A three-dimensional ES-FEM for fracture mechanics problems in elastic solids

• ES-FEM is extended to 3D fracture mechanics to obtain SIFs and simulate crack growth.
• A 3D singular ES-FEM with singular fields of arbitrary order is developed.
• The singular element employs a simple PIM to construct shape functions.
• The stiffness matrix involves only the shape functions (not their derivatives).
• Several numerical examples are illustrated and compared to published references.

The edge-based smoothed finite element method (ES-FEM) using triangular mesh was recently proposed to model the fracture problems in 2D solids. This paper contains the following ingredients: (1) the ES-FEM is extended to three-dimensional (3D) ES-FEM using tetrahedral elements to compute the stress intensity factors and simulate crack propagation in 3D elastic solids; (2) to model the singular fields of arbitrary order near the crack front, a layer of specially designed seven-noded crack-tip element is constructed; (3) the displacement is then enriched with ease to reproduce the necessary order of stress singularity; and (4) the enrichment is done without losing the essential properties of partition-of-unity and the linear function reproduction. Because the singular ES-FEM uses the strain smoothing technique and it is a typical weakened weak (W2) formulation, the system stiffness matrix is computed employing only the shape function values on the surface of the smoothing domains created based on the edges of elements. No derivatives of the shape functions are needed, and thus no mapping and integration for the W2 form is required. Several numerical examples are presented to validate the effectiveness of the proposed method.