Frequency analysis of a sandwich conical shell with nanocomposite face layers and a honeycomb core with oriented cells

Abstract

In this research, the free vibration analysis is investigated for a sandwich conical shell with two nanocomposite face layers and either a hexagonal honeycomb (HH) or a re-entrant honeycomb (RH) core oriented in an arbitrary direction. Both HH and RH cores are orthotropic structures, but the RH is an auxetic structure and the HH is a non-auxetic one. The nanocomposite face layers are fabricated of a polymeric matrix strengthened with uniformly distributed agglomerated either carbon nanotubes (CNTs) or graphene nanoplatelets (GNPs). The sandwich shell is modeled via Murakami’s zig-zag theory, and the governing equations and boundary conditions are derived through Hamilton’s principle. The influences of various parameters on the natural frequencies are investigated including orientation, wall thickness, and inclined angle of the cells in the honeycomb core, thickness of the honeycomb core, mass fraction and type of the nanofibers, agglomeration intensity, and boundary conditions. It is concluded that in each vibrational mode, there are optimum values for the orientation and wall thickness of the cells and thickness of the honeycomb core which result in the highest natural frequency.

Keywords

Vibration sandwich conical shell honeycomb nanocomposite zig-zag theory

Introduction

Structures with low density and high endurance against shear loading are the best options for use as a core in a sandwich structure with rigid face layers. Such a composition brings about high values of load capacity and rigidity along with low density. Since the core of a sandwich structure is not supposed to have high strength and rigidity, honeycomb structures which are fragile but extremely light can be chosen as the core in a three-layered sandwich structure. Apart from the low density of these orthotropic structures, honeycomb structures benefit from high energy absorption capability which has led to their use in various engineering fields like aerospace, mechanical, civil, and transportation engineering. In recent years, a fair number of works have been presented to estimate the elastic constants and density of honeycomb structures. Moreover, several works have been presented regarding the mechanical analysis of sandwich structures with honeycomb cores integrated with various types of face layers.

Hexagonal ones are the most well-known kind of honeycomb structure and have been extensively used in various engineering fields. The density and elastic constants of these orthotropic structures and their influences on the mechanical behaviors of sandwich structures have been fairly investigated in recent decades.^1–6 Due to the orthotropic behavior of hexagonal honeycomb structures, their Poisson’s ratios can exceed values higher than 0.5 but are always positive values. One disadvantage of hexagonal honeycombs is that when they are bent out of the plane, they create a saddle-shaped curve owing to their positive Poisson’s ratios. However, some novel kinds of honeycomb structures have been invented in recent years whose Poisson’s ratios are negative values and are called auxetic honeycomb structures. Apart from the research which focused on the estimation of the density and elastic constants of auxetic honeycomb structures, a fair number of works have been presented which have studied the influences of utilizing auxetic honeycomb cores on the mechanical behaviors of sandwich structures. Duc et al.⁷ inspected the vibration analysis of a doubly-curved sandwich panel with an auxetic honeycomb core covered with two face layers fabricated from aluminum. They inspected the impacts of geometric parameters of the auxetic honeycomb core on the natural frequencies of the panel and its dynamic response to blast load. The time-dependent deflection and stress distributions in a sandwich cylindrical shell with an auxetic honeycomb core subjected to moving pressure were analyzed by Eipakchi and Naserkani.⁸ The effects of the geometric parameters of the auxetic honeycomb core on the dynamic deflection and the critical velocity of the moving pressure were inspected by them. Xu et al.⁹ examined the in-plane vibration behavior and multi-objective optimization of an auxetic honeycomb structure with curved sinusoidal walls. They explored the amplitude of vibration and the energy absorption rate of the structure. Cong and Duc¹⁰ presented an exact solution for the nonlinear dynamics analysis of a sandwich doubly-curved panel with an auxetic honeycomb core subjected to blast and thermo-mechanical loading. The influences of geometric parameters of the auxetic honeycomb core on the dynamic response of the structure were studied by them. In another research, they presented a nonlinear analysis for the thermo-mechanical buckling and post-buckling behaviors of a sandwich doubly-curved shell panel with an auxetic honeycomb core whose material properties vary with temperature change.¹¹ It was assumed by them that the shell is subjected to combinations of external pressure, temperature elevation, and axial load. They explored the impacts of the geometric parameters of the auxetic honeycomb core on the thermo-mechanical stability regions of the shell. Pham et al.¹² inspected the free vibration behavior of a sandwich rectangular plate with an auxetic honeycomb core and two nonhomogeneous face layers fabricated from functionally graded material (FGM). The effects of the geometric parameters of the auxetic honeycomb core on the natural frequencies were inspected by them. A parametric study was provided by Cong et al.¹³ to study the nonlinear vibration analysis of a sandwich doubly-curved panel with an auxetic honeycomb core and CNT-reinforced face layers. The temperature-dependency of the material properties was included to investigate the impacts of the geometric parameters of the auxetic honeycomb core on the nonlinear deflection and natural frequencies. Quan et al.¹⁴ investigated the nonlinear dynamic analysis of a sandwich rectangular plate with an auxetic honeycomb core and piezoelectric face layers. The effects of the geometric parameters of the auxetic honeycomb core on the dynamic deflection and natural frequencies of the plate were examined by them.

The nonlinear vibration analysis of a sandwich rectangular plate with an auxetic honeycomb core and two electro-magneto-elastic face layers exposed to blast load was investigated by Dat et al.¹⁵ They inspected the impacts of the geometric parameters of the auxetic honeycomb core on the natural frequencies and dynamic deflection of the plate. The bending analysis of a sandwich rectangular plate with a tunable auxetic honeycomb core and two FGM face layers was investigated by Liu et al.¹⁶ The impacts of the geometric parameters of the auxetic honeycomb core on the deformation and stress distribution were inspected by them. Through a finite element method (FEM)-based analysis via the ANSYS software, Necemer et al.¹⁷ studied the fatigue resistance characteristics of various types of auxetic honeycomb structures. Five types of auxetic honeycomb structures were investigated by them including re-entrant, star-shaped, S-shaped, chiral, and double arrowhead structures. The dynamic response of a sandwich panel with an auxetic honeycomb core under a blast load was studied by Rai et al.¹⁸ They tried to optimize the energy absorption capacity of such a sandwich structure. Sadikbasha and Pandurangan¹⁹ examined the crashworthiness of hexachiral auxetic honeycomb structures exposed to in-plane load. They studied the effects of using such an auxetic honeycomb core on the energy absorption capacity of sandwich structures. Through an FEM-based analysis via the ABAQUS software, Yu et al.²⁰ studied the failure behavior of an auxetic honeycomb structure under a compressive load. They compared their results with some experimental results to verify the correctness of their model. Ha et al.²¹ inspected the chaotic and dynamic analysis of a sandwich cylindrical panel with an auxetic honeycomb core exposed to thermal loading. They inspected the impacts of geometric parameters of the auxetic honeycomb core on the chaotic and dynamic behaviors of the panel. Haq et al.²² inspected the dynamic response of a sandwich rectangular plate with an auxetic honeycomb core exposed to blast load. They tried to improve the blast performance of the plate by enhancing the energy absorption of the plate and minimizing its dynamic deflection. Thang et al.²³ studied the free vibrational behavior of a sandwich barrel-shaped shell with an auxetic honeycomb core and FGM face layers. The prepared benchmark results for the natural frequencies of sandwich barrel-shaped shells with various values of curvature radii. Biglari et al.²⁴ utilized an auxetic honeycomb core to improve the dynamic response of a sandwich panel subjected to a low-velocity impact. They examined the impacts of the geometric parameters of the auxetic honeycomb core on the dynamic deflection of the panel due to the low-velocity impact. Amirabadi et al.²⁵ inspected the aeroelastic stability analysis of a sandwich cylindrical shell with an auxetic honeycomb core made of functionally graded materials and nanocomposite face layers enriched with CNTs, GNPs, or graphene oxide powders (GOPs). They observed that in the same mass fractions, the GNPs have a better reinforcing effect. In another work, Amirabadi et al.²⁶ studied the free vibrational analysis of a sandwich conical shell with an auxetic honeycomb core fabricated from FGM. They examined the relationships between the natural frequencies and the geometric parameters and material gradation in the auxetic honeycomb core. The aeroelastic stability analysis of a ring-stiffened sandwich conical shell with an auxetic honeycomb core fabricated from FGM was studied by Abasi et al.²⁷ The influences of the geometric parameters and material gradation in the auxetic honeycomb core on the critical aerodynamic pressure were investigated by them.

In this article, the free vibrational behavior of a sandwich conical shell with a honeycomb core integrated with nanocomposite face layers is investigated. It is assumed that the honeycomb core is of either re-entrant or hexagonal types and the face layers are fabricated from a polymeric matrix enriched with uniformly distributed agglomerated either CNTs or GNPs. In all of the works associated with the mechanical behaviors of the sandwich structures with honeycomb cores, it is assumed that the cells of the honeycomb core are arranged in specified directions parallel with the edges of the structure. To expand the investigation to a more general case of study, in the current work, it is assumed that the cells of the honeycomb core are arranged in an arbitrary orientation which can be considered as the novelty of the work. Considering the arbitrary orientation of the cells of the honeycomb core enables designers to change the dynamic behavior of the shell by adjusting the orientation of the cells of the honeycomb core alongside the geometric parameters of the cells to meet the desired design requirements. For example, since conical shells have an extended usage in aerospace structures like missiles and airplanes, an optimum orientation can be found for the cells of the honeycomb core to bring about the highest critical aerodynamic pressure in the flutter (aeroelastic stability) analysis of such a structure. As another example, owing to the extended usage of cylindrical and conical shells in rotating machinery like centrifuges, the orientation of the cells of the honeycomb core can be adjusted to prevent the shell from rotating near the critical rotational speeds and subjecting it to resonance phenomenon. The sandwich shell is modeled based on the zig-zag theory. Due to the high values of the strength-to-mass and stiffness-to-mass ratios of such a structure, the results of this work can be used in the design, analysis, and optimization of mechanical, civil, transportation, and aerospace structures.

Modeling

The mathematical modeling of the shell is conducted in this section to derive the governing equations and boundary conditions.

Description of the problem

As illustrated in Figure 1, a sandwich conical shell of semi-vertex angle ψ, small radius a, length L, and constant thickness h is considered. The sandwich shell consists of a honeycomb core (either HH or RH) of thickness h_c oriented in an arbitrary direction (ϕ). The geometric parameters of the cells of the honeycomb core are shown in Figure 2 which include dimensions d₀ and l₀, wall thickness t₀, and inclined angle β. Moreover, the nanocomposite face layers have a polymeric matrix enriched with uniformly distributed agglomerated CNTs or GNPs.

Figure 1.

A three-layered sandwich conical shell with a honeycomb core and nanocomposite face layers.

Figure 2.

Geometrical parameters of the cells.

It is noteworthy that honeycombs are usually manufactured in two methods including corrugation which is used for metallic materials (usually aluminum), and expansion which is used for non-metallic materials. After manufacturing a thin honeycomb sheet, it can be cut and rolled to fabricate a honeycomb structure oriented in an arbitrary direction. This orientation can be achieved by adjusting the cutting line in the honeycomb sheet.

Material properties

The material properties of the core and the face layers are calculated in this section.

