New Mathematical Methods for Fracture Evolution

本研究项目涉及断裂力学中的基本数学问题，这是材料和结构工程中的一个重要领域。尽管最近在断裂和裂纹扩展模型的数学分析方面取得了很大进展，但材料缺陷的成核和扩展，特别是断裂，仍然知之甚少，但它们的准确预测在许多材料科学应用中非常重要。该项目旨在开发新的数学方法，以解决这一领域的一些主要挑战。这些包括显示存在的解决方案的断裂演化的数学模型类，改进动态断裂模型，并分析动态断裂解决方案的属性，特别强调探索裂纹分支及其后果。在断裂力学的数学分析中，如何证明准静态内聚断裂演化的存在性，如何证明动态断裂数学模型的存在性，以及如何建立动态断裂解的定性性质是一个重要的挑战。已经被用来显示准静态格里菲斯演化的存在的方法，现在已知的是失败的内聚断裂。主要的困难来自于历史在这些解决方案的定义中所起的微妙作用。该项目将继续开发新的方法来分析这个和其他准静态问题，基于高阶能量近似使用历史，只有有限数量的前一个时间。动态格里菲斯断裂也是非常微妙的，由于复杂的相互作用之间的弹性奇点和（先验未知的）不断变化的裂纹集。基于爆破技术的新方法将被开发用于分析这些演变。

This research project concerns fundamental mathematical questions in fracture mechanics, an area of importance in materials and structural engineering. Despite substantial recent progress in mathematical analysis of models for fracture and crack propagation, nucleation and propagation of material defects in general, and fracture in particular, remain poorly understood, yet their accurate prediction is of great importance in many materials science applications. This project aims to develop new mathematical methods for addressing some of the major challenges in this area. These include showing existence of solutions to classes of mathematical models for fracture evolution, improving dynamic fracture models, and analyzing properties of dynamic fracture solutions, with a particular emphasis on exploring crack branching and its consequences. Showing existence of quasi-static cohesive fracture evolutions, showing existence for mathematical models of dynamic fracture, and establishing qualitative properties of dynamic fracture solutions are major challenges in the mathematical analysis of fracture mechanics. The methods that have been used to show existence for quasi-static Griffith evolutions are now known to fail for cohesive fracture. The main difficulty arises from the delicate role that history plays in the definition of these solutions. This project will continue the development of new methods for analyzing this and other quasi-static problems, based on higher order energy approximations using history at only a finite number of prior times. Dynamic Griffith fracture is also very delicate, due to complex interactions between elastic singularities and the (a priori unknown) evolving crack set. New methods based on blow-up techniques will be developed for analyzing these evolutions.