The remote stress field shown in Figure 1 has principal stresses M+S and M-S, imagined as compressions. Eventually we shall want to consider the interface to be coherent and an overall compression M helps to make this plausible. But such an overall compression has no effect on the deformation or the associated stress variations; these can be fully discussed with remote principal stresses of S and –S, without the M. Additionally, for algebraic purposes it is convenient to have elongation rates as positive and hence to have tensile stresses as positive also; thus we consider a tension S to east and west and a compression –S from north and south. We use polar coordinates (r,q) with the interface at r = 1 and the direction of maximum tension as q = 0.

A solution of the corresponding elastic problem in absence of diffusion is given by Muskhelishvili (1963) and followed by Jaeger and Cook (1979), but their powerful general method is not readily extended to conditions that include diffusion. By contrast the following method is of narrow application but can be extended so as to shed light on diffusive behavior.

Assume that the stress field can be described by a series of terms of the form A.r^N.cos mq; then for reasons of symmetry, m must be an even integer. Also, stress magnitudes must not become infinite at r = 0, so that inside the inclusion, no term can have N negative; similarly, stress magnitudes must not become infinite at remote points () so that outside the inclusion, no term can have N positive.

To narrow the range of possibilities farther, it is convenient to look at a different term B.rⁿ.cos mq and to name it f. The convenience lies in the fact that if we put

we automatically conserve momentum and describe steady flows, with no accelerations. Specifically for f = B.rⁿ.cos mq,

where e_r and e_q are linear strain rates and g is the engineering shear strain rate (twice the tensor shear strain rate). For a Newtonian material of viscosity N,

In particular, for variation specifically with cos 2q, m = 2 and n = 4, 2, 0 or –2.

where a, b, d, f and g are coefficients yet to be determined. At once, d is fixed by the stress state as r becomes infinite: at q = 0, s_r(inf) = S so, from equation [2b], we need d = -S/2. The remaining coefficients a, b, f and g can be chosen according to whatever conditions we wish to satisfy at the interface.

The classical conditions at the interface are:

equal stresses, (s_r)_i = (s_r)_e and t_i = t_e, [7a,b]

and equal velocities (to maintain coherence),

u_i = u_e radial and v_i = v_e tangential [7c,d].

These yield

[8a,b,c,d]

where R is the ratio (viscosity)_i/(viscosity)_e. Putting these expressions into (6a) and (6b) gives the same stress functions as are derived by Muskhelishvili (1963). In particular, the value a = 0 corresponds with the notable fact that the stress field throughout the inclusion is homogeneous, and this value of a is derived specifically from the interface conditions given. Any other interface conditions are likely to yield a non-zero value for a and an inhomogeneous stress field in the inclusion.

From the main text, we would like to find interface conditions that would allow for the material diffusing away from the north and south quadrants of the interface and accumulating at the east and west quadrants. To allow the radial velocities u_i and u_e to be unequal is an obvious choice; then at the east point, for example, a gap opens at the rate u_e – u_i and we can seek a balance between this rate and the rate at which material is arriving by diffusion. But once the process of deposition at the interface gets under way, it is not at all clear how the other interface conditions (7a, b and d) would change. For present purposes, we focus attention on just the moment when deposition begins, when the deposited film or sliver is infinitesimally thin, and propose that at this moment conditions (7a, b and d) can still be applied. This is clearly just a first step toward a more realistic analysis.

With u_i and u_e unequal, we can introduce b where u_e = b.u_i and now find:

If we use these values to estimate and close to the interface and assume that the diffusivities of inclusion and matrix are also in the ratio R (K_e/K_i = R = N_i/N_e), we derive for any interface point:

The product 4K_iN_i = L² where L is a characteristic length of the material of the inclusion. Thus for example for R = 10, b = 5/4 if L = 1/6, or b = 2 if L = 1/3. Diffusion not only enables the matrix to deform more freely, it reduces the height of the stress maxima; for example, in the condition (R =10, L = 1/3 and b = 2), the radial compression at the north point is about 0.8 of its magnitude without diffusion.

For larger values of b or R, the exact form [10c] has to be used, and as R tends to infinity and u_i tends to zero,

For example, if L² = 0.2, u_e = S cos 2q/6N_e or specifically at the east point, u_e = S/6N_e. (For comparison, if we replaced a rigid inclusion by material homogeneous with the matrix, then at the east point of a circle of radius 1 we should find u_e = S/2N_e). Again, diffusion allows the matrix to deform as if the inclusion were much less viscous than it is. In fact algebraically, if we put L² = 1 into equation (13), u_e = S/2N_e as for a homogeneous material. But Appendix 2 shows that treating a material with microstructure as a continuum becomes increasingly unrealistic as L² increases above 0.1 or 0.2; it is more the qualitative trend in these results that is of value.