A first-order in thickness model for flexural deformations of geometrically non-linear shells

The shallow shells, characterized by deflections 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 differs essentially from a membrane, are not negligible only for the short-wave-length deformations, when the deflections 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 differentiation in the shell space are revisited. It is shown that the geometrical non-linearity constributes terms of the same order of magnitude as the momentum stresses. For the flexural deformations and equation of Boussinesq type is derived containing fourth-order dispersion and cubic non-linearity.