Vibration analysis of a sandwich cylindrical shell in hygrothermal environment

The sandwich structures are threeor multilayered structures such that their mechanical properties are better than each single layer. In the current research, a three-layered cylindrical shell including a functionally graded porous core and two reinforced nanocomposite face sheets resting on the Pasternak foundation is used as model to provide a comprehensive understanding of vibrational behavior of such structures. The core is made of limestone, while the epoxy is utilized as the top and bottom layers’ matrix phase and also it is reinforced by the graphene nanoplatelets (GNPs). The pattern of the GNPs dispersion and the pores distribution play a crucial role at the continuous change of the layers’ properties. The sinusoidal shear deformation shells theory and the Hamilton’s principle are employed to derive the equations of motion for the mentioned cylindrical sandwich shell. Ultimately, the impacts of the model’s geometry, foundation moduli, mode number, and deviatory radius on the vibrational behavior are investigated and discussed. It is revealed that the natural frequency and rotation angle of the sandwich shell are directly related. Moreover, mid-radius to thickness ratio enhancement results in the natural frequency reduction. The results of this study can be helpful for the future investigations in such a broad context. Furthermore, for the pipe factories current study can be effective at their designing procedure.


Introduction
Structural analyses attracted higher levels of scholars' attention nowadays. Among them, sandwich structures are well-known due to their high level of functionality. These structures are broadly implemented in ships, aerospace vehicles, and cargo containers. The porous type of sandwich structures meets extensive applications due to their lower weight to stiffness ratio. For instance, in the aircraft industries, aircraft with higher speed and stiffness and lower weight is accessible by the means of sandwich panels [1][2][3][4]. Depending on their application, sandwich structures may include three or more layers. Catania et al. [5] studied the mechanical damping behavior of multilayer components due to their wide range of applications. In their work, the results are validated by an experimental test. Beside this, plasma techniques are used by Rongong et al. [6] to generate constrained layer damping coatings on the metallic substrates. Also, in another work, Yu et al. [7] discussed the damping efficiency of the coating system based on the Reuss model and Hashin-shtrickman equation. In their work, the theoretical results showed that for a coating system, there is an optimum thickness of the coating layer that causes the coating structure obtain the best balance between the strength and the damping capacity. In two different works, Amir and his coresearchers [8,9] presented their findings of vibrational behavior of three-layered circular and annular plates, in which a rheological fluid core was put between two magnetostrictive face sheets. More recent sandwich structures exploit a core that is constituted of porous material to gain a lighter structure. Furthermore, functionally graded materials (FGMs) are a new branch of elaborated engineering composite materials [10,11] that attracted the scientists' attention. The layers constituted from FGMs are fully metallic at the bottom surface and fully ceramic at the top surface or vice versa. FGMs are first introduced by Koizumi [12]. Liu et al. [13] broadened our knowledge about FGMs by working on the wave propagation using a numerical technique. In another work, Han et al. [14] numerically studied FGMs' transient wave propagation using another method. In 2002, Najafizadeh and Eslami [15] examined buckling the behavior of a circular FGM plate under radial loading. Arshid and Khorshidvand [16] examined the vibrational behavior of a saturated porous FG circular plate patched to piezoelectric face sheets using a differential quadrature method (DQM). One of the first works on the sandwich cylindrical shells dates back to 1995, in which Huang and Dasgupta [17] investigated the composite cylindrical shells to explain their free vibration behavior. After that, laminated cylindrical shells have been considered as a model to investigate static bending behavior under various boundary conditions using the differential cubature method [18]. In more recent years, the bending behavior of cylindrical sandwich panels has been evaluated by Pydah and Batra [19], who succeed to present an analytical solution for the mentioned bending behavior. In 2019, Sun et al. [20] experimentally investigated the different dynamic responses of the sandwich panels. They aimed to present a thorough guideline for the fabrication of sandwich panels with a higher ratio of blast protection capacity to weight. Electromechanical systems are exposed to different electrical and mechanical loadings and require a comprehensive study due to their important parameters. By the growth of the application of sandwich plates, more accurate plate theories are needed to obtain responses with higher accuracy. In the past decades, the FG plates and shells have been often analyzed using the first order shear deformation theory (FSDT) or higher order theories [21][22][23][24]. As