Relationship between fracture surface pattern, crack geometry and fracture toughness in brittle materials

摘要

Mecholsky et cd. [1] experimentally investigated the relationship between the fractal dimension of the fracture surface profile, D' (lD2) and the fracture toughness, Kc, in alumina and glass ceramics. They found that the increment of fracture toughness (Kc = Kic - Ko) was in proportion to the increase in the square root of the fractional part of the fractal dimension (D* = D' - 1), such that Kc = A(D*)1/2, where Ko is the toughness of the materials for a smooth (Euclidean) fracture surface and i is a constant. The author [2] analysed the relationship between the fracture toughness and the fractal dimension of the crack (E>, 1Dg 2) in brittle materials on the basis of a fractal geometry model, and obtained an equation: lnKic = l/ 2(ln(2r(l - v2)) - (E> - 1)In rL] (E = Young's modulus: v = Poisson's ratio; T = the work done in creating a unit crack surface; p: rmjn/Vmax, the ratio of the lower limit (rm) and the upper limit (rmax) of the scale length (r), between which a crack exhibits a fractal nature). A similar functional dependence can be expected between the fracture toughness, Kc, and the fractal dimension of the fracture surface, D (2SDi 3), in brittle materials, although the generalized analysis based on the fractal geometry has not been made on the relationship between them. Let us consider a fracture surface in brittle materials (Fig. 1). The fracture surface area, S, can be measured by covering with squares of side length, r [3]. The number of squares covering the fracture surface, TV, is generally correlated through the fractal dimension of the fracture surface, D (2 =5 D =s 3), with the side length r [4].