Convergence of equilibria for planar thin elastic beams
Stefan G. Müller, Maria Giovanna Mora, Maximilian G. Schultz · Indiana University Mathematics Journal · 2007
We consider a thin elastic strip Ω h = (0,L)×(-h/2, h/2), and we show that stationary points of the nonlinear elastic energy (per unit height) E h (v) = (1/h) ∫ Ωh (W(∇v)- h 2 g(x 1 )·v) dx whose energy is bounded by Ch 2 converge to stationary points of the Euler-Bernoulli functional J 2 (y) = ∫ L 0 (1 24 EK 2 - g ·y) dx 1 where y: (0,L) → R 2 , with y' = ( cosθ sinθ ), and where K = 0'. This corresponds to the equilibrium equation - 1 12Eθ + g ·( -sinθ cosθ ) = 0, where 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.