Dislocations Emerging at Planar Boundaries

d«AF(
π

to eq. (37) gives, for ΰ(φ), ο(φ)=--Β-ΗΦ)[Βτ(φ)Σ5(φ) π \

άφΣΑ{φ) +π ο sin^ - φ)

(39)

2.1.2. The displacement boundary-value problem According to eq. (35) the tangential components of the displacement gradient have the form (ίχν)®α(ρ)

= ^ (

Μ φ

® ^ ^ - η

φ

® Α \

A.Yu. Belov

402

and must coincide with eq. (31). Hence, δ(φ) ζηά/(φ)

Γ

c

diA^®[S((AW>A)-Q(^)/(>A)] sin(0 — φ)

are obtainable from

+ πηΐφ ® ο(φ)

dB(o)

= -ηφ®Α+ηιφ®-^.

(40)

The dyadic equation, eq. (40), can be considered as a vector equation,

i.

άφοοϊ(φ - φ)(3(φ)Β{φ)

- α(φ)ί(φ))

0 together with a supplementary condition,

dB(d>) + πδ{φ) = - ^ , °Φ

\άφ($(φ)Ηφ) - &{φ)/(φ)) = A. Jo Applying the Gilbert transform to eq. (41), in the form Ρ(Φ) =

π

άφοοΧ(φ-φ)/(φ\ o

/(φ)=--2 π

(41)

(42)

άφοοΙ(φ-ψ)Ρ(φ), 0

we find that the only solution of eqs. (41)-(42) is

ΛΛ-ΐ6-.,«[.,«(5«!)* 1

+ -

diA cot((/> - ^)

— ^

- A .

(44)

π Thus, an arbitrary elastic field in a semi-infinite solid without sources of internal strain or body forces, and which decreases inversely proportional with the distance from some point in the surface, can be represented as the elastic field of a fan-shaped distribution of infinite straight dislocations and line forces crossing at that point. 2.2. Dislocations inclined to a free surface; elastic anisotropy Now consider a straight dislocation with Burgers vector b and with sense t' meeting the surface of an anisotropic half-space x 3 > 0 at an arbitrary angle Θ. The coordinate systems xt and x\ (i = 1, 2, 3) with unit basis vectors (m, n, t) and (m\ n\ t') are chosen so that x2 = x'2 and so that the dislocation coincides with the positive x'3-axis, see fig. 3. In view of eq. (28), the free surface condition, eq. (1), takes the form of eq. (34), σ ί 3 = -tf? 3 | r s = o,

( 45 )

Dislocations emerging at planar boundaries

403

Fig. 3. Coordinate systems x, = (x, y, z) and x\ for the description of a dislocation inclined to a free surface. Three mutually orthogonal orientations of the Burgers vector bx \\m\ b2 \\n and b3 \\ t' are shown.

with the stress σ^3|ζ = 0 = ΣΪ{Φ)ΙΡ having an anti-symmetrical orientational fac­ tor Σι(φ). Hence, one can make the x1 x 2 -plane free of tractions by introducing in this plane a pure dislocation fan centered on an infinite straight dislocation as shown in fig. 4. The distribution of the Burgers vector ο(φ) in the fan is directly obtainable from eq. (39). However, calculations may be significantly simplified if one makes use of the integral representation for
= - - Γ άφ 2

2π }0

C*k,(n
(46)

where integration is performed over the azimuth φ in the χΊχ 2 -plane. The basis (ηιφ, ηφ, t') is obtained from (m'9 n\t') by rotation about t' by an angle φ, and m((p) = τηφ,η(φ) = ηφ. The so-called planar elastic stiffness tensor for the /in­ direction is given by C*ki(n
Cij^nnWA-^nJnpWCspM,

where Λ^η {ηφ) denotes the tensor inverse of the Christoffel tensor = (n
Λ^Ηφ)

Formula (46) synthesizes the two-dimensional stress σ^(χΊ,χ 2 ) fr°m o n e _ dimensional fields which depend on the coordinate r · ηφ along ηφ. However, we would like to have this stress in the form of a superposition of one-dimensional functions of r*n^. When the point r lies in the x ^ - p l a n e , this problem can be solved in the following manner. Relate the variables φ and φ so that ηφ will be in the plane formed by ηΦ and t. In this case, ηφ must be perpendicular to m^ and,

A.Yu. Belov

404

Fig. 4. Replacing a dislocation emerging at a free surface by a dislocation in an infinite medium and a dislocation fan.

Fig. 5. (a) The relation between polar coordinates (ρ', φ) in the Xix'2-plane and coordinates (p, φ) in the x1x2-plane. (b) The choice of m^ in the plane of (m^, t'\ ηψ points out of the paper.

hence, the latter should be chosen to be in the plane formed by the unit vectors τηφ and t' as shown in fig. 5. Now, from fig. 5, it follows that m,n = ηω

=

1 sin a 1 sin a

(ηΐψ — cos af'), (47) (cos Θ sin ψ t + sin Θ n^),

wherefrom we derive the coordinates r · ηφ and r · ηψ to be related by sinö sin α

ψ

(48)

Dislocations emerging at planar boundaries

405

Further, sin0 , cos φ = - — cos ψ, sin a

. sin i/^ sin φ = ——, sin a

/A^ (49)

and, finally, the required interrelation between φ and φ is given by cot φ = sin 0 cot φ. Note also that the points (p, φ) in the plane ΧχΧ2 and (ρ', φ) in the plane which are related by p' = p sin a,

x\x2,

cot φ = cot φ sin 0,

are equally far from the dislocation line and lie in the same plane containing the dislocation, see fig. 5. Then, changing the integration variable φ to φ with the help of άφ

= $^-άψ, sura

(50)

it is possible in eq. (46) to pass over to integration over φ in the plane ΧχΧ2 and, thus, to present σ ·)(/>) in the form of a superposition of one-dimensional functions of the coordinate ρ'Πφ. Substituting now the fan stresses, eq. (32), and those of the straight disloca­ tion, eq. (46), into eq. (45), we obtain the integral equation, 1

%π

ΛΙ

o

Β(ψ)Β(ψ) ρηψ

1 f* άφ ifm^Xb 2nzJ0smoc

ρηψ

where [(ab)*]jk = aiCfjkl(n)bi. Comparison of the integrands on the left- and right-hand sides of eq. (51) yields (Lothe et al. 1982)

'Μ-ιί^ί*^'·

(52)

The formulae of sections 1.1 and 2.1 together with eq. (52) form the necessary background for numerical investigations of the effect of anisotropy on the dislocation field near a free surface. However, in the general case, such calcu­ lations require considerable efforts. So, in order to illustrate the effects due to anisotropy, we limit ourselves to some special examples which are tractable analytically. In these examples some components of the elastic field are inti­ mately connected with the analytically obtainable symmetrical part of the surface Green function of the anisotropic half-space with the surface parallel to a mirror plane, see Appendix I.

A.Yu. Belov

406

2.3. Electron microscopic images of end-on edge dislocations; cubic crystals 2.3.1. Surface relief due to an edge dislocation in the case of normal incidence In the presence of some elements of crystal symmetry, surface distortions uZt x and uZt y are obtainable in an analytical form. Let the surface coincide with a mirror plane and let the dislocation of sense t run normally to it. From eq. (31), it follows that _8_ dp

__8_ dp

1

di//nll,®S(il/)b(il/) sin((/> — φ) 2πρ Jo

ηφ ® Ηφ)

(53)

In the case of normal incidence,

Α0Α) = - Β - ^ ) ( ^ ) * , Α ,

(54)

π

where the Burgers vector of the edge dislocation lies along m and B ~* (ψ) is obtainable from eq. (A25). When ΧχΧ2 is a mirror plane, the matrices of elastic constants relative to the coordinate system xf have the form

0

mn

_

W 3

0 <-15

_ 0

0

W 2

W 3

0

^22

<-23

0

^25

0

<-23

<-33

0

^35

0

0

0

^44

0

^46

<-25

<-35

0

<-55

0

0

0

^46

0

Ί

r° J (55)

rco ^11 CO »3 i2 0 CO __ mn &mn —

ύ

13

CO ^12 cO ^22 CO ^23

CO ^13

0

CO ^23

0

cO ^15 cO ^25 ύCO 35

0

CO

sh

0

0

0

0

0

cO ^44

CO ^15

cO ^25

Sis

0

s°55

0

0

cO ^46

0

- 0

"1

0 0 ^46

1

0 cO 66

ύ

_| —■

Let us now apply eq. (55) to evaluate b{\j/). On account of the property of the planar elastic stiffness tensor that C*(n)n = 0, the vector (tm^^m reduces to (tm^^m = cos^(rm^)*^/ffy. This vector has to change its direction under reflection in the mirror plane XiX2· Hence, it lies along f, namely (ίτηψ)*ψηί = at,

Dislocations emerging at planar boundaries

407

where coefficient a is obtainable from the relations

F

(ηφΠφ) =

0

r°

Ί

W5

0

0 /^0 /^O

"^66^15

66^55

o

c?1c?5-c?I

o

""^66^15

0

^66^11

(fluffy) * =

(ίηΐψ)ιηψ = C 2 3^,

1

(nlj,mli,)m}j, = C 13 /ty + C35/ffy.

(56)

Finally, we find that

*(
frxCOS^ πΒ§ 2 rO

' - ' 1 2 1 ^ S S W 3 ~ W 5 W S . ) ~ ^ 2 5 ^ 11 ^ 35 ~

L

W5W3J

citCSs-ic?,)«

_T (57)

Since 6(1^) is along f, the integrand on the right-hand side of eq. (53), and, thus, the surface distortions uZiX and uZtV due to the edge dislocation become sin φ uz, x( ρ9 φ) = - 2ρ ^ - bz(4)9

cos φ uZt y( ρ,φ) = - -^-2ρ M
(58)

where we have taken into account that owing to the presence of the mirror plane u\ = 0. Now, remembering that, for every direction πιφ in the mirror plane, pa are expressed by radicals, we can prove that the surface distortions, given in eq. (58), can be evaluated without numerical calculations. This way purely analytical solutions of three-dimensional dislocation boundary problems have been found by Belov (1987a). 2.3.2. Electron micrographs In absorbing crystals under conditions of anomalous transmission, the electron micrograph of a dislocation is a direct image of its surface distortions (Indenbom and Chamrov 1980a) d(g'u)/gdz, where g is the diffraction vector and z is the beam direction. According to Indenbom and Chamrov (1980a), if the specimen is in the Bragg orientation, the contrast <50 of bright-field micrographs

A.Yu.

408

electron

i

I

4

Φ

beom

ψ

m

///

Belov

+

V

+

*

777^"

[ΙΪ0]

tin]

t +

Dislocat I o n

f.c.c.

[no]

[110]

1

asj^

/ +

V -r Cu

X.

\ +

/+ '*_

C-f-\

/^Π

Au .

(+/ AV

Fig. 6a, b.

