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theta plus left-parenthesis StartFraction 3 kappa Subscript m Baseline Over kappa Subscript eff Baseline plus 2 kappa Subscript m Baseline EndFraction minus 1 right-parenthesis StartFraction alpha b cubed cosine theta Over r squared EndFraction comma b less-than r less-than infinity period"/>(3.9)

      If the cluster in Figure 3.1(a) is represented accurately by the effective medium shown in Figure 3.1(b), then, at large distances from the cluster, the temperature distributions (3.8) and (3.9) should be identical, leading to the fourth step, where the perturbation terms in relations (3.8) and (3.9) are equated, so that

(3.10)

      which is a ‘mixtures’ relation for the quantity 1/(κ+2κm). On using (3.1), the effective thermal conductivity may be estimated using

(3.11)

      For multiphase composites, Hashin and Shtrikman [5, Equations (3.21)–(3.23)] derived bounds for magnetic permeability, pointing out that they are analogous to bounds for effective thermal conductivity. Their conductivity bounds may be expressed in the following simpler form, having the same structure as the result (3.10) derived using Maxwell’s methodology

(3.12)

      where κmin is the lowest value of conductivities for all phases, whereas κmax is the highest value.

      3.3 Bulk Modulus and Thermal Expansion Coefficient

      3.3.1 Spherical Particle Embedded in Infinite Matrix Subject to Pressure and Thermal Loading

Consider an isolated particle of radius a perfectly bonded to an infinite surrounding matrix, subject to a pressure p applied at infinity and a uniform temperature change ΔT from the stress-free temperature at which the stresses and strains in particle and matrix are zero. The displacement field in the particle and surrounding matrix is purely radial so that displacement components (ur,uθ,uϕ), referred to a set of spherical polar coordinates (r, θ, ϕ) with origin at the particle centre, are of the form

      (3.13)

      The corresponding strain field obtained from (2.142) is then given by

      (3.14)

      The stress field follows from stress-strain relations expressed in the form (see (2.160) for the Cartesian equivalent)

(3.15)

      where λ and μ are Lamé’s constants and where α is now the linear coefficient of thermal expansion. On using the equilibrium equations (2.130)–(2.133), it can be shown that, within the spherical particle of radius a, the resulting bounded displacement and stress fields are given by

      (3.16)

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