The honeycomb core

In a linear elastic deformation of a thin sandwich structure, the components of the stress (σ_ij) inside the honeycomb core are presented in terms of normal and shear components (ε_ij, γ_ij) of the strain as follows²⁸:

{\begin{cases} σ_{x x} \\ σ_{θ θ} \\ σ_{θ z} \\ σ_{x z} \\ σ_{x θ} \end{cases}}^{(c)} = [\begin{array}{l} Q_{11}^{(c)} (z) & Q_{12}^{(c)} (z) & 0 & 0 & 0 \\ Q_{12}^{(c)} (z) & Q_{22}^{(c)} (z) & 0 & 0 & 0 \\ 0 & 0 & Q_{44}^{(c)} (z) & 0 & 0 \\ 0 & 0 & 0 & Q_{55}^{(c)} (z) & 0 \\ 0 & 0 & 0 & 0 & Q_{66}^{(c)} (z) \end{array}] {\begin{cases} ε_{x x} \\ ε_{θ θ} \\ γ_{θ z} \\ γ_{x z} \\ γ_{x θ} \end{cases}},

(1)

where

\begin{array}{l} Q_{11}^{(c)} = Q_{11}^{0} \cos^{4} ϕ + 2 (Q_{12}^{0} + 2 Q_{66}^{0}) \sin^{2} ϕ \cos^{2} ϕ + Q_{22}^{0} \sin^{4} ϕ, \\ Q_{22}^{(c)} = Q_{22}^{0} \cos^{4} ϕ + 2 (Q_{12}^{0} + 2 Q_{66}^{0}) \sin^{2} ϕ \cos^{2} ϕ + Q_{11}^{0} \sin^{4} ϕ, \\ Q_{12}^{(c)} = Q_{12}^{0} (\sin^{4} ϕ + \cos^{4} ϕ) + (Q_{11}^{0} + Q_{22}^{0} - 4 Q_{66}^{0}) \sin^{2} ϕ \cos^{2} ϕ, \\ Q_{66}^{(c)} = (Q_{11}^{0} + Q_{22}^{0} - 2 Q_{12}^{0} - 2 Q_{66}^{0}) \sin^{2} ϕ \cos^{2} ϕ + Q_{66}^{0} (\sin^{4} ϕ + \cos^{4} ϕ), \\ \begin{array}{l} Q_{44}^{(c)} = Q_{44}^{0} \cos^{2} ϕ + Q_{55}^{0} \sin^{2} ϕ, & Q_{55}^{(c)} = Q_{55}^{0} \cos^{2} ϕ + Q_{44}^{0} \sin^{2} ϕ, \end{array} \end{array}

(2)

in which

\begin{array}{l} \begin{array}{l} Q_{11}^{0} = \frac{E_{11}^{(c)}}{1 - ν_{21}^{(c)} ν_{12}^{(c)}}, & Q_{22}^{0} = \frac{E_{22}^{(c)}}{1 - ν_{21}^{(c)} ν_{12}^{(c)}}, & Q_{12}^{0} = ν_{12}^{(c)} Q_{22}^{0}, \\ Q_{44}^{0} = G_{23}^{(c)}, & Q_{55}^{0} = G_{13}^{(c)}, & Q_{66}^{0} = G_{12}^{(c)}, \end{array} \end{array}

(3)

where the density (ρ^(c)) of the honeycomb core, and the elastic moduli (

E_{i j}^{(c)}

), shear moduli (

G_{i j}^{(c)}

), and Poisson’s ratio (

ν_{i j}^{(c)}

) of the honeycomb core in 1-2-z coordinates are defined in terms of the geometric parameters of an individual cell depicted in Figure 2 as follows^4,29:

\begin{array}{l} Hexagonal honeycomb (HH) : \\ \begin{array}{l} \frac{E_{11}}{E_{0}} = \frac{(1 - f_{2}^{2} \cot^{2} β) \cos β}{(f_{1} + \sin β) \sin^{2} β} f_{2}^{3}, & \frac{E_{22}}{E_{0}} = \frac{[1 - f_{2}^{2} (f_{1} \sec^{2} β + \tan^{2} β)] (f_{1} + \sin β)}{\cos^{3} β} f_{2}^{3}, \end{array} \\ \begin{array}{l} \frac{G_{12}}{E_{0}} = \frac{(f_{1} + \sin β) f_{2}^{3}}{(1 + 2 f_{1}) f_{1}^{2} \cos β}, & \frac{G_{13}}{G_{0}} = \frac{f_{2} \cos β}{f_{1} + \sin β}, & \frac{G_{23}}{G_{0}} = [\frac{f_{1} + \sin β}{1 + 2 f_{1}} + \frac{f_{1} + 2 \sin^{2} β}{2 (f_{1} + \sin β)}] \frac{f_{2}}{2 \cos β}, \end{array} \\ \begin{array}{l} ν_{21} = \frac{[1 - f_{2}^{2} (1 + f_{1}) \sec^{2} β] (f_{1} + \sin β) \sin β}{\cos^{2} β}, & ν_{12} = \frac{E_{11} ν_{21}}{E_{22}}, & \frac{ρ}{ρ_{0}} = \frac{(1 + 0.5 f_{1}) f_{2}}{(f_{1} + \sin β) \cos β}, \end{array} \end{array}

\begin{array}{l} Re - entrant honeycomb (RH) : \\ \begin{array}{l} \frac{E_{11}}{E_{0}} = \frac{f_{2}^{3}}{(f_{1} - \sin β) (f_{2}^{2} + \tan^{2} β) \cos β}, & \frac{E_{22}}{E_{0}} = \frac{(f_{1} - \sin β) f_{2}^{3}}{[1 + f_{2}^{2} (\tan^{2} β + f_{1} \sec^{2} β)] \cos^{3} β}, \end{array} \\ \begin{array}{l} \frac{G_{12}}{E_{0}} = \frac{η_{2}^{3}}{f_{1} (1 + 2 f_{1}) \cos β}, & \frac{G_{13}}{G_{0}} = \frac{f_{2} \cos β}{f_{1} - \sin β}, & \frac{G_{23}}{G_{0}} = (\frac{f_{1} - \sin β}{1 + 2 f_{1}} + \frac{0.5 f_{1} + \sin^{2} β}{f_{1} - \sin β}) \frac{f_{2}}{2 \cos β}, \end{array} \\ \begin{array}{l} ν_{21} = - \frac{(f_{1} - \sin β) (1 - f_{2}^{2}) \sin β}{[1 + f_{2}^{2} (f_{1} \sec^{2} β + \tan^{2} β)] \cos^{2} β}, & ν_{12} = \frac{E_{11} ν_{21}}{E_{22}}, & \frac{ρ}{ρ_{0}} = \frac{(1 + 0.5 f_{1}) f_{2}}{(f_{1} - \sin β) \cos β}, \end{array} \end{array}

(4)

in which f₁ = d₀/l₀ and f₂ = t₀/l₀. In equation (4), ρ₀, E₀, ν₀, and G₀ = 0.5E₀/(1+ν₀) show the density, elastic modulus, Poisson’s ratio, and shear modulus of the isotropic material used to fabricate the honeycomb core, respectively.

To make a fair comparison between these two types of honeycomb core, the aspect ratios (η₁) can be related to each other to bring about the same density of the core. As a result, by using equation (4), the following equation can be found between the aspect ratio in the re-entrant honeycomb core ( $f_{1}^{R}$ ) and the aspect ratio in the hexagonal honeycomb core ( $f_{1}^{H}$ ):

f_{1}^{R} = \frac{2 f_{1}^{H} + (f_{1}^{H} + 4) \sin β}{2 - \sin β} .

(5)

The nanocomposite face layers

The relation below describes the stress tensor inside the nanocomposite face layers²⁸:

{\begin{cases} σ_{x x} \\ σ_{θ θ} \\ σ_{θ z} \\ σ_{x z} \\ σ_{x θ} \end{cases}}^{(f)} = [\begin{array}{l} Q_{11}^{(f)} & Q_{12}^{(f)} & 0 & 0 & 0 \\ Q_{12}^{(f)} & Q_{22}^{(f)} & 0 & 0 & 0 \\ 0 & 0 & Q_{44}^{(f)} & 0 & 0 \\ 0 & 0 & 0 & Q_{55}^{(f)} & 0 \\ 0 & 0 & 0 & 0 & Q_{66}^{(f)} \end{array}] {\begin{cases} ε_{x x} \\ ε_{θ θ} \\ γ_{θ z} \\ γ_{x z} \\ γ_{x θ} \end{cases}},

(6)

in which

\begin{array}{l} Q_{11}^{(f)} = Q_{22}^{(f)} = \frac{E^{(f)}}{1 - {[ν^{(f)}]}^{2}}, & Q_{12}^{(f)} = ν^{(f)} Q_{11}^{(f)}, & Q_{44}^{(f)} = Q_{55}^{(f)} = Q_{66}^{(f)} = G^{(f)} . \end{array}

(7)

The face layers are fabricated from a polymeric matrix enriched with agglomerated either CNTs or GNPs. The nanofibers are distributed uniformly within the matrix and their volume fraction (F_nf) is described in terms of the mass fraction of nanofibers (w_nf), the density of the nanofibers (ρ_nf), and the density of the polymeric matrix (ρ_m) as follows³⁰:

F_{n f} = \frac{1}{1 + \frac{ρ_{n f}}{ρ_{m}} (\frac{1}{w_{n f}} - 1)} .

(8)

As described in the rule of mixture, the density of the nanocomposite face layers is calculated as follows³⁰:

ρ^{(f)} = F_{m} ρ_{m} + F_{n f} ρ_{n f},

(9)

in which F_m = 1-F_nf stands for the volume fraction of the polymeric matrix.

Owing to their low flexural rigidity, CNTs and GNPs tend to cluster which attenuates their reinforcing effects and is known as agglomeration.^31,32 When the polymeric matrix is enriched with nanofibers, a certain amount of nanofibers appears in cluster form and the other ones scatter within the polymeric matrix. In such a case, the effective elastic constants of the nanocomposite such as can be calculated utilizing the Eshelby-Mori-Tanaka approach.^33,34 To incorporate the influences of agglomeration, a two-parameter model presented by Shi et al.³⁵ is used in this paper. According to this model, the agglomeration is described with the dimensionless parameters below:

\begin{array}{l} μ = \frac{V_{c}}{V_{f}}, & η = \frac{V_{a}}{V_{n f}} \end{array},

(10)

where V_c shows the volume of the clusters, V_f is the volume of the nanocomposite face layer, V_a stands for the volume of agglomerated nanofibers, and V_nf indicates the volume of the nanofibers. The elastic modulus (E) and the Poisson’s ratio (ν) of the nanocomposite are related to the shear (G) and bulk (K) moduli as

\begin{array}{l} ν = \frac{3 K - 2 G}{6 K + 2 G}, & E = \frac{9 K G}{3 K + G}, \end{array}

(11)

in which shear and bulk moduli are described as³⁶

\begin{array}{l} G = G_{o u t} [1 + \frac{μ (\frac{G_{i n}}{G_{o u t}} - 1)}{1 + \frac{2 (1 - μ)}{15} \frac{4 - 5 ν_{o u t}}{1 - ν_{o u t}} (\frac{G_{i n}}{G_{o u t}} - 1)}], & K = K_{o u t} [1 + \frac{μ (\frac{K_{i n}}{K_{o u t}} - 1)}{1 + \frac{1 - μ}{3} \frac{1 + ν_{o u t}}{1 - ν_{o u t}} (\frac{K_{i n}}{K_{o u t}} - 1)}], \end{array}

