Instabilities in Materials Science

研究人员Grabovsky和他的合作者研究了几个重要的现象，其中主要的潜在特征是不稳定，或者对测量误差、形状缺陷或载荷非常敏感。三个项目被不稳定的一般概念统一起来。其中之一涉及了解与马氏体相变有关的非线性弹性不稳定性，从而在实验上观察到尖锐的相界。这种相变导致了合金的形状记忆效应，以及某些材料表现出的超磁致伸缩效应，仅举几个例子。另一个项目应用新的工具来分析轴向压缩的圆柱壳的屈曲应力对缺陷的极端敏感性--圆柱壳是从粮仓到飞机和宇宙飞船的各种结构中的基本结构部件。第三个项目专注于定量理解赫格尔茨函数的刚性和灵活性--一种看似矛盾的性质组合--的众所周知的融合，这些函数在从材料科学到核物理的应用中无处不在。例如，研究人员的研究量化了由于实验误差而导致的复介电常数估计中的不确定性，复介电常数描述了一种材料如何与从无线电波到可见光再到伽马射线等各种频率的电磁波相互作用。这个项目是由一名研究生在研究人员的指导下进行的博士研究。马氏体形变中最显著的特征，尤其是形状记忆效应，是相界的存在--将不同的马氏体变种分开的尖锐界面。这种相边界的稳定性可以完全用拟凸性的概念来刻画。考虑到对拟凸性的刻画通常被认为是无法实现的，研究人员和他的合作者将重点放在相边界稳定性的显著可计算的代数条件上，并通过具体的例子来评估它们的“强度”，这些例子表明代数条件离真正的答案有多远，这些答案可以通过解析或数值计算得到。另一个项目涉及圆柱壳的屈曲，这是力学中被广泛研究但知之甚少的问题之一，由于载荷和形状的初始缺陷，经典公式不能用来预测实验中观察到的屈曲应力。该项目建立了细长物体屈曲的严格数学理论，能够解释为什么形状和载荷的初始缺陷对某些结构的屈曲应力有很大影响，而对另一些结构的影响可以忽略不计。第三个项目研究赫格尔茨函数有些自相矛盾的行为，这些函数同时非常严格，同时又出人意料地灵活。它的理解有助于数学和物理的几个领域，并回答了许多重要的问题。解决了数据验证和外推的问题。本项目是调查者研究生的博士生研究。

1714287Grabovsky The investigator and his collaborators study several important phenomena where the main underlying feature is instability, or extreme sensitivity to measurement errors, imperfections of shape, or load. Three projects, unified by the general idea of instability, are pursued. One of them deals with understanding instabilities in nonlinear elasticity associated with martensitic phase transitions, whereby sharp phase boundaries are observed experimentally. Such phase transitions are responsible for shape memory effects in alloys, and for a giant magnetostrictive effect exhibited by some materials, to name just a few examples. Another project applies new tools to analyze the extreme sensitivity to imperfections of the buckling stress of axially compressed circular cylindrical shells -- an essential structural component in a vast variety of structures, from grain silos to airplanes and space ships. The third project focuses on quantitative understanding of a well-known confluence of rigidity and flexibility -- a seemingly contradictory combination of properties -- of Herglotz functions that are ubiquitous in applications from materials science to nuclear physics. For example, the investigator's research quantifies the uncertainty due to experimental errors in the estimates of complex electromagnetic permittivity, which describes how a material interacts with electromagnetic waves of various frequencies from radio waves through visible light to gamma rays. This project constitutes Ph.D. research of a graduate student working under the investigator's guidance. The most salient feature in martensitic shape transformations, which in particular is responsible for shape memory effect, is the presence of phase boundaries -- sharp interfaces separating different martensitic variants. Stability of such phase boundaries can be completely characterized in terms of the concept of quasiconvexity. With the understanding that characterization of quasiconvexity is often regarded as unachievable, the investigator and his collaborators focus on eminently computable algebraic conditions of stability of phase boundaries and assess their "strength" through specific examples that show how far off algebraic conditions are from true answers, which could be computed either analytically or numerically. Another project deals with buckling of cylindrical shells -- one of the extensively studied, yet poorly understood problems in mechanics, where the classical formula cannot be used to predict buckling stress observed in experiments due to initial imperfections of load and shape. This project creates a mathematically rigorous theory of the buckling of slender bodies that is capable of explaining why initial imperfections of shape and load have strong influence on the buckling stress in some structures, while having negligible effect in others. The third project studies the somewhat paradoxical behavior of Herglotz functions, that are at the same time extremely rigid and surprisingly flexible. Its understanding contributes to several areas of mathematics and physics and answers many important questions. The problems of data validation, and extrapolation are addressed. This project constitutes Ph.D. research of the investigator's graduate student.