Mathematical Sciences: Global Continuation Methods in Nonlinear Elasticity

亲爱的Hariharan博士：从今天开始，我们部门的财务助理正在根据您的建议制定新的预算。在我们签字之前，她会在明天某个时候寄出去给你非官方的批准，等等。文本格式可以吗，还是传真给你更好？鉴于预算减少，我准备尽最大努力完成原来的工作范围。这里是您要求的摘要：************* DMS-9625830：非线性弹性的全局延续方法P.I: t.j.h healey摘要。我们计划开展非线性弹性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涉及相变的问题上（弱解和椭圆性损失）。在非常一般的水平上对这种模型的分析是理解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。还将考虑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。在经典问题（橡胶弹性）中，由Healey和他的同事在以前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、全局延拓和奇异极限的全新方法。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)为分析工程实践中的难题提供新的数学工具，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。*********让我知道，如果这是不可接受的-我会很高兴做出改变。谢谢你，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