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.


Introduction
The static behaviour of a thin cylindrical shell exhibits remarkable features arising from virtually negligible shear and circumferential extension of its mid-surface. This is especially important for sufficiently long shells, for which the simplest membrane model fails, and some of the moments (stress couples) have to be retained in the equilibrium equations. The associated semi-membrane (semi-momentless) formulation was initially developed in Ref. [1] using an ad hoc engineering approach. Later, it was asymptotically justified within the framework of 2D classical shell theory, see [2]. The semi-membrane theory for cylindrical shells is also mentioned in more recent books, e.g. see [3,4].
The aforementioned geometric constraints also support the family of lowest cut-off frequencies, e.g. see [5,6] with their values tending to zero at the thin shell limit. In this case, the near-cut-off behaviour is very peculiar also because of the presence of two small parameters. The first of them is the ratio of the shell thickness and its mid-surface radius, while the second one is given by the ratio of the latter and a typical wavelength. The associated shortened equations of motion are derived in Ref. [6] using a straightforward asymptotic procedure in the framework of the Sanders-Koiter version of 2D shell theory. Earlier, similar ad hoc equations were established in [7], see also [8,9], inspired by modelling of elongated carbon nanotubes, e.g. see [10,11]. The results in Ref. [6] were extended to an anisotropic cylindrical shell in Ref. [12], see also [13] dealing with a functionally graded material.
This work was completed with the support of our T E X-pert. Previously, near-cut-off asymptotic analysis of elastic shells was focussed on the high-frequency domain with the cut-off frequencies corresponding to thickness resonances, e.g. see books [14,15] and journal publications [16,17]. We also mention related publications studying high-frequency trapped modes, e.g. see [18,19]. In contrast to the current context, the effect of the ratio of the radius and wavelength is not that sophisticated for high-frequency near-cut-off behaviour.
Asymptotic validation of the considerations in [6] starting from the 3D set-up seems to be of an obvious interest. The point is that the general asymptotic scaling underlying dynamic behaviour of a shell of arbitrary shape, see [14] and references therein, degenerates for a thin cylinder due to specific constraints on a part of mid-surface deformations. As might be expected, the related cancellation of leading-order terms is especially pronounced near the smallest cut-off frequencies. In the latter case, higher-order terms retained in [6] within the 2D framework may hypothetically be negligible in comparison with the truncation in the equations of motion in 3D elasticity. This motivates the derivation of the low-frequency equations for a cylindrical shell from the original 3D formulation.
Below, we start from the scaling generalising that in [6] to the 3D case, since, as it was already mentioned, the previous considerations for an arbitrary shell cannot be adapted. The developed asymptotic procedure appears to be very technical. For the sake of simplicity, we study only the scenario, in which the ratios of the thickness to the radius and the radius to a typical wavelength are of the same order. Nevertheless, the derivation of a leading-order shortened equation necessitates operating with four-term expansions of 3D equations.
It is surprising, in a sense, that most of the coefficients in the established low-frequency equation are identical to their counterpart in [6]. The exception is the essential correction to the leading-order estimation for the cut-off frequency, which, however, coincides with that obtained in [20,21] from the related plane strain problem. At the same time, the leading-order expressions for the mid-surface circumferential extension and the tangential shear stress resultants are different from those in [6], whereas the tangential normal stress resultants and mid-plane shear deformation are the same.
The paper is organised as follows. The statement of the problem in terms of 3D elasticity is presented in Sect. 2. A single circumferential mode is studied. The asymptotic scaling is determined in Sect. 3. Special arrangements are made to incorporate the generation of the deformations corresponding to midsurface shear and circumferential extension. The leading-order estimation for the lowest cut-off frequency is obtained in Sect. 4. Section 5 completes the derivation of the sought-for equation of motion. Section 6 is concerned with comparisons of the established results with previous developments, including the coefficients in the above-mentioned equation, as well as the asymptotic formulae for stress resultants and mid-surface deformations.

