## Abstract

A new asymptotic approximation of the dynamic equations in the two-dimensional classical theory of thin-elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wavenumber, the latter corresponds to the eigenfrequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two-dimensional shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set-ups, including the assumption on vanishing of shear and circumferential mid-surface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin-walled structures, such as carbon nanotubes.

## 1. Introduction

The theory of thin-elastic shells, as an important branch of theoretical solid mechanics, widely exploits various approximate formulations. These are based on appropriate physical assumptions or originate from asymptotic expansions, e.g. [1–5] and references therein. Each of the approaches has its own advantages. Sensible ad hoc assumptions are often attractive owing to the possibility of an immediate insight. At the same time, their ranges of applicability are not always well defined. The asymptotic methodology is better mathematically justified but its implementation relies on a substantial preliminary analysis aiming to determine a correct scaling and often cannot be completely formalized. Both these approaches were extensively used within the framework of shell theory and it might be an impression that there is little or no room for new developments in this area. In particular, it might be expected that all possible shortened forms of the classical two-dimensional dynamic equations have already been derived [6]. However, recent achievements in nanotechnology [7–11] inspire revisiting of this problem. In contrast to macroscopic shells used in aerospace, mechanical and civil engineering, carbon nanotubes may have a large aspect ratio, characterizing the relationship between the length and radius. The macroscopic shells with such aspect ratio are usually treated as one-dimensional beams because of considerable overall (beam-type) flexibility. Design of modern nanodevices can require modelling of one-dimensional beam-type vibrations with rigid cross-sections as well as two-dimensional shell-type vibrations over a broad frequency band [12,13].

Thorough analysis of the state of art in this area indicates that dynamic asymptotic theory for elongated cylindrical shells is not yet established. In the limiting set-up of an infinite shell the most pronounced long wave vibrations at a fixed circumferential wavenumber occur near the lowest cut-off frequency coinciding with the eigenfrequency of a thin almost inextensible ring. In this case, the dispersion relation associated with a consistent approximate long wave theory has to coincide with the appropriate near-cut-off expansion of the exact dispersion relation in the full classical shell theory. The well-known semi-membrane shell theory [1,2,14], originally attributed to statics, violates this condition. The aim of the paper is to derive an asymptotic theory governing dynamic behaviour of a circular cylindrical shell in the vicinity of the lowest cut-off frequency.

It is interesting that near-cut-off asymptotic expansions were previously studied only for various set-ups of high-frequency thickness shell vibrations [4,5,15–17]. The developed long wave approximations have been applied to trapped mode analysis [18–20].

The paper is organized as follows. The problem is formulated in §2 starting from the Sanders–Koiter version of the classical shell theory [21]. In §3, the associated dispersion relation is derived along with its asymptotic expansion valid in the vicinity of the lowest cut-off frequency. The latter has two limiting forms depending on the ratio of the longitudinal wavelength and the relative thickness of the shell. In §4, a straightforward asymptotic procedure is adapted for the Sanders–Koiter equations of motion. The established fourth-order ordinary differential equation for the transverse displacement has two shortened forms. The first of them is nothing more than the semi-membrane theory, whereas the second one is given by a second-order equation typical of near-cut-off approximations [4]. The dispersion relations for these two shortened equations coincide with the expansions of the exact dispersion relations in §3. It is shown in §5 that the asymptotic considerations of §4 suggest rather sophisticated geometric hypotheses underlying the obtained equation. They refine the assumption of zero shear and circumferential mid-surface deformations underlying the semi-membrane theory. It is also demonstrated in this section that the earlier proposed ad hoc equation [22], based on the hypotheses in the semi-membrane theory but also taking into account some extra terms in the related energy functional, is virtually identical to its asymptotic counterpart to within the values of constant coefficients.

