The Stress-Intensity Factor for nonsmooth fractures in antiplane elasticity

TL;DR

This paper analyzes the asymptotic behavior of the energy near nonsmooth cracks in antiplane elasticity, establishing the existence of the stress intensity factor for such fractures using advanced mathematical techniques.

Contribution

It extends the understanding of stress intensity factors to nonsmooth, connected cracks in 2D elasticity, employing Bonnet's monotonicity and Gamma-convergence methods.

Findings

Blow-up limit of solution near crack points is the cracktip function.

Stress intensity factor is well-defined for nonsmooth cracks.

Method combines monotonicity formula with Gamma-convergence.

Abstract

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 $\sqrt{r}\sin(\theta/2)$, with a well-defined coefficient (the "stress intensity factor" or SIF). The method relies on Bonnet's monotonicity formula \cite{b} together with $\Gamma$-convergence techniques.