A Path-following technique implemented in a Lagrangian formulation to model quasi-brittle fracture

Abstract

A path-following strategy based on an energy release criterion is proposed and implemented in an augmented Lagrangian formulation to model localized cohesive cracks. The model is formulated in a rigourous variational framework of continuous spaces. A procedure for the solution of the proposed formulation in a continuous spaces context is also presented. The discrete form of the problem for the implementation within the finite element method is also exposed. The ability and robustness of the proposed method to track equilibrium paths including snap-back phenomenon are analyzed and discussed by means of several numerical tests. The restriction based on an energy release criterion is introduced in the formulation by using a augmented Lagrangian method. The effect of the interpolation order for the Lagrange multiplier (so-called Lagrange field), its influence on the proposed constraint equation and the mesh density dependency are studied and illustrated in the numerical simulations. Different crack patterns are analyzed and some recommendations concerning the interpolation degree that should be used for the Lagrangian field to avoid spurious energy release are also included.