Mathematical Sciences: Global Continuation Methods in Nonlinear Elasticity

6/3/96 亲爱的 Hariharan 博士： 我们部门的财务助理现在正在根据您今天的建议制定新的预算。 她将在明天某个时间将其邮寄给您，以便您在我们签署之前进行非官方批准等。文本格式可以吗？或者传真给您会更好吗？ 鉴于预算减少，我准备尽最大努力完成最初的工作范围。 以下是您请求的摘要： ************* DMS-9625830：非线性弹性中的全局连续方法 P.I.：T.J.Healey 摘要。 拟开展非线性弹性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在涉及相变的问题（弱解和椭圆度损失）。 在非常一般的层面上对此类模型进行分析对于理解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。 还将考虑经典非线性弹性的一些有趣问题。 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。 在经典问题（橡胶弹性）中，Healey 及其同事在之前 NSF 资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、全局连续和奇异极限的全新方法。 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，具有在制造工程和非被动或“智能”结构设计中的潜在应用。 ********* 如果这是不可接受的，请告诉我 - 我很乐意做出改变。 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