Free Vibration of Graphene Nanoplatelet-Reinforced Porous Double-Curved Shells of Revolution with a General Radius of Curvature Based on a Semi-Analytical Method

Abstract

Based on domain decomposition, a semi-analytical method (SAM) is applied to analyze the free vibration of double-curved shells of revolution with a general curvature radius made from graphene nanoplatelet (GPL)-reinforced porous composites. The mechanical properties of the GPL-reinforced composition are assessed with the Halpin–Tsai model. The double-curvature shell of revolution is broken down into segments along its axis in accordance with first-order shear deformation theory (FSDT) and the multi-segment partitioning technique, to derive the shell’s functional energy. At the same time, interfacial potential is used to ensure the continuity of the contact surface between neighboring segments. By applying the least-squares weighted residual method (LWRM) and modified variational principle (MVP) to relax and achieve interface compatibility conditions, a theoretical framework for analyzing vibrations is developed. The displacements and rotations are described through Fourier series and Chebyshev polynomials, accordingly, converting a two-dimensional issue into a suite of decoupled one-dimensional problems. The obtained solutions are contrasted with those achieved using the finite element method (FEM) and other existing results, and the current formulation’s validity and precision are confirmed. Example cases are presented to demonstrate the free vibration of GPL-reinforced porous composite double-curved paraboloidal, elliptical, and hyperbolical shells of revolution. The findings demonstrate that the natural frequency of the shell is related to pore coefficients, porosity, the mass fraction of the GPLs, and the distribution patterns of the GPLs.

1. Introduction

A double-curved shell of revolution structure is a non-traditional geometric design extensively utilized in the aerospace and other industries because of its superior strength, load-carrying ability, and efficiency [1,2,3]. These structures often operate under challenging environmental conditions, leading to significant vibration and fatigue damage [4,5,6]. Therefore, understanding the vibration properties of these structures is crucial for preventing destructive resonance effects in practical engineering applications.

Currently, methods to optimize the structural mechanical characteristics of new composite materials have become a hot issue for scholars in China and internationally. For example, a GPL-reinforced porous composite is known as a new advanced lightweight material with different GPL configurations along the thickness direction and distributions of different porosities. Research shows that GPL-reinforced composites can greatly improve the mechanical properties of structures [7,8,9]. In terms of free vibrations, Kitipornchai [10] and Xu [11] investigated the effects of GPL distribution pattern, weight fraction, and geometry on the free vibration of beams. Anirudh [12] examined the effects of a curved beam’s curvature radius on the features of free vibration. Hung [13] analyzed the dynamic characteristics of pore coefficient and boundary support on the free vibration of sandwich beams. Yang [14] performed elaborate parametric research on the influence of GPL weight fraction, geometric parameters, and pore coefficient to determine the optimal approach to increasing the antivibration performance of plates. Reddy [15] explored the free vibration behavior of variable-thickness functionally graded plates. Pourjabari [16] explored the influence of porosity on the free vibration characteristics of nanoshells. The literature above demonstrates the extensive research on free vibration in GPL-reinforced porous beams, plates, and shells.

However, in certain specialized engineering fields, a double-curved shell plays a pivotal role due to its unique geometric shape and mechanical properties. In recent decades, many scholars worldwide have carried out numerous studies on double-curved shells with a fixed radius of curvature based on various methods. Chakravorty and Leissa [17,18] used the eight-noded curved quadrilateral isoparametric FEM approach to examine the free vibration of a double-curved parabolic shell. Kang [19] proposed an analysis in three dimensions to examine the free vibration of parabolic shells of non-revolution with completely variable wall thickness. Using C0 finite element formulations, Thakur [20] explored the response of hyperbolic paraboloidal shells. Using FEM, Patel [21] analyzed the free vibration of functionally graded elliptic cylindrical shells by approximating the thickness of in-plane displacement and transverse displacement. Kim [22] proposed a method involving Haar wavelet discretization to analyze the free vibration of a composite elliptic–cylindric–elliptical shell. Tornabene [23] presented a generalized differential quadrature method (DQM) for analyzing the free vibrations of cylinders with oval and elliptic cross-sections. Irie [24] employed a transfer matrix method approach to study the free vibration of a tapered conical shell featuring non-uniform thickness. The effect of different boundary conditions on the properties of a conical shell of revolution was investigated by Lam [25] using the Galerkin method. Safarpour [26] applied DQM and elasticity theory to analyze the free vibration of a GPL-reinforced truncated conical shell.

