The energy balance theory for the fracture propagation in brittle and quasi-brittle materials is discussed in a toroidal region around the crack periphery. A simple form requiring the dynamic solution valid only around the crack front and the knowledge of the fracture energy is presented. The results are applied to the fracture of an infinite medium containing a central through crack and subjected to anti-plane shear loads at infinity.
INTRODUCTION
IN DISCUSSING
the fracture of solids under a single application of the load, one may differentiate three types of material response. One would be the so-called crystalline shatter of a perfect crystal with ideally uniform geometry and under ideally uniform external loads. In this case, the fracture may be a complete shatter of the material or an instantaneous rupture along a plane. Another ideal situation may arise if the geometry and loading conditions are such that the fracture nucleation and growth, that is, the formation and propagation of microcracks take place uniformly and simultaneously along a certain plane. In this case, even though the actual fracture velocities may be somewhat limited, due to the multiplicity of fracture nuclei, the time for the solid to rupture may be very short. A carefully grooved homogeneous sheet under uniform tension may come close to satisfying these conditions. A more realistic and common material response however is the propagation of a dominant flaw in the material-the response which is termed in literature as the fracture or the crack propagation.
Depending on the structure of the material, the type of the external loading and the environmental conditions, the growth of the dominant flaw, or the fracture propagation may be either ductile or brittle. Ductile fracture is usually associated with large deformations, very high rates of energy dissipation and slow fracture velocities. Brittle fracture on the other hand, is a low energy failure, and takes place in a catastrophic manner, that is, the fracture velocities are usually very high. The results of the experimental studies indicate that in such low energy type failures, the fracture velocities. in some cases. may be of the order of magnitude of the elastic wave velocities in the solid. Hence, in studying the problem, it becomes necessary to take into account the dynamic nature of the phenomenon. Basically, the problem in this case is the following:
Let a given solid be subjected to a system of time-dependent external loads, generally consisting of any combinations of surface tractions, surface displacements and body forces, and contain an initial imperfection which serves as a fracture nucleus. If the external loads are increased beyond a critical level, fracture propagation will ensue. Let A be the portion of the surface of the solid created as a result of fracture (Fig. 1). Knowing the material characteristics and the environmental conditions, the question is then the determination of the size and the shape of the fracture area, A,