Mathematical Sciences: Nonlocal Equations Modeling Fine- scale Structures in Solids

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Ren 9703727 Martensite transformations are phase transformations that produce a change of shape and a change of crystal symmetry. Shape-memory materials are materials that are extremely malleable in the martensite phase below a transformation temperature, but that return to a `remembered' original shape when heated above the transformation temperature. Coherent phase transitions of crystalline solids lead to mixtures of distinct phases or phase variants with characteristic fine-scale structures. The main issues addressed in this project are the characteristic scales, the generation and the propagation of the microstructures. A nonlocal theory involving integro-differential evolutionary equations is proposed to answer these questions. This theory lies between the traditional microscopic quantum theory which uses Schodinger's equation and the macroscopic elasticity theory which uses elliptic and hyperbolic partial differential equations. In the study of the characteristic scales, a periodic solution of a stationary nonlocal equation is needed. But the equation seems to have multiple solutions. The `right' periodic solution should be selected by a suitable least energy principle. Another intriguing question is to find a `generalized' traveling wave solution of a dynamic nonlocal equation, which describes the propagation of microstructures. This solution should have an oscillatory structure at one end, constant value at the other end, and as time increases the uniform-to-oscillatory transition region should advance in a periodic manner. Such a solution can also be viewed as a heteroclinic orbit in an infinitely dimensional function space connecting a periodic stationary solution to a constant stationary solution. Materials and processing are critical to the success of industries such as the aerospace, automotive, biomaterials, chemical, electronics, energy, metals, and telecommunications industries. This project is concerned with shape memory materials and martensite transformations. The mathematical theory in this project is aimed at understanding the characteristic scales, the generation and the propagation of the microstructures during shape memory materials' phase transformations. The study of stationary periodic solutions ald uniform-to-oscillatory waves of the integro-differential evolutionary equations used in this nonlocal mathematical theory is at the cutting edge of mathematical analysis and computation.