Mathematical Sciences: Phase Transitions, Defects and Nonconvex Variational Problems

数学科学：相变、缺陷和非凸变分问题

• 批准号: 9201215
• 负责人: Irene Fonseca
• 金额: $ 11.4万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-06-15 至 1995-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201215&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Phase; Transitions; Defects

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.

该项目的一般目标是研究材料不稳定性，例如在连续力力学框架中，流体和弹性固体和具有缺陷的晶体的亚稳态平衡状态。 该程序所涉及的理论工具是连续力学，几何测量理论，部分微分方程，热弹性动力学和变异的计算。 将获得非凸和界面能量的弛豫和较低的半内态性能，并将将单数扰动方法用于非线性弹性，以解决平衡的非唯一性。 这些结果将有助于我们理解并预测经过热或机械处理的晶体的表面结构和几何形状。 当未达到最小能量时，将研究表面张力在稳定振荡中所起的作用，并研究微观结构的动态创建及其在广义测量溶液框架内的演变。 当考虑到物理变量（例如组成）时，将考虑使用的相干两相合金的热化学平衡。 这项研究将使能够预测平衡相成分对整体组成和体积分数的依赖性。 分析缺陷相互作用，位错，域的不同类型和排列以及缺陷效应的模型将得到解决。 特别是，将分析具有缺陷的弹性晶体亚稳态平衡的变异配方。 该项目解决的数学问题 - 流体和合金的相变，缺陷的晶体平衡状态，缺陷相互作用和缺陷的全球效应 - 是当代材料科学的强烈工作领域。 直到最近，它们才以系统的方式进行数学解决。 涉及的问题逃脱了古典数学理论的框架。 因此，实现这些目标将需要操纵一些最近的数学工具，并可能引入新的工具。 解决此类问题可能会对行业和技术产生重大影响，因为它将有助于预测经过热或机械处理的晶体的表面结构和几何形状，而且通常，它将促进对智能材料的更好理解。