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

研究者使用变分方法研究材料损伤，重点研究损伤区域的性质及其时间演变。损伤的例子包括脆性裂纹、疲劳开裂和薄膜脱粘。工程学界存在一些损伤演化的离散时间模型，但这些模型通常是特别的，并且不知道与有限的连续时间模型相对应。研究了极限模型的存在性，损伤区域的基本性质，以及将静态问题的数值方法推广到准静态连续问题的方法的理由。材料的损伤在技术上具有重要意义，不仅因为它会导致材料失效，而且因为它在构建某些纳米结构中起着重要作用，例如通过选择性地剥离薄膜。该项目旨在通过改进数学模型和研究其解的基本性质，以及通过开发和证明数值方法，极大地提高我们对这些材料的理解。数学博士生也包括在这个项目中，并在重要的跨学科问题的数学建模和分析方面进行训练。

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.