Large amplitude vibration of doubly curved FG-GRC laminated panels in thermal environments

1 Introduction

Doubly curved panels have many important engineering applications. During their service life, these panels may be subjected to different combinations of loading and environmental conditions which can cause the panels to experience large amplitude vibration with the panel deflection being in the order of the panel thickness. Therefore, it is necessary to fully understand the nonlinear vibration behaviors of doubly curved panels in thermal environments in engineering design and practice. Many studies have been carried out on the nonlinear vibration behavior of isotropic and composite laminated curved panels [1, 2, 3, 4, 5, 6, 7, 8]. It is observed that an acceptable agreement for flat plates can be achieved by different researchers. However, the results for a curved panel exist large discrepancies which may be due to the hardening behavior (i.e. increase in vibration amplitude leads to the increase of the nonlinear frequency) or the softening behavior of the panel (i.e. increase in vibration amplitude leads to the decrease of the nonlinear frequency) [1, 2, 3, 4, 5, 6, 7, 8].

Advanced composite materials possess unique features that many of the conventional materials do not have. We have witnessed the increased use of advanced composite materials, including the functionally graded material (FGM) [9], as key structural members in various engineering applications. Shen [10] first proposed to use FGM concept to nanocomposite structures in order to fully utilize the effect of nano filler reinforcement in composite structures. Shen and his co-authors and other research teams further studied the linear and nonlinear vibration characteristics of FGM curved panels [11, 12, 13, 14, 15, 16, 17] and FG carbon nanotube reinforced composite (CNTRC) curved panels [18, 19, 20, 21, 22] subject to temperature changes and/ or resting on elastic foundations. Since the discovery of graphene by Geim and Novoselov in 2004 [23], extensive studies on graphene have been conducted by many researchers and the extraordinary material properties of graphene have been widely reported [24, 25, 26, 27, 28]. Due to these remarkable properties, graphene has become one of the ideal reinforcement agents in creating advanced polymer composites [29]. For graphene-based nanocomposites, one kind

is graphene platelet reinforced composite (GPLRC) where both the polymer matrix and graphene platelets (GPLs) are assumed to be isotropic and independent of temperature. In essence, GPL reinforced composites belong to particle reinforced composites. The GPLRC model is relatively simple and was adopted by many researchers, for example, the vibration analysis for doubly-curved panels reinforced by GPLs was reported by Wang et al. [30, 31] and Fazelzadeh et al. [32]. It is noted that graphene sheets have anisotropic and temperature dependent material properties [24, 25, 26, 27, 28] and it is possible to align graphene sheets in polymer matrix that can result in better reinforcement effect for the graphene-based composites [33, 34, 35]. Shen et al. [36] first proposed a functionally graded graphene reinforced composite (GRC) model where aligned graphene reinforcements are anisotropic and the material properties of both the polymer matrix and graphene sheets are assumed to be temperature dependent. As reported by Lei et al. [37] the GRC model is more accurate than GPLRC model and was adopted by many researchers [38, 39, 40, 41, 42, 43].

This paper will investigate the nonlinear free vibration behavior of doubly curved GRC laminated panels resting on elastic foundations in thermal environments. The panels with the piece-wise functionally graded GRC laminar layer pattern are considered in the study. The novelty of this study lies in the account of both the functionally graded material configurations and the temperature dependent properties in the nonlinear vibration analyses of FG-GRC laminated doubly curved panels. The extended Halpin-Tsai micromechanical model is applied to estimate the material properties of the GRC layers. The Reddy’s third order shear deformation shell theory is employed to derive the motion equations for the GRC laminated panels. Note that the motion equations of the panels also include the effects of the von Kármán geometric nonlinearity, the foundation support and the temperature variation. The boundary conditions of the panels are assumed to be simply supported. A two-step perturbation approach is employed to determine the nonlinear frequencies of doubly curved GRC laminated panels. The large amplitude vibration behavior of doubly curved FG-GRC laminated panels subject to the influence of foundation support and temperature variation is discussed in detail.

