Probability distribution of fracture elongation, strength and toughness of notched rectangular blocks with lognormal Young's modulus

We find analytical approximations to the probability distribution of fracture properties of one-dimensional rods and thin two-dimensional plates when Young's modulus varies spatially as an isotropic lognormal field. The properties considered are the elongation, strength, and toughness modulus at fracture initiation and at ultimate failure. This is an extension of a previous study that, under the same conditions, dealt with the distribution of the bulk elastic moduli (Dimas et al., 2015). For all quantities at fracture initiation our approach is analytical in 1D and semi-analytical in 2D. For ultimate failure, we quantify the random effects of fracture propagation and crack arrest by fitting regression models to simulation data and combine the regressions with the distributions at fracture initiation. The results are validated through a series of Monte Carlo simulations. Through parametric analysis, we gain insight into the strengthening/weakening roles of the Euclidean dimension and size of the specimen and the variance and correlation function of the log-modulus field.