A Griffith-Euler-Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics

TL;DR

This paper derives an effective Griffith-Euler-Bernoulli model for thin brittle beams from nonlinear fracture mechanics, capturing elastic bending and fracture behavior in the limit of small aspect ratio.

Contribution

It introduces a rigorous derivation of a simplified beam model incorporating fracture effects from complex nonlinear variational fracture mechanics models.

Findings

Derivation of a Griffith-Euler-Bernoulli functional for brittle beams.

The model includes elastic bending and fracture discontinuities.

Validation of the model in the small aspect ratio limit.

Abstract

We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The elastic part of this functional is the classical Euler-Bernoulli functional for thin beams in the bending dominated regime in terms of the curve's curvature. In addition there also emerges a fracture term proportional to the number of discontinuities of the curve and its first derivative.