New Mathematical Methods for Dynamic Fracture Evolution

动态断裂演化的新数学方法

• 批准号: 1909991
• 负责人: Christopher Larsen
• 金额: $ 25万
• 依托单位: Worcester Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1909991&HistoricalAwards=false
• 关键词: New; Mathematical; Methods; Dynamic; Fracture

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.

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。然而，我们预测这种故障的能力受到建模的限制，这在某种程度上是特设的，并且数学可用于制定和分析模型，以及证明数值方法。这些问题在动态问题中最为严重，例如当负载快速变化时的冲击。该项目的主要目标是发展新的动态裂缝演化的数学方法。特别是，主要研究者（PI）将扩展方法，定期裂纹路径更现实的路径，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一般的环境中取得进一步的进展是必要的。最后，PI将研究断裂的相场近似，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本数学模型的质量以及对解决方案的基本属性的理解。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，当它们不是时，到目前为止唯一的方法是将路径视为更规则路径的限制。这里的主要技术问题是，相应的弹性动力学的强收敛是必要的能量平衡，以及解决方案的其他属性，但这种收敛在许多情况下仍然开放。另一个基本问题是对于一个给定的裂纹路径的弹性动力学解的唯一性。研究者将在某些设置中显示唯一性，并探索一般的结果，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。虽然在工程界非常受欢迎，但许多属性，包括它们是否接近正确的表面能，或满足最大耗散条件，仍然存在疑问。该奖项反映了NSF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。

项目成果

Variational fracture with boundary loads

边界载荷下的变分断裂

• DOI: 10.1016/j.aml.2021.107437
• 发表时间: 2021
• 期刊: Applied Mathematics Letters
• 影响因子: 3.7
• 作者: Larsen, Christopher J.

Variational phase-field fracture with controlled nucleation

受控成核的变分相场断裂

• DOI: 10.1016/j.mechrescom.2023.104059
• 发表时间: 2023
• 期刊: Mechanics Research Communications
• 影响因子: 2.4
• 作者: Larsen, Christopher J.