(12)

where the subscripts “in” and “out” indicate inside and outside of the clusters, respectively, and³⁶

\begin{array}{l} \begin{array}{l} G_{i n} = G_{m} + \frac{η (η_{n f} - 2 G_{m} β_{n f}) F_{n f}}{2 [μ + η (β_{n f} - 1) F_{n f}]}, & G_{o u t} = G_{m} + \frac{(1 - η) (η_{n f} - 2 G_{m} β_{n f}) F_{n f}}{2 [1 - μ + (β_{n f} - 1) (1 - η) F_{n f}]}, \end{array} \\ \begin{array}{l} K_{i n} = K_{m} + \frac{η (δ_{n f} - 3 K_{m} α_{n f}) F_{n f}}{3 [μ + η (α_{n f} - 1) F_{n f}]}, & K_{o u t} = K_{m} + \frac{(1 - η) (δ_{n f} - 3 K_{m} α_{n f}) F_{n f}}{3 [1 - μ + (α_{n f} - 1) (1 - η) F_{n f}]}, & ν_{o u t} = \frac{3 K_{o u t} - 2 G_{o u t}}{6 K_{o u t} + 2 G_{o u t}}, \end{array} \end{array}

(13)

in which G_m = E_m/(2 + 2ν_m) and K_m = E_m/(3-6ν_m) are the bulk and shear moduli of the polymeric matrix, respectively. Moreover, α_r, β_r, η_r, and δ_r are defined as follows³⁶:

\begin{array}{l} CNT - reinforced nanocomposite : \\ \begin{array}{l} α_{n f} = \frac{l_{n f} + k_{n f} + 3 (G_{m} + K_{m})}{3 (k_{n f} + G_{m})}, & δ_{n f} = \frac{1}{3} [2 l_{n f} + n_{n f} + \frac{(G_{m} + 3 K_{m} - l_{n f}) (l_{n f} + 2 k_{n f})}{k_{n f} + G_{m}}], \end{array} \\ β_{n f} = \frac{1}{5} [\frac{l_{n f} + 2 k_{n f} + 4 G_{m}}{3 (G_{m} + k_{n f})} + \frac{4 G_{m}}{p_{n f} + G_{m}} + \frac{4 G_{m} (3 K_{m} + 4 G_{m})}{G_{m} (G_{m} + 3 K_{m}) + m_{n f} (7 G_{m} + 3 K_{m})}], \\ η_{n f} = \frac{1}{5} [\frac{2 (n_{n f} - l_{n f})}{3} + \frac{8 G_{m} p_{n f}}{G_{m} + p_{n f}} + \frac{2 (k_{n f} - l_{n f}) (2 G_{m} + l_{n f})}{3 (G_{m} + k_{n f})} + \frac{8 m_{n f} G_{m} (3 K_{m} + 4 G_{m})}{3 K_{m} (G_{m} + m_{n f}) + G_{m} (7 m_{n f} + G_{m})}], \\ GNP - reinforced nanocomposite : \\ \begin{array}{l} α_{n f} = \frac{3 K_{m} + 2 n_{n f} - 2 l_{n f}}{3 n_{n f}}, & δ_{n f} = \frac{3 K_{m} (n_{n f} + 2 l_{n f}) + 4 (k_{n f} n_{n f} - l_{n f}^{2})}{3 n_{r}}, \end{array} \\ \begin{array}{l} β_{n f} = \frac{4 G_{m} + 7 n_{n f} + 2 l_{n f}}{15 n_{n f}} - \frac{2 G_{m}}{5 p_{n f}}, & η_{n f} = \frac{2}{15} (k_{n f} + 6 m_{n f} + 8 G_{m} - \frac{l_{n f}^{2} + 2 G_{m} l_{n f}}{n_{n f}}), \end{array} \end{array}

(14)

in which k_nf, l_nf, m_nf, n_nf, and p_nf are called Hill’s elastic constants which are presented in Table 1 along with the density for several single-walled carbon nanotubes and GNPs,

Table 1.

Mechanical properties of the GNPs.^36,37

	k_nf (GPa)	l_nf (GPa)	m_nf (GPa)	n_nf (GPa)	p_nf (GPa)	ρ_nf (kg/m³)
SWCNT (5,5)	536	184	132	2143	791	1400
SWCNT (10,10)	271	88	17	1089	442
SWCNT (15,15)	181	58	5	726	301
SWCNT (20,20)	136	43	2	545	227
SWCNT (50,50)	55	17	0.1	218	92
GNP	850	6.8	369	102,000	102,000	1060

Stress, strain, deformation

According to the zig-zag theory, the displacement field inside the shell can be considered as follows^38,39:

{\begin{cases} u_{1} (t, z, x, θ) \\ u_{2} (t, z, x, θ) \\ u_{3} (t, z, x, θ) \end{cases}} = {\begin{cases} u (t, x, θ) \\ v (t, x, θ) \\ w (t, x, θ) \end{cases}} + z {\begin{array}{c} α_{x} (t, x, θ) \\ α_{θ} (t, x, θ) \\ 0 \end{array}} + f (z) {\begin{array}{c} s_{x} (t, x, θ) \\ s_{θ} (t, x, θ) \\ 0 \end{array}},

(15)

where u₁, u₂, and u₃ respectively show displacement along meridional (x), circumferential (θ), and transverse (z) directions; α_x and α_θ sequentially are rotations around θ- and x-axes; s_x and s_θ do not represent physical concepts; and f(z) is considered to incorporate the zig-zag effect and is defined as follows^38,39:

f (z) = \sum_{j = 1}^{P} \frac{2 {(- 1)}^{j}}{h_{j}} [z - 0.5 (z_{j} + z_{j + 1})] [H (z - z_{j}) - H (z - z_{j + 1})] .

(16)

in which h_j is the thickness of the jth layer, z_j and z_j+1 respectively show the transverse position of the bottom and top of the jth layer, P is the number of layers, and H stands for the well-known Heaviside function.

It should be noted that by discarding f(z), s_x, and s_θ, the well-known first-order shear deformation theory (FSDT) is obtained. As the main difference between the zig-zag theory and the FSDT, it should be noted that in the FSDT, all layers of a sandwich structure experience the same rotation but, in the zig-zag theory, layers of the sandwich structure experience different rotations. Thus, the definitions of displacement fields are different for each layer of the sandwich structure.

The components of strain are presented as follows^40,41:

\begin{array}{l} ε_{x x} = \frac{\partial u_{1}}{\partial x}, & ε_{θ θ} = \frac{\sin ψ}{r} u_{1} + \frac{1}{r} \frac{\partial u_{2}}{\partial θ} + \frac{\cos ψ}{r} u_{3}, & ε_{z z} = \frac{\partial u_{3}}{\partial z}, \\ γ_{x θ} = \frac{1}{r} \frac{\partial u_{1}}{\partial θ} + \frac{\partial u_{2}}{\partial x} - \frac{\sin ψ}{r} u_{2}, & γ_{x z} = \frac{\partial u_{1}}{\partial z} + \frac{\partial u_{3}}{\partial x}, & γ_{θ z} = \frac{\partial u_{2}}{\partial z} - \frac{\cos ψ}{r} u_{2} + \frac{1}{r} \frac{\partial u_{3}}{\partial θ}, \end{array}

(17)

where r = r(x) = a+xsinψ stands for the radius of the shell measured from its center to the middle of the auxetic honeycomb core.

By inserting equation (15) into equation (17), the components of strain tensor can be represented as follows:

\begin{array}{l} ε_{x x} = \frac{\partial u}{\partial x} + z \frac{\partial α_{x}}{\partial x} + f \frac{\partial s_{x}}{\partial x}, \\ \begin{array}{l} ε_{θ θ} = \frac{\sin ψ}{r} u + \frac{1}{r} \frac{\partial v}{\partial θ} + \frac{\cos ψ}{r} w + z (\frac{\sin ψ}{r} α_{x} + \frac{1}{r} \frac{\partial α_{θ}}{\partial θ}) + f (\frac{\sin ψ}{r} s_{x} + \frac{1}{r} \frac{\partial s_{θ}}{\partial θ}), & ε_{z z} = 0, \end{array} \\ γ_{x θ} = \frac{1}{r} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x} - \frac{\sin ψ}{r} v + z (\frac{1}{r} \frac{\partial α_{x}}{\partial θ} + \frac{\partial α_{θ}}{\partial x} - \frac{\sin ψ}{r} α_{θ}) + f (\frac{1}{r} \frac{\partial s_{x}}{\partial θ} + \frac{\partial s_{θ}}{\partial x} - \frac{\sin ψ}{r} s_{θ}), \\ \begin{array}{l} γ_{x z} = \frac{\partial w}{\partial x} + α_{x} + g s_{x}, & γ_{θ z} = - \frac{\cos ψ}{r} v + \frac{1}{r} \frac{\partial w}{\partial θ} + α_{θ} + g s_{θ}, \end{array} \end{array}

(18)

where

g (z) = \frac{d f (z)}{d z} = \sum_{j = 1}^{P} \frac{2 {(- 1)}^{j}}{h_{j}} [H (z - z_{j}) - H (z - z_{j + 1})] .

(19)

It is noteworthy that the components of the strain presented in equation (19) are presented by considering the assumptions of shallow shells as described in equation (18):

\begin{array}{l} 1 - \frac{z \cos ψ}{r} \approx 1, & g - \frac{f \cos ψ}{r} \approx g, \end{array}

(20)

Hamilton’s principle

As described in Hamilton’s principle, one can attain the governing equations and boundary conditions associated with the dynamic analysis of a structure through the equation below⁴²:

\int_{t_{1}}^{t_{2}} (δ T - δ U_{s} + δ W_{n . c .}) d t = 0,

(21)

where δ indicates the variational operator, [t₁,t₂] show an arbitrary time interval, T, U_s, and W_n.c. are the kinetic energy, the strain energy, and the work done by non-conservative loads.

The relation below provides the kinetic energy:

T = \frac{1}{2} \underset{V}{∭} ρ [{(\frac{\partial u_{1}}{\partial t})}^{2} + {(\frac{\partial u_{2}}{\partial t})}^{2} + {(\frac{\partial u_{3}}{\partial t})}^{2}] d V,

(22)

where

\underset{V}{∭} () d V = \iint_{S} \int_{- \frac{h}{2}}^{\frac{h}{2}} () d z d S,

(23)

According to equations (15) and (23), equation (22) can be rewritten as

T = \frac{1}{2} \iint_{S} {I_{0} [{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2}] + I_{2} [{(\frac{\partial α_{x}}{\partial t})}^{2} + {(\frac{\partial α_{θ}}{\partial t})}^{2}] + 2 I_{4} (\frac{\partial α_{x}}{\partial t} \frac{\partial s_{x}}{\partial t} + \frac{\partial α_{θ}}{\partial t} \frac{\partial s_{θ}}{\partial t}) + I_{5} [{(\frac{\partial s_{x}}{\partial t})}^{2} + {(\frac{\partial s_{θ}}{\partial t})}^{2}]} d S,

(24)

in which the relation below provides the inertia terms of the shell:

{\begin{cases} I_{0} \\ I_{2} \\ I_{4} \\ I_{5} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (z) {\begin{array}{c} 1 \\ z^{2} \\ z f (z) \\ f^{2} (z) \end{array}} d z .

(25)

It is noteworthy that due to the symmetry of the material properties ρ(-z) = ρ(z), the integrals of density multiplication in odd functions (z and f(z)) are equal to zero.

