LEAPS-MPS: Time-Discrete Regularized Variational Model for Brittle Fracture in Novel Strain-Limiting Elastic Solids

对于应用数学家、物理学家和工程师来说，脆性弹性固体中演化裂缝的研究一直是一个具有挑战性的课题，而且对档案文献的兴趣也在迅速增加。脆性骨折的瞬时冲击和生长对各种机械系统（如大型结构、船舶、加压飞机机舱和人体骨科植入物）具有固有的危险性。因此，对材料因断裂而失效的基本理解对于应用是必不可少的。该项目的中心主题是使用适用于弹性固体脆性断裂的新型隐式本构关系开发模型。为新的裂缝理论和演化裂缝的模拟开发收敛的数值格式是本研究的另一个重要目标。该项目将为博士后和研究生提供数学和科学计算方面的研究培训机会。在过去的几十年里，在建立理想弹性固体中准静态和动态裂缝扩展的数学模型方面取得了实质性进展。然而，这些方法中的许多都使用线性本构关系，这种关系允许有充分记录的逻辑不一致的奇点。另一种方法是使用本构关系，在整个材料体中均匀地限制应变值。该项目的首要目标是使用后者来研究弹性材料的行为并预测实验观察到的现象。研究者将开发一种准静态和弹性动力裂缝演化的全离散有限元方法。两者都将被表述为正则化变分模型的约束能量最小化，该模型由两项组成，一项代表由应变极限本构关系定义的非线性弹性体能，另一项代表断裂的表面能。本课题的另一个目标是开发正则相场断裂模型的并行有限元求解器，并对数值解进行分析。计算框架将在固体力学的经典基准案例中进行测试，并使用制造的解决方案进行收敛分析。由此产生的数学模型和有限元技术将在计算力学中具有更广泛的适用性，例如，它们将在研究高强度材料（如口香糖金属和钛合金）的断裂方面具有实际意义。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。

The study of evolving fractures in brittle elastic solids has been and still is a challenging topic for applied mathematicians, physicists, and engineers, and its interest to the archival literature continues to increase rapidly. The instantaneous impact and growth of brittle fractures are inherently dangerous to a variety of mechanical systems such as large structures, ships, pressurized airplane cabins, and orthopedic implants in the human body. Therefore, a fundamental understanding of the failure of materials due to fracture is essential for applications. A central theme of this project is the development of models using novel implicit constitutive relations applicable to brittle fracture in elastic solids. Developing convergent numerical schemes for the new fracture theory and simulation of evolving fractures is another crucial objective of this research. The project will provide research training opportunities in mathematics and scientific computing for postdoctoral associates and graduate students.In the last few decades, substantial progress has been made in formulating mathematically well-posed models for quasi-static and dynamic propagation of fractures in ideally elastic solids. However, many of these approaches use a linear constitutive relation which admits a well-documented logically inconsistent singularity. An alternative approach is to use constitutive relations which limit the strain values uniformly in the entire material body. The overarching objective of this project is to use the latter to study the behavior of elastic materials and to predict experimentally observed phenomena. The investigator will develop a fully discrete finite element method for quasi-static as well as elastodynamical fracture evolution. Both will be formulated as a constrained energy minimization of a regularized variational model consisting of two terms, one representing the nonlinear elastic bulk energy defined by virtue of strain-limiting constitutive relationship, the other the surface energy of a fracture. Another objective of this project is to develop parallel finite element solvers for the regularized phase-field fracture model and to analyze the numerical solutions. The computational framework will be tested on classical benchmark cases in solid mechanics and convergence analysis will be performed using a manufactured solution. The mathematical models and the finite element techniques emerging from this research will have broader applicability in computational mechanics, for example they will be of practical interest in the study of fractures in high-strength materials such as gum metals and titanium alloys.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.