Global Continuation Methods in Nonlinear Elasticity

9704730 Healey 计划开展非线性弹静力学偏微分方程全局非线性分析研究。 拟议工作的主要重点将集中于辛普森和我本人最近开发的新通用存在工具的应用（来自过去部分 NSF 支持），即全局连续和分叉的 Leray-Schauder 度的推广。 在具体问题的背景下，我们寻求先验边界、对称性/正性属性等，所有这些都是为了获得整个类别材料的有意义的本构限制。 我们的研究将包括传统（强椭圆）问题和涉及相变（椭圆度损失）的问题。 对于后者，提出了一种基于更高梯度正则化、全局连续和奇异极限的新方法。 在非常一般的层面上对此类模型进行分析对于理解传统工程材料/结构以及在许多先进工程合金中观察到的马氏体转变和形状记忆效应至关重要。这项工作有两个主要目标：（i）获得新的定性结果并发现具有数学和物理意义的新现象； (ii) 在 2 维和 3 维弹性问题中获得新的全局连续（存在）结果 - 包括涉及相变的问题。 从广义上讲，所提出的工作将为结构和机械工程等传统工程领域以及材料科学等现代领域中出现的困难非线性问题提供重要的数学基础。这项工作有潜力：（i）为分析工程实践中的难题提供新的数学工具，最终实现更安全、更优化​​的结构设计； (ii) 更好地理解某些工程合金的非线性材料行为，在制造工程和非被动或“智能”结构的设计中具有潜在的应用。

9704730 Healey We plan to carry out research in global nonlinear analysis of partial differential equations of nonlinear elastostatics. A major thrust of the proposed work will be focused on applications of a new general existence tool developed recented by Simpson & myself (from past partial NSF support), viz., a generalization of the Leray-Schauder degree for global continuation and bifurcation. In the context a concrete problems, we seek a-prior bounds, symmetry/positivity properties, etc., all with a view toward obtaining meaningful constitutive restrictions for entire classes of materials. Our study will include both traditional (strongly elliptic) problems and those involving phase change (loss of ellipticity). For the latter, a new approach based upon higher-gradient regularization, global continuation and singular limits is being proposed. The analysis of such models at a very general level is fundamental to the understanding of traditional engineering materials/structures and martensitic transformations and shape-memory effects, which are observed in many advanced engineering alloys. The work has two major goals: (i) To obtain new qualitative results and detect new phenomena - of both mathematical and physical significance; (ii) To obtain new global-continuation (existence) results in problems of 2 and 3-dimensional elasticity - including problems involving phase transformations. Broadly speaking,the proposed work will provide important mathematical underpinnings to difficult nonlinear problems arising in traditional engineering fields like structural & mechanical engineering and also in more modern areas like materials science. The work has the potential to: (i) deliver new mathematical tools for the analysis of difficult problems of engineering practice,leading ultimately to safer and more optimal design of structures; (ii) lead to a better understanding of the nonlinear material behavior of certain engineering alloys, with potential application s to manufacturing engineering and the design of non-passive or "smart" structures.