SUMMARYA displacement-based continuous-Galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics. The polyhedra can have an arbitrary number of vertices or faces. The faces of the polyhedra can have an arbitrary number of edges and can be nonplanar. The polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex-averaged centroid. Conforming shape functions are constructed using harmonic functions defined on the undeformed configuration, thus requiring the use of a total-Lagrangian finite element formulation in large deformation applications. For nonlinear applications with computationally intensive constitutive models, it is important to minimize the number of element quadrature points. For this reason, an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices. As a first step toward verifying the element behavior in the general nonlinear setting, several linear verification examples are presented using both random Voronoi meshes and distorted hexahedral meshes. For the hexahedral meshes, results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation. The element behavior in the nearly incompressible regime is also examined.
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.
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