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

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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.