Mathematical Sciences: Microstructure and Macroscopic Behavior

Kohn相干相变导致具有特征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，观测到的细尺度结构和宏观效应可以用弹性能量最小化来解释。从数学上讲，问题的本质是“多井”结构下非凸弹性能量的最小化。这个项目带来了各种各样的工具，其中许多是相对较新的，包括变分问题的松弛、平移方法、杨测度极限、h测度和奇异摄动。能量最小化微结构可以看作是具有极端有效行为的复合材料，因此均质化和复合材料分析的方法也是相关的。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具体问题。一个目标是理解为什么一些形状记忆合金在多晶形式下保持其形状记忆行为，而另一些则不能。另一个是解释CuAlNi单晶孪晶和迟滞的实验观察。

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.