Algebra, Geometry, and Computations of Exact Relations for Effective Moduli of Composites
In this paper we will review and extend the results of , which covered the case of 3D thermopiezoelectric polycrystals. In that context the settings of conductivity, elasticity, pyroelectricity, piezoelectricity, thermoelectricity, and thermoelasticity 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 . For example, the Hall effect for conduction in a weak magnetic field is described by a nonsymmetric conductivity tensor. We explain the step-by-step process of finding all exact relations for the simple example of the 2D Hall effect. The paper concludes with a discussion of new algebraic and geometric questions posed by the theory of exact relations.
KeywordsHall Effect Jordan Algebra Weak Magnetic Field Hall Coefficient Conductivity Tensor
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