Extended finite simulations of cohensive crack propagation under mode-3 loading

Abstract

Modeling of propagating discontinuities (cracks) is a challenging problem in the field of computational mechanics. Numerical techniques such as Finite Difference Method (FDM), Boundary Element Method (BEM), and Finite Element Method (FEM) have been widely used in the literature to model propagating cracks. Solutions obtained from the above said numerical tools are mesh sensitive. In a FE based simulation, crack faces are modeled as boundaries of elements to satisfy the continuity requirements. Partition of Unity Method (PUM) (Melenk and Babuska 1]) relaxes these stringent requirements by extending the basis of FE interpolation shape functions. In this technique, the standard displacement approximation is enriched with local functions that describe the behavior of the solution around the crack tip. Belytschko and Black [2] modified the classical FEM by enriching the nodes around the crack by local functions and this technique was termed as Extended Finite Element Method (X-FEM ). The rate of convergence is faster in XFEM compared to classical FEM. The enrichment functions are obtained from the asjmaptotic singular crack tip fields and represent the local behavior o f the solution near the crack tip. The stress singularity prevailing at the crack tip is only a mathematical artifact, which was eliminated by introducing a cohesive zone model (Dugdale [3] and Barenblatt [4]). Cohesive zone models are a phenomenological description of the constitutive relation between the tractions and their corresponding displacement jumps. In order to incorporate the cohesive zone modeling into the X-FEM framework, the variation of the displacement field around the cohesive zone tip should be known a priori. We use a spatially-dependent cohesive zone model that yields a closed form solution for the displacement field surrounding the cohesive zone tip. The new enrichment functions are developed based on the closed form solution to model cohesive cracks. The cohesive based X-FEM developed in this study can be used to model intersonic crack propagation, where the stress singularity is a function of the propagation velocity. This numerical tool can also be used in modeling crack propagation in inelastic media. We have demonstrated the applicability of the technique to simulate steady state rectilinear cohesive crack propagation