This interaction arises not from the strain field of other defects or from applied loads but is caused by the changing strain field of the point defect itself as it approaches an interface or a free surface of the finite solid. We have shown in Section 1.01.4 that the strain energy associated with a point defect is given by

r _ 2Kmo /Vrel2 _ 2m(1 + v) (Vrel)2 U0 = 3K + 4m Q = 9(1 — v) Q-

when the defect is in the center of a spherical body with isotropic elastic properties or when the defect is sufficiently far removed from the external surfaces of a finite solid. This strain energy has been obtained by integrating the strain energy density of the defect over the entire volume of the solid, and since this density diminishes as r-6, where r is the distance from the defect center, it is concentrated around the defect. Nevertheless, close to a free surface, the strain field of the defect changes, and with it the strain energy. This change is referred to as the image interaction energy Uim, and the actual strain energy of the defect becomes

U (h) = U0 + Uim(h) [64]

Here, h is the shortest distance to the free surface. The strain energy of the defect, U(h), changes with h for two reasons. First, as the defect approaches the surface, the integration volume over regions of high strain energy density diminishes, and second, the strain field around the defect becomes nonspherical and also smaller. The evaluation of both of these contributions requires advanced techniques for solv­ing elasticity problems.

Eshelby 1 has shown that the strain energy of a defect, modeled as a misfitting inclusion of radius r0, in an elastically isotropic half-space, is given by

(1 + n) £0 4 h3

where h is the distance from the center of the defect to the surface. Equation [65] clearly demonstrates that the strain energy of the defect decreases as it approaches the surface. The minimum distance h0 is obviously that for which U(h) = 0, and it is given by (1 + n)

Another case for the image interaction has been solved by Moon and Pao,32 namely when a point defect approaches either a spherical void of radius R or, when inside a solid sphere of radius R, approaches its outer surface.

For a defect in a sphere, its strain energy changes with its distance r from the center of the sphere according to 1 + n гЛ3

4 RJ (n + 1)(n + 1)(2n + 1)(2n + 3) r

n2 + (1 + 2n)n + 1 + n while the strain energy of a defect at a distance r from the void center is given by 1 + n Г0 3

4 R)

1 n(n — 1)(2n — 1)(2n + 1) ^2n+2 [68]

n 2 n2 + (1 — 2n)n + 1 — n r

Again, at a distance of closest approach to the void, h0(R), the strain energy of the defect vanishes. The numerical solutions of Us(R + h0) = 0 and of UV(R—h0) = 0 gives the results for h0/r0 shown in Figure 17. There is a modest dependence on the radius of curvature of the surface. Approximately, however, the defect strain energy becomes zero about halfway between the top and first subsurface atomic layer, assuming that r0 is equal to the atomic radius.