Mathematical Sciences: Phase Transitions, Defects and Nonconvex Variational Problems

这个项目的总体目标是研究材料 不稳定性，例如流体和弹性体的相变 固体和亚稳平衡状态的晶体， 在连续统力学框架中， 理论 在这个程序中涉及的工具是连续介质力学，几何 测度论，偏微分方程， 热弹性动力学和变分法。 非凸的松弛与下连续性 将获得体能和界面能， 扰动方法将用于非线性弹性， 解决均衡的非唯一性。 这些结果将帮助我们 了解和预测的表面结构和几何形状 经过热处理或机械处理的晶体。 当未达到最小能量时， 表面张力稳定振荡， 微观结构的动态产生及其在 广义测度值解的框架，将是 研究了 共格两相合金的热化学平衡 当物理变量，如成分，被纳入 帐户将被考虑。 这项研究将有助于预测 平衡相组成对总的 组成和体积分数。 用于分析缺陷相互作用、位错、 不同类型和安排的领域和全球影响 缺陷将得到解决。 特别是，变 弹塑性晶体亚稳平衡公式 缺陷将被分析。 在这个项目中解决的数学问题-阶段 流体和合金的转变， 缺陷晶体，缺陷相互作用和整体效应 缺陷--是当代材料中的高强度工作领域 科学 直到最近，它们才被数学地解决 以系统的方式。 所涉及的问题脱离了框架 经典数学理论的一部分 因此，实现 这些目标将需要操纵一些最近的 数学工具和可能的新工具的引入。 解决这些问题可能会对世界产生重大影响。 工业和技术，因为它将有助于预测表面 晶体的结构和几何形状受到热或 机械治疗，一般来说，它会促进更好的 对智能材料的理解。

The general objective of this project is to study material instabilities, such as phase transitions for fluids and elastic solids and metastable equilibrium states for crystals with defects, in a Continuum Mechanics framework. The theoretical tools involved in this program are continuum mechanics, geometric measure theory, partial differential equations, thermoelastodynamics and the calculus of variations. Relaxation and lower semicontinuity properties of nonconvex bulk and interfacial energies will be obtained and singular perturbation methods will be used in nonlinear elasticity to resolve nonuniqueness of equilibria. These results will help us understand and predict the surface structures and geometries of crystals subjected to thermal or mechanical treatments. When the minimum energy is not attained, the role played by the surface tension in stabilizing the oscillations, and the dynamical creation of the microstructure and its evolution within the framework of generalized measure-valued solutions, will be studied. Thermochemical equilibria for coherent two-phase alloys when physical variables, such as composition, are taken into account will be considered. This study will enable predicting the dependence of equilibrium phase composition on the overall composition and volume fraction. Models to analyze defect interaction, dislocations, different types and arrangements of domains and global effects of defects will be addressed. In particular, variational formulations for metastable equilibria of elastoplastic crystals with defects will be analyzed. The mathematical problems addressed in this project -- phase transitions for fluids and alloys, equilibrium states for crystals with defects, defect interaction and global effects of defects -- are areas of intense work in contemporary material science. Only recently they have been addressed mathematically in a systematic way. The questions involved escape the framework of classical mathematical theories. Therefore, accomplishing these goals will require manipulation of some very recent mathematical tools and possibly the introduction of new ones. Addressing such issues may have a significant impact in the industry and technology as it will help to predict the surface structures and geometries of crystals subjected to thermal or mechanical treatments and, in general, it will promote a better understanding of smart materials.