Macroscopic Properties of Heterogeneous Media and Development of the Applied Mathematics Curriculum

Temple University, Philadelphia PA

Investigators

Abstract

NSF Award Abstract - DMS-0094089 Mathematical Sciences: CAREER: Macroscopic properties of heterogeneous media and development of the applied mathematics curriculum Abstract 0094089 Grabovsky The unifying idea of this project's research is the general Hilbert space approach to homogenization due to Milton. There are three major issues based on this approach that will be investigated. One concerns the exploration of the convexity properties of G-closures with the goal of developing new numerical approximations to the elusive set. This approach may lead to the new analytic results in several basic cases, where the G-closure is still not known. The second issue is to make use of the new formula for effective tensors in terms of the W-transformation of Milton for studying effective behavior of the random media. That formula provides a new series expansion for the fields and effective tensors of the random composites. The new expansion is shown to converge rapidly even for a relatively high contrast media. Finally, we will develop a new and exciting idea of space-time composites proposed recently by Lurie. The space time composites refer to composites whose electromagnetic properties change rapidly in time. These temporal oscillations may be a result of high frequency vibrations of the composite specimen or of an active nature of the material itself. This project's research is centered on the investigation of properties of complex and smart materials. Examples of such materials include composites (used in spacecraft, airplanes, cars, skies, golf clubs, etc.), random media like rock, clay or bone, novel active materials that are used in sensors and actuators. This grant provides funding for a variety of research and educational activities that includes a small elasticity demonstration lab. The main purpose of the lab will be to heighten students' interest in mathematics through the inclusion of experiments and demonstrations. The lab will also facilitate involvement of undergraduates in mathematical research and interdisciplinary interaction throughout the College of Science and Technology at Temple University. The grant will also support a multi-prong applied research program unified by the common mathematical tools used in this research. One of the goals of this project is the prediction of properties of composite materials. Success in this direction may make it possible to reduce or eliminate expensive and time-consuming measurements of elastic properties of composite materials by providing good theoretical predictions. Another application is to the study of high-contrast disordered media, of interest in hydrology for prediction of the permeability of large volumes of sedimentary rock based on local measurements. The permeability is a crucial property of soil that determines how ground water, oil or pollutants propagate underground. Yet another application is to the novel field of space-time composites: composite materials whose components are in rapid relative motion or whose components have the ability to change their properties rapidly in time without mechanical motion. The success of this part of the project will be essential for designing and understanding the new generation of smart materials that are able to respond quickly to a changing environment.