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控制的固体连续力学中的各种问题。所提出的工作的主要推力将是在形状记忆合金的二维模型中，由于相变引起的微观结构的静态形态和动态演变。与众所周知的绝对，最小能量方法静力学问题相反，我们计划采用对称破缺，分岔理论的现代技术。后者能够系统地确定亚稳态，我们相信我们随后的动力学研究将显示与滞后有关。这项工作的另一个方面将涉及将广义拓扑度（先前由Healey & Simpson开发）应用于非线性弹性的全局分岔问题-这将是三维弹性的第一个这样的结果。这项工作的总体目标是检测新的非线性现象，获得对一般材料类别在物理和数学上合理的本构限制，并开发新的工具/方法来解决这些非线性问题-分析和计算。该方法综合了固体连续介质力学、材料科学、非线性分析、对称和群论方法和计算。在非常一般的水平上对非线性材料模型的分析是理解某些先进工程合金以及更传统的工程材料/结构中的形状记忆效应的基础。拟议的研究是跨学科的，结合了力学、材料科学和现代数学的模型和技术。总的来说，这项工作将为结构/机械工程和材料科学中出现的困难非线性问题提供新的数学方法。这有可能导致对材料行为和设计的更好理解——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.