Variational Methods for Material Damage: Fracture, Fatigue, and Debonding

研究者使用变分方法来研究材料损伤，重点是损伤区域的性质及其时间演化。 损坏的实例包括脆性裂纹、由于疲劳引起的开裂和薄膜的剥离。 一些离散时间模型的损伤演化存在于engineeringcommunity，但这些通常是特设的，并不知道对应于有限的连续时间模型。 本文研究了极限模型的存在性、损伤区域的基本性质以及将静力问题的数值方法扩展到准静力、连续时间问题的方法的合理性。 对材料的损伤具有重要的技术意义，不仅因为它会导致材料失效，还因为它在构建某些纳米结构中起着重要作用，例如通过选择性剥离薄膜。 该项目旨在通过改进数学模型和研究其解决方案的基本性质，以及通过发展和证明数值方法来大大提高我们对这些材料的理解。 数学博士学生被包括在该项目中，并在重要的跨学科问题的数学建模和分析方面受到培训。

The investigator uses variational methods to study materialdamage, with an emphasis on properties of damaged regions andtheir time evolution. Examples of damage include brittle cracks,cracking due to fatigue, and debonding of thin films. Somediscrete-time models for damage evolution exist in the engineeringcommunity, but these are generally ad hoc and not known tocorrespond to limiting continuous-time models. The investigatorstudies the existence of the limiting models, basic properties ofthe damage regions, and the justification for extending numericalmethods for static problems to methods for the quasi-static,continuous-time problems. Damage to materials is of great technological importance notonly because it results in material failure, but also because itplays an important role in building certain nanostructures, forexample by selectively debonding thin films. The project seeks togreatly improve our understanding of these materials by improvingthe mathematical models and studying basic properties of theirsolutions, as well as by developing and justifying numericalmethods. Mathematics Ph.D. students are included in the project,and are trained in the mathematical modeling and analysis ofimportant interdisciplinary problems.