'i\

\+\

X

Dislocations emerging at planar boundaries

409

Fe

Ca

W

Fig. 6. (a) Electron microscope imaging of an edge dislocation normal to the surface. Surface coordinates (x, y, z) are along (m, #i, t). The y-axis points out of the paper. In fee crystals m || [1 T 0], /i|| [001], f|| [110], A = i[lT0], in bec crystals m || [Ϊ 11], n|| [1 T 2], f||[110], A = i [ T l l ] . Cal­ culated dark-field images of the edge dislocation in (b) fee Cu, Au, AI and (c) bec Fe, Cr, W crystals. Contours of equal contrast ög(x, y) = 0.25 for diffraction vectors g = [220] and g = [002] (fee), and g = [222] and g = [112] (bec) are shown. The half-extinction distance \ξβ is indicated.

is approximately given by a sum, and the contrast bq of dark-field micrographs by a difference between the slopes w = ^(d/dz)(^«w) of the reflecting planes at the entrance (x 3 = — /) and exit (x 3 = /) surfaces of the specimen, s0(g) = w(*> y> - 0 ± w(*> y> 0 ·

( 59 )

Equation (59) determines only the contrast observed at distances of the order of the extinction length ξ9 from the image of the dislocation line and, of course, is not applicable to the simulation of dislocation core micrographs. Let us now apply eq. (59) to end-on edge dislocations, normal to a free surface and a mirror plane. In this case, the shear stresses and the strains are related by — 2(C45syz + C55sxz),

oyz — 2(C4^eyz 4- C 4 5 e x z ),

(60)

where Cm„ are the elastic stiffness matrix elements referred to the coordinate

A.Yu. Belov

410

system xt ( x t x 2 is the mirror plane). Then, from the free surface boundary condition, eq. (1), it follows that the surface strains εχζ and syz are equal to zero. This allows us to relate the surface distortions determining the contrast and the above-introduced derivatives of the surface relief, eq. (58); WJC, z = — uz, x^uy,z—

—uz,y. Finally, we have

w(x,j>, ± 0 = - i , ( * P ) u , ( x , y , ± / ) .

(61)

In case of a mirror plane, the symmetry of displacements asserts that uz(x,yj) = — uz(x> y, —l) and, thus, according to eq. (59), the end-on edge dislocations may be observed only on the dark-field micrographs, the contrast then being determined by the following expression; δ0(ρ, φ) = ig{g%n^)bz(^)jp with frz(0) from eq. (57) (Belov and Chamrov 1987). Calculated isolines bQ = 25% for the dislocations in bcc (W, Cr, Fe) and fee (Al, Au, Cu) metals are reproduced in fig. 6b and c, respectively. In bcc metals, calculations were performed for the glide dislocation {lT2}^a[Tl 1] and in fee for the sessile dislocation {111}^α[1ΪΟ]. In both cases, the dislocation direction coincides with [110]. Figure 6 allows us to trace the influence of anisotropy on the surface distortions from practically isotropic W and Al to the quite anisotropic Fe, Cr, Au and Cu. 2.4. Surface induced interaction of end-on screw dislocations with point defects; cubic crystals Let us consider another analytically tractable effect, namely that of the inter­ action of point defects and screw dislocations normal to a free surface and a mirror plane of a cubic crystal. As a model of a point defect, we adopt a spherical dilatation center. In the case of cubic crystals, the elastic interaction energy of a dilatation center with the dislocation field sik(r) may be written as Eint=-K8Vskk(r),

(62)

where K = j ( c n + 2c 12 ) is the bulk modulus, cnm the elastic constants referred to the crystallographic axes and δ V the change in the crystal volume due to the point defect. Let a screw dislocation with sense t and Burgers vector b = bzt run normally to the surface of the half-space x 3 > 0. If the elastic anisotropy is neglected, the strains due to the dislocation are pure shear strains and, hence, skk = 0. In the presence of a mirror plane normal to the dislocation, the strains in an infinite solid also obey the condition skk = 0. Therefore, the interaction between such a dislocation and point defects results from the simultaneous action of the free surface and elastic anisotropy. As Eint = — K8Vskk is determined only by the relaxational part of the dilatation, which increases with 1/r, the interaction becomes significant only close to the exit point. Thus, in order to estimate £ i n t , it is sufficient to know the dilatation in the crystal surface.

Dislocations emerging at planar boundaries

411

Regarding the determination of the distribution of the Burgers vector in the dislocation fan, wefirstnotice that a reflection in the mirror plane xxx2 does not change the direction of the vector (ίηίψ)*φί. The most general form of such a vector is {tm^)*nJ = αηφ + dm^. On account of (ίηΐψ)ί = C 46 /fy + C 44 iw^,

(ηψΐΠψ)ΐ =

C46t,

we find that a = 0 and d = (C^C^6 - CU)/C%6 = 1/S24, and from what follows that

4W =


(63)

The surface dilatation is determined by the contraction of the distortions, given in eq. (30),

ε {ρ}

"

= - i JlriE^j(* SW,) + ' Σ ± *M.® ^ )*<*) -^ίιΐψ + ί £pa^®£« W ) ·

< 64 >

Applying the sum rules, given in eqs. (21) and (23), to eq. (64), one can obtain the surface dilatation in the form

-γίηφ-*(ίί)-ι(ίηφ)]Ηφ)

eu(p)=

--Γ

άψ {ηφέ(ψ) - ί(ίί) _1 (Β + (tfv)S)}AWr). sin( — ι/0

(65) Next, we again take advantage of the equivalence of a mirror plane and a two­ fold axis. Since the dislocation and its Burgers vector lie along the two-fold axis, the symmetry of the elastic problem assures us that the surface dilatation is symmetric: ekk(—p) = skk(p). However, according to eq. (29), ο(φ + π) = —ο(φ) and, hence, only the first term in the right-hand side of eq. (65) is symmetric. The second one, expressed by an integral, is anti-symmetric and, therefore, must vanish. Indeed, as follows from eq. (A24), the vector 5{φ)δ(φ) lies along t and its convolution with the vectors ηφ and t(tt)~Λ(ίπψ), lying in the mirror plane, is equal to zero. Similarly, one can establish that the convolution of ο(φ) with the vector ί(η)~1Β(φ) normal to the mirror plane vanishes. Now, taking into account the relations (ίηφ)ιηφ = Cist,

(ίηφ)ηφ = C ? 2 i ,

A.Yu. Belov

412

and 0

(tt) =

^46

0

c° *_46 -i

c§2 o

o cl.

^22^44

(«)

_1

=

0

0

C^^Cßß

^22^46

— C22^46

0

'46

^22(^44^66

υ

^22^66

J

—

WoJ

(66)

we find that titty1

(tn+)m+ = C°25/C°22,

t{tty

x

(tnt)ni = C°12/C°22,

and the surface dilatation becomes tkkip) = ~

ö~ll(-22 — ^12J^13

2npS 4o 4 ^r2 2

— ^25^33

)·

(67)

The elements of the matrix Β(φ) are, as in the previous section, expressed by the roots of the bi-cubic equation, eq. (13), and, thus, the surface dilatation in this problem is obtainable without numerical calculations (Belov 1987a). Figure 7 represents the contours of equal dilatation \skk\ = 10~ 3 calculated by eq. (67) for crystals with the rock salt structure: LiF, KBr, NaCl. Axes x, y and z are along [110], [001] and [T 10], respectively, the unit of p is 10b. We recall that the anisotropy of cubic crystals is determined by Zener's ratio A = lehnen — c12). Its values in the case of LiF, KBr, NaCl are 1.9, 0.35 and

.LIIOl

777"

Dislocation

tUO]

Fig. 7. Contours of equal dilatation |efcjfc| = 10"3 for a screw dislocation of Burgers vector b = ί [T 10] in (a) LiF, (b) KBr, and (c) NaCl. Surface coordinates (x, y, z) are shown. The y-axis points out of the paper; m || [110], n || [0 01], /1| [T 10].

Dislocations emerging at planar boundaries

413

0.70, respectively. In crystals of LiF and KBr, the anisotropy becomes appre­ ciable and the surface dilatation is of the same order as that caused by edge dislocations in isotropic materials with the Poisson ratio v = ^. 2.5. Dislocations meeting the surface normally; hexagonal crystals Let the free surface of a semi-infinite hexagonal crystal be parallel to the basal plane and consider a dislocation running normally to the surface. In this case, the dislocation is along six-fold axis and may belong to a (1 ΤθΟ)^α<000 1> or a (1 Τ00)α[0001] slip system. By the method of Lothe et al. (1982), one can obtain the displacements due to dislocations of this type not only at the surface but also below it. Let the x- and z-axes coincide with the crystallographic axes X and Z (fig. 8). Then, for every direction m^ in the basal plane, the elements of B(i^) are given by BOA)

(^13-^13)^44 T/2(|_(^13 + C13 + 2c 2c44 )c333 3 J 4 4)c Γ

X (ity ®91φ + y/c33/Cut

+ c

v

® t) + x /c 4 4 C 6 6 /W^ ®

tffy,

(68)

where c 1 3 = ( c n c 3 3 ) 1 / 2 and cmn refer to the crystallographic system. According to eqs. (57), (63) and (68), the distribution of Burgers vector in the fan takes the form: (a) in the case of an edge dislocation of Burgers vector b = bxm, ,/ ix _ bx Cj 3 (cn — c 12 ) π

Ci3 + ^ 1 3

C13 + c i3 + 2c4 | _ ( c 1 3 - c 1133 ) C 1 1 C 4 4 .

U/2

cost/ff;

(69)

(b) in the case of a screw dislocation of Burgers vector b = bzt, (70)

bM^ictt/cee)1'2^.

In what follows, we also need the displacements due to a fan dislocation of Burgers vector b(\j/) and sense ηιφ. The well-known result for the dislocation

*00

D'islo cation

Fig. 8. Coordinate systems (x, y, z) and (X, Y, Z) for the description of a dislocation in the basal hexagonal half-space; m||[l 120], n||[lT00], f||[0001].

