Failure initiation at V-notch tips in quasi-brittle materials

At V-notched tips in specimens made of quasi-brittle materials a small damaged or plastic zone is evident that cannot not be neglected in terms of dissipated energy and stress state, although it is small. Herein, to predict the failure initiation at the notch tip, we extend the finite fracture mechanics (FFM) coupled criterion, which requires a simultaneously fulfillment of an energy and a stress criteria. In the small damaged zone, a damage model is introduced so to decrease the effective Young's modulus in a power law in terms of the distance to the notch tip in such a way that the stress field remains bounded. It seems particularly suited to quasi-brittle materials, since no diffuse damage can occur. This damage zone is coupled to the FFM criterion to provide the necessary condition for failure initiation. Under the assumption that the damaged zone and the virtual crack extension are small, matched asymptotic expansions are used. It is shown that the damaged zone grows first, proportionally to the square of the applied load and then, above a threshold, a virtual crack of a given length simultaneously satisfies the energy and stress criteria, and failure occurs. The approach allows taking into account varying tensile strength and material toughness in the damaged zone, as may reasonably be expected. Moreover, it is shown that the same coupled stress-energy criterion can directly be applied to quasi-brittle materials by appropriately using the actual material toughness as measured on a cracked specimen.