Koiter Estimate Revisited

We prove a general adimensional energy estimate between the solution of the three-dimensional Lame system on a thin clamped shell and a displacement reconstructed from the solution of the classical two-dimensional Koiter model. This estimate only involves the thickness parameter e, constants attached to the mid-surface, the two-dimensional energy of the solution of the Koiter model and ``wave-lengths'' associated with this latter solution. This bound is in the same spirit as Koiter's heuristic estimate (1970) and can be viewed as an a posteriori estimation of the modeling error by means of the two-dimensional solution. It is general with respect to the geometry of the mid-surface which is an arbitrary smooth manifold with boundary. Taking boundary layer terms into account, we prove that our estimates are sharp in the cases of plates and elliptic shells.

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