an analysis of equilibrium dislocation...

Acta metall, mater. Vol. 41, No. 2, pp. 625~42, 1993 0956-7151/93 $6.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1993 Pergamon Press Ltd

AN ANALYSIS OF EQUILIBRIUM DISLOCATION DISTRIBUTIONS

V. A. LUBARDA, J. A. BLUME and A. NEEDLEMAN Division of Engineering, Brown University, Providence, RI 02912, U.S.A.

(Received 13 December 1991; received for publication 28 July 1992)

Abstract--Equilibrium distributions of collections of discrete dislocations are analyzed, with the dislocations modelled as line defects in a linear elastic medium. The dislocated equilibrium configuration is determined by finding a minimum potential energy configuration, with respect to variations in the dislocation positions, for a fixed number and type of dislocations. Numerical results are presented for finite and infinite bodies with distributions of edge dislocations under plane strain conditions. Calculations involving doubly periodic arrays of cells, within which there is a single set of parallel slip planes, show a strong tendency for sharp dislocation walls to form. Perturbations of the wall structure due to the presence of pinned dislocations, vacant slip planes and free surfaces are illustrated. The stress fields due to the dislocation walls are calculated and large shear stress values are found away from any dislocation core. Pileups involving dislocations on two sets of intersecting slip planes are found to give rise to equilibrium configurations involving dislocation free regions. The response of dislocation patterns in an infinite medium to an imposed shear stress is also analyzed.

Rtsmnt--On analyse des distributions ~ l'tquilibre de collections de dislocations discrttes, les dislocations 6rant modtlistes comme des dtfauts lintaires dans un milieu 61astique lintaire. La configuration l'tquilibre est dtterminte en trouvant une configuration d'tnergie potentielle minimale par rapport aux variations de position des dislocations, pour un nombre fixe et un type donn6 de dislocations. Des rtsultats numtriques sont prtsentts pour des corps finis et' infinis avec des distributions de dislocations coins en dtformation plane. Des calculs impliquant des arrangements doublement ptriodiques de cellules l'inttrieur desquelles il y a un seul systtme de plans de glissement paralltles, montrent une forte tendance

former des parois 6troites de dislocations. Des perturbations de la structure en parois dues fi la prtsence de dislocations 6pingltes, de plans de glissements vides et de surfaces libres sont illustrtes. Les champs de contrainte dus aux patois de dislocations sont calcults et on trouve de grandes valeurs pour le cisaillement loin du coeur des dislocations. On trouve que des empilements qui impliquent des dislocations sur deux systtmes de glissement qui s'interseetent donnent naissance fi des configurations d'tquilibre impliquant des rtgions exemptes de dislocations. La rtponse de rtseaux de dislocations clans un milieu infini fi une contrainte de cisaillement imposte est aussi analyste.

Zusammenfassung--Gleichgewichtsverteilungen von Ansammlungen diskreter Versetzungen werden analysiert, wobei die Versetzungen als lineare Defekte in einem linear elastischen Medium angesehen werden. Die Gleichgewichtsanordnung der versetzungen wird bestimmt, indem die Anordnung minimaler potentieller Energie fiir eine feste anzahl von Versetzungen bestimmten Typs dutch Variation der Versetzungsorte aufgesucht wird. Numerische Ergebnisse werden fiir endliche und unendliche Festk6rper mit Verteilungen von Stufenversetzungen unter ebenen Dehnungsbedingungen vorgelegt. Berechnungen, die doppeltperiodische Zellanordnngen mit einem einzigen parallelen Gleitebenensatz unfassen, weisen auf eine starke Neigung zur Bildung lokalisierter Versetzungsw/inde hin. St6rungen der Wandstruktur dutch verankerte Versetzungen, freie Gleitebenen und freie oberflfi~hen werden dargestellt. Die Spannungsfelder der Versetzungsw/inde werden bereehnet; es ergeben sich groBe Scherspannungen weitab von den Versetzungskernen. Aufstauungen, gebildet aus zwei S~itzen von Versetzungen in sich schneidenden Ebenen, ffihren zu Gleichgewichtsanordnungen mit versetzungsfreien Bereichen. Das Verhalten der Versetzungsstrukturen in einem unendlichen Medium unter einer iiberlagerten Scherspannung wird auBerdem analysiert.

1. INTRODUCTION

It is commonly observed tha t dis locat ions in struc- tura l metals tend to be dis t r ibuted in organized pat terns . Accordingly, a basic p rob lem in dislocat ion theory is to determine the equi l ibr ium a r rangement of dis locat ions under their mutua l interact ions and under the act ion of an applied stress field. This p rob lem has been addressed f rom a variety of perspectives and a comprehens ive review of the l i terature has been given in A m o d e o [1]. Examples

of the con t inuum approaches t aken include: analytical and numerical solut ions for equi l ibr ium con- f igurat ions of discrete dislocations, e.g. Eshelby et al. [2] and Head [3]; the deve lopment of and solut ion to approximate dis locat ion based models, as in Kuh lmann-Wi l sdo r f and Nine [4] and Hol t [5]; solut ions for con t inuous dis t r ibut ions of dislocations, Walgraef and Aifant is [6]; and compu- t a t ion of relaxed configurat ions f rom dislocat ion dynamics, N e u m a n n [7], GuUuoglu et aL [8] and A m o d e o [1].

625

626 LUBARDA et al.: EQUILIBRIUM DISLOCATION DISTRIBUTIONS

Here, as in Eshelby et al. [2] and Head [3], equilibrium configurations of dislocations considered as line defects in a linear elastic continuum are analyzed. The number and type of dislocations in the body are prescribed and the dislocations are constrained to move along specified slip planes. The general formulation is given for a finite body free of external tractions and for the simpler situation of an infinite solid. In either case, the dislocation arrangement that renders the potential energy stationary with respect to variations in the dislocation positions is sought. The finite body formulation gives a perspective on the relation of the dislocation distributions obtained for infinite solids to boundary value problem solutions for a finite dislocated solid.

The numerical examples concern distributions of edge dislocations under plane strain conditions. Some simple configurations are analyzed to illustrate characteristic features of the problem, which include long range interactions between dislocations and the possible existence of multiple equilibrium configurations. Doubly periodic cells subject to periodic boundary conditions are considered. The potential energy is minimized with respect to changes in the dislocation positions using a conjugate gradient method. For an infinite body, a cut-off procedure is used to reduce integrals over an infinite domain to finite domain integrals. Convergence issues related to this finite cut-off are discussed. The development of characteristic dislocation patterns is shown, such as dislocation walls and multiple-dipole configurations. Dislocation pinning and its effect on equilibrium patterns is also illustrated. The evolution of the obtained dislocation structures under increasing applied shear stress is analyzed. We also consider dislocation distributions in a finite body with traction free boundaries and compare these with corresponding distributions in an infinite medium.

The framework presented here is one step toward the development of a discrete dislocation based analysis of plastic deformation in crystalline solids. The formulation is geared toward the use of numerical methods to solve problems involving large num- bers of dislocations and to account for boundaries, interfaces and free surfaces. The present study focuses on the equilibrium arrangement of a given number and type of dislocations, which is a problem of elasticity theory. The eventual goal is to use the traction free finite body configurations as initial configurations in an analysis of dislocation evolution that accounts for dislocation nucleation and annihilation as well as for lattice resistance to dislocation motion. Rules describing these phenomena have recently been developed, in a full three dimensional context, by Kubin et al. [9].