Equation (26) provides the variation of the kinetic energy:

\begin{array}{l} δ T = \iint_{S} [I_{0} (\frac{\partial u}{\partial t} \frac{\partial δ u}{\partial t} + \frac{\partial v}{\partial t} \frac{\partial δ v}{\partial t} + \frac{\partial w}{\partial t} \frac{\partial δ w}{\partial t}) + I_{2} (\frac{\partial α_{x}}{\partial t} \frac{\partial δ α_{x}}{\partial t} + \frac{\partial α_{θ}}{\partial t} \frac{\partial δ α_{θ}}{\partial t}) \\ + I_{4} (\frac{\partial α_{x}}{\partial t} \frac{\partial δ s_{x}}{\partial t} + \frac{\partial s_{x}}{\partial t} \frac{\partial δ α_{x}}{\partial t} + \frac{\partial α_{θ}}{\partial t} \frac{\partial δ s_{θ}}{\partial t} + \frac{\partial s_{θ}}{\partial t} \frac{\partial δ α_{θ}}{\partial t}) + I_{5} (\frac{\partial s_{x}}{\partial t} \frac{\partial δ s_{x}}{\partial t} + \frac{\partial s_{θ}}{\partial t} \frac{\partial δ s_{θ}}{\partial t})] d S, \end{array}

(26)

The equation below shows the variation of the strain energy⁴²:

δ U_{s} = \underset{V}{∭} (σ_{x x} δ ε_{x x} + σ_{θ θ} δ ε_{θ θ} + σ_{x θ} δ γ_{x θ} + σ_{x z} δ γ_{x z} + σ_{θ z} δ γ_{θ z}) d V,

(27)

which can be rewritten according to equation (23) as

\begin{array}{l} δ U_{s} = \iint_{S} [N_{x x} \frac{\partial δ u}{\partial x} + M_{x x} \frac{\partial δ α_{x}}{\partial x} + P_{x x} \frac{\partial δ s_{x}}{\partial x} + \frac{N_{θ θ} \sin ψ}{r} δ u + \frac{N_{θ θ}}{r} \frac{\partial δ v}{\partial θ} + \frac{N_{θ θ} \cos ψ}{r} δ w \\ + M_{θ θ} (\frac{\sin ψ}{r} δ α_{x} + \frac{1}{r} \frac{\partial δ α_{θ}}{\partial θ}) + P_{θ θ} (\frac{\sin ψ}{r} δ s_{x} + \frac{1}{r} \frac{\partial δ s_{θ}}{\partial θ}) + \frac{N_{x θ}}{r} \frac{\partial δ u}{\partial θ} + N_{x θ} \frac{\partial δ v}{\partial x} \\ - \frac{N_{x θ} \sin ψ}{r} δ v + M_{x θ} (\frac{1}{r} \frac{\partial δ α_{x}}{\partial θ} + \frac{\partial δ α_{θ}}{\partial x} - \frac{\sin ψ}{r} δ α_{θ}) + P_{x θ} (\frac{1}{r} \frac{\partial δ s_{x}}{\partial θ} + \frac{\partial δ s_{θ}}{\partial x} - \frac{\sin ψ}{r} δ s_{θ}) \\ + Q_{x z} (\frac{\partial δ w}{\partial x} + δ α_{x}) + R_{x z} δ s_{x} + Q_{θ z} (- \frac{\cos ψ}{r} δ v + \frac{1}{r} \frac{\partial δ w}{\partial θ} + δ α_{θ}) + R_{θ z} δ s_{θ}] d V, \end{array}

(28)

where

\begin{array}{l} {\begin{cases} N_{i j} \\ M_{i j} \\ P_{i j} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} σ_{i j} {\begin{cases} 1 \\ z \\ f (z) \end{cases}} d z, & {\begin{cases} Q_{q q} \\ R_{q q} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} σ_{q q} {\begin{cases} 1 \\ g (z) \end{cases}} d z, & \begin{array}{l} i, j = 1, 2, 6 . \\ q = 4, 5 . \end{array} \end{array}

(29)

By substituting equations (1), (5), and (19) into equation (29), one can present the equation below:

\begin{array}{l} N_{x x} = A_{11} \frac{\partial u}{\partial x} + A_{12} (\frac{\sin ψ}{r} u + \frac{1}{r} \frac{\partial v}{\partial θ} + \frac{\cos ψ}{r} w), \\ M_{x x} = D_{11} \frac{\partial α_{x}}{\partial x} + H_{11} \frac{\partial s_{x}}{\partial x} + D_{12} (\frac{\sin ψ}{r} α_{x} + \frac{1}{r} \frac{\partial α_{θ}}{\partial θ}) + H_{12} (\frac{\sin ψ}{r} s_{x} + \frac{1}{r} \frac{\partial s_{θ}}{\partial θ}), \end{array}

\begin{array}{l} P_{x x} = H_{11} \frac{\partial α_{x}}{\partial x} + T_{11} \frac{\partial s_{x}}{\partial x} + H_{12} (\frac{\sin ψ}{r} α_{x} + \frac{1}{r} \frac{\partial α_{θ}}{\partial θ}) + T_{12} (\frac{\sin ψ}{r} s_{x} + \frac{1}{r} \frac{\partial s_{θ}}{\partial θ}), \\ N_{θ θ} = A_{12} \frac{\partial u}{\partial x} + A_{22} (\frac{\sin ψ}{r} u + \frac{1}{r} \frac{\partial v}{\partial θ} + \frac{\cos ψ}{r} w), \\ M_{θ θ} = D_{12} \frac{\partial α_{x}}{\partial x} + H_{12} \frac{\partial s_{x}}{\partial x} + D_{22} (\frac{\sin ψ}{r} α_{x} + \frac{1}{r} \frac{\partial α_{θ}}{\partial θ}) + H_{22} (\frac{\sin ψ}{r} s_{x} + \frac{1}{r} \frac{\partial s_{θ}}{\partial θ}), \\ P_{θ θ} = H_{12} \frac{\partial α_{x}}{\partial x} + T_{12} \frac{\partial s_{x}}{\partial x} + H_{22} (\frac{\sin ψ}{r} α_{x} + \frac{1}{r} \frac{\partial α_{θ}}{\partial θ}) + T_{22} (\frac{\sin ψ}{r} s_{x} + \frac{1}{r} \frac{\partial s_{θ}}{\partial θ}), \\ N_{x θ} = A_{66} (\frac{\partial v}{\partial x} + \frac{1}{r} \frac{\partial u}{\partial θ} - \frac{\sin ψ}{r} v), \\ M_{x θ} = D_{66} (\frac{1}{r} \frac{\partial α_{x}}{\partial θ} + \frac{\partial α_{θ}}{\partial x} - \frac{\sin ψ}{r} α_{θ}) + H_{66} (\frac{1}{r} \frac{\partial s_{x}}{\partial θ} + \frac{\partial s_{θ}}{\partial x} - \frac{\sin ψ}{r} s_{θ}), \\ P_{x θ} = H_{66} (\frac{1}{r} \frac{\partial α_{x}}{\partial θ} + \frac{\partial α_{θ}}{\partial x} - \frac{\sin ψ}{r} α_{θ}) + T_{66} (\frac{1}{r} \frac{\partial s_{x}}{\partial θ} + \frac{\partial s_{θ}}{\partial x} - \frac{\sin ψ}{r} s_{θ}), \\ \begin{array}{l} Q_{θ z} = A_{44} (- \frac{\cos ψ}{r} v + \frac{1}{r} \frac{\partial w}{\partial θ} + α_{θ}) + Y_{44} s_{θ}, & R_{θ z} = Y_{44} (- \frac{\cos ψ}{r} v + \frac{1}{r} \frac{\partial w}{\partial θ} + α_{θ}) + Z_{44} s_{θ}, \end{array} \\ \begin{array}{l} Q_{x z} = A_{55} (\frac{\partial w}{\partial x} + α_{x}) + Y_{55} s_{x}, & R_{x z} = Y_{55} (\frac{\partial w}{\partial x} + α_{x}) + Z_{55} s_{x}, \end{array} \end{array}

(30)

where

\begin{array}{l} {\begin{cases} A_{i j} \\ D_{i j} \\ H_{i j} \\ T_{i j} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{i j} (z) {\begin{array}{c} 1 \\ z^{2} \\ z f (z) \\ f^{2} (z) \end{array}} d z, & {\begin{cases} A_{q q} \\ Y_{q q} \\ Z_{q q} \end{cases}} = \int_{- \frac{h}{2}}^{\frac{h}{2}} Q_{q q} (z) {\begin{array}{c} 1 \\ g (z) \\ g^{2} (z) \end{array}} d z, & \begin{array}{l} i, j = 1, 2, 6 . \\ q = 4, 5 . \end{array} \end{array}

(31)

It is noteworthy that due to the symmetry of the elastic constants Q_ij(-z) = Q_ij(z), the integrals of Q_ij multiplication in odd functions (z and f(z)) are equal to zero.

By inserting equations (26) and (28) into equation (21) and considering W_n.c. = 0 for the free vibration analysis, the governing equations are attained as follows:

\begin{array}{l} \frac{\partial N_{x x}}{\partial x} + \frac{\sin ψ}{r} (N_{x x} - N_{θ θ}) + \frac{1}{r} \frac{\partial N_{x θ}}{\partial θ} - I_{0} \frac{\partial^{2} u}{\partial t^{2}} = 0, \\ \frac{1}{r} \frac{\partial N_{θ θ}}{\partial θ} + \frac{\partial N_{x θ}}{\partial x} + \frac{2 \sin ψ}{r} N_{x θ} + \frac{\cos ψ}{r} Q_{θ z} - I_{0} \frac{\partial^{2} v}{\partial t^{2}} = 0, \\ \frac{\partial Q_{x z}}{\partial x} + \frac{\sin ψ}{r} Q_{x z} + \frac{1}{r} \frac{\partial Q_{θ z}}{\partial θ} - \frac{\cos ψ}{r} N_{θ θ} + P_{\infty} - I_{0} \frac{\partial^{2} w}{\partial t^{2}} = 0, \\ \frac{\partial M_{x x}}{\partial x} + \frac{\sin ψ}{r} (M_{x x} - M_{θ θ}) + \frac{1}{r} \frac{\partial M_{x θ}}{\partial θ} - Q_{x z} - I_{2} \frac{\partial^{2} α_{x}}{\partial t^{2}} - I_{4} \frac{\partial^{2} s_{x}}{\partial t^{2}} = 0, \\ \frac{1}{r} \frac{\partial M_{θ θ}}{\partial θ} + \frac{\partial M_{x θ}}{\partial x} + \frac{2 \sin ψ}{r} M_{x θ} - Q_{θ z} - I_{2} \frac{\partial^{2} α_{θ}}{\partial t^{2}} - I_{4} \frac{\partial^{2} s_{θ}}{\partial t^{2}} = 0, \\ \frac{\partial P_{x x}}{\partial x} + \frac{\sin ψ}{r} (P_{x x} - P_{θ θ}) + \frac{1}{r} \frac{\partial P_{x θ}}{\partial θ} - R_{x z} - I_{4} \frac{\partial^{2} α_{x}}{\partial t^{2}} - I_{5} \frac{\partial^{2} s_{x}}{\partial t^{2}} = 0, \end{array} \frac{1}{r} \frac{\partial P_{θ θ}}{\partial θ} + \frac{\partial P_{x θ}}{\partial x} + \frac{2 \sin ψ}{r} P_{x θ} - R_{θ z} - I_{4} \frac{\partial^{2} α_{θ}}{\partial t^{2}} - I_{5} \frac{\partial^{2} s_{θ}}{\partial t^{2}} = 0 .