A.Yu. Belov

414 lying in the basal plane is v bz(il/)[c13-c13 (r Wr(C, z) = A _

4TCC13|_ vx + v2

^ qt c 1 3 + c 1 3 ^ ζ2 + zj In . Ύ . — — - I n -~2 j2 sinV

2(vj - v2)

wz(C, z) = — — arctan -^ 4π L C - z ^

C + z

7—2 ^ l(vi-vi)

=

c13c44

ln

7~ ί

π

*„vs,*,ctan -- , u„(C,z) = - - ^- — ^ a rarctan (71) 2π V z3 where q2 = C2 + i£(*i - z 2 ) + ZiZ2, t2 = ζ2 - ΐζ(ζ! - z 2 ) + ζ ^ , ζ7· = zv,·, and where the parameters v,· are determined by =

f(Cl3 ~ Cl3)(Cl3 + Cl3 + 2 c 4 4 ) T / 2

L

4c 33 c 44

J

R e u + c 13 )(c 13 - c13 - 2c 44 )~j 1/2

L =

4c 33 c 44

J

|"(Cl3 ~ Cl3)(Cl3 + Cl3 + 2c 4 4 )~j 1 / 2

L

4c 33 c 44

J

_ Γ (Cl3 + Cl3)(Cl3-Cl3 ~ 2c 4 4 ) "j 1 / 2

L

V3 =

4c 33 c 44

J

(C66/C44)1/2·

The parameters v1 and v2 may be either real (e.g., in the case of Mg v1 = 1.41, v2 = 0.69) or complex conjugates (e.g., in the case of Zn v ls 2 = 107 ± i0.67). Integrating the two-dimensional field, given in eq. (71), over φ, we see that the relaxation displacements become (a) in the case of an edge dislocation of Burgers vector b = bxm, (i) in the absence of degeneracy (vx Φ v2), ur = Ux

bxcl3{cu -c12)xyT v_2 In 2 C l l c 4 4 (vi - v2) |_(1 + fci)(ri + z x ) 2

vi "1 (1 + k2)(r2 + z 2 ) 2 J '

bx c 1 3 ( c n - c 1 2 ) «1 = 2π2ο ο (ν - v ) 11 44 1 2

x

:a

+ *i)L (l + /c2)L

(r.+z,)2]

r,+Zl r 2 + z2

(r2 + z2)

2

JJ (73)

^ c 1 3 ( c n -c12)v1v2y / fcx w* = 2π c 1 1 c 4 4 ( v i - v 2 ) V(! + k1)(rl + Zl)

/c2 (1 + fc2)(r2 + z 2 ) / '

Dislocations emerging at planar boundaries

415

where kj = (cii/v? - c 4 4 )/(c 1 3 + c 44 ); (ii) in the presence of degeneracy [vx = v2 = v0 = (cii/c 3 3 ) 1 / 4 ] } fcxC13(cn r wL = 4π Cuic^

-c12) xy ( 3c 13 + c 1 3 z0 :— r 1 + c 13 ) r0(r0 + z 0 ) \ c 1 3 - c 1 3 r0 + z0

i>xC 13 (c n - c 1 2 ) f y2 ln(r + z ) + ■ 0 0 4 i r c n ( c 1 3 + c 13 ) t r 0 (r 0 + z 0 )

+

3c l3+c

" z; [ 1 - - ^ ] } , (74)

f

c

C i 3 - c 1 3 r 0 + z 0 |_ ^o(^o + ^o)JJ bx c 1 3 (cn - c 12 )v 0 y / 3 c 1 3 + c 1 3 r0 - z0 4π Cn(c 13 + Ci3) r 0 \ ^13 - C13 r0 + z0 (b) in the case of a screw dislocation of Burgers vector b = bzt, 2πν 3 r 3 + z 3

2πν 3 r 3 + z 3

where z, = ν,ζ (;' = 0, 1, 2, 3) and r] = p2 + z). For the sake of convenience, we also reproduce the displacements due to a dislocation in an infinite hexagonal crystal. When the dislocation is parallel to the hexad axis, its displacements u\ have the same form as in an isotropic material with a Poisson ratio of vh = c12/(cil + c 1 2 ); (a) in the case of an edge dislocation of Burgers vector b = bxm

«* = ¥η{φ +

4^)ύη2φ

«J = - ^ ( ( 1 - 2v h )lnp - sin 2 φ)/(l - v h ); (b) in the case of a screw dislocation of Burgers vector b = bzt,

■*-£♦· 2.6. Dislocations arbitrarily inclined to the surface; elastic isotropy Now consider a dislocation inclined to the surface of a semi-infinite elastically isotropic solid of shear modulus G and Poisson ratio v. The three independent orientations of its Burgers vector bx = b\m\ b2 = b'2n\ b3 = b'3t' we denote as bj (j = 1, 2, 3) and we adopt the convention not to use the summation rule over repeated indices j and m.

A.Yu. Belov

416

In the case of an elastically isotropic material, 2v

Cipki — G

l-2v

<>ipSki + δΛδρι + δαδρΙί

(76)

,

the matrix Β(φ) for a direction m^ in the fan is determined by B(«A) =

(77)

(I - ν / Μ ^ / Μ ψ ) .

1—v By substitution of eqs. (76) and (77) we see that the general anisotropic formula, eq. (52), for ο(φ) becomes (a) in the case of an edge dislocation of Burgers vector b± = b\m' (j = 1) or b2 = b'2n (j = 2), 2fy(sin0cost/O 2 " j sin J ' *M0 = π(1 + c 2 sin2 ι/Ο

ι

φ

c2 cos 2 φ v + -1 + c 2 sin 2 t/^ (t — c sin φ ηψ) — c cos φ m^

}

(78)

(b) in the case of a screw dislocation of Burgers vector b3 = b'3t'9 1 + c2 sin2 φ m^

71

(1 — v)ccos^ (t — csini^»ψ)\> 1 4- c2 sin2 φ

(79)

where c = cotö and Cj = 1 + c 2 . Next, we write the displacements of a fan dislocation of Burgers vector b{ij/) and of sense /ηφ in the form 1

«(U) = 2^

I-

+ 2(1

1

,

, 51

2(1 - v) (f (g) t — ηφ ® n^)z —

ozj ^[(l-2v)(„,ψ®ί-ί®ηφ)

arctan

-f z

5 + (ί®ηφ + ηψ® i)z-, -I^P|sin^| - AW, (80) 9zJ""

:J |sin ^ where # 2 = ζ2 + z2. Substitution of eq. (80) into eq. (15) together with eqs. (78) and (79), gives the integral representation of uT{r). By introducing the notations 1

C klm

—

άφ cos1 φύ^φ

,/,\* o (1 4- c 2 sin221/0

arctan

d φ cos* φ sinm φ R In: 2 2 o (1 + c sin 1/0* | sin ^ |

(81)

c

(82)

Dislocations emerging at planar boundaries

417

for the integrals, the relaxation displacements ur(r) may be written in the form (Belov and Chamrov 1987); (a) in the case of an edge dislocation of Burgers vector b± = b^m'ij = 1) or b2 = b'2n (j = 2), jci"2(1 - v) + z ^ l [(v - l)Cf3~q) + C!C5 (3 - e) ]

ui = -D){-\T

- Γ 1 - 2v + z £ | [ ( v - l)Lf2-q) + ClLf-q)^ + (-l)m2c(l - v ) ^ 3 " " » 8~ [(v dz

ι/3=£]{Γ2(1-ν)

+ c\ 1 - 2 v - z

(

m = l,2

(83)

- lJC?-·»«-1' + c C f * " " " ] (84)

dz

(b) in the case of a screw dislocation of Burgers vector b3 = b'3t', u'm = Ds(-l)m\c\

2 ( l - v ) + z— C f 3 - " " -

l - 2 v +, z; .

r m(2 — m)

+ (-l)m[2C(02-m)(m-1)-c1C(12-m)(m-1)]/cj) m = 1,2

ί)8 2(1 ν) ζ €ί0 + 1 2ν ζ

{[ - - έ]

1

( - - έ]^ }'

(85) (86)

where D] = &}(8ΐη0)2"72π(1 - v),Ds = fc'3cos0/27r, g = m + 1 -j,p = 5-j-m. Note that the functions Cl™ and L[m are harmonic. The integrals in eqs. (81) and (82) are obtainable by the residue theorem and have the simple form

clc

C\° =

-ωί9

C?1 = sin 2 0cu 2 ,

ω,

CP=^| 1

c \

1/

, fli ?

- ωχ + — 2\ 2 fl r+z sin 0 CY = ^ — sin 2 0co 2 +

q2

x

a r +z αγ

r+ z

2?

r+z

1 2c2

y

a r + z/ sin 2 0a> 2

a2 x a r+z

A.Yu. Betov

418

1 QAsinld) Li 1 = - ^-^arctan— arctan, ^ „ " ,- — ^ ,, c 1 — Qylcos20

. .x. L\ LJ 1 =

/Isin0sin2
(87)

2 o _ l - ß 'Γΐη(Γ | + ζ) + ^ 1 η β 1 ,

if° = Li° =

2

sinö

ln(r + z) + - l n a + ^

L?2=^-^sinÖ 2 02

,

ln(r + z)

sin 3 0

, ln(r + z) + :-Ina 2 a _\ where the following notations have been used ^2

—

Γ

1 cy ω χ = - arctan — , c a2(r + z) „

1-sinö 1 + sin 0

A

ω2 =

1 xcosö arctanh — , cosu a^yr + z)

r-z r+ z

1 + QA 1+ 6

I - QA 1— Q

with a 3 = (a - a1a2)/cos2e9 α = (1 + Q2A2 - 2QAcos2(l))/(l - Qf. The rest of the functions are linearly dependent on eq. (87): L°° = L° 2 + L20, Cl° = Cl° - Cl2, C£3 = C? 1 - Ci1. Equations (83)-(86) represent relaxation displacements in terms of coordinates x, y, z connected only with the surface, and they provide an useful alternative to the Yoffe-Shaibani-Hazzledine expres­ sions (Shaibani and Hazzledine 1981). For the sake of convenience, we reproduce here displacements uc caused by the straight dislocation in an infinite isotropic solid. It has the form (a) in the case of an edge dislocation of Burgers vector b = b\m' + b'2ri, Λ e

tJ L,

r

Sill2(

£'

«.

T

1

~2V i

,

COS 20' 1

(88)

(b) in the case of a screw dislocation of Burgers vector b3 = b'3t\ 2πιιβ = Α 0 \

(89)

2.7. Concluding remarks Finally, we will mention some three-dimensional elastic solutions for dis­ locations of other shapes near the free surface of a semi-infinite solid. Isotropic materials. Infinitesimal dislocation loops close to the surface of an isotropic half-space were first considered in a paper of Steketee (1958). Explicit

Dislocations emerging at planar boundaries

419

expressions for the displacements due to infinitesimal loops were derived by Bacon and Groves (Groves and Bacon 1970, Bacon and Groves 1970). They used a system of force dipoles as loop model which allowed them to employ the Mindlin-Cheng results (Mindlin 1936, Mindlin and Cheng 1950) for concen­ trated forces and their dipoles in a semi-infinite solid. The force on the infinitesi­ mal loop was also investigated by Tikhonov (1967) and Jager et al. (1975). The stresses and the displacements of loops of finite size (circular prismatic loops parallel to the surface) were derived by Bastecka (1964) and Ohr (1978), respectively. The elastic fields of polygonal dislocation loops near a free surface may be derived from the results of Maurissen and Capella (1974a, b) for the dislocation rays and from the results of Comninou and Dundurs (1975) or Kiselev and Shmatov (1979) for angular dislocations with their apex below the surface. In Maurissen and Capella (1974a, b), Li's expressions (Li 1964) for the stresses due to semi-infinite 'ray' in an infinite medium were expanded to rays parallel to the surface and perpendicular to it. On the basis of the results of Maurissen and Capella (1974a, b), one can construct the elastic fields due to an arbitrary polygonal loop parallel to the surface or due to a rectangular loop perpendicular to the surface. In the papers by Comninou and Dundurs (1975), Kiselev and Shmatov (1979), the displacements were derived for an angular dislocation which apex is in the bulk and which arms go into the bulk, one arm normal to the surface and the other one inclined to the surface. Combining the results of Maurissen and Capella (1974a, b) and those of Comninou and Dundurs (1975), Kiselev and Shmatov (1979), we can obtain the elastic field of an inclined ray 'terminating' in the bulk and of a polygonal loop possessing inclined segments. Anisotropie materials. Displacements due to an infinitesimal loop close to a free surface of an anisotropic half-space were derived in a paper of Belov and Kaganer (1987) in the form of an integral over two-dimensional solutions. Finite rectangular prismatic loops parallel to the surface and basal plane of a hexag­ onal crystal were considered by Berry ad Sales (1962), who used the stress function method.

