Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hill's formulae for the effective bulk modulus of a two-phase mixture when the phases have equal shear moduli, Levin's formulae linking the effective thermal expansion coefficient and effective bulk modulus of two-phase mixtures, and Dykhne's result for the effective conductivity of an isotropic two-dimensional polycrystalline material. 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 matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three-dimensional polycrystalline materials. We present complete lists of exact relations for three-dimensional thermoelectricity and for three-dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases.