An examination of the effects of some idealized models of fracture on accelerating cracks

Abstract

A new representation, based on the solution of a diffraction problem given by F.G. Friedlander, is presented for the stress field produced by a semi-infinite accelerating crack in a longitudinal shear field. This is employed to give an explicit expression for the field at any point generated by an incident plane stress wave and also to obtain asymptotic expressions for the components of strain and velocity near the crack tip when the crack is loaded in any prescribed manner. The relationship of the Griffith-Irwin energy balance to the Dugdale-Barenblatt model of small-scale yielding is next examined, and it is concluded that the Griffith-Irwin energy balance is acceptable for all but the most rapid loading conditions, provided the yielding is small-scale. Finally, some effects of large-scale yielding are investigated by adopting the Dugdale model, with a plastic zone of finite size, the motion of a crack subjected to a step load being investigated in detail. An important simplifying approximation, which is adequate in this case and likely also to be so in most practical contexts, is noted, which very drastically reduces the complexity of the mathematical analysis. It is concluded that a crack with a finite plastic zone accelerates less rapidly than one for which the yielding is small-scale, although the accelerations predicted are still unrealistically high. It is expected that this defect will be remedied by the future inclusion of rate-dependence into the model.