3. Disclinations

in a semi-infinite

solid

While a dislocation in an elastic body results from cutting along a surface S and a rigid translation of one face of the cut relative to the other face by the Burgers vector A, a disclination results when the faces of the cut experience a rigid rotation by the Frank vector Ω. Hence, the difference between the displacements below and above S is [MiM] =

*iq*Oq(x*-xi),

(90)

A.Yu. Belov

420

where r is a point in S and r° is the position of the rotation axis. As a conse­ quence of eq. (90), the rotation vector also experiences a jump across S [ ω , ( Γ ) ] = Κ . [ « , ( ι · ) ] „ = Ω,.

(91)

According to Weingarten's theorem, the stresses and the elastic deformations due to a disclination satisfy the St Venant compatibility conditions everywhere in the body except for the disclination line. This property unites dislocations and disclinations in one class of Volterra dislocations. For a comprehensive dis­ cussion of various aspects of the disclination theory and in particular the elastic theory of disclinations in infinite anisotropic solids, we refer to the paper of de Wit (1973). Here, we shall be concerned only with straight wedge disclinations which Frank vector is along the disclination direction. Such disclinations produce the same displacements and stresses as a homogeneous plane distribu­ tion of parallel edge dislocations bounded from one side by the disclination line, see fig. 9. 3.1. Wedge disclinations inclined to a free surface; elastic anisotropy The problem of a straight disclination inclined to a free surface of a semi-infinite solid is another example where the relaxation field u[(r) may be synthesized from the two-dimensional solutions, given in eq. (14), in a way allowing simple physical interpretation. Consider an anisotropic half-space x3 > 0 and, arbitrarily inclined to its surface, a wedge disclination of sense t\ Frank vector Ω = Q'^t' and with rotation axis position r° = 0, fig. 10. The designations for the inclination angle and the coordinate systems and basis vectors connected with the surface and the disclination line are the same as we have adopted in the case of the inclined dislocation. We seek the stress field due to the disclination in the half-space in the form given by eq. (28) where σβ means the stresses of the straight disclination of sense t' in an infinite medium and which is determined by the integral

Fig. 9. An elastic wedge disclination and its model in the form of a distribution of parallel edge dislocations.

Dislocations emerging at planar boundaries

421

Fig. 10. Coordinate systems x, = (x, y, z) and xj for the description of a disclination inclined to a free surface. The y-axis points out of the paper.

expression of Alshits et al. (1975), _θ3_Γ* r-n(n (92) 2 When approaching infinity the stresses, given in eq. (92), asymptotically diverge because of the logarithmic singularity and, thus, one should bear in mind eq. (92) actually may be applied only to self-screened disclination systems such as dipoles, quadrupoles, etc. The stresses associated with the self-screened systems are expressed by logarithmic terms of dimensionless arguments. At the surface x 3 = 0, the divergent part of the tractions σ*3|ζ = 0 behaves as In p. Therefore, in order to cancel these tractions, following Belov (1987a) we introduce in a plane x 3 = 0 the distribution of semi-infinite dislocation walls consisting of parallel straight dislocations and which gives rise to tractions having this type of singularity. A wall of dislocations of sense m^ homogeneously distributed in the half-plane (x? > 0, x 3 ) is illustrated by fig. 11. Rotating this wall around the x3-axis and varying its Burgers vector, we obtain a distribution of walls which is characterized by the vector άΩ = Ω(φ)άφ. This is the total Burgers vector per unit length of the dislocation walls which boundaries lie within the angular interval άφ. The displacements due to a wall of sense m^ are given by eq. (27). Adding the linear displacement to eq. (27) we have

2π ν

~πι ^—f 2πί

e|sin^|

(93)

Next, integrating eq. (93) over φ, we find the relaxation displacements in the bulk x 3 > 0, (φ)(ζ +

wir) ^

a= 1

Ρχζ)Ιη

elsini/Ί'

(94)

A.Yu. Belov

Fig. 11. Homogeneous distribution of parallel dislocations with sense m^ in the (x? > 0, XaJ-plane. t points into the paper.

which give the following tractions at the surface x3 = 0; ta*(p) = -

1 f«.. ^ Γ # £ 2πί

.

/,

±La®Laß(
\ρ·ηψ + sin^

ίπθ(ρ-ηψ) (95)

Here, 0(ζ) is the step function 0 ( 0 = 1 , C>0, 0(0 = 0, C < 0 . Taking into account the closure relations, eq. (20), and the definition given in eq. (22) of the matrix B, we can transform eq. (95) into

1 Γ π ώ;/Β(.//)Ωθ//)Ιη Ρ'"ψ

w

" -s.

(96)

simj/

Now substituting eqs. (96) and (92) into the free surface condition, eq. (45), and passing from integration over φ in the χΊχ'2-plane to integration over φ in the plane x 3 = 0 by means of the relation given in eq. (50), we obtain the equation for fl(^) 2π Jo

άψΒ(ψ)Ω(ψ)\η

ρηΦ

ύηψ

t

2π2)0

. sin Θ .

P -"ψ

sin a

sint/^

for which the solution is (Belov 1987a) Ω3 sinö

Ä. Β

(97) ^)(tm9)*m9 π sin a If the vector Ω(φ) is along f, then, in accordance with eq. (27), the wall corresponds to a wedge disclination of sense m^ and Frank vector ηψ χ Ω(φ). Analysis of eq. (97) analogous to the analysis in sections 2.3 and 2.4 shows us Ω(ψ)··

Dislocations emerging at planar boundaries

423

that this case takes place when, e.g., the disclination is normal to the surface and there exists a mirror plane parallel to the surface. For such disclinations, the relaxation field may be represented as the elastic field of a fan-shaped distribu­ tion of wedge disclinations. 3.2. Wedge disclinations inclined to the surface; elastic isotropy Now, consider the wedge disclination inclined to the surface of an isotropic solid. In this case, the relaxation displacements are obtainable explicitly. In the isotropic limit, eq. (93) reduces to

«(U»4

ζΐ - ζ(ηψ t +

1 ® ηφ) arctan 1—v R l-2v ζ{ηφ®ίf (x)#fy)ln e|sin^| 2(1 - v)

+ζ|ϊ+

*®?-'®'ι1η 2(1 — v)

/

<

R lsint/Ί J

In

and the distribution of the Burgers vector in the system of dislocation walls becomes

Ω(φ) =

1 + c 2 sin 2 ^ 1 - v -

1 + c sin φ

]·

(t — c sin φ ηφ) + c cos φ m^

(99)

where Dw = Ω'3 sin θ/π and the other designations have been introduced in section 2.6. Integrating the displacements, given in eq. (98) over the parameter φ and using the notations given in eqs. (81) and (82) for the integrals, one can represent the relaxation displacements in the form (Belov 1987a) 2

i4 = D " ( - I f £ [<5fcl(v - 1) + ^ c j

x{{cr'

3 - -2v 2(1 - v )

-l)(2-m)

1 - 2v s=l

L

2(1 - v) v

]

L^2-p)-cCl°

2

+ CDA zL{?-m){m~l) + X -zDwcos0

c£(m-l)(3-m)

"]}

(-l)sxsC[3~q)q

. 1 vsin0 --—- Sml9 2 1 + sin Θ

m=l,2,

(100)

A.Yu. Belov

424 2

ur3 = - D

w

X [ M v - l ) + <5,2Cl]

[2(1-v)

+ zD

w

*

1-v

*

l_(l-2v)sin2 2sin0

(101)

where p = m + s — 2, q = m — s + 1, and where the functions Ll™ are related to the harmonic functions in eq. (82) as Ll™(r) = Lfcm(r/e). The displacements due to the wedge disclination of sense t' in an infinite solid are 2nue = [Ω x rW + * 2 \ ^ ' a ( In - P sinö 2(1 - v)

1

«Ω;

2(1-v)·

(102)

3.3. Wedge disclinations normal to the surface When Θ tends to ^π, expressions (100)—(101) for the relaxation displacements a very simple form, 2π«3 „ 1 — 2v , / -pT = 2r-zln (r + z), νΩ3 1— v 2πι4 νΩ3 "

' Xm

2z

_rTI

+

1 - 2v

2(l-v)

(103) , r + z ( - l ) m + yl In + '-

m= 1,2. (104)

Note that the solution in the case of normal incidence and elastic isotropy was first found by Romanov (1982) by Yoffe's method. For a review of the other three-dimensional elastic solutions for disclinations close to planar boundaries and for an analysis of the disclination-surface interaction, we refer to the book of Vladimirov and Romanov (1986) and a special review by Romanov (1984).

4. Dislocations in bi-crystals The effect of a planar interface on the elastic fields of dislocations of various shapes in bi-crystals have been studied by a number of authors, and mostly for isotropic materials. The displacements and stresses caused by a straight screw

Dislocations emerging at planar boundaries

425

dislocation normal to the interface of an isotropic bi-crystal were found by Hsieh and Dundurs (1973). The general isotropic case of a dislocation refracting at the interface and forming a dislocation bend with the apex at the interface was investigated by Dixon (1966). For an anisotropic bi-crystal, this problem was solved in a paper of Belov et al. (1983), where the elastic field was obtained in the form of a superposition of two-dimensional solutions. In the rest of the papers (Wu et al. 1974, Salamon and Comninou 1979, Dundurs and Salamon 1972, Salamon and Dundurs 1977, Bonnet 1983, Vagera 1970), dislocation loops parallel to the interface and also interface loops have been considered. 4.1. Dislocations arbitrarily piercing an interface; elastic anisotropy The dislocation-line force fans, introduced in section 2.2, allow us to solve most simply the problem of a dislocation bend with its apex at the interface of an anisotropic bi-crystal. Let the plane x 3 = 0 coincide with the interface, let the apex of the bend be at the origin of the coordinate system xh and let the senses of the branches be indicated as ti and t2, see fig. 12. Here, we shall restrict ourselves to the case of a bend in the plane ΧχΧ3 [the extension to the case of arbitrarily directed tx and t2 does not cause additional difficulties, see Belov et al. (1983)]. The unknown three-dimensional displacements in the two half-spaces p = 1, 2 may be represented as

uf = u*{p) + u]{p\

(105)

{p)

where u] means the displacements due to the straight dislocation of sense tp'm the infinite medium of elastic stiffness tensor C-J^, and the second term is the accommodation field and it should be obtained from the boundary conditions,

Fig. 12. A dislocation bend iri the x^-plane with its apex at the interface. Coordinates x, are shown.

