Driving Forces and Boundary Conditions in Continuum Dislocation Mechanics

Transcription

1 Driving Forces and Boundary Conditions in Continuum Dislocation Mechanics Amit Acharya Det. of Civil and Environmental Engineering Carnegie Mellon University, Pittsburgh, PA 1513, U.S.A. Summary As a guide to constitutive secification, driving forces for dislocation velocity and nucleation rates are derived for a field theory of dislocation mechanics and crystal lasticity roosed in Acharya (001, J. Mech. Phys. Solids). A condition of closure for the theory in the form of a boundary condition for dislocation density evolution is also derived. The closure condition is generated from a uniqueness analysis in the linear setting for artial differential equations controlling the evolution of dislocation density. The boundary condition has a simle hysical meaning as an inward flux over the dislocation inflow art of the boundary. Kinematical features of dislocation evolution like initiation of bowing of a inned screw segment, and the initiation of cross-sli of a single screw segment are discussed. An exact solution reresenting the exansion of a olygonal dislocation loo is derived for a quasilinear system of governing artial differential equations. The reresentation within the theory of some hysical features of dislocation mechanics and lastic deformation like local (dislocation level) Schmid and non-schmid behavior, unloaded, stress free and steady microstructures, and yielding are also discussed. 1. Introduction This aer aims at advancing the theory of continuum dislocation mechanics resented in Acharya (001) (referred to as [I] henceforth) with resect to roviding general guidelines for the formulation of constitutive equations and the secification of boundary conditions for dislocation density evolution. The associated discussion also considers the reresentation within the theory of such hysical behaviors as exansion of a olygonal dislocation loo and the increase of the norm of dislocation density from a rescribed initial state urely due to the kinematics of dislocation density evolution; kinematics of the initiation of bowing of a inned screw segment and the (concetually similar) initiation of cross-sli of a single screw segment; local (dislocation level) Schmid and non-schmid behavior; unloaded, stress free (zero energy) and steady microstructures; and yielding. Tel. (41) ; Fax. (41) ; amita@andrew.cmu.edu 1

2 The theory develoed in this aer, and its redecessor [I], is a continuum theory in the sense that it is a mathematical idealization of the stressing and deformation of crystalline materials. It deals with a mathematical continuum of oints, endowed with hysically motivated attributes, and such a continuum serves as an idealization of a crystalline body. In their concetion, its ingredients are not exlicitly built on rigorous averages of discrete hysical quantities (e.g. dislocation densities) however, the hoe is that its solutions model observable hysical quantities related to dislocation mechanics in crystals, and its success, or failure, is to be judged urely by the closeness of such solutions to actual hysical behavior. As a result of this interretation, we shall not be unduly concerned about the hilosohical imlications of sometimes trying to model individual dislocation behavior with this theory, where an individual dislocation is modeled as a suitable variation of the dislocation density field, e. g. a non-zero distribution in a cylinder along a curve in sace. In fact, at every aroriate oortunity we aly this continuum theory of dislocations to the mechanics of single dislocations in order to test its soundness and glean imortant modeling information from such efforts. To some extent, this rogram has been carried out in [I], and we use this idea reeatedly in the aer. In the concetion of a well-osed, non-equilibrium continuum theory suitable for the study of stress, work-hardening, and ermanent deformation of dislocation distributions, this work is motivated by the works of Kröner (1981), Mura (1963, 1970), Willis (1967), Kosevich (1979), Fox (1966), Head et al. (1993), Aifantis (1986), and van der Giessen and Needleman (1995). A recent thrust in continuum crystal lasticity modeling has been to account for geometrically necessary dislocations, while using ideas from sli-based conventional crystal lasticity to model the hysical effects of statistically stored ones. Since the idea of geometrically necessary and statistically stored dislocations is necessarily related to a scale of satial resolution, it may also be useful to view the roblem of dislocation mechanics, and the associated crystal lasticity, at a sufficiently small hysical scale (inter-dislocation sacing) so that all dislocations are geometrically necessary. There is a large class of current and emerging technological roblems for which such a theory would be directly alicable. Moreover, if a continuum framework for the nonequilibrium behavior of dislocation distributions can be develoed with such a fine scale of resolution, then an effort can be made to average such a theory (e. g. Muncaster, 1983 a,b) to obtain others

3 with more macroscoic scales of resolution that have naturally built into them the geometric and hysical attributes, and effects, of statistically stored as well as geometrically necessary dislocation distributions. The oints mentioned above serve as artial motivation for the theory being develoed in this aer.. Notation and some esults from Potential Theory The symbol is shorthand for belongs to ;, that for for all ; stands for subset and for imlies. sgn( a ) denotes the sign of the argument a. A suerosed dot on a symbol reresents a material time derivative. The statement a: = b is meant to indicate that a is being defined to be equal to b. The summation convention is imlied excet for indices aearing between arenthesis or when exlicitly mentioned to the contrary. We denote by Ab the action of the second-order (thirdorder, fourth-order) tensor A on the vector (second-order tensor, second-order tensor) b, roducing a vector (vector, second-order tensor). A reresents the inner roduct of two vectors, a : reresents the trace inner roduct of two second-order tensors (in rectangular Cartesian comonents, A: B= AB ij ij ) and matrices. The symbol AB reresents tensor multilication of the second-order tensors A and B. The curl oeration and the cross roduct of a second-order tensor and a vector are defined in analogy with the vectorial case and the divergence of a second-order tensor: for a second-order tensor A, a vector v, and a satially constant vector field c, In rectangular Cartesian comonents, T T ( A v) c= ( A c) v c T T ( curl A) c= curl( A c) c. ( A v ) = emjk A im ijvk ( curl A ) = e A,, im where e mjk is a comonent of the third-order alternating tensor Χ. The definition for the curl of a second order tensor field above follows that of Cermelli and Gurtin (001) u to a transose. The notation A// reresents the orthogonal rojection i of the second-order tensor field A on the null sace of the oerator curl defined in the weak sense [I]. Without mjk ik j (1) () i Usually, the terminology rojection refers to the rojection oerator in linear algebra. Here we refer to call the oerator the rojector, and the result of its action on a member of its domain as the rojection of the element being rojected. 3

