Mathematical Sciences: Global Continuation Methods in Nonlinear Elasticity

6/3/96尊敬的哈里哈兰博士：从今天起，我们部门的财务助理正在根据您的建议编制新的预算。她会在明天的某个时候邮寄给你，让你非官方批准，然后我们再签字，等等。文本格式可以吗，还是传真给你更好？鉴于预算减少，我准备尽最大努力满足最初的工作范围。这是您所要求的摘要：*DMS-9625830：非线性弹性力学中的整体延拓方法P.I.：T.J.Healey摘要。我们计划开展非线性弹性偏微分方程组的整体非线性分析研究。拟议工作的主要重点将集中在涉及相变(弱解和椭圆性损失)的问题上。在非常一般的水平上对这些模型的分析是理解马氏体相变和形状记忆效应的基础，这些效应在许多先进的工程合金中都可以观察到。文中还将讨论经典非线性弹性力学中一些有趣的问题。这项工作有两个主要目标：(I)获得新的定性结果并发现新的现象--具有数学和物理意义；(Ii)在二维和三维弹性问题中获得新的整体延拓(存在)结果--包括涉及相变的问题。在经典问题(橡胶弹性)中，Healey和同事在以前由NSF赞助的工作中开发的工具将发挥重要作用。对于相变问题，提出了一种基于高梯度正则化、全局延拓和奇异极限的全新方法。一般而言，拟议的工作将为传统工程领域(如结构和机械工程)以及更现代的领域(如材料科学)中出现的复杂非线性问题提供重要的数学基础。这项工作有可能：(I)为工程实践中的难题分析提供新的数学工具，最终导致结构的更安全和更优化的设计；(Ii)使人们更好地理解某些工程合金的非线性材料行为，这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。*如果这是不可接受的，请让我知道-我很乐意进行更改。谢谢，蒂姆·希利

6/3/96 Dear Dr. Hariharan: Our Department Financial Assistant is now working up a new budget in accordance with your recommendations from today. She will mail it out sometime tomorrow for your non-official approval, before we sign it, etc. Is a Text format OK for this, or would it be better to fax it to you? I am prepared to make a best effort attempt to meet the original scope of the work, given the reduced budget. Here is the abstract that you requested: ************* DMS-9625830: Global Continuation Methods in Nonlinear Elasticity P.I.: T.J.Healey Abstract. We plan to carry out research in global nonlinear analysis of partial differential equations of nonlinear elasticity. A major thrust of the proposed work will be focused on problems involving phase change (weak solutions and loss of ellipticity). The analysis of such models at a very general level is fundamental to the understanding of martensitic transformations and shape-memory effects, which are observed in many advanced engineering alloys. Some interesting problems of classical nonlinear elasticity will also be considered. 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. In classical problems (rubber elasticity), tools developed by Healey and co-workers in previous NSF-sponsored work will play an important role. For phase-change problems, an entirely new approach based upon higher-gradient regularization, global continuation and singular limits is being proposed. 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 scien ce. The work has the potential to: (i) provide new mathematical tools for the analysis of hard 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 applications to manufacturing engineering and the design of non-passive or "smart" structures. ********* Let me know if this is not acceptable - I'll be happy to make changes. Thanks, Tim Healey