FE modeling of fracture in quasi-brittle material

A new approach for simulation of fracture in quasi-brittle material is proposed. The structure is discretized using classical triangular finite elements (FE). Only mode I fracture is considered. Cracks are able to develop along the element edges while elements behave elastically. The inelastic deformations are localized at the element nodes, in which crack control is carried out in terms of traction force. At every node, the inelastic displacement (i.e. crack opening) and the traction force are related by means of a single branch softening curve. The structural formulation is based on the Colonnetti approach. The non-linear response is obtained as the superposition of the linear-elastic responses to given external actions and to plastic strains conceived as unknown imposed strains. The unknown independent variables of the corresponding mathematical models are the nodal displacements of the elastic model (as standard in FE method) plus the inelastic non-compatible displacements. From a computational point of view, the problem is posed as a parametric linear complementarity problem (PLCP) and it is solved using mathematical programming algorithms. A new procedure is developed such that, automatically, the load parameter is inverted in sign when the mathematical problem is no longer semi definite positive. The procedure terminates or because a solution is found for the given loads, or an unbounded solution (i.e. collapse) has been reached or the load factor has been decreased down to zero during the softening phase. A numerical example from literature is presented to validate the method.