The Euler-Bernoulli limit of thin brittle linearized elastic beams

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