TOUGHENING BY CRACK BRIDGING IN HETEROGENEOUS CERAMICS
The toughening of a ceramic by crack bridging is considered, including the heterogeneity caused simply by spatial randomness in the bridge locations, The growth of a single planar crack is investigated numerically by representing the microstructure as an array of discrete springs with heterogeneity in the mechanical properties of each spring, The stresses on each microstructural element are determined, for arbitrary configurations of spring properties and heterogeneity, using a lattice Green function technique. For toughening by (heterogeneous) crack bridging for both elastic and Dugdale bridging mechanisms, the following key physical results are found: (i) growing cracks avoid regions which are efficiently bridged, and do not propagate as selfsimilar penny cracks; (ii) crack growth thus proceeds at lower applied stresses in a heterogeneous material than in an ordered material; (iii) very little toughening is evident for moderate amounts of crack growth in many cases; and (iv) a different R-curve is found for every particular spatial distribution of bridging elements. These results show that material reliability is determined by both the flaw distribution and the ''toughness'' distribution, or local environment, around each flaw. These results also demonstrate that the ''microstructural'' parameters derived from fitting an R-curve to a continuum model may not have an immediate relationship to the actual microstructure; the parameters are ''effective'' parameters that absorb the effects of the heterogeneity. The conceptual issues illuminated by these conclusions must be fully understood and appreciated to further develop microstructure-property relationships in ceramic materials.