Superscript 5 Baseline zero width space zero width space plus left-parenthesis 2 plus 2 nu Subscript m Baseline right-parenthesis upper D Subscript m Baseline left-parenthesis a slash r right-parenthesis cubed right-bracket sine theta cosine 2 phi comma EndLayout right-brace midline-horizontal-ellipsis a less-than r less-than infinity period"/>(3.38)
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
and it can then be shown that
As Cp=0, it follows from (3.35) and (3.36) that both the strain and stress distributions in the particle are uniform.
3.4.2 Application of Maxwell’s Methodology
To apply Maxwell’s methodology to a cluster of N particles embedded in an infinite matrix, the stress distribution in the matrix at large distances from the cluster is considered. The perturbing effect in the matrix at large distances from the cluster of particles is estimated by superimposing the perturbations caused by each particle, regarded as being isolated, and regarding all particles to be located at the origin. The properties of particles of type i will again be denoted by a superscript (i).
The stress distribution at very large distances from the cluster is then given by the following generalisation of relations (3.38)
where from (3.40), for i = 1, …, N,
For the isolated sphere of radius b having the effective properties of the particulate composite cluster as illustrated in Figure 3.1(b), it follows that the stress field in the matrix at large distances is described exactly by relations of the type (3.38) where the coefficient Dm is replaced by D¯m having the value determined by the relation