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

相变不稳定性、壳屈曲和反问题的研究

• 批准号: 2305832
• 负责人: Yury Grabovsky
• 金额: $ 32.54万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305832&HistoricalAwards=false
• 关键词: Study; Instabilities; Phase; Transitions; Shell

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.

能量最小化原理通常可用于解释在自然界中观察到的现象，例如不稳定 - 环境中的微小变化会导致大量甚至质变的现象。该项目考虑使用能量方法来分析与应用相关的三种类型的不稳定性。第一类涉及固体中的相变，它是形状记忆效应和智能材料其他应用的基础，重点是相界的稳定性，以提高在相变开始时识别临界应变的能力。第二种是屈曲，即在超过临界应力阈值后，细长结构突然发生失效。细长结构在技术世界中发挥着越来越重要的作用，提供轻质且功能强大的设备。然而，力学和工程领域尚未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通过揭示形状和载荷的小缺陷对临界应力产生巨大影响的机制，阐明轴向压缩圆柱壳的屈曲。分析的第三种不稳定性是数值类型，其中材料特性（例如电磁介电常数或电阻抗谱）测量的精度并不能转化为在比现有频率高得多或低得多的频率下对其响应的同样精确的预测。数据。受放射学和遥感应用的启发，其目的是量化这些不稳定性，为创建新算法提供信息，从而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。为研究相界稳定性而发展的数学代表了对变分微积分的贡献，其中几乎难以处理的拟凸性概念发挥了核心作用。该研究项目将开发深入了解拟凸包络结构所需的工具。尽管准凸性一般来说无法抵抗任何有意义的全面攻击，但这项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弛豫，并将提供一套可证明的完整的实际可访问的信息。对圆柱壳屈曲的研究将创建一组新的理论预测，以便与实验进行比较。这些预测的关键特征是对造成经典屈曲载荷与实验观察到的载荷之间差异的不稳定机制的定量描述。与更一般的外壳有关的理论问题也将得到解决。 斯蒂尔切斯函数形成了一类特殊的分析函数，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一类重要函数的匮乏知识。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 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。