Deformations Without Bending: Explicit Examples

Abstract

Here we consider an interesting class of free of bending deformations of thin axial symmetric shells subjected to uniform normal pressure. The meridional $k_{\mu }$ and the parallel $k_{\pi }$ principal curvatures of the middle surfaces of such shells obey the non-linear relationship $k_{\mu }=2a{k}_{\pi }^2+3k_{\pi }$, $a=const$. These non-bending shells depend on two arbitrary parameters, which are the principal radii $r_{\mu }$ and $r_{\pi }$ of some fixed parallel of the shell. Besides, these surfaces have no closed form description in elementary functions. Our principle aim here is to present such a parameterization of the whole class of non-bending closed surfaces by making use of the canonical forms of the elliptic integrals. The obtained explicit formulas are then applied for the derivation of the basic geometrical characteristics of these surfaces.