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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.