Universal size effect law and effect of crack depth on quasi-brittle structure strength

In cohesive fracture of quasi-brittle materials such as concrete, rock, fiber composites, tough ceramics, rigid foams, sea ice, and wood, one can distinguish six simple and easily modeled asymptotic cases: the asymptotic behaviors of very small and very large structures, structures failing at crack initiation from a smooth surface and those with a deep notch or preexisting deep crack, the purely statistical Weibull-type size effect, and the purely energetic (deterministic) size effect. Size effect laws governing the transition between some of these asymptotic cases have already been formulated. However, a general and smooth description of the complex transition between all of them has been lacking. Here, a smooth universal law bridging all of these asymptotic cases is derived and discussed. A special case of this law is a formula for the effect of notch or crack depth at fixed specimen size, which overcomes the limitations of a recently proposed empirical formula by Duan, 2003, 2004, 2006.