A.Yu. Belov

426

eqs. (2)-(3) which on account of eq. (105) take the form fx[ßr] = - f x [ ß e ] ,

(106)

*[*]=-*[*],

(107)

where the symbol \_X~\ indicates a jump of X across the interface: [ X ] = Xa) — X{2\ The jump of the stresses on the right-hand side of eq. (107) is obtainable from eq. (46). Passing from integration over φ(ρ) in the planes perpendicular to tp to integration over φ in the plane x 3 = 0 according to the relations cot φip) = cot φ sm Θ,,

sin φ(ρ) = - ^ - , sina p we find from eq. (46) the stress jump to become

,m.·.r_*

άφ(ρ) = -^=-2- diA, surap

Γ<=ώ»ι.

2πρ Jo sin(0 — yr)|_

sina

J

(UW

To determine the right-hand side of eq. (106), we make use of the integral expression for the elastic distortions of the straight dislocation in an infinite medium (Indenbom and Alshits 1974). In the designations of eq. (46), we have

«.<">- " 2 ?

'" d /5»K)^)\ o

ν·ηφ

(109)

where the planar compliance tensor is defined by S**/K) = <5»<$/z - M P M i ^ K K C w w .

(110)

Taking into account the relation π Jo

sin (0 — φ)

the elastic distortions, given in eq. (109), may be reduced to 1

1

f*

άφ

The angles (φ', α', φ) are related to each other in the same way as (φ, α, ψ), thus, sinö . /ιώ = - — - /!<* + cos 0 sin φ t. sin a Now applying eq. (Ill) to β)\Χ) and β)\2\ and next passing again from integra­ tion over φ{ρ) to integration over ψ, we obtain for the right-hand side of eq. (106)

Dislocations emerging at planar boundaries

421

the following expression, rx[ße] =

inO 1 sin nVj sin

2πρ

1 Γπ_###ν® fsinö

(112)

2n2p J 0 sin(0 — i^)|_sin 2 a Decomposing the vector m^ as m^ = m^cos((/) - ψ) - ηφ$ϊη(φ - ψ), one can transform eq. (112) to

t*m =

m®b Γ sing 1

2πρ

|_sin 2 a'J

-■ζ-2-Μφ® ^π Ρ

Jo

άφοο{(φ -φ)\ άφ\

|_snra

^-^-(ηφηφ) |_sin a

-τ-2-(ηφηφ)~1(ηφιηφ)Β

1

(ηφνηφ)ο

J (113)

The jump t x [ ß e ] means that there exists a distribution of interface dislocations of the dislocation density tensor a = t x [ ß e ]. In our case, ά decreases as 1/p and also ά(φ + π) = ά(φ). Hence, ά consists of coaxial dislocation loops centered at the origin [the third term on the right-hand side of eq. (113)] and of emerging from the origin semi-infinite rays of the vanishing integrated Burgers vector [the first and the second terms in eq. (113)]. As indicated in fig. 13, in order to compensate the jumps given in eqs. (108) and (113) of stresses and elastic distortions, let us introduce in the interface x3 = 0 the fan-shaped distributions of straight dislocations and line forces of densities ο(1)(φ) and /{1)(φ) in the case of the medium of elastic constants

Straight dislocations

Fig. 13. Replacing a dislocation bend in a bi-crystal by straight dislocations and dislocation-line force fans in an infinite homogeneous media.

428

A.Yu. Belov

C$i and of densities δ(2)(φ) and/ (2) (^) in the case of medium C^.By eqs. (31) and (32), dislocation-line force fans cause the jump of the elastic distortions,

^x[Pr]=?®(^(1)(0) + ^(0)) 2p

2πρ

Jo

1 2πζρ

#[S(W*W-OW/Wl

(114)

0

and the jump of the stresses ^[Α Γ ]=-^-(/ ( 1 ) (0)+/ ί 2 ) (0)) 2p

π — ί 2πρ ίπρ Jo s i n ( 0 - ^ ) #

Γ β , Λ , , Λ

* T ,

Since the right-hand sides of eqs. (108) and (113) are anti-symmetric under inversion, the symmetric terms in eqs. (114) and (115) must vanish. Thus, we have #ι\ψ) = - *(2>(^) = Ηφ),

/Μ(ψ) = -/Μ(φ)

=W).

(116)

On account of eq. (116), the boundary conditions, eqs. (106)-(107), transform to a system of integral equations which, by the methods of section 2.1, reduces to an algebraic system for δ(ψ) and /(ψ), (S U ) + S(2))AW>) - (Q(1> + Q.i2))f(4>)

=

Η[^ (ηψ%Γΐ(% ' Μψ ' ) *]"πΙο #ί:θ1(
': (B(1> + Βα))Ηφ) - (ST(1) + ST(2>)/(4>) = sin a' π sin r The integration in eq. (117) may be performed explicitly,

.

J' '

(us)

lfV, ,, sinθ . cos2Θ /± αψcot(0 — ψ) . 0 = cos φΛ sin φ1 . , , , 7rJo sura sura and as a result solving of the system of eqs. (117)—(118) reduces to evaluation of the inverse matrix (N(1) + N (2) ) _1 , where for each medium the six-dimensional matrix N(p) is formed from the matrices B, S, Q;

N^ = ( -

B{p)

r

ST(P)

Dislocations

emerging at planar

boundaries

Some useful properties of the matrix (N ( 1 ) + N (2) ) et al. (1983), and Kirchner and Lothe (1987).

1

429

were established by Belov

4.2. Dislocations piercing the interface normally; elastic isotropy The general anisotropic solution obtained in the preceding section may be used to determine the elastic field of a dislocation bend in an isotropic bi-crystal. Here, however, we restrict ourselves to the simplest case of a straight dislocation piercing the interface normally, seefig.14. In the isotropic limit, blocks of the matrix (N (1) + N (2) ) depend only on the direction of ηιφ, S (1) + S (2) = s(e#^), B (1) + B (2) = b\ + ο~ιηφ ® ιηφ,

Qd) + Q(2) = q\ + qm4®m^

whereft,b, q, q are constants, from what follows that the system given in eqs. (117) and (118) significantly simplifies, and takes the form ι

ΞΟ(φ)χηιφ - qf((f)) - gm*(nty ·/(<£)) = - [(ηφηφ) Μ(φ) + bm^ Here,

(ηφιηφ)ο~],

· Β(φ)) + 8/(φ) χ*ηφ = - [(im 0 ) *b]. π

(119) (120)

l - 2 v J_ + _LI2v2_1 i) 2 ( l - v 2 ) J ' 2(1 1-V! q

q =

1-V

L

2

1 _ V

1

1 _ V

2 j

Γ fci , k2 1 L4Gi(l-Vi) 4G 2 (l-v 2 )J' ~ [4G1(1 - vj

+

4G2(l-v2)J· Dislocatvon

\,\,V ////

-► x

Fig. 14. Coordinate system (x, y, z) for the description of a dislocation normal to the interface of an isotropic bi-crystal. The y-axis points out of the paper.

A. Yu. Belov

430

In the case of an edge dislocation of Burgers vector b = bxm, 1

r<+ x* AT ν>»φ -> π^

2i?xcos(/>*/ G1v1 v l - Vl π 7Γ

π

G2v2 l y i - Vj

1 - V2

and the system given in eqs. (119)—(120) may be satisfied by a solution in the form A(

bx

cos0

£ = — 7i

77;

bx

v ßt,

π (1 - ν χ )(1 - ν 2 ) The constants β and a are determined by a_~n 0 - 2(1 -

W1

Vl)(l

cos<£

ΆΦ) = — 71

-T71

v «Λφ ·

(121)

π (1 - ν χ )(1 - ν 2 )

,fc2ViGi + (vi ~ V2)GiG 2 ~ fciV2G2 - v 2 ) feiG2 + ( 1 + f e i k 2 ) G i G 2 + k 2 G 2 >

4GiG 2 a = fciGf+ (1 +/c fe )G G + fc Gi 1 2 1 2 2 xKvi-v.XGia-v^ + G.a-vJ) + (v1G1(l-v2)-v2G2(l-v1)) x((l-2v1)(l-v2) + (l-2v2)(l-v1))}, where fcf = 3 — 4vf. According to eq. (121), when averaging over the interface, the function /(φ) does not tend to zero and, thus, there exist non-vanishing net tractions on the interface from each of the two half-spaces. In the case of a dislocation parallel to an interface, a similar effect has previously been pointed out by Dundurs and Sendeckyj (1965) for isotropic bi-crystals and by Barnett and Lothe (1974) for the anisotropic case. The presence of non-vanishing integrated tractions <σίζ> in fact means that to the solution obtained there should be added an infinitesimal constant stress field to eliminate net tractions. In the case of a screw dislocation of Burgers vector b = bzt9 1

-ί{ηφηφ)-1{ηφπιφ^ n

1

=0,

£i

-[(im*)* A] = — ψ n η

f

l

bzm(t>,

and the system of eqs. (119)—(120) has the following solution;

*(*) = ^ r 1 I r 2 m » ' π Ux + CJ2

/W>) = 0 ·

( 122 )

Integrated tractions are, in this case, equal to zero. However, there exists a net moment on the interface from each of the two half-spaces.

Dislocations emerging at planar boundaries

431

In the bulk of a bi-crystal dislocations cause the following displacements: (a) an edge dislocation of Burgers vector b = bxm9 z > 0:

sin 20 Mx = -^· Φ + 4(1 - vx) J In

.]

