A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry

Details A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry

2009-07-07

arxiv.org/abs/0907.1290v1

Mathematics

Analysis of PDEs

15 pages

Scientific paper

We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness $h$ and around the mid-surface $S$ of arbitrary geometry, converge as $h\to 0$ to the critical points of the von K\'arm\'an functional on $S$, recently derived in \cite{lemopa1}. This result extends the statement in \cite{MuPa}, derived for the case of plates when $S\subset\mathbb{R}^2$. We further prove the same convergence result for the weak solutions to the static equilibrium equations (formally the Euler- Lagrange equations associated to the elasticity functional). The convergences hold provided the elastic energy of the 3d deformations scale like $h^4$ and the external body forces scale like $h^3$.

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