Nonlinear Free Vibration Analysis of Functionally Graded Porous Conical Shells Reinforced with Graphene Nanoplatelets

The nonlinear vibration analysis of functionally graded reinforced with graphene platelet (FG-GRC) porous truncated conical shells surrounded by the Winkler-Pasternak elastic foundation is presented in this paper. An improved model for evaluating the material properties of porous composites is proposed. Three types of porous distribution and three patterns of graphene nanoplatelets (GPLs) dispersion are estimated. Coupled with the effect of the Winkler-Pasternak elastic foundation, the nonlinear governing equations are developed by using the Hamilton principle. The Galerkin integrated technique is employed to obtain the linear and nonlinear frequencies of the shells. After the present method is validated, the effects of the pores, GPLs, the Winkler-Pasternak foundation, and the semi-vertex are investigated in detail. The results show that the linear frequency can be raised by increasing the values of the mass volume of the GPL and foundation parameters. In contrast, the ratio of nonlinear to linear frequency declines as the mass volume of the GPLs and foundation parameters rises. Furthermore, it is found that the minimum ratio of nonlinear to linear frequency can be obtained as the semi-vertex angle is about 55º, and the effect of porosity distribution on the linear and linear frequencies might be neglected.


INTRODUCTION
Because the nonlinear dynamic behaviour of structures inevitably appears in engineering applications, the nonlinear characteristics of the structures have attracted the attention of many researchers.For example, Lu et al.
[1] and [2] and Hao et al. [3] developed a novel model to analyse the nonlinear vibration of isolation systems with a high-static-low-dynamic stiffness.They validated the analytical results by using direct time integration and experiments.The inherent vulnerability of nonlinear vehicle platoons was studied by Wang et al. [4].They proposed a vibration-theoretic approach to compute the platoon's resonance frequency.Zhou et al. [5] presented a high-order nonlinear friction model to investigate the nonlinear hysteresis characteristics of metal rubbers.They found that the nonlinear friction hysteresis dynamic model has a high prediction accuracy and signal-to-noise ratio.Yang and Kai [6] built the nonlinear coupled Schrodinger equation in fibre gratings, and the periodic solution was presented.For porous composite plates reinforced with graphene platelets, Huang et al. [7] presented a two-step perturbation technique to analyse the nonlinear vibration of the plates.In their studies, the effects of the pores, graphene platelets, and elastic foundations on the nonlinear to linear frequency ratio were discussed in detail.
Due to the advantages of withstanding severe high temperatures while maintaining structural integrity, functionally graded materials (FGMs) are widely used in engineering fields such as aerospace, nuclear, mechanical, and civil engineering.The nonlinear static and dynamic characteristics of FGM shell structures have gained much attention.Chan et al. [8] to [10] studied the nonlinear buckling and vibration of FGM truncated conical shells, and Duc et al. [11] and Vuong et al. [12] investigated the nonlinear stability of FGM toroidal shells.
Applying the concept of FGMs, a new type of functionally graded nanocomposites, in which the carbonaceous nanofilters such as graphene platelets (GPLs) and carbon nanotubes (CNTs), are gradually distributed in the thickness direction of the polymeric matrix, has been developed.Because nanocomposites are one of the most promising materials in composite structures, research work has been devoted to examining the linear vibration of functionally graded truncated conical shells reinforced by GPLs and CNTs.Wang et al. [13] studied the free vibration of the composite conical shells reinforced with GPLs.They found that the fundamental frequency is greatly affected by the distribution patterns of GPLs.Afshari [14] and [15] examined the vibration characteristics of truncated conical shells reinforced with graphene nanoplatelets and discussed the influences of the boundary conditions and constant angular velocity on the natural frequencies.For the joined conicalconical shells made of epoxy-enriched with graphene nanoplatelets, Damercheloo et al. [16] studied free vibration and examined the influences of the boundary conditions, the length-to-small radius ratio, and semi-vertex angle in two shell segments.Using the finite element method in conjunction with a higherorder shear deformation theory, Singha et al. [17] investigated the free vibration behaviour of rotating pre-twisted sandwich conical shell panels with functionally graded graphene-reinforced composite face sheets and homogenous cores in a uniform thermal environment.In their studies, the influence of graphene distribution patterns on the fundamental frequencies is discussed with an emphasis on triggering parameters like pre-twist angle, cone lengthto-thickness ratio, core-to-face sheets thickness ratio, and dimensionless rotational speed.Adab et al. [18] and [19] presented a vibrational analysis on truncated conical sandwich microshells.For the vibration behaviour of the composite conical shell reinforced with CNTs, Yousef et al. [20] and [21] investigated the effect of CNTs agglomeration on the vibration characteristics of three-phase CNT/polymer/fibre laminated truncated conical shells.The results show that the subjoining of CNTs leads to a remarkable rise in the natural frequency.Employing the first-order shear deformation theory and the Eshelby-Mori-Tanaka scheme along with the rule of mixture, Afshari and Amirabadi [22] conducted the free vibration analysis of rotating truncated conical shells reinforced with CNTs and the effects of different parameters on the forward and backward frequencies of the shells are investigated.Moreover, the aeroelastic stability of polymeric truncated conical shells reinforced with CNTs and under supersonic fluid flow was studied by Afshari et al. [23].However, the published literature on the nonlinear vibrations of functionally graded