## 2. Equations of motion

Consider linear vibrations of a circular cylindrical shell (figure 1) starting from the Sanders–Koiter version of the two-dimensional classical theory [21]. This version allows elegant variational formulation along with natural generalization for nonlinear problems, see also [23] for more detail. Let us introduce the dimensionless coordinates *ξ* and *φ*, where *ξ*=*x*/*L* is the longitudinal coordinate related to typical wavelength (or length) *L* and *φ* is the azimuthal angle. Denote by *u*,*v* and *w* longitudinal, circumferential and radial mid-surface displacements related to the mid-surface radius *R*. In addition, we define dimensionless time using the scaling *E* is the Young modulus, *ρ* is the mass density, *ν* is the Poisson ratio and *h* is the thickness of the shell.

The inverse aspect ratio *α*=*R*/*L* and the relative thickness *β*=*h*/*R* are two major geometric parameters governing shell vibrations. The parameter *β* is small in all two-dimensional thin-shell theories. The parameter *α* is also small for elongated shells. In the following, we always assume that

The Hamilton function within the framework of the Sanders–Koiter shell theory can be written as
*ε*_{ξ}, *ε*_{φ} and *ε*_{ξφ} are longitudinal, circumferential and shear deformations of the mid-surface; *κ*_{ξ} and *κ*_{φ} are curvatures of the mid-surface in the longitudinal and circumferential directions; *κ*_{ξφ} is torsion of the mid-surface. Here dot denotes a partial derivative with respect to the dimensionless time *τ*=*t*/*t*_{0}.

Kinematic relations expressing the deformations, curvatures and torsion in terms of the displacements are

Variation of the functional *H* with respect to the displacements *u*,*v* and *w* after substitution of (2.3) into (2.2) leads to three partial differential equations

On separating the azimuthal and time variables in (2.4) by the formulae
*n* is circumferential wavenumber, *ω* is vibration frequency, and *U*(*ξ*), *V* (*ξ*) and *W*(*ξ*) are vibration amplitudes, we arrive at ordinary differential equations

## 3. Dispersion relation

For an infinite shell or in the case of periodic boundary conditions *V* (0)=*V* (2*π*)=*W*(0)=*W*(2*π*)=*U*_{ξ}(0)=*U*_{ξ}(2*π*)=0, we may set (*k* is the normalized longitudinal wavenumber)

This shows that the frequency spectrum of the shell consists of three branches, e.g. see [4,5] for more detail. The lowest of them is associated with bending of the mid-surface strongly affected by shell curvature, see also [6]. These low-frequency bending vibrations are the main focus of the paper. Two other branches correspond to extension, and shear motions along the mid surface and are less affected by curvature. As an illustration, three dispersion curves are shown in figure 2 for *n*=2 and *ν*=0.2; in this figure, *k*_{α}=*αk*. The curve of interest is plotted with a bold line.

The equations for the cut-off frequency of the aforementioned lowest branch follow from (2.6) at *U*=*dV* /*dξ*=*dW*/*dξ*=0. They are

The latter govern free in-plane vibrations of an almost inextensible thin ring (figure 3) with the eigenfrequency *ω*=*ω*_{*} given by
*k*=0.

Exact analysis of (3.2) is a rather sophisticated problem. At the same time for the lowest branch of interest, the asymptotic behaviour of the related root of (3.2) under assumptions (2.1) is expressed by the relatively simple formula (*k*∼1 and *n*∼1)
*β*≪1. It is
*α*≪*β*, we get the leading-order, near-cut-off expansion
*α*≫*β*, (3.9) may be simplified to

It is worth noting that at *α*∼*β*^{1/2} all the terms in (3.11) are of the same asymptotic order as it is usually assumed in the dynamic semi-membrane theory [24].

Numerical results are presented in figure 4 for *n*=2 and *n*=3. The logarithmic deviation from the cut-off frequency *α*/*β*=*R*^{2}/*Lh* at *k*=1 and *ν*=0.2. The values Δ for the full dispersion relation at *β*=0.01 (3.2) and its asymptotic expansions (3.8), given by solid line, are not distinguishable in this figures. The curves corresponding to formulae (3.10) and (3.11) are plotted with dotted and dash-dotted lines, respectively. As might be expected, the semi-membrane theory (see formula (3.11)) is valid outside a narrow vicinity of the cut-off frequency, in which formula (3.10) demonstrates a better accuracy. At the same time, the two-term formula (3.8) picks up uniform near-cut-off behaviour.

