A derivation of linearized Griffith energies from nonlinear models

TL;DR

This paper derives linearized Griffith fracture energies from nonlinear models using Gamma-convergence, providing a rigorous link between complex nonlinear fracture mechanics and simplified linearized models.

Contribution

It introduces a novel derivation of linearized Griffith energies from nonlinear, frame-indifferent energies in brittle fracture through a Gamma-convergence approach.

Findings

Convergence of nonlinear energies to linearized Griffith energies.

Limiting configurations include partitions and piecewise rigid deformations.

The limiting functional incorporates both fracture energy and segmentation energy.

Abstract

We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in brittle fracture via Gamma-convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional comprises also the segmentation energy which is necessary to disconnect the parts of the specimen.