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Convergence of equilibria of planar thin elastic beams. (English) Zbl 1125.74026

Summary: 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) \[ E^h(v)=(1/h)\int_{\Omega_h}(W(\Delta v)-h^2g(x_1)\cdot v)\,dx \] whose energy is bounded by \(Ch^2\) converge to stationary points of the Euler-Bernoulli functional \[ J_2(\overline y)= \int^L_0\left(\frac{1}{24}{\mathcal E}\kappa^2-g\cdot\overline y\right)\,dx_1, \] where \(\overline y:(0,L)\to\mathbb{R}^2\), with \(\overline y'={\cos\theta\choose\sin \theta}\), and where \(\kappa=\theta'\). This corresponds to the equilibrium equation \(-\frac {1}{12}{\mathcal E}\theta''+\widetilde g\cdot{-\sin\theta\choose\cos \theta} =0\), where \(\widetilde g\) is the primitive of \(g\). The proof uses the rigidity estimate for low-energy deformations and a compensated compactness argument in a singular geometry. In addition, possible concentration effects are ruled out by a careful truncation argument.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B20 Nonlinear elasticity
74G65 Energy minimization in equilibrium problems in solid mechanics
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