Bifurcation Analysis and Computation in Elliptic and Multiphase Problems of Nonlinear Elasticity

T.Healey和P.Rosakis计划研究由非线性偏微分方程组支配的固体连续力学中的各种问题。这项工作的主要内容是研究形状记忆合金二维模型相变引起的微观结构的静态形态和动态演化。与众所周知的绝对最小能量静力学问题的方法不同，我们计划使用对称破缺、分叉理论等现代技术。后者使系统地确定亚稳态成为可能，我们相信我们随后的动力学研究将表明这与磁滞有关。这项工作的另一个方面将涉及将广义拓扑度(由Healey&Amp；Simpson先前开发的)应用于非线性弹性的整体分叉问题--这将是三维弹性中的第一个此类结果。这项工作的总体目标是发现新的非线性现象，获得对一般材料类别在物理和数学上都合理的本构限制，并开发解决此类非线性问题的新工具/方法-无论是分析还是计算。该方法融合了固体连续介质力学、材料科学、非线性分析、对称性和群论方法和计算方法，在非常一般的水平上分析非线性材料的模型是理解某些先进工程合金以及更传统的工程材料/结构中形状记忆效应的基础。拟议的研究具有很强的跨学科性质，结合了力学、材料科学和现代数学的模型和技术。总的来说，这项工作将为解决结构/机械工程和材料科学中出现的困难的非线性问题提供新的数学方法。这有可能导致更好地理解材料的行为和设计--包括“智能”结构的设计。

T. Healey and P. Rosakis plan to study various problems from solid continuummechanics governed by nonlinear partial differential equations. A majorthrust of the proposed work will be on the static morphology and dynamicevolution of microstructure due to phase transitions in two-dimensionalmodels of shape-memory alloys. In contrast to well-known absolute,minimum-energy approaches to the statics problem, we plan to employ moderntechniques of symmetry-breaking, bifurcation theory. The latter enablessystematic determination of metastable states, which we believe our ensuingdynamical studies will show to be associated with hysteresis. Anotheraspect of the work will involve applications of a generalized topological degree(developed previously by Healey & Simpson) to global bifurcation problemsof nonlinear elasticity - these will be the first such results inthree-dimensional elasticity. The overall goals of the work are to detectnew nonlinear phenomena, obtain constitutive restrictions that are bothphysically and mathematically reasonable for general classes of materials,and to develop new tools/approaches to such nonlinear problems - bothanalytical and computational. The proposed approach comprises a blend ofsolid continuum mechanics, materials science, nonlinear analysis, symmetryand group-theoretic methods and computation.The analysis of models of nonlinear materials at a very general level is fundamental to the understanding of shape-memory effects in certain advanced engineeringalloys as well as more traditional engineering materials/structures. Theproposed investigation is strongly interdisciplinary, combining models andtechniques from mechanics, materials science and modern mathematics.Broadly speaking the work will provide new mathematical approaches todifficult nonlinear problems arising in structural/mechanical engineeringand materials sciences. This has the potentia of leading to a better understanding ofmaterial behavior and design - including the design of "smart" structures.