This work will improve our understanding of creep in composite materials, through establishing bounds on the total creep and identifying microstructures that exhibit the maximum or minimum possible creep. It will also identify microstructures which are best for guiding stress in linear elasticity. These composites should be useful for distributing or concentrating the force in a structure. Additionally, microstructures will be identified with an anomalous Hall coefficient, which could potentially be useful in reducing stray electric fields in circuit design when there is a magnetic field present. In a second part of the work three fundamental problems will be studied: characterizing the surfaces in tensor space associated with sets of effective tensors of hierarchical laminate geometries; exploring the connection between rank-one convexity and quasiconvexity; and developing new tools for bounding the effective moduli of composites and for bounding the quasiconvexification of a function. These problems are important for making progress on the question of what responses of composites are or are not possible. Their solution could, in a wide variety of cases, make numerically tractable the search for the possible macroscopic responses of composites.
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 defence, automative, 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 modelling), to understanding the macroscopic response of biological materials (such as tissues, bones, shells and tendons). In the past many useful composites were obtained by trial and error, or by mimicking composites found in nature, or by using intuition. In this century we will have greater flexibility to produce designer composites tailor made to meet specific needs. Improving our understanding of the response of composites will facilitate the construction of these designer composites.
