Variational Fracture with Loads

对于从民用基础设施到国防的广泛应用，了解材料的失效至关重要。 预测故障的能力取决于建模以及可用于制定和分析模型以及证明数值方法的数学方法。 在过去的二十年里，在这一领域，特别是在断裂力学方面，有了重大的数学进展，但这些结果在很大程度上限于没有施加力的模型。然而，包括施加的力对于应用是相关的。该项目旨在改进目前的数学公式与施加的力平衡的材料中的断裂，将此公式扩展到材料演变的模型，并研究这些演变的某些功能与施加的力的存在。 该项目为博士生提供培训研究机会。 准确预测故障的能力取决于缺陷的基础数学建模的质量，以及对解决方案基本属性的理解。 直到最近，静态和准静态断裂的变分模型一直限于Dirichlet边界条件，因为不存在对包括Neumann边界条件的看似最自然的公式的解，即，边界载荷该项目的目的是改进最近引入的具有边界载荷的变分断裂的静态公式，该公式可以有解，将该静态模型扩展到准静态情况并显示解的存在性，并研究了在准核中存在能降的可能性该奖项反映了美国国家科学基金会的法定使命，并被认为是值得通过使用基金会的智力价值进行评估来支持的和更广泛的影响审查标准。

For a large range of applications, from civil infrastructure to national defense, understanding the failure of materials is critical. The ability to predict failure depends on modeling and on the mathematics available to formulate and analyze models, and to justify numerical methods. Over the last twenty years, there have been significant mathematical advances in this area, particularly in fracture mechanics, but these results are largely limited to models without applied forces. However, the inclusion of applied forces is relevant for applications. This project aims to improve the current mathematical formulation for fracture in materials in equilibrium with applied forces, to extend this formulation to models for material evolution, and to study certain features of these evolutions related to the presence of applied forces. The project offers training research opportunities for doctoral students. The ability to accurately predict failure depends on the quality of the underlying mathematical modeling of defects, as well as on understanding fundamental properties of solutions. Until recently, variational models for static and quasi-static fracture have been limited to Dirichlet boundary conditions, since there do not exist solutions to the seemingly most natural formulation that includes Neumann boundary conditions, i.e., boundary loads. The aim of the project is to improve on a recently introduced static formulation for variational fracture with boundary loads which can have solutions, to extend this static model to the quasi-static case and show existence of solutions, and to study the possibility of existence of energy drops in quasi-static models.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.