Stability and macroscopic properties of heterogeneous media

Temple University, Philadelphia PA

Investigators

Abstract

Grabovsky DMS-1008092 The investigator studies problems related to composite materials, martensitic phase transformations, and morphological stability in materials. First, he considers exact relations and links for effective tensors of fiber-reinforced elastic composites, by applying the general theory developed by him and his collaborators. The general theory provides a strategy for finding every relation and link. However, the actual execution of that strategy is far from trivial. The case of fiber-reinforced elastic composites is both important for applications and incredibly challenging technically, because the microstructure is two-dimensional while the properties of the composite are represented by three-dimensional fully anisotropic elastic tensors. This topic builds on the successful solution of a similar, simpler problem in the context of Hall-effect conductivity. A second problem concerns morphological stability of phase boundaries in materials capable of undergoing martensitic phase transformations. The investigator studies the possibility of a configuration in which the only mode of instability is the global motion of the phase boundary. Mathematically this corresponds to the failure of the generalized second variation to stay positive. The difficulty is that all other stability criteria are required to be satisfied. The resolution of this issue opens the way to a complete understanding of all instabilities of configurations with smooth phase boundaries. In the context of periodic composites, morphological instabilities may occur locally within a period cell. Their interaction with the effective behavior of a composite is investigated, following prior work for structural and material instabilities. Heterogeneous media, or media with internal structure, are of great importance in applications. Composite materials, which have by now become ubiquitous, are one example. Another example is shape memory alloys and other smart materials. These two groups of materials are very different, yet have some important common features. A well-known connection between the two theories can be used to shed light on the less well-understood theory of martensitic phase transformations responsible for much of the shape memory behavior. One aim of this project is to understand elastic properties of fiber-reinforced elastic composites. Such composites may combine two cheap but imperfect materials to produce a new material that is either rare or not found in nature at all. For example, modern skis use composites to create a material that bends easily but is incredibly stiff with respect to torsion. However, there are limitations to what can be achieved by composites. The investigator maps out those instances where the benefits of creating composites are dramatically reduced due to the presence of "exact relations," i.e. relations between material moduli that cannot be altered by making composites, no matter how hard one tries. The present work focuses on a very widely used class of fiber-reinforced composites. Another aim of the project is to understand instabilities in models of non-linear elasticity and shape memory alloys. One widely known type of elastic instability is buckling. Martensitic phase transitions responsible for shape memory behavior are another kind of instability. These examples show how important instabilities are in applications. The investigator classifies all possible instabilities in a systematic way, with the secondary goal of solving one of the long-standing mathematical problems: finding "correct" sufficient conditions for stability.