Variational Methods for Some Problems in Materials Science

BOcean DMS-0806789该项目的动机是材料科学的两个不同领域中的几个问题。该项目的第一部分致力于发展一种普遍的理论，该理论允许通过适当的无穷大变分原理来有效地刻画多晶的屈服集。一种新的变分形式自然而然地引出了L无穷远中正在兴起的变分领域中的几个问题，当要优化的泛函是涉及无散度场的某些表达式的本质上确界时。在这里，自然的方法是最小化某些相关量的最大值，而不是它们的平均值。研究人员在这个方向上研究选定的问题，重点是那些可能对我们理解可塑性产生影响的问题。其中，研究以Aronson方程形式出现的偏微分方程组与新的无穷大变分原理是特别有意义的。该项目的第二部分涉及从三维弹性出发的非线性膜理论的推导，以寻求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源的减水剂的结构，并指出了适当的最小化顺序。该项目的目标是建立在严格的数学基础上，一方面建立工程师和材料科学家使用的一些传统模型，另一方面探索新的变分原理，以便人们想知道某个量在其最大值时可以达到多大，而不仅仅是平均。这个问题出现在许多科学和工程领域，在弹性力学、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。研究人员在项目的第一部分研究了这个问题。第二部分，从三维弹性力学出发，推导了非线性薄膜理论。这在一定程度上是因为需要更好地了解和预测马氏体材料薄膜中界面的形成，这一需求源于重要的技术应用。该项目揭示了研究人员最近提出的一种新的有效薄膜能量的最小化的结构，并指出了适当的最小化序列，这些序列有望为设计技术上有用的活性材料结构提供一些见解，这些结构比现有的薄膜理论所建议的结构更复杂。

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.