Algebra, geometry and computations of exact relations for effective moduli of composites. Contribution to a Birkhauser volume “Advances in Multifield Theories of Continua with Substructure” edited by Capriz and Mariano

In this paper we will review and extend the results of [21], which covered the case of 3D thermo-piezoelectric polycrystals. In that context the settings of conductivity, elasticity, pyroelectricity, piezoelectricity, thermo-electricity and thermo-elasticity can be viewed as particular cases. We will consider a class of composites more general than polycrystals, where the set of allowable materials is not constrained in any way. In addition, the tensors of material properties are not assumed to be symmetric—an assumption we made in [21]. For example, the Hall effect for conduction in a weak magnetic field is described by a non-symmetric conductivity tensor. We explain the step-by-step process of finding all exact relations for the simple example of 2D Hall effect. The paper concludes with a discussion of new algebraic and geometric questions posed by the theory of exact relations.