This section presents a sampling of general techniques for using solutions of the conductivity or elasticity equations in a given microgeometry to generate associated solutions in related microgeometries. In addition to the methods discussed here, it should be mentioned that conformal transformations are useful for finding exact solutions for the fields, in particular, two-dimensional geometries: See Berdichevskii (1985), who found the exact solution for the fields in a regular checkerboard geometry; Obnosov (1996, 1999), who obtained the explicit solution for rectangular and triangular checkerboards, as well as for certain rectangular arrays of rectangles at a volume fraction of 1/4; the many papers of Vigdergauz referred to in section 23.9 on page 481, who found various periodic microstructures where the field was constant within one phase; and Reuben, Smith, and Radchik (1995).

Modifying the material moduli so the field is not disturbed

Consider the formula (7.13) for the effective bulk modulus k* of the coated sphere assemblage. It does not depend on the shear modulus µ1 of the core material. The physical reason for this is quite clear: Under an externally applied hydrostatic loading the displacement field (7.11) in the core of each coated sphere is a pure dilation with no shear component, and so the effective bulk modulus is not influenced by the shear modulus of phase 1. This is an example of a more general principle, which says that if we modify the material moduli in any way that leaves the field undisturbed, then the response of the composite to that average field will remain unchanged.