graphene-reinforced composite (FG-GRC) plates and conical shells is limited.Using the 2-D differential quadrature method, arc-length continuation technique, and harmonic balance technique, Jamalabadi et al. [24] studied the nonlinear vibration of FG-GRC truncated conical shells and investigated the effects of the mass volume and distribution of the GPLs, semi-vertex, and foundation parameters on the ratio of linear to nonlinear frequency.They found that the rising valve of the GPL mass volume can raise the linear frequency.In contrast, the ratio of the nonlinear to linear frequency declines.Using the variational differential quadrature and finite element method, Ansari et al. [25] investigated the nonlinear vibration characteristics of FG-GRC conical panels with arbitrary-shaped cutouts.The results show that the natural frequency rises as the weight fraction of the GPLs increases.Yang et al. [26] employed the Galerkin and harmonic balance methods to obtain the nonlinear frequencies of FG-GRC conical shells and studied the periodic and chaotic motions.To investigate the nonlinear dynamic characteristics of FG-GRC conical shells, Wang et al. [27] employed the Galerkin method and fourth-order Runge-Kutta technique to obtain the frequency response.They found that both the mass volume and distribution of GPLs significantly affect the resonance response of the shells.Ding and She [28] obtained the nonlinear dynamic response of FG-GRC truncated conical shells and discussed the effects of the mass fraction of the GPLs, geometrical parameters, and the position of the external load on the response.Additionally, Bidgoli and Arefi [29] presented the nonlinear vibration analysis of sandwich plates with graphene nanoplatelet-reinforced face sheets.They found that the nonlinear to linear frequency ratio can be affected by the weight fraction and geometric parameters of graphene nanoplatelets.
In the literature above, the conical shells were regarded as the perfect structures without pores.However, internal pores may appear inside composite materials [30].Thus, it is necessary to investigate the effect of internal pores on the dynamic behaviour of porous structures.Some researchers studied the vibration characteristics of porous isotropic [31] and [32], sandwich [33] and [34].For the functionally graded porous (FGP) truncated conical panels with piezoelectric actuators in thermal environments, Chan et al. [35] investigated the nonlinear dynamic response and free vibration.In their studies, the effect of the porosity distribution on the natural frequency and the deflection amplitudes were discussed.Considering the two types of porosity distribution, the buckling and vibration of sigmoid functionally graded material shells were studied by Huang et al. [36].The results show both the porosity volume fraction and distribution have significant effects on the buckling pressures and lowest frequency.Nevertheless, little work has been done for the porous truncated conical shells reinforced by nano filters [37].Bahhadini et al. [38] and Yan et al. [39] studied the linear vibration of porous FG-GRC nanocomposite sandwich conical shells and found that the natural frequencies were increased with the rising material length-scale parameter and porosity coefficient.The vibration behaviour of porous sandwich truncated conical shells with FG face sheets and a saturated FGP core was studied by Rahmani et al. [40] and Esfahani et al. [41].The effect of internal pores on the natural frequency was also discussed in detail.Applying the isogeometric analysis, Le et al. [42] obtained the three-dimensional solution of the free vibration and buckling for functionally graded porouscellular conical shells.Sobhani et al. [43] studied the vibrational behaviour of porous nanocomposite joined hemispherical-cylindrical-conical shells and examined the effects of the porosity distributions and geometric properties.The results revealed that the natural frequency inclines with the increase of the porosity factor.Additionally, the free vibration of FG-joined conical-cylindrical shells reinforced with graphene nanoplatelets was investigated by Kiarasi et al. [44].It was found that the volume fraction and distributions of the internal pores have an insignificant impact on the natural frequencies of FG-GRC porous conical shells.
Unlike the Winkler elastic foundation model, the Winkler-Pasternak model incorporates the shear interaction between structures and foundations and has been widely used to examine the dynamic behaviour of FGM-truncated conical shells [45] to [47].However, the model was used less to investigate the vibration of FG-GRC conical shells.Safarpour et al. [48] studied the free vibration of perfect and imperfect FG-GRC conical shells resting on Pasternak foundations.Their results show that the effect of the elastic foundation becomes more dominant as the foundation parameters increase.Eyvazian et al. [49] computed the natural frequencies and the corresponding mode shapes of FG-GRC conical panels and discussed the influences of boundary conditions, GPLs volume fraction, and foundation parameters.Furthermore, the frequency responses of rotating two-directional FG-GPLs conical shells on elastic foundations were presented by Amirabadi et al. [50].They found that the increasing value of the GPL mass fraction can raise the natural frequency.The pattern of GPLs scattering near the inner surface has a higher effect on the frequency.
As reviewed above, the published literature on the free vibration of FG-GRC porous truncated conical shells remains limited.Most of the open literature focused on the case of the linear problem.The model used for evaluating the material properties of porous nanocomposites was based on the three assumptions about Young's elastic modulus, shear modulus, and mass density, which is too complicated.Hence, the present work attempts to solve these problems, that is, to propose an improved model for estimating the material properties and present the analytical solution for the nonlinear free vibration of FG-GRC porous truncated shells.Also, the effects of the internal pores, GPL, and elastic foundation on the linear and nonlinear frequencies are investigated.

