Dynamic and Nonlinear Static Problems in Periodic and Random Composites

This research deals with problems arising in the mathematical modeling of strongly heterogeneous materials, primarily composites. A composite is a mixture of several different constituent materials or phases so that it combines the most useful features of each. For example, a high thermal conductivity composite consists of ceramic filler beads in a polymeric matrix. The goal is to maximize thermal conductivity. Ceramic is a very good thermal conductor while the polymer is not. It is not practical to use pure ceramic since it is too brittle. However, the polymer/ceramic composite is a good thermal conductor and it has desirable mechanical properties. The problem in this context is to optimize strength and thermal conductivity by the correct selection of the filler identity, size, shape, and the random size distribution. Tools from the mathematical theory of homogenization will be used to attack these questions. The three problem areas of the proposed work are the optimization of dielectric and mechanical properties of epoxy/ceramic composites, with applications to the design of capacitors; an investigation of dynamical problems and frequency dependent effects in transducer materials, with applications to sensors and transmitters of acoustic signals; and a study of problems from superconductivity, superfluidity, and liquid crystals, with the goal of understanding mathematical issues caused by the presence of vortices, nonlinearity, and nonstandard boundary conditions. The planned work lies at the frontier between mathematics and materials science, and close collaborations with materials scientists from universities and private industry will be carried out. Mathematics can help to develop new materials with superior properties for various industrials needs, verify the reliability of experimental data, and devuise efficient ways to compute properties of designer composites. The results of the work in the first of the three areas (epoxy/ceramic composites) will provide guidance for future directions in manufacturing and materials development in various important industrial applications. Two typical examples are the design of new 'packages' for integrated circuits which remove heat from the electronics more efficiently and significant enhancement in functionality of capacitors which are used in various electronics products. The planned work on transducer materials has immediate applications in underwater sonar and medical imaging. The work in the third area (superconductivity) will help to clarify which of the existing physical models provide the best way of calculating effective properties of superconducting thin films with a large number of vortices, and it will lead to better models for these phenomena.