an example, Trabelsi et al. [25] used modified FSDT to examine the thermal buckling of the FG plates and cylindrical shells. Based on the nonlocal elasticity theory, Ke et al. [26] took a nano piezoelectric cylindrical shell into consideration and investigated its vibrational behavior. They employed Love's theory which is usually used for thin shells and also used DQM for solving the obtained differential equations. In a similar study, Razavi et al. [27] conducted a research on electromechanical vibrational response of nano cylinders. They derived the governing equations with the aid of the energy method and Hamilton's principle and captured the size effect via the couple stress theory. As another instance, Khoa et al. [28] employed a cylindrical panel model which was reinforced by singlewalled carbon nanotubes (SWCNTs) to study the vibrations of the functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical shells. For integrity, they applied the thermal environment on the entire structures and used higher-order shear deformation theory (HSDT). Nanoparticles and especially CNTs and graphene nanoplatelets (GNPs) specifications are discussed in numerous researches, recently [29][30][31][32][33]. Single-layered graphene sheets' mechanical behavior and especially their buckling response, regarding the scale-effect, is considered by Fattahi et al. [34,35]. They attained the governing equations based on classical theory, FSDT, and HSDT. As another attempt, forced and free vibrational behaviors of a viscoelastic core equipped with GNPs-reinforced face sheets are examined by Mohseni and Shakouri [36]. They assumed face sheets are FG due to different types of GNPs distributions. A developed HSDT was applied to provide an analytical solution for the buckling capacity of sandwich plates containing cores with and without the GNPs reinforcement phase [37]. Moreover, free vibration and buckling analysis of cylindrical shells and sandwich plates is addressed by Kumar and Srinivasa [38]. After that, Ibrahim et al. [39] used FSDT to analyze the mechanical responses of FGM panels. Furthermore, Mehar and Panda [40] employed the finite element method to evaluate the temperature dependency of vibrational behavior of sandwich curved panel reinforced by functionally graded carbon nanotubes (FG-CNTs). In their paper, HSDT is used to derive displacement field equations. More recently, Karimiasl et al. [41] applied HSDT to assess doubly curved flexible core integrated with piezoelectric layers as a model to address the nonlinear vibration analysis. They concluded that an increment in the shaped memory alloys (SMA) volume fraction declined the frequency. Furthermore, the elasto-dynamic behavior of cylindrical tubes is evaluated by Ramezani and Mirzaei [42]. They assumed their model under moving pressures and different boundary conditions. In 2014, a new quasi-3D hyperbolic shear deformation theory for vibrational responses is proposed by Hebali et al. [43] to get more accurate results. This theory accounts for the stretching and shear deformation effects and eliminates their dependency on shear correction factor. Mirjavadi et al. [44] carried out a study on forced vibration of annular sector plates which are composed of the nanocomposite. They employed More-Tanaka approach to determine the material properties. In another study [45], they provided similar research, but for conical shells, and investigated the effect of different parameters on the results. Three types of porosity distributions are considered by Safaei [46] to analyze a multilayered sandwich plate whose core is made from porous materials. He used finite element method to obtain the results. Next, Sahmani et al. [47] provided a closed-form solution for examining vibrational response of small-scaled beams that are made from GNPs-reinforced composites. They derived the motion equations via HSDT to take the shear deformations into account. Moreover, they conducted another study [48] to consider the effect of size on the post-buckling behavior of plates that were reinforced by GNPs. Different types of GNPs dispersion patterns are considered by them. Most recently, Arshid and Amir [49] provided an investigation on the thermal buckling of porous-reinforced nanocomposite structures and concluded that adding GNPs to the structure may increase or decrease the critical buckling temperature, depending on their dispersion patterns. Beside this, Fan et al. [50] used modified couple stress theory (MCST) and examined the nonlinear oscillations of porous FG plates in nanoscale. They used NURBS-based isogeometric approach to gain the results. Wang et al. [51] performed nonlinear vibration analysis of porous shells that were reinforced with GNPs. They employed an improved Donnell nonlinear shell theory to formulate their model. A general approach is provided for the free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite (FG-CNTRC) cylindrical shells with arbitrary boundary conditions by Qin et al. [52] based on FSDT. Also, a comparison study is presented by Qin et al. [53] on free vibrations of cylindrical shell with various boundary conditions using numerical methods. Shen et al. [54] considered effect of thermal environment on nonlinear vibration of functionally graded graphene nanoplatelets (FG-GNPs)-reinforced laminated cylindrical shells.
