The stress intensity factor for non-smooth fractures in antiplane elasticity

Motivated by some questions arising in the study of quasistatic growth in brittle fracture, we investigate the asymptotic behavior of the energy of the solution u of a Neumann problem near a crack in dimension 2. We consider non smooth cracks K that are merely closed and connected. At any point of density 1/2 in K, we show that the blow-up limit of u is the usual “cracktip” function ${Csqrt{r}sin(theta/2)}$ , with a well-defined coefficient (the “stress intensity factor” or SIF). The method relies on Bonnet’s monotonicity formula (Bonnet, Variational methods for discontinuous structures, pp. 93–103. Birkhäuser, Basel, 1996) together with Γ-convergence techniques.