8π(1 - V l ) 2 ( l - v2)

^['(^rH)-^-;)]· ™ 2π

[

sin2 Ί

1 — 2vx

fr,

8π(1 -

χ|)?Γ(1-2ν 1 )) ln(r + z) + x

1-

r(r + z)

r(r + z)

- 4«(1- )2(1-ν )Γ νι 2

2

(l - ν 2 )

+ (3-2vt)

Γ+ Ζ

"£['■ln(r + z) + 4(l + fci

uz=

Vl)

r 7 (r + z)'

vx)

r+ z

]}·(124)

i

U^ -"^]-0·

(125)

z < 0:

wx = — In

sin 20

x sin 2φ

M

,]

Φ+ 4(l-v )J 2

bx\

8π(1-νι)(1-ν2)2

Η^-

- 2v2

1-2v2

--2^L2(r^j

lnp

« A2 2G2 V 2

z

sin2<£ Ί

-2(r^)J-

8π(1 -

z r

(126)

*>* 2 V l )(l - v 2 )

x |/?Γ(1 - 2v2) (ln(r - z) + ^ r ^ j ) - (3 - 2v2) - i x

1-

r(r - z)

•H['

fc2ln(r-z)-4(l-v2)+ \ k2 +

r—z

rj(r-z)2 (127)

A.Yu. Belov

432

Uz

4π(1-ν1)(1-v2)2r-z

'■{,[:2(l-v )-2

+ ^r-y,

r

(128)

2G2rj'

(b) a screw dislocation of Burgers vector b = bzt, b

z < 0:

z>0: -

y

bz Gi

~

Gi

y

2πG1 + G2r + z

_bz Gx - G2 x 2π Gi + G2 r + z9

-

^ bz

(129) G

i ~

G

2

y

In Gx + G2 r — z'

,

_ fe2 Gx - G2 x ~2π G1 + G2r-z'

Μ)7

(130)

(131)

5. Dislocations in plates In investigating the elastic field due to a straight dislocation close to the point of emergence at a free surface, the solution obtained in section 2 with the half-space approximation is quite adequate. This solution takes account of the most significant part of the relaxation field, possessing a singularity in the form of a pole at the point of emergence. However, when evaluating the dislocation field in a thin plate at distances of the order of the plate thickness from the dislocation, one should make use of a more exact solution which pays due regard to the stress relaxation at both surfaces. The elastic field of a screw dislocation normal to the free surfaces of an isotropic plate was first found by Eshelby and Stroh (1951). Saito et al. (1972) considered a periodic distribution of straight dislocations inclined to the plate surfaces, and in the limiting case, when the period tends to infinity, they obtained the elastic field of a dislocation normal to the surfaces. However, the edge dislocation case contained misprints and their solution required cor­ rections [for the displacements, deformations and stresses, see a paper by Kolesnikova and Romanov (1986), and for the stress functions a paper by Belov (1987b,c)]. Some numerical results for the elastic field of edge and screw dislocations in plates are presented by Indenbom and Chamrov (1980b) and Kawamoto and Shibata (1965), respectively. The more difficult case of a disloca­ tion inclined to the isotropic plate surfaces was solved by Belov (1987a). Straight dislocations meeting the surfaces of a basal hexagonal plate normally were considered in a paper by Belov (1987b,c).

Dislocations emerging at planar boundaries

433

5.7. Dislocations normal to the free surfaces; basal hexagonal plates Although close to the points of emergence the field of a straight dislocation becomes three-dimensional, one can point out a particular case where the boundary problem involved actually is two-dimensional. Such a situation occurs for dislocations normal to the free surfaces and the basal plane of a hexagonal plate, see fig. 15. These are dislocations of the type that have been considered in section 2.2 in the case of a semi-infinite hexagonal crystal. It will be shown here that the determination of the relaxation stresses a\k for such dislocations reduces to solving axially symmetrical problems in the theory of elasticity. Suppose that the x r and x3-axes coincide with the axes X and Z of the crystallographic system, and that the plane xx x2 coincides with the middle plane of a plate of the thickness t = 21. For dislocation stresses in the form of eq. (24), the free surface condition has the form σ\ζ=-σ!ζ\ζ

= ±ι,

(132)

where the field σ\ζ possesses only one non-vanishing cylindrical component, which in the case of a screw dislocation with Burgers vector b = bzt is axially symmetrical, ζφ — C44-

σ

Inp'

(133)

and which in the case of an edge dislocation with Burgers vector b = bxm can be expressed by an axially symmetric function, bx . Q ci3(cn - c 1 2 ) bx -—sin = - ^-lnp. (134) Cn 2πρ oy cn 2π Thus, the boundary condition, eq. (132), is axially symmetrical. Consider now the equilibrium equation, eq. (4), in the absence of body forces. As was shown by Hu (1953), in this case, for hexagonal crystals, the general solution, given in eq. <*\z=

Ci3(cn-c12)

Fig. 15. A dislocation normal to the plate surfaces. The coordinate system (x, y, z) is shown. In basal hexagonal plates m\\ [1120], n\\ [1 TOO], 11| [000 1].

A.Yu. Belov

434

(4), can be expressed in terms of two stress functions Φ and Ψ, which satisfy the axially symmetrical equations,

_?!_ _?!_ _?! (a2

82

dx2

+

dy2

+

dz22

(135)

Φ = 0,

82 \

(136)

where z, = zv,· and v, are given by eq. (72). By means of Φ and Ψ, one can represent the relaxation displacements as r

_

ay 3y'

82Φ

3x3z

&Φ dydz 32 Φ 3^2/

Cll

u' =

c 1 3 + c 44 \ 3 x

dW dx '

2

(137)

c 44 32Φ c 1 3 + c 44 3z2 '

(138)

The displacements given in eqs. (137)—(138) correspond to stresses, r

c 44

Γ

- 2 c , 66

2

/8

33Φ

32
2

3x3y

3y 3z

C44 C

13 +

c

-2c66

σ' = =

ö"*v

r zy

°

C44 C

13 +

c

44

+

c

c

13 +

l\

32

r _-> 83Φ 2C66 ^" o^878^

8x

/ δ2 \dx~2

Cii

+

C

3x 2

44

[-,.(

'3z 2

dxdyj9

-llc33

_ c44 Γ ~ cl3 + c 4 4 |_~

- C i

\3χ2+3y2

3x 2 3z -

Ci3

Cll

44

3822 ΊΊ3Φ 3z

82\

2 +

+

By2 2

dy ) 92

+

/8 2 !Ρ C 6 6 l87

+ Cl3

\

2

df )

+ Cl3

33c44

9

c 1 3 + c 4 4 8z 2

a

2

8z

2

+ C4.4

82"|8Φ 3? J 87 "

C44

8Φ

(139)

8z~' 82y 8y8z'

52Ψ 8^8i'

82y 8x2

5.1.1. Screw dislocations In the case of a screw dislocation, the displacement field u\ has only one non­ zero cylindrical component ητφ which is obtainable from the stress function Ψ, Ψ =

2πν3

dA

sinhAz3_ Ä^ÄIiIJo(Ap)-

(140)

Dislocations emerging at planar boundaries

435

Here /3 = Zv3 and J0 is a Bessel function. Using the stress function, eq. (140), and the well-known property of Bessel functions that J'0(x)= —Ji(x\we obtain 8y

bz

f°°dAsinh/lz3

τ /η x

Note that unlike the stress function, eq. (140), the displacements given by eq. (141) are expressed by a convergent integral. 5.1.2. Edge dislocations In the case of an edge dislocation, only the stress function Φ is non-vanishing. The form of this function largely depends on whether the degeneracy ^i3 — c i3 — 2c 44 = 0 is present or not: (i) in the presence of the degeneracy, Φ is a bi-harmonic function of the variables x, y, z 0 Λ

Φ=

c 1 3 (cn - c 1 2 ) bx . άλ Ji(/lp) sin φ o P(sinh2>Uo + 2λ10) CuCa 2πν0 x

L

sinh λζ0 ( λΙΌ cosh λ10 H

V

— sinh Λ/0 λζ 0 cosh λζ0

^ V - sinh λ10 I

c13 + c13

)

,

(142)

where Z0 = Zv0, v0 = (cii/c 3 3 ) 1 / 4 , see section 2.5; (ii) in the absence of the degeneracy (vx # v2), the stress function Φ, being a solution of an equation of the fourth order, eq. (135), can be expressed in terms of two stress functions φχ and φ2, 8Φ ö ^ = 0i + 02,

(143)

which satisfy the second order equations (Elliott 1948) - 22 + 3dzf -2 1 ^ = 0 ' ,6x 22 + 3dy

ί=1

>2'

( 144 )

and which are determined by the relations φ

2=

c 1 3 ( c n - c 1 2 ) v2 fr, CnC44 1 -h/c! 2π c 1 3 (cn - c 1 2 ) ^11^44

p^sinh^coshlz, Jo / S(A)

vi ft* 1 , / ^-sm(t> 1 + κ2 2π

f00^

άλ

Jo

sinh J ^ cosh Λζ2 J T277T\ iVW)> /i s(/i) (146)

A.Yu. Belov

436

where l( = vj (i = 1,2) and s(X) = v 2 sinh/l/ 2 cosh/l/ 1 — v ^ i n h A / i C O s h ^ . By means of the stress function φί and 0 2 one can derive the displacements *χ = 1(Φΐ+Φ2). τζ{φι+φ2),

ιι; = = -(φ^1+φ2), <4

(147)

«4 = ^ ( Μ ι + Μ 2 ) ,

(148)

and the stresses 82

θχ 8

2

δ2 \ / δ2^ d2d>2 + C12 τ~Ί. )(Φΐ + Φΐ) + Cl3 ^1 -TIT2 + kz2 1J

-"a^y

2

2

8 \

' ™

,

^

'

v

/,

x

θζ

'

2

δ 0!

8z

, 92<£' 2

*L = (/c l C 3 3 - c 1 3 / v 2 ) - g ^ r + (/c fec2 c3333 - cC1133 /v§) /Vi) - - ^^ , 92i

σ

ζχ —

C

44

σ

ζν —

C

44

8202

820x

(Cll -012)^-^-{Φΐ

82<£2

(149)

]■

]■

+ 02).

Using the stress functions, eqs. (145)—(146), it is possible, by going to the limit / -► oo, to obtain the stress functions for the in section 2.5 described edge dislocation normal to the surface of a semi-infinite hexagonal crystal. F o r these functions, the following expressions are valid, , c13(cn-c12) v 2 bx Γ ζ1 Ί 15 0i = ~ : \TTl~^~y\ i + ln ri + z i ) > ° r z 2c 11 c 44 (v 1 - v2) 1 +/Ci 2π | _ i + i J , ci3(cn-c12) Vi bx [ z2 Ί 02 = - ~ : -\T^~T^~y\ — I — + ln(r 2 + z 2 ) , (151) 2c 11 c 44 (v 1 - v2) 1 + k2 2π |_r2 + z2 J and these lead to the same displacements as have been obtained in another way in section 2.5. 5.2. Dislocations normal to the free surfaces; isotropic

materials

Starting from the solutions, given in eqs. (140) and (142), one can derive the stress functions for dislocations in an isotropic plate. When going to isotropic

Dislocations emerging at planar boundaries

437

materials, for which the relations c 4 4 = c66 and c x l = c 3 3 are valid, one finds that v0 = v3 = 1. 5.2.1. Screw dislocations In the case of a screw dislocation the stress function has the form

h. r°° d ^ s i n h ^ Μλρ),

(152)

In Jo λ2 cosh A/

and eq. (141) for the displacements reduces to the well-known result of Eshelby and Stroh (1951), bz f-dAsinhAz

=

^ -2^J 0 T^hI/ J l ( ^·

(153)

We also reproduce the associated expressions for the stresses Gbjl

f00

-=^LHo

<*τώ

=

coshAz

dA

^

Aτ (/ ηA

Ί

4

„ ^

(i54)

Gftz Γ00 „sinhAz , ,„ , ,«^λ dl J (155) 0 The stress σζφ determines the force between screw dislocations and the stress σρφ the force on an edge dislocation from the screw dislocation. It is interesting to investigate the asymptotic behavior of the interaction force. When p/l> 1, 2Gbzfl\1/2 ,2l Gbz2z iAc^

^ = ^J

^hI/ ^·

where σζφ is an average stress over the plate thickness. Thus, the force between screw dislocations decreases exponentially (Eshelby and Stroh 1951), while the force on an edge dislocation from a screw dislocation only decreases inversely with the second power of distance, the average force in this case vanishing. 5.2.2. Edge dislocations In the isotropic limit, the stress function Φ is given by the following expression; vb* · , f°°dA f°°cU J^kp) ±Λ Φ=— -sin0 3 Jo A (sinh2) + 2μ) π(1 — v) λ3 (sinh2^ Λ

x [sinh>lz(^cosh^ + 2v sinh^) — sinh^AzcoshAz],

(157)

where μ = λΐ. In a cylindrical coordinate system, οροζ

ροφ oz

4=-2(1-ν)ΔΦ +ν

.

