Mathematical Sciences: Mathematical Problems from Materials Sciences

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9404376 Kohn Coherent phase transformations lead to mixtures of different phases or phase variants with characteristic fine scale structures. The modelling of these microstructures is crucial for understanding macroscopic phenomena such as shape-memory behavior and hysteresis, as well as microscopic phenomena such as the morphology of twinning. In some situations the observed fine scale structures and macroscopic effects can be explained on the basis of elastic energy minimization. Mathematically, the essence of the matter is the minimization of nonconvex elastic energies with "multi-well structure. This project brings to bear a variety of tools, many of them relatively new, including relaxation of variational problems, the translation method, Young measure limits, H-measures, and singular perturbations. Energy minimizing microstructures can be viewed as composite materials with extremal effective behavior, so methods from homogenization and the analysis of composite materials are also relevant. The goals of this project include developing new mathematical tools, and also applying these tools to specific problems from materials science. One objective is to understand why some shape-memory alloys maintain their shape-memory behavior in polycrystalline form while others do not. Another is to explain experimental observations concerning twinning and hysteresis in single crystals of CuAlNi.