Computer simulation and prediction of rapid fracture processes presents many inherent challenges.  Under fast crack growth, fracture is considered pervasive since cracks nucleate and propagate dynamically in complex patterns that branch and coalesce in arbitrary directions.  Pervasive fracture is a strongly nonlinear phenomenon: in addition to modeling contact, complex constitutive behavior must be accurately captured, including material softening, crack nucleation, and crack growth. The current state-of-the-art computational method for dynamic fracture is the cohesive Galerkin finite element method. This approach requires the insertion of evolving fracture surfaces into a pre-determined finite element discretization of the domain of interest, which allows crack nucleation and growth to be outcomes of the analysis. However, the element shapes in the standard finite element method are restricted to triangles and quadrilaterals in two dimensions (tetrahedra, prisms and hexahedra in three dimensions). This is a drawback when used in pervasive fracture, since they limit the possible fracture network and tend to bias the topology of cracks, potentially leading to non-natural crack shapes. Over the past decade, the development of generalized barycentric coordinates (referred to as basis functions when used in Galerkin methods) has permitted more general polygonal and polyhedral element shapes to be valid within the finite element method. With polygonal and polyhedral element formulations, an unlimited number of convex and nonconvex element shapes are available, which reduces mesh bias induced by element selection. Random element shapes also provide a non-preferential fracture network and allow for more natural, unbiased cracks to propagate in a continuum.

In this project, we have developed and implemented a new approach for the modeling of pervasive fracture with polygonal and polyhedral finite elements by combining the simplicity of cohesive inter-element surfaces with the relaxed meshing requirements of generalized barycentric coordinates. Some of the advantages of this approach include: (1) a more realistic inter-element fracture network using random polygonal and polyhedral elements; (2) reduction in bias of crack growth direction compared to fracture modeling using inter-element surfaces on traditional finite elements; (3) accurate dissipation of energy required to form new crack surfaces through the framework of cohesive surfaces; and (4) general compatibility with existing finite element paradigms, which provide the ability to leverage proven finite element technology. A standalone C++ code has been developed that provides two- and three-dimensional dynamic fracture capabilities. A total Lagrangian description of the balance laws of a continuum is used. Large-scale fracture processes create new surfaces unpredictably, and furthermore, all of these surfaces may come into contact with each other. Therefore contact detection and enforcement is needed, which is accomplished using a node-to-surface algorithm. The cohesive surfaces are described by traction-separation constitutive relationship. The maximal Poisson sampling algorithm is used to generate the point-distribution, and the Voronoi tessellation of these points (nodes) is used to generate the meshes.

Polygonal and polyhedral finite element methods show a great deal of flexibility in solving some of the outstanding challenges in computational mechanics, and in this project, their merits for dynamic fracture simulations were established.  The fracture simulations on polygonal meshes were in agreement with benchmark dynamic fracture experiments (Kalthoff-Winkler), and were also able to replicate fracture surfaces from a dynamic fracture notched glass experiment. Furthermore, the numerical simulations result in crack branching, which are in correspondence with the experiments. This project has demonstrated the viability and promise of using polygonal and polyhedral finite elements to model pervasive fracture.  Broader impacts of this project have been on a few different fronts. It has led to further advancing polygonal and polyhedral finite element technology for numerical simulations, and in particular for the modeling of dynamic fracture. The findings from this project have been disseminated to the broader research community through journal publications and research presentations at conferences, national laboratories and academic institutions. This project has fostered research collaborations with scientists at Sandia National Laboratories and Los Alamos National Laboratory. The PI, together with Professor Kai Hormann (University of Lugano, Switzerland), has developed the first book that provides a modern perspective on generalized barycentric coordinates with applications in computer graphics and solid mechanics. The edited book is entitled: ``Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics,'' and was published by CRC Press in October 2017.

Last Modified: 12/11/2017
Modified by: Natarajan Sukumar