Asymptotic derivation of refined dynamic equations for a thin elastic annulus

Abstract

Low-frequency vibrations of a thin elastic annulus are considered. The dynamic equations of plane strain are subjected to asymptotic treatment beyond the leading-order approximation. The main peculiarity of the considered problem is a specific degeneration associated with the effect of the almost inextensible midline of the annulus, resulting in a few unexpected features of the mechanical behaviour. In particular, it is discovered that the leading-order even component of the circumferential stress is not uniform across the thickness, as is usually assumed, and can be determined only at the next order. The derived refined equations also govern vibrations of a cylindrical shell at the lowest cut-off frequencies. The two-term asymptotic formula obtained for the latter fully agrees with the expansion of the transcendental dispersion relation for plane strain but does not coincide in the second term with the prediction of the Kirchhoff–Love theory for thin shells.