Quasi-static crack propagation in plane and plate structures using set-valued traction-separation laws

We introduce a numerical technique to model set-valued traction-separation laws in plate bending and also plane crack propagation problems. By using of recent developments in thin (Kirchhoff-Love) shell models and the extended finite element method, a complete and accurate algorithm for the cohesive law is presented and is used to determine the crack path. The cohesive law includes softening and unloading to origin, adhesion and contact. Pure debonding and contact are obtained as particular (degenerate) cases. A smooth root-finding algorithm (based on the trust-region method) is adopted. A step-driven algorithm is described with a smoothed law which can be made arbitrarily close to the exact non-smooth law. In the examples shown the results were found to be step-size insensitive and accurate. In addition, the method provides the crack advance law, extracted from the cohesive law and the absence of stress singularity at the tip.