On reviewing the existing research reports, we observed that most methods or techniques are only applied to specific types of shell with a fixed radius of curvature, such as paraboloidal shells, elliptical shells, and hyperbolical shells. However, there are limited computational approaches available for dealing with hyperbolic shells of revolution that have a variable radius of curvature. Choe [27,28], Guo [29], and Zhao [30] employed a unified Jacobi–Ritz method for laminating an axisymmetric double-bending shell of revolution made of composite materials with different radii of curvature. A new SAM was proposed by Qu [31,32,33,34] for the static and free vibration analysis of GPL-reinforced porous double-curved shells of revolution with a general radius of curvature. The functional energy of the shell was obtained using a multi-segment partitioning technique. An interfacial potential was employed to maintain continuity across the contact surfaces of adjacent segments. A theoretical framework for vibration analysis was established by employing MVP and LWRM to ensure interface compatibility. Rotations and displacements were described through Chebyshev polynomials and Fourier series, thereby transforming a two-dimensional issue into a suite of decoupled one-dimensional problems.

Based on the above SAM, the free vibration of GPL-reinforced porous double-curved shells of revolution with a general radius of curvature under different boundary conditions are given in accordance with FSDT. The properties of GPL-reinforced composite are evaluated by the Halpin–Tsai model. To determine the applicability and accuracy of SAM, the results gained from FEM and other results are compared with present solutions for double-curved shells of revolution. A suite of numerical instances is used to study the dynamic analysis of structures with material parameters. Distinct porosity distributions and GPL distribution patterns are also taken into account. The effect of certain crucial parameters on the free vibration properties of GPL-reinforced porous composite double-curved shells of revolution is indicated.

2. Theoretical Formulations

Figure 1a demonstrates the coordinate system and shape of GPL-reinforced porous composite double-curved shells of revolution. This system, comprising coordinates α, β, and z, is situated on the reference plane (z = 0) of the shell. R_α and R_β stand for the main curvature radius values of the reference plane on the α and β axes, correspondingly.

In engineering applications, the double-curved shells of revolution with different geometric shapes are in accordance with different curvature features, such as double-curved paraboloidal, hyperbolical, and elliptical shells of revolution. However, the shells are simplified with parabolic, elliptic, and hyperbolic meridian curves to better present their features.

2.1. Explanation of Double-Curved Shells of Revolution

The relevant geometric characteristic expressions for the aforementioned meridian curves are illustrated:

$$R_{\alpha }\left(\alpha \right)={\displaystyle \frac{k}{2{\cos }^3\alpha }},R_{\beta }\left(\alpha \right)={\displaystyle \frac{k}{2\cos \alpha }},$$

(1)

for the parabolic meridian of the shell shown in Figure 2, k stands for characteristic parameter.

$$R_{\alpha }\left(\alpha \right)={\displaystyle \frac{a^2{b}^2}{\sqrt{{\left(a^2{\sin }^2\alpha +b^2{\cos }^2\alpha \right)}^3}}},R_{\beta }\left(\alpha \right)={\displaystyle \frac{a^2}{\sqrt{a^2{\sin }^2\alpha +b^2{\cos }^2\alpha }}},$$

(2)

in Figure 3, a and b, respectively, indicate the length of semi-major and semi-minor axes of the elliptic meridian.

$$R_{\alpha }\left(\alpha \right)={\displaystyle \frac{-a^2{b}^2}{\sqrt{{\left(a^2{\sin }^2\alpha -b^2{\cos }^2\alpha \right)}^3}}},R_{\beta }\left(\alpha \right)={\displaystyle \frac{a^2}{\sqrt{a^2{\sin }^2\alpha -b^2{\cos }^2\alpha }}}+{\displaystyle \frac{R_s}{\sin \alpha }},$$

(3)

for the hyperbolic meridian of the shell shown in Figure 4, a and b, respectively, represent the length of the semi-transverse axes and the semi-conjugate axes.

2.2. Preliminaries

Based on the FSDT, the displacement $\left(\overline{u},\overline{v},\overline{w}\right)$ of a specified point within a double-curved shell of revolution is represented as follows according to the displacements and rotations of a reference plane:

$$\begin{array}{l}\overline{u}\left(\alpha ,\beta ,z,t\right)=u\left(\alpha ,\beta ,t\right)+z{\psi }_{\alpha }\left(\alpha ,\beta ,t\right),\\ \overline{v}\left(\alpha ,\beta ,z,t\right)=v\left(\alpha ,\beta ,t\right)+z{\psi }_{\beta }\left(\alpha ,\beta ,t\right),\\ \overline{w}\left(\alpha ,\beta ,z,t\right)=w\left(\alpha ,\beta ,t\right),\end{array}$$

(4)

where u, v, and w show the displacements of a datum plane in α, β, and z orientations, individually; ψ_α and ψ_β denote the transverse rotations of a datum plane about the α and β axes, individually. t is the time variable.