2. PROBLEM FORMULATION

The problem addressed here is the equilibrium arrangement, consistent with imposed boundary con-

ditions, of a given number of interacting dislocations. The problem is formulated within the context of linear elasticity theory, i.e. a linear constitutive relation and infinitesimal displacement gradient kin- ematics. The elastic theory of dislocations interacting with each other and with a boundary is described, for example, in Nabarro [10], Hirth and Lothe [11] and Eshelby [12]. The formulation here is aimed at providing a basis for the use of modern numerical methods, such as finite element or boundary element methods, to solve mixed boundary value problems for dislocated solids.

" The analysis proceeds in several stages. First, for a finite body with a given number and type of dislocations, an equilibrium configuration is sought for which the boundary tractions vanish. Due to the presence of dislocations, this configuration is not stress free, but has an equilibrated residual stress distribution. For an infinite body, the traction free condition is not relevant and the determination of the equilibrium configuration is simplified. This dislocated configuration can be used as an initial configuration, which is then subject to some prescribed loading history.

The governing equations are formulated in three dimensions and diadic notation is used: vectors and tensors are denoted by bold-face symbols, • denotes the inner product and • the trace product. For example, with respect to a Cartesian basis ei ( i = 1 , 2 , 3 ) , a ' b = a i b i , A : B = A u B j i and (L:B)g= LuktB~k, with summation implied over repeated indices.

2.1. Governing equations

The stress field in the initial, dislocated equilibrium configuration is written as the superposition of two fields, a = # + 0, see Fig. l(a). One, #, is the infinite body equilibrium stress field associated with n dislocations, which gives rise to surface tractions ~'. The other stress field, &, is the equilibrium stress field that cancels these surface tractions. In particular, for a given spatial distribution of n dislocations,

= ~7= i ai, in which a~ is the equilibrium stress field associated with the ith dislocation in an infinite medium; # is determined as the solution to the following traction boundary value problem for the dislocation free body

V ' 8 = 0 , 8 = L : g in V, v ' # = - l ~ o n S (1)

where V is the gradient operator, L is the tensor of elastic moduli, V and S are, respectively, the volume and surface of the body and v is the surface normal. Also, the infinite body dislocation solutions, a~, are presumed to be based on the tensor of moduli L, so that o~ = L : ~ a n d ~ = L:g.

A distribution of dislocations is sought that pro- vides a local minimum in the total potential energy of the body, subject to the constraints that the number, type and length of the dislocations are fixed and that the position of each dislocation (as a rigid entity) can

LUBARDA et al.: EQUILIBRIUM DISLOCATION DISTRIBUTIONS 627

only vary along a given slip plane. In other words, each dislocation has in its slip plane at most 3 d.f. Certain dislocations can be specified to have a fixed position, and their positions do not vary during the minimization process, but their presence affects the equilibrium arrangement of the remaining dislocations.

In order to write the potential energy for the body, we define V, as the volume V excluding a small core region C, surrounding the ith dislocation line, and let S, be the surface of this core; V. stands for the volume exterior to all of the core regions. Since the exterior boundary S of the body is traction-free, the potential energy H is defined as the strain-energy in the volume V. that excludes the core volumes, and the core energies are taken into account by the inclusion of the work of tractions on the inner surfaces S, of V,

l f v ~ 1 I 17 = ~ o:~ dV + T ' u d S (2) , i=l~dSI

with T and u standing for the traction and displacement on each surface Si. Recalling that the total stress and strain fields (o, ~) are the superposition of the infinite-medium dislocation fields (#, g) and the smooth fields (& g), and that (#, g) are further com- prised of the sum of the individual dislocation fields (oi, ~-i), one has

11 d:g dV + ~vd:g dV H = ~ jv

+tEl i= 1 2 fv °i:~j dV

-~- '~1 I1 ~ ' = 2 i ~i*,(.i dV-~-l~2j S̀ "ri,uidS] . (3)

Here , T i and ui are the traction and displacement on the surface S, due to the corresponding dislocation. In arriving at the above, use is made of the fact that the singularity in o, on the corresponding dislocation line is weak enough to allow

fs T,.r, dS = fc, O,:e dV (4)

for any displacement field fi that is smooth on C, and has the corresponding strain field i. The third term in (3) is the interaction energy of the infinite-medium dislocation fields within the volume V, and is denoted by U~t, while the last two terms represent the self energy.

In order to compute the variation in the potential energy due to changes in positions of the dislocations, it is convenient to use the relation

fvO,:~,dV= fe'ri:~,dV

-- I a~:~,dV, ( i = 1 , 2 . . . . n) (5) d~

in which E, is the unbounded region exterior to the small core volume surrounding the ith dislocation line, and V is the exterior of V. If the material is elastically isotropic, the integral over E, is independent of the position of the ith dislocation, as is the integral over S, in (3). If the material is anisotropic and the dislocations are restricted to translate on slip planes, preserving a fixed orientation with respect to the anisotropy axes, the above integrals again remain constant with respect to variation in the dislocation positions.

When the dislocation segment positions vary, so do the ( ' ) fields. Accordingly, the variation of potential energy is

6n= fva:rt dV + ;v[a:St +&~:tldV

+ oi:&jdV -- _~,:&idV. (6) /ffilj~l i=l JV

If 5 u, is the displacement-field variation in an infinite medium associated with an infinitesimal shift in the position of the ith dislocation, this field is continuous on the exterior V of V. Since V" 5o, = 0 on V, one has

fv °':~'dV = - I T''~a'dsJs (7)

in which T, is the traction on the material in V due to o,. Further, because "i" = -"i" on S

v ~ :Sg dV = fs ~£.6fi dS

= -f~.aadS

= -- fv~:fg dV (8)

with 611 the variation in displacement associated with the strain variation 5¢. The identity 5d :g = d :SZ, along with (7) and (8), enables the potential-energy variation to be written as

j~l

+ iffil ~ ;S ti'~uidS" (9)

The divergence theorem gives

fv':~dV=fs"~adS-=~fL, r,'b, rs, dl (10)

in which 6fi is the variation in the displacement associated with the infinite-medium dislocation fields, while L,, b, and ~s, respectively stand for the dislocation line, Burgers vector and infinitesimal perpendicular advance of the points along the ith dislocation line; I"i is the traction due to d on the slip plane of the ith dislocation.

628 LUBARDA et aL: EQUILIBRIUM DISLOCATION DISTRIBUTIONS

Similarly

ffr,:&:dV= fsT:6ujdS- f~ T,:bj6sjdt; (l l)

T 0. is the traction induced by ai on the slip plane of the j th dislocation.

