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

Abstract

We prove that the critical points of the $3$d nonlinear elasticity functional over a thin shell of arbitrary geometry and of thickness $h$, as well as the weak solutions to the static equilibrium equations (formally the Euler Lagrange equations associated to the elasticity functional) converge, in the limit of vanishing thickness $h$, to the critical points of the generalized von Karman functional on the mid-surface, recently derived in [14]. This holds provided the elastic energy of the $3$d deformations scale like $h^4$ and the magnitude of the body forces scale like $h^3$.