Mathematical Sciences: Phase Transitions, Defects and Nonconvex Variational Problems

该项目的总体目标是在连续介质力学框架内研究材料不稳定性，例如流体和弹性固体的相变以及含有缺陷的晶体的亚稳平衡态。本程序涉及的理论工具有连续介质力学、几何测度论、偏微分方程组、热弹性力学和变分法。得到了非凸体能和界面能的松弛和下半连续性质，并在非线性弹性力学中用奇异摄动方法解决平衡点的不唯一性问题。这些结果将有助于我们理解和预测经过热或机械处理的晶体的表面结构和几何形状。当不能达到最小能量时，我们将在广义测值解的框架内研究表面张力在稳定振动中所起的作用，以及微观结构的动态产生和演化。当考虑到组成等物理变量时，将考虑相干两相合金的热化学平衡。这项研究将能够预测平衡相组成对总组成和体积分数的依赖关系。将讨论用于分析缺陷相互作用、位错、不同类型和排列的磁区以及缺陷的全局效应的模型。特别是，将分析具有缺陷的弹塑性晶体亚稳态平衡的变分公式。该项目中涉及的数学问题--流体和合金的相变、具有缺陷的晶体的平衡态、缺陷相互作用和缺陷的整体效应--是当代材料科学中紧张工作的领域。直到最近，它们才得到了系统的数学解决。所涉及的问题脱离了经典数学理论的框架。因此，实现这些目标将需要操纵一些最新的数学工具，并可能引入新的工具。解决这些问题可能会对行业和技术产生重大影响，因为这将有助于预测受热或机械处理的晶体的表面结构和几何形状，总体上将促进对智能材料的更好理解。

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.