Energy release rate for non smooth cracks in planar elasticity

TL;DR

This paper characterizes the energy release rate for non-smooth cracks in planar elasticity, analyzing displacement blow-up limits and deriving the rate as the elastic energy derivative with respect to crack growth.

Contribution

It introduces a novel analysis of the energy release rate for cracks with minimal regularity, including blow-up limits and derivative characterization.

Findings

Displacement blow-up limits are established as positively 1/2-homogeneous functions.

The energy release rate is derived as the elastic energy derivative with respect to crack extension.

The study provides a framework for understanding crack propagation in non-smooth crack geometries.

Abstract

This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with density $1/2$ at the tip. First, the blow-up limit of the displacement is analyzed, and the convergence to the corresponding positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate is obtained as the derivative of the elastic energy with respect to an infinitesimal additional crack increment.