## 4. Asymptotic derivation

For the sake of simplicity, we set in (2.6) *α*=*β*≪1. In this case, both terms in the right-hand side of the asymptotic expansion (12) are of the same order. Let us specify asymptotic series

At leading order, we get from second and third equations in (2.6)
*ξ*. From (4.2) and (4.3), we immediately obtain

At first order, we have from second and third equations in (2.6)
*ω*_{0}=*Ω*_{0}, where *Ω*_{0} is given by (3.6). As a result, each of them takes the form

Now, we consider the first equation in (2.6), which, at first order, is given by
*M*(*w*_{0}) is expressed as

At second order, we obtain from second and third equations in (2.6), respectively

Then, on substituting (4.11) into (4.12), we arrive at
*δ*,*γ* and *ω*_{*} are given by (3.6) and (3.9). In the original variables, *x* and *t*, equation (4.14) becomes

Its range of validity is

The semi-membrane shell dynamic equation [24]
*α*∼*β*^{1/2} (*R*∼*L*^{2/3}*h*^{1/3}).

## 5. Geometric hypotheses

The derivation in §4, see (4.6), (4.7) and (4.9), (4.10), suggests that the equation (4.15) should also follow from the geometric hypotheses

On introducing (5.1) into the Hamiltonian (2.2) taking into account formula (2.3) and integrating over the angle *φ*, we get by varying it in *W* an equation identical to (4.15) to within asymptotically secondary terms in *α* and *β*. It is interesting that a similar manipulation with the simplified relations (5.2) results at leading order in the equation
*Ω*_{1}, coincide at *ν*=0. Equation (5.3) is asymptotically identical to that in [22].

## 6. Concluding remarks

An asymptotic theory valid in the vicinity of the lowest cut-off frequency is derived for a circular cylindrical shell. For a fixed circumferential wavenumber, its shortened forms are given by the conventional fourth-order equation in the semi-membrane theory and a second-order equation typical for near-cut-off approximations.

Consistency of the proposed theory is illustrated by comparison with the dispersion relation in the Sanders–Koiter theory. The asymptotic behaviour of shell displacements established in the paper enables refining the traditional assumption on zero shear and circumferential mid-surface deformations underlying both the semi-membrane theory and the more general ad hoc theory in [22].

The developed approach allows various extensions and generalizations. In particular, asymptotic derivation of the long wave theory in question from the three-dimensional dynamic equations in linear elasticity seems to be of obvious interest. The effect of geometric nonlinearity also can be incorporated in order to justify the analogues ad hoc nonlinear equation in [22]. Finally, we mention applications dealing with localized vibrations, including those caused by shape imperfections as well as shell edge waves, see review article [25] and references therein.

The proposed theory appears to be of interest for predicting the vibration spectra of nanotubes over a broad range of geometric parameters for various boundary conditions specified at the edges. The small parameter *α* expressing the inverse aspect ratio of a nanotube may approach the values of order 0.01, here and below, see [26]. Another small parameter *β* is evaluated as the ratio of extensional and bending nanotube stiffness rather than relative thickness of a macro shell. Its values are usually within the interval 0.05÷0.1. Typical for nanotubes boundary conditions include, in particular, a free edge as well as an edge cupped with a fullerene hemisphere. In this case, the fourth-order differential equation (4.15) is subject to two conditions at each edge. For example, conditions on longitudinal and shear stress resultants are imposed at a free edge [1,2].

## Ethics

This work does not involve any aspects such as collection of human data or such.

## Data accessibility

This work does not have any experimental data.

## Authors' contributions

The idea of a low-frequency near-cut-off approximation was originated by L.I.M. J.K. proceeded with asymptotic derivation. V.V.S. produced all the computations. The text was composed jointly by J.K. and L.I.M.

## Competing interests

We have no competing interests.

## Funding

The research of L.I.M. and V.V.S. was supported by the Russian Foundation for Basic Research (grant no. 14-01-00284a).

- Received October 30, 2015.
- Accepted April 5, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.