Combining (9), (10), (l l) and rearranging the summation terms yields

fS[ ÷,~1 --~1 ] t~I'I-- "r.$~l Ti'c~u j d S • = j =

But

so that

i~lj~=l Ti" 1~ nJl dS

= fs( + dS =0

6H= -i=I ~ , "r,+ ~iTji= "bi6sidL (14)

j#t

Since Ti=#'ni and T:i=a:'ni with ni the normal vector to the slip plane of the ith dislocation, the stationarity condition 6H = 0 gives

bi a "n, ,idt=O. (15) i=1 I "= J # t f

Each dislocation can rigidly rotate and translate in its slip plane, so that the infinitesimal normal advance 6si of the ith dislocation in its slip plane is given by

6si = m: {6~ini x x + 6e~- (ni' ~ei)ni},

( i = l , 2 . . . . n). (16)

Here, mi is the outer unit normal to the ith dislocation loop in the slip plane, while &o i and 6e i are respectively the infinitesimal rotation and translations of the ith dislocation; x is the position vector of points on the dislocation line. Because &o i and 6ci are independent for each dislocation, (15) and (16) imply

O aj • n i [ (m i x x)" nil d l = 0 .=

( i = 1 , 2 . . . . n) (17)

and, since ni-m i - 0

b i" O 6y "nimidl = 0 (i = 1,2 . . . . n ) .

j # i

Equations (17) and (18) give three equilibrium conditions for each dislocation. Also, since bi lies in the slip plane with the normal ni, bi" # 'n i is proportional to the corresponding resolved shear stress in that plane and therefore depends only on the deviatoric part of &

For dislocation distributions in an infinite solid, the condition of traction free boundaries no longer pertains and # can be taken to vanish. In this special case (17) and (18) simplify to

fL ~ b:a/ni[(m i x x)'n,] dl =0 i j = l

j # i

( i = 1 , 2 . . . . n) (19)

and

fL ~b i ' a j ' n im, d l = 0 ( i = 1 , 2 . . . . n). (20) i j = l

j # i

In the interior of a solid, well away from any traction (13) free boundary, it is expected that a solution to (19)

and (20) is a good approximation to a solution to (17) and (18). An iterative method is used to solve (17) and (18) that involves first obtaining a dislocation distribution from (19) and (20). The negative of the resulting boundary tractions is imposed on the body to estimate ~. This perturbs the dislocation distribution through (17) and (18). A new estimate of 0 is then obtained and so on. Using (1) to obtain the (^) quantities involves solving a standard boundary value problem for an elastic solid, which we solve using finite elements.

When attention is confined to the infinite solid case, the (^) fields vanish, and the potential energy reduces to the interaction energy; see (3) and (6). Dislocation distributions obtained from (19) and (20) give rise to stationary values of the interaction energy with respect to variations in dislocation positions. Those that correspond to a local minimum in the interaction energy are termed stable, while other stationary values are termed unstable.

The configuration obtained by solving (17) and (18) or (19) and (20) can serve as the initial configuration for an imposed loading history. The initial configuration has a residual stress field, given by a = 0 + ~, that satisfies equilibrium, since # and ~ are each equilibrium stress fields. If the body is finite, the external surface is also traction free.

The formulation presented above is, in principle, valid for dislocation loops of arbitrary shape and curvature. However, since analytical expressions for the equilibrium stress fields associated with curved dislocations in an infinite medium are largely unavail- able, the energy-minimization procedure outlined is primarily useful for straight dislocations and dislocations consisting of collections of straight segments. It should further be noted that there is no guarantee that a solution exists for a given number, strength and

(18) length of dislocations, and, for finite bodies, for a

LUBARDA et al.:

given body size and shape. Moreover, if a solution does exist, it is not necessarily unique.

2.2. Edge dislocations in plane-strain

The numerical examples will concern distributions of edge dislocations in an isotropic elastic solid under plane strain conditions. In the case of an infinite medium, the formulation, in spirit, parallels that of Head [3]. However, increased computational capa- bility permits consideration of more complex configurations and bounded media.

The x t - x 2 plane is taken to be the plane of deformation. When all slip planes are parallel to the Xl -axis, as will be assumed in most of the calculations here, the interaction energy (per unit length of dislo-

EQUILIBRIUM DISLOCATION DISTRIBUTIONS 629

cation line) for two edge dislocations positioned at (X~, X~) and (X~, X J E), respectively, is [ll]

F [ A x i J ' l 2 .~- ( A x i J ' t 2 - - i j l l . K 11 ~. 21 Ut~t= --K /2In

(AXe)2 7, #b'bJ (21) + (AX~J) 2 + (AXe) 2] ka = 27r(1 - v )

where A X ~ = X { - X ~ (~ = 1,2), b ~ is the signed Burgers vector, # is the shear modulus and v is Poisson's ratio, while R 0 is an arbitrary length.

If the dislocations are on two slip planes which intersect at an arbitrary angle [Fig. l(b)], their interaction energy is conveniently obtained by dissociating dislocation j into two partial dislocations, one on a

(a)

\ 4 ,, . L /

)

)

)÷ im

< •

$i

(b) Fig. 1. (a) Superposition of two solutions, one corresponding to the infinite medium dislocation solution and the other arising from the stress field that cancels the infinite medium surface tractions. (b) Definition of geometric quantities entering the expression for the interaction energy (22) for edge dislocations on slip

planes that intersect at an angle.


parallel and one on an orthogonal slip plane, and summing the interaction energies of dislocation i and the two components of dislocation j.

If s ~ and s j are the respective distances from the point of intersection of the slip planes to the i and j dislocation, the interaction energy can be written as

Uint = 1 (COS q~) In ~02 + (sin 2 q~) (22)

where R 2 = (si) 2 + (s02 - 2sisJ cos ~b. For parallel slip planes, the glide force ff~J (per unit

length of dislocation line) on dislocation i due to dislocation j is

P~ = ov,'J.,= _ k, jAX~q(AX'~) ~ (axe)2] l

~xi [(AX?) 2 + (AXe)2] 2

(23)

Note that although the value of U~nt depends on the arbitrary length R0, the value of P~ does not. For the case of intersecting slip planes, the glide force on dislocation i due to dislocation j is obtained as

~.. OUSt k/j F~ Os, = R-4 [(sj - s ' cos ~b)(R 2 cos 4~

- - 2SiS j sin 2 q~) + R2s i sin 2 $]. (24)

For more dislocations and more slip planes, the total interaction energy is the sum of the interaction energies between all pairs of dislocations. Specifically, for the case of slip planes all parallel to the xj-axis, the glide force on dislocation i due to its interaction with other dislocations becomes

ff~ = ~ ff~ (25) )#i

and (15) reduces to

(ff~ + 612) 6X~ = 0 (26) i

which implies P] + #/2 = 0 for each dislocation that is not pinned. The component of force out of the slip plane, the climb force, does not necessarily vanish. For a pinned dislocation, 6X~ = 0 and the reaction glide force is obtained from (25).