τ 8?

(159)

A.Yu. Belov

438

Substituting the stress function, eq. (157), in eqs. (158) and (159), we obtain r

vbx . .[™άλ J.ilp) sin0 π(1 — v) Jo λ (sinh2^ + 2μ) Jo x {sinh λζ [ μ cosh μ + 2(1 — vjsinh^u] — λζ cosh λζ sinh μ}, vbx

r

άλ

cos φ

(160)

J^kp)

o A2 (sinh 2μ + 2μ) x {coshλζ\_μcoshμ — (1 — 2v)sinh^] — λζsinhλζsinhμ}, vf?x sin 0 π(1 — v) p

(161)

dA[ApJ 0 (/lp)-Ji(Ap)] A2 (sinh 2μ + 2μ)

x {coshΛ,ζ[μ coshμ — (1 — 2v)sinh/z] — Az sinh λζ sinhμ} .

(162)

The expressions obtained by Saito et al. (1972) for the displacements urz and u\ coincide with eq. (160) and eq. (161), respectively. However, their result for uTp contains misprints. In conclusion, we note that in isotropic and in basal hexagonal plates the elastic problems of the relaxation field u\ for an edge dislocation and a line force normal to the surface (load is parallel to the surface) are equivalent. In the isotropic case, this was indicated by Eshelby (1979). The relaxation field for a line force in an isotropic plate was found in a paper by Green and Willmore (1948), who used two harmonic functions instead of one biharmonic function Φ. For a hexagonal plate, in the case vx φ ν 2 , the solution of this problem was suggested by Shield (1951). The stress functions obtained by Shield (1951) differ from those of eqs. (145), (146) only by coefficients. 5.3. Edge dislocations in an isotropic plate; average stresses and the plane stress solution In a number of practical problems, knowledge of the average stresses of an edge dislocation over the plate thickness is required. When averaging, only the planar components of the stress tensor give non-vanishing results, Λ^

σ„-σΦΦ

2

σ

ρρ + σφφ

2

G

1/ 8

/9

2

V9p2

Gpp +

1 \ 6 8Φ 19

p9p

(163)

1 82 \ 8 Φ

(164)

p 2 8(AV8z'

σ

φφ

2 ΛίΛ

Λ

, / 82

18

1 82 \

"

>>

2V

8 2 8Φ 1

8?

8F'

(165)

Dislocations emerging at planar boundaries

439

where the relation σρρ = σ\φ is taken into account. According to eqs. (163)—(165) the average stresses are expressed by Φ(χ, y91) and 9 2 Φ(χ, y, z)/dz2\z = l and have the form 4v2Gbx cos0 f00° ° du d wcosh c o s hww-- l1 / p\ ρψ ρφ π ^ _ yJ p jJoo Wu22 S sinhw \ t i n h W ++ Wu 00 2v Gbx sincj) f dwcoshw — 1 2 π(1 — v) p Jo u sinhu + w

2J2( u^)-u^J1(u^\]9

(167)

2 2 GfcGfc dudu cosh u - 1 <*pp + σ ^ _ σρρ + σ ^ 2v2v f °° x xs isni0n 0 f °° -/1 Λ x π ( 1 - ν -) ί J o *u■ sinhu + u v

^ t (168)

If p ^> t, the average stresses, given in eqs. (166)—(168), must transform into those of a dislocation in a plate of infinitesimal thickness and they must agree with the solution for a state of plane stress. In order to establish this, one should evaluate the asymptotics of the stresses taking into account only terms decreasing with 1/P, (1 + v)Gbx sin φ . (1 + v)Gbx cos(/> σρρ = σΦΦ= —, ** = — ^ —· (169) The solution for the state of plane stress is, as would be expected, obtainable from that of plane strain, Gbx G*PP = θφψ =

~ ττ-Γί

sin<£ ϊ

Gbx

e

'

σ

ρφ

=

ΤΤΛ

cos ΐ

>

1 7 0

2π(1 — ν) ρ 2π(1 — ν) ρ by replacement of the Poisson ratio v by v/(l + v). In conclusion, we note that there exists a set of papers concerning the evaluation of elastic fields and energies of circular dislocation loops parallel to the free surfaces of isotropic (Chou 1963, Kubo et al. 1976, Bushueva et al. 1980) and basal hexagonal plates (Chou 1964, Khzardzhyan et al. 1982).

Appendix I Concentrated forces on an anisotropic half-space 1. The degeneracy problem The Stroh method (referred to as an algebraic method) for treating two-dimen­ sional problems has the disadvantage that the characteristic solutions, given in

A.Yu.

440

Belov

eq. (14), must be modified in the case of equal roots pa. This fact should be taken into account when constructing three-dimensional solutions by formula (15), since it becomes invalid even if the degeneracy occurs for only one value of ψ. However, in some problems the difficulties associated with degeneracy can be removed by transformation from the algebraic representation to the so-called integral representation. The theory for this purpose is given by Barnett and Lothe (1973) (see chapter 4, this volume). We will illustrate the transform to the integral representation by the example of a concentrated force applied to the free surface of an elastically anisotropic half-space x 3 > 0. This problem has been considered by Willis (1966) and by Barnett and Lothe (1975) by an algebraic method. The displacements uk(r) satisfy eq. (10) and the stresses aik obey the boundary condition σί3Ιζ = ο = ~fi(p)> (A1) with fi(p) =fiS(p), where δ(ρ) is the Dirac delta-function. Let us seek a solution for uk(r) in the form given in eq. (15) with uk(£, z) caused by the one-dimensional force^(C), namely the Radon transform offi{p\

fi{p)= fi{

\l

Μρ)=\Μ)άφ.

(Α2)

The inverse transform has the form (Gel'fand and Shilov 1964) άζ' d

MO=-^-22?

ζ'-ζάζ'

άη'Μρ'),

(A3)

where p' — η'ηΐψ + ζ'ηφ. Substituting the delta function into eq. (A3), we obtain the one-dimensional generalized function / ( 0 = - ί 3J_ 7ϊ· 2π2ζ2 Thus, the boundary condition in the two-dimensional problem is *»(C,0)=-/ £ (C),

(A4)

(A5)

where/(C) is determined by eq. (A4). Now take a two-dimensional solution (assumed as a generalized function) in the form

a= 1

where Β(φ) is the unknown vector function to be determined from the boundary condition given in eq. (A5). Comparing eq. (A6) with eq. (26) we see that eq. (A6) corresponds to a dipole of straight dislocations of sense ηιψ. Further, using the

Dislocations emerging at planar boundaries

441

definition, given in eq. (16), of I«, one can find the tractions associated with the dipole ίσί α=1

(ί + v.zf

In view of 1 ιΙί 7ϊ±ίπδ'(ζ), (ζ±ϊθγ~:χ= Ρ?^ "^>>

(A8)

and the closure relations, given by eq. (20), the surface tractions are derived to be fÄ(C 0) =

Β(ψ)ϋ(φ) 2πζ2

(A9)

Finally, (A 10) and the displacements due to the concentrated force in the half-space take the same form as if they were caused by a fan of dislocation dipoles distributed in the plane x3 = 0(fig. 16), 6

. A„ ® L„ \ - . *>-Η{&*'τ%)*-'»>'«-

(All) Thus, the solution of the problem under consideration is represented as the integral over two-dimensional solutions with a simple physical interpretation. Note also that, by virtue of C±iO

■ = ρ):-(±)ίπδ(ζ), ζ

(A12)

dD=D(SD
Fig. 16. Distribution of dislocation dipoles in the form of a fan centered at the origin. A beam of dipoles of width άψ and of total moment d/> is indicated, t points out of the paper.

A.Yu. Belov

442

eq. (All) immediately reduces to the well-known result of Barnett and Lothe (1975) for the surface displacements,

«(/» = ^ B - W

+ 2 ; tc 2 Pjo

sin (φ — φ)

(A13)

at point p with the polar coordinates (p, φ). Let us now obtain the displacements in a form valid in the case of degeneracy. For this purpose, one should consider the transformation properties of ξα and pa with respect to rotation of the basis (m0,n0) by ω about t0. The basis obtained by rotation we denote as (#ηω, ηω). According to Barnett and Lothe (1973), 3ω

= -(1+Ρα2),

(A14)

wherefrom

£-0.

(A15) ceo and thus the eigenvectors are invariant under such rotations. In turn, the eigenvalues ρα(ω) have the mean value 1

ρα{ω)άω = ± i,

(A16)

and hence, integrating the sum rules, given in eq. (21), over ω, we find, in view of eqs. (A 15) and (A 16), an integral representation for the matrices given in eq. (22) which is valid in the case of equal pa, 1_

B= - — In

((τηωηω){ηωηω)

S= -

{ηωηω)~1{ηωπιω)άω,

2π

1 Γ2π Q= - — (ηωηω)

>^ω)-^ω^ηω))αω, (A17)

χ

άω.

Equation (A 13) together with eq. (A 17) gives the surface displacements in a form stable with respect to degeneracy. Now it is our purpose to extend this result to the displacements below the surface. In order to obtain the dipole displacements in the integral form, we need the distortions associated with a straight dislocation. From eq. (26), it follows F e„p

*y ^ 2πι *-;

+

ft'Q®^·»), m(af-\-p0.nO3r

(A18)

443

Dislocations emerging at planar boundaries

The coordinate system (/η ω , ηω) in the plane of the two-dimensional problem may be chosen so that mü}r = R,

(A19)

/ι ω .|· = 0,

where R2 = ζ2 + z2 and ω = arccot(£/z), see fig. 17. Then, on account of the sum rules (23), the distortions, given in eq. (A 19), take the form V®"D = ^;(-'n(0®S + Λω®("ω"ωΓ1[Β +(n„mJS])A. (A20) 2nR Finally, since the displacements associated with the dipole and the dislocation are related by u(C,z) = —h(m0V)uD9 where h is the distance between the dislocations forming the dipole, it follows from eq. (A20) that u(C,z) = —-(cos 2nR

X

coS -ϊϊηω(ηωηω)

(A21)

[B + ( ^ m J S ] ) / ) ,

with D = hb. Thus, the following expression, 1 «(') =

o^22 ÜT

o Ä

(^ωβ-^ηωίη^Γ^Β+ίη^^^Β-

1

/

(Α22)

solves the problem of the concentrated force in the case of degeneracy. In this chapter, we shall mainly use the algebraic method. Nevertheless, in view of the Barnett-Lothe method, many of the results to be obtained are also applicable in the case of degeneracy, e.g., for elastically isotropic materials. 2. The surface relief in the case of normal loading; the surface parallel with a mirror plane The further analysis of the displacements associated with the concentrated force requires the solution of an infinite number of sex tic equations, eq. (13), or

C

?.
Fig. 17. Rotation of the basis (m0, n0) by the angle ω. /ηω is directed to the point r with cylindrical coordinates (p, φ, z) and polar coordinates (R, ω).

