Energy of dislocation networks

Transcription

1 Energy of dislocation networks ictor L. Berdichevsky Mechanical Engineering, Wayne State University, Detroit MI 48 USA (Dated: February, 6)

2 Abstract It is obtained an expression for energy of a random set of dislocation lines. It is used to determine the characteristics of dislocation ensembles which can appear in continuum theory of dislocations. Keywords: continuum theory of dislocations, homogenization, energy of dislocation line I. INTRODUCTION One of the basic problems in modeling of plasticity of metals is an identi cation of state variables. The problem is not simple, because dislocation networks controlling the process of deformation are extremely complex random sets. Among recent works on this subject note the papers by aiser (5), Hochrainer (5, 6), Le (6), Mohamed et al. (5), Hochrainer et al. (4), Poh et al. (3), Geers et al. (3), Mesarovic et al. (), aiser et al. (6), El-Azab (), Ghoneim et al. (). In this paper, some intrinsic probability characteristics of dislocation networks are introduced, and energy is found in terms of these characteristics from homogenization reasoning. The key role is played by dislocation density correlation tensor. Remarkably, it has a singularity. This simpli es the choice of approximations. A simple approximation of dislocation density correlation tensor is considered. It yields the formula for energy and the corresponding set of macroscopic state variables. An interesting outcome is a con rmation of the assertion (Berdichevsky 6, 6a) that averaged dislocation density tensor can enter into continuum theory of dislocations without a small parameter. A consequence is that applications of the theory are much broader than just description of size e ects in strain gradient plasticity. We begin the consideration with a formula for energy of one dislocation line (section ), then introduce intrinsic probability characteristics (section 3), express energy in terms of these characteristics (section 4), introduce correlation tensor and it s approximations (section 5), obtain the corresponding relations for energy (section 6), and discuss an alternative mechanism for dependence of energy on averaged dislocation density tensor in section 7.

3 II. ENERGY OF ONE DISLOCATION LOOP Consider homogeneous anisotropic elastic body occupying an unbounded threedimensional space R 3 : Let ij (x) be dislocation density tensor, Latin indices run through values,, 3, x denotes points in R 3 : Then elastic energy of the body has the form (see, e.g. Mura (99)), E = R 3 R 3 x ~x Hmnpq jx ~xj jx ~xj pm (x) qn (~x) d x d ~x : () Here d x and d ~x are volume elements in x and ~x variables, summation over repeated indices is implied, H mnpq is a tensor de ned on unit vectors and expressed in terms of Green s tensor. H mnpq are even functions of : The explicit form of H mnpq is not essential for what follows. For one dislocation loop with Burgers vector b i ; t i is tangent vector to ; ( ) delta-function of ; ( ) = 3 (x r (s)) ds; ij (x) = b i t j ( ) ; () x = r (s) the parametric equations of ; s arc length on ; 3 (x) = (x ) (x ) (x 3 ) ; x i are components of x; is one-dimensional delta-function. Note that j j being the length of : R 3 ( ) d x = j j ; (3) Substitution of () in () yields a diverging integral, and a regularization of () is needed. As such we use the following construction: we embed into a tube a with a circular crosssection of radius a in such a way that passes through the centers of cross-sections, and replace ( ) in () by ( a )/ a ; where ( a ) is the characteristic function of region a ; i.e. ( a ) = inside of a and ( a ) is zero outside of a: So, we set ij (x) = b i t j ( a )/ a : (4) 3

4 Obviously, for a R 3 ( a ) a d = j j : Parameter a plays the role of dislocation core radius. It is supposed to be chosen in such a way that elastic energy of the region a coincides with energy of the dislocation core. Alternative ways of regularization have been reviewed and discussed by Cai et al. (6), see also Lazar et al. (5, 4), Aifantis (9), Po et al. (4). Here From () and (4) E = a a x ~x Hmn jx ~xj jx H mn = H mnpq b p b q ; ~xj t m (s) t n (s ) dsds d X d X : (5) X are dimensionless Cartesian coordinates in cross-sections chosen in such a way that crosssection is determined by the inequality X X 6 : arious asymptotically equivalent approximations of energy for a can be used (see Cai et al. 6). We will employ the following relation, which does not seem to have been known, E = r (s ) Hmn j r (s ) r (s) t m (s) t n (s ) r (s)j ca + j r (s ) dsds (6) r (s)j where c = e =8 = :44: Formula (6) follows from (5) and can be derived in the same way as the similar expression for kinetic energy of vortex lament in ideal incompressible uid (see Berdichevsky 8, 9, p ). Note that the above-mentioned choice of a making elastic energy of region a coinciding with energy of dislocation core yields the dependence of a on b i ; t i (s) and their orientations with respect to crystal lattice. Then parameter a in (6) is a function of s: If one neglects the di erence of dislocation core energies for screw and edge dislocations, then a in (6) can be viewed as a constant. In contast to vectors associated with R 3, like x and ~x in (5), all vectors associated with dislocation lines are provided by arrow, like r (s) and r (s ) in (6). Parameter a in dynamics of vortex laments changes in the course of motion, therefore it was convenient to use a di erent asymptotically equivalent form of energy when a is kept outside of integrals. 4

