Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

In this study, the first-order shear deformation theory (FSDT) is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite (FG-GPLRC). The vibration analyses of the FG-GPLRC truncated conical shell are presented. Considering the graphene platelets (GPLs) of the FG-GPLRC truncated conical shell with three different distribution patterns, the modified Halpin-Tsai model is used to calculate the effective Young’s modulus. Hamilton’s principle, the FSDT, and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical shell. The Galerkin method is used to obtain the ordinary differential equations of the truncated conical shell. Then, the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance method. The effects of the weight fraction and distribution pattern of the GPLs, the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are discussed. This study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.


Introduction
The outstanding material properties of graphene composites, e.g., excellent thermal, mechanical, physicochemical, and electronic properties, have attracted the attention of many researchers. For example, Young's modulus of graphene can be as high as 1 TPa, the ultimate strength can be as high as 130 GPa [1] , the thermal conductivity is 5 000 W · m −1 ·K −1 [2] , and the electrical conductivity of graphene is 6 000 S · cm −1 [3] . Rafiee et al. [4] found that the critical buckling load of graphene-reinforced composite material is increased by 52% by adding 0.1% weight fraction of graphene sheets to an epoxy resin matrix. The conical shell is a well-known important structural unit in many engineering fields, e.g., rocket propulsion systems, aircraft, pressure, and piping vessels, and ship structures [5][6] . Because the graphene-reinforced composite conical shell structure has extensive application prospects in many engineering fields, it is necessary to predict and discover nonlinear vibration characteristics.
Researchers have increasingly paid attention to graphene-reinforced composite structures because of their exceptional mechanical properties. Yang and his group conducted numerous studies to assess the bending, vibration, and buckling of beam and plate structures of the functionally graded graphene platelet-reinforced composite (FG-GPLRC) [7][8][9][10][11] . Zhao et al. [12] reviewed recent studies dedicated to characterizing the mechanical and structural behavior of FG-GPLRC beams, plates, and shells with various loadings or boundary conditions, and discussed the challenges associated with and the future directions for FG-GPLRC structures. Shen et al. [13][14] studied the nonlinear vibration and postbuckling of an FG graphene reinforced composite cylindrical panel in thermal environments. The size-dependent postbuckling characteristics of FG-GPLRC multilayer microtubes containing initial geometrical imperfection were investigated by Lu et al. [15] . Their work could provide theoretical guidelines for the safety assessment and optimal design of GRC tubular structures. Yas and Rahimi [16] presented a study on the thermal vibration of FG porous nanocomposite beams reinforced by graphene platelets (GPLs) based on the Timoshenko beam theory. Blooriyan et al. [17] proposed an analytical approach to investigate the postbuckling behavior of FG GPL reinforced polymer composite circular cylindrical shells subject to lateral pressure and axial compression. Nguyen et al. [18] developed a computational approach to investigate the dynamic responses, free vibrations, and active control behavior of smart FG metal foam plates reinforced with GPLs. Gao et al. [19] derived the governing equations for the FG metal foam plates by using the Hamilton principle based on three different plate theories, and investigated the effects on the wave dispersion relations of the geometrical and material properties of the plates. Wang et al. [20] employed the Hamilton principle and the improved Donnell nonlinear shell theory to formulate a model for a metal foam cylindrical shell reinforced by GPLs and investigated the nonlinear vibration characteristics. Liu et al. [21] studied the free vibrations, forced vibrations, and transient responses of the GPLs-FG plates.
A number of papers have reported the investigations of the vibrational characteristics of conical shell structures. A review on the buckling and vibrational characteristics of functionally graded material (FGM) conical shells was published by Sofiyev [22] . Song et al. [23] studied the free vibration of the conical shells subject to an elastic constraint and inertia force under arbitrary boundary conditions. Based on the assumptions of uncoupled thermo-elasticity, Akbari et al. [24] analyzed the thermal vibrations of an FGM conical shell. Ansari et al. [25] employed the higher-order shear deformation shell theory to study the free vibrations and large-amplitude forced vibrations of a composite conical shell reinforced by FG carbon nanotubes. Chan et al. [26][27] investigated the nonlinear dynamic behavior of the truncated eccentrically stiffened FG conical shell and truncated FG conical panel. The Haar wavelet method was used to derive the frequencies of a truncated rotating conical shell [28] . Rahmani et al. [29] investigated the vibrations of temperature dependent sandwich truncated conical shells under various thermal conditions. Hao et al. [30][31] analyzed the nonlinear flutter characteristics of the truncated FG conical panels and shells.
