Search

MiS Preprint Repository

Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.

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.

Received:
May 13, 2006
Published:
May 13, 2006
MSC Codes:
74K10

Related publications

inJournal
2007 Repository Open Access
Maria Giovanna Mora, Stefan Müller and Maximilian G. Schultz

Convergence of equilibria of planar thin elastic beams

In: Indiana University mathematics journal, 56 (2007) 5, pp. 2413-2438