4 loss of generality, in this aer we modify slightly the weak definition of the oerator curl and its null sace. In [I], the roblem curl W = α was defined weakly as follows: Wrk ekjiqri, j dv= α riqri dv for all Qri in T, (3) where all comonents are with resect to a rectangular Cartesian basis. T is the set of continuous test functions on the domain with vanishing tangential comonent on the boundary of, and at least iece-wise continuous first derivatives in. From hereon, we consider all test functions belonging to the set T to vanish on the boundary of the body instead of only their tangential comonents. With this minor change in the definition of the set T, all other definitions and results in [I] remain in force. In articular, the null sace, N( curl ), of the oerator curl is defined as the set of all square integrable matrix fields W on that satisfy W : curl Q dv = 0, Q T. (4) In other words, N( curl ) could be referred to as the set of all weakly irrotational matrix fields. In this aer we shall also need the following results that are obvious extensions of the work of Weyl (1940) to the matrix case. Let D be the set of all square integrable matrix fields on the domain, endowed with the standard L inner roduct the inner roduct of two matrix fields A, B is given by A: Bdv. Then, any A D can be uniquely written as the sum A= A + A // (5) where N curl D. i. // A belongs to ( ) { } ii. A belongs to N ( curl) B D such that B : V dv 0, V N ( curl) ( curl) = =. The sace N is the closure of the set of all tensor fields of the tye curl Q on, Q T, and Q sufficiently smooth for curl Q to make sense. iii. A: B dv = A// : B// dv A : B dv +. iv. For every A D there exists a WA D that satisfies WA : curl Q dv = A : Q dv Q T, with W A = 0 on. (6) W may be determined by solving A 4

5 curlwa = A on divwa = 0 W = 0 on, A or the weak equivalent of (7) when A is not sufficiently smooth. (7) 3. Field Equations For ease of resentation of the main hysical ideas without the necessary subtleties that arise from a consideration of finite deformations, this aer will deal only with the theory for small deformations. Most of the ideas resented are, however, generalizable to the finite deformation case, and such a generalization for all of the ideas resented herein will be the focus of subsequent work. For reasons that will become clear when the boundary condition/uniqueness analysis for the evolution of dislocation density is resented, we associate only one dislocation velocity vector to each sli system dislocation density tensor. This may be viewed as a secial case of the tye of dislocation velocity descrition allowed in [I]. With this understanding the field equations of geometrically linear continuum dislocation mechanics are [I]: curl U = α (8) U// = U // (9) 1 T 1 T T = C ( ε ε ) ; ε : = ( U+ U ) ; ε : = ( U + U ) (10) div T = 0 (11) ( ) ( ) ( ) ( ) α = curl( α V ) + s ; ( ) α = α (1) ( ) ( ) U = α V. (13) In the above, U is the dislacement gradient, I is the second-order identity and e U, U, and U are measures of small elastic, sli, and lastic deformation resectively (second-order tensors), and C is the fourth-order tensor of linear ( ) elastic moduli. Also, T is the (symmetric) stress tensor, α and α the total and sli system dislocation density tensors resectively, and V ( ) the th sli system dislocation velocity vector. s ( ) reresents the dislocation source on the th sli system. The reason for denoting the sli deformation rate as in (13) is based on the kinematics of the sli deformation increment roduced by dislocation motion. Indeed, let b l reresent a discrete dislocation dyad with b as the true Burgers vector of the dislocation and l the (ositive) line direction of the dislocation in the th 5

6 dislocated lattice (for conventions, see Willis, 1967). If v is the dislocation velocity, then the local shear increment roduced in the time interval t, of the material b l t v. Associated with each sli system around the dislocation line, is given by ( ) are three unit vectors that remain fixed in the small deformation idealization: the sli lane normal ( i ) 1, the sli direction ( i ), and the other unit vector in the sli lane that forms an orthonormal triad given by ( ) ( ) ( i := i i ) In order to understand why one might succeed in solving the system (8)-(13), we think of the sli system dislocation densities as the state variables of the theory. Given the body and an initial dislocation density state, (8), (10), and (11) are solved first to determine the state of initial stress and the initial condition on U [I]. The evolution equations (1)-(13) rovide the forcing functions for the solution of U from (8)-(9). The solution for the U field and (10)-(11) rovide the solution for the total dislacement. Of course, aroriate boundary conditions are also required to erform the above calculations uniquely. The utility of (9) is that it renders the weak solution of (8) and (9) unique. The main idea behind the roof of this assertion may be understood readily from an analogy with the matrix case if a square matrix A is singular then a solution of the matrix equation Ax = b is non-unique u to addition of any vector from the null sace of A. If the matrix equation is now augmented by the additional requirement that the null sace comonent of the solution x be a secific vector from the null sace of A then, of course, the non-uniqueness is eliminated from the solution of the augmented system. Of course, in the matrix case the linear oerator in question has a finite-dimensional (and, consequently, comlete) linear sace as its domain, so seaking of a null sace comonent of an element in the oerator s domain through an orthogonal rojection does not require any additional concerns; in the case when the domain is an infinite-dimensional function sace, comleteness of the function sace and the oerator s null sace is not guaranteed, and a weak formulation of the roblem is required. In roceeding further with this matrix analogy, the condition for existence of a weak solution to (8) is that the field α be weakly solenoidal (Weyl, 1940) in the matrix case this is reresented by the fact that a solution to Ax = b exists only if the vector b is in the column sace of the oerator A. 4. Kinematic Comonent of the Sli System Source Terms Observations of crystal dislocations suggest that dislocation lines lie on sli lanes of the crystal. Their Burgers vectors may be in arbitrary directions, with the most

