New Mathematical Methods for Dynamic Fracture Evolution

对于从民用基础设施到国防的大范围应用，了解材料的失效至关重要。然而，我们预测这种失败的能力受到建模（这在某种程度上是临时的）和用于制定和分析模型以及证明数值方法的数学的限制。当加载变化很快时，这些问题在动态问题中最为严重，例如冲击。该项目的主要目标是开发新的动态裂缝演化数学方法。特别是，首席研究员（PI）将扩展常规裂缝路径的方法到更现实的路径，有扭结和分支。第二个目标是解决基本的数学问题，这些问题对于在完全通用的环境中取得进一步进展是必要的。最后，PI将研究裂缝的相场近似，这已经成为工程界非常流行的工具，但仍然知之甚少。准确预测故障的能力取决于缺陷的基础数学模型的质量，以及对解决方案基本特性的理解。当裂纹路径是规则的时，可以用数学方法来研究这些演化。然而，当它们不是，到目前为止，唯一的方法是考虑路径是更规则路径的极限。这里的主要技术问题是，相应弹性动力学的强收敛对于能量平衡以及解的其他性质是必要的，但这种收敛在许多情况下仍然是开放的。另一个基本问题是给定裂纹路径的弹性动力解的唯一性。研究者将在某些设置中显示独特性，并探索一般后果，例如裂纹速度的界限。该项目的最终目标是分析裂缝的相场模型。虽然在工程界非常流行，但一些性质，包括它们是否近似正确的表面能，或满足最大耗散条件，仍然是悬而未决的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

For a large range of applications, from civil infrastructure to national defense, understanding the failure of materials is critical. Yet, our ability to predict this failure is limited by both modeling, which is somewhat ad hoc, and the mathematics available to formulate and analyze models, as well as to justify numerical methods. These issues are most severe in dynamic problems, such as impacts, when loading changes quickly. The main goal of this project is the development of new mathematical methods for dynamic fracture evolution. In particular, the principal investigator (PI) will extend methods for regular crack paths to more realistic paths, with kinking and branching. A second goal is to address fundamental mathematical issues that are necessary for further progress in completely general settings. Finally, the PI will study phase-field approximations of fracture, which have become very popular tools in the engineering community but remain poorly understood.The ability to accurately predict failure depends on the quality of the underlying mathematical models of defects as well as on understanding fundamental properties of solutions. When crack paths are regular, mathematical methods are available to study these evolutions. However, when they are not, the only methods so far involve considering the paths to be limits of more regular paths. The main technical issue here is that strong convergence of the corresponding elastodynamics is necessary for energy balance, as well as for other properties of solutions, but this convergence remains open in many situations. Another fundamental issue is uniqueness of elastodynamic solutions for a given crack path. The investigator will show uniqueness in certain settings, and explore general consequences, such as bounds on crack speed. The final goal of the project is to analyze phase-field models for fracture. While very popular in the engineering community, a number of properties, including whether they approximate the correct surface energy, or satisfy a maximal dissipation condition, remain open questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.