A POROUS FG-GPLS TRUNCATED CONICAL SHELL
As shown in Fig. 1, a porous FG-GPLs truncated conical shell surrounded by Winkler-Pasternak elastic medium is considered.The curvilinear coordinate system (x, θ, z) is located on the middle surface of the shell.s 1 and s 2 are the distances from the vertex to the small and large ends, respectively.L, h and α denote the length, thickness, and semi-vertex of the shell.Three types of porous distribution in the thickness direction, denoted by P-1, P-2, and P-3, are taken into account, as depicted in Fig. 2.
It is assumed all the internal pores are tiny and the total volume fraction of the pores is small.Unlike other models for evaluating the material properties [30] and [39], in which Young's modulus, shear modulus, and mass density need to be assumed, the present model is based on the following assumption about the volume fraction V P (z) of the internal pores: here, e 0 denotes the porosity coefficient.Young's elastic modulus E(z) and mass density ρ(z) for various porosity distributions can be expressed as [39].
where E 1 and ρ 1 denote the maximum values of Young's modulus and mass density, and e m is the mass coefficients.Furthermore, the coefficients e 0 , e m and the Poisson ratio μ(z) are calculated as follows [39]: It is noted that the present model in Eqs.
(1) to (3) can be used not only for FG-GRCs but also for other composites in which the foam metal is the matrix.
In the present study, three patterns of GPL dispersion, denoted by G-1, G-2, and G-3, are considered, as shown in Fig. 3.The volume fractions V GPL (z) for the three patterns are expressed as follows [43]: where f i1 , f i2 and f i3 are the maximum values of various GPL distributions, calculated by [39]: In Eqs.
(2) and (3), E 1 , ρ 1 , and μ 1 can be determined by using the Halpin-Tsai micromechanics model and the rule of the mixture as follows [39]: in which the geometric parameters of GPL ξ a , ξ b, η a and η b can be stated as here, E GPL , ρ GPL and μ GPL denote the Young's modulus, mass density and Poisson ratio of GPL, and E m , ρ m and μ m are the corresponding values of the matrix.a GPL , b GPL and h GPL are the length, width, and thickness of the GPL.

Governing Equations
It is assumed that the shell is thin and has a large deformation.According to the classic shell theory with the geometrical nonlinearity, the normal strains (ε 1 , ε 2 ) and shear strain γ 12 can be expressed as [51] H in which the strains H H J and twist κ 12 in the middle surface are defined by [51] H where u, v and w are the displacements along the directions of x, θ and z axes, respectively.According to Hooke's law, the normal stresses σ 1 and σ 2 , and the shear stress σ 12 can be stated as Here, Q ij (i, j = 1, 2, 6) are the reduced stiffness coefficients, defined by [51] Q The membrane forces (N 1 , N 2 , N 12 ) and moments (M 1 , M 2 , M 12 ) are expressed as [55] N Substituting Eq. ( 12) into ( 14), the following equations can be obtained [51]: The stiffness constants A ij , B ij , and D ij , can be calculated by [51] The interaction force between the Winkler-Pasternak medium and shell is assumed to be , in which k w and k p are the parameters of Winkler and Pasternak foundation, and ∇ 2 is the Laplace operator.Using the Hamilton principle, the nonlinear dynamic Equilibrium equations of the conical shell surrounded by the Winkler-Pasternak medium can be derived as [51]: where the mass inertia is U U .Substituting Eqs. ( 11) and ( 15) into (17), the nonlinear vibrational equations of the shell can be derived as follows [51]: where the linear operators L ij (i, j = 1, 2, 3) have been given by Duc et al. [51].The nonlinear operators L i4 (i, j = 1, 2, 3) are listed in Appendix.According to Volmir's assumption [26], the inertia forces w w 2 2 can be neglected.