For the later reference the in-plane stresses at a point (xl,x2) due to a dislocation at (X~,X~) are listed here (see, for example, Nabarro [10])

0"11 ( X l , X2)

]-~b i (X 2 -- X~) [3(Xl - X~ )2 + (x2 -- Xi)2]

2n (1 - v) [(xl - X~ )2 + (x2 - X~ )02

(27)

~2~(Xl, x : )

/zb' (x2 - X~ ) [(x, - X~ )2 _ (x 2 _ X~)2]

2 ~ (1 - v ) [(xl - x ~ ) 2 + (x2 - x ; )212

(28)

~12(Xl , X2)

u b ' ( x , - - X ~ ) [ (x , - - X~ )2 __ (x2 - - X ; ) 2 ]

2n (1 - v) [(x I - X~ )2 + (x 2 _ X~)212

(29)

We will confine our attention to dislocations of equal magnitude, so that b ~ = - t - b (i = 1, 2 . . . . n). The parameter k 'J appearing in (21) and (22) then has the same magnitude for every i and j

ub 2 [k°[ = k - - - for all i, j . (30)

2n(1 - v)

Subsequently, dislocations for which b ~ = b > 0 are termed positive dislocations and are represented by a + in various figures showing dislocation distributions. Similarly, dislocations for which b ~ = - b are termed negative dislocations and are represented by a - - .

3. SIMPLE SOLUTIONS

In this selection, solutions are presented for two simple dislocation configurations that illustrate characteristic features of this class of problems, namely long range interaction effects and the possibility of multiple equilibrium configurations.

First, consider a dislocation wall consisting of identical, uniformly spaced edge dislocations along x l = 0 . The shear stress, crn, at an arbitrary point along the slip plane x 2 = 0 can be obtained

(a)

- - - 101 dislocation wall 3 \ - - Infinite dislocation wall

% \ ~ 2

\ 0 Illll I I l l ~

10_ l 2 3 100 2 3 10 z 2 3 ]02 2 3

Log (Xl/h)

54~ I- (h) 20,001 dislocation wall

P Infinite dislocation wall

0 I J l l l I l l I I i i i id J i l l I i iiiJl~ i i I1L IA I ,~ INIB I I Jl l l l l [ I I I I I i i i i l ~ l l [ [ ] I IHJ

10_12 1002 1012 zo 22 zo 32 1042 1052

Log (Xl/h)

Fig. 2. Shear stress along x 2 = 0 vs distance from a dislocation wall at x I =0. (a) 101 dislocations in the wall; (b) 20,001 dislocations in the wall. The shear stress is normalized by k/bh, where b is the Burgers vector of the wall dislocations, h is the wall dislocation spacing and k is given

by (3O).


obtained by adding the contributions from all dislocations in the wall

.ffil (~2 + n2) ------------~ + (31)

where ~ = x~/h, with h being the uniform dislocation spacing and N the number of dislocations in the wall. For an infinite wall, (31) can be summed to give

nk 2rr~ Crl2(Xl) = bh cosh(2n~) - 1 ' (32)

The shear stress in (32) is unbounded at xl = 0 and decays exponentially as x~ increases.

Figure 2(a) shows the shear stress variation along the slip plane of the middle dislocation from a finite wall o f 101 (uniformly-distributed) dislocations, which is obtained from (31) by taking N = 50. I f Xl is sufficiently close to the wall (say, x~ < h), the curve essentially coincides with that obtained from the infinite wall solution (32). Further away from the wall, the behavior changes. After reaching a mini-

m u m at x~ = 1.43h, the shear stress increases until a local maximum of try2 = k/bh is attained at x~ = 50h, which is half the wall height. This maximum value is the same as the shear stress very near the wall, at x~ ~ 0.49h. The shear stress then decays as a single dislocation with a magnified Burgers vector, and approaches zero asymptotically at large x~. The behavior of a wall consisting of 20,001 dislocations is shown in Fig. 2(b). In this case, a local minimum is reached at x~ = 2.35h, while the local maximum shear stress of k/bh occurs when x I = 10,000h. The shear stress maximum occurs for any finite wall; as N ~ oo, the location of the maximum approaches x~ = oo.

This illustrates significant long range interaction effects. Accordingly, modelling a finite wall as an infinitely long wall is satisfactory only for represent- ing the behavior near the wall, caused by that wall. It should also be noted that the possibility of a disturbance in dislocation separation near the ends of the wall, due to the large climb force there, has not been considered.

l & l

&

&

1

&

& t ..................

& &

& &

& l

& &

& l

< L >

Xl

h

(a) (b)

- 1 2 . 8 . . . . . . . . . . . . . . . . . . .

- 1 2 . 9

-~3.o \ / " " ~ . . . . . . . . . . - " < 1 ,, \ . \ / /

D'~ -13.1 \ ~ / .. ' \ / • . h = L / l O "'. \ / h=L/7

-13.2 - " \ / ' "-- ' - - h=L/5 ....... h=L/4

i -13.3 . . . . . . . . . . , ,

0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0

Xl/L

Fig. 3. (a) Doubly periodic array of square cells populated by edge dislocations on parallel slip planes. (b) A cell containing two positive dislocations (denoted by the + signs) on parallel slip planes. The dislocation at x I = 0.2L is pinned. (c) Interaction energy vs dislocation separation for the doubly periodic array shown in Fig. 3(b). h is the slip plane spacing and the interaction energy is normalized by nk, where n is the number of dislocations per cell and k is given by (30). The transition from a single interaction

energy minimum to two minima is seen as the separation between the slip planes is decreased.


We next consider the variation in the interaction energy with dislocation spacing for a doubly periodic array of cells, Fig. 3(a). In the particular case considered, there are two slip planes in each cell and one has a dislocation pinned at xi = 0.2L, while the dislocation on the other slip plane is free, as sketched in Fig. 3(b). The two dislocations each have the same sign. Figure 3(c) shows interaction energy vs dislocation separation curves for various slip plane spacings. There is a change from a single minimum to two minima for a slip plane spacing of about L/6. For more closely spaced slip planes, a stable equilibrium configuration with a dislocation spacing of L/2 emerges.

If the two dislocations are of opposite sign, the sign of the interaction energy in Fig. 3(c) changes, but the magnitude remains the same. The change in sign of the interaction energy means that the minima in

Fig. 3 becomes maxima and vice versa. In this case, for widely spaced slip planes the sole energy minima corresponds to an equilibrium configuration with a dislocation spacing of half the cell side. For closely spaced opposite-signed dislocations, the tendency is for the horizontal dislocation spacing to approach the vertical separation between the slip planes and thus form a dipole configuration. In such a configuration, the replica dislocations are far away, and the effects of their interaction with the dislocations within the cell become insignificant.

4. NUMERICAL RESULTS

4.1. Dislocation distributions in an infinite solid

We first consider doubly periodic square cells populated by a given density of edge dislocations, which lie on parallel slip planes [see Fig. 3(a)]. Unless

+44"4 '

, H ' 4 . +

' e " l ~ +

÷4 -1 '÷

~ . ' H - t

4 ' 4 " H '

~ . ~ . ÷

÷ , t < . +

4 . , t - t . t

~ ' , M ' ÷

÷ ' H . +

÷ 4 " t - ÷

÷ ' t ' ~ ÷

+ 4 , + +

÷ . l ' t ÷

÷ ' t '+ ' t .

4.,(,,,I.,(,

, l ' 4 "k÷

(o)

,~4. + 4 .

÷ 4 " , I , +

4 .4 . ,I,~,

• I . + + 4 .

