Brittle fracture of solids with arbitrary cracks

Abstract

A B S T R A C TOne of the problems of fracture mechanics is the prediction of the propagation of cracks in solids. Thepresent paper deals mainly with linear fracture mechanics which owes its origin to the works erA. A. Griffith [1,2] and studies the development of cracks under sufficiently low loads when the behaviour of the material within a region sufficiently remote from the edges of cracks may he regarded as linearly elasti~ At present, linear fracture mechanics [3] is restricted mainly to special kinds of loading geometry, with the crack extending rectilinearly (in a plane case) or in its plane (in a three-dimensional case). The main problem here is to establish a relationship between the dimensions of cracks and the loads applied. Within the framework of linear fracture mechanics the fracture itself and other non-linear phenomena that precede it are assumed to take place only within local regions which are small compared to the dimensions of cracks. The possibility that such a situation exists is associated with the fact that when the crack dimensions are sufficiently large the characteristic dimension of the end region is fully determined by a certain intrinsic dimension of the material structure. Therefore, if the material does not exhibit time dependency, the state of the end region at the moment of rupture becomes fully independent of the loads applied and the geometry of the solid, i.e. autonomous. The notion of autonomy [4] leads to the formulation of this theory as one of limit equilibrium.If the conditions of rectilinear extension of the crack (or those of the crack extension in its plane) are disturbed, there arises a problem of determining not only the dimensions of the crack, but also the path of the crack extension under such conditions of loading that a slow, quasi-static crack development is possible. This problem can be actually subdivided into two: (1) Criteria for the determination of the dimensions and paths of the crack extension, and (2) Expressions for the characteristics of the stress-strain state which are constituents of these criteria through the geometry of solid with cracks and the loads applied.As regards (1), there have been many assertions, and the connections between them are not quite clear, at present. The first of the suggested criteria, namely that of local symmetry for the plane problem formulated by Barenblatt and Cherepanov [5,6] and by Erdogan and Sih [7] can be within certain limits substantiated and generalized for the threedimensional case. The guiding principle here is the treatment of the theory of cracks from the standpoint of the method of inner and outer expansions or that of singular perturbations [8]. The concept of the stress intensity factor which is basic in linear fracture mechanics is decisive in matching inner and outer expansions to find the main term of the asymptotic solution of the complete problem. Actually the construction of the theory of equilibrium cracks [4] implicitly employs this technique for a certain specific model. More explicit indicat...

Recent developments in dynamic fracture: some perspectives

“…In 2D and under quasi-static conditions, v c R , it has been shown that the Principle of Local Symmetry [93] offers good approximations for crack paths [94][95][96][97][98][99][100]. This approximation suggests that under quasi-static conditions, and under the assumption that no intrinsic length scales exist near the tip, crack propagation along smooth paths will annihilate any shear (mode II) component of the fields ahead of the tip.…”

Theory of dynamic crack branching in brittle materials

“…Griffith's energy criterion (Griffith, 1920;Freund, 1990;Broberg, 1999) and the principle of local symmetry (Gol'dstein and Salganik, 1974;Leblond, 1989) imply…”

“…As shown in (Adda-Bedia, 2004b), the paths of the branches can be determined if each crack tip satisfies the principle of local symmetry during propagation (Gol'dstein and Salganik, 1974). Imposing this (Eq.…”

“…The principle of local symmetry states that the crack advances in such a way that in-plane shear stresses always vanish in the vicinity of the crack tip. This rule was first proposed for quasi-static cracks (Gol'dstein and Salganik, 1974;Leblond, 1989), and generalized to rapidly moving cracks (Adda-Bedia et al, 1999). It was shown in (Adda-Bedia et al, 1999) that the two criteria arise from the same physical origin.…”

Biaxial stress controlled three-dimensional helical cracks

“…If cracks are under mixed-mode loading, 8 such as tensile and shear stresses, they often propagate in a curved or kinked manner. [9][10][11][12] Under certain conditions, the kinking paths may continue regularly to form various crack patterns. 13,14 For example, two-dimensional spiral or sinusoidal cracks can arise in the dried crusts of aqueous suspensions 15 or in glass slides under gradient temperature fields.…”