The following represents the linear strain–displacement relationships within shells:

$$\begin{array}{l}\left\{\begin{array}{c}{\epsilon }_{\alpha }\\ {\epsilon }_{\beta }\\ {\epsilon }_{\alpha \beta }\end{array}\right\}=\left\{\begin{array}{c}{\epsilon }_{\alpha }^0\\ {\epsilon }_{\beta }^0\\ {\epsilon }_{\alpha \beta }^0\end{array}\right\}+z\left\{\begin{array}{c}{\chi }_{\alpha }\\ {\chi }_{\beta }\\ {\chi }_{\alpha \beta }\end{array}\right\},\\ \left\{\begin{array}{c}{\epsilon }_{\alpha z}\\ {\epsilon }_{\beta z}\end{array}\right\}=\left\{\begin{array}{c}{\psi }_{\alpha }-{\displaystyle \frac{u}{R_{\alpha }}}+{\displaystyle \frac{1}{A}}{\displaystyle \frac{\partial w}{\partial \alpha }}\\ {\psi }_{\beta }-{\displaystyle \frac{v}{R_{\beta }}}+{\displaystyle \frac{1}{B}}{\displaystyle \frac{\partial w}{\partial \beta }}\end{array}\right\},\end{array}$$

(5)

where the membrane strains of the reference plane are represented by ${\epsilon }_{\alpha }^0$, ${\epsilon }_{\beta }^0$, and ${\epsilon }_{\alpha \beta }^0$; ${\epsilon }_{\alpha z}$ and ${\epsilon }_{\beta z}$ indicate the transverse shear strain; χ_α, χ_β, and χ_αβ denote the curvature changes in the shell. Lamé parameters are labeled as quantities A and B.

If h/R_α and h/R_β (h is the uniform thickness of a double-curved shell of revolution) are almost negligible as compared with unity, the curvatures and membrane strains of the reference plane are represented as:

$$\begin{array}{l}\left\{\begin{array}{c}{\epsilon }_{\alpha }^0\\ {\epsilon }_{\beta }^0\\ {\epsilon }_{\alpha \beta }^0\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \frac{1}{A}}{\displaystyle \frac{\partial u}{\partial \alpha }}+{\displaystyle \frac{v}{AB}}{\displaystyle \frac{\partial A}{\partial \beta }}+{\displaystyle \frac{w}{R_{\alpha }}}\\ {\displaystyle \frac{1}{B}}{\displaystyle \frac{\partial v}{\partial \beta }}+{\displaystyle \frac{u}{AB}}{\displaystyle \frac{\partial B}{\partial \alpha }}+{\displaystyle \frac{w}{R_{\beta }}}\\ {\displaystyle \frac{B}{A}}{\displaystyle \frac{\partial }{\partial \alpha }}\left({\displaystyle \frac{v}{B}}\right)+{\displaystyle \frac{A}{B}}{\displaystyle \frac{\partial }{\partial \beta }}\left({\displaystyle \frac{u}{A}}\right)\end{array}\right\},\\ \left\{\begin{array}{c}{\chi }_{\alpha }\\ {\chi }_{\beta }\\ {\chi }_{\alpha \beta }\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \frac{1}{A}}{\displaystyle \frac{\partial {\psi }_{\alpha }}{\partial \alpha }}+{\displaystyle \frac{{\psi }_{\beta }}{AB}}{\displaystyle \frac{\partial A}{\partial \beta }}\\ {\displaystyle \frac{1}{B}}{\displaystyle \frac{\partial {\psi }_{\beta }}{\partial \beta }}+{\displaystyle \frac{{\psi }_{\alpha }}{AB}}{\displaystyle \frac{\partial B}{\partial \alpha }}\\ {\displaystyle \frac{B}{A}}{\displaystyle \frac{\partial }{\partial \alpha }}\left({\displaystyle \frac{{\psi }_{\beta }}{B}}\right)+{\displaystyle \frac{A}{B}}{\displaystyle \frac{\partial }{\partial \beta }}\left({\displaystyle \frac{{\psi }_{\alpha }}{A}}\right)\end{array}\right\},\end{array}$$

(6)

