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

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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.