(32)

Meanwhile, the relations below describe the boundary conditions:

\begin{array}{l} \begin{array}{l} N_{x x} δ u = 0, & N_{x θ} δ v = 0, & Q_{x z} δ w = 0, \end{array} & \begin{array}{l} M_{x x} δ α_{x} = 0, & M_{x θ} δ α_{θ} = 0, & P_{x x} δ s_{x} = 0, & P_{x θ} δ s_{θ} = 0, \end{array} \end{array}

(33)

In this article, three classic types of conditions are considered. In conjunction with equation (33), the mathematical modeling of these conditions is defined as

\begin{array}{l} Clamped (C) : & \begin{array}{l} u = 0, & v = 0, & w = 0, & α_{x} = 0, & α_{θ} = 0, & s_{x} = 0, & s_{θ} = 0 . \end{array} \\ Simply supported (S) : & \begin{array}{l} N_{x x} = 0, & v = 0, & w = 0, & M_{x x} = 0, & α_{θ} = 0, & P_{x x} = 0, & s_{θ} = 0 . \end{array} \\ Free (F) : & \begin{array}{l} N_{x x} = 0, & N_{x θ} = 0, & Q_{x z} = 0, & M_{x x} = 0, & M_{x θ} = 0, & P_{x x} = 0, & P_{x θ} = 0 . \end{array} \end{array}

(34)

Solution

In this section, a semi-analytical solution is presented which consists of an analytical solution in the circumferential direction followed by an approximate solution in the meridional direction.

The circumferential direction (analytical solution)

Due to the continuity of all geometric and physical parameters at θ = 0 and θ = 2π, the following solution can be considered:

\begin{array}{l} {\begin{cases} u (t, x, θ) \\ w (t, x, θ) \\ α_{x} (t, x, θ) \\ s_{x} (t, x, θ) \end{cases}} = {\begin{cases} U (x) \\ W (x) \\ X (x) \\ S_{x} (x) \end{cases}} \cos (n θ) \exp (i ω t), & {\begin{cases} v (t, x, θ) \\ α_{θ} (t, x, θ) \\ s_{θ} (t, x, θ) \end{cases}} = {\begin{cases} V (x) \\ Θ (x) \\ S_{θ} (x) \end{cases}} \sin (n θ) \exp (i ω t), \end{array}

(35)

in which i² = −1, n = 0,1,2,3,… is known as the circumferential wave number, and ω represents the natural frequency of the shell. By substituting equation (30) into equation (32) and applying equation (35), one can write the relations below as the governing equations:

\begin{array}{l} ω^{2} I_{0} U + A_{11} U^{″} + A_{11} r^{- 1} \sin ψ U^{'} - (A_{22} \sin^{2} ψ + n^{2} A_{66}) r^{- 2} U + (A_{12} + A_{66}) n r^{- 1} V^{'} \\ - (A_{22} + A_{66}) n r^{- 2} \sin ψ V + A_{12} r^{- 1} \cos ψ W^{'} - 0.5 A_{22} r^{- 2} \sin 2 ψ W = 0, \end{array}

\begin{array}{l} ω^{2} I_{0} V - n (A_{12} + A_{66}) r^{- 1} U^{'} - (A_{22} + A_{66}) n r^{- 2} \sin ψ U + A_{66} V^{″} + A_{66} r^{- 1} \sin ψ V^{'} \\ - (n^{2} A_{22} + A_{44} \cos^{2} ψ + A_{66} \sin^{2} ψ) r^{- 2} V - (A_{22} + A_{44}) n r^{- 2} \cos ψ W + A_{44} r^{- 1} \cos ψ Θ + Y_{44} r^{- 1} \cos ψ S_{θ} = 0, \end{array}

\begin{array}{l} I_{0} ω^{2} W - A_{12} r^{- 1} \cos ψ U^{'} - 0.5 A_{22} r^{- 2} \sin 2 ψ U - n (A_{22} + A_{44}) r^{- 2} \cos ψ V \\ + A_{55} W^{″} + A_{55} r^{- 1} \sin ψ W^{'} - (A_{22} \cos^{2} ψ + n^{2} A_{44}) r^{- 2} W + A_{55} X^{'} \\ + A_{55} r^{- 1} \sin ψ X + n A_{44} r^{- 1} Θ + Y_{55} S_{x}^{'} + Y_{55} r^{- 1} \sin ψ S_{x} + n Y_{44} r^{- 1} S_{θ} = 0, \end{array}

\begin{array}{l} ω^{2} (I_{2} X + I_{4} S_{x}) - A_{55} W^{'} + D_{11} X^{″} + D_{11} r^{- 1} \sin ψ X^{'} - [A_{55} + (D_{22} \sin^{2} ψ + n^{2} D_{66}) r^{- 2}] X \\ + n (D_{12} + D_{66}) r^{- 1} Θ^{'} - n (D_{22} + D_{66}) r^{- 2} \sin ψ Θ + H_{11} S_{x}^{″} + H_{11} r^{- 1} \sin ψ S_{x}^{'} \\ - [Y_{55} + (H_{22} \sin^{2} ψ + n^{2} H_{66}) r^{- 2}] S_{x} + n (H_{12} + H_{66}) r^{- 1} S_{θ}^{'} - n (H_{22} + H_{66}) r^{- 2} \sin ψ S_{θ} = 0, \end{array}

\begin{array}{l} ω^{2} (I_{2} Θ + I_{4} S_{θ}) + A_{44} r^{- 1} \cos ψ V + n A_{44} r^{- 1} W - n (D_{12} + D_{66}) r^{- 1} X^{'} - n (D_{22} + D_{66}) r^{- 2} \sin ψ X \\ + D_{66} Θ^{″} + D_{66} r^{- 1} \sin ψ Θ^{'} - [A_{44} + (n^{2} D_{22} + D_{66} \sin^{2} ψ) r^{- 2}] Θ - n (H_{12} + H_{66}) r^{- 1} S_{x}^{'} \\ - n (H_{22} + H_{66}) r^{- 2} \sin ψ S_{x} + H_{66} S_{θ}^{″} + H_{66} r^{- 1} \sin ψ S_{θ}^{'} - [Y_{44} + (n^{2} H_{22} + H_{66} \sin^{2} ψ) r^{- 2}] S_{θ} = 0, \end{array}

\begin{array}{l} ω^{2} (I_{4} X + I_{5} S_{x}) - Y_{55} W^{'} + H_{11} X^{″} + H_{11} r^{- 1} \sin ψ X^{'} - [Y_{55} + (H_{22} \sin^{2} ψ + n^{2} H_{66}) r^{- 2}] X \\ + n (H_{12} + H_{66}) r^{- 1} Θ^{'} - n (H_{22} + H_{66}) r^{- 2} \sin ψ Θ + T_{11} S_{x}^{″} + T_{11} r^{- 1} \sin ψ S_{x}^{'} \\ - [Z_{55} + (T_{22} \sin^{2} ψ + n^{2} T_{66}) r^{- 2}] S_{x} + n (T_{12} + T_{66}) r^{- 1} S_{θ}^{'} - n (T_{22} + T_{66}) r^{- 2} \sin ψ S_{θ} = 0, \end{array}

\begin{array}{l} ω^{2} (I_{4} Θ + I_{5} S_{θ}) + Y_{44} r^{- 1} \cos ψ V + n Y_{44} r^{- 1} W - n (H_{12} + H_{66}) r^{- 1} X^{'} - n (H_{22} + H_{66}) r^{- 2} \sin ψ X \\ + H_{66} Θ^{″} + H_{66} r^{- 1} \sin ψ Θ^{'} - [Y_{44} + (n^{2} H_{22} + H_{66} \sin^{2} ψ) r^{- 2}] Θ - n (T_{12} + T_{66}) r^{- 1} S_{x}^{'} \\ - n (T_{22} + T_{66}) r^{- 2} \sin ψ S_{x} + T_{66} S_{θ}^{″} + T_{66} r^{- 1} \sin ψ S_{θ}^{'} - [Z_{44} + (n^{2} T_{22} + T_{66} \sin^{2} ψ) r^{- 2}] S_{θ} = 0 . \end{array}

(36)

Inserting equation (30) into equation (34), applying equation (35), and performing some simplifications for the simply supported conditions, the equations below describe the boundary conditions:

\begin{array}{l} \begin{array}{l} C : & \begin{array}{l} U = 0, & V = 0, & W = 0, & X = 0, & Θ = 0, & S_{x} = 0, & S_{θ} = 0 . \end{array} \end{array} \\ \begin{array}{l} S : & A_{11} U^{'} + A_{12} r^{- 1} \sin ψ U = 0, & V = 0, & W = 0, & D_{11} X^{'} + D_{12} r^{- 1} \sin ψ X + H_{11} S_{x}^{'} + H_{12} r^{- 1} \sin ψ S_{x} = 0, \end{array} \\ \begin{array}{l} Θ = 0, & \begin{array}{l} H_{11} X^{'} + H_{12} r^{- 1} \sin ψ X + T_{11} S_{x}^{'} + T_{12} r^{- 1} \sin ψ S_{x} = 0, & S_{θ} = 0 . \end{array} \end{array} \\ \begin{array}{l} F : & A_{11} U^{'} + A_{12} r^{- 1} \sin ψ U + n A_{12} r^{- 1} V + A_{12} r^{- 1} \cos ψ W = 0, \end{array} \\ \begin{array}{l} - n A_{66} r^{- 1} U + A_{66} V^{'} - A_{66} r^{- 1} \sin ψ V = 0, & A_{55} W^{'} + A_{55} X + Y_{55} S_{x} = 0, \end{array} \\ D_{11} X^{'} + D_{12} r^{- 1} \sin ψ X + n D_{12} r^{- 1} Θ + H_{11} S_{x}^{'} + H_{12} r^{- 1} \sin ψ S_{x} + n H_{12} r^{- 1} S_{θ} = 0, \\ - n D_{66} r^{- 1} X + D_{66} Θ^{'} - D_{66} r^{- 1} \sin ψ Θ - n H_{66} r^{- 1} S_{x} + H_{66} S_{θ}^{'} - H_{66} r^{- 1} \sin ψ S_{θ} = 0, \\ H_{11} X^{'} + H_{12} r^{- 1} \sin ψ X + n H_{12} r^{- 1} Θ + T_{11} S_{x}^{'} + T_{12} r^{- 1} \sin ψ S_{x} + n T_{12} r^{- 1} S_{θ} = 0, \\ - n H_{66} r^{- 1} X + H_{66} Θ^{'} - H_{66} r^{- 1} \sin ψ Θ - n T_{66} r^{- 1} S_{x} + T_{66} S_{θ}^{'} - T_{66} r^{- 1} \sin ψ S_{θ} = 0 . \end{array}

(37)

The meridional direction (numerical solution)

It seems that there is a low and weak possibility to find an exact solution for the governing equation (36). Thus, in this section, a numerical solution is presented via the differential quadrature method (DQM). Based on the main idea of the DQM, the solution domain is discretized into a pre-selected set of grid points consisting of N points. At each of these points, derivatives of a function like T(x) are approximated in terms of values of the function as the relation below⁴³:

{\frac{d^{s} T}{d x^{s}}} = [A^{(s)}] {T},

(38)

where, with the definitions below, [A^(s)] is called the weighting coefficient matrix of the sth-order derivative⁴³:

A_{i j}^{(1)} = {\begin{cases} \begin{array}{l} \sum_{\begin{array}{l} m = 1 \\ m \neq i \end{array}}^{N} \frac{1}{x_{i} - x_{m}} & i = j \\ \frac{\prod_{\begin{array}{l} m = 1 \\ m \neq i, j \end{array}}^{N} (x_{i} - x_{m})}{\prod_{\begin{array}{l} m = 1 \\ m \neq j \end{array}}^{N} (x_{j} - x_{m})} & i \neq j \end{array} & i, j = 1, 2, . . ., N \end{cases},

\begin{array}{l} [A^{(s)}] = [A^{(1)}] [A^{(s - 1)}], & s = 2, 3, 4, \dots . \end{array}

(39)

As far as the dispersion of the grid points is concerned, in this work, a non-uniform distribution pattern is utilized. According to this pattern, the interval 0 ≤ x ≤ L can be discretized as follows⁴³:

x_{i} = \frac{L}{2} [1 - \cos (\frac{i - 1}{N - 1} π)], i = 1, 2, . . ., N .