In the present paper, a sandwich cylindrical structure including FG porous core integrated by the GNPsreinforced composites (GNP-RC) layers is considered as model to provide a comprehensive insight on its vibrational behavior. In the previously published papers, the sandwich cylindrical model with such exclusive configuration is not examined, yet. The sandwich structure is rested on the Pasternak foundation and the whole model is placed in a hygrothermal environment. As sinusoidal shear deformation shells theory (SSDST) is more accurate than lower-order theories, it is hired to gain more reliable results. Based on the Hamilton's principle and variational approach, the equations of motion are derived and analytically solved. Finally, the influences of various parameters as porosity coefficient index and their distribution patterns, GNPs volume fraction index, and their dispersion types, geometry, mode number, and the moduli of foundation on the natural frequency of presented sandwich cylindrical shell are evaluated. The results could help in obtaining a deeper understanding of these structures which can be helpful in different industries, such as, automobiles, micro electro mechanical system (MEMS) processes, and aerospace.

Mathematical modeling
According to the cross-sectional view in Figure 1, the proposed model is a cylindrical sandwich shell including an FG porous core confined by two GNP-RC face sheets with mid-radius of R, length of L, and rotation angle of φ r resting on Pasternak elastic foundation. The structure is entirely subjected to hygrothermal loading. Also, h c , h t , and h b denote representatives of the z direction width of the porous core, top, and bottom skins. Moreover, the sum of the height of different parts serves as h representing the total height of the sandwich model. In the intermediate-plane of the shell, the cylindrical coordinate system (x, θ, z) is placed at the corner point in which x, θ, and z are axial, circumferential, and thickness directions of the model, respectively.
Strain components can be obtained by SSDST. Accordingly, the displacements of an arbitrary point either in porous core or in GNPs-RC face sheets can be presented as [55]: where u, v, and w are the movement components in the longitudinal, radial, and thickness direction, respectively. Moreover, u 0 , v 0 , and w 0 represent the movement components on the intermediate-plane; λ x and λ θ denote the rotation about θ and x-axes, respectively. Furthermore, f(z) and g(z) can be determined by SSDST as: Based on the Von Karman's assumptions, strain field can be derived as [56,57]: The stresses related to the core and face sheets in the hygrothermal environment can be presented by ref. [58,59]: (4) where C denotes the elastic coefficients and subscript c and f represent the porous core and GNP-RC face sheets, respectively. Moreover, α and β represent the coefficients related to thermal expansion and moisture expansion. ΔT and ΔH are the temperature alternation and moisture changes, one after another. A porous layer, made of Limestone, is used whose elastic coefficients can be written as [43]: in which: To determine the elastic coefficients for the FG-GNP-RC face sheets, the following relations can be represented Top and bottom physical properties are also variable through their thicknesses. To determine the effective values of these parameters for different models, Halpin-Tsai and MR micromechanical models are employed. Halpin-Tsai model uses the following relation to predict the effective Young's modulus [64]: in which E M denotes matrix part Young's modulus; while the volume fraction of GNPs is shown by V GNP . Furthermore, ζ L , ζ W , η L , and η W state properties of the GNPs, which is related to its geometry, and can be defined as [65]: Here, l GNP denotes the GNPs' length, their thickness is shown by t GNP , and their width is represented by w GNP . Moreover, E GNP is Young's modulus of GNPs reinforcements. Noteworthy, the GNPs' volume fraction added to volume fraction of matrix portion should be 1 and that of GNPs can be obtained via the following equation [66]: In the abovementioned relation, ρ is the density. The subscripts GNP and M are related to the GNPs reinforcements and matrix. Also, g GNP refers to the dispersion pattern of the reinforcement phase weight fraction and can be presented as [67]: Parabolically dispersed GNPs through the matrix: Linearly dispersed GNPs through the matrix Uniformly dispersed GNPs through the matrix μ P , μ L , and μ U are the GNPs' gradient indices. The subscript P, L, and U are representatives of the words parabolic, linear, and uniform, respectively, where they are addressing different dispersion patterns of GNPs. For more information, it should be mentioned that the total GNPs content percentage plays a crucial role in GNPs dispersion pattern variations. For 0, 1/3, and 1 percentage of GNPs, the values of μ P are 0, 1, and 3, respectively. The values of μ L for 0, 1/3, and 1 percentage of GNPs are 0, 2/ 3, and 2, respectively. Also, for 0, 1/3, and 1 percentage of GNPs, the values of μ U are 0, 1/3, and 1, respectively [68].
