Mathematical Sciences: Nonlocal Equations Modeling Fine- scale Structures in Solids

马氏体相变是一种产生形状变化和晶体对称性变化的相变。形状记忆材料是在马氏体相中具有极强延展性的材料，但当加热到高于转变温度时，会恢复到“记忆”的原始形状。结晶固体的相干相变导致具有特征精细结构的不同相或相变体的混合物。本项目主要研究的问题是特征尺度、微观结构的产生和扩展。提出了一种涉及积分-微分进化方程的非定域理论来回答这些问题。该理论介于传统的微观量子理论和宏观弹性理论之间，前者采用的是薛定谔方程，后者采用的是椭圆型和双曲型偏微分方程。在特征尺度的研究中，需要一个平稳非局部方程的周期解。但这个等式似乎有多个解。“正确的”周期解应根据合适的最小能量原理来选择。另一个有趣的问题是找到描述微观结构传播的动态非局部方程的“广义”行波解。该解的一端应具有振荡结构，另一端应具有恒定值，并且随着时间的增加，均匀到振荡的过渡区域应以周期性的方式推进。这样的解也可以看作是无限维函数空间中的异斜轨道，它连接了一个周期平稳解和一个常数平稳解。材料和加工对航空航天、汽车、生物材料、化学、电子、能源、金属和电信等行业的成功至关重要。该项目涉及形状记忆材料和马氏体转化。本项目的数学理论旨在理解形状记忆材料相变过程中微观结构的特征尺度、产生和传播。非局域数学理论中使用的积分-微分演化方程的平稳周期解和均匀-振荡波的研究处于数学分析和计算的前沿。

Ren 9703727 Martensite transformations are phase transformations that produce a change of shape and a change of crystal symmetry. Shape-memory materials are materials that are extremely malleable in the martensite phase below a transformation temperature, but that return to a `remembered' original shape when heated above the transformation temperature. Coherent phase transitions of crystalline solids lead to mixtures of distinct phases or phase variants with characteristic fine-scale structures. The main issues addressed in this project are the characteristic scales, the generation and the propagation of the microstructures. A nonlocal theory involving integro-differential evolutionary equations is proposed to answer these questions. This theory lies between the traditional microscopic quantum theory which uses Schodinger's equation and the macroscopic elasticity theory which uses elliptic and hyperbolic partial differential equations. In the study of the characteristic scales, a periodic solution of a stationary nonlocal equation is needed. But the equation seems to have multiple solutions. The `right' periodic solution should be selected by a suitable least energy principle. Another intriguing question is to find a `generalized' traveling wave solution of a dynamic nonlocal equation, which describes the propagation of microstructures. This solution should have an oscillatory structure at one end, constant value at the other end, and as time increases the uniform-to-oscillatory transition region should advance in a periodic manner. Such a solution can also be viewed as a heteroclinic orbit in an infinitely dimensional function space connecting a periodic stationary solution to a constant stationary solution. Materials and processing are critical to the success of industries such as the aerospace, automotive, biomaterials, chemical, electronics, energy, metals, and telecommunications industries. This project is concerned with shape memory materials and martensite transformations. The mathematical theory in this project is aimed at understanding the characteristic scales, the generation and the propagation of the microstructures during shape memory materials' phase transformations. The study of stationary periodic solutions ald uniform-to-oscillatory waves of the integro-differential evolutionary equations used in this nonlocal mathematical theory is at the cutting edge of mathematical analysis and computation.