Towards a Complete Theory of Exact Relations

Louisiana State University, Baton Rouge LA

Investigators

Abstract

Sage DMS-0606300 Typically, physical properties of composite materials are strongly dependent on microstructure. However, in exceptional situations, exact relations exist that are microstructure-independent. These express fundamental invariances in a given physical setting. Exact relations have been extensively studied, but the classical approach has been heavily dependent on the physical context. In the late 1990's, an abstract theory of exact relations was originated by Grabovsky and greatly extended by Grabovsky, Milton, and the investigator. It has proved to be enormously powerful. Indeed, by reducing the search for exact relations to a purely algebraic problem involving group representation theory, it has led to complete lists of all rotationally invariant exact relations for three-dimensional thermopiezoelectric composites, which include all exact relations for elasticity, thermoelasticity, and piezoelectricity as special cases. This new approach has been responsible for a great leap forward in this area of material science. However, the theory is by no means complete. The purpose of this project is to complete the theory of exact relations by addressing the three remaining major open questions. First, previous work has given complete lists of exact relations only in relatively simple physical contexts. The investigator studies highly coupled problems with the ultimate goal of finding an explicit parameterization of exact relations for the physical problem with any number of temperature, electric, and elastic fields. The second project is the exploration of the relationship between lamination and homogenization for exact relations. The basic question is whether there exist exact relations that are stable under lamination, but not under homogenization. Third, the investigator undertakes a detailed study of the algebraic structures associated with exact relations. In particular, questions about exact relations may be reformulated in terms of algebraic objects called Jordan algebras. This approach seems very promising, and it is considered to be crucial for progress on the two problems described previously. Composite materials are everywhere in the modern world. They are used in the manufacture of products ranging from skis to airplanes and from tennis rackets to cell phones. It is accordingly of great technological importance to understand how physical properties (such as conductivity and elasticity) of a composite are related to the properties of its constituents. This is difficult in general because the way in which the constituents are put together strongly influences the end result. For example, take two materials, one soft and one hard. If the hard material is embedded in the soft substance, the composite will be compressible. On the other hand, if the soft material lies in a matrix of the hard material, the composite will be rigid. However, in certain situations, this typical variability is greatly reduced. The goal of this project is to understand and classify these situations. From a practical point of view, the project establishes "impossibility theorems" in engineering, i.e. results showing that a composite with desired properties cannot be constructed from a given set of starting materials. The proposal thus has significant implications for industrial research and development.