Global Continuation Methods in Nonlinear Elasticity

小行星9704730 我们计划开展非线性弹性静力学偏微分方程的整体非线性分析研究。 拟议工作的一个主要重点将集中在辛普森本人最近开发的一种新的通用存在工具的应用上（来自过去的部分NSF支持），即，推广了Leray-Schauder度的全局延拓和分支。 在具体问题的上下文中，我们寻求先验界，对称性/正性等，所有这些都是为了获得对整个材料类别的有意义的本构限制。 我们的研究将包括传统的（强椭圆）问题和那些涉及相变（椭圆度损失）。 对于后者，提出了一种基于高梯度正则化、全局延拓和奇异极限的新方法。 在一个非常普遍的水平上分析这样的模型是理解传统的工程材料/结构和马氏体相变和形状记忆效应的基础，这在许多先进的工程合金中观察到。这项工作有两个主要目标：（一）获得新的定性结果，并检测新的现象-数学和物理意义;（二）获得新的全球连续（存在）的结果问题的2和3维弹性-包括问题涉及相变。 从广义上讲，拟议的工作将提供重要的数学基础，以困难的非线性问题，在传统的工程领域，如结构机械工程，也在更现代的领域，如材料科学。这项工作有可能：（i）提供新的数学工具，用于分析工程实践中的困难问题，最终导致更安全和更优化的结构设计;（ii）导致更好地理解某些工程合金的非线性材料行为，并有可能应用于制造工程和非被动或“智能”结构的设计。

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.