On fracture initiation toughness and crack sharpness for Mode II cracks

Fracture toughness of brittle materials is calibrated in experiments where a sample with a preexisting crack is loaded up to a critical point of the onset of static instability. Experiments with ceramics, for example, exhibit a pronounced dependence of the toughness on the sharpness of the crack/notch: the sharper is the crack the lower is the toughness. These experimental results are not entirely compatible with the original Griffith theory of brittle fracture and Linear Elastic Fracture Mechanics which both ignore the crack sharpness.To explain the experimental observations qualitatively we earlier considered Mode I cracks [Volokh KY, Trapper P. Fracture toughness from the standpoint of softening hyperelasticity. J Mech Phys Solids 2008;56:2459-72.] and in the present work we extend our considerations to Mode II cracks. We simulate pure shear of a thin plate with a small crack of a finite and varying sharpness. In simulations we introduce the failure energy as a limiter for the stored energy of the Hookean solid. The energy limiter induces softening, indicating material failure. Thus, elasticity with softening allows capturing the critical point of the onset of static instability of the cracked plate, which corresponds to the onset of the failure propagation at the tip of the crack. In numerical simulations we find that the magnitude of the fracture toughness can not be determined uniquely because it depends on the sharpness of the crack: the sharper is the crack the lower is the toughness.Based on the obtained results we argue that a stable magnitude of the toughness of brittle materials can only be reached when a characteristic size of the crack tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance etc. In other words, the toughness can be calibrated only under conditions where the hypothesis of length-independent continuum fails.