In the past few years, a large number of scholars have focused on the buckling and post-buckling behaviors of conical shell structures. The dynamic instability of the truncated FG conical shell and the FG orthotropic conical shell was studied by Deniz and Sofiyev [32] and Sofiyev et al. [33] who used the first-order shear deformation theory (FSDT). Hoa et al. [34] studied the nonlinear thermo-mechanical buckling and post-buckling behaviors of a truncated eccentric stiffener-FGM conical shell. The effects of the foundations, stiffeners, geometric dimensions, temperatures, and material properties on the stability of the truncated eccentric stiffener-FGM conical shell were determined. With the classical shell theory, Chan et al. [35][36] analyzed the nonlinear buckling of the truncated stiffened FG conical shell and a conical shell reinforced by FG carbon nanotubes under an axial compressive load. Kiani [37] conducted a buckling analysis for a composite laminated conical shell reinforced by graphene sheets. Jiao et al. [38] investigated the dynamic buckling behaviors of an FG carbon nanotube composite cylindrical shell subject to a dynamic displacement load by using a semi-analytical approach. Dung et al. [39] investigated the buckling characteristics of an imperfect thin conical plate subject to the simply supported condition. Bich et al. [40] studied the buckling characteristics of an FGM conical plate with external and axial loads.
To date, a large number of scholars have therefore analyzed the vibrational and buckling characteristics of conical shell structures. Nevertheless, few scholars have paid attention to graphene-reinforced composite truncated conical shell structures. This motivates our study, which involves nonlinear vibrational analyses of the FG-GPLRC truncated conical shell. Our study considers three distribution patterns of the GPLs in the FG-GPLRC truncated conical shell, and the effective Young's modulus is calculated by utilizing the modified Halpin-Tsai model. Based on the von-Karman nonlinear relations, FSDT, and Hamilton principle, we establish a dynamic model of the FG-GPLRC truncated conical shell, which can be expressed as ordinary differential equations via the Galerkin method. Then, the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by using the harmonic balance method. The effects of the distribution pattern of the GPLs and the weight fraction, the ratio of the radius to the thickness as well as the ratio of the length to the radius of the FG-GPLRC truncated conical shell on the nonlinear vibrational behavior are discussed. This study leads to the discovery of the chaotic and periodic motions of the FG-GPLRC truncated conical shell.
2 Formulation of a model of the FG-GPLRC truncated conical shell An FG-GPLRC truncated conical shell with the length L, the thickness h, the minor inner radius r 1 , and the semi-vertex angle β is considered as shown in Fig. 1. In addition, the radius of the arbitrary point can be computed as R = r 1 + x sin β along the meridional direction. In Fig. 1, the curvilinear coordinate system (x, θ, z) is located on the mid-surface of the FG-GPLRC truncated conical shell along the axial, circumferential, and radial directions, respectively. The displacement components u, v, and w indicate the displacements of an arbitrary point of the FG-GPLRC truncated conical shell in the directions x, θ, and z, respectively. The transverse excitation F cos(Ωt) is loaded on the outside surface of the FG-GPLRC truncated laminated conical shell along the z-direction.
GPLs are introduced into the truncated conical shell structures to improve the performance. Three different distribution patterns of the GPLs in the FG-GPLRC truncated conical shell are considered, namely, the GPL-U pattern, the GPL-X pattern, and the GPL-O pattern, as shown in Fig. 2. For the GPL-U pattern, the volume fractions of the GPLs maintain the same value along the thickness direction. The exterior and interior surfaces of the FG-GPLRC truncated conical shell are rich in the GPL-X pattern, whereas the GPL-O pattern prevails in the central region of the FG-GPLRC truncated conical shell, as shown in Fig. 2.
The volume fractions of the GPLs for the FG-GPLRC truncated conical shell change linearly Fig. 1 Model and curvilinear coordinate system of an FG-GPLRC truncated conical shell (color online)

