Mathematical Sciences: Nonconvex Variational Problems and Material Instabilities CMU Proposal 06823

DMS-9500531 Fonseca该项目的总体目标是研究与材料不稳定性有关的变分问题。解决的现象将包括相变，亚稳平衡状态和晶体中的位错和其他缺陷的存在，微观结构的发病和浓度的创建，断裂和损伤之间的相互作用，以及液晶的行为和缺陷。他们的研究脱离了经典数学理论的框架，集中于材料科学中非常当代的研究问题。这一计划的成功将需要操纵最近开发的数学工具，引入新的数学工具，它可能会对工业和技术的进步产生相当大的影响，澄清材料科学和智能材料中的问题。 该计划涉及的领域是连续介质力学，几何测量理论，偏微分方程，热弹性动力学和变分法。在基本的数学问题中，可以找到约束变分问题，相边界的演化，界面能，松弛技术和广义测度值解。

DMS-9500531 Fonseca The general objective of this project is the study of variational problems in connection with material instabilities. The phenomema addressed will include phase transitions, metastable equilibrium states and the presence of dislocations and other defects in crystals, the onset of microstructure and the creation of concentrations, interaction between fracture and damage, and the behavior and defects of liquid crystals. Their study escapes the framework of classical mathematical theories, and it concentrates on very contemporary research issues in material science. The success of this program will require manipulation of recently developed mathematical tools, the introduction of new ones, and it may have a considerable impact in the advance of industry and technology, clarifying problems in material sciences and smart materials. The areas involved in this program are Continuum Mechanics, Geometric Measure Theory, Partial Differential Equations, Thermoelastodynamics, and the Calculus of Variations. Among the underlying mathematical questions, one can find constrained variational problems, evolution of phase boundaries, interfacial energies, relaxation techniques, and generalized measure-valued solutions.