The Solution of the Governing Equations
In this study, simply supported boundaries are considered.The boundary conditions are written as In the present case, the asymmetric solution is taken into account.The solution satisfying the boundary conditions is assumed to be [51] where m and n are the numbers of half-waves along the generator and parallel circle, respectively.In Eq. ( 22), if the terms of sin(nθ/2) and cos(nθ/2) are eliminated, the asymmetric solution is transformed into the corresponding symmetric solution.
For the sake of convenience in integration, multiplying Eqs. ( 18) and ( 19) by x and Eq. ( 20) by x 2 , then applying the Galerkin method for the resulting equations, Eq. ( 18) to (20) can be developed as follows [51]: in which ' ' Substituting Eqs. ( 22) and ( 24) into Eq.( 23), the following ordinary differential equations can be derived: where a i (i, j = 1, 2, 3, 4) are integral coefficients.According to Eq. ( 25), the linear frequency is obtained as Z L a a 2 1 / , and the nonlinear frequency can be derived as [52]: Here, the dimensionless vibrational amplitude A is w max / h, in which w max is the maximum dynamic deflection.

Comparison Studies
In this subsection, two examples are given to validate the accuracy of the present method.
Example 1.In Fig. 4, the curves of the dimensionless linear frequency λ versus the porosity coefficient e 0 for a porous FG-GRC truncated conical shell are depicted.The material properties of the matrix are E m = 130 GPa, ρ m = 8960 kg/m³ and μ m = 0.34.The pattern of GPL dispersion is G -1.The material and geometrical parameters are E GPL = 1.01 TPa, ρ GPL = 1062.5kg/m³, μ GPL = 0.86, a GPL = 2.5 µm, b GPL = 1.5 µm, and h GPL = 1.5 nm.The geometrical parameters of the shell are R 2 / h = 200 cos α, L = s 1 and α = 10°.The dimensionless linear frequency is λ = ω[ρ t h 2 L 2 / D 1 ] 0.5 .The figure reveals that the present results are in good agreement with those given by Bahaadini et al. [38].It is noted the natural frequency decreases with the rising porous coefficient e 0 .Although both the stiffness and mass density decline with the increase of the coefficient, the decreasing speeds are different.As the decreasing speed of the mass density is higher than that of stiffness, the natural frequency reduces.In the opposite case, the natural frequency increases.
Example 2. The fundamental linear and nonlinear frequencies for an isotropic truncated conical shell surrounded by a Winkler-Pasternak foundation are calculated and listed in Table 1.The geometrical parameters of the shell are R 1 / h = 300, L = 2R 1 , γ = 30°.The vibrational amplitude is A = 3.The dimensionless frequencies are Strojniški vestnik -Journal of Mechanical Engineering 70(2024)3-4, [181][182][183][184][185][186][187][188][189][190][191][192][193] It is seen that the present results agree well with those given by Najafov and Sofiyev [36].The variation of the linear frequency ϖ L with the generatrix half-wave number m and circumferential half-wave number n is shown in Fig. 5.As demonstrated by other literature [37] to [39], the number m has a different impact on the frequency with that of the genetrix number n.The frequency is raised with the rising value of m.If n < 13, the frequency is monotonously decreased.However, if n > 13, the frequency is increased.This is because the mass density declines more significantly than the stiffness when n < 13.In contrast, the mass density declines more slowly than the stiffness when n > 13.Hence, the fundamental frequency can be obtained when m ≈ 1, n ≈ 13.The following linear and nonlinear frequencies are calculated at the vibration mode (1,13).Tables 2 to 4 list the dimensionless linear frequencies ϖ L of the shell with the different parameters of pores, GPL, and elastic foundations.Because the Winkler-Pasternak foundation can raise the effective stiffness of the conical shell, the frequency inclines with the increases of foundation parameters k w and k p .Furthermore, the effect of this is more significant than that of the parameter because it can enhance the effective stiffness more significantly k w .The three tables show that the frequency increases with the increasing mass fraction of the GPLs, which is due to the fact that the elastic modulus of the GPL is higher than that of the matrix.Also, the tables reveal that the effect of the porosity coefficient e 0 is related to the parameters of the elastic foundation.If the values of k w and k p are not zero, the frequency reduces with the increasing value of the porosity coefficient e 0 .In contrast, if the values of k w and k p are zero Table 2. Dimensionless linear frequencies ϖ L for a porous FG-GRC conical shell distributed with G-1 GPLs and rested on an elastic foundation k w  Table 3. Dimensionless linear frequencies ϖ L of a porous FG-GRC conical shell distributed with G-2 GPLs and rested on an elastic foundation k w  Table 4. Dimensionless linear frequencies ϖ L of a porous FG-GPLs conical shell distributed with G-3 GPLs and rested on an elastic foundation k w  (k w = 0, k p = 0), the frequency rises.Additionally, it can be observed that the effects of the pattern of GPL dispersion and the type of porosity distribution on the linear frequency can be neglected.The influences of the type of GPL dispersion and mass fraction of the GPL on the ratio of nonlinear to linear frequency ϖ ϖ NL L / are shown in Fig. 6.Among the three patterns of GPL dispersion, the ratio for G-2 is slightly larger than those for G-3 and G-1.That is because the nonlinear frequency for G-2 increases faster than that for G-3 and G-1.Moreover, the mass fraction of the GPLs W GPL rises with the increase of the ratio.For instance, when W GPL changes from 0 % to 0.5 %, the ratio increases by about 9 % at A = 5.The influences of the porosity distribution and porosity coefficient on the frequency ratio ϖ ϖ NL L / are shown in Fig. 7.It is found that the porosity distribution has an insignificant effect on the ratio.When A = 5, the ratio for P-1 is only higher by 1.5 % than that for P-3.Also, it is found that the ratio can be raised by increasing the porosity coefficient.
Fig. 8 shows the effects of parameters k w and k p on the frequency ratio ϖ ϖ NL L / .It is seen that the effects are very significant.For example, when the parameter k p changes from 0 kN/m to 75 kN/m, the ratio declines by 98.9 %.Finally, the effects of the semi-vertex angle α on the linear frequency ϖ L and the ratio of linear to nonlinear frequency ϖ ϖ NL L / are shown in Fig. 9.It is shown that the linear frequency declines when the semi-vertex angle increases.This is because the effective stiffness is raised with the increase of the semi-vertex angle.The figure also shows that the ratio reduces as the angle increases from 15º to 55º.In contrast, it is increased as the angle changes from 55º to 75º.Thus, the minimum value of frequency ratios can be obtained at α ≈ 55º.