Moreover, other properties of the face sheets such as their Poisson's ratio, density, and hygrothermal expansion coefficients can be determined via the MR model as [69]:

Motion equations
Hamilton's principle is used to extract the motion equations of the aforementioned sandwich shell [70]: in which Λ, K, and Π are whole strain energy, kinetic energy, and external work applied on the sandwich shell, respectively. Also, t is the time. Strain energy is due to the two face sheets and core and can be presented as: The FG porous core strain energy is derived as: , , Therefore, the variations of the total strain energy can be presented as: where: Moreover, the definition of kinetic energy can be added as [35]: Using the variational approach, variations of the kinetic energy can be stated as: Pasternak foundation is assumed for the sandwich shells. Therefore, external work due to the Pasternak foundation can be determined by ref. [72]: where K w is the Winkler spring coefficient and K g shows the shear layer coefficient.
The hygrothermal external work applied on the whole sandwich structure due to the hygrothermal load as a result of the temperature and moisture difference can be determined as follows [73,74]:  Finally, whole external work's variations may be evaluated by summing the presented two works as: By substituting equations (21), (24), and (30) into equation (17), the motion equations can be derived, setting the coefficients of δu 0 , δv 0 , δw 0 , δλ x , and δλ θ to zero.

Analytical solution procedure
To solve the obtained differential equations, Navier's solution approach is applied. Based on this scheme, the geometrical boundary conditions for the simply supported type can be satisfied using the below functions for the displacement vectors where U, V, W, Υ x , and Υ θ denote the coefficients which are unknown. Also, α = nπ/L and β = mπ/φ r in which wavenumbers along x and θ directions are represented by n and m, respectively, which are integers. In the end, the motion equations can be obtained in a matrix form as: The arrays of stiffness [K] and mass [M] matrices can be determined by substituting the functions of equation (31) into the governing motion equations which are obtained in the previous section. Solving equation (32) yields the natural frequencies of the studied structure.

Results and discussions
The current section is aimed to provide the results, in numerical and graphical form, to examine the vibrational behavior of the aforementioned sandwich shell under different conditions. The results will be reliable when the main code becomes validated with previous papers in this context. In this paper, the natural frequencies (in Hz) are listed for a single-layer orthotropic cylindrical shell with the following mechanical and geometrical specifications: The cylindrical panel's different mode numbers' frequencies are obtained and compared with refs. [56,76] in Table 1. The results of reference papers are similar to the results acquired by the current code. Some partial errors may be in response to different displacement field utilizations, and also, different solution methods.
As another attempt to validate the results, the outcomes of this study are compared to those of Razavi et al. [27] and Ke et al. [26]. In these studies, the dimensionless natural frequency is achieved via ( − )/ ωR ρ ν E 1 2 in which L/R = 20, h/R = 0.01, and ν = 0.3.
Therefore, the validity of the proposed method is confirmed. The vibrational behavior of the current sandwich cylindrical shell in response to different variable changes will be addressed in the following.
As previously mentioned, FG porous core is made of limestone, while the faces are fabricated from the GNPreinforced epoxy. It is noted that for the limestone, E 0 = 25 GPa, ρ 0 = 2,700 kg/m 3 , α 0 = 1.23 × 10 −5 1/K, β 0 = 2.43 × 10 −5 1/K, and ν = 0.25 [77]. Furthermore, the properties of GNPs reinforcement and the epoxy matrix are listed below Also, it must be noted that all the following results, except otherwise mentioned, are obtained with a porosity coefficient of 0.2 and for uniform GNPs dispersion and even pores distribution patterns ( Table 2). Figure 2 depicts natural frequency variations of the sandwich shell versus the porosity index. As it is evident, the higher porosity coefficients led to lower rigidity. As another expression and to provide a more physical point of view, it should be noted that higher values of porosity coefficients mean more holes and free spaces inside the core and cylindrical structures which is responsible for the lower stiffness and natural frequency, and higher flexibility. According to Figure 2, for the constant thickness of the sandwich cylindrical model, mid-radius enhancement decreased the natural frequency. Figure 3 illustrates the variation range of natural frequency versus length to thickness ratio (L/h) of the cylindrical sandwich shell. An increase in the mentioned ratio for constant total shell thickness implied an enhancement in the model length which can lead to the reduction of natural frequency and stiffness. Figure 3 also offers valuable information about the relationship of the porosity distribution patterns with the natural frequency of sandwich shell. It is clear that uniform and