The macroscopic response of composites

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DMS Award AbstractAward #: 0108626PI: Milton, GraemeInstitution: University of Utah Program: Applied MathematicsProgram Manager: Catherine MavriplisTitle: The macroscopic response of compositesThis proposal aims to investigate the macroscopic response of materials, focusing on four topics. The first topic concerns creep in a two phase composite given that one knows the behavior of the constituent phases. The problem is to obtain bounds on the range of creep that might occur, and to identify microstructures having the maximum and minimum creep. The second topic concerns the complex dielectric constant of homogeneous or heterogeneous materials which governs how a material reflects, transmits and absorbs radiation. Given measurements of the complex dielectric constant over a range of frequencies, the problem is to say something definite about the response of the material over an interval of frequencies outside the measured range. The third topic, which represents a continuation of previous work, is to explore the range of values the average strain can take in a composite which is subject to a fixed average stress, as the microstructure is varied. The objective is to bound this range, and to identify optimal structures that generate strains at the boundary of the range of admissible values. Such structures should be useful as ``stress guides'' for channeling stress to desired locations. The particular question of what microstructures make the best possible hydrostatic compression to shear converters will be investigated. The fourth topic is to explore the properties of a rather exotic class of microstructures, called partial differential microstructures. These may turn out to be the best microstructures for solving certain optimal design problems. A better understanding of the macroscopic response of materials is of central technological importance. This importance stretches across the board, from understanding the macroscopic response of engineered materials (of critical importance to the defense, automotive, and aerospace industries), to understanding the macroscopic response of polycrystalline and porous rocks (relevant to earthquake prediction and to the oil industry), to understanding the macroscopic response of sea ice (important to climate modeling), to understanding the macroscopic response of biological materials (such as tissues, bones, shells and tendons). This proposal will enhance our understanding in ways that will (1) facilitate the development of new materials having unusual properties that are achieved by tailoring the microstructure; (2) provide limits on the response of composite materials, that could be essential for assessing the safety of such materials in desired applications; and (3) predict certain aspects of the response of materials to radiation outside frequencies where the response has been measured, which could be important if the material is biological and if one wants to know if radiation could be harmful outside frequencies where its effects are known.Date: May 30, 2001