5 III. INTRINSIC PROBABILITY MEASURES OF DISLOCATION NETWORKS Following homogenization theory of random structures (see Kozlov 977, 978, Jikov et al. 994, Berdichevsky 9), consider an ensemble of dislocations placed randomly over R 3. The dislocations are of two types: closed loops and open loops. In any bounded region an open loop has a nite length, which goes to in nity when volume j j of region increases inde nitely. We will use intrinsic characteristics of dislocations, i.e. the characteristics which can be measured directly on the curves without going to the embedding space. The simplest intrinsic characteristic of dislocation lines is the scalar dislocation density : if is the set of dislocation lines in region and j j is the total length of dislocation lines in region, then the scalar dislocation density is the limit = j j lim j j j j : (7) The limit in (7) has the meaning of space average: for any function of space point ' (x) it s space average h'i is de ned as the limit h'i = lim j j j j ' (x) d: Scalar dislocation density is space average of function ( ) : Another important characteristic is probability density of tangent vectors f t : Let d t be area of a small region on the unit sphere near the unit vector t ; in this case we write t d t t : Denote by the subset of where t d t t ; and by t t the length of : Then f is de ned as the limit Roughly, the length of segments of f t d t = lim j j t j j where t d t is j j f : (8) t d t : It is convenient to introduce average over dislocation lines hi. If ' (s) is a function on dislocation lines, then by de nition h'i = lim j j j j 5 ' (s) ds:

6 Let us introduce also the delta-function on the two-dimensional unit sphere t = which is concentrated at the point t t t : We denote it by t : The delta-function t t is de ned by the relation: for any smooth function ' on the unit sphere t t ' t d t t = ' : t In particularly, for ' = ; t t d t = : In these terms, probability distribution of tangent vectors is average of t t (s) over dislocation lines, t t E f = D t (s) The two-point statistics is characterized by joint probability distribution of tangent vectors at points s and s ; t (s) and t (s ) ; separated by space vector r = r (s ) r (s): t f ; r ; t = lim j j j j : t t (s) 3 ( r r (s ) + t r (s)) t (s ) dsds : It is assumed that dislocation ensemble is such that any double integral can be computed by successive limit transition. In particular, in formula (9) one can rst x s and compute the limit F r (s) + r ; t and then take the limit lim j j j j = lim j j 3 ( r r (s ) + t r (s)) t (s ) ds ; t t (s ) F r (s) + r ; t ds: Not any positive function of t ; r ; t can be a two-point distribution. A simple necessary condition is that integral of f ; t r ; t over r must diverge. Indeed, d 3 t r j j t (s) 3 ( r r (s ) + t r (s)) t (s ) dsds () 6 (9)

7 = j j t t (s) ds t t (s ) ds : t The linear integrals in the right hand side of () are of the order j j f and t j j f ; respectively, thus the right hand side is of order j j and tend to in nity as j j : t Formula (9) seems suggesting that f ; r ; t can be viewed as a joint probability density of t ; r and t : However, the above-mentioned divergence of integral of t f ; r ; t over r excludes such an interpretation. It is more convenient another form of (9) written in terms of r = j r j and unit vector = r. j r j : Since for any r 6= ; we have t f ; r; ; t 3 ( r a ) = r (r j a a j) j ; () a j = r lim j j j j t t (s) (r j r (s ) r (s)j) () r (s ) r (s) t j r (s ) t (s ) dsds : r (s)j Note that f( t t ) and t t o are dimensionless, (r) and f ; r; ; t have dimensions length and length ; respectively. Formula () allows one to nd easily the asymptotics of two-point probability density as r : Indeed, for r, the points s; s at which the integrand is nonzero are getting close, js sj : Therefore, j r (s ) r (s)j = t (s) (s s) = js sj : Besides, r (s ) j r (s ) r (s) = r (s)j t (s) (s s) js sj = t (s) sgn (s s) : In the leading approximation t (s ) = t (s) : Then integral over s is easily calculated, (r js sj) t (s) sgn (s t s) t (s) ds = t = t (s) + t (s) t (s) : (3) 7

