Study of Instabilities in Phase Transitions, Shell Buckling, and Inverse Problems

能量最小化原理通常可以用来解释在自然界中观察到的东西，例如不稳定性-环境中的微小变化导致大量数量甚至质量变化的现象。本项目考虑能量方法来分析与应用相关的三种不稳定性。第一种类型涉及固体中的相变，这是形状记忆效应和智能材料的其他应用的基础，重点是相边界的稳定性，以提高在相变开始时识别临界应变的能力。第二种是屈曲，在超过临界应力阈值后，细长结构突然发生破坏。细长结构在技术世界中发挥着越来越重要的作用，提供了轻质和高功能的设备。然而，在力学和工程上还没有完全理解它们对缺陷的极端敏感性。该项目的一个具体目标是通过揭示允许形状和负载的小缺陷对临界应力产生巨大影响的机制，来阐明轴向压缩圆柱壳的屈曲。所分析的第三种不稳定性是数值类型的，其中测量材料特性的精度，如电磁介电常数或电阻抗谱，不能转化为与现有数据相比在更高或更低频率下同样精确的预测。受放射学和遥感应用的启发，其目的是量化这些不稳定性，以创建新的算法，以可证明的最优性重建材料响应特性。该项目为研究生提供研究指导和培训机会。研究相边界稳定性的数学发展是对变分学的贡献，在变分学中，几乎难以处理的拟凸性概念起着中心作用。该研究项目将开发工具，以深入了解拟凸包络的结构。尽管准自凸性通常不受任何有意义的一般攻击，但本研究将提供的信息将具有直接的实际重要性，允许特定能量的精确或近似松弛，并将提供一套可证明的完整的实际可访问信息。对圆柱壳屈曲的研究将建立一套新的理论预测，并与实验进行比较。这些预测的关键特征是对导致经典屈曲载荷与实验观测载荷之间差异的失稳机制的定量描述。与更一般的shell相关的理论问题也将被讨论。Stieltjes函数构成了一类特殊的解析函数，在物理学中几乎无处不在。对这些函数的定量理解将有助于重建材料响应的实用算法，例如介电介质的复杂电磁介电常数和电路的复杂阻抗。对完全单调函数性质的新研究将补充和扩展目前关于这类重要函数的缺乏知识。预计将应用于更有效地处理放射学和遥感数据。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Energy minimization principles can often be used to explain what is observed in nature, such as instabilities - phenomena where small changes in the environment cause large quantitative or even qualitative changes. This project considers energy methods to analyze three types of instability relevant to applications. The first type concerns phase transformations in solids, which underly shape memory effects and other applications of smart materials, with a focus on stability of phase boundaries, to improve the ability of identifying critical strains at the onset of phase transitions. The second is buckling, where the failure of a slender structure occurs abruptly, after the critical stress threshold has been crossed. Slender structures play an increasingly important role in the technological world, delivering light-weight and highly functional devices. However, a full understanding of their extreme sensitivity to imperfections is not yet available in both mechanics and engineering. One specific goal of the project is to shed light on the buckling of axially compressed cylindrical shells, by revealing the mechanisms that allow small imperfections of shape and load to have a dramatic effect on the critical stress. The third instability analyzed is of a numerical type, where the precision of measurements of properties of materials, such as electromagnetic permittivity or electrical impedance spectrum, does not translate to equally precise prediction of their response at much higher or much lower frequencies than in the available data. Inspired by applications to radiology and remote sensing, the aim is to quantify these instabilities to inform the creation of new algorithms that reconstruct the material response characteristics with provable optimality. The project provides research mentoring and training opportunities for graduate students. Mathematics being developed to study stability of phase boundaries represents a contribution to Calculus of Variations, where the almost intractable concept of quasiconvexity plays a central role. The research project will develop tools needed to gain insight into the structure of quasiconvex envelopes. Notwithstanding the fact that quasiconvexity in general defies any meaningful general attack, the information this research will deliver will be of direct practical importance, permitting exact or approximate relaxations of specific energies, and will provide a provably complete set of practically accessible information. The investigation of buckling of cylindrical shells will create a new set of theoretical predictions to be compared with experiment. The key feature of these predictions is the quantitative description of mechanisms of instability responsible for the discrepancy between the classical buckling load and the experimentally observed ones. Theoretical issues pertaining to more general shells will also be addressed. Stieltjes functions form a special class of analytic functions virtually ubiquitous in physics. Quantitative understanding of such functions will contribute to practical algorithms for reconstructing material responses, such as complex electromagnetic permittivity of dielectric media and complex impedance of electrical circuits. A new investigation into the properties of completely monotone functions will complement and expand the current scant knowledge about this important class of functions. Applications to more efficient processing of radiology and remote sensing data are anticipated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.