A First-order in Thickness Model for Flexural Deformations of Geometrically Non-linear Shells

The shallow shells, characterized by deeections of the order of unity, small deformations and still smaller curvatures, have most thoroughly been studied in the literature. However, the momentum terms, due to which the shell diiers essentially from a membrane, are not negligible only for the shortwave length deformations, when the deeections are small, the deformations | of the order of unity and the curvatures | of the order of the inverse of the small parameter. In order to treat consistently the case of momentum supporting shells, the formulas for covariant diierentiation in the shell space are revisited. It is shown that the geometrical non-linearity contributes terms of the same order of magnitude as the momentum stresses. For the exural deformations an equation of Boussinesq type is derived containing fourth-order dispersion and cubic non-linearity.