7 easily movable dislocations having their Burgers vector in a sli direction on the sli lane on which the dislocation line lies. The evolution equations for the sli system dislocation densities (1) do not a riori guarantee that ( α ) i ( ) = 0 at all times for hysically aroriate choices of the sli system dislocation velocity and the source. This fact motivates a artial characterization of the sli system sources of a kinematic nature; we choose them to be of the form where ( ) S ij { ( )} ( ) ( ) ( ( ) ( ) ) ( ) ( ) 3 3 ( ) ( ) ( ) : = curl α + fij + Sij i j i= 1 j= s V n n i i, (14) reresent crystallograhic dislocation nucleation rates (to be secified constitutively) and the f ( ) ij arise from a redistribution of the normal art of the sli system dislocation density increments. The latter will be secified as follows. Denote the sum of all the normal sli system dislocation increments as ( ) ( ) { ( )} ( curl ) ( υ:= α V n n ). (15) Let Κ be the total number of sli systems in the material. Consider the linear transformation I from the vector sace 6Κ of all 6Κ -tules of real numbers f ( ) δ to the sace of second-order tensors denoted symbolically by ( ij ) 3 3 ( ) ( ) ( ) ( δ ) δ i i = : I δ f f f. (16) i= 1 j= ij i j We assume that the range sace of I is the whole sace of second-order tensors Κ ( ) ( ) is endowed with the standard inner roduct ( δ f, δg) δ fij δgij i= 1 j= 1 = for all δ f and δ g, so that it is ossible to seak of an orthogonal rojection of any 6Κ - tule f f aearing in (14) δ on the null sace of the oerator I. The 6Κ -tule ( ) is now uniquely determined as the solution of I f = υ (17) subject to the additional condition that the rojection of ( f ) on the null sace of the oerator I be the null 6Κ -tule. The requirement of a vanishing null sace comonent of the solution may be interreted as requiring that all the redistributed crystallograhic dislocation density increments be geometrically necessary (Arsenlis and Parks, 1999), which is aroriate since we have in mind a theory whose satial resolution is required to be adequate to account for henomena at the scale of inter-dislocation sacing. 7

8 The rocedure described above may be summarized as a deterministic rule for assigning the forest dislocations arising due to dislocation activity on a system (e.g. cross sli), to other systems whose sli lanes can accommodate such dislocations. A similar mathematical rocedure has also been used by Arsenlis and Parks (1999) ii in a related but different hysical context our mathematical formulation is also intended to amlify directly the role of all redistributed dislocation density rates being geometrically necessary. There are also obvious similarities in the general idea used here and the one behind the formulation of (9), viewed in the context of delivering uniqueness of solutions to (8). We view the secial case of a material for which the set of dislocation dyads aearing in (16) do not san the sace of second-order tensors as a situation where ( ) ( ) the sli system dislocation velocities have to be constrained so that α i = 0. This is an unlikely situation in the case of most crystal structures, since six linearly indeendent dyads are rovided just by one sli system, but it is ossible, e.g., for a material with only one sli lane. 5. Driving Forces To identify the driving forces for the sli system dislocation velocity vectors and crystallograhic nucleation rates, we examine the global (mechanical) dissiation in the theory, D : = T : U dv ψ dv (18) where ψ is the free energy er unit volume of the body. The free energy, like the e stress, is assumed to deend only on the elastic strain, ε, ψ = ψˆ ( ε ε ). (19) Consequently, ψˆ ψˆ D = : dv + : dv e e T ε ε ε ε. (0) We assume hyerelasticity of the material and adot ψˆ T=. (1) e ε The dissiation now takes the form, 1 ii Whether the rocedures are equivalent or not requires further analysis our rocedure does not exlicitly require a minimum norm solution. 8

9 ψˆ : : D = ε dv = dv e T ε. () ε For the urose of the resent discussion, we assume that the stress field T is smooth. Then D = T : dv = T + T : U dv = T : U dv + curlw : U dv, (3) ( ) ε // // // T where (9) has been used. Now, using (8), (1), and (13), curlw : U dv = W : curlu dv = W : α dv T ( ) V T V T V ( ) = W : curlu s dv = T : U dv W : sdv, T V V V so that (3) and (4) together imly D = T : U dv : dv WT s, (5) ( ) where s= s. Sinceυ =I f, 3 3 ( ) ( ) ( ) ( ) Sij i j i= 1 j= and using (13), D takes the form T (4) s = s = i i, (6) ( ) ( ) 3 3 ( ) ( ) ( ) D = : dv : S T ij i j dv α V WT i i. (7) i= 1 j= We now recall [I] that a necessary condition for the existence of solutions to (8) is that divα = 0 (where we assume for the resent discussion that α is sufficiently smooth). This requirement may be fulfilled by seeking the total sum of the crystallograhic dislocation nucleation rates to be of the form 3 3 i= 1 j= S ( ) ( ) ( ) ij i j i i = curlω (8) for a second order tensor Ω, as can be seen from (1) and (6). Once an aroriate Ω is secified, then we use exactly the same rocedure as the one used to ( ) ( ) determine from the tensor υ to determine from curl Ω. With this ( f ij ) ( S ij ) understanding (7) imlies D = ( ) Χ Tα V dv + -T : Ω dv. (9) ( ) ( ) We refer to the tensor Ω as the nucleation rate otential. Based on the form of (9), we now define the driving forces for the theory: th ( ) The driving force for the sli system velocity vector ( V ) is 9

