Remarks on Crack-Bridging Concepts

The article draws upon recent work by us and our colleagues on metal and ceramic matrix composites for high temperature engines. The central theme here is to deduce mechanical properties, such as toughness, strength and notch-ductility, from bridging laws that characterize inelastic processes associated with fracture. A particular set of normalization is introduced to present the design charts, segregating the roles played by the shape, and the scale, of a bridging law. A single material length, δ₀E/σ₀, emerges, where δ₀ is the limiting-separation, σ₀ the bridging-strength, and E the Young’s modulus of the solid. It is the huge variation of this length—from a few nanometers for atomic bond, to a meter for cross-over fibers—that underlies the richness in material behaviors. Under small-scale bridging conditions, δ₀E/σ₀ is the only basic length scale in the mechanics problem and represents, with a pre-factor about 0.4, the bridging zone size. A catalog of small-scale bridging solutions is compiled for idealized bridging laws. Large-scale bridging introduces a dimensionless group, a/(δ₀E/σ₀), where a is a length characterizing the component (e.g., hole radius). The group plays a major role in all phenomena associated with bridging, and provides a focus of discussion in this article. For example, it quantifies the bridging scale when a is the unbridged crack length, and notch-sensitivity when a is hole radius. The difference and the connection between Irwin’s fracture mechanics and crack bridging concepts are discussed. It is demonstrated that fracture toughness and resistance curve are meaningful only when small-scale bridging conditions prevail, and therefore of limited use in design with composites. Many other mechanical properties of composites, such as strength and notch-sensitivity, can be simulated by invoking large-scale bridging concepts.