We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

48/2006

Convergence of equilibria of planar thin elastic beams

Maria Giovanna Mora, Stefan Müller and Maximilian G. Schultz

Abstract

We consider a thin elastic strip $\Omega_h = (0,L) \times (-h/2, h/2)$, and we show that stationary points of the nonlinear elastic energy (per unit height) $E^h(v) = \frac{1}{h} \int_{\Omega_h} (W(\nabla v) - h^2 g(x_1) {\,\cdot\,} v) \, dx$ whose energy is bounded by $C h^2$ converge to stationary points of the Euler-Bernoulli functional $J_2(\bar{y}) = \int_0^L (\frac{1}{24} E \kappa^2 - g{\,\cdot\,} \bar{y}) \, dx_1$ where $\bar{y}: (0,L) \to \mathbb{R}^2$, with $\bar{y}' = \binom{ \cos \theta}{\sin \theta}$, and where $\kappa = \theta'$. This corresponds to the equilibrium equation $-\frac{1}{12} E \theta'' + \tilde{g}{\,\cdot\,} \binom{-\sin \theta}{\cos \theta} = 0$, where $\tilde{g}$ is the primitive of $g$. The proof uses the rigidity estimate for low-energy deformations [4] and a compensated compactness argument in a singular geometry. In addition, possible concentration effects are ruled out by a careful truncation argument.