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.

A Poisson scaling approach to backward wave propagation in a tube

“…An important feature of our method is the explicit dependence on parameters which it yields in all formulae. Thus it is potentially applicable to parametric studies of tunable metamaterials and nanotubes, for example [12,13], and also to a possible type of pressure pulsation damper which operates at vanishingly small or negative Poisson's ratio [17]. Our results suggest that if a driving frequency is increased through the critical frequency, the onset of wave propagation will be rapid when Poisson's ratio is small, and a wide range of wavenumbers will be excited almost simultaneously; alternatively, under broadband forcing one may expect a sharper resonance than normal near the critical frequency.…”

“…Although we concentrate on the frequencies and wavenumbers for which backward propagation occurs, the method of Poisson scaling is in fact general; it may be used at any frequency and wavenumber, and also in nonlinear elastic problems [10,11]. Potential applications of the work are to modern materials such as homogenized media and composites with tunable parameters, including nanotubes [12,13] and metamaterials [14][15][16]. One recent application is to the design of a pressure pulsation dampener made of a material with zero or negative Poisson's ratio [17].…”

A Wronskian method for elastic waves propagating along a tube

“…The horizontal scale of the figure shows that the frequency is almost constant over a wide range of wavenumbers. Physically, this means that, near the ring frequency, a wide range of axial wavenumbers can be excited simultaneously, making possible a localized field which may need a nonlinear correction in order to be represented accurately [20,21]. This is even more pronounced when ν = ν ± ( ϵ ), because the coefficient of K 2 in (6.5) is then zero, and the dispersion curve becomes even flatter near the ring frequency, having a quartic form in K rather than its usual parabolic form.…”

“…nanotubes [20,21]. A possible starting point would be to include transverse shear and rotational inertia, as in [22], for example.…”

“…Promising applications are to metamaterials and nanotubes [20,21], including extensions to account for transverse shear and rotational inertia [22], and nonlinearity. In particular, the concluding section of [21] is explicit that current shell theory still has unresolved difficulties near the ring frequency, because of the difficulty in framing suitable kinematic hypotheses to account for the near-inextensibility of the mid-surface of the tube wall, and the parabolic form of the even part of the circumferential stress.…”