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

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。 特别感兴趣的是当材料响应于环境变化而经历突然变化时的材料不稳定性。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使原始相（材料的结构）不稳定并产生可具有不同体积的不同结晶相，从而导致原始材料样品的形状改变。这种材料可以用于纳米级器件中的开关、杠杆或计量器。转化的材料是由相边界分隔的相的混合物。细长结构在受压下的屈曲是另一个不稳定的例子。轴向压缩圆柱壳的屈曲是数学分析中特别感兴趣的问题，因为理论计算的理想圆柱壳的屈曲载荷比实验中观察到的高5倍以上。这项研究将提供分析工具，确定稳定的配置和不稳定性。它将促进我们对特定机制的定量理解，通过这些机制，形状和载荷的微小缺陷可以对屈曲强度产生巨大影响。因果性和无源性的基本物理原理经常用对参数的特殊解析依赖性来数学表达。调查员将研究属性的相关类的解析函数从恢复他们从实验测量的角度来看。这样的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的行星和卫星等恶劣环境中的材料特性时，它们与遥感有关。无损检测的一个相关问题将受益于利用复合材料研究中的数学进步，从边界测量预测非均匀介质的结构特征。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是对变分法的贡献，在变分法中，更好地理解准凸性可能对形状记忆合金或超磁致伸缩材料的亚稳性和磁滞现象的力学产生影响。这种经历马氏体相变的材料用于传感器和致动器以及日常设备，如牙齿矫正器。圆柱壳屈曲的研究旨在建立一个数学上严格的细长体屈曲理论。特别令人感兴趣的是，在屈曲开始之前，特别是在存在形状缺陷的情况下，偏离线弹性变形的可忽略性的严格理由。斯蒂尔捷斯函数是一类重要的特殊解析函数，用它可以描述电路的复阻抗或材料的复电磁介电常数。它们也出现在信号处理、天线设计和粒子物理学中。这项研究将解决的基本问题是严格的数学理论，他们从噪声测量恢复在一组离散的点或曲线上的复杂的上半平面。可证明的最佳重建算法产生可证明的有效数据将是本研究的一个具体目标。复合材料的精确关系理论将用于识别狄利克雷到诺依曼映射的属性，这些属性对介质的内部结构不敏感。该奖项反映了NSF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。

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.