The constitutive equations are given in matrix form, which connect force and moment resultants with the strains and curvatures of the datum plane:

$$\left[\begin{array}{c}{N}_{\alpha }\\ N_{\beta }\\ N_{\alpha \beta }\\ M_{\alpha }\\ M_{\beta }\\ M_{\alpha \beta }\end{array}\right]=\left[\begin{array}{cccccc}{A}_{11}& A_{12}& 0& B_{11}& B_{12}& 0\\ A_{12}& A_{11}& 0& B_{12}& B_{11}& 0\\ 0& 0& A_{66}& 0& 0& B_{66}\\ B_{11}& B_{12}& 0& D_{11}& D_{12}& 0\\ B_{12}& A_{11}& 0& D_{12}& D_{11}& 0\\ 0& 0& B_{66}& 0& 0& D_{66}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\alpha }^0\\ {\epsilon }_{\beta }^0\\ {\epsilon }_{\alpha \beta }^0\\ {\chi }_{\alpha }\\ {\chi }_{\beta }\\ {\chi }_{\alpha \beta }\end{array}\right],$$

(7)

$$\left[\begin{array}{c}{Q}_{\alpha }\\ Q_{\beta }\end{array}\right]=k_s\left[\begin{array}{cc}{A}_{66}& 0\\ 0& A_{66}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\alpha z}\\ {\epsilon }_{\beta z}\end{array}\right],$$

(8)

where N_α, N_β, and N_αβ represent the in-plane force resultants per unit length; M_α, M_β, and M_αβ indicate the bending and twisting moment resultants; Q_α and Q_β are the transverse shear force resultants. The parameter k_s is the shear correction factor, taken as 5/6 [35,36]. A_ij is the extensional stiffness; B_ij is the extensional-bending coupling stiffness; D_ij is the bending stiffness. They can be shown as:

$$\left[\begin{array}{c}{A}_{ij}\\ B_{ij}\\ D_{ij}\end{array}\right]=\begin{array}{cc}{\displaystyle {\int }_{-h/2}^{h/2}{Q}_{ij}\left[\begin{array}{c}1\\ z\\ z^2\end{array}\right]dz}& i,j=1,2,6\end{array}.$$

(9)

The integrals used in this manuscript are in the sense of Riemann. where the elastic coefficients Q_ij are functions of z, expressed as:

$$\left\{Q_{11},Q_{12},Q_{66}\right\}=\left\{{\displaystyle \frac{E\left(z\right)}{1-{\mu }^2\left(z\right)}},{\displaystyle \frac{\mu \left(z\right)E\left(z\right)}{1-{\mu }^2\left(z\right)}},{\displaystyle \frac{E\left(z\right)}{2\left[1+\mu \left(z\right)\right]}}\right\}.$$

(10)

2.3. Energy Functional of Each Segment

A double-curved shell of revolution is evenly split into the N shell segments along the meridian direction. Then, the vibration of the shell is interpreted by MVP, which involves finding the minimum of the variational functional:

$$\Pi ={\displaystyle {\int }_{t_0}^{t_1}{\displaystyle \sum _{i=1}^N\left(T_i-U_i\right)dt}}+{\displaystyle {\int }_{t_0}^{t_1}{\displaystyle \sum _{i,i+1}{\Pi }_B}dt},$$

(11)

where U_i and T_i, respectively, represent the elastic strain energy and kinetic energy of the i-th shell segment, using t₀ and t₁ as the specified times. Π_B indicates the interface potential on a shared boundary between shell segments i and i + 1. The maximum kinetic energy of the i-th shell segment based on FSDT is expressed as:

$$\begin{array}{l}{T}_i={\displaystyle \frac{1}{2}}\left\{{\displaystyle {\iint }_{S_i}{\overline{\rho }}_0\left[{\left(\frac{\partial u_i}{\partial t}\right)}^2+{\left(\frac{\partial v_i}{\partial t}\right)}^2+{\left(\frac{\partial w_i}{\partial t}\right)}^2\right]+2{\overline{\rho }}_1\left(\frac{\partial u_i}{\partial t}\frac{\partial {\psi }_{\alpha ,i}}{\partial t}+\frac{\partial v_i}{\partial t}\frac{\partial {\psi }_{\beta ,i}}{\partial t}\right)}\right.\\ \left.+{\overline{\rho }}_2\left[{\left({\displaystyle \frac{\partial {\psi }_{\alpha ,i}}{\partial t}}\right)}^2+{\left({\displaystyle \frac{\partial {\psi }_{\beta ,i}}{\partial t}}\right)}^2\right]\right\}ABd\alpha d\beta ,\end{array}$$

(12)

where S_i is the datum plane area of the i-th shell segment. ${\overline{\rho }}_0$, ${\overline{\rho }}_1$, and ${\overline{\rho }}_2$ are inertia terms, expressed as:

$${\overline{\rho }}_0={\displaystyle \sum _{k=0}^N{\displaystyle {\int }_{z_k}^{z_{k+1}}\rho \left(z\right)}dz,}{\overline{\rho }}_1={\displaystyle \sum _{k=0}^N{\displaystyle {\int }_{z_k}^{z_{k+1}}z\rho \left(z\right)}dz},{\overline{\rho }}_2={\displaystyle \sum _{k=0}^N{\displaystyle {\int }_{z_k}^{z_{k+1}}{z}^2\rho \left(z\right)}dz},$$

(13)

where $z_k=-{\displaystyle \frac{h}{2}}+{\displaystyle \frac{k\cdot h}{N}}$ and $k=0,1,\cdots N$.

