Stability and macroscopic properties of heterogeneous media

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GrabovskyDMS-1008092 The investigator studies problems related to compositematerials, martensitic phase transformations, and morphologicalstability in materials. First, he considers exact relations andlinks for effective tensors of fiber-reinforced elasticcomposites, by applying the general theory developed by him andhis collaborators. The general theory provides a strategy forfinding every relation and link. However, the actual executionof that strategy is far from trivial. The case offiber-reinforced elastic composites is both important forapplications and incredibly challenging technically, because themicrostructure is two-dimensional while the properties of thecomposite are represented by three-dimensional fully anisotropicelastic tensors. This topic builds on the successful solution ofa similar, simpler problem in the context of Hall-effectconductivity. A second problem concerns morphological stabilityof phase boundaries in materials capable of undergoingmartensitic phase transformations. The investigator studies thepossibility of a configuration in which the only mode ofinstability is the global motion of the phase boundary. Mathematically this corresponds to the failure of the generalizedsecond variation to stay positive. The difficulty is that allother stability criteria are required to be satisfied. Theresolution of this issue opens the way to a completeunderstanding of all instabilities of configurations with smoothphase boundaries. In the context of periodic composites,morphological instabilities may occur locally within a periodcell. Their interaction with the effective behavior of acomposite is investigated, following prior work for structuraland material instabilities. Heterogeneous media, or media with internal structure, areof great importance in applications. Composite materials, whichhave by now become ubiquitous, are one example. Another exampleis shape memory alloys and other smart materials. These twogroups of materials are very different, yet have some importantcommon features. A well-known connection between the twotheories can be used to shed light on the less well-understoodtheory of martensitic phase transformations responsible for muchof the shape memory behavior. One aim of this project is tounderstand elastic properties of fiber-reinforced elasticcomposites. Such composites may combine two cheap but imperfectmaterials to produce a new material that is either rare or notfound in nature at all. For example, modern skis use compositesto create a material that bends easily but is incredibly stiffwith respect to torsion. However, there are limitations to whatcan be achieved by composites. The investigator maps out thoseinstances where the benefits of creating composites aredramatically reduced due to the presence of "exact relations,"i.e. relations between material moduli that cannot be altered bymaking composites, no matter how hard one tries. The presentwork focuses on a very widely used class of fiber-reinforcedcomposites. Another aim of the project is to understandinstabilities in models of non-linear elasticity and shape memoryalloys. One widely known type of elastic instability isbuckling. Martensitic phase transitions responsible for shapememory behavior are another kind of instability. These examplesshow how important instabilities are in applications. Theinvestigator classifies all possible instabilities in asystematic way, with the secondary goal of solving one of thelong-standing mathematical problems: finding "correct" sufficientconditions for stability.