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

小行星950578 这个跨学科的小组将研究非线性材料的行为和相关问题的分析和计算。其重点是经历马氏体和磁性相变的材料，特别是形状记忆和超磁致伸缩材料。作者正在开发的理论和计算方法将应用于新的致动器材料和复合材料的设计。作者建议设计一种复合材料，表现出很强的双向形状记忆效应，设计小规模，大的力与低磁滞致动器，并制定了一项战略，寻找磁记忆材料，材料将结合联合收割机的形状记忆（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在于非线性分析的框架内，特别是在非凸电位的上下文中，其平衡是高度振荡跨越广泛变化的长度尺度。将开发创新的数值方法和代码，以计算复杂微结构的动态响应各种环境变化。 活性材料，即当受到热或电磁场等外部刺激时其结构发生显著变化的材料，具有巨大的潜力，可用作从航空航天到通信到医学的各种技术的致动器和传感器。该项目致力于基础研究，旨在改进和优化这些材料并发现新材料。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。研究生和博士后助理的跨学科培训是该项目的一个组成部分。

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.