4 . + ÷ ÷

4 .4 . < .s t

,~4. t t

+ ' 4 , , I , #

+ 4 . ,~,1,

4 . + @ ÷

,e.+ ÷ +

I I I I ,

4 , ' , . , I , ÷

÷ ÷ 4"~P

+ 4 . + 4 '

J i l l / t

÷ 4. 4. +

4" + + ,I.

4. 4. + ,t.

÷ ÷ ÷ +

<' + 4. +

4. ÷ + +

+ 4. ,I. +

÷ + + ,4.

+ 4. 4, ~,

÷ ÷ ÷ ,~

4. ,~ ÷ ÷

4. , . ,~ +

÷ 4. <. ~.

4. 4. 4. +

÷ <, ~. +

4. ~ 4. +

, t ,(. 4. 4.

" Il l l i i l l

(b) (c) Fig. 4. Three stable equilibrium configurations for a distribution with four like-sigued dislocations on each of twenty slip planes. Each configuration has sharply defined dislocation walls, but the spacings differ.

+ denote a positive dislocation. The configuration shown in (a) has the lowest energy of the three.


otherwise stated, the ratio of the cell side-length L to the Burgers vector b is L/b = 4000. The number of slip planes per cell and the number of dislocations per slip plane are specified. The spacing between slip planes is taken to be uniform, although in some cases certain slip planes are taken to be dislocation free. An arbitrary, more or less random, starting distribution of dislocations on a slip plane is prescribed or, to explore the effect of the starting distribution on the equilibrium pattern obtained, a more controlled starting distribution may be specified. In general, this starting distribution is not, of course, an equilibrium distribution. The interaction energy defined by (25) is summed for each pair of dislocations within a cell and for several surrounding replicas of that cell and then is minimized by a conjugate gradient method using the algorithm described in Press et al. [13].

We find that more replica cells are needed in a calculation if the spacing between slip planes is relatively broad. However, when the slip planes are closely spaced, one or two surrounding replicas are sufficient to obtain converged results. The equilibrium dislocation distributions on closely-spaced slip planes are calculated using replicas that encom- pass two cell shifts in each direction, so that 25 total cells are included in the calculation. For several cases, the equilibrium distribution calculated in this manner was taken as the starting configuration for a compu- tation using four cell shifts and then eight cell shifts. The computed equilibrium configurations were found to be essentially independent of the number of surrounding cells.

The stable equilibrium configurations in Fig. 4 illustrate the tendency of like-signed dislocations to form walls. With four like-signed dislocations on each of twenty equally spaced slip planes in a cell, the three equilibrium wall structures shown occur. The particular equilibrium configuration obtained depends on the starting dislocation distribution chosen. The lowest value of the energy, - 26,074k, is associated with the configuration shown in Fig. 4(a). This is the configuration in Fig. 4 that has the greatest separation between walls, although it gives rise to the closest dislocation spacing within the wall. The dislocation spacing in Fig. 4(a) is ~0.02L, which is still much larger than the Burgers vector, b = 0.00025L. The configurations in Fig. 4(b) and Fig. 4(c) have energies of - 25,750k and - 25,676k, respectively. In Fig. 4(c) where four walls form, the two inner walls are closer to each other than to the outside walls.

Figure 5 shows equilibrium configurations where two like-signed dislocations lie on each slip plane within the cell, with alternating dislocation signs from one plane to another. With 20 slip planes of equal spacing in a cell, two pairs of dislocation walls form, as shown in Fig. 5(a). The separation between the dislocations in the positive and the negative walls is 0.048L, while the mean separation between walls is L/2. However, if the starting dislocation distribution

4 . . ÷

4 4"

,I. #,

,#, ,I.

.41. 3,

'I" 4"

(a)

4.

4"

4 .

41.

4. .1.

4-

4 .

.I. 4 .

'¢" 4.

4~

4 .

÷

(b) Fig. 5. Equilibrium configurations of two like-signed dislocations per slip plane, with dislocation sign alternating from one slip plane to the next. The particular configuration arrived at depends upon the positions of the dislocations at

the start of the energy-minimization process.

is close to one with diagonal multi-dipoles, the equilibrium distribution shown in Fig. 5(b) is obtained.

Figure 6 shows stable equilibrium distributions of single like-signed dislocations on each of the slip planes within the square cell. In Fig. 6(a) there are 30 dislocations on 30 equally spaced slip planes, while Fig. 6(b) shows 20 dislocations on 30 slip planes, with the intermediate l0 planes free from dislocations. Dislocation wall structures again form. Equilibrium dislocation patterns are also obtained with some dislocations fixed (pinned) within the cell. For example, Fig. 6(c) shows an equilibrium configuration for like-signed dislocations, one on each of 30 slip planes, when the dislocation on the slip plane


(o)

÷

+

÷

r

(b)

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

÷

d,

(c) Fig. 6. Stable equilibrium configurations illustrating dislocation wall structures for like-signed dislocations. + denotes a positive dislocation. (a) One dislocation on each of 30 equally spaced slip planes per cell. (b) One dislocation on each of 20 slip planes per cell with the intermediate 10 slip planes dislocation free. (c) One dislocation on each of 30 equally spaced slip planes per cell with one dislocation

pinned at xl = 0.2L, x 2 = 0.

along x2 = L/30 is pinned at xl = 0.2L. A dislocation wall structure is formed by most of the dislocations, although the equilibrium positions of several dislocations are out of the wall. This distribution is the minimum energy configuration found from a random starting dislocation distribution. There may be other stable equilibrium configurations.

Equilibrium configurations with the slip planes containing dislocations of alternating sign are shown in Fig. 7. In Fig. 7(a), positive and negative dislocation walls form that are separated by L/2, while in Fig. 7(b), where the intermediate 10 slip planes are dislocation free, occasional dislocation dipoles interrupt the wall structure. The equilibrium configurations are local minima and depend on the starting

dislocation distribution. For example, the equilibrium configurations shown in Fig. 7(c, d) were obtained from starting distributions relatively close to a diagonal multi-dipole structure. The interaction energy associated with the distribution in Fig. 7(a) is -252k , while the distribution in Fig. 7(c) has an interaction energy equal to -188k.

Figure 8(b) shows the formation of variable thickness dislocation walls, which are obtained from the random starting distribution of positive and negative dislocations shown in Fig. 8(a). There are 30 equally spaced slip planes in Fig. 8 with a total of 250 dislocations per cell. The average separation between the two walls in Fig. 8Co) is 0.SL. The dislocation density in Fig. 8 is 2.5 x 10~°/cm 2, if b is identified

LUBARDA et al.:

÷

÷

÷

÷

÷

÷

÷

÷

÷

1o)

EQUILIBRIUM DISLOCATION DISTRIBUTIONS

÷

÷

635

÷

÷

÷

4,

÷

4.

÷

÷

÷

÷

4"

÷

÷

°

,$.