uneven distributions of porosity provide the lowest natural frequency and stiffness, respectively, due to pores' placement and its impact on the stiffness of the structure. The effects of W GNP and h c /h f on natural frequency are displayed in Figure 4. At constant total thickness, an increase in the core thickness (i.e., a decline in the face sheets thickness) decremented the natural frequency and rigidity. Importantly, this reduction got steadier for higher values of porous core   thickness. In another conclusion, at constant GNPs length, higher values of W GNP led to less rigid cylindrical sandwich shell structure. Moreover, Figure 5 presents the effect of different GNPs dispersion on natural frequency. Based on this figure, for similar GNPs dispersion coefficients, the parabolic dispersion of GNPs offered a stiffer structure. It means the highest natural frequency can be observed in the structures involving parabolic GNPs dispersion. Furthermore, comparing the epoxy curvature highlights the importance of GNPs presence as the reinforcing phase within the epoxy matrix. Figure 6 depicts natural frequency versus mid-radius to thickness ratio (R/h) of the shell for different temperature gradients. At the constant thickness of the sandwich cylindrical model, mid-radius enhancement decreased the natural     frequency. Such effect is visible for temperature difference enhancement and natural frequency, and consequently, stiffness reduction occurred due to the higher temperatures. The physical point of view is tangible when such structure initiates to melt due to the higher values of temperature differences. Natural frequency variation against humidity difference of surroundings is plotted in Figure 7 to illustrate the decline in the natural frequency, stiffness, and stability due to the humidity difference enhancement. The effect of different levels of moisture in the environment is negligible on the natural frequency of the whole cylindrical sandwich structure, but it can cause a crucial difference in the design of sensitive devices. Figure 8 presents the natural frequency versus GNPs geometrical dimensions. At constant GNPs     width, natural frequency and stability increased by increasing the GNPs' length. Tables 3 and 4 are presented to provide a complete understanding of the effects of the mode number, pores dispersion, GNPs dispersion, and geometrical shape of the sandwich shell on natural frequency. For each condition, mode numbers of (2,2) and (1,1) led to the highest and lowest natural frequencies, respectively. Moreover, even dispersion of pores and parabolic dispersion of GNPs caused the highest magnitudes of natural frequency, stiffness, and stability for each mode. Table 3 is listed for R = 20 h and uniform GNPs dispersion pattern, while Table 4 is for uniform porosity distribution. Figure 9 investigates the influence of porous core thickness and rotation angle alternations on the cylindrical sandwich structure frequency. The whole structure thickness is kept constant and equal to 1 cm. Figure 10 shows the dependence of the natural frequency on the rotation angle of the cylindrical sandwich shell. As it is clear, the natural frequency and stability of the model deal with the rotation angle in the direct direction. As another expression, allocating lower values to the rotation angle led to lower natural frequency and instability for any radius to thickness ratio. By comparing Figures 9 and 10 what is novel in Figure 9 is the decline in the natural frequency due to the porous core thickness enhancement which resulted in stability reduction as well. Finally, in the last figure of this section, a 3-D figure is provided in Figure 11 to generate a comprehensive analysis of natural frequency variation versus foundation parameters. The stiffness and rigidity of the system decreased due to a reduction in the spring and shear layer constants.

Conclusion
SSDST, Navier's methods, and Hamilton's principle are used to derive and solve the governing equations of motion to assess the vibrational responses of a cylindrical sandwich shell including FG porous core and two FG-GNPRC face sheets. Different figures and tables are presented to address the physical properties such as stiffness, flexibility, and rigidity. It is observed that an enhancement in the GNPs' length and a decline in the porosity coefficient, humidity, and temperature differences can increment the natural frequency. The lowest natural frequency is observed in non-reinforced samples. At the constant shell thickness, lower natural frequencies, and consequently, stability are observed in higher shell radii. On the other hand, the rotation angle of the shell has a direct relationship with stiffness. These changes in the stiffness get intensified at the lower values of the shell radius and rotation angle of the shell. Based on these results, this paper tried to use the proposed cylindrical sandwich shell as a model for developing lighter and stiffer structures to use in the corresponding industries. Scholar Priority Discipline Talent Group program funded by the Shan Dong Province, and the first-class discipline project funded by the Education Department of Shandong Province.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.