Effect of stress singularity magnitude on scaling of strength of quasi-brittle structures

Engineering structures are often designed to have complex geometries, which could introduce stress singularities that are weaker than the conventional21=2 crack-tip singularity. Extrapolating the results of small-scale laboratory tests to predict the response of a full-scale structure comprised of quasi-brittle materials requires an understanding of how the weak stress singularities modify the classical energetic and statistical scaling theories of quasi-brittle fracture. Through a theoretical and numerical study, a new scaling law for quasi-brittle fracture is derived, which explicitly relates the nominal structural strength to the structure size and the magnitude of the stress singularity. The theoretical analysis is based on a generalized weakest-link model that combines the energetic scaling of fracture with the finite weakest-link model. The model captures the transition from the energetic scaling to statistical scaling as the strength of the stress singularity diminishes. The new scaling law is in close agreement, for the entire range of stress singularities, with the size effect curves predicted through finite-element simulations of concrete beams containing an arbitrary-angle V-notch under Mode-I fracture.