Linear and non-linear elasticity: Study of exact relations and instabilities

该项目围绕着两个中心主题：材料和结构的稳定性和异质性，从不同的角度审视它们。这种非均质性可以是人造的，如在复合材料中，也可以是自然自发产生的，如在形状记忆合金中或在一些圆柱形外壳的屈曲模式中。在前一种情况下，材料的宏观性能对其微观结构的依赖是本研究的主题，其目标是揭示所有可以在组成材料和复合材料的性能之间建立精确公式的实例，而不管微观结构如何。本科生研究是这个项目的重要组成部分。在后一种情况下，微观结构的产生是由最小能量原理所解释的不稳定性引起的。从数学上严格确定是否有可能降低一个构型的能量是一个重要的问题，特别是当构型已经是异质的时候。不稳定性的另一个重要例子是圆柱壳的屈曲。这些结构特别有趣，因为理论上预测的屈曲压缩可能比实验观察到的高5倍，这是由于对初始缺陷的高度敏感性。一种新发展的屈曲理论（研究者是该理论的合著者）有望对屈曲载荷对缺陷的极端敏感性给出新的解释。已知的纤维增强弹性复合材料的少数确切关系分散在文献中。该项目的目标是创建所有精确关系和链接的完整列表，为参与复合材料研究和设计的机械工程师提供有用的资源。精确的关系可以作为衡量实际和理想复合材料性能差异的基准，因为它们不依赖于微观结构——任何复合材料中最不可控的变量。该项目的另一个目标是了解能够经历马氏体相变的材料的弹性不稳定性。在相边界存在的情况下，理解整体二次变分正性条件与整体局部拟凸性条件之间相互作用的问题需要新的认识。所提出的具有大量弱局部极小值的能量密度函数的例子可以用来理解强局部极小值的必要条件的逻辑层次。细长体的屈曲是压缩作用下局部弱稳定性的普遍失效。轴向压缩圆柱壳是特别有趣的，因为经典的启发式渐近屈曲分析不仅预测了临界载荷的值比实验观察到的高5倍，而且还预测了屈曲载荷作为壳厚度函数的不正确比例。该项目研究了一种新发现的模式切换现象，即小缺陷激活具有不同标度定律的“潜在”屈曲模式。

This project revolves around two central themes: stability and heterogeneity of materials and structures, examining them from different perspectives. The heterogeneity can be artificial, or man-made, as in composite materials, or it can be created spontaneously by nature as in shape memory alloys or in some of the buckling patterns of cylindrical shells. In the former case the dependence of the macroscopic properties of materials on their microstructure is the subject of this study, where the goal is to uncover all instances where exact formulas can be established between the properties of constituent materials and the composite, regardless of the microstructure. Undergraduate research is an essential part of this project. In the latter case the creation of microstructure is caused by instabilities that are explained by the principle of minimum energy. Mathematically rigorous determination of whether it is possible to lower the energy of a configuration is an important question, especially when the configuration is already heterogeneous. Another important example of instability is buckling of cylindrical shells. These structures are especially interesting because theoretically predicted buckling compression can be up to five times higher than the experimentally observed one due to high sensitivity to initial imperfections. A newly developed theory of buckling, of which the investigator is a coauthor, shows promise in giving a new explanation of extreme sensitivity of the buckling load to imperfections. The few exact relations that are known for fiber-reinforced elastic composites are scattered across the literature. The goal of this project is the creation of a complete list of all exact relations and links, providing a useful resource for mechanical engineers involved in the study and design of composite materials. Exact relations can be used as benchmarks for measuring the discrepancies between actual and ideal composites' properties, since they do not depend on the microstructure -- the least controllable variable in any composite. Another goal of this project is understanding of elastic instabilities in materials capable of undergoing martensitic phase transitions. In the presence of phase boundaries the problem of understanding the interplay between the global condition of positivity of second variation and local quasiconvexity conditions in the bulk requires new insights. Proposed examples of energy density functions with a large number of weak local minimizers can be used to understand the logical hierarchy of necessary conditions for strong local minima. Buckling of slender bodies represents a universal failure of weak local stability under compression. Axially compressed cylindrical shells are of special interest, since the classical heuristic asymptotic analysis of buckling predicts not only the value of the critical load that is some five times higher than experimentally observed, it also predicts incorrect scaling of the buckling load as a function of shell's thickness. This project examines a newly discovered mode-switching phenomenon, whereby small imperfections activate "latent" buckling modes with different scaling laws.