A.Yu. Belov

444

alternatively an infinite number of numerical integrations given in eq. (A 17), necessary for subsequent integration over ψ. For this purpose, computer pro­ grams must be developed. However, a simple particular example can be treated analytically. This is the problem of the surface displacement uz (referred to as the surface relief) caused by a normal force f=fzt acting on a surface parallel to a mirror plane. Remember that the elastic properties of a crystal with a mirror plane are the same as those of a crystal with a two-fold axis normal to this plane. Matrices B and S. The mirror plane simplifies the matrices B and S. Indeed, owing to the equivalence between a mirror plane and a two-fold axis, the components of B and S are invariant under a rotation by 180° about x2. Then, in view of the relations, B(-f0)=B(f0),

§(-fo)=-S(f0),

following from eq. (A 17), the only possible form for these matrices is Β(ψ) = Β^ηψ ® ηψ + B°22t ® t + Β033ηιφ ® m^ + Β°13(ηφ ® tffy + /ffy ® ηφ),

(Α23)

SOA) = Ξ°ί2ηφ ® t + S°21t ® n+ + S°23t ® m^ + S?2/*fy ® t,

(Α24)

where the components £°„ and S°„ are referred to the coordinate system xf. The inverse of B is given by RO

Β~ί(Φ) =

DU / D U

r>22(rfllD33

Du

RO

^22^33

1 —

RÜ2\

n13)

0 RO RO -#22^13

0 D02 RO R° " #13 #11^33 -

0

—

RO DO "I D22D\3

0 RO

RO

#22^11

J

(A25) On account of eq. (A25), the general expression for eq. (A 10) in the case of normal loading reduces to D(il,) = ^ r t , nB22

(A26)

with rS(iA)z>0A) = o. Therefore, the surface relief is Mp^) = . i (A27) ( . V 2πρΒ22(φ) Now, we notice that, for every direction ιηφ in the mirror plane, eq. (13) is bi-cubic and, hence, its roots pa are expressed by radicals. This means the elements of the matrix Β(φ) and the surface relief given in eq. (A27) can be

Dislocations

emerging at planar

boundaries

445

evaluated without numerical calculations. Other analytical results of this kind will be presented in sections 2.3 and 2.4.

References Alshits, V.l., V.L. Indenbom and E. Kossecka, 1975, Phys. Status Solidi B 70, K25. Bacon, D.J., and P.P. Groves, 1970, in: Fundamental Aspects of Dislocation Theory, Vol. 317, Spec. Publ., eds J.A. Simmons, R. de Wit and R. Bullough (Natl. Bur. Stand., Washington, DC) p. 35. Barnett, D.M., and J. Lothe, 1973, Phys. Norv. 7, 13. Barnett, D.M., and J. Lothe, 1974, J. Phys. F 4, 1618. Barnett, D.M., and J. Lothe, 1975, Phys. Norv. 8, 13. Bastecka, J., 1964, Czech. J. Phys. B 14, 430. Belov, A.Yu., 1987a, Boundary Problems in the Anisotropie Theory of Dislocations and Disclinations, Ph.D. Thesis (Institute of Crystallography, Moscow) (in Russian). Belov, A.Yu., 1987b, Sov. Phys.-Crystallogr. 32, 550. Belov, A.Yu., 1987c, Kristallografiya 32, 1072. Belov, A.Yu., and VA. Chamrov, 1987, Metallofizika 9, 68 (in Russian). Belov, A.Yu., and V.M. Kaganer, 1987, Metallofizika 9, 79 (in Russian). Belov, A.Yu., VA. Chamrov, V.L. Indenbom and J. Lothe, 1983, Phys. Status Solidi B 119, 565. Berry, D.S., and T.W Sales, 1962, J. Mech. & Phys. Solids 10, 73. Bonnet, R., 1983, Philos. Mag. A 47, 529. Brown, L.M., 1967, Philos. Mag. 15, 363. Bushueva, G.V, A.A. Predvoditelev, R.D. Frolova and S.M. Khzardzhyan, 1980, Prikl. Mat. & Mekh. 44, 760 (in Russian). Chou, Y.T., 1963, Acta Metall. 11, 829. Chou, Y.T., 1964, Acta Metall. 12, 305. Comninou, M., and J. Dundurs, 1975, J. Elast. 5, 203. de Wit, R., 1973, J. Res. Nat. Bur. Stand. Sec. A 77, 49. Dixon, D.A., 1966, Dislocation piercing a bimetallic interface, Ph.D. Thesis (Northwestern Univer­ sity, Evanston). Dundurs, J., and N.J. Salamon, 1972, Phys. Status Solidi B 50,125. Dundurs, J., and G.P. Sendeckyj, 1965, J. Appl. Phys. 36, 3353. Elliott, H.A., 1948, Proc. Cambridge Philos. Soc. 44, Part 4, 522. Eshelby, J.D., 1979, in: Dislocations in Solids, Vol. 1, ed. F.R.N. Nabarro (North-Holland, Amster­ dam) p. 167. Eshelby, J.D., and A.N. Stroh, 1951, Philos. Mag. 42,1401. Eshelby, J.D., W.T. Read and M. Shockley, 1953, Acta Metall. 1, 251. Filippov, A.P., G.N. Gaidukov and S.K. Maksimov, 1985, Phys. Status Solidi A 90, 215. Filippov, A.P., G.N. Gaidukov and S.K. Maksimov, 1986, Phys. Status Solidi A 94, 53. Gel'fand, I.M., and G.E. Shilov, 1964, Generalized Functions, Vol. 1 (Academic Press, New York). Green, A.E., and T.M. Willmore, 1948, Proc. R. Soc. London A 193, 229. Groves, P.P., and D J . Bacon, 1970, Philos. Mag. 22, 83. Hsieh, C.F., and J. Dundurs, 1973, Int. J. Eng. Sei. 11, 933. Hu, H.C., 1953, Acta Phys. Sin. 9,130. Humble, P., 1985, Philos. Mag. A 51, 335. Indenbom, V.L., 1960, Sov. Phys.-Dokl. 4,1125. Indenbom, V.L., and VI. Alshits, 1974, Phys. Status Solidi B 63, K125. Indenbom, V.L., and VA. Chamrov, 1980a, Sov. Phys.-Crystallogr. 25, 268.

446

A.Yu.

Belov

Indenbom, V.L., and V.A. Chamrov, 1980b, Metallofizika 2, 3 (in Russian). Indenbom, V.L., and S.S. Orlov, 1968, Sov. Phys.-Crystallogr. 12, 849. Jager, W, M. Ruhle and M. Wilkens, 1975, Phys. Status Solidi A 31, 525. Kawamoto, M., and T Shibata, 1965, Mem. Fac. Eng. Kyoto Univ. 27, 408. Khzardzhyan, A.A., S.M. Khzardzhyan and A.A. Predvoditelev, 1982, Izv. Akad. Nauk ArmSSR, Ser. Fiz. 17, 260 (in Russian). Kirchner, H.O.K., and J. Lothe, 1987, Philos. Mag. A 56, 583. Kiselev, VV, and V.T. Shmatov, 1979, Fiz. Met. & Metalloved. 47, 639 (in Russian). Kolesnikova, A.L., and A.E. Romanov, 1986, Preprint FTI No. 1019 (Izd. FTI, Leningrad) (in Russian). Kubo, R., H. Ishii and K. Saito, 1976, Trans. Jpn. Soc. Mech. Eng. 42, 359. Lekhnitskii, S.G., 1963, Theory of ELasticity of an Anisotropie Body (Holden Day, San Francisco, CA). Li, J.C.M., 1964, Philos. Mag. 10, 1097. Lothe, J., V.L. Indenbom and V.A. Chamrov, 1982, Phys. Status Solidi B 111, 671. Maurissen, Y., and L. Capella, 1974a, Philos. Mag. 29,1227. Maurissen, Y., and L. Capella, 1974b, Philos. Mag. 30, 679. Mindlin, R.D., 1936, Physics 7, 195. Mindlin, R.D., and D.H. Cheng, 1950, J. Appl. Phys. 21, 926. Nishioka, K., and J. Lothe, 1972, Phys. Status Solidi B 51, 645. Ohr, S.M., 1978, J. Appl. Phys. 49, 4953. Orlov, S.S., 1971, Phys. Status Solidi B 47, K21. Pertsev, N.A., 1984, in: Experimental Research and Theoretical Description of Disclinations, eds VI. Vladimirov and A.E. Romanov (PTI, Leningrad) (in Russian) p. 208. Romanov, A.E., 1982, Poverkhnost 12,121 (in Russian). Romanov, A.E., 1984, in: Experimental Research and Theoretical Description of Disclinations, eds VI. Vladimirov and A.E. Romanov (PTI Leningrad) (in Russian) p. 110. Saito, K., R.O. Bozkurt and T. Mura, 1972, J. Appl. Phys. 43, 182. Salamon, N.J., and M. Comninou, 1979, Philos. Mag. A 39, 685. Salamon, N.J., and J. Dundurs, 1977, J. Phys. C 10, 497. Shaibani, S.J., and P.M. Hazzledine, 1981, Philos. Mag. A 44, 657. Shield, R.T., 1951, Proc. Cambridge Philos. Soc. 47, 401. Steketee, J.A., 1958, Can. J. Phys. 36,192. Stroh, A.N., 1958, Philos. Mag. 3, 625. Stroh, A.N., 1962, J. Math. Phys. (Cambridge) 41, 77. Tikhonov, L.V, 1967, Fiz. Met. & Metalloved. 24, 577 (in Russian). Vagera, I., 1970, Czech. J. Phys. B 20, 702. Vladimirov, V.l., and A.E. Romanov, 1986, Dislocations in Crystals (Nauka, Leningrad) (in Rus­ sian). Willis, J.R., 1966, J. Mech. & Phys. Solids 4, 163. Wu, J.B.C., K.K. Shih and J.C.M. Li, 1974, Mater. Sei. & Eng. 14, 15. Yoffe, E.H., 1961, Philos. Mag. 6,1147.