10 ( ) ( ) ( ) ξ : = Χ Tα iii. (30) The driving force for the nucleation rate otential ( Ω ) is Θ : = T. (31) We also note for later use that for arbitrary vector and second-order tensor ( ) valued functions v and ω, the definitions ( ) ( ) ( ) ( ) V : = sgn ( ) dv Χ Tα v v D 0. (3) Ω: = sgn ( -T : ωdv) ω 6. Guidelines for Constitutive Equations arising from Theory and some Consequences We now examine ξ ( ) and Θ with a view towards defining the sli system dislocation velocities and crystallograhic nucleation rates. Following conventional wisdom, we adot the oint of view that constitutively secified kinetic variables be ( ) functions of their driving force fields. We intend to secify v and ω subject only to restrictions arising from known facts about dislocations, ositive dissiation being guaranteed by the final form of constitutive equations given by (3). 6.1 Dislocation Velocity Using the symmetry of the stress tensor, ( ) ( ) ( ) ( ) ( ) ( ) ({ } )( ) ( ) ( ) ( ) ( ) ( ) ( ) = α jk j k = j α jk k ξ Χ Ti i i i i Ti i i, (33) ( ) ( ) ( ) ij i j ( ) j1 0 where α = i α i. Since α =, (33) imlies ( ) ( ) ( ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) ( ) ( ) ( i ) j Ti i α j3 i3 ij Ti3 i3 α ji ( ) ( ) ( ) = j 1 1 α j + α j3 3 ξ i Ti i i i + +. We examine the hysical content of (34) by rewriting it in the form (34) iii Mura (1970) derives essentially the same exression in the context of a continuum without crystalline structure. His derivation, however, is not generalizable to the case when dislocation sources are included. Also, the lastic strain rate cannot be derived uniquely in his theory in terms of the dislocation density and velocity, and the same non-uniqueness translates to the driving force. 10

11 where ( ) ( ) ( ) T ij = i j { } ( ) ( ) { ( ) ( ) ( ) ( ) } ( ) ( ) ( ) ( ) { } ( ) ( ) ( { ) ( ) ( ) ( + T ) α3 T3 α i1 T1 α3 i α i3 } ( ) ( ) ( ) ( ) { T } ( ) ( ) { ( ) ( ) ( ) ( ) 3 α33 T33 α3 i1 T31 α33 i α3 i3 } ( ) ( ) ( ) ( ) ( ) = T1 α13 T13 α1 1 T11 α13 α1 3 ξ i i i +, (35) i Ti. On considering the (common) situation when the dislocation ( ) state at a given oint can be exressed as ( ) ( ) ( ) ( ) ( ) ( α = i α i + α i ), i.e. an 3 3 infinitesimal dislocation with Burgers vector in the sli direction and line direction in the sli lane, only the second line of (35) is non-vanishing and it indicates that the in-lane comonent of the driving force deends on the state of stress only through the resolved shear stress (Schmid stress) on the relevant sli lane. The direction of the in-lane driving force is also seen to be erendicular to the line direction ( ) ( ) ( ) ( ) ( ) ( ) 1 α + α3 3 i i i, which is exactly in accord with the direction of the Peach-Koehler force on a single dislocation of classical dislocation theory. The stress-deendence of the driving force is on the total local stress which includes contributions from the stress field of the dislocation distribution as well as alied loads iv - this indicates that a dislocation velocity descrition based on a deendence ( ) on ξ would have dislocation interactions, in the context of a linear elastic material, incororated naturally. ( ) Continuing with the same dislocation state, i.e. ( ) ( ) ( ) ( ) ( ) ( α = i α i + α i ), we 3 3 note that the out-of-lane comonent of the driving force contains stress comonents other than the Schmid stress and the deendence of the dislocation velocity on such comonents may be construed as giving rise to non-schmid effects. Indeed, the out-of-lane comonent is believed to be a necessary condition for dislocation climb and cross-sli, the latter requiring additionally a gradient in stress or dislocation density. This can be seen most simly by considering the dislocation state to consist of only a screw dislocation with Burgers vector in the sli direction. Assuming for simlicity that the velocity is roortional to the driving force, a art of the instantaneous dislocation density evolution is ( ) ( ) ( ) ( ) ( ) ( ) ({ α } T 3 α 1 ) curl i i i (u to the roortionality constant) which may be exressed as iv This is to be contrasted with a deendence on the so-called defect stress (Gurtin, 001). 11

