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

研究人员使用变分方法来研究材料损伤，重点研究了损伤区域的性质及其时间演化。损坏的例子包括脆性裂纹、疲劳引起的破裂和薄膜脱粘。在工程领域中存在一些用于损伤演化的离散时间模型，但这些模型通常是特别的，并且不是已知的对应于极限连续时间模型。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、连续时间问题方法的合理性。材料损伤在技术上具有重要意义，不仅因为它会导致材料失效，而且还因为它在构建某些纳米结构中发挥着重要作用，例如通过选择性地使薄膜脱粘。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来极大地提高我们对这些材料的理解。该项目包括数学博士生，他们接受重要跨学科问题的数学建模和分析方面的培训。

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.