Simulating the pervasive fracture of materials and structures using randomly close packed Voronoi tessellations

Abstract

Under extreme loading conditions most often the extent of material and structural fracture is pervasive in the sense that a multitude of cracks are nucleating, propagating in arbitrary directions, coalescing, and branching. Pervasive fracture is a highly nonlinear process involving complex material constitutive behavior, material softening, localization, surface generation, and ubiquitous contact. A pure Lagrangian computational method based on randomly close packed Voronoi tessellations is proposed as a rational and robust approach for simulating the pervasive fracture of materials and structures. Each Voronoi cell is formulated as a finite element using the Reproducing Kernel Method. Fracture surfaces are allowed to nucleate only at the intercell faces, and cohesive tractions are dynamically inserted. The randomly seeded Voronoi cells provide a regularized random network for representing fracture surfaces. Example problems are used to demonstrate the proposed numerical method. The primary numerical challenge for this class of problems is the demonstration of model objectivity and, in particular, the identification and demonstration of a measure of convergence for engineering quantities of interest.

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

“… indicated that ‘ an element can be crossed by a discontinuity in such a way that one of the two resulting parts of the element becomes so small that the critical time increment for a stable solution procedure will become almost infinitesimal ’. In this general context, according to Bishop , ‘ once crack branching and crack coalescence phenomena appear , the prospect of modeling a multitude of arbitrary three‐dimensional intersecting cracks quickly becomes untenable ’.…”

Uniform Random Voronoi Meshes

“…For two dimensional meshing problems, this angle may be randomized by moving the mesh vertex on the boundary [2], but in higher dimensions this leads to non-planar facets.…”

“…"Voronoi froths" because of their similar geometry to soap bubbles.) See Bishop [2] for an overview of fully Lagrangian fracture simulations over Voronoi froths, including element formulation and cell movement.…”

“…Traversing an element may involve walking dynamic datastructures. For performing analysis, developing shape functions for Voronoi elements is non-trivial [2]. Many codes are available for computing Voronoi tessellations, and related structures, for point sets.…”

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

“…Conforming polygonal finite elements [1][2][3][4] and finite elements on convex polyhedra [5][6][7] require the integration of nonpolynomial basis functions. The integration of polynomials on irregular polytopes arises in the non-conforming variable-element-topology finite element method [8,9], discontinuous Galerkin finite elements [10], finite volume element method [11] and mimetic finite difference schemes [12][13][14].…”