2 Large amplitude vibration of doubly curved GRC laminated panels

A doubly curved GRC laminated panel with two radii of curvature R₁ and R₂, as shown in Figure 1, is considered in this study. The panel is made of N laminated GRC layers of different graphene volume fractions to form five different functionally graded (FG) patterns, i.e. UD, FG-Λ, FG-V, FG-X and FG-O. The panel with the UD pattern consists of GRC layers of the same graphene volume fraction. The panel with the FG-O pattern has the maximum graphene volume fraction in the GRC layers at the mid-plane and the minimumgraphene volume fraction at the top and bottom GRC layers with a step change of graphene volume fractions between the surface and the mid-plane GRC layers, while the panel with the FG-X pattern has an inverse graphene volume fraction arrangement as that of the FG-O pattern. The FG-V panel consists of graphene-rich top GRC layers and graphene-poor bottom GRC layer, while the FG-Δ panel has an inverse graphene volume fraction arrangement as that of the FG-V panel. The panel has length a, width b and thickness h and is located in a coordinate system (X, Y, Z) as shown in Figure 1. Note that X and Y are in the directions of the curvature lines on the middle surface of the panel and Z is in the direction of the inward normal to the middle surface of the panel. The panel is supported by a Pasternak-type foundation with the panel-foundation pressure being defined by p₀ = K̄₁W̄ − K̄₂∇²W̄, where W̄ is the displacement of the panel in the Z direction, K̄₁ is the transverse foundation stiffness (Winkler stiffness) and K̄₂ is the shearing layer stiffness of the foundation, and ∇² = ∂²/∂X² + ∂²/∂Y² is the Laplace operator.

Figure 1

Coordinate system and geometry for a doubly curved panel supported by a Pasternak-type elastic foundation

One of the key issues in structural analysis of graphene reinforced composites is the thermomechanical property evaluation of the composite. The Halpin-Tsai micromechanical model [44] is employed to estimate the effective material properties of the GRC layers in this study as we assume that the graphene sheets are aligned in the polymer matrix to form aligned 2D reinforcement agents. Due to incomplete stress transfer between graphene sheets and polymer matrix resulting from surface effect, strain gradients effect and intermolecular effect, the Halpin-Tsai model needs to be modified to account for these effects [45]. In the present study, the graphene reinforcement is either zigzag (refer to as 0-ply) or armchair (refer to as 90-ply). Based on the extended Halpin-Tsai model, the effective Young’s moduli and the shear modulus of the GRC layer can be expressed as [36]

in which a_G, b_G and h_G are the length, the width and the effective thickness of the graphene sheet, and

where E^m and G^m are the elasticity modulus and shear modulus of the polymer matrix. Besides, E11G,E22Gand G12Gindicate the elasticity moduli and shear modulus of graphene sheet. We can see that the only difference between Eq. (1) and the conventional Halpin-Tsai model is the presence of efficiency parameters η_j (j=1,2,3). These parameters are obtained by matching the data which are evaluated by MD simulations [46] and the Halpin-Tsai model. In Eq. (1), V_G and V_m are the volume fractions of graphene and matrix, which should satisfy the partition of unity condition V_G + V_m = 1.

Poisson’s ratio v₁₂ and the mass density ρ of the GRC layer may easily be expressed according to the conventional rule of mixtures

where ν12G, ρ^G and v^m, ρ^m are the Poisson’s ratios and mass densities of the graphene and matrix, respectively. They are assumed to be weakly dependent on temperature variation.

Note that the material properties of the GRC layers estimated in Eq. (1) are temperature dependent as the material properties of the graphene sheets and the polymer matrix are both temperature dependent. According to Schapery model [47], the thermal expansion coefficients of the GRC layers can be expressed by

in which α₁₁ and α₂₂ are the longitudinal and transverse thermal expansion coefficients of the GRC layers, α11G,α22Gand α^m are thermal expansion coefficients, respectively, of the graphene and matrix.

