A non-standard numerical method for finite element modelling of tensile cracks in quasi-brittle material

In this paper, a non-standard finite element (FE) method, coupled with a mathematical programming procedure, is presented to simulate initiation and propagation of in-plane tensile cracks in quasi-brittle material. Potential crack lines are defined and limited by the FE edges. The cracks are considered as localized plastic deformations imposed at the nodes along the potential crack paths. Crack opening is detected at each node by a limit value of tensile force normal to the crack direction. When the limit value is reached, a cohesive crack starts, following a linear softening branch. The mechanical model is such that no remeshing neither the introduction of interfaces is necessary to describe the discontinuity between elements, as usually required in standard FE approaches. The non-linear solution of the structural problem is obtained using a mathematical programming procedure based on the solution of a parametric linear complementarity problem (PLCP). The main advantage of the PLCP formulation is its ability to manage structural configurations in which instability and multiplicity of solutions are possible, providing one solution even at points characterized by a not-positive-semidefinite Hessian matrix. Simple numerical simulations are presented with the aim to test the method and the solution procedures.