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

脆性弹性固体中演化断裂的研究一直是并且仍然是应用数学家、物理学家和工程师的一个具有挑战性的课题，其对档案文献的兴趣继续迅速增加。脆性骨折的瞬时冲击和增长对各种机械系统（例如大型结构、船舶、加压飞机机舱和人体内的矫形植入物）具有固有的危险。 因此，对材料断裂失效的基本理解对于应用至关重要。 这个项目的一个中心主题是使用新的隐式本构关系适用于弹性固体脆性断裂模型的发展。本研究的另一个重要目标是为新的裂缝理论和模拟发展裂缝开发收敛的数值方案。该项目将为博士后助理和研究生提供数学和科学计算方面的研究培训机会。在过去几十年中，在理想弹性固体中裂缝准静态和动态扩展的数学适定模型方面取得了重大进展。然而，许多这些方法使用的线性本构关系，承认一个有据可查的逻辑不一致的奇点。另一种方法是使用本构关系，该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。研究人员将开发一种完全离散的有限元方法，用于准静态以及弹性动力学断裂演化。这两个将被制定为一个约束的能量最小化的正则化变分模型，由两个条款组成，一个代表的非线性弹性体能量定义的凭借应变限制本构关系，其他的表面能的断裂。本计画的另一个目标是开发正则化相场裂缝模型的并行有限元求解器，并分析数值解。计算框架将在固体力学的经典基准案例上进行测试，并使用制造的解决方案进行收敛分析。从这项研究中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，它们将在高强度材料（如胶质金属和钛合金）的断裂研究中具有实际意义。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.