8 Plugging (3) in () we obtain the asymptotics of two-point probability density: t f ; r; ; t = t r t + t t t f as r : (4) We assume that at large distances tangent vectors are statistically isotropic, i.e. for r t f ; r; ; t does not depend on t : Then the asymptotics of f ; r; ; t is t f ; r; ; t t t = f f as r : (5) This is seen, if (9) is integrated over a spherical layer R 6 r 6 R + R : t f ; r; ; t r drd t = lim j j j j ds t (s) R6j r (s ) r (s)j6r+r t t (s ) ds : t For su ciently large R the integral over s is equal to f R R; and does not depend t t on s. Then the right hand side of (6) is f f R R; while the left hand side is t f ; r; ; t R R: Thus, we arrive at (5). (6) I. ENERGY OF DISLOCATION NETWORK Under some assumptions on statistics of dislocation lines, energy of dislocation ensemble is the sum of elastic energy and energy of microstructure (Berdichevsky 6). For energy of microstructure per unit volume U m we have U m = x ~x j j j j Hmnpq jx ~xj jx ~xj ( pm (x) qn (~x) pm qn ) A d x d ~x ; where pm is averaged dislocation density (7) pm = M pm (x): (8) M stands for mathematical expectation. Due to the assumed ergodicity, pm = h pm i ; while formula (7) can be written also as U m = Hmnpq jj jj B mpnq () d 3 ; (9) 8

9 where B mpnq is the dislocation density correlation tensor, B mpnq = M pm (x) qn (x + ) pm qn = M pm(x) qn (x + ) ; () pm being uctuation of dislocation density tensor, pm = pm pm : All dislocations can be split in subsets of dislocations with the same Burgers vector. Further we consider only one such subset. Incorporation of several subsets complicates the picture but does not bring new issues. For dislocations with the same Burgers vector t pm = b p ht m ( )i = b p T m ; T m = t m f d t : () Length of vector T i may serve as a measure of curvature of dislocation lines: If dislocations are straight lines with unit tangent vector t t T ; then f is delta-function T. ectors T and T coincide, thus, T i T i =. If T is not a unit vector, dislocation lines are not straight. Waviness of dislocations lines increases with decrease of T i T i : Using the results of section and formula (), equation (7) can be rewritten as B r (s ) r (s) U m = j j j j Hmn j t m (s) t n (s ) r (s ) r (s)j ca + j r (s ) dsds r (s)j x ~x Hmn jx ~xj jx ~xj d xd ~x T m T n A : () Both integrals in () go to in nity as j j increases. One can show that their di erence has a nite limit, U m = Hmn tm t n ca + r t f ; r; ; t f t t f r drd d t d t : (3). CORRELATION OF DISLOCATION DENSITY TENSOR AND ITS APPROX- IMATIONS Formula () for correlation of dislocation density tensor (or, brie y, correlation tensor) is meaningful for smooth elds of dislocation density tensor, for which (7) and (9) were 9

10 obtained. For delta-type elds, () does not make sense as it includes products of deltafunctions. This does not cause problems in (7) and (9), where integrals of correlation tensor are used, but in order to deal with correlation tensor outside of energy relations, a regularization of () is needed. As such we will use the relation which makes equation (3) analogous to equation (9). We set B mn r; t = t m t n f ; r; ; t Then from (3) U m = f t t f d t d t : (4) Hmn ca + r B mn r; r drd : (5) Plugging (4) in (4) we obtain the asymptotics of correlation tensor: B mn r; = r m n f + f as r : (6) Correlation tensor has a singularity: correlation tensor increases inde nitely as r approaches zero. The asymptotics is di erent along di erent directions : Presence of various types of singularities is a characteristic feature of correlation tensors for other random structures as well (see Berdichevsky 6). For large r the correlation tensor must vanish, B mn r; as r : The convergence of integral (5) requires that the rate of decay must be faster than r : We take it to be r 3. To make an approximation of correlation tensor we need a more detailed description of dislocation line statistics. We will distinguish closed and open dislocation loops and introduce scalar dislocation densities for closed and open loops, cl and o ; and probability distributions t t of tangent vectors for closed and open dislocation loops, f cl and f o ; Here f o t = lim j j o and cl o o o = lim j j o j j ; o cl = lim j j j j ; t t (s) ds; f cl t = lim j j cl are sets of open and close dislocation loops contained in. cl t t (s) ds: (7)

