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的各种问题。 一个主要的推力，拟议的工作将是静态形态和动态演变的微观结构，由于相变的二维模型的形状记忆合金。 与众所周知的绝对，最小能量的方法来解决静力学问题，我们计划采用现代技术的破缺，分叉理论。 后者使亚稳态，我们相信我们的ensuingdynamic研究将显示与滞后系统的测定。 另一方面的工作将涉及应用的广义拓扑度（以前开发的希利辛普森）的全局分叉问题的非线性弹性-这些将是第一个这样的结果在三维弹性。 这项工作的总体目标是检测新的非线性现象，获得本构限制，这是物理和数学上合理的一般类别的材料，并开发新的工具/方法，这样的非线性问题-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、非线性理论和群论方法以及计算，对非线性材料模型的一般性分析是理解某些先进工程合金和传统工程材料/结构中形状记忆效应的基础。 这项研究具有很强的跨学科性，结合了力学、材料科学和现代数学的模型和技术，从广义上讲，这项工作将为结构/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.