Variational Methods for Some Problems in Materials Science

材料科学中一些问题的变分法

• 批准号: 0806789
• 负责人: Marian Bocea
• 金额: $ 6.65万
• 依托单位: North Dakota State University Fargo
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806789&HistoricalAwards=false
• 关键词: Variational; Methods; Some; Problems; Materials

BoceaDMS-0806789 The project is motivated by several questions in twodistinct areas of materials science. The first part of theproject undertakes the development of a general theory thatallows for efficient characterizations of the yield set of apolycrystal by means of suitable variational principles inL-infinity. A new variational formalism leads naturally toseveral problems in the emerging area of calculus of variationsin L-infinity for the case where the functionals to be optimizedare the essential supremum of some expression involvingdivergence-free fields. Here the natural approach is to minimizethe maximum of certain relevant quantities instead of theiraverage. The investigator studies selected issues in thisdirection, with an emphasis on those that can have an impact onour understanding of plasticity. Among these, the study of thesystems of partial differential equations that arise as Aronssonequations associated to the new variational principles inL-infinity is of particular interest. The second part of theproject deals with the derivation of nonlinear membrane theoriesfrom three-dimensional elasticity, seeking to better understandthe formation of interfaces in thin films of martensiticmaterials. The project sheds light on the structure of theminimizers for a new effective thin film energy, and indicatesappropriate minimizing sequences. The project aims to set on rigorous mathematical groundssome of the traditional models used by engineers and materialscientists and, on the other hand, to explore new variationalprinciples in the case where one wants to know how big somequantity can be at its maximum value, not merely on average. This question arises in many areas of science and engineering andis of fundamental importance in such applications as elasticity,image processing, damage and fracture mechanics, and plasticity. The investigator studies this question in the first part of theproject. In the second part he deals with the derivation ofnonlinear membrane theories from three-dimensional elasticity. This is motivated in part by the need, originating from importanttechnological applications, to better understand and predict theformation of interfaces in thin films of martensitic materials. The project sheds light on the structure of the minimizers for anew effective thin film energy recently proposed by theinvestigator, and indicates appropriate minimizing sequences thatare expected to provide some insight into the design oftechnologically useful active material structures, more complexthan those suggested by existing thin film theories.

该项目是由材料科学的两个不同领域的几个问题所驱动的。该项目的第一部分是开发一种通用理论，该理论允许通过l -∞中合适的变分原理有效地表征多晶的产率集。一种新的变分形式很自然地导致了l -∞变分微积分这一新兴领域的几个问题，在这种情况下，要优化的泛函是一些涉及无散度场的表达式的本质上的最大值。在这里，自然的方法是最小化某些相关数量的最大值，而不是最小化它们的平均值。研究者在这个方向上选择研究问题，重点是那些可能对我们对可塑性的理解产生影响的问题。其中，对与l无穷新变分原理相关的以aronsson方程形式出现的偏微分方程系统的研究是特别有趣的。该项目的第二部分涉及从三维弹性中推导非线性膜理论，以更好地理解马氏体材料薄膜中界面的形成。该项目揭示了一种新的有效薄膜能量最小化器的结构，并指出了适当的最小化序列。该项目旨在将工程师和材料科学家使用的一些传统模型建立在严格的数学基础上，另一方面，在人们想知道某个量的最大值（而不仅仅是平均值）有多大的情况下，探索新的变分原理。这个问题出现在许多科学和工程领域，并且在诸如弹性、图像处理、损伤和断裂力学以及塑性等应用中具有重要的基础意义。研究者在项目的第一部分研究这个问题。在第二部分中，他讨论了从三维弹性出发的非线性膜理论的推导。这在一定程度上是由于来自重要技术应用的需要，以便更好地理解和预测马氏体材料薄膜中界面的形成。该项目揭示了研究者最近提出的新有效薄膜能量最小化器的结构，并指出了适当的最小化序列，这些序列有望为技术上有用的活性材料结构的设计提供一些见解，这些结构比现有薄膜理论提出的结构更复杂。