article

Title: Convergence of equilibria for planar thin elastic beams

Authors: Stefan Muller, Maria Giovanna Mora and Maximilian G. Schultz

Issue: Volume 56 (2007), Issue 5, 2413-2438

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) \[ E^h(v) = (1/h) \int_{\Omega_h} (W(\nabla v) - h^2g(x_1) \cdot v) \mathrm{d}x \] whose energy is bounded by $C h^2$ converge to stationary points of the Euler-Bernoulli functional \[ J_2(\bar{y}) = \int_{0}^{L} \left(\frac{1}{24} \mathcal{E}\kappa^2 - g \cdot \bar{y}\right) \mathrm{d}x_1 \] 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 $-\frac{1}{12} \mathcal{E}\theta'' + \tilde{g} \cdot \binom{-\sin\theta}{\cos\theta} = 0$, where $\tilde{g}$ is the primitive of $g$. The proof uses the rigidity estimate for low-energy deformations [G. Friesecke, R.D. James, and S. M\"uller, \textit{A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity}, Comm. Pure Appl. Math. \textbf{55} (2002), No. 11, 1461--1506] and a compensated compactness argument in a singular geometry. In addition, possible concentration effects are ruled out by a careful truncation argument.$