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

任9703727马氏体相变是产生形状变化和晶体对称性变化的相变。形状记忆材料是在相变温度以下的马氏体相中具有极强延展性的材料，但当加热到相变温度以上时，这些材料会恢复到“记忆”的原始形状。晶体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。本项目所涉及的主要问题是微结构的特征尺度、生成和传播。提出了一种涉及积分-微分发展方程的非局部理论来回答这些问题。该理论介于使用肖丁格方程的传统微观量子理论和使用椭圆型和双曲型偏微分方程组的宏观弹性理论之间。在特征尺度的研究中，需要一个平稳的非局部方程的周期解。但这个方程式似乎有多个解。应根据适当的最小能量原则选择“正确的”周期解。另一个有趣的问题是找到一个描述微结构传播的动态非局部方程的“广义”行波解。这种解的一端应该具有振荡结构，另一端应该是恒定值，并且随着时间的增加，均匀到振荡的过渡区应该以周期性的方式前进。这样的解也可以看作是无穷维函数空间中连接周期定常解和常定定解的异宿轨道。材料和加工对航空航天、汽车、生物材料、化工、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相变。该项目的数学理论旨在了解形状记忆材料相变过程中微结构的特征尺度、生成和传播过程。这一非局部数学理论中所使用的积分-微分发展方程的定常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.