respectively, such that τ=2μγ where μ is the shear modulus of an isotropic material. The principal values of the stress field are along (tension) and perpendicular to (compression) the line x2=x1.
A single spherical particle of radius a is now placed in, and perfectly bonded to, an infinite matrix, where the origin of spherical polar coordinates (r, θ, ϕ) is taken at the centre of the particle. The system is then subject only to a shear stress applied at infinity. At the particle/matrix interface the following perfect bonding boundary conditions must be satisfied:
A displacement field equivalent to that used by Hashin [5], based on the analysis of Love [8, Equations (5)–(7)] that leads to a stress field satisfying the equilibrium equations and the stress-strain relations (3.15) with ΔT=0, can be used to solve the embedded isolated sphere problem (see Appendix A). The displacement and stress fields in the particle are bounded at r = 0 so that
In the matrix the displacement field and stress field (stresses bounded as r→∞) have the form
The representation is identical in form to that used by Christensen and Lo [9] although they used a definition of ϕ that differs from that used here by an angle of π/4. This difference has no effect on the approach to be followed. It follows from (3.35)–(3.38) that the continuity conditions (3.34) are satisfied if the following four independent relations are satisfied
