Mathematical Sciences: Microstructure and Macroscopic Behavior

数学科学：微观结构和宏观行为

• 批准号: 9402763
• 负责人: Robert Kohn
• 金额: $ 125万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9402763&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Microstructure; Macroscopic; Behavior

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.

9404376科恩相干相变导致不同相或相变体的混合物具有特征的精细尺度结构。这些微观结构的模拟对于理解形状记忆行为和滞后等宏观现象以及孪晶形态等微观现象至关重要。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最小化来解释。从数学上讲，问题的实质是具有“多井结构”的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，因此从均匀化和分析复合材料的方法也是相关的。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中的特定问题。一个目标是了解为什么一些形状记忆合金保持其多晶形式的形状记忆行为，而另一些则不是。另一种是解释关于CuAlNi单晶中孪晶和磁滞的实验观察。