. 3. Academic Press, New York, pp. 79–144. Fox, N., 1966. A continuum theory of dislocations for single crystals. J. Inst Math. Appl. 2, 285–298. Hutchinson, J., 2000. Plasticity at the micron scale. Int. J. Solids Struct. 37, 225–238. Kosevich, A.M., 1979. Crystal dislocations and the theory of elasticity. In: Nabarro, F.R.N. (Ed.), Dislocations in Solids. North-Holland Publishing Company, Amsterdam, pp. 33–141. Milstein, F., 1982. Crystal elasticity. In: Hopkins, H.G., Sewell, M.J. (Eds.), Mechanics of Solids, The Rodney Hill 60th Anniversary Volume. Pergamon Press, Oxford, pp. 417–452. Mura, T., 1963. Continuous distribution of moving dislocations. Philos. Mag. 89, 843–857. Nye, J.F., 1953. Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162. Ortiz, M., Phillips, R., 1999. Nanomechanics of defects in solids. Adv. Appl. Mech. 36, 1–79. Rosakis, P., Rosakis, A.J., 1988. The screw dislocation problem in incompressible 4nite elastostatics: a discussion of nonlinear e3ects. J. Elasticity 20, 3–40. Teodosiu, C., Soos, E., 1981a. Non-linear elastic models of single dislocations. Rev. Roum. Sci. Tech. Ser. Mec. Appl. 26, 731–745. Teodosiu, C., Soos, E., 1981b. Non-linear e3ects in the elastic 4eld of single dislocations. I. Iteration scheme for solving non-linear problems. Rev. Roum. Sci. Tech., Ser. Mec. Appl. 26, 785–793. Teodosiu, C., Soos, E., 1982. Non-linear e3ects in the elastic 4eld of single dislocations. II. Second-order elastic e3ects of an edge dislocation. Rev. Roum. Sci. Tech. Ser. Mec. Appl. 27, 15–35. Weyl, H., 1940. The method of orthogonal projection in potential theory. Duke Math. J. 7, 411–444. Willis, J.R., 1967. Second-order e3ects of dislocations in anisotropic crystals. Int. J. Eng. Sci. 5, 171–190.