÷

(c) (d)

÷

÷

Fig. 7. Equilibrium configurations for the situation in which the slip planes contain dislocations of alternating signs. (a) Positive and negative walls form, separated by a horizontal distance of L/2. (b) The middle 10 slip planes are set taken be dislocation free. Dislocation dipoles interrupt the wall structures. (c) and (d) Equilibrium configurations for the same populations as those in (a) and (b), respectively. The starting positions of the dislocations were different from those leading to (a) and (b), and thus led to these

alternate equilibrium patterns.

with the Burgers vector for Cu, b = 2.5 x 10 -l° m and L = 4000b = 10-6m. We also considered a square cell with Lib = 2000, populated by 62 dislocations on 15 equally spaced slip planes, which give rise to essentially the same dislocation density. Two variable thickness opposite-signed dislocation walls form, with the mean wall separation again equal to half the cell size.

Contour plots of normalized shear stress are shown in Fig. 9, which are calculated taking Poisson's ratio, v, to be 0.324. Figure 9(a) pertains to the case of two opposite-signed walls, with one dislocation lying on each of the ten equally spaced slip planes within the cell. The stress level between the opposite signed walls in Fig. 9(b), which gives shear stress contours for the configuration shown in Fig. 8(b), is about an order of magnitude greater than that in Fig. 9(a). With

L = 10-6m, the dislocation density in Fig. 9(a) is 1 x 109/cm 2, compared with 2.5 x 101°/cm 2 in Fig. 9(b). In Fig. 9(b), shear stresses of the order of 1% of the elastic shear modulus occur within the cell at points away from any dislocation core. These high stresses may limit the dislocation densities attainable in such wall structures. Near each dislocation, of course, the divergent behavior of the elastic self-stress dominates and the stress increases without bound.

4.2. Rearrangement of dislocation distributions due to a superposed shear stress

In this section we consider the response of dislocation configurations in an infinite solid to a monotonically increasing shear stress. The only non- vanishing component of the stress field ~ in (18) is #12,

AM 4L /2 - -U

636 LUBARDA e t al.: EQUILIBRIUM DISLOCATION DISTRIBUTIONS

+ + ÷ ÷ ÷ ÷ ÷ ÷

4- 4. ÷ 4. 4.

-. - ..

÷ ÷ ÷ ÷ ÷ ÷ .¢.

÷ ÷ 4. ÷ ÷

• ~- ÷ 4. ÷ ÷÷ 4. 4.(. ÷ ÷ . ÷

if ÷ ÷ ÷ ,¢.÷" ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷

• I. ÷ ÷÷ ÷ ÷ ÷

÷ ÷ 44. ÷

÷ ÷ ÷ ÷÷ 4- ÷ ,¢. ÷ ÷

÷÷ ÷ ÷

÷ ÷

÷ ÷ ÷

÷ 4-

- - ° °

.l. .l. $ ÷ 4- ÷ ÷

÷ ÷ 4. ÷ ÷

4. ÷ ÷.l-

÷ ÷ ÷ ÷

.. ...

÷ ÷ ÷ ÷ ÷ ÷

. .

(o1

• ° . . . . . .

÷.H-H-÷

+mH,.¢-÷

4:::::::::: I-

+ : : : : : : : I- +.H-I.-~÷ .~ .= .= : .~ : ÷

"t-H~+

4 :::::::: : .t.

÷ ~ ÷

(b) Fig. 8. (a) A random initial distribution of 250 dislocations on 30 equally spaced slip planes. Co) The stable equilibrium configuration obtained starting from (a). + denotes a positive dislocation while - denotes a negative dislocation.

which is taken to be spatially uniform. A monotonically increasing sequence of values of dJ2 is prescribed.

For parallel walls, there is no resistance to simul- taneous and equal translation of the walls, so that any small imposed shear stress would cause their parallel motion. To prevent such motion, the dislocations at the cell ends are pinned. Figure 10 shows the shear response for the 250 dislocations, whose initial equilibrium configuration is shown in Fig. 8(b). With all dislocations at both ends of the positive and negative walls pinned and with the remaining 212 dislocations in the cell free, virtually no change in the equlibrium positions of the free dislocations occurs until an applied shear stress of about 0.5 x 10-3#/(1 - v ) is reached. Then, the positive dislocations move to the

(a)

-3. " 3. ~ ~ 3.

3. : . ~ : : ; :

) (b)

Fig. 9. Contours of constant shear stress in stable equilibrium configurations with sharply defined dislocation walls. The shear stress values shown are normalized by 10-3/~, where # is the elastic shear modululs. + denotes a positive dislocation while - denotes a negative dislocation. (a) Ten slip planes per cell with positive and negative dislocations on alternating slip planes. Co) For the equilibrium configuration of 250 dislocations shown in

Fig. 8(b).

fight, the negative dislocations move to the left and the equilibrium pattern shown in Fig. 10(a) is obtained at a shear stress of 2 x 10-3/~/(1 - v). The limiting shear stress is about 5 x 10-3/~(1 - v ) , and the dislocations break away as shown in Fig. 10(b). In Fig. 10(b), due to periodicity, the dislocations leaving the right hand boundary of the cell can be


+,,.+-~-+÷

÷ * ~ ÷ ÷ ÷

÷++~.++ +

÷ ' H -+ +

÷ : : : : : : :

+ ÷ ÷ + + +

+,l-~m¢+ +

,H-M-÷

: : : : : : : : : : ÷

÷ ÷ ÷ ÷ 4 . ÷ ÷ ÷ ÷ ÷ ÷

~ ÷ ÷

(o)

÷ + ÷ ÷ +

+-H-..-+ +

t ÷ 4 + +

+.1-1-~++÷ + ÷ ÷ ÷ ÷ ÷ ÷

+4-~- I -+÷

+ 4 - ~ - +

t : : : : ; : : ; ;

~l- ÷÷.~-÷ ÷ ÷

÷ ÷ ÷ ÷

÷'~-(~4-¢'÷ +

(b) Fig. 10. The effect of a superposed shear stress on the configuration in Fig. 8(b). The end dislocations of both walls are pinned. + denotes a positive dislocation while - denotes a negative dislocation. (a) The equilibrium configuration with a superposed shear stress of 2 x 10-3#/(1 - v ) . (b) Dislocation "break-away" from the

wall equilibrium structure.

seen entering the cell on the left. A similar calculation was carried out for a configuration having a dislocation density a factor of 20 smaller than in Fig. 10. In that case, the limiting shear stress was reduced by about an order of magnitude. However, the limiting shear stress depends on the pinned dislocations as well as on the number of dislocations in the walls. With fewer pinned dislocations the resistance to dislocation mot ion decreases and so does the limiting shear stress.

The incremental calculations here fail to find an equilibrium configuration when the superposed shear stress exceeds a certain value. One possibility

is that equilibrium solutions do not exist when the shear stress exceeds a certain value. How- ever, another possibility is that equilibrium configurations still exist above that value, but that these configurations are not near enough to the initial configuration for the numerical scheme to find them.

Figure 11 shows a case where there are 128 dislocations populating 2 families of parallel slip planes which intersect at an angle of 60 ° . Each family consists of eight slip planes. The dislocations are all of the same sign and the symbols o and + are used to distinguish dislocations associated with the two sets of slip planes. Obstacles that prevent the dislocations from leaving the cell are placed at the edges of a square L x L region. Figure 1 l(a) shows one of the possible equilibrium configurations in an infinite medium. Figures 11 (b-d) portray the evolution of the dislocation configuration with an imposed shear stress. The imposed shear stress is 35k/bL, 70k/bL and 140k/bL in Fig. 1 l(b), (c) and (d), respectively. Here, all dislocations are pressed downward, and the lower obstacle forces are greatly increased. For example, the force on the bottom left corner obstacle increases from 445k/L in Fig. l l (a ) to 1568k/L in Fig. 1 l(d).