12 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ({ α i i } 3 α i 1 ) ijk { α 3} δr δk 3 ri, j ( ) ( ) ( ) ( ) ( ) ( ) ( ) ({ α i i } 3 α i 1 ) { α 3 }. 1, curl T = e T curl T = T The above suggests a reassuring similarity with the hysical icture of cross sli occurring from a gliding screw dislocation that meets a air of obstacles along its line direction (reflected in the stress gradient required along the line direction in (36), assuming a straight, cylindrical dislocation with no gradient in density along the line direction) and then forms a air of edge jogs on the cross sli lane (the develoment of 1 comonent in the dislocation density rate in (36) ). It is indeed temting here to ostulate that a necessary condition for cross sli is the alignment of the driving force with a sli lane of the material and the resence of gradients in the T 3 comonent along the sli direction in the sli system basis. Other non-schmid behaviors can also be seen to arise whenever the dislocation state at a oint may be reresented as ( ) 3 ( ) 3 α ( ) = k i i l i i i i= 1 i i=, where k, i l i are scalars. This state reresents a dislocation with line direction in the sli lane and Burgers vector in a direction not necessarily in the sli lane. Such a dislocation state may arise in the modeling of immobile or relatively immobile dislocations, e.g Lomer-Cottrell lock, screw dislocations in bcc materials at low temeratures with core structure v. For a screw dislocation with its core sread out so that it has some edge comonent, the driving force can be seen to contain the stress comonent T which is believed to give rise to the orientation deendence of yield in some intermetallic comounds. The driving force in any of these cases contains non- Schmid stress comonents. It is also worthy of note here that in all instances the driving force incororates a deendence on the dislocation state naturally. Since the dislocation density is a solution variable of the theory, such a deendence does not have to be further henomenologically modeled, the latter being the tyical case in sli based conventional crystal lasticity where dislocation density is not a solution variable. All of the above features aear to be desirable attributes of a theory of continuum dislocation mechanics. In analogy with conventional ideas related to the (36) ( ) 31 v Of course, dealing with core structure also requires, in addition to the above kinematics, a consideration of nonlinear crystal elasticity incororating lattice symmetries and eriodicity. 1

13 motion of discrete dislocations, we would now want to adot the rule that the inlane and out-of-lane comonents of the sli system dislocation velocity vector be functions of the corresonding comonents of the driving force, i.e. ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) ( V ) n : = V i1 i1 = function ξ i1 i1 (37) ( ) ( ) ( ) ( ) ( ) V V = function ξ ξ i i. n ( { } ( ) 1 1 ) However, before taking such a ste we have to ensure that the theory reresents two fundamental facts about dislocations: a straight discrete dislocation does not move under its own stress field in an infinite homogeneous medium; and a straight dislocation in such a medium should move with constant velocity under a satially uniform alied stress field - of course, only in the geometrically linear theory with a linear elastic constitutive law for stress does it make sense to seak of an alied stress field. In the resent continuum theory, a straight dislocation may be modeled as a cylinder of a certain core radius r 0 (to be secified external to the theory) with a nonvanishing dislocation density in it [I]. The integral of the dislocation density field over the cross-section of the cylinder reresents the strength of the dislocation. Let the dislocation velocity be chosen according to (37), with the functional relationshi being a simle roortionality between velocity and driving force. Then, using the solution for a screw dislocation of strength b derived in [I] for an otherwise arbitrary axisymmetric density distribution in the core cylinder, a simle calculation reveals that under the above constitutive assumtion for dislocation velocity the dislocation cylinder disintegrates under its own stress field, the driving force tending to sread it out in the domain vi. Such a result clearly is hysically unaccetable, and this observation leads us to modify our definition of the dislocation velocity. ( ) For fixed,, i j, let P ij be the sum of terms in ξ ( ) (35) that are linear in ( α ) ij with direction orthogonal to ( ) ( ) i 1. Let N ij be the sum of terms linear in ( α ) ij with direction arallel to ( i ). We denote by ˆ P x ; y and ˆ N x; y the values of the ( ) ij P and ( ) ij 1 ij ( ) ( ) ( ) ( ) N fields at the oint x of an infinite, homogeneous medium of crystal symmetry given by that of the actual crystal at the oint y, and generated from the resence of a straight cylindrical dislocation of core radius r 0 with axis assing ij vi As known to Nabarro (1987) 4. 13

14 through y whose line oints in the direction corresonding to the indices ( j, ) and whose Burgers vector is in the direction corresonding to the indices ( i, ). These fields can be generated as solutions to (8), (10), and (11) for the infinite domain and dislocation density distribution defined above. Let A ( ) j ( y ) be the set of oints contained in the closed disc of radius r 0 with center y, erendicular to the direction corresonding to the indices( j, ). We now define ξ ( ) ˆ( ) ( ) ( ) ˆ ˆ ( ) ( ) ( ) ˆ = da ; = da ij ( ) ( A ( ) ij ij ) j y Aj ( y) ij ( ) 3 3 ( ) ( ( ) ˆ ) ( ) ( ) ˆ = da da ( ) ( A ( ) ij ij + ) ( ) ij ij 1 j y A i= j= j y P y P N y N, (38) ( ) y P P y N N ( y ), (39) ( ) ( ) ( ) v : = f ξ i i ; f 0 = 0 ( ) ( ) 1 1 ( ) n n n { } ( ) ( ) { } ( ) ( ) ( ) ( ) ξ ξ i ( ) ( ) ( ) ( ) ( { } ( ) 1 i 1 f ξ ξ 1 1 ) f ( ) ( ) ( ) ξ ξ i1 i1 v : = i i ; 0 = 0 ( ) ( ) ( ) = n + V v v, with V to be defined according to (3). We note here that while the hysical ( ) ( ) dimensions of ξ is a force er unit volume, ξ has units of force er unit length. Essentially, the above choice for the constitutive equation has been designed to ensure that a single, straight dislocation in an infinite medium remains stationary under its own stress field vii and, under the action of a homogeneous alied stress field, moves as a rigid body. That the latter is a consequence of the theory can be deduced by considering the solution for the motion of a single straight dislocation under satially uniform velocity field derived in [I] and noting that (40) dictates that the driving force at every oint of the cylinder reresenting a straight dislocation under a homogeneous alied stress field is identical. The functions f and f n will, in general, also deend uon a dislocation drag coefficient, temerature, and the sli system dislocation density tensor. The latter may be necessary to henomenologically model additional resistance to dislocation motion if the dislocation state contains Burgers vectors off of the sli lane. As a (40) vii While the above rocedure is essential to adot in a theory where the elasticity is linear, it is reasonable to exect that a roer accounting for nonlinear crystal elasticity would make the above construction unnecessary (Nabarro, 1987). 14