The doubly curved GRC laminated panel is subjected to a transverse dynamic load q(X, Y, t̄) in a thermal environment. Within the framework of the Reddy’s third order shear deformation shell theory [48] and considering the effects of the von Kármán geometric nonlinearity, the panel-foundation interaction and the temperature variation, we can derive the motion equations for the doubly curved panel as follows

in which

where a comma denotes partial differentiation with respect to the corresponding coordinates, and W̄ is the transverse displacement, Ψ¯xand Ψ¯yare the rotations of the normals to the middle surface with respect to the Y - and X - axes, F̄ is the stress function defined by N̄_x = ∂²F̄/∂Y², N̄_y = ∂²F̄/∂X² and N̄_xy = −∂²F̄/∂X∂Y.L~ij() and L~() are the linear and nonlinear operators as defined in Shen [49], and L~() contains the geometric nonlinearity terms in the von Kármán sense, and can be given by

The panel is assumed to be in a constant temperature field at an isothermal state. The terms associated with the superscript T in Eqs. (5)-(8) contain the effect of temperature variation, N̄^T are the thermal forces, M̄^T are the thermal moments and P̄^T are the higher order thermal moments. The effect of the panel-foundation interaction is included in the terms associated with K̄₁ and K̄₂ in Eq. (5). The terms I_j,Ȋ_j and Ĩ_j as well as N̄^T, M̄^T and P̄^T are given in detail in Appendix A. We can use Eqs. (5)-(8) to analyse the case for GRC laminated cylindrical panels by setting R₂ = R and R₁ =∝ and for GRC laminated square spherical panels by setting R₁ = R₂ = R.

Besides the governing equations (5)-(8), it is necessary to deal with different boundary conditions to solve the boundary-value problem. In the current study, the four curved edges of the panel are assumed to be simply supported, and the associate boundary conditions are

where M̄_x and M̄_y are the bending moments and P̄_x and P̄_y are the higher order moments as defined in Reddy and Liu [48].

Two in-plane boundary conditions, i.e. movable and immovable, are considered. For movable in-plane boundary conditions, one has

and for immovable in-plane boundary conditions, one has

where Ū and V̄ are the plate displacements in the X and Y directions.

The movable conditions of Eqs. (11c) and (11d) can be imposed in the average sense as

Also, the immovable conditions of Eqs. (11e) and (11f) are fulfilled in the average sense as

or

In the above equations, the reduced stiffness matrices [Aij∗],[Bij∗],[Dij∗],[Eij∗],[Fij∗]and[Hij∗]are defined in Appendix B.

It should be noticed that, in the current study, either governing equations (5)-(8) or boundary conditions (11a)-(11f) are different from that used in [50]. For nonlinear problems the superposition principle is no longer valid. Hence, each nonlinear boundary value problem with different governing equations or boundary conditions should be solved separately.

3 Solution procedure

The nonlinear vibrations of flat or cylindrical or doubly curved panels are different nonlinear problems. A two-step perturbation approach was developed by Shen [49] and was successfully to solve different kinds of nonlinear problems of beams, plates and shells by many research teams [51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. To use this two-step perturbation approach to solve the large amplitude vibration problem of doubly curved FG-GRC laminated panels, the motion equations (5) to (8) can be expressed in dimensionless forms as

with

and the other dimensionless linear operators L_ij( ) are given in Shen [49]. In Eqs. (15)-(19), the non-dimensional parameters are defined by

in which E₀ and 𝜌₀ are the material properties of the polymer matrix E^m and ^m at room temperature (T₀=300 K), and the terms AxT,DxT,FxT,etc. are defined by

Based on Eq. (20), the boundary conditions of Eqs. (11a) and (11b) can be written in non-dimensional forms as

and the movable in-plane boundary conditions of Eqs. (11c) and (11d) become

and the immovable in-plane boundary conditions of Eqs. (11e) and (11f) become

where ϒ_ijk are defined in Shen [49].

Equations (15) to (18) can be separated into two sets of differential equations which are then solved in sequence. The first set of differential equations is for the nonlinear thermal bending problem and can be solved using the same method as reported in [62], and the second set of differential equations are used to obtain the homogeneous vibration solution on the initial deflected panel. A two-step perturbation technique is applied to determine this homogeneous solution. We assume that the perturbation equations for the displacements and the forces with a small perturbation parameter ε which has no physical meaning in the first step are given by

We introduce τ = ε t to improve the perturbation solution process for solving the large amplitude vibration problem. In order to satisfy the simply supported boundary conditions in the space domain, the first order solution of the panel is assumed to have the form

where (m, n) is the number of waves of vibration mode in the X and Y directions. The initial conditions are assumed to be

