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

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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.