(40)

It should be noted that this distribution pattern is called the Chebyshev-Gauss–Lobatto pattern.

By applying equation (38), the governing equation (36) can be represented in the algebraic form below:

[K] {y} = ω^{2} [M] {y},

(41)

where, as described in Appendix A,

{y}

, [M], and [K] are the displacement vector, mass matrix, and stiffness matrix, consecutively.

By applying equation (38), any combinations of the boundary conditions (37) are described in the following algebraic form:

[R] {y} = {0},

(42)

where, as presented in Appendix B, the matrix [R] is the boundary conditions matrix.

Simultaneous solutions of the set of algebraic equations (41) and (42) lead to an inconsistency in the numbers of unknown variables and algebraic equations.⁴⁴ This challenge can be overcome by dividing the set of grid points into two sets: the points at the edges of the shell (x₁ and x_N) known as the boundary points, and the other interior ones (x₂-x_N-1) called the domain points.⁴⁰ If one neglects satisfying the algebraic form of the governing equations at the boundary points, this relation can be presented as follows:

[K_{0}] {y} = ω^{2} [M_{0}] {y},

(43)

in which the subscript 0 indicates corresponding non-square truncated stiffness and mass matrices. By separating the columns of each matrix associated with the domain (d) and boundary (b) points in the truncated stiffness and mass matrices (equation (43)) and the boundary conditions matrix (equation (42)), these algebraic relations are represented as⁴⁰

{[K_{0}]}_{b o} {y}_{b o} + {[K_{0}]}_{d o} {y}_{d o} = ω^{2} ({[M_{0}]}_{b o} {y}_{b o} + {[M_{0}]}_{d o} {y}_{d o}),

(44)

{[R]}_{b o} {y}_{b o} + {[R]}_{d o} {y}_{d o} = {0},

(45)

Inserting equation (45) into equation (44) leads to the final eigenvalue equation presented in equation (46) as

[K^{*}] {y}_{d o} = ω^{2} [M^{*}] {y}_{d o},

(46)

in which

\begin{array}{l} [K^{*}] = {[K_{0}]}_{d o} - {[K_{0}]}_{b o} {[R]}_{b o}^{- 1} {[R]}_{d o}, & [M^{*}] = {[M_{0}]}_{d o} - {[M_{0}]}_{b o} {[R]}_{b o}^{- 1} {[R]}_{d o} . \end{array}

(47)

Through the solution of the eigenvalue equation (46), one can obtain the natural frequencies and the vibrational mode shapes respectively as the eigenvalues and the eigenvectors.

Numerical results

The numerical results and associated physical explanations are presented and discussed in the current section. To mention the boundary conditions, two capital letters are utilized referring to equation (34) which respectively show conditions at x = 0 and x = L. Except for the cases that are stated otherwise, numerical examples are presented for a CS shell of a = 0.25, ψ = 30°, L/a = 3, and h/a = 0.1. The honeycomb core is of re-entrant type fabricated from Aluminum (E₀ = 70 GPa, ν₀ = 0.35, and ρ₀ = 2700 kg/m³) of h_c/h = 0.75, $f_{1}^{H} = 2$ ( $f_{1}^{R} \approx 3.65$ ), f₂ = 0.02, and β = 20° oriented at ϕ = 45°. In the face layers, the polymeric matrix (E_m = 3 GPa, ν_m = 0.34, ρ_m = 1200 kg/m³) is reinforced with GNPs of w_nf = 0.01, and the agglomeration parameters are considered as η = 0.8 and μ = 0.2. Results are prepared in a dimensionless form in which the dimensionless natural frequency is defined as follows:

λ_{n l} = ω_{n l} a \sqrt{\frac{ρ_{m}}{E_{m}}} .

(48)

In equation (48), the subscript n = 0,1,2,… is the circumferential wave number defined in equation (35), and the subscript l = 1,2,3,… is utilized to indicate the sequence of mode shapes in the meridional direction.

Convergence and verification

The convergence of the provided numerical solution applied in the meridional direction is examined in Table 2 for n = 3 and l = 1,2,3,4. As observed, the presented numerical solution in the meridional direction is convergent, and reliable results can be attained using N = 17. Also, the corresponding mode shapes are illustrated in Figure 3.

Table 2.

The convergence analysis of the presented numerical solution.

	N = 7	N = 9	N = 11	N = 13	N = 15	N = 17	N = 19
l = 1	0.3120	0.3126	0.3127	0.3128	0.3128	0.3129	0.3129
l = 2	0.6197	0.6215	0.6207	0.6204	0.6202	0.6201	0.6201
l = 3	0.9438	0.9427	0.9388	0.9389	0.9391	0.9393	0.9393
l = 4	1.3108	1.3021	1.3002	1.2975	1.2974	1.2972	1.2972

Figure 3.

Mode shapes of the CS shell for n = 3 and l = 1,2,3,4.

So as to check the accuracy of the presented work, an SS single-layer conical shell of ψ = 30°, Lsinψ/b = 0.25, h/b = 0.01, elastic modulus E, Poisson’s ratio ν = 0.3, and density ρ is assumed. The dimensionless natural frequencies defined as ω^* = ωb[ρ(1-ν²)/E]^0.5 are presented in Table 3 for n = 1,2,3,..,9 and l = 1 against those predicted by Dai et al.⁴⁵ As observed, a high agreement can be observed in the results presented in this research with those reported by Dai et al.⁴⁵ The small differences between the results presented in this table can be elucidated by the differences between the employed shell theories which is the zig-zag theory in the present work and a classical shell theory called Kirchhoff-Love shell theory in Ref⁴⁵. It should be noted that for a single-layer structure, the zig-zag theory is simplified to the FSDT.

Table 3.

Dimensionless natural frequencies (ω^*) of an SS single-layer conical shell (ν = 0.3, h/b = 0.01, Lsinψ/b = 0.25, l = 1).

	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
Present	0.5922	0.7907	0.7281	0.6348	0.5524	0.4940	0.4640	0.4628	0.4871
Dai et al⁴⁵	0.5922	0.7909	0.7282	0.6349	0.5525	0.4941	0.4643	0.4633	0.4879

Parametric study

For l = 1,2,3,4, Figure 4 shows the natural frequencies associated with several values of the circumferential wave number (n = 0,1,2,…,10). As shown, for each meridional mode number, there is a specific circumferential wave number which results in the lowest natural frequency. Figure 4 shows that for the selected case study, the four lowest natural frequencies are associated with (n,l)=(3,1), (n,l)=(2,1), (n,l)=(4,1), and (n,l)=(0,1), respectively. In the following results, the numerical examples are presented to examine the influences of several parameters on the natural frequencies associated with these vibrational modes. Also, the associated mode shapes are depicted in Figure 5.

Figure 4.

The influences of the circumferential wave number on the natural frequencies.

Figure 5.

Mode shapes of the CS shell for n = 3,2,4,0 and l = 1.

Figure 6 is dedicated to examining the effects of the thickness-to-radius ratio and the inclusion of the zig-zag effect on the natural frequencies. As observed, the natural frequencies increase by increasing the thickness of the shell which can be explained by higher enhancement in the stiffness of the shell compared to its inertia. This figure shows that for thin shell there are negligible differences between the results predicted by inclusion of the zig-zag effect and those predicted by discarding of the zig-zag effect (the FSDT). However, as the thickness of the sandwich shell grows, the importance of inclusion of the zig-zag effect increases and it is more essential to incorporate the zig-zag effect. Figure 6 reveals that the natural frequencies predicted by the inclusion of the zig-zag effect are always smaller than those predicted by discarding of the zig-zag effect which can be elucidated by additional flexibility created at each layer of a sandwich structure by the inclusion of the zig-zag effect.

Figure 6.

The influences of inclusion of the zig-zag effect on the natural frequencies.

The influences of the orientation of the cells in the honeycomb core on the natural frequencies are inspected in Figure 7. As observed, for each vibrational mode, an optimum orientation can be found for the orientation of the cells in the honeycomb core which brings about the highest natural frequency. Figure 7 shows that the differences between the minimum and maximum values of the natural frequencies in the first to fourth vibrational modes are about 5%, 10%, 2.5%, and 15%, respectively. These values reveal that the orientation of the cells in the honeycomb has a remarkable effect on the natural frequencies.

Figure 7.

The influences of the orientation of the honeycomb core on the natural frequencies.

By assuming a specified value for the total thickness, Figure 8 shows the influences of the core-to-shell thickness ratio on the natural frequencies. In comparison with the nanocomposite face layers, the honeycomb core benefits from lower density and suffers from smaller elastic constants. Thus, as the thickness of the honeycomb core increases, both the rigidity and inertia of the shell decrease which have opposite influences on the natural frequencies. As a result, as shown in Figure 8, for each vibrational mode, there is an optimum ratio between the thickness of the honeycomb core and the thickness of the shell which leads to the highest natural frequency. According to Figure 8, the differences between the minimum and maximum values of the natural frequencies in the first to fourth vibrational modes are about 22%, 14%, 31%, and 29%, respectively. These values show that the core-to-shell thickness ratio plays a significant role in determining the natural frequencies.

Figure 8.

The influences of the core-to-shell thickness ratio on the natural frequencies.

Figure 9 is prepared to investigate the influences of the inclined angle of the cells in the honeycomb core on the natural frequencies. An increase in the inclined angle of the cells results in a higher number of cells and consequently higher density of the honeycomb core. However, the inclined angle has opposite influences on the elastic constants of the honeycomb core.⁴⁶ Consequently, as observed in Figure 9, the natural frequencies show different trends as the inclined angle of the cells increases. Figure 9 shows that the differences between the minimum and maximum values of the natural frequencies in the first to fourth vibrational modes are about 12%, 25%, 3.4%, and 10.6%, respectively. These values confirm the high dependency of the natural frequencies on the inclined angle of the cells.

Figure 9.

The influences of the inclined angle of the cells in the honeycomb core on the natural frequencies.

The effects of the aspect ratio of the cells in the honeycomb core on the natural frequencies are examined in Figure 10. By increasing the aspect ratio of the cells in the honeycomb core, the density of the core decreases which results in lower inertia of the shell. Thus, as shown in this figure, the natural frequencies increase by increasing the aspect ratio of the cells in the honeycomb core. According to Figure 10, as the aspect ratio of the cells in the honeycomb core increases from one to 4, natural frequencies in the first to fourth vibrational modes experience about 1.3%, 1.3%, 1.1%, and 2.8% increases, respectively. Thus, it can be concluded the natural frequencies are weakly dependent on the aspect ratio of the cells in the honeycomb core.

Figure 10.

The influences of the aspect ratio of the cells in the honeycomb core on the natural frequencies.

