Limit Process Expansions and Homogenization

The use of limit process expansions and asymptotic matching is illustrated for a simple problem of homogenization. Rayleigh's problem of calculating the effective thermal conductivity keffof a regular cubic array of spheres of conductivity kIin a matrix of conductivity kMis considered for small volume fractions fI. An inner expansion is found near each sphere. This is matched to an outer expansion in which the sphere shrinks to a singularity. Rayleigh's result that the expansion for keffcontains terms fI, f2 I, f3 I, f4 I, f13/3is reconstructed. In addition an independent verification of Rayleigh's summability method for an infinite array of dipoles is given. Comments are made on related problems and the random case where similar summability difficulties appear.