Statement of the problem
Consider free harmonic vibrations of a thin elastic cylindrical shell of thickness 2h with the mid-surface radius R for which η = h/R is a small geometrical parameter, i.e. h R, see Fig. 1. Take the traditional three orthogonal coordinates along the mid-surface α i , i = 1, 2, 3, in the form where ξ is the longitudinal coordinate (−∞ < ξ < ∞), θ is the circumferential coordinate (0 ≤ θ < 2π), and ζ is the transverse coordinate (−1 ≤ ζ ≤ 1). Also, define the dimensionless frequency by where ω is the original frequency, ρ is the mass density, E is the Young's modulus and ν is the Poisson's ratio; in what follows, the factor e iωt , where t is time, is omitted. The first formula (1) defines a typical wavelength L of order R 2 /h which is much greater not only than the shell thickness (h/L ∼ η 2 1 ) but also than the shell radius ( R/L ∼ η 1). Assuming, in addition, Ω ∼ 1 we restrict ourselves to the lowest vibration modes investigated in [6] within the framework of 2D shell theory.
The aim of the paper is to reduce the equations above to a 1D form over the long-wave, low-frequency domain defined in this section.

Scaling of stresses and displacements
Let us introduce the dimensionless displacement and stress components in the form and In addition, we set e = ηRe * , s= ηRs * (17) reflecting the specific behaviour of a cylindrical shell related to its asymptotically small circumferential extension and tangential shear, see [2] for greater detail.
In the formulae above, all starred quantities are assumed to be of order unity. The adapted scaling is motivated by the asymptotic consideration in [6] within the framework of Sanders-Koiter version of 2D classical shell theory.

Cut-off frequency
First, at leading order, integrating Eqs. (24), (25) and (23) with respect to thickness variable ζ, we obtain, respectively, u where U (0) , V (0) and W (0) are unknown 1D functions. On account of (27), the latter are related to each other by from which we also deduce that Next, integrating Eq.
demonstrating that normal circumferential stress resultant disappears at leading order. As a result, the associated stress couple appears to be more significant. This observation nicely illustrates the basic idea of the semi-membrane shell theory, see [2] We also have from (20) The relations above do not allow calculating the leading-order term Ω 2 0 in the asymptotic series (30). This necessitates proceeding to the next order approximation.
We now integrate Eqs. (24), (25) and (23) with respect to ζ at first order, yielding, respectively In this case, the relations (27) give and Then, we integrate (22) through the thickness, using (38) along with (35), we first obtain and, hence Similarly, Equations (34) and (36), due to (41), become and Now, insert (26) with (27) 2 in (18). The solvability of the latter, taking into account homogeneous boundary conditions (28), yields and, consequently Integrating, again, (18) with respect to ζ, and satisfying (28), we finally get In the same manner, we derive from (19) and (20) together with (28), respectively, s and s (1) As a result, the solvabilities of (19) and (20) with (28) are given by On comparing the latter, we arrive at At a given n, this is the expression of the lowest cut-off frequency within the classical 2D shell theory, e.g. see [5] and [6] along with the related plane strain set-up in Fig.2. For n = 0 and n = 1, we have Ω 2 0 = 0. In this case, the assumptions underlying the asymptotic consideration above are violated. These two modes correspond to the so-called bar (or beam)-type vibration of a cylindrical shell and assume a special treatment.
Although we determined the frequency, we do not yet have the sought-for equation of motion. Therefore, we should proceed to higher orders.
Thus, we need to proceed to the next asymptotic order. First, integrating Eqs. (24), (25) and (23) with respect to ζ, we obtain and u Also, and Then, integrating (22) and incorporating (66), we have  (2) . (73) Now, substituting (73) into (63) and (64), we get and At this order, the solvability of (19) together with the boundary conditions (28) is given by Similarly, for (20) and (28) ZAMP Asymptotic corrections to the low frequency Page 11 of 16 43 Comparison of the last two formulae leads to Finally, on using the relations (30), (32) and (56) in the last equation, we arrive at
The solutions of Eq. (80) can be substituted into the formulae in the previous two sections in order to derive the expressions for stress and strain components. As an example, we present the asymptotic estimations for the quantities e and s, see (13), at the mid-surface ζ = 0. We stress once more that these quantities are key ones for a cylindrical shell. Starting from formulae (56)   We also present the expressions for the tangential stress resultants in terms of W . These become where i = j = 1, 2. Let us set, according to (29) kl + · · · , k,l = 1, 2.