Mathematical Sciences: Microstructure and Macroscopic Behavior

小行星9404376 相干相变导致具有特征精细尺度结构的不同相或相变体的混合物。这些微观结构的建模对于理解宏观现象（如形状记忆行为和滞后）以及微观现象（如孪晶形态）至关重要。在某些情况下，所观察到的精细尺度结构和宏观效应可以解释的基础上的弹性能量最小化。从数学上讲，问题的实质是具有“多阱结构”的非凸弹性能的最小化。这个项目带来承担各种工具，其中许多是相对较新的，包括放松变分问题，翻译方法，杨措施限制，H措施，奇异摄动。能量最小化的微结构可被视为复合材料 具有极端有效行为，因此均匀化方法和复合材料分析也是相关的。 该项目的目标包括开发新的数学工具，并将这些工具应用于材料科学的特定问题。一个目标是了解为什么一些形状记忆合金保持其形状记忆行为的多晶形式，而其他人没有。 另一个是解释 实验观察， 结对 以及CuAlNi单晶中的磁滞现象。

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.