Fig. 2 Distribution patterns of three GPLs in FG-GPLRC materials (color online)
along the thickness direction [41] , as shown in Fig. 3, and they are expressed as follows: where V * is the total volume fraction of the GPLs of the FG-GPLRC truncated conical shell, and can be obtained from the weight fraction of the GPLs W GPL as follows: in which ρ GPL and ρ M are the mass densities of the GPLs and the matrix, respectively. The experiment proved that Young's modulus can be determined more accurately with the modified Halpin-Tsai model, and the rule of mixture enables the other properties to be computed accurately [42][43] . To reduce the amount of calculation and ensure calculation efficiency and accuracy, we use a modification of the Halpin-Tsai model for determining the effective Young's modulus, and the rule of mixture for the other properties.
According to the modified Halpin-Tsai model [43][44] , the effective Young's modulus E of the FG-GPLRC truncated conical shell is assumed to be where E M is Young's modulus of the matrix, and E GPL , h GPL , w GPL , and l GPL are Young's modulus, thickness, width, and length of the GPLs, respectively. The effective mass density ρ, the effective Poisson's ratio ν, and the coefficient of the thermal expansion α of the FG-GPLRC truncated conical shell are determined with the rule of mixture as follows: where ν GPL and ν M are Poisson's ratios of the GPLs and the matrix, respectively. The thermal expansion coefficient of the FG-GPLRC truncated conical shell does not change considerably over a certain temperature range [45][46] . Thus, we consider the thermal expansion coefficient to be constant. According to the FSDT [47] , the displacement fields of the FG-GPLRC truncated conical shell are assumed to be where u 0 , v 0 , and w 0 denote the mid-plane displacements in the x-, θ-, and z-directions, respectively. In addition, ϕ x and ϕ θ represent the rotations of the radial normal in the θand x-directions, respectively. Substituting the displacement field equation (8) into the von Karman nonlinear strains-displacement relationships, the nonlinear strains of the FG-GPLRC truncated conical shell are rewritten as where ε x and ε θ are the principal strains, and γ xθ , γ θz , and γ xz denote the shear strains. Furthermore, = 1/R in the above equations. The temperature increment of the FG-GPLRC truncated conical shell in the radial direction varies linearly, where T o represents the external temperature of the FG-GPLRC truncated conical shell, and T i is the internal temperature. The constitutive relation of the FG-GPLRC truncated conical shell is written as [42][43] where Q ij (i, j = 1, 2, 4, 5, 6) represent the stiffness coefficients of the FG-GPLRC truncated conical shell, and they are denoted as Hamilton's principle is used to obtain the system of nonlinear partial differential equations of the FG-GPLRC truncated conical shell, which is expressed as where δU and δK denote the virtual strain energy and virtual kinetic energy, respectively. The virtual strain energy δU is derived as follows: The virtual kinetic energy δK is obtained as The virtual work δW resulting from transverse excitation is expressed as Substituting Eqs. (14)- (16) into Eq. (13) and utilizing Hamilton's principle enable a system of nonlinear partial differential equations of the FG-GPLRC truncated conical shell to be derived as follows: where one overdot represents the first-order derivative with respect to time, and two overdots denote the second-order derivative. I η is the mass moment of inertia, expressed as The stress resultants and moment resultants of the FG-GPLRC truncated conical shell can be calculated as where K = 5/6 is the shear correction coefficient.
In the above equations, the tensile rigidity A ij , the bending-tensile coupling rigidity B ij , and the bending rigidity D ij of the FG-GPLRC truncated conical shell are expressed as Substituting Eqs. (17)-(23) into Eq. (16), the nonlinear partial differential governing equations can be expressed by the generalized displacements u 0 , v 0 , w 0 , ϕ x , and ϕ θ of the FG-GPLRC truncated conical shell, as shown in Appendix A.
In this study, the boundary condition of the FG-GPLRC truncated conical shell is considered to be simply supported at both ends, and is expressed by The displacements u 0 , v 0 , w 0 , ϕ x , and ϕ θ of the FG-GPLRC truncated conical shell, which is subject to the simply supported boundary condition, are expressed as the double Fourier sine series, where u mn (t), v mn (t), w mn (t), ϕ xmn (t), and ϕ θmn (t) are the time varying coefficients, and m and n are the generatrix half wave and circumferential wave numbers, respectively. The transverse excitation F can also be expanded as The effects of the inertia terms of u 0 , v 0 , ϕ x , and ϕ θ on the nonlinear vibration characteristics of the FG-GPLRC truncated conical shell are much smaller than those of the radial inertia term [48][49] . Hence, the inertia terms u 0 , v 0 , ϕ x , and ϕ θ can be omitted from Eq. (16). Employing the Galerkin method and neglecting the inertia terms of u 0 , v 0 , ϕ x , and ϕ θ , the general rotation and in-plane displacements can be represented as functions of the transverse displacement. Furthermore, a system of second-order ordinary differential equations for the FG-GPLRC truncated conical shell can be yielded as where ω l is the linear natural frequency of the conical shell, and ς is the coefficient of the cubic term of the system. This system is the Duffing differential equation.

