The stretched surface of a cavity can relieve its resid­ual strain e* to some degree by reducing the surface area at the expense of creating stresses in the sur­rounding material.

In the absence of externally applied stresses, this creates a spherically symmetric deformation field that can be derived from a radial displacement function

u(r) = A/r2 [119]

where r is the distance from the cavity center. The following bulk strains and stresses can then be obtained:

err 2A/ r ; eyy e» A/r

ffrr = -4mMA/r3, See = S» = 2mMA/r3 [120]

The surface stresses, on the other hand, are deter­mined from

g = gee = g" = 2(ms T As)(e;T a/r3) [121]

The constant A is chosen such as to satisfy the bound­ary condition for the radial bulk stress at the cavity surface:

Srr (R)=2g — P [122]

This boundary condition replaces the incorrect one stated in eqn [110]. The cavity must always satisfy this mechanical equilibrium condition expressed by eqn [122], regardless of whether the thermodynamic equilibrium condition is satisfied or not. In other words, thermodynamic equilibrium and mechanical equilib­rium obey two different and separate conditions.

From eqns [121] and [122] it follows that the surface strains eyy and are both equal to

r A = pR — 4(ms + As)e* r3 4mMR + 4(ms + 1s)

4mMR + 4(ms + 1s)

Using this result we can determine from eqn [118] the surface energy of the cavity as a function of its radius R:

g(e(R)) — g0 + 2 (mS + 1S)[2e* + e(R)]e(R) [124]

Associated with the surface strain e(R) is a stress and a strain field in the surrounding material given by eqn [120]. It gives rise to the strain energy

Sj Ejd 3r — 8pR3mMe2(R)

The reference cavity radius R defined by eqn [106] undergoes a small change as the surface strains adjust to their mechanical equilibrium values given by eqn [120]. As a result, the change in cavity volume, its relaxation volume, is

A Vr(R) — 4tcR2«(R) — 4pR3 e(R) [126]

When gas is present in the cavity at a pressure p, it performs the work —pAV~ when the surface relaxes. Therefore, the total free energy associated with the creation of a cavity is

Fc (R) — 4pR2y(e(R)) + 8nR3 mMe2 (R) — 4pR3pe(R) [127]

Fq(R(N)) replaces now the surface free energy Fs(N) used in eqn [105] to arrive at the cavity surface tension 2g0/R. The latter is now given by Fc(N + 1) — Fc(N). It will be evaluated in the next section and compared with 2g0/R.