The Euler-Bernoulli limit of thin brittle linearized elastic beams

TL;DR

This paper proves that the energy of thin brittle elastic beams converges to a one-dimensional Euler-Bernoulli beam model, providing a sharp compactness result with bounded jumps, advancing the mathematical understanding of brittle beam limits.

Contribution

It establishes a rigorous Gamma-convergence and a sharp compactness result for brittle elastic beams, extending previous work with new convergence and rigidity insights.

Findings

Energy convergence to Euler-Bernoulli beam model

Sharp compactness with bounded jumps

Weak convergence after subtracting piecewise rigid motions

Abstract

We show that the linear brittle Griffith energy on a thin rectangle $\Gamma$-converges after rescaling to the linear one-dimensional brittle Euler-Bernoulli beam energy. In contrast to the existing literature, we prove a corresponding sharp compactness result, namely a suitable weak convergence after subtraction of piecewise rigid motions with the number of jumps bounded by the energy