For the i-th shell segment, the strain energy equation is provided as follows:

$$\begin{array}{l}{U}_i={\displaystyle \frac{1}{2}}{\displaystyle {\iint }_{S_i}\left(N_{\alpha }{\epsilon }_{\alpha }^0+N_{\beta }{\epsilon }_{\beta }^0+N_{\alpha \beta }{\epsilon }_{\alpha \beta }^0+M_{\alpha }{\chi }_{\alpha }+M_{\beta }{\chi }_{\beta }\right.}\\ \left.+M_{\alpha \beta }{\chi }_{\alpha \beta }+Q_{\alpha }{\epsilon }_{\alpha z}+Q_{\beta }{\epsilon }_{\beta z}\right)ABd\alpha d\beta ,\end{array}$$

(14)

The fundamental interface continuity limitations are achieved by means of MVP in combination with LSWRM [33]. The interface potential Π_B is expressed as:

$$\begin{array}{l}{\Pi }_B={\displaystyle {\int }_{l_i}\left({\varsigma }_u{N}_{\alpha }{\Theta }_u+{\varsigma }_v{N}_{\alpha \beta }{\Theta }_v+{\varsigma }_w{Q}_{\alpha }{\Theta }_w+{\varsigma }_{\eta }{M}_{\alpha }{\Theta }_{\alpha }+{\varsigma }_{\vartheta }{M}_{\alpha \beta }{\Theta }_{\beta }\right)}dl\\ -{\displaystyle \frac{1}{2}}{\displaystyle {\int }_{l_i}\left({\varsigma }_u{\kappa }_u{\Theta }_u^2+{\varsigma }_v{\kappa }_v{\Theta }_v^2+{\varsigma }_w{\kappa }_w{\Theta }_w^2+{\varsigma }_{\eta }{\kappa }_{\eta }{\Theta }_{\alpha }^2+{\varsigma }_{\vartheta }{\kappa }_{\vartheta }{\Theta }_{\beta }^2\right)dl},\end{array}$$

(15)

where the first term integral of Equation (15) is introduced by using MVP to relax the constraint of interface continuity, and second term integral of Equation (15) is introduced by the LSWRM to ensure the convergence and stability of the calculation and deal with the non-classical boundary of the shell. Θ_u, Θ_v, Θ_w, Θ_α, and Θ_β are the basic continuity equations between shared interfaces, represented as: Θ_u = u_i − u_i+1, Θ_v = v_i − v_i+1, Θ_w = w_i − w_i+1, Θ_α = ψ_α,i − ψ_α,i+1, and Θ_β = ψ_β,i − ψ_β,i+1. κ_σ(σ = u,v,w,η,ϑ) are preassigned the weighted parameters. ς_σ(σ = u,v,w,η,ϑ) are the parameters defining distinct interface continuity conditions, as shown in

Table 1. Values of ς_σ(σ = u,v,w,η,ϑ) for distinct boundary conditions.

Boundary Set	Conditions	${\mathit{\varsigma }}_{\mathit{u}}$	${\mathit{\varsigma }}_{\mathit{v}}$	${\mathit{\varsigma }}_{\mathit{w}}$	${\mathit{\varsigma }}_{\mathit{\eta }}$	${\mathit{\varsigma }}_{\mathit{\vartheta }}$
Free (F)	\	0	0	0	0	0
Simply supported (S)	$v=w=0$	0	1	1	0	0
Clamped (C)	$u=v=w={\psi }_{\alpha }={\psi }_{\beta }=0$	1	1	1	1	1

.

2.4. Motion Equations and Solution Methodology

Considering the symmetry of the double-curved shells of revolution and setting β = θ to be the circumferential coordinate, the rotations and displacements of the segment are represented using the Chebyshev polynomial and the Fourier series. Conforming to the harmonics of the Fourier series, subscript i is neglected here, and the expression can be set out in forms [34]:

