Global Continuation Methods in Nonlinear Elasticity

我们计划开展非线性弹性静力学偏微分方程的全局非线性分析研究。所提议的工作的一个主要推力将集中在辛普森本人最近开发的一个新的一般存在工具的应用上（来自过去的部分国家科学基金会支持），即Leray-Schauder度对全局延拓和分岔的推广。在具体问题的背景下，我们寻求先验边界，对称性/正性性质等，所有这些都是为了获得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（椭圆性损失）的问题。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对这些模型的分析是理解传统工程材料/结构、马氏体相变和形状记忆效应的基础，这些在许多先进的工程合金中都观察到。这项工作有两个主要目标：(i)获得新的质量结果和发现新的现象- -具有数学和物理意义；（ii）在二维和三维弹性问题（包括涉及相变的问题）中获得新的全局延拓（存在）结果。总的来说，这项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非线性难题提供重要的数学基础。这项工作有潜力：(i)提供新的数学工具来分析工程实践中的难题，最终导致更安全和更优化的结构设计；（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.