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

该项目围绕两个中心主题：材料和结构的稳定性和异质性，从不同的角度对其进行研究。 这种不均匀性可以是人为的，或者是人造的，如在复合材料中，或者它可以是自然自发产生的，如在形状记忆合金中或在圆柱壳的某些屈曲模式中。 在前一种情况下，材料的宏观性能对其微观结构的依赖性是本研究的主题，其目标是揭示所有情况下，可以在组成材料和复合材料的性能之间建立精确的公式，而不管微观结构如何。 本科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种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。 另一个重要的不稳定性例子是圆柱壳的屈曲。 这些结构是特别有趣的，因为理论上预测的屈曲压缩可以高达五倍以上的实验观察到的，由于初始缺陷的高灵敏度。 一个新开发的理论屈曲，其中的研究人员是一个合著者，显示承诺在给予一个新的解释极端敏感性的屈曲载荷的缺陷。纤维增强弹性复合材料已知的几个精确关系式分散在文献中。 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。 精确的关系可以用作测量实际和理想复合材料性能之间差异的基准，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理解能够经历马氏体相变的材料的弹性不稳定性。 在相边界的存在下，理解二次变分的正性的全局条件和局部拟凸性条件之间的相互作用的问题需要新的见解。 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。 细长体的屈曲是一种普遍的局部失稳破坏。 轴向压缩圆柱壳是特别感兴趣的，因为经典的启发式渐近屈曲分析预测的临界载荷值，这是约5倍高于实验观察到的，它也预测不正确的比例的屈曲载荷作为外壳的厚度的函数。 这个项目研究了一个新发现的模式切换现象，即小的缺陷激活“潜在的”屈曲模式与不同的标度律。

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.