Harmonic balance method
In this section, the nonlinear natural frequency of the FG-GPLRC truncated conical shell is analyzed by using the harmonic balance method. For natural frequency analysis, the transverse excitation f mn can be neglected. We assume the approximated periodic solution of the Duffing differential equation (26) as where ψ = ω nl t + β 0 , in which ω nl is the nonlinear natural frequency of the conical shell. Introducing Eq. (27) into the Duffing differential equation (26) and combining the coefficients of the cosine functions yield the following equation: By using the harmonic balance method, setting the coefficients of the constant terms and cos ψ in Eq. (28) to zero, a series of algebraic equations can be obtained as follows: Compared with the amplitude A 1 of the FG-GPLRC truncated conical shell, the shift A 0 is very small. Thus, A 2 0 and A 3 0 in Eq. (29) can be neglected. For the nonlinear free vibration of the FG-GPLRC truncated conical shell, the initial condition provided in Eq. (26) can be set as follows: where w max is the maximum amplitude of the FG-GPLRC truncated conical shell. Setting A 1 = w max in Eq. (29) to zero, the natural frequency of the FG-GPLRC truncated conical shell can be obtained as Thus, the frequency ratio ω nl /ω l of the FG-GPLRC truncated conical shell is indicated as

Free vibrations
To evaluate the aforementioned formulations and the programs we developed, the dimensionless natural frequency Ω n = ω n R (1 − ν 2 )ρ/E is calculated. The dimensionless natural frequency of the pure metal truncated conical shell and the results of Najafov and Sofiyev [50] , Kerboua et al. [51] , and Liew et al. [52] are compared in Table 1. The geometric relations (β = π/6, R/h = 100, and L = 0.25R sin β) of the pure metal truncated conical shell are considered. Specifically, our results are in good agreement with the previous results in Refs. [50]- [52] for the pure metal truncated conical shell. Table 1 Dimensionless natural frequency of the pure metal truncated conical shell n Najafov and Sofiyev [50] Kerboua et al. [51] Liew et al. [52] Present study The method proposed in this paper is validated by comparing the nonlinear to linear frequency ratios of an isotropic square plate. The geometrical parameters of the isotropic square plate are a/b = 1 and a/h = 10. It should be noted that these nonlinear to linear frequency ratios are in excellent agreement with the results in Refs. [53]- [54].   [42] . The temperature of the inner surfaces of the FG-GPLRC truncated conical shell is T i = 300 K. The thermal expansion coefficients are α GPL = 5 × 10 −6 K −1 and α M = 60 × 10 −6 K −1 [46] . Table 3 lists the linear natural frequencies of the FG-GPLRC truncated conical shell for the three GPL distribution patterns when the ratio of the radius to the thickness is r 1 /h = 50, the ratio of the length to the radius is L/R = 2, and the weight fraction of the GPLs is W GPL = 0.01%. Consequently, the linear natural frequencies of the FG-GPLRC truncated conical shell increase with an increase in both m and n for the three GPL distribution patterns.

Linear natural frequencies
The minimum frequency appears when m = 1 and n = 1, and it is the fundamental natural frequency.  Figure 4 illustrates the effects of the temperatures of the outer surfaces and the weight fraction of GPLs on the fundamental natural frequency of the FG-GPLRC truncated conical shell for the three GPL distribution patterns. Three different colored lines represent three different GPL distribution patterns. The fundamental natural frequencies of the FG-GPLRC truncated conical shell are observed to increase significantly when the distribution pattern of the GPLs increases from 0 to 1%. In addition, the fundamental natural frequencies of the FG-GPLRC truncated conical shell change gradually as the distribution pattern of the GPLs changes. The GPL-X and GPL-O patterns give rise to the highest and lowest frequencies, respectively. The fundamental natural frequencies of the FG-GPLRC truncated conical shell of the GPL-U pattern are always between the highest and lowest values. The GPL-X pattern introduces the maximum stiffness to the system, whereas the GPL-O pattern results in the minimum stiffness of the system.

