Eigendamage: an Eigendeformation model for the variational approximation of cohesive fracture -- a one-dimensional case study

TL;DR

This paper introduces an eigendeformation model for approximating cohesive fracture in a one-dimensional setting, demonstrating that as the approximation parameter tends to zero, the model converges to a cohesive zone model.

Contribution

It proposes a novel eigendeformation-based approximation scheme for cohesive fracture and proves its convergence to a cohesive zone model as the approximation parameter vanishes.

Findings

The approximation functionals $ ext{Gamma}$-converge to the cohesive zone model.

The model captures inelastic responses via a non-local energy functional.

The approach provides a rigorous link between eigendeformation models and cohesive fracture theories.

Abstract

We study an approximation scheme for a variational theory of cohesive fracture in a one-dimensional setting. Here, the energy functional is approximated by a family of functionals depending on a small parameter $0 < \varepsilon \ll 1$ and on two fields: the elastic part of the displacement field and an eigendeformation field that describes the inelastic response of the material beyond the elastic regime. We measure the inelastic contributions of the latter in terms of a non-local energy functional. Our main result shows that, as $\varepsilon \to 0$, the approximate functionals $\Gamma$-converge to a cohesive zone model.