We use matching asymptotic expansions to treat the anti-plane 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. That requires in particular to separate 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 non trivial 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.