15 ractical device, the ( ) ˆij P and ˆ ( ) N ij terms may also be modeled by henomenological resistances. As for the question of work-hardening due to short range interactions, we note that individual dislocation stress fields in this theory are not singular for smooth dislocation density distributions within the core, as shown in [I], while reroducing the classical elastic fields outside the core region. Consequently, even for dislocations at very close roximity, the theory would be caable of reresenting the effects of elastic interactions of dislocations, at least in rincile, u to the sohistication in the elastic constitutive law. Of course, all of the ideas resented have their analogues in a theory emloying nonlinear crystal elasticity and finite deformations, and in such a theory a more recise accounting of such interactions would be ossible. In ractice, say in the context of numerical comutation, it would erhas be unreasonable to demand mesh resolution on the order of the interatomic sacing when dealing with a hysical situation reresentative of a large collection of dislocations viii. In such a case, the constitutive equation for dislocation velocity may be modified for the effects of comutationally unresolved short-range interactions by modeling insights as resented in the treatise of Kocks, Argon, and Ashby (1975). It may also be hoed that a comlete, i.e. including field equations and constitutive equations, coarse scale theory may be develoed by the Method of Invariant Manifolds (Muncaster, 1983 a,b) by averaging the fine-scale theory roosed in this work and [I], in which case the effects of short range interactions in the macroscoic theory should be accounted for naturally. This is an exciting and romising research direction since the Method of Invariant Manifolds was develoed as a generalization of the rocedure used to derive, in a recise and secial sense, the field equations and constitutive equations of fluid dynamics from those of the kinetic theory. 6. Nucleation ate As oosed to the dislocation velocity, much less detail seems to be aarent from theory for the constitutive equation for nucleation rate s. The simlest ossibility is to assume Ω = ct, (41) viii This may be reasonable in the case of dealing with a collection of a few dislocations, as may occur in studies of semiconductor thin films grown eitaxially. 15

16 where c is a scalar arameter that is required on dimensional grounds. If c is assumed to be a material constant, the choice (41) imlies s= ccurlt, (4) which is true due to the fact curl T// = 0. While (4) is a sound idea from the thermodynamic oint of view, it does not aear to have a simle mechanically intuitive interretation related to dislocation nucleation. One mechanical imlication is the following (essentially) force sum snda = c curl da = c A Tn A Tdx, (43) C for any surface A bounded by the closed curve C, but how this may be recisely related to dislocation nucleation is not clear. This in itself is not necessarily a shortcoming of a constitutive equation since thermodynamic driving forces do not always have interretations that aeal to mechanically guided intuition that a screw dislocation, viewed as mechanical entity, should move erendicular to itself under the action of an alied shear stress in the direction of its line defies all mechanical notions of alied mechanical force and resulting motion. Be that as it may, much exerimental work is required before acceting (4) as a legitimate constitutive equation for nucleation. For instance, it would imly that dislocation nucleation could occur in the resence of an inhomogeneous hydrostatic stress field which is not curl free whether dislocation nucleation can haen at all under hydrostatic stress fields is a fact that can erhas be used to check the validity of (4). We indulge in this seculative vein only to oint out that the curl of the stress field may have a connection to dislocation nucleation, esecially since it emerges from the same thermodynamic rocedure that delivers a driving force for dislocation velocity that is in accord with a number of hysically observed facts about dislocations. Before ending this section, we note that (4) may be interreted as indicating that the stress for dislocation nucleation is nonlocal in the dislocation nucleation rate. A rate-indeendent counterart of such a conclusion (which is achievable with a suitable rate-indeendent choice in lace of the arameter c ) seems to be in suerficial accord with the broad conclusion of an atomistic-dislocation theory study (Miller et al., 1998) that indicates that a nonlocal-in dislocation-density stress criterion for nucleation is a better model for nucleation than one based on a local interlanar otential, but a more recise study in this regard is warranted. 16

17 6.3 Some Consequences We now discuss the notions of steady dislocation microstructures and yielding within the context of this theory, without further commitment to articular constitutive assumtions beyond (40) and (4). Steady (constant in time) dislocation density distributions under no alied loads are imortant redictions since they can be ut to test against exerimental observations. The imortance of the concet of yielding needs no elaboration for an audience well-versed in lasticity. A steady microstructure is a state of the body where ( α ) 0,. The field equations of the theory indicate that it is ossible to obtain stressed, steady microstructures with time-varying total deformation under no alied loads. This can haen as follows: let there be no alied loads but a non-trivial stress field due to the resence of dislocations. Additionally, let this stress field and the dislocation distribution be such that we have a steady microstructure instantaneously. Even though a non-trivial stress field results in the evolution of the U field in general, it is ossible for the instantaneous steady microstructure to ersist in time since it can be seen that the solution for the stress field in (8)-(11) under no alied loads does not vary in time with the U field, the only evolution being that of the total deformation which makes u for the comatible art of the evolving U field. Additionally, we observe that the stress field in the case of a steady microstructure under no alied loads is a functional only of the geometry and elasticity of the body along with the total dislocation density field on it, and consequently with the assumed constitutive structure given by (40) and (4), the conditions for a steady microstructure can be entirely exressed as a set of nonlinear, (satial) integroartial differential equations in the sli system dislocation density fields. The set of all ossible solutions to this set of equations on a given body characterizes the entire class of steady microstructures under no alied loads for that articular body. Such a characterization may be undertaken by the methods of Lie Grou theory. It is natural to ask at this oint as to what rogress can be made on the above question without attemting an exhaustive classification. A large class of solutions are contained in dislocation density distributions that result in no stress in the body in addition, these are also equilibrium solutions of theory. It is easy to see from the field equations that, in the absence of alied loads, whenever the total dislocation density field on the body can be reresented as a curl of a skew 17