$$\begin{array}{l}u\left(\xi ,\theta ,t\right)={\displaystyle \sum _{q=0}^Q{\displaystyle \sum _{n=0}^N{T}_q\left(\xi \right)\left[\cos \left(n\theta \right){\tilde{u}}_{qn}\left(t\right)+\sin \left(n\theta \right){\overline{u}}_{qn}\left(t\right)\right]=U\left(\xi ,\theta \right)u\left(t\right)}},\\ v\left(\xi ,\theta ,t\right)={\displaystyle \sum _{q=0}^Q{\displaystyle \sum _{n=0}^N{T}_q\left(\xi \right)\left[\cos \left(n\theta \right){\tilde{v}}_{qn}\left(t\right)+\sin \left(n\theta \right){\overline{v}}_{qn}\left(t\right)\right]=V\left(\xi ,\theta \right)v\left(t\right)}},\\ w\left(\xi ,\theta ,t\right)={\displaystyle \sum _{q=0}^Q{\displaystyle \sum _{n=0}^N{T}_q\left(\xi \right)\left[\cos \left(n\theta \right){\tilde{w}}_{qn}\left(t\right)+\sin \left(n\theta \right){\overline{w}}_{qn}\left(t\right)\right]=W\left(\xi ,\theta \right)w\left(t\right)}},\\ {\psi }_{\alpha }\left(\xi ,\theta ,t\right)={\displaystyle \sum _{q=0}^Q{\displaystyle \sum _{n=0}^N{T}_q\left(\xi \right)\left[\cos \left(n\theta \right){\tilde{\psi }}_{\alpha ,qn}\left(t\right)+\sin \left(n\theta \right){\overline{\psi }}_{\alpha ,qn}\left(t\right)\right]={\mathsf{\Psi }}_{\alpha }\left(\xi ,\theta \right){\psi }_{\alpha }\left(t\right)}},\\ {\psi }_{\beta }\left(\xi ,\theta ,t\right)={\displaystyle \sum _{q=0}^Q{\displaystyle \sum _{n=0}^N{T}_q\left(\xi \right)\left[\cos \left(n\theta \right){\tilde{\psi }}_{\alpha ,qn}\left(t\right)+\sin \left(n\theta \right){\overline{\psi }}_{\beta ,qn}\left(t\right)\right]={\mathsf{\Psi }}_{\beta }\left(\xi ,\theta \right){\psi }_{\beta }\left(t\right),}}\end{array}$$

(16)

where T_q(ξ) is the q-order Chebyshev polynomial, n is the positive integer and circumferential wave number. [α_i, α_i+1] changes linearly to ξ $\in$ [−1, 1] and N and Q represent the maximum degrees within the series/polynomial. ${\tilde{u}}_{qn}$, ${\tilde{v}}_{qn}$, ${\tilde{w}}_{qn}$, ${\tilde{\psi }}_{\alpha ,qn}$, ${\tilde{\psi }}_{\beta ,qn}$, ${\overline{u}}_{qn}$, ${\overline{v}}_{qn}$, ${\overline{w}}_{qn}$, ${\overline{\psi }}_{\alpha ,qn}$, and ${\overline{\psi }}_{\beta ,qn}$ denote the generalized coordinate variables. U(ξ,θ), V(ξ,θ), W(ξ,θ), Ψ_α(ξ,θ), and Ψ_β(ξ,θ) are the admissible function vectors. u, v, w, ψ_α, and ψ_β are the coordinate variables. For antisymmetric displacement expansion, the sin and cos symbols of Fourier series need to be exchanged.

By replacing Equations (13) and (15)–(17) into Equation (12), the free vibration equation of the shell of revolution is gained as:

$$M\ddot{\mathit{q}}+\left[K-K_{\lambda }+K_{\kappa }\right]\mathit{q}=0.$$

(17)

assuming q = $\overline{q}$sin(ωt + φ), where $\overline{q}$ is the generalized coordinate amplitude. By substituting it into Equation (17), the following form is expressed as:

$$-{\omega }^{2}M\overline{\mathit{q}}+\left[K-K_{\lambda }+K_{\kappa }\right]\overline{\mathit{q}}=0.$$

(18)

If the above formula has a non-zero solution, the determinant |−ω²M + K − K_λ + K_κ| = 0 is obtained. Its eigenvalue is matched with the natural frequency of the shell of revolution when free vibration occurs.

3. Numerical Results and Discussion

In this paper, a comparative study of a double-curved parabolic shell of revolution with a C-C boundary is accomplished to prove the effectiveness of the above method. Its geometric parameters and material properties are shown in Appendix A. A comparative study of the dimensionless frequencies of this structure based on a different order of region decomposition and axial wave number is shown in

Table 2. The dimensionless natural frequencies of a double-curved parabolic shell of revolution with a C-C boundary.

