Variational Methods for Some Problems in Materials Science

BoceaDMS-0806789 该项目的动机是在材料科学的两个不同领域的几个问题。 该项目的第一部分承担了一个通用理论的发展，允许有效的表征的屈服集的一个多晶体通过适当的变分原理在L-无穷大。 一个新的变分形式主义自然导致几个问题的新兴领域的变分法在L-无穷的情况下，泛函被优化的本质上确界的一些表达式涉及divergence自由领域。 在这里，自然的方法是最小化某些相关量的最大值，而不是它们的最大值。 研究者在这个方向上研究选定的问题，重点是那些可能影响我们对可塑性的理解的问题。 其中，研究的偏微分方程系统，产生的Aronssonequations相关的新的变分原理在L-无穷大是特别感兴趣的。 该项目的第二部分涉及非线性膜理论从三维弹性的推导，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量最小化器的结构，并指出了适当的最小化序列。 该项目的目的是建立在严格的数学基础上的一些工程师和材料科学家使用的传统模型，另一方面，探索新的变分原理的情况下，一个人想知道有多大的一个数量可以在其最大值，而不仅仅是平均值。这个问题出现在科学和工程的许多领域，在弹性力学、图像处理、损伤和断裂力学以及塑性力学等应用中具有根本的重要性。研究者在项目的第一部分研究了这个问题。 在第二部分中，他论述了从三维弹性理论导出非线性薄膜理论。这部分是由于重要的技术应用需要更好地理解和预测马氏体材料薄膜中界面的形成。该项目揭示了最近提出的新的有效薄膜能量的最小化的结构，并指出适当的最小化序列，预计将提供一些深入了解技术上有用的活性材料结构的设计，比现有的薄膜理论所建议的更复杂。

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.