Collaborative Research: Mathematical Sciences; Transitions and Defects in Ordered Materials

950578 Kinderlehrer这个跨学科小组将在分析和计算中研究非线性材料行为和相关问题。它的重点是经历马氏体相变和磁性相变的材料，特别是在形状记忆和超磁致伸缩材料方面的应用。作者正在开发的理论和计算方法将应用于新的驱动器材料和复合材料的设计。作者建议设计一种具有强双向形状记忆效应的复合材料，设计具有低滞后的小尺寸、大作用力的致动器，并制定寻找磁记忆材料的策略，这种材料结合了形状记忆(大应变)和磁致伸缩(由磁场驱动)材料的最有用的方面。这些现象的定量描述依赖于非线性分析的框架，特别是在非凸势的背景下，其平衡在大范围变化的长度尺度上是高度振荡的。将开发创新的数值方法和代码来计算复杂微结构的动力学，以响应各种环境变化。活性材料，即其结构在受到热或电磁场等外部刺激时发生剧烈变化的材料，在从航空航天到通信再到医学的各个技术领域都具有巨大的用作致动器和传感器的潜力。该项目致力于对这些材料的改进和优化以及发现新材料的基础研究。它们的微观结构和宏观性质之间的相互作用是其开发的基础，对这些性质的研究导致了非线性理论、非线性分析和科学计算的前沿。对研究生和博士后助理进行跨学科培训是该项目不可或缺的一部分。

950578 Kinderlehrer This interdisciplinary group will investigate nonlinear material behavior and related issues in analysis and computation. Its focus is materials which undergo martensitic and magnetic phase transitions, with particular application to shape memory and giant magnetostrictive materials. The theories and computational methods under development by the authors will be applied to the design of new actuator materials and composites. The authors propose to design a composite that exhibits a strong two way shape memory effect, design small scale, large-force actuators with low hysteresis, and formulate a strategy for finding a magneto-memory material, a material which would combine the most useful aspects of a shape memory (large strain) and magnetostrictive (actuated by a magnetic field) material. The quantitative description of these phenomena rests in the framework of nonlinear analysis, in particular within the context of non-convex potentials whose equilibria are highly oscillatory across widely varying length scales. Innovative numerical methods and codes will be developed to compute the dynamics of complex microstructure in response to a variety of environmental changes. Active materials, that is, materials whose structure transforms dramatically when subjected to external stimuli like heat or electromagnetic fields, have enormous potential for use as actuators and sensors across the spectrum of technology, from aerospace to communications to medicine. This project addresses basic research directed toward the improvement and optimization of these materials and the discovery of new ones. The interplay between their microstructure and macroscopic properties is fundamental to their exploitation and the study of these properties leads to the frontiers of nonlinear theory, nonlinear analysis, and scientific computation. The interdisciplinary training of graduate students and postdoctoral associates is an integral part of this project.