11 Since j j = o + cl ; = o + cl ; f t = o f o t + cl f cl t : (8) For a closed dislocation line t i (s) ds = ; therefore, t i f cl t d t = : For open dislocation lines, the similar integral is not zero. Denote it by T i ; t T i = t i f o d t : It can be expressed in terms of vector T i introduced by (): T i = o T i olume average of dislocation density tensor () depends only on o ; b i and T i : ij h ij i = b i ht j ( )i = b i lim t j (s)ds = b i T j = j j j j o b itj : (9) The two-point statistics is characterized by two-point probability densities, t f cl ; r; ; t t and f o ; r; ; t for closed and open loops, respectively, and by mixed two-point probability density, (r j r (s ) t f ocl ; r; ; t = lim r j j r (s ) r (s)j) j r (s ) o o cl t t (s) r (s) t t (s ) dsds : (3) r (s)j Accordingly, there are three correlation tensors for open and closed loops and correlations of open and closed loops Bmn o r; = o t m t n t f o ; r; ; t t t o f o f o d t d t Bmn cl r; = cl t t m t n f cl ; r; ; t d t d t (3)

12 Bmn ocl r; = o For r ; f o t ; r; ; t to (4) t t m t n f ocl ; r; ; t d t d t : t and f cl ; r; ; t have the asymptotics that are similar t f o ; r; ; t = t r t + t t t f o t f cl ; r; ; t = t r t + t t t f cl : (3) t The asymptotics of f ocl ; r; ; t is di erent, because open and closed loops are at some nite distances: t f ocl ; r; ; t as r : This determines the asymptotics of correlation tensors B o mn; B cl mn and B ocl mn as r : B o mn r; = o Bmn cl r; = cl r m n f o + f o r m n f cl + f cl ; = : B ocl mn (33) Correlation tensors vanish as r : Microstructure energy is expressed in terms of correlation tensors: U m = Hmn ca + r Bo mn r; r drd + Hmn ca + r Bcl mn r; r drd + H mn Bmn ocl r; rdrd : (34) Interaction energy of open and closed loops is converging as a ; and the last integral of (34) is the limit value of interaction energy as a : Macroparameters are introduced by approximations of correlation tensors. We consider the simplest one, which is consistent with (33) B o mn r; = Bmn cl r; = r r o p + c o o r m n f o cl p + c cl cl r m n f cl (35)

13 Bmn o cl r; = : Here it is used that f o + f o and f cl + f cl can be replaced by f o and f cl ; respectively, due to evenness of H mn ; c o and c cl are some dimensionless phenomenological constants, which control the decay rate of correlation tensors. Relations (35) correspond to the following two-point probability densities t f o ; r; ; t = t f cl ; r; ; t = r r p + c o o r t p o r t t + + p o r o f o f o p + c cl o r t p cl r t + + p cl r cl f cl t + t t t f o t + t t t f cl t f o : (36) arious functions of r in the last terms of (36) can be taken; they do not a ect energy. Note more sophisticated extrapolations of (3) to nite r: It is clear that delta-functions in (3) transform to some smooth functions for nite r: As such one can take a Gauss-type function t on the unit sphere, G t ; which becomes the delta-function concentrated at the point t as r : For example, t f ; r; ; t = p G t + G t G t t + t f : r + c r t In this case, correlation tensor will include the variances introduced by G t : Another possibility is to improve (36) by including the next terms of asymptotics as r : They involve the joint probability density of tangent vector and curvature. In this paper we discuss only the simplest approximation (36). I. ENERGY OF DISLOCATION NETWORKS (CONTINUED) Substitution of (35) in (34) yields energy expression, U m = o ln cc o a p H mn m n f o d + cl o ln cc cl a p H mn m n f cl d : cl (37) 3