The effects of the wall thickness of the cells in the honeycomb core on the natural frequencies are inspected in Figure 11. As the wall thickness of the cells increases both elastic constants and mass of the shell increase which has opposite influences on the natural frequencies. Therefore, the natural frequencies experience different trends as the wall thickness of the cells in the honeycomb core increases. However, in each vibrational mode, an optimum value can be found for the wall thickness of the cells in the honeycomb core which brings about the highest natural frequency. Figure 11 shows that the difference between the minimum and maximum values of the natural frequencies in the first to fourth vibrational modes are about 4%, 5.7%, 8.7%, and 22%, respectively. These values reveal that the effect of the wall thickness of the cells in the honeycomb core on the natural frequencies increases in higher vibrational modes.

Figure 11.

The influences of the wall thickness of the cells in the honeycomb core on the natural frequencies.

In Figure 12, the dependency of the natural frequencies on the mass fraction of the nanofibers is investigated. As Table 1 shows, in comparison with the polymeric matrix, the nanofibers have comparable densities and remarkably higher elastic moduli. Thus, subjoining even a small amount of the nanofibers to the polymeric matrix results in a significant enhancement in the rigidity of the face layers which provides higher natural frequencies. According to Figure 12, as the mass fraction of the nanofibers increases from zero to 0.5%, the natural frequencies in the first to fourth vibrational modes experience about 95%, 87.5%, 96.7%, and 120% enhancements, respectively. Such increases confirm the high reinforcing effect of nanofibers.

Figure 12.

The influences of the mass fraction of the nanofibers on the natural frequencies.

Figures 13 and 14 are dedicated to studying the influences of the agglomeration parameters (η and μ) on the natural frequencies. Equation (10) shows that an increase in the agglomeration parameters η means a higher amount of nanofibers are agglomerated, and an increase in the agglomeration parameters μ means a lower intensity of agglomeration inside clusters. Thus, the reinforcing effect of nanofibers diminishes by increasing the agglomeration parameters η and decreasing the agglomeration parameters μ. As a result, as shown in Figures 13 and 14, the natural frequencies decrease by increasing the agglomeration parameters η and reducing the agglomeration parameters μ. Figure 13 shows that as the agglomeration parameter η increases from 0.2 to 0.9, the natural frequencies in the first to fourth vibrational modes experience about 21%, 20%, 21%, and 24%, reductions, respectively. Figure 14 shows that as the agglomeration parameter μ increases from 0.2 to 0.9, the natural frequencies in the first to fourth vibrational modes experience about 26.5%, 25.5%, 26.7%, and 32.3% enhancements, respectively. According to these values, it can be observed that it is of high importance to incorporate the agglomeration of nanofibers.

Figure 13.

The influences of the agglomeration parameter η on the natural frequencies.

Figure 14.

The influences of the agglomeration parameter μ on the natural frequencies.

A comparison is made in Table 4 between the natural frequencies in 12 cases including two types of honeycomb cores and six types of nanofibers. As observed, the natural frequencies increase by decreasing the chiral indices of SWCNTs. The reason behind this is the smaller distances between the carbon atoms in the SWCNTs with lower chiral indices which bring about higher elastic constants (Table 1) and consequently better reinforcing effect. Table 4 shows that in comparison with SWCNTs, the GNPs bring about a better reinforcing effect and consequently higher natural frequencies. To explain this privilege of the GNPs, it should be noted that compared with SWCNTs, the GNPs benefit from higher elastic constants, lower density, and bigger specific surface area which creates a stronger bonding between them and the polymeric matrix which provides a more effective reinforcing effect. Table 4 shows that in most vibrational modes, the sandwich shell with RH core has higher natural frequencies in comparison with the sandwich shell with HH core of the same mass. It reveals the better dynamic performance of RH structures compared to HH ones.

Table 4.

The natural frequencies for the sandwich shells with various types of honeycomb cores and several types of nanofibers.

		SWCNT					GNP
		(5,5)	(10,10)	(15,15)	(20,20)	(50,50)	GNP
RH	λ₃₁	0.2423	0.2528	0.2575	0.2655	0.2836	0.3129
	λ₂₁	0.2859	0.2971	0.3022	0.3108	0.3305	0.3626
	λ₄₁	0.2870	0.3001	0.3059	0.3159	0.3383	0.3744
	λ₀₁	0.3227	0.3402	0.3479	0.3611	0.3911	0.4391
HH	λ₃₁	0.2305	0.2415	0.2464	0.2548	0.2737	0.3041
	λ₂₁	0.2533	0.2659	0.2715	0.2811	0.3027	0.3375
	λ₄₁	0.2861	0.2993	0.3052	0.3152	0.3379	0.3743
	λ₀₁	0.3318	0.3491	0.3568	0.3699	0.3996	0.4471

Table 5 is dedicated to investigating the effects of the boundary conditions on the natural frequencies. As this table shows, the lowest natural frequencies usually belong to the CF shell and the highest ones belong to the fully clamped (CC) one. Thus, it can be concluded that utilizing more restricted conditions leads to higher natural frequencies. A comparison between the natural frequencies of the SC and CS shells and a similar comparison between the natural frequencies of the FC and CF shells reveals that to attain higher natural frequencies, it is more effective to utilize the more restricted condition at the larger radius of the shell. The reason behind this is the higher perimeter of the shell at its larger radius which provides a larger boundary.

Table 5.

The natural frequencies for various types of edge conditions at both ends of the shell.

	CC	SC	CS	SS	FC	CF
λ₄₁	0.3678	0.3505	0.3129	0.2960	0.3083	0.1157
λ₅₁	0.4063	0.3869	0.3626	0.3447	0.2659	0.0885
λ₆₁	0.4251	0.4189	0.3744	0.3696	0.4132	0.1901
λ₇₁	0.6560	0.6560	0.4391	0.1292	0.6555	0.2740

Conclusions

This article was presented to investigate the free vibrational characteristics of a sandwich conical shell with a honeycomb core with the cells oriented in an arbitrary direction covered with two nanocomposite face layers. It was assumed that the honeycomb core is of either re-entrant or hexagonal types, and the face layers are fabricated of a polymeric matrix enriched with uniformly distributed agglomerated either CNTs or GNPs. The main findings and conclusions of this article are described as follows:

• The natural frequencies predicted by the inclusion of the zig-zag effect are always smaller than the corresponding ones predicted by discarding the zig-zag effect. This difference grows by raising the thickness of the shell.

• For each vibrational mode, there is an optimum orientation of the cells in the honeycomb core which provides the highest natural frequency.

• For each vibrational mode, an optimum ratio can be found between the thickness of the honeycomb core and the total thickness of the shell which provides the highest natural frequency.

• There is no specific trend for the variations of the natural frequencies versus the variation of the inclined angle of the cells in the honeycomb core.

• To attain higher natural frequencies, it is better to use a honeycomb core with the cells with higher values of aspect ratio.

• For each vibrational mode, there is an optimum value of the wall thickness of the cells in the honeycomb core which brings about the highest natural frequency.

• Subjoining a small amount of the nanofibers (CNTs or GNPs) to the polymeric matrix provides a remarkable enhancement in the natural frequencies.

• The highest natural frequencies belong to the shell with an auxetic honeycomb core and GNP-reinforced face layers.

• Utilizing the SWCNTs with lower chirality indices results in higher natural frequencies.

• An increase in the agglomeration intensity leads to lower natural frequencies.

• Utilizing more restricted boundary conditions (especially at the larger radius of the conical shell) brings about higher natural frequencies.

Due to the extended usage of conical shells in aerospace structures (missiles and airplanes) and rotating machinery (centrifuges), supersonic flutter analysis of a sandwich conical shell with a honeycomb core with oriented cells and the vibration analysis of a rotating sandwich conical shell with a honeycomb core with oriented cells can be considered as two interesting topics for future works. The main goals in these works can be finding the optimum orientation for the cells in the honeycomb core to achieve the highest aero elastic stability in the flutter analysis and the highest critical rotational speeds in the free vibration analysis of a rotating sandwich conical shell.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Hossein Amirabadi

Hassan Afshari

Appendix A

(A.1)

\begin{array}{l} {y} = {\begin{cases} {U} \\ {V} \\ {W} \\ {X} \\ {Θ} \\ {S_{x}} \\ {S_{θ}} \end{cases}}, & [M] = - [\begin{array}{l} I_{0} I & [0] & [0] & [0] & [0] & [0] & [0] \\ [0] & I_{0} I & [0] & [0] & [0] & [0] & [0] \\ [0] & [0] & I_{0} I & [0] & [0] & [0] & [0] \\ [0] & [0] & [0] & I_{2} I & [0] & I_{4} I & [0] \\ [0] & [0] & [0] & [0] & I_{2} I & [0] & I_{4} I \\ [0] & [0] & [0] & I_{4} I & [0] & I_{5} I & [0] \\ [0] & [0] & [0] & [0] & I_{4} I & [0] & I_{5} I \end{array}], \end{array} [K] = [\begin{array}{l} k_{11} & k_{12} & k_{13} & [0] & [0] & [0] & [0] \\ k_{21} & k_{22} & k_{23} & [0] & k_{25} & [0] & k_{27} \\ k_{31} & k_{32} & k_{33} & k_{34} & k_{35} & k_{36} & k_{37} \\ [0] & [0] & k_{43} & k_{44} & k_{45} & k_{46} & k_{47} \\ [0] & k_{52} & k_{53} & k_{54} & k_{55} & k_{56} & k_{57} \\ [0] & [0] & k_{63} & k_{64} & k_{65} & k_{66} & k_{67} \\ [0] & k_{72} & k_{73} & k_{74} & k_{75} & k_{76} & k_{77} \end{array}],

where I and [0] show the identity and zero matrices of order N, and k₁₁-k₇₇ are defined as

\begin{array}{l} k_{11} = A_{11} [A^{(2)}] + A_{11} \sin ψ [a_{1}] [A^{(1)}] - (A_{22} \sin^{2} ψ + n^{2} A_{66}) [a_{2}], \\ k_{12} = (A_{12} + A_{66}) n [a_{1}] [A^{(1)}] - (A_{22} + A_{66}) n \sin ψ [a_{2}], \\ k_{13} = A_{12} \cos ψ [a_{1}] [A^{(1)}] - 0.5 A_{22} \sin 2 ψ [a_{2}], \\ k_{21} = - (A_{12} + A_{66}) n [a_{1}] [A^{(1)}] - (A_{22} + A_{66}) n \sin ψ [a_{2}], \\ k_{22} = A_{66} [A^{(2)}] + A_{66} \sin ψ [a_{1}] [A^{(1)}] - (n^{2} A_{22} + A_{44} \cos^{2} ψ + A_{66} \sin^{2} ψ) [a_{2}], \\ \begin{array}{l} k_{23} = - n (A_{22} + A_{44}) \cos ψ [a_{2}], & k_{25} = A_{44} \cos ψ [a_{1}], & k_{27} = Y_{44} \cos ψ [a_{1}], \end{array} \\ \begin{array}{l} k_{31} = - A_{12} \cos ψ [a_{1}] [A^{(1)}] - 0.5 A_{22} \sin 2 ψ [a_{2}], & k_{32} = - n (A_{22} + A_{44}) \cos ψ [a_{2}], \end{array} \\ k_{33} = A_{55} [A^{(2)}] + A_{55} \sin ψ [a_{1}] [A^{(1)}] - (A_{22} \cos^{2} ψ + n^{2} A_{44}) [a_{2}], \\ \begin{array}{l} k_{34} = A_{55} [A^{(1)}] + A_{55} \sin ψ [a_{1}], & k_{35} = n A_{44} [a_{1}], \end{array} \\ \begin{array}{l} k_{36} = Y_{55} [A^{(1)}] + Y_{55} \sin ψ [a_{1}], & k_{37} = n Y_{44} [a_{1}], \end{array} \\ \begin{array}{l} k_{43} = - A_{55} [A^{(1)}], & k_{44} = D_{11} [A^{(2)}] + D_{11} \sin ψ [a_{1}] [A^{(1)}] - A_{55} I - (D_{22} \sin^{2} ψ + n^{2} D_{66}) [a_{2}], \end{array} \\ k_{45} = n (D_{12} + D_{66}) [a_{1}] [A^{(1)}] - n (D_{22} + D_{66}) \sin ψ [a_{2}], \\ k_{46} = H_{11} [A^{(2)}] + H_{11} \sin ψ [a_{1}] [A^{(1)}] - Y_{55} I - (H_{22} \sin^{2} ψ + n^{2} H_{66}) [a_{2}], \\ k_{47} = n (H_{12} + H_{66}) [a_{1}] [A^{(1)}] - n (H_{22} + H_{66}) \sin ψ [a_{2}], \\ \begin{array}{l} k_{52} = A_{44} \cos ψ [a_{1}], & k_{53} = n A_{44} [a_{1}], & k_{54} = - n (D_{12} + D_{66}) [a_{1}] [A^{(1)}] - n (D_{22} + D_{66}) \sin ψ [a_{2}], \end{array} \\ k_{55} = D_{66} [A^{(2)}] + D_{66} \sin ψ [a_{1}] [A^{(1)}] - A_{44} I - (n^{2} D_{22} + D_{66} \sin^{2} ψ) [a_{2}], \\ k_{56} = - n (H_{12} + H_{66}) [a_{1}] [A^{(1)}] - n (H_{22} + H_{66}) \sin ψ [a_{2}], \end{array}

