Exact relations for e ective tensors of composites : Necessaryconditions and su cient conditions

Typically, the elastic and electrical properties of composite materials are strongly microstruc-ture dependent. So it comes as a nice surprise to come across exact formulae for eeective 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 be also regarded as fundamental invariances existing in a given physical context. Classic examples include, Hill's formulae for the eeective bulk mod-ulus of a two-phase mixture when the phases have equal shear moduli, Levin's formulae linking the eeective thermal expansion coeecient and eeective bulk modulus of two-phase mixtures, and Dykhne's result for the eeective 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 eeective shear modulus of a class of three-dimensional polycrystalline materials. We present complete lists of exact relations for 3D thermo-electricity and for 3D thermo-piezo-electric composites which includes all exact relations for elasticity, thermo-elasticity and piezo-electricity as particular cases.