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

简体中文 虽然材料中的缺陷在材料失效中起着根本性的作用，但它们的分析仍然是应用数学中的一个重大挑战。 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表面和奇点的困难。 调查员扩展最近的成功，在分析全球最小化和局部最小化的准静态演化的局部稳定的准静态演化和动态演化。 一个目标是开发和研究新的模型内聚断裂和塑性软化，局部稳定性的基础上，而不是全球最小值（这是数学上的问题）。 研究人员还研究存在和分析动态断裂解决方案的基本属性，根据他以前制定的模型。 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本数学模型的质量，以及对解决方案基本性质的理解。 在这些领域，无论是在制定健全的模型，还是在分析解决方案的定性行为方面，都仍然存在着巨大的挑战。 研究人员试图在这些方面取得根本性进展，通过开发新的模型，这些模型在数学上是良好的，并且比现有模型在物理上更加逼真，并进行必要的数学分析以评估其准确性。

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.