Thermoelectric materials have been used for cooling and heating systems for over a hundred years. Today practical applications of thermoelectric devices include cooling car seats, power generation, and refrigeration. Thermoelectric materials are special for their ability to convert temperature imbalances into electricity. Their applications can inform the discourse about the transition to renewable energy sources---something that our Earth most desperately needs. The goal of this dissertation is to describe how the effective tensors of two-dimensional thermoelectric composites made from two isotropic materials depend on thermoelectric parameters of the constituents. Using the theory of exact relations and links developed by Grabovsky and his collaborators, we describe all equations satisfied by the thermoelectric effective tensor of a composite without the explicit knowledge of its microstructure. In some special cases, the effective tensor can be determined completely. Even in the general case, four out of 10 components of the two-dimensional thermoelectric tensor can be expressed in terms of the remaining 6, regardless of the microstructure. We started with special cases and worked our way up to the more general ones. / Mathematics

Predicting the effective elastic properties of a composite material based on the elastic properties of the constituent materials is extremely difficult, even when the microstructure is known. However, there are cases where certain properties in constituents always carry over to a composite, regardless of the microstructure of the composite. We call such instances exact relations. The general theory of exact relations allows us to find all of these instances in a wide variety of contexts including elasticity, conductivity, and piezoelectricity. We combine this theory with ideas from representation theory to find all exact relations for fiber-reinforced polycrystalline composites. We further extend these ideas to the concept of links. When two composites have the same microstructure but different constituent materials, their effective tensors may be related. We use the theory of exact relations to find such relations, which we call links. In this work we describe a special set of links between elasticity tensors of fiber-reinforced polycrystalline composites. These links allow us to generalize certain results from specific examples to generate new information about this widely-used class of composites. In particular, we apply the link to obtain information about composites made from two transversely isotropic materials and polycrystals made from one orthotropic material. / Mathematics