4.3. Dislocation distributions in a finite body

In this section we consider equilibrium arrangements of edge dislocations in a finite, square region with traction-free boundaries. The iterative procedure described in Section 2 is used to find equilibrium distributions. At each iteration, the auxiliary stress field d corresponding to boundary tractions l" = - "r, is determined by finite element calculations. Rectangular four noded elements are used to compute the stresses at the dislocation positions. In most calculations a finite element mesh with 40 x 40 elements is used. Calculations were carried out which showed that the resolution provided by this mesh was in good agreement with that obtained using a 50 × 50 mesh.

Figure 12 illustrates the effect of free surfaces on the dislocation pattern for ten parallel pileups of spacing L/IO, so that the pileups occupy an L x L region. The pileups are placed in a 2L x 2L region having traction free boundaries. In Fig. 12(a) the array of pileups is symmetrically positioned, while in Fig. 12(b) and (c) it is shifted to within L/5 of the upper and left boundaries, respectively. Finally, in Fig. 12(d) the pileup array is a distance L/5 from both the upper and left boundaries. The dislocation pattern for the centrally positioned case, Fig. 12(a), differs from the pattern in an infinite solid by having a smaller dislocation free zone in the center. The equilibrium distribution, not shown here, for the same array of pileups in a 3L x 3L region has a larger dislocation free zone than in Fig. 12(a), but still differs noticeably from the infinite medium distribution.


%" %, ~* %/ ~ /

~o *0 "0 • +.o o f 4 0 4O +

#0 0#

• +* 5b

/ % .- %*

,., X ;', °-

%* ~ * oo~ %** oo÷.~ • .% ÷o .÷

o ~ o .~ • '~ o Oo ~

I o o**

5J

o÷* °o °o ",+

(a) {'b)

• o + o + o +

S°o~ o.S

/

/ o

° / ° °o. % / I~, /% /% t% ,,%

o

% o+

/

% o

/ AAA;XX

(c) (d)

Fig. I I. (a) An externally unstressed equilibrium dislocation distribution in an infinite medium. The distribution is subjected to a shear stress of: (b) 35k/bL; (c) 70k/bL and (d) 140k/bL. The dislocations

are pushed down toward the lower obstacles.

We note that there are dislocation populations that cannot be arranged in an equilibrium configuration in an infinite medium, but that do have equilibrium configurations in a finite body. For example, Fig. 13(a) shows an equilibrium arrangement of 68 dislocations on 15 slip planes of vertical spacing h in a square region of dimensions 31h x 31h. The signs of the dislocations on any slip plane are the same, but the sign alternates from one slip to another. There are 5 positive and 4 negative dislocations on the alternate slip planes, and positive and negative dislocation walls form. The inner walls of this distribution resemble a Taylor-type lattice structure [4], although there is some deviation at the outer walls due to the effects of the nearby free boundaries. For the same dislocation population, however, an equilibrium arrangement in an infinite medium was not found. The calculations here are carded out assuming a vanishing lattice friction stress. If a non-vanishing lattice

friction stress were to be accounted for, the dislocation distribution would be in equilibrium when the force on each dislocation was less than the lattice friction stress. Hence, it may be that certain dislocation structures require a non-vanishing lattice friction stress to exist away from free boundaries or other sources of inhomogeneous stress fields. Figure 13(b) shows contour plots of shear stress a~2 for the dislocation distribution of Fig. 13(a). In Fig. 13(b) a~: is normalized by k/bL. With L/b =4000 and using v = 0.324, k/bL ~ 0.059 x 10-3/J.

Shear stress distributions for ten pileups of ten positive dislocations are shown in Fig. 14(a, b). The dislocation configuration in Fig. 14(a) is the one shown in Fig. 12(a), while that in Fig. 14(b) is for the same array of pileups in a 3L x 3L region. The magnitude of the shear stress (away from the dislocation cores) is much larger than in Fig. 13(b). Figure 14(c, d) show the two components of the shear stress


(a) (b)

e • • • • • • • • •

4 1 + • • • • • • e

. m ~ . • • 4. • •

e . ~ • • • • t

. m ~ • • • • e

¢ M . ~ • • • • • •

~ • 4 • • •

O~0.o • • • • ~

a • ~ o • • • • •

Co) (d) Fig. 12. Variation of dislocation distribution with pileup location in a 2L x 2L region with traction free boundaries. (a) Centrally positioned. (b) Positioned L/5 from the upper boundary. (c) Positioned L/5

from the left boundary. (c) Positioned L/5 from both the left and upper boundaries.

• ¢s

1 0 ~ .0

- I 0

Cb) Fig. 13. (a) Equilibrium arrangement of 68 dislocations within 15 slip planes symmetrically positioned in a square region of dimensions 2L x 2L. There are 5 positive and 4 negative dislocations on alternating slip planes, whose vertical spacing is 2L/31. The dislocations form wall structures, which resemble a Taylor lattice. (b) Contour plots of shear stress, a~2, normalized by k/bL, for the dislocation distribution in (a).


J

)

(c)

f

(a) (b)

5 ¸ ' | "

J

(d) Fig. 14. Contour plots of shear stress, normalized by k/bL. (a) The total shear stress, trj2, for ten pileups of length L and vertical spacing L/IO, with ten positive dislocations per pileup, in a 2L x 2L region with

• traction free boundaries. (b) The total shear stress, try2, for the same array of pileups as in (a), but in a 3L x 3L region. (c) The shear stress distribution, a~2, due to the dislocations alone for the dislocation arrangement in Fig. 13(a). (d) The shear stress distribution, #~2, set up by the boundary tractions and determined by the finite element calculations. The sum of (c) and (d) gives the total shear stress distribution

in Fig. 13(b).

distribution, #12 from the dislocations themselves, Fig. 14(c), and 612, from the traction free boundaries, Fig. 14(d), making up the total shear stress plotted in Fig. 13(b). The boundary induced shear stress distribution, Fig. 14(d), exhibits the smoothly varying nature of the fields associated with the (^) field quantities which permits these fields to be well- resolved by the finite element discretization.

The behavior of 250 dislocations randomly placed on 30 parallel slip planes bounded by obstacles is shown in Fig. 15. The obstacle horizontal spacing is L and the vertical slip plane spacing is L/25. Case (a) shows the equilibrium arrangement attained in an

infinite medium, while case (b) corresponds to square region of dimensions 2L x 2L• There are approxi- mately the same number of positive as negative dislocations in this population and the distributions differ only slightly• In Fig. 15(c), the slip planes have been placed closer to the upper boundary, at distance 2L/25 from it. The difference from the infinite medium solution is most pronounced at the upper- most slip planes•

Figure 16(a) shows one of the possible equilibrium configurations in an infinite medium for 128 dislocations on 2 families of parallel slip planes which intersect at an angle of 60 ° and are confined to an

- o o - - o - " ~ ' ~ - - -

. . . . . . . . . .