N	Q	Present	Pang [37]	FEM
1	1	0.7059	0.6682	0.6651
	2	0.9922	1.0382	1.0132
	3	0.9681	1.2655	1.1721
2	1	0.4976	0.4604	0.4562
	2	0.7506	0.8183	0.8058
	3	1.0628	1.1102	1.0445
3	1	0.3326	0.3863	0.3822
	2	0.6712	0.6880	0.6745
	3	0.9736	1.0034	0.9427

. As N and Q change, it is confirmed that the dimensionless natural frequency of the parabolic shell is approximately in agreement with the findings obtained by Pang [37] and FEM.

In the following investigation, the free vibration behaviors of three kinds of GPL-reinforced porous composite double-curved shells of revolution are examined and analyzed: the double-curved parabolic shell, the double-curved elliptic shell, and the double-curved hyperbolic shell. The effects of different GPL distribution patterns, mass fractions, pore coefficients, and pore distributions on the mechanical behavior of the shells are analyzed and discussed. Unless specified otherwise, the geometrical parameters and material properties are referenced in Appendix A.

The influences of different pore coefficients on the dimensionless natural frequency of three kinds of double-curved shells of revolution with two GPL distribution patterns are tabulated in

Table 3. The dimensionless natural frequencies with different pore coefficients (Porosity-II S-S boundary conditions).

Pore Coefficients	Paraboloidal Shell		Elliptical Shell		Hyperbolical Shell
e0	O	X	O	X	O	X
0.1	2.03569	4.27235	9.00359	18.74518	1.88184	3.95104
0.2	1.98195	4.15957	8.76872	18.27182	1.82875	3.83957
0.3	1.92347	4.03682	8.51282	17.75373	1.76827	3.71259
0.4	1.85912	3.90176	8.23100	17.18060	1.69853	3.56617
0.5	1.78733	3.75110	7.91633	16.53779	1.61692	3.39483
0.6	1.70574	3.57985	7.55840	15.80341	1.51962	3.19054
0.7	1.61054	3.38005	7.14057	14.94238	1.40068	2.94083

. As the pore coefficient increases, the dimensionless natural frequency of the three kinds of GPL-reinforced porous composite double-curved shells of revolution decreases. This is due to the enlargement of the pores, which in turn reduces the shell stiffness.

Figure 5 and

Table 4. The dimensionless natural frequencies with different mass fractions (Porosity-II S-S boundary conditions).

Mass Fraction	Paraboloidal Shell		Elliptical Shell		Hyperbolical Shell
WG	O	X	O	X	O	X
0	1.41053	1.41053	6.25059	6.25059	1.27599	1.27599
0.002	1.49349	2.09767	6.61758	9.28613	1.35105	1.89775
0.004	1.57210	2.61029	6.96515	11.54369	1.42217	2.36174
0.006	1.64696	3.03806	7.29610	13.42173	1.48991	2.74903
0.008	1.71857	3.41304	7.61258	15.06288	1.55471	3.08861
0.01	1.78733	3.75110	7.91633	16.53779	1.61692	3.39483
0.012	1.85354	4.06145	8.20876	17.88762	1.67684	3.67603

show the dimensionless natural frequency of three double-curved shells of revolution with two GPL distribution patterns and different GPL mass fractions. Obviously, the dimensionless natural frequency of all shells increases along with the increase in GPL mass fraction due to the increasing stiffness. Combining this with Table 4, it is observed that the elliptical shell consistently exhibits the highest dimensionless natural frequency, followed by the parabolic shell, and then the hyperbolic shell, irrespective of the GPL distribution pattern or pore coefficient. No matter how GPL mass fraction or porosity changes, the dimensionless natural frequency and its change rate for the X-pattern double-curved shell of revolution are always higher than those of the O pattern. Such a difference can be explained by an increase in the expansion terms of the number of circumferential variables, which leads to a different rate of change in the dimensionless natural frequency of the double-curved hyperbolic shell compared to the double-curved paraboloid and elliptical shells. The number of expansion terms is related to the curvature-induced changes in the dimensionless natural frequency of the GPL-reinforced porous composite double-curved shell of revolution.

The effects of different pore coefficients on the dimensionless natural frequency of a double-curved paraboloidal shell of revolution with two GPL distributions, under three different boundary conditions, are presented in Figure 6 and

Table 5. The dimensionless natural frequencies with different pore coefficients (Porosity-II).

Pore Coefficients	O			X
e0	F-F	C-C	S-S	F-F	C-C	S-S
0.1	0.65595	2.84330	1.88184	1.36224	5.96430	3.95105
0.2	0.63782	2.76306	1.82875	1.32469	5.79599	3.83957
0.3	0.61732	2.67166	1.76827	1.28203	5.60426	3.71259
0.4	0.59368	2.56627	1.69853	1.23301	5.38319	3.56617
0.5	0.56626	2.44294	1.61692	1.17590	5.12448	3.39483
0.6	0.53365	2.29589	1.51962	1.10841	4.81601	3.19054
0.7	0.49442	2.11613	1.40069	1.02693	4.43894	2.94083

. It is evident that with the increase in pore coefficient, the dimensionless natural frequency decreases due to a reduction in shell stiffness. Under the same boundary conditions, the dimensionless natural frequency of the X-pattern shell is larger than that of the O pattern. Therefore, increasing the GPL content of internal and external surfaces of the shell will effectively improve the stiffness of the shell.