Nonlinear natural frequencies
This section presents our study of the nonlinear natural frequency of the FG-GPLRC truncated conical shells. This is based on Eq. (27), which is used to obtain the fundamental frequency ratio ω nl /ω l of the FG-GPLRC truncated conical shell. Figure 5 depicts the effects of the maximum amplitude of the FG-GPLRC truncated conical shell on the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l with the distribution patterns of three GPLs. Figure 5 indicates that the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l increases with an increase in the maximum amplitude of the FG-GPLRC truncated conical shell. However, for the three GPL distribution patterns, the order in which the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l increases is as follows: GPL-O pattern > GPL-U pattern > GPL-X pattern. Figure 6 shows the effects of the weight fraction of the GPLs W GPL the FG-GPLRC truncated conical shell on the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l with three GPL distribution patterns. Clearly, the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l stays the same as the weight fraction of the GPLs increases. This is a highly interesting finding. As the weight fraction of the GPLs increases, the values of the fundamental nonlinear and linear frequencies increase simultaneously. Thus, the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l remains almost the same.
The effects of the ratio of the radius to the thickness r 1 /h of the FG-GPLRC truncated conical shell on the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l are shown in Fig. 7. It is seen that the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l increases as the ratio of the radius to the thickness r 1 /h increases. In addition, this phenomenon exists in any of the GPL distribution patterns of the FG-GPLRC truncated conical shell. Figure  fundamental frequency to the linear fundamental frequency ω nl /ω l versus the ratio of the length to the radius L/r 1 of the FG-GPLRC truncated conical shell. For each of the GPL distribution patterns, the ratio of the nonlinear fundamental frequency to the linear fundamental frequency ω nl /ω l decreases with the increase in the ratio of the length to the radius.

Periodic and chaotic motions
This section presents the investigation of the nonlinear forced vibration of the FG-GPLRC truncated conical shell. To analyze both the chaotic and periodic motions of the system, the bifurcation diagram, the time history diagrams as well as the phase portraits are obtained by using the Runge-Kutta algorithm. For the first mode of the FG-GPLRC truncated conical shell, the initial condition of Eq. (26) is selected as w 11 = −0.001 andẇ 11 = −0.001 in the following numerical simulation. Figure 9 depicts the bifurcation diagram of the transverse excitation versus the amplitude of the first mode of the FG-GPLRC truncated conical shell with the GPL-U pattern when the transverse excitation increases from f = 5 × 10 5 to f = 10 × 10 5 . This diagram enables us to visualize the chaotic and periodic motion of the FG-GPLRC truncated conical shell. In general, the motion law of the FG-GPLRC truncated conical shell that corresponds to an increase in the transverse excitation from f = 5 × 10 5 to f = 10 × 10 5 is given as chaotic motion → periodic motion → chaotic motion → periodic motion → chaotic motion → periodic motion → chaotic motion → periodic motion → chaotic motion. Thus, the periodic motion and chaotic motion alternate. However, a small chaotic window can be found in the periodic regions, with excitations such as f = 5.496 × 10 5 and f = 5.548 × 10 5 . On the other hand, a small periodic window can be found in the chaotic regions with excitations such as f = 6.083 × 10 5 and f = 9.008 × 10 5 . Thus, the nonlinear dynamic behavior of the FG-GPLRC truncated conical shell is highly complex. The periodic motion (see Fig. 10) and chaotic motion (see Fig. 11) of the FG-GPLRC truncated conical shell are visually contrasted for different values of the transverse excitation (f = 6.3 × 10 5 and f = 6.6 × 10 5 ). Each of these figures includes the phase portrait and the time history diagram. The amplitude and velocity of the chaotic motion are larger than those of the periodic motion. .

Conclusions
This study focuses on the analyses of the nonlinear vibrations of the FG-GPLRC truncated conical shell. Hamilton's principle, the FSDT, and the von-Karman type nonlinear geometric relationship are used to derive a system of partial differential equations for the FG-GPLRC truncated conical shell. The ordinary differential equations of the FG-GPLRC truncated conical shell are obtained according to the Galerkin method. Then, the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by utilizing the harmonic balance method. The proposed method is validated by comparing its performance with that of existing methods in the literature. A parametric study is proposed to assess the effects of the GPL weight fraction, the GPL distribution pattern, the ratio of the radius to the thickness, and the ratio of the length to the radius of the FG-GPLRC truncated conical shell on the nonlinear free vibration behavior. In addition, the chaotic and periodic motions of the FG-GPLRC truncated conical shell are discovered. It can be concluded as follows.
(i) The GPL distribution pattern, the weight fraction, the ratio of the radius to the thickness as well as the ratio of the length to the radius have significant effects on the free vibration characteristics of the FG-GPLRC truncated conical shell.
(ii) The GPL-X pattern maximizes while the GPL-O pattern minimizes the stiffness of the system.
(iii) An increase in the weight fraction of the GPLs is accompanied by a concomitant increase in the values of the fundamental nonlinear and linear frequencies, and the ratio of the fundamental nonlinear frequency to the linear frequency ω nl /ω l remains almost the same.
(iv) The nonlinear dynamic behavior of the FG-GPLRC truncated conical shell is highly complex, and the periodic and chaotic motions are discovered.

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