Stochastic model of an R-curve due to crack bridging in fibrous and textured ceramics

A stochastic model, previously formulated to analyze crack tip shielding from an applied load in nontransforming polycrystalline ceramics with equiaxial grains, has been extended to include non-equiaxial and oriented grains. The aspect ratio of fibrous grains has been varied from one to 10, and the fraction of fibrous grains oriented normally to the crack plane from 0.33 (representing random orientation) to 0.8 for textured materials. The role of the distribution of grain size, aspect ratio, degree of texturing, and strength of grains and interfaces in the development of the crack bridging is analyzed and numerically evaluated. It is assumed that the crack closure stress arises predominantly due to the frictional pullout of grains, i.e. the contribution from elastic bridges is neglected. The model parameters are chosen to represent ceramics such as aluminum oxide or silicon nitride, as extensive experimental data are available for various microstructural variants of these materials. The principal parameters resulting from the model include a maximum increment of the resistance to fracture Delta G due to the frictional bridging of long cracks, Delta G(m), and a slope of Delta C vs crack extension within the first 100 mu m of the crack growth, V-G = d(Delta G)/dx. As long cracks will almost never have a chance to develop in ceramic components under tensile or bending service stresses, it is argued that the study of V-G = d(Delta G)/dx could be more informative as far as reliability of ceramics is concerned. Variation of both parameters is extensively mapped as a function of the microstructural characteristics, such as grain aspect ratio, grains orientation, grain and interfacial strength, and pullout stress due to grain surface roughness. The model outcome confirms that long, strong, oriented grains would result in the most effective toughening of R-curve exhibiting ceramics. However, an optimum grain aspect ratio that results in the largest values of saturated toughness is predicted. In most cases the optimum aspect ratio is 4:8, but can be close to one for some microstructures. The general range of variation of the saturated toughness as a function of varying model parameters is similar to 100 J/m(2)- similar to 350 J/m(2). Weak interfacial shear strength is desirable in most cases to achieve large final toughening. An optimum grain aspect ratio of 4:8 is required to maximize the toughening gradient VG The highest toughening gradient V-G is expected for strong grains which experience large stresses due to resistance to pullout. These conditions would not simultaneously generate the largest saturated toughness. It is concluded, therefore, that the routes to achieve the large ultimate toughening and the high reliability in ceramics are different. It is predicted that as long as the pullout stress due to interfacial roughness is maintained high, high-reliability ceramics could be produced even with equiaxial grains. It appears that the grain aspect ratio of about four for strong, oriented grains with rough interfaces which are easy to separate is a necessary condition for achieving high reliability of R-curve exhibiting ceramics. (C) 1997 Acta Metallurgica Inc.