The effects of different GPL mass fractions on the dimensionless natural frequency of a double-curved paraboloidal shell of revolution with two GPL distributions, under three distinct boundary conditions, are demonstrated in Figure 7 and

Table 6. The dimensionless natural frequencies with different mass fractions (Porosity-II).

Mass Fraction	O			X
WG	F-F	C-C	S-S	F-F	C-C	S-S
0	0.44735	1.92803	1.27599	0.44735	1.92803	1.27599
0.002	0.47355	2.04141	1.35105	0.66375	2.86695	1.89775
0.004	0.49833	2.14883	1.42217	0.82404	3.56718	2.36174
0.006	0.52196	2.25113	1.48991	0.95683	4.15130	2.74903
0.008	0.54454	2.34899	1.55471	1.07244	4.66317	3.08861
0.01	0.56626	2.44294	1.61692	1.17590	5.12448	3.39483
0.012	0.58702	2.53341	1.67684	1.27026	5.54785	3.67603

. An increasing trend in the dimensionless natural frequency is observed as the GPL mass fraction rises.

Combining Figure 6 and Figure 7 with Table 5 and Table 6, it can be seen that irrespective of the GPL distribution pattern or pore coefficient, the shell with a C-C configuration exhibits the highest dimensionless natural frequency and the quickest growth rate, followed by the S-S and F-F configurations.

The effects of different pore coefficients on the dimensionless natural frequency of a double-curved paraboloidal shell of revolution with three GPL distributions under two porosity distributions are shown in Figure 8 and

Table 7. The dimensionless natural frequencies with different pore coefficients.

Pore Coefficients	Porosity-I S-S Boundary Conditions			Porosity-II S-S Boundary Conditions
e0	X	O	U	X	O	U
0.1	4.04194	1.92513	3.16400	3.95105	1.88184	3.09452
0.2	4.03394	1.92132	3.15774	3.83957	1.82875	3.01282
0.3	4.02591	1.91750	3.15146	3.71259	1.76827	2.92390
0.4	4.01785	1.91366	3.14515	3.56617	1.69853	2.82606
0.5	4.00975	1.90980	3.13880	3.39483	1.61692	2.71691
0.6	4.00160	1.90592	3.13243	3.19054	1.51962	2.59285
0.7	3.99343	1.90203	3.12603	2.94083	1.40069	2.44809

. One can note that under the same GPL distribution pattern and pore coefficient, the dimensionless natural frequency of a Porosity-I shell is higher than that of a Porosity-II shell. Further, the dimensionless natural frequency of the shell can be increased by reducing the aperture size of the exterior surface of the shell and increasing density. Therefore, increasing the GPL on the shell’s external and internal surfaces can effectively enhance its stiffness and Young’s modulus.

4. Conclusions

In this investigation, the free vibration of three GPL-reinforced porous composite double-curved shells of revolution with a general radius of curvature was examined using the FSDT and the MVP methods. The governing equations were derived from the principle of energy conservation and transformed into ordinary differential equations. These equations were then solved by a semi-analytical method based on domain decomposition to determine the dimensionless natural frequencies. The findings indicate that:

(1) For the free vibration of three shells of GPL-reinforced porous composite, the reliability of the semi-analytical method based on the domain decomposition has been confirmed.

(2) Pore coefficient significantly affects the free vibration behavior of GPL-reinforced porous composite shells. This effect is integrally linked to the distribution patterns of GPL and the porosity distribution.

(3) The mass fraction of the GPL plays an important role in determining the free vibration of GPL-reinforced porous composite shells, primarily due to changes in the stiffness resulting from the variations in the GPL content.

(4) The dimensionless natural frequencies of the shells with distinct pore coefficients ascend with the rising mass fraction of GPL. Among the patterns, the X-pattern hyperbolic shell of revolution exhibits the highest increase in dimensionless natural frequency, followed by the U-pattern and O-pattern shells. However, as the pore coefficients increase, the rate of increase in the dimensionless natural frequencies decrease. Nonetheless, the X-pattern hyperbolic shell of revolution maintains the highest rate of increase, followed by the U-pattern and O-pattern shells. The numerical results have a certain theoretical guiding significance in engineering practice.