A mixed extended finite element for the simulation of cracks and heterogeneities in nearly incompressible materials and metal plasticity

This paper describes a mixed extended finite element (XFEM) formulation that can handle nearly incompressible material behaviour for fracture mechanics problems as well as for heterogeneous materials under small and finite elastic and elastoplastic deformations without showing locking behaviour. Typical applications are the simulation of cracks and their propagation in rubber materials, microstructure simulations of rubber and simulations of heterogeneities in J(2) and crystal plasticity. The developed finite element technique is an extension of the well established mixed Q1P0 formulation. Herein, enrichments as they are commonly used within the XFEM are considered to capture discontinuities within the displacement and/or strain field. The volumetric stress is discontinuous across a crack face or an interface between two materials. For that reason the piecewise constant ansatz used in the Q1P0 formulation for the volumetric stress needs to be enriched with suitable enrichment functions as well. Therefore, the developed mixed extended finite element formulation is named XQ1XP0. With several examples it is shown that the new formulation does not show locking effects even for almost incompressible materials and for significant plastic deformations and that it shows superior convergence behaviour compared to a standard formulation.