Exact relations for composites: Towards a complete solution

Abstract

Typically, the electrical and elastic properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for (or linking) effective tensor elements that are universally valid no matter what the microstructure is. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for tensor subspaces satisfying certain algebraic conditions. One new exact relation is for the effective shear modulus of a class of three-dimensional polycrystalline materials.