Suppose that in a spherical domain O = 4pa3 the transformation strain is uniform while it vanishes outside this domain. Then, in the above eqns [B20] to [B22], the transformation strain tensor can be taken outside the integral, and it remains to solve integrals of the following type:

d3 RRaRb • • • Rin

O

A A A dfflR Ra Rj • • • R2n

Here, Ra = Ra/|R| is the a-th component of the unit vector of R, and the remaining integral in eqn [B24] is over the surface of the unit sphere. The value for these surface integrals can be found from the general formula

d°RRaRb. •••R2n dab • •• d2n— 1,2n [B25]

4p (2n + 1)!!

The sum extends over all possible combinations of the indices, and hence it contains (2n-1)!! terms. The double factorial is defined as

(2n + 1)!! = 1*3*5*- • •* (2n + 1)

From these relations, one then finds the following mul­tipole tensors for a spherical inclusion:

Pjk O Cjkmn emn

a2„

Pjkpq = ~5djk Ppq

a4

Pjkpqrs 35 (djk + djp dkp + dkp ) Prs [B26]

We see that all multipoles tensors of higher rank than two are given in terms of the dipole tensor, and all tensors with an odd rank are zero.

When the transformation strain is that associated

Inclusion

Self-interstitials may aggregate into planar clusters with their dumbbell axes aligned in parallel. We may

view them as plate-like inclusions of thickness h and with a normal vector n. If we transform this plate by displacing one of its faces by b relative to the other face, then the transformation displacement field throughout the unconstrained plate is

«T = X-h-b, n, [B29]

where x j are the components of the position vector. The transformation strain is obtained by differentia­tion and found to be

etj = 2( + «£■) = + hn<) [B30]

Inserting these transformation strains into eqn [B21], we find for this plate-like inclusion the dipole tensor

Pjk CjkmnbmnnA [B31]

where A is the area of the plate. Note that this result is independent of the thickness h and of the shape of the plate. However, to evaluate the higher order multi­pole tensors, one must specify the shape of the plate.

Let us consider a circular plate of radius c, vanish­ing thickness h!0, and with its normal unit vector n pointing in the X3-direction.

Then, all tensors Pjkpq… vanish that have one or more indices that are equal to 3. For all other cases, the tensor components can be obtained with the formula

f fx2Mx2N, , c2(M+N+1) (2M — 1)!!(2N- 1)!!

JJXl X2 dX1dX2 = 2(M + N + 1) 2(M+N)(m + N) [B32]

All multipole tensors of odd rank vanish, and the remaining can again be expressed in terms of the dipole tensor. For example, the quadrupole tensor can be written as

Pjhpq = Pjk Qpq

where

c2 1 0 0

Qpq = 4 0 1 0

4 0 0 0

For an infinite isotropic material, the far-field dis­placement components of this circular platelet are given by « (r) ffi — G“ (r)Pjk

A1

_________

8%(1 — n) r2

3

+ 3 Xi r3

where the distance vector r is from the center of the platelet. A comparison of this displacement field with
the one for a circular dislocation loop by Kroupa36 reveals that the dipole approximation gives an accu­rate representation for the dislocation loop for dis­tances r 2 c.