9.2. K-Version of Smeared-Tip Method for Cohesive Fracture

In the smeared-tip method, one superposes the solutions of the given body for various lengths of sharp (LEFM) cracks, the tips of which are continuously distributed (smeared) along the crack path. Such superposition is used to represent the solution of a cohesive crack (Fig. 9.1). Any opening profile and any stress profile can be represented in this manner.

For a single LEFM mode I crack,

(9.5)

where P applied load or a parameter of a system of distributed and concentrated loads; b body thickness; K _I = SIF for Mode I and crack tip located at ?; k( ?) = = dimensionless SIF = SIF value for D = b = P = 1, which is a function of the relative crack length ? = a/D for a body of given geometry; g( ?) = dimensionless energy release function; a = actual crack length; and D = characteristic size or dimension of the structure (Fig. 9.1). The stresses ? on the crack line ahead of the crack tip and the crack opening width w behind the crack tip are, according to LEFM,

(9.6)

(9.7)

(9.8)

(9.9)

where E' = E/(1 - v ²) for plane stress, E' = E for plane strain, E = Young's modulus, v = Poisson's ratio; and b ₁ , b ₂, ..., c ₁,