Stochastic model of an R-curve due to crack bridging

A stochastic model is formulated to analyse crack tip shielding from the applied load, as a function of microstructural parameters and loading conditions, in nontransforming polycrystalline ceramics. The model recognizes the random nature of the microstructural elements, such as grains, inclusions or fibers, which are traversed by the propagating crack. The role of distribution of grain size, and strength of grains and interfaces in the development of the crack interface bridging is emphasized, and numerically evaluated. The standard model parameters are chosen to represent aluminium oxide, as an extensive experimental data base is available for this material. Quantitative predictions of toughening and closure stresses within the bridging process zone are in agreement with experimental data quoted in the literature. It is found that a typical coarse-grained alumina with geometric average grain size of 10 μm and geometric standard deviation of 1.3 exhibits a 5 mm long bridging zone, with the maximum closure stress of 86 MPa, and the maximum toughening due to crack bridging of 90 J/m². The R-curve has been confirmed to depend both on the average grain size and on the grain size distribution, as well as on the level of residual stresses, single grain strength, interfacial roughness and the grain boundary strength. The validity of the relatively simple Monte Carlo model proposed in this work opens up a possibility for optimization of microstructures of monolithic and composite ceramics for maximum resistance to fracture.