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现象，例如不稳定性-环境中的微小变化会导致大量的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类型的不稳定性。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，细长结构在临界应力阈值被跨越后突然发生失效。细长结构在技术世界中发挥着越来越重要的作用，提供重量轻，功能强大的设备。然而，在力学和工程学中还没有充分了解它们对缺陷的极端敏感性。该项目的一个具体目标是揭示轴向压缩圆柱壳的屈曲，通过揭示允许形状和载荷的小缺陷对临界应力产生显着影响的机制。分析的第三种不稳定性是数值型的，其中材料特性的测量精度，例如电磁介电常数或电阻抗谱，并不转化为在比可用数据高得多或低得多的频率下对其响应的同样精确的预测。受放射学和遥感应用的启发，目的是量化这些不稳定性，以创建新的算法，重建具有可证明的最优性的材料响应特性。该项目为研究生提供研究指导和培训机会。数学正在发展研究相边界的稳定性是对变分法的贡献，其中几乎难以处理的准凸性概念起着核心作用。该研究项目将开发深入了解拟凸包络结构所需的工具。尽管事实上，一般的拟凸性蔑视任何有意义的一般攻击，这项研究将提供的信息将是直接的实际重要性，允许精确或近似的比能量松弛，并将提供一个可证明的完整的一套实际上可访问的信息。对圆柱壳屈曲的研究将产生一套新的理论预测，以便与实验进行比较。这些预测的主要特点是负责经典屈曲载荷和实验观察到的不稳定性之间的差异的机制的定量描述。有关更一般的壳的理论问题也将得到解决。 斯蒂尔吉斯函数是物理学中普遍存在的一类特殊的解析函数。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，例如电介质的复电磁介电常数和电路的复阻抗。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知识。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.