14 Let us evaluate integrals over for isotropic material. In this case, 4 H mn = + b sb j e srm e jtn rt + mn (b s s ) + b m n + 4 ( sjb s b j mn + b s sm b j jn ) ; rt = rt r t ; e srm being Levi-Civita symbols, shear modulus, Poisson coe cient. Then H mn m n = b b i i ( ) Assume that open loops are "almost straight", i.e. f o is "almost" delta-function concentrated at point ^T i ; and neglect the dependence of dislocation core energy on b s ^T s thus accepting that a is a constant. Then the rst term of (37) becomes b ( ) o ln cc o a p o s b s ^T : (38) Formula (38) describes the familiar dependence of dislocation elastic energy on Burgers vector: if Burgers vector is orthogonal to dislocation lines on average b s ^T s = ; i.e. edge components of dislocations dominates, we get elastic energy of edge dislocations, b ( ) o ln cc o a p o : The logarithmic factor ln =cc o p ao describes the contribution brought by the dislocation ensemble with density o, dislocation core radius a, and the decay rate of correlations at in nity c o : If b s is directed along ^T s ; then we get energy of predominantly screw dislocations, Formula (38) describes the mixed case. b o ln cc o a p : o An immediate consequence of (38) is that averaged dislocation density enters microstructure energy. Indeed, the scalar product b s ^T s can be expressed in terms of averaged dislocation density: from (9) i i = o b i T i ; ij ij = ob Ti T i ; and the relation ^T i = T i. q Tj T j ; 4

15 we have Then (38) takes the form s b s ^T = b (m m) : (39) mn mn b o ( ) ln cc o a p (m m) : (4) o mn mn Interestingly, averaged dislocation density tensor enters in microstructure energy without small parameters, in contrast to what is assumed in strain gradient plasticity. Suppose that all closed loops are statistically homogeneous over : Since r t d = 3 rt we obtain for microstructure energy U m = b ( ) o ln cc o a p (m m) o mn mn + b 3 ( ) cl ln cc cl a p cl (4) The coe cients in (4) are speci ed by the asymptotics of correlation tensor. In the approximation considered the dislocation network is described by three characteristics: scalar dislocation densities of open and closed loops, o and cl ; and averaged dislocation density tensor ij : II. INEQUALITY FOR DISLOCATION DENSITY AND ANOTHER MECHA- NISM FOR DEPENDENCE OF MICROSTRUCTURE ENERGY ON AERAGED DISLOCATION DENSITY TENSOR It follows from () that averaged dislocation density tensor ij density of open loops o are linked by the inequality 3 and scalar dislocation ij ij 6 b o: (4) 3 In case of the ensemble with several Burgers vectors b ; :::; b n ; in (4) b o should be replaced by () o (n) b + ::: + o () b n ; where o ; :::; (n) o are the scalar dislocation densities of open loops with Burgers vectors b ; :::; b n. 5

16 The uniquality cannot be improved, i.e. the equality of both sides is possible. In continuum theory of dislocations ij and o are independent kinematic variables. In search for equilibrium dislocation con gurations energy must be minimized over o : Inequality. (4) shows that the minimum possible value of o is ( ij ij ) b: Plugging this to (4) we obtain b ( ) ij ij = ln ( ij ij ) =4 cc o a Energy (43) is large enough to be included in macroscopic theory. specimen size, ij ` ; c o ; a b: Then energy (43) is of the order (m m) : (43) mn mn Indeed, let ` be the b ` ln ` b : (44) Compare this number with elastic energy density, which is of the order =" (e); for elastic deformation " (e) of order 4 : Then (44) is of the order of elastic energy, if `=b 6 8 : For b :5 9 m; the characteristics length ` is about :3m: For smaller ` microstructure energy is larger than elastic energy. Note that energy (43) has the same order as microstructure energy of linear phenomenological theory (Berdischevsky, Sedov 967), (characteristic length) ij ij ; (45) if the characteristic length in (45) is chosen to be of the order p o : Other choices of the characteristic length, b or p ; would make energy (45) negligible in macroscopic theory. References Aifantis, E.C., 9. Non-singular dislocation elds, Mater. Sci. Eng., 3, 6. El-Azab, A.,. Statistical mechanics treatment of the evolution of dislocation distributions in single crystals, Phys. Rev. B, 6, 956. Berdichevsky,.L., and L.I. Sedov, 967. Dynamic theory of continously distributed dislocations. Its relation to plasticity theory, PMM, 3 (6), 98- (English translation: 967. J. Appl. Math. Mech. (PMM), 989-6). Berdichevsky,.L., 6. On thermodynamics of crystal plasticity, Scripta Materialia, 54,

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