CAREER: Lattice models of martensitic phase transitions

经历马氏体相变的材料的一个特征是在循环载荷作用下表现出的滞后现象，这一过程涉及到精细层状微观结构的形成和演化，而这些微观结构的形成和演化在很大程度上取决于载荷的方向和历史.能量耗散的起源、相形核和相边界动力学是马氏体和其他活性材料模拟中重要的开放性问题。该研究计划旨在从介观和微观框架的角度推进对这些现象的理解。它将集中在一系列的原型晶格模型的日益复杂的捕获相成核，界面动力学和相关的滞后的基本特征的目标。该模型有两个主要组成部分：（i）最近邻之间的非凸相互作用，允许存在两个或多个稳定的均匀状态和（ii）长程相互作用。在这项研究的成果将是一个界面动力学定律的推导和预测的形核微观结构模式。这些结果将与实验观察结果进行比较。特别是，拟议的工作将宏观观察，如尺寸和其他功能的磁滞回线的材料的微观特性，这项研究的结果将是重要的新兴民用，航空航天和工业应用，需要显着的passivedamping马氏体。这些包括复合结构的损伤和振动控制，以及建筑物和桥梁中地震和风引起的振动的衰减。这项工作将影响许多其他科学领域，其中遇到类似的离散系统，包括位错理论，断裂力学，生物学，图像识别和数值分析。该计划的更广泛的影响也将通过跨学科研究计划和各种教育和推广活动中的研究生和本科生的培训来实现。这些措施包括开发新的课程，指导女研究生和运行初中和高中学生的方案。这些活动将促进数学和科学一般作为一个有趣的智力追求和未来的职业道路。

A characteristic feature of materials undergoing martensitic phasetransitions is hysteresis they exhibit in response to cyclic loading.The process involves formation and evolution of finely layeredmicrostructures which depend in a crucial way on the orientation andhistory of the loading. The origin of energy dissipation, phase nucleation and kinetics of phase boundaries are important open problems in modeling of martensites and other active materials. This research program seeks to advance the understanding of these phenomena from the perspective of mesoscopic and microscopic frameworks. It will focus on a series of prototypical lattice models of increasing complexity with a goal of capturing the essential features of phase nucleation, interface kinetics and the associated hysteresis. The models have two main ingredients: (i) nonconvex interactions between nearest neighbors allowing for existence of two or more stable homogeneous states and (ii) the long-range interactions. Among the outcomes of this research will be derivation of an interface kinetic law and prediction of nucleated microstructural patterns. These results will be compared to experimental observations. In particular, the proposed work will relate the macroscopic observations such as the size and other features of a hysteresis loop to the microscopic characteristics of the material.Results of this research will be important for emerging civil, aerospace and industrial applications which require significant passivedamping exhibited by martensites. These include damage and vibrationcontrol in composite structures and attenuation of earthquake and wind-induced vibrations in buildings and bridges. The work will impact many other scientific areas where similar discrete systems are encountered, including dislocation theory, fracture mechanics, biology, image recognition and numerical analysis. The broader impact of this program will be also achieved through training of graduate and undergraduate students in an interdisciplinary research program and a variety of educational and outreach activities. These include developing new courses, mentoring women graduate students and running programs for middle and high school students. These activities will promote mathematics and science in general as an interesting intellectual pursuit and a prospective career path.