18 symmetric tensor field we have a zero-stress dislocation density distribution. In articular a homogeneous total dislocation density distribution satisfies this condition, as can be seen by solving (8) corresonding to this density distribution by the method of solution of exterior differential equations illustrated in [I] and observing that the distortion field so obtained has a comatible strain field. Alternatively, Kröner s solution method for the elastic theory of dislocations (Kröner, 1981) indicates directly that non-trivial stress fields are obtained only if the dislocation density field is inhomogeneous. As an aside, it would be interesting to understand how an equilibrium homogeneous dislocation density field can be rationalized as being a limit of some distribution of discrete, cylindrical dislocation curves in the body. It is imortant to note here that observed dislocation cell structures, i.e. three dimensional regions free of dislocation density searated by dislocation walls, can be modeled as a rigorous stress-free microstructure in the resent theory as regions of uniform orientation reresented by locally uniform skew symmetric elastic distortion field searated by layers over which the skew symmetric elastic distortion field transitions from one orientation to another. This observation was first made by Head et al. (1993) in their study of an equilibrium theory of dislocation mechanics. Of course, it is of equal, if not more, interest to know how such a microstructure develos under any articular loading rogram in a general nonequilibrium rocess. The resent theory is equied to deal with this question in dislocation mechanics too albeit, erhas, only with the aid of aroximate numerical comutations. Finally, the idea that a reasonable model of macroscoic yielding may well be achievable within the theory can be inferred from the arguments resented in Head et al. (1993) that also alies to the resent discussion. Essentially, if one considers a stress free cellular dislocation microstructure as described above as an initial state and considers the alication of loads to the body, for small loads the cellular structure ersists and dislocation motion (sli deformation evolution in the theory) takes lace only in the walls. Since the walls form a very small volume fraction of the total material, macroscoically this wall lasticity is not discernible until a oint where the alied load is high enough to make the wall structure disintegrate and the dislocation density evolution takes lace even in the cell interiors. The critical load at which this transition takes lace may be interreted as the yield oint of the material. 18

19 Aart from roviding guidance on the nature of some redictions, the rigorous conclusions and lausible conjectures resented in this subsection form recise targets for comutational aroaches based on the theory. 7. Boundary Conditions on Dislocation Density and Closure Condition In [I], it is shown through a simle examle that boundary conditions are required for (1) on finite domains, if we demand that an adequately osed general theory should rovide for unique solutions to the equation for dislocation density evolution in the uncouled case, i.e. the dislocation velocity is not a function of stress and, in articular, when it is a constant. elying on heuristic arguments to obtain some idea of conditions required to have a closed theory, we observe that (8)-(9) has at most one solution, by design, if the dislocation density field is secified. From exerience with the mathematical structure of henomenological lasticity we know that (10)-(11) has at most one solution under the usual traction/dislacement/mixed boundary conditions for the equilibrium equations when U is known. Noticing that (13) is essentially an ordinary differential equation, we roceed on the assumtion that secifying initial conditions for it is sufficient for a well-defined evolution. This leaves conditions of closure for (1) to be derived. That this is the only condition that is required may be further substantiated in the case when the dislocation velocities and nucleation rates are considered to be secified functions of sace and time. One otion in deriving such a condition is to consider the secification of initial and boundary conditions. It is natural to think of secifying initial conditions on the dislocation density fields; however, the recise nature of any boundary conditions that may be required is not obvious. Closure can also be ensured in the case of some artial differential equations without secifying boundary conditions,e.g. (8)-(9). However, it has to be made sure that the conditions rescribed are not overly restrictive so as to reclude hysical behaviors. In the context of dislocation density evolution, one such behavior we have to be concerned about is the increase in dislocation line length in the body. Frank-ead sources and exanding dislocation loos are believed to be some of the main mechanisms behind the increase in total dislocation density by several orders of magnitude in a cold worked material. It is imortant that the general theory be caable of redicting such behavior, and any uniqueness condition that is secified not reclude such growth in the dislocation density. 19

20 With the above ideas in mind, we begin with two simle examles that illustrate the caability of the theory to model growth in dislocation line length, and consequently in the magnitude of the dislocation density, in the body. We then attack the roblem of uniqueness of solutions through the secification of initial conditions and aroriate boundary conditions. 7.1 Initiation of Bowing of a Screw Segment In [I], the equations governing the evolution of a dislocation density field of the form b t, where b is a satially uniform vector field (Burgers vector) and t is a vector field that lies on a sli system (the line direction field), have been derived under the simlifying assumtion that there exists only one sli system. These equations are t 1 = 0 t = V t (44) { },3 t = { V t } 3, where comonents and coordinates are with resect to the sli system basis, V is the dislocation velocity magnitude, and the dislocation velocity is assumed to be in the sli lane and normal to the direction t. Suose the fields V and t vary at most with x at the initial instant, and let the initial condition reresent a straight, cylindrical screw dislocation in the sli direction with no variation of dislocation density in the cylinder along x. Under these circumstances, only the last of (44) is non-trivial and takes the form t = V t. (45) 3, If we now think of a inned segment, i.e. a dislocation velocity variation along the axis of the dislocation that vanishes outside a certain segment and is a symmetric arabola within the segment, then the center of the segment remains screw in character while the maximum increments in edge character aear at the inning oints with oosite signs, just as hysically exected in the bowing of a screw segment in a Frank-ead source. There is a great deal of similarity in the kinematics of the above situation and that of cross sli discussed in Section 6.1, u to the orientation of the lane in which bowing takes lace. One imortant difference is that if the velocity was assumed to be roortional to the driving force, then bowing in the sli lane occurs due to a gradient in the resolved shear stress ( T 1 comonent) in the x direction, whereas cross sli requires a gradient in the T 3 comonent in the x direction., 0

