Asymptotic analysis of small defects near a singular point in antiplane elasticity, with an application to the nucleation of a crack at a notch

Abstract

We use matching asymptotic expansions to treat the antiplane elastic problem associated with a small defect located at the tip of a notch. In a first part, we develop the asymptotic method for any type of defect and present the sequential procedure which allows us to calculate the different terms of the inner and outer expansions at any order. This requires in particular separating in each term its singular part from its regular part. In a second part, the asymptotic method is applied to the case of a crack of variable length located at the tip of a given notch. We show that the first two nontrivial terms of the expansion of the energy release rate are sufficient to well approximate the dependence of the energy release rate on the crack length in the range of values of the length which are sufficient to treat the problem of nucleation. This problem is considered in the last part where we compare the nucleation and the propagation of a crack predicted by two different models: the classical Griffith law and the Francfort-Marigo law based on an energy minimization principle. Several numerical results illustrate the interest of the method.

Stress gradient effects on the nucleation and propagation of cohesive cracks

“…Besides these first results on shape or size effects, [10,11] treated the nucleation and propagation of a cohesive crack problem at the notch tip in the same context of Dugdale's model. In such a situation, since the notch induced a stress singularity if the response is purely elastic, there exists no elastic phase in the loading process and a cohesive crack is created at the notch tip as soon as a load is applied.…”

“…The nucleation and growth of a crack at a regular point. The goal of the present work is to consider the same problem as in [10,11] but for another type of structure. In particular, we assume that the body contains neither a notch nor any corner which would induce elastic singularities.…”

“…(i) The elastic branch starts at t = 0 and finishes at t = t e given by (11). All along this branch, there is no crack, a(t) = 0.…”

A generalisation to cohesive cracks evolution under effects of non-uniform stress field

“…Besides these first results on shape or size effects, [17,18] treat the problem of the nucleation and the propagation of a cohesive crack at the tip of a notch, still in the context of Dugdale's model. In such a situation, since the notch would induce a singularity of the stress if the response was purely elastic, there exists no elastic phase in the loading process, a cohesive crack is created at the tip of the notch as soon as a load is applied.…”

“…The goal of the present work is to consider the same problem as in [17,18] but for another type of structure. Specifically, we assume that the body contains neither a notch nor any corner which would induce elastic singularities.…”