(A.2)

\begin{array}{l} k_{57} = H_{66} [A^{(2)}] + H_{66} \sin ψ [a_{1}] [A^{(1)}] - Y_{44} I - (n^{2} H_{22} + H_{66} \sin^{2} ψ) [a_{2}], \\ \begin{array}{l} k_{63} = - Y_{55} [A^{(1)}], & k_{64} = H_{11} [A^{2}] + H_{11} \sin ψ [a_{1}] [A^{(1)}] - Y_{55} I - (H_{22} \sin^{2} ψ + n^{2} H_{66}) [a_{2}], \end{array} \\ k_{65} = n (H_{12} + H_{66}) [a_{1}] [A^{(1)}] - n (H_{22} + H_{66}) \sin ψ [a_{2}], \\ k_{66} = T_{11} [A^{(2)}] + T_{11} \sin ψ [a_{1}] [A^{(1)}] - Z_{55} I - (T_{22} \sin^{2} ψ + n^{2} T_{66}) [a_{2}], \\ k_{67} = n (T_{12} + T_{66}) [a_{1}] [A^{(1)}] - n (T_{22} + T_{66}) \sin ψ [a_{2}], \\ \begin{array}{l} k_{72} = Y_{44} \cos ψ [a_{1}], & k_{73} = n Y_{44} [a_{1}], \end{array} \\ k_{74} = - n (H_{12} + H_{66}) [a_{1}] [A^{(1)}] - n (H_{22} + H_{66}) \sin ψ [a_{2}], \\ k_{75} = H_{66} [A^{(2)}] + H_{66} \sin ψ [a_{1}] [A^{(1)}] - Y_{44} I - (n^{2} H_{22} + H_{66} \sin^{2} ψ) [a_{2}], \\ k_{76} = - n (T_{12} + T_{66}) [a_{1}] [A^{(1)}] - n (T_{22} + T_{66}) \sin ψ [a_{2}], \\ k_{77} = T_{66} [A^{(2)}] + T_{66} \sin ψ [a_{1}] [A^{(1)}] - Z_{44} I - (n^{2} T_{22} + T_{66} \sin^{2} ψ) [a_{2}], \end{array}

in which (A.3)

\begin{array}{l} [a_{p}] = [\begin{array}{l} r^{- p} (x_{1}) & 0 & \dots & 0 \\ 0 & r^{- p} (x_{2}) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & r^{- p} (x_{N}) \end{array}], & p = 1, 2 . \end{array}

Appendix B

(B.1)

[R] = [\begin{array}{l} R_{11} & R_{12} & \dots & R_{17} \\ R_{21} & R_{22} & \dots & R_{27} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ R_{141} & R_{142} & \dots & R_{147} \end{array}],

in which R₁₁-R₇₇ describe the condition at x = 0 and are defined in Eqs. (B.2)-(B-4):

Clamped: (B.2)

R_{i j} = {\begin{array}{c} I_{1} & i = j \\ {0}_{1 \times N} & i \neq j \end{array}

Simply supported: (B.3)

R_{i j} = {\begin{array}{c} I_{1} & i = j = 2, 3, 5, 7 \\ A_{11} A_{1}^{(1)} + A_{12} a^{- 1} \sin ψ I_{1} & i = j = 1 \\ D_{11} A_{1}^{(1)} + D_{12} a^{- 1} \sin ψ I_{1} & i = j = 4 \\ H_{11} A_{1}^{(1)} + H_{12} a^{- 1} \sin ψ I_{1} & i j = 46, 64 \\ T_{11} A_{1}^{(1)} + T_{12} a^{- 1} \sin ψ I_{1} & i = j = 6 \\ {0}_{1 \times N} & else \end{array}

Free: (B.4)

\begin{array}{l} \begin{array}{l} R_{11} = A_{11} A_{1}^{(1)} + A_{12} a^{- 1} \sin ψ I_{1}, & R_{12} = n A_{12} a^{- 1} I_{1}, & R_{13} = A_{12} a^{- 1} \cos ψ I_{1}, & R_{21} = - n A_{66} a^{- 1} I_{1}, \end{array} \\ \begin{array}{l} R_{22} = A_{66} A_{1}^{(1)} - A_{66} a^{- 1} \sin ψ I_{1}, & R_{33} = A_{55} A_{1}^{(1)}, & R_{34} = A_{55} I_{1}, & R_{36} = Y_{55} I_{1}, \end{array} \\ \begin{array}{l} R_{44} = D_{11} A_{1}^{(1)} + D_{12} a^{- 1} \sin ψ I_{1}, & R_{45} = n D_{12} a^{- 1} I_{1}, & R_{46} = H_{11} A_{1}^{(1)} + H_{12} a^{- 1} \sin ψ I_{1}, & R_{47} = n H_{12} a^{- 1} I_{1}, \end{array} \\ \begin{array}{l} R_{54} = - n D_{66} a^{- 1} I_{1}, & R_{55} = D_{66} A_{1}^{(1)} - D_{66} a^{- 1} \sin ψ I_{1}, & R_{56} = - n H_{66} a^{- 1} I_{1}, & R_{57} = H_{66} A_{1}^{(1)} - H_{66} a^{- 1} \sin ψ I_{1}, \end{array} \\ \begin{array}{l} R_{64} = R_{46}, & R_{65} = R_{47}, & R_{66} = T_{11} A_{1}^{(1)} + T_{12} a^{- 1} \sin ψ I_{1}, & R_{67} = n T_{12} a^{- 1} I_{1}, \end{array} \\ \begin{array}{l} R_{74} = R_{56}, & R_{75} = R_{57}, & R_{76} = - n T_{66} a^{- 1} I_{1}, & R_{77} = T_{66} A_{1}^{(1)} - T_{66} a^{- 1} \sin ψ I_{1}, \end{array} \\ R_{14} = R_{15} = R_{16} = R_{17} = R_{23} = R_{24} = R_{25} = R_{26} = R_{27} = R_{31} = R_{32} = R_{35} = R_{37} = \\ R_{41} = R_{42} = R_{43} = R_{51} = R_{52} = R_{53} = R_{61} = R_{62} = R_{63} = R_{71} = R_{72} = R_{73} = {0}_{1 \times N} . \end{array}

where the subscript 1 implies the first row of the matrices.

Also, R₈₁-R₁₄₇ describe the condition at x = L and are defined in Eqs. (B.5)-(B-7):

Clamped: (B.5)

R_{(i + 7) j} = {\begin{array}{c} I_{N} & i = j \\ {0}_{1 \times N} & i \neq j \end{array}

Simply supported: (B.6)

R_{(i + 7) j} = {\begin{array}{c} I_{N} & i = j = 2, 3, 5, 7 \\ A_{11} A_{N}^{(1)} + A_{12} b^{- 1} \sin ψ I_{N} & i = j = 1 \\ D_{11} A_{N}^{(1)} + D_{12} b^{- 1} \sin ψ I_{N} & i = j = 4 \\ H_{11} A_{N}^{(1)} + H_{12} b^{- 1} \sin ψ I_{N} & i j = 46, 64 \\ T_{11} A_{N}^{(1)} + T_{12} b^{- 1} \sin ψ I_{N} & i = j = 6 \\ {0}_{1 \times N} & else \end{array}

Free: (B.7)

\begin{array}{l} \begin{array}{l} R_{81} = A_{11} A_{N}^{(1)} + A_{12} b^{- 1} \sin ψ I_{N}, & R_{82} = n A_{12} b^{- 1} I_{N}, & R_{83} = A_{12} b^{- 1} \cos ψ I_{N}, & R_{91} = - n A_{66} b^{- 1} I_{N}, \end{array} \\ \begin{array}{l} R_{92} = A_{66} A_{N}^{(1)} - A_{66} b^{- 1} \sin ψ I_{N}, & R_{103} = A_{55} A_{N}^{(1)}, & R_{104} = A_{55} I_{N}, & R_{106} = Y_{55} I_{N}, \end{array} \\ \begin{array}{l} R_{114} = D_{11} A_{N}^{(1)} + D_{12} b^{- 1} \sin ψ I_{N}, & R_{115} = n D_{12} b^{- 1} I_{N}, & R_{116} = H_{11} A_{N}^{(1)} + H_{12} b^{- 1} \sin ψ I_{N}, & R_{117} = n H_{12} b^{- 1} I_{N}, \end{array} \\ \begin{array}{l} R_{124} = - n D_{66} b^{- 1} I_{N}, & R_{125} = D_{66} A_{N}^{(1)} - D_{66} b^{- 1} \sin ψ I_{N}, & R_{126} = - n H_{66} b^{- 1} I_{N}, \end{array} \\ \begin{array}{l} R_{127} = H_{66} A_{N}^{(1)} - H_{66} b^{- 1} \sin ψ I_{N}, & R_{134} = R_{116}, & R_{135} = R_{117}, & R_{136} = T_{11} A_{N}^{(1)} + T_{12} b^{- 1} \sin ψ I_{N}, \end{array} \\ \begin{array}{l} R_{137} = n T_{12} b^{- 1} I_{N}, & R_{144} = R_{126}, & R_{145} = R_{127}, & R_{146} = - n T_{66} b^{- 1} I_{N}, & R_{147} = T_{66} A_{N}^{(1)} - T_{66} b^{- 1} \sin ψ I_{N}, \end{array} \\ R_{84} = R_{85} = R_{86} = R_{87} = R_{93} = R_{94} = R_{95} = R_{96} = R_{97} = R_{101} = R_{102} = R_{105} = R_{107} = \\ R_{111} = R_{112} = R_{113} = R_{121} = R_{122} = R_{123} = R_{131} = R_{132} = R_{133} = R_{141} = R_{142} = R_{143} = {0}_{1 \times N} . \end{array}

where the subscript N implies the Nth (last) row of the matrices.

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