21 7. Exansion of a Polygonal Loo We would now like to derive a two-dimensional solution to (44) as oosed to the solution for the motion of a straight dislocation under constant velocity [I] which was essentially one-dimensional. It turns out that one of the simlest solutions that can be derived corresonds to the exansion of a olygonal dislocation loo. We assume that t 1 0 and V a constant. We define the variable s= Vt. (46) Even though the field t can deend uon x 1, and necessarily does when modeling a dislocation segment on the sli lane by a cylinder, such a deendence is only arametric and we consider the roblem in x x 3 s sace. With resect to these variables, (44) takes the form t, s = ϕ,3 (47) t = ϕ 3, 3, s, where ϕ t + t. If we now consider a vector field T = ( t, t, ϕ) in this sace and demand that 3 ( ϕ ϕ ) ( ) curl T : = t, t, t + t = 0,0,0, (48),3, s, 3, s, 3,3 then we have a solution to (47). But if curl T vanishes, then it can be reresented as a gradient of a scalar field: θ, = t3 θ,3 = t (49) θ = ϕ, s. Clearly, if we can find a scalar function θ satisfying (49), then (48) is satisfied and hence (47). Assuming there exists a solution θ of (49), we note that ( θ ) ( θ ) ( θ ), s,,3 = +. (50) If we further assume that θ is of the form θ ( x, x3, s) = f ( mx+ m3x3 cs a), (51) where f is a function of a real variable, and m, m 3, c, aare constants, then (50) imlies c=± m + m. (5) 3 We are now in a osition to exlore the evolution of a olygonal dislocation loo. For initial conditions, we assume the lane of the loo to be arallel to the x x 3 1

22 lane. The loo is visualized as straight cylindrical segments joined in the shae of a regular olygon. The olygon formed by joining the axes of these segments is assumed to lie in the lane x 1 = 0, with center at the origin. We assume that the olygon is n -sided, and arbitrarily number its n triangular sectors consecutively from 1 to n. Let the direction cosines, with resect to the x and x 3 directions, of the in-lane normal to the axis of the dislocation in the th i i m, m. The i sector be ( 3) sense of the normal is meant in the outward direction w.r.t. the origin. The initial condition is now defined as i i i t( x1, x, x3,0 ) = m3 β ( x1, mx+ m3x3 a) for ( x i i i, x 3) in the i th sector. (53) t3( x1, x, x3,0 ) = m β ( x1, mx+ m3x3 a) The function β ( x, 1 i ) reresents the variation of the dislocation density within the core cylinder for fixed x 1. It is assumed to be smooth in (, ), and zero everywhere in (, ) excet in the interval wx ( 1), wx ( 1). We now define i i i t( x1, x, x3, t) = m3 β ( x1, mx+ m3x3 Vt a) for ( x i i i, x 3) in the i th sector, (54) t3( x1, x, x3, t) = m β ( x1, mx+ m3x3 Vt a) and note that (54) is a solution to (44) in the interior of the sectors, assuming t 1 0 for all times. It is easily seen that the solution corresonds to an exanding loo with the straight dislocation segment in each sector translating outwards, w.r.t. the origin, with constant velocity V. A slight comlication arises if we interret the governing equations in the strong form - for fixed x 1, the solution is not differentiable on the lanes that divide the n sectors in x x3 s sace. We get around this difficulty by osing the roblem curl T = ( 0,0,0) (55) in variational form with continuous test functions that are iecewise smooth. A consequence of this is that if T is smooth and satisfies the strong form (55) in regions other than a finite number of internal surfaces, and on these surfaces its tangential comonent is continuous, then the weak form of the roblem is satisfied. It is clear that the required condition in the interior of the sectors is satisfied. As for the continuity condition on the internal surfaces dividing the sectors, we note that if we define the function

23 ( i i mx+ mx s a) x x x s x d 3 3 λ(,,, ) β(, ) = for (, ) x x in the i th sector (56) then, for fixed x 1, T = gradλ (in x x3 s sace) for ( x, x 3) in the interior of each sector. (57) Since λ is a continuous function, for fixed x 1 its tangential derivative is necessarily continuous on any surface in x x3 s sace (by a theorem of Maxwell), and we are done with roving that (54) is a legitimate weak solution to (44) with initial conditions defined by (53) and t Uniqueness of dislocation density evolution allowing growth We now concern ourselves with deducing a sufficient condition for assuring uniqueness of solutions to a system of the form µ = curl( µ V) + r; µ ( x,0 ) = µ 0 ( x ) on ;. (58) where µ and r are second-order tensor fields reresentative of the sli system dislocation density tensors and sli system source, and V is a vector field reresenting the sli system dislocation velocity. V and r are assumed to be rescribed functions of sace and time so that the system is linear (with variable coefficients). The latter assumtion is far from hysical reality as we have already seen, but we roceed on the belief arising from exerience with other artial differential equations that the nature of boundary conditions do not change in making the transition from the variable coefficient linear case to a quasilinear or fully nonlinear artial differential equation that may arise due to comlexity in coefficients. Let a µ exist that satisfies (58). Then Let µµ = µ ( µ ) + µ ( µ ), + µ m dv V dv V dv r dv ij ij im im n in im n ij ij, n 1 1 = µ µ V dv µ µ V dv+ µ µ V dv im im n im im n, n in im n, m, n + µ µ Vdv+ µ rdv. in im, m n ij ij i be the inflow art of the boundary on which V n < 0 and o be the outflow art on which V n > 0, where n is the outward unit normal to the boundary. Then (59) imlies (59) 3