Energy-Driven Instabilities in Nonlinear Elasticity and Other Questions from Materials Science

非线性弹性中能量驱动的不稳定性以及材料科学中的其他问题

基本信息

批准号：
2005538
负责人：
Yury Grabovsky
金额：
$ 30.8万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005538&HistoricalAwards=false
关键词：
Energy Driven Instabilities Nonlinear Elasticity

项目摘要

This project is devoted to the application of the fundamental physical principle of energy minimization to understanding and utilization of several important phenomena arising in materials science. Of special interest are materials instabilities when the material experiences sudden changes in response to the changes in environment. One type of instability is a phase transition in crystalline solids, where changing loading or temperature can make the original phase (structure of the material) unstable and give rise to a different crystalline phase that could have a different volume leading to the shape change of the original material sample. Such materials can be used in switches, cantilevers, or gauges in nano-scale devices. The transformed material is a mixture of phases, separated by phase boundaries. Buckling of slender structures under compression is another example of an instability. Buckling of axially compressed cylindrical shells is of special interest for mathematical analysis, since the theoretically computed buckling load of a perfect circular cylindrical shell is more than 5 times higher than what is observed in experiments. This investigation will provide analytical tools for identifying both stable configurations and instabilities. It will advance our quantitative understanding of specific mechanisms through which small imperfections of shape and load can have a dramatic effect on buckling strength. Fundamental physical principles of causality and passivity are very often expressed mathematically in terms of special analytic dependence on parameters. The investigator will study properties of relevant classes of analytic functions from the point of view of recovering them from experimental measurements. Such questions emerge from a wide spectrum of disciplines from materials science to particle physics. They are relevant for remote sensing when one wants to measure material properties in inhospitable environments, from the Arctic to planets and moons in the Solar system. A related problem of nondestructive testing will benefit from harnessing mathematical advances in the study of composites to predicting structural features of heterogeneous media from boundary measurements. A junior scientist will be trained while contributing to the research described above.Mathematics being developed to study stability of phase boundaries represents a contribution to Calculus of Variations, where a better understanding of quasiconvexity can have implications for mechanics of metastability and hysteresis in shape memory alloys or giant magnetostrictive materials. Such materials undergoing martensitic phase transitions are used in sensors and actuators, and in everyday devices, like dental braces. The investigation of buckling of cylindrical shells aims to create a mathematically rigorous theory of buckling of slender bodies. Of special interest is the rigorous justification of negligibility of departure from linear elasticity of deformations before the onset of buckling, especially in the presence of shape imperfections. Stieltjes functions is an important special class of analytic functions in terms of which one can describe the complex impedance of electrical circuits or complex electromagnetic permittivity of materials. They also arise in signal processing, antenna design and particle physics. The fundamental question that will be addressed by this research is the rigorous mathematical theory of their recovery from noisy measurements at either a discrete set of points or on a curve in the complex upper half plane. Provably optimal reconstruction algorithms producing certifiably valid data will be a specific target of this research. The theory of exact relations for composites will be used to identifying properties of the Dirichlet to Neumann map that are insensitive to the internal structure of the medium.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.

该项目致力于应用能量最小化的基本物理原理来理解和利用材料科学中出现的几个重要现象。特别令人感兴趣的是当材料因环境变化而发生突然变化时材料的不稳定性。一种类型的不稳定性是结晶固体中的相变，其中改变负载或温度可能会使原始相（材料的结构）不稳定并产生不同的结晶相，该结晶相可能具有不同的体积，从而导致材料的形状发生变化。原始材料样品。此类材料可用于纳米级设备中的开关、悬臂或仪表。转变的材料是由相界分隔的相的混合物。细长结构在压缩下的屈曲是不稳定性的另一个例子。轴向压缩圆柱壳的屈曲对于数学分析特别重要，因为理论计算的完美圆柱壳的屈曲载荷比实验中观察到的高 5 倍以上。这项研究将为识别稳定构型和不稳定性提供分析工具。它将促进我们对特定机制的定量理解，通过这些机制，形状和载荷的小缺陷会对屈曲强度产生巨大的影响。因果关系和被动性的基本物理原理通常用对参数的特殊分析依赖性来数学表达。研究者将从实验测量中恢复相关解析函数类别的性质来研究它们。这些问题来自从材料科学到粒子物理学的广泛学科。当人们想要测量从北极到太阳系中的行星和卫星等恶劣环境中的材料特性时，它们与遥感相关。无损检测的相关问题将受益于利用复合材料研究中的数学进步，通过边界测量来预测异质介质的结构特征。一名初级科学家将在为上述研究做出贡献的同时接受培训。为研究相界稳定性而开发的数学代表了对变分微积分的贡献，其中对拟凸性的更好理解可以对形状记忆合金中的亚稳态和磁滞力学产生影响或超磁致伸缩材料。这种经历马氏体相变的材料可用于传感器和执行器以及牙套等日常设备中。对圆柱壳屈曲的研究旨在创建数学上严格的细长体屈曲理论。特别令人感兴趣的是在屈曲发生之前，特别是在存在形状缺陷的情况下，对变形偏离线性弹性的可忽略性的严格论证。 Stieltjes 函数是一类重要的特殊解析函数，可以用它来描述电路的复阻抗或材料的复电磁介电常数。它们还出现在信号处理、天线设计和粒子物理学领域。这项研究将解决的基本问题是从一组离散点或复杂上半平面曲线上的噪声测量中恢复的严格数学理论。可证明最佳重建算法产生可证明有效的数据将是本研究的具体目标。复合材料的精确关系理论将用于识别对介质内部结构不敏感的狄利克雷到诺依曼图的属性。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值进行评估，被认为值得支持以及更广泛的影响审查标准。