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

这个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。特别值得注意的是，当材料随着环境的变化而突然发生变化时，材料就不稳定了。一种类型的不稳定性是晶体固体中的相变，在这种情况下，改变载荷或温度会使原始相(材料的结构)不稳定，并产生不同的晶相，该不同的晶相可能具有不同的体积，从而导致原始材料样品的形状变化。这种材料可用于纳米级设备中的开关、悬臂或量规。变换后的材料是由相边界分隔的相的混合物。细长结构在压缩下的屈曲是另一个不稳定的例子。轴心受压圆柱壳的屈曲是数学分析中特别感兴趣的问题，因为理想圆柱壳的理论计算屈曲载荷是实验观测的屈曲载荷的5倍以上。这项研究将为确定稳定构型和不稳定构型提供分析工具。这将促进我们对具体机制的定量理解，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显著影响。因果性和被动性的基本物理原理经常以对参数的特殊解析依赖关系的数学形式来表示。研究者将从从实验测量中恢复解析函数的角度来研究相关解析函数类的性质。这样的问题出现在从材料科学到粒子物理的广泛学科中。当人们想要测量从北极到太阳系行星和卫星等不适宜居住的环境中的材料特性时，它们与遥感相关。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来根据边界测量预测非均匀介质的结构特征。初级科学家将在为上述研究做出贡献的同时接受培训。正在开发的研究相界稳定性的数学是对变分演算的贡献，在该演算中，对准凸性的更好理解可能对形状记忆合金或超磁致伸缩材料的亚稳性和磁滞的力学产生影响。这种经历马氏体相变的材料被用于传感器和执行器，以及日常设备，如牙套。对圆柱壳屈曲的研究旨在建立一种数学上严格的细长物体屈曲理论。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，可以忽略变形的线弹性的严格理由。Stieltjes函数是一类重要的特殊解析函数，可以用来描述电路的复阻抗或材料的复介电常数。它们还出现在信号处理、天线设计和粒子物理领域。这项研究将解决的基本问题是从离散点集或复杂上半平面上的曲线上的噪声测量中恢复它们的严格数学理论。产生可证明有效数据的可证明最优重建算法将是本研究的具体目标。复合材料的精确关系理论将被用来识别Dirichlet到Neumann映射中对介质内部结构不敏感的属性。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势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.