New Variational Methods for Quasi-static and Dynamic Material Defect Evolution

准静态和动态材料缺陷演化的新变分方法

• 批准号: 1313136
• 负责人: Christopher Larsen
• 金额: $ 34.4万
• 依托单位: Worcester Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-15 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1313136&HistoricalAwards=false
• 关键词: New; Variational; Methods; Quasi; static

Larsen1313136 While defects in materials play a fundamental role in material failure, their analysis remains a major challenge in applied mathematics. This is partly due to the difficulty of formulating precise mathematical models, and partly due to the difficulty of analyzing the free surfaces and singularities involved. The investigator extends recent successes in the analysis of globally minimizing and locally minimizing quasi-static evolutions to both locally stable quasi-static evolutions and dynamic evolutions. One goal is to develop and study new models for cohesive fracture and plasticity with softening, based on local stability rather than global minimality (which is mathematically problematic). The investigator also studies existence and analyzes fundamental properties of dynamic fracture solutions, based on models he formulated previously. The failure of materials rests on the nucleation and evolution of defects such as cracks, plastic regions, and damage. The ability to accurately predict failure depends on the quality of the underlying mathematical models of these defects, as well as on understanding fundamental properties of solutions. Substantial challenges remain in these areas, both in formulating sound models and in the analysis of qualitative behavior of solutions. The investigator seeks to make fundamental progress on these fronts, by developing new models that are both mathematically well-posed and significantly more physically realistic than existing models, and performing the mathematical analysis necessary to assess their accuracy.

虽然材料中的缺陷在材料失效中起着重要作用，但它们的分析仍然是应用数学中的一个主要挑战。这部分是由于制定精确的数学模型的困难，部分是由于分析所涉及的自由曲面和奇点的困难。研究者将最近在全局最小化和局部最小化准静态演化分析中的成功扩展到局部稳定准静态演化和动态演化。一个目标是基于局部稳定性而不是全局最小值（这在数学上是有问题的），开发和研究具有软化的内聚断裂和塑性的新模型。研究者还研究了动态裂缝解的存在性，并基于他之前制定的模型分析了动态裂缝解的基本性质。材料的失效取决于裂纹、塑性区和损伤等缺陷的成核和演化。准确预测故障的能力取决于这些缺陷的基础数学模型的质量，以及对解决方案基本属性的理解。在这些领域，无论是在制定合理的模型还是在分析解决方案的定性行为方面，都存在着重大的挑战。研究者试图在这些方面取得根本性的进展，通过开发新的模型，这些模型在数学上是良好的，并且比现有的模型在物理上更加真实，并进行必要的数学分析来评估它们的准确性。