Visco-energetic solutions for a model of crack growth in brittle materials

TL;DR

This paper introduces visco-energetic solutions as a new approach to modeling crack growth in brittle materials, extending existing variational models by incorporating viscous corrections and proving their existence and compliance with Griffith's criterion.

Contribution

It develops the concept of visco-energetic solutions for crack growth, providing existence results and linking to Griffith's criterion in brittle fracture modeling.

Findings

Existence of visco-energetic solutions with initial cracks.

Solutions satisfy Griffith's criterion under certain conditions.

Extension of variational models with viscous corrections.

Abstract

Visco-energetic solutions have been recently advanced as a new solution concept for rate-independent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco-energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introduced by Almgren, Taylor, and Wang in the study of mean curvature flows. With our main result we prove the existence of a visco-energetic solution with a given initial crack. We also show that, if the cracks have a finite number of tips evolving smoothly on a given time interval, visco-energetic solutions comply with Griffith's criterion.