Asymptotic corrections to the low-frequency theory for a cylindrical elastic shell

The general scaling underlying the asymptotic derivation of 2D theory for thin shells from the original equations of motion in 3D elasticity fails for cylindrical shells due to the cancellation of the leading-order terms in the geometric relations for the mid-surface deformations corresponding to shear and circumferential extension. As a consequence, a cylindrical shell as an elastic waveguide supports a small cut-off frequency for each circumferential mode. The value of this cut-off tends to zero at the thin shell limit. In this case, the near-cut-off behaviour is strongly affected by the presence of two small parameters associated with the relative thickness and wavenumber. It is not obvious whether it can be treated within the 2D theory. For the first time, a novel special scaling is introduced, in order to derive an asymptotically consistent formulation for a cylindrical shell starting from 3D framework. Comparisons with the previous results obtained using the popular 2D Sanders-Koiter shell theory are made. Asymptotic corrections are deduced for the fourth-order equation of low-frequency motion and some of other relations, including the formulae for tangential shear stress resultants.