CONCLUSIONS
In this study, an improved method for evaluating the material properties of porous FG-GRCs was proposed.An analytical method to investigate the nonlinear vibration of composite such as FGM, FG-GRC, and FG-CNT porous truncated conical shells was presented.The effects of the pore, GPL, elastic foundation, and semi-vertex angle on the linear frequency and the ratio of nonlinear to linear frequency were discussed.Compared with other methods for investigating the nonlinear vibration behaviour of FG-GRC porous conical shells, the present method is simpler and briefer.However, it is not more accurate because the inertia forces U t u t w w 2 2 and U t v t w w 2 2 are neglected.Moreover, some interesting conclusions can be drawn from the parametric studies as follows: 1.As the values of the mass fraction of the GPLs is raised.In contrast, the frequency ratio reduces.
Among the three patterns of GPLs dispersion, the frequency ratio for G-1 is the lowest, and that for G-2 is the highest.2. The natural frequency is not monotonously varied with the rise of the porous coefficient 0 e .If the mass density decreases faster than the stiffness, the natural frequency increases.In contrast, the natural frequency declines.3. The frequency ratio can be decreased by increasing the porosity coefficient.However, the effect of porosity distribution on the frequency ratio might be negligible.

Fig. 2 .
in which μ 1 is the Poisson ratio of the composite without pores, and p is the coefficient of the Poisson ratio.Three types of porosity distribution: a) symmetric distribution (P-1), b) asymmetric distribution (P-2), and c) even distribution (P-3) a) b) c) Fig. 3. Three patterns of GPLs dispersion: a) symmetric distribution (G-1), b) asymmetric distribution (G-2), and c) even distribution (G-3)

Fig. 4 .
Fig. 4. Comparison of the fundamental frequency for a porous FG truncated conical shell

Fig. 5 .
Fig. 5. Variation of the linear frequency with a) the generatrix halfwave number m and b) the circumferential half-wave number n

LFig. 6 .
Fig. 6.Influence of the GPLs on the frequency ratio; a) the pattern of GPL dispersion, and b) the mass fraction of the GPLs

ePϖFig. 9 .
Fig. 9. Influence of the semi-vertex angle on a) the linear frequency and b) the frequency ratio