MiS Preprint Repository

We consider a thin elastic strip ${\mathrm{\Omega }}_h=(0,L)\times (-h/2,h/2)$, and we show that stationary points of the nonlinear elastic energy (per unit height) $E^h(v)=\frac{1}{h}{\int }_{{\mathrm{\Omega }}_h}(W(\mathrm{\nabla }v)-h^{2}g(x_1)\cdot v)dx$ whose energy is bounded by $C{h}^2$ converge to stationary points of the Euler-Bernoulli functional $J_2(\overline{y})={\int }_0^L(\frac{1}{24}E{\kappa }^2-g\cdot \overline{y})d{x}_1$ where $\overline{y}:(0,L)\to {\mathbb{R}}^2$, with ${\overline{y}}^{\prime }=(\genfrac{}{}{0ex}{}{\cos \theta }{\sin \theta })$, and where $\kappa ={\theta }^{\prime }$. This corresponds to the equilibrium equation $-\frac{1}{12}E{\theta }^{″}+\tilde{g}\cdot (\genfrac{}{}{0ex}{}{-\sin \theta }{\cos \theta })=0$, 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.