. . . . : . . . . .

. . . . . : : . . : **** ° ° : . . . . o° .

: ~ : • • •

( a )

. . . - , - ° - - , - ; . -

• . o * , " : ° , , : :

: ' . . . ° . ° . . . . ."

. : . : . : - . . : '. .:

( b )

- o o - - - • o - - - -

• • ~ " ~ " ~ " "~ "~ ~ " -

• . ; " , ' , " " , " , " . -

, • • ~ 4 ' 4. 4 ' 4 ' 4" " 4 ' '~ • . ° ° . . ° . ° •

. - . - . : : . . :

"..';. i '. ".," 0° . ° . ° . . . . .

qb* q J ° q l # • ~' q%#

~o .~ • • 44$ 40 'O 4,

q b ° ° •

# o • #

q b . 4 , ~ O. ° ~. ed.

- . . :

(a)

~o * • ° °o*° 5** eo.~' 1 0 , 0 , " 0 . o , o • 5 /

g o 0 4 ,4 " 4 ,

o

I o * o l

qb~ ° o o o o

o,o, •

I % * * *%. . o %** ° . . l q , °*,o .°00 . . .5,,, , ~

( b )

%* %¢ ..,¢ ,I,.* ,t,.s t ' o . o *° °° • **%** ~jj 4.4, • 4 . • 04J ,l~ • • 4'4' 4. qJ

# 0 . 0 . ° o # . *o , ."~

~ 4 ' 4' 04' 0 4 ,

t , , ,4,o.. , , . . . , , . . ,

#*5 **5 **•o * 5 °oQ

(c)

Fig. 15. Pileup configurations o f 250 dislocations with variable sign, arbitrarily distributed on 30 parallel slip planes of length L and with vertical spacing L/25: (a) in an infinite medium; (b) in the center of a finite region of dimensions 2L x 2L with traction free boundaries; (c) at a distance 2L/25 from the upper boundary of a 2L x 2L

region with traction fre¢ boundaries.

(c)

Fig. 16. Equilibrium configuration o f 128 dislocation on 8 pairs o f slip planes intersecting at 60 ° with obstacles confining the dislocations to an L x L square region: (a) in an infinite medium; (b) symmetrically located in a square region with side 2L; (c) closer to the upper- most traction-free boundary in a square region with

side 2L.

6 4 1


L x L region by obstacles. Another infinite medium equilibrium configuration of the same dislocation array is shown in Fig. 1 l(a). Figure 16(b, c) show equilibria in a finite region of dimensions 2L x 2L. The obstacles are centered in the region in Fig. 16(b), and are closer to the upper boundary in Fig. 16(c). The dislocation free zone seen in the infinite medium case is reduced in Fig. 16(b, c).

5. CONCLUDING REMARKS

We have analyzed the equilibrium arrangement of collections of dislocations in highly idealized circumstances---edge dislocations under plane strain conditions. Stable equilibrium states are determined as stationary values of the potential energy with respect to variations in dislocation position along specified slip planes, with the number and size of the dislocations fixed. Within this context, full account is taken of the long range interaction between dislocations. The variational procedure involves the interaction energy of the dislocations, but not the self-energy. Hence, determination of the equilibrium configurations does not require any assumptions concerning a core radius. Characteristic features of the resulting equilibrium patterns include a strong tendency for the formation of highly organized patterns separated by essentially dislocation free regions and the occurrence of several distinct equilibrium distributions with similar energies. These features are expected to hold in more general circumstances.

When distributions including both positive and negative dislocations are considered, equilibrium states involving both walls of like-signed dislocations and walls of dipoles are found. The dipole type structures appear to be relatively robust with regard to perturbations, such as vacant slip planes [Fig. 7(d)] and nearby free surfaces (Fig. 15). Patterns of like- signed dislocations are more sensitive to such perturbations, as seen in Figs 6, 12 and 16. Even though the walled structure in Fig. 7(a) has a lower energy than the dipole structure in Fig. 7(c), the walled structure is more affected by vacant slip planes [compare Figs 7(b) and (d)].

The quasi-static results obtained here have possible implications for the dynamic simulation of dislocation distributions. The dynamical simulations of Amodeo [1] led to the development of sharper dislocation walls than those of Gulluoglu et aL [8]. The multiplicity of "nearby" equilibrium configurations found here suggests that the relaxed configurations

obtained in such dynamical simulations may be sensitive to details of the relaxation procedure.

The focus in this work has been on fully accounting for the elastic interaction between a large number of dislocations (i.e. a larger number than can be considered analytically) and between dislocations and free surfaces. The present formulation, can, however, be generalized to incorporate lattice friction stress, line tension effects, and dislocation creation and annihilation, along the lines described in Kubin et al. [9], as well as more general geometries of finite bodies. Equilibrium dislocation configurations of the type determined here can then serve as initial configurations for the analysis of dislocation motion. By coupling consideration of discrete dislocations with the finite element (or boundary element) method for enforcing boundary conditions, initial/boundary value problem for dislocated solids can be solved.

Acknowledgements--V.A.L. expresses gratitude to the Fulbright Commission for the fellowship support provided to carry out this research. J.A.B. acknowledges the support of the Brown University Materials Research Group on the Micromechanics of Failure-Resistant Materials, funded by the National Science Foundation. A.N. is grateful for support provided by an Alcoa research grant to Brown University. The computations reported on here were carried out on a Stardent Computer GS2000 workstation.

REFERENCES

1. R. J. Amodeo, Res. Mech. 30, 5 (1990). 2. J. D. Eshelby, F. C. Frank and F. R. N. Nabarro, Phil.

Mag. 42, 351 (1951). 3. A. K. Head, Phil. Mag. 4, 295 (1959). 4. D. Kuhlmann-Wilsdorf and H. D. Nine, d. appl. Phys.

38, 1683 (1967). 5. D. L. Holt, d. appl. Phys. 41, 3197 (1970). 6. D. Walgraef and E. C. Aifantis, J. appl. Phys. 58, 688

(1985). 7. P. D. Neumann, Acta metall. 19, 1233 (1971). 8. A. N. Gulluoglu, D. J. Srolovitz, R. LeSar and P. S.

Lomdahl, Scripta metall. 23, 1347 (1989). 9. L. P. Kubin, G. Canova, M. Condat, B. Devincre, V.

Pontkis and Y. Brechet, Non-Linear Phenomena in Materials Science H (edited by G. Martin and L. P. Kubin), in press.

10. F. R. N. Nabarro, Theory of Crystal Dislocations. Oxford Univ. Press, Oxford (1967).

11. J. P. Hirth and J. Lothe, Theory of Dislocations. McGraw-Hill, New York (1968).

12. J. D. Eshelby, Mechanics of Solids, The Rodney Hill 60th Anniversary Volume (edited by H. G. Hopkins and M. J. Sweell), p. 185. Pergamon Press, Oxford (1982).

13. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes--The Art of Scien- tific Computing. Cambridge Univ. Press, Cambridge (1987).

an analysis of equilibrium dislocation...

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