Taking into consideration of Eq. (23), a set of perturbation equations are obtained by collecting the terms of the same order of ε in Eqs. (15)-(18). Eq. (24) is then applied as the first step solution to the perturbation equations and following a step by step approach, we can obtain the 4^th order asymptotic solutions as

It is worth noting that the perturbation series is a divergent series. Which order solution is closer to the real solution needs to be determined by experimental verification or by comparing with the theoretical exact solution. Contrary to Zhang’s conclusion [63], there is no such thing as a

higher order perturbation solution being more correct than a lower order solution.

In Eqs. (26)-(30), τ is replaced back by t. It is noted that the small perturbation parameter ε is replaced by (εA11(1))in the second step. For free vibration analysis, applying Galerkin procedure to Eq. (30), we have

in which the terms g_ij are given in Appendix C. Eq. (31) can be solved to obtain the nonlinear frequency of the panel as follows

where ω_L = [g₃₁/g₃₀]^1/2 is the dimensionless linear frequency, and A=Wmax=W¯max/[D11∗D22∗A11∗A22∗]1/4is the dimensionless maximum amplitude of the panel. It is worth noting that Eqs. (31) and (32) are similar in form to those of the cylindrical panels [50], but have different contents, as shown in Appendix C.

4 Numerical results and discussion

In this section, numerical results for the nonlinear vibration of doubly curved GRC laminated panels resting on elastic foundations and under different thermal environmental conditions are obtained. It is noted that the material properties of graphene sheets are anisotropic [24, 25, 26] and temperature dependent [27]. We select zigzag (refer to as 0-ply) graphene sheets with effective thickness h_G = 0.188 nm and ρ_G = 4118 kg/m³ as reinforcement agents. Lin et al. [46] performed a molecular dynamics simulation to evaluate the material properties of graphene sheets at different temperatures. These material properties at three different temperature levels are provided in Table 1. It must be pointed out that the Young’s modulus of single-layer graphene sheet is not a constant which depends on the value of its effective thickness [64]. For example, it has been reported that Young’s modulus of single-layer graphene sheet is estimated to be about 1 TPa. This is due to the fact that the effective thickness of graphene sheet is taken to be 0.34 nm [65, 66], otherwise the Young’s modulus may reach 2.47 TPa which is significantly larger than that reported in [65, 66], when the effective thickness of graphene sheet is only 0.129 nm [27]. In the MD work of Lin et al. [46], the calculated effective thickness of the graphene sheet is 0.188 nm according to the research findings of Shen et al. [27]. Therefore, the predicted Young’s moduli of the graphene sheet (see in Table 1) reach around 1.8 TPa. The graphene efficiency parameters η₁, η₂ and η₃ used in the extended Halpin-Tsai micromechanical model are given in Table 2 which are obtained by comparing the GRC moduli from the MD simulations and from the Halpin– Tsai model, as previously reported in Shen et al. [36]. We assume that G₁₃ = G₂₃ = 0.5G₁₂. We select Poly (methyl methacrylate), referred to as PMMA, for the matrix. The material properties of PMMA are assumed to be ρ^m = 1150 kg/m³, v^m = 0.34, α^m = 45(1+0.0005ΔT)×10⁻⁶/K and E^m = (3.52-0.0034T) GPa, inwhich T = T₀ + and T₀ = 300 K (room temperature). Hence, we have α^m = 45.0×10⁻⁶/K and E ^m = 2.5 GPa when T = 300 K.

Comparison studies are carried to verify the correctness of the present solution method. The fundamental frequencies from the present method and from Pouresmaeeli and Fazelzadeh [67] using the Galerkin method for CNTRC doubly curved panels with a/b = 1, a/h = 20 and a/R₁ = b/R₂ = 0.5 are presented in Table 3. Note that Pouresmaeeli and Fazelzadeh [67] employed the first order shear deformation theory with the shear correction factor of 5/6 in their analysis. The dimensionless frequency is defined by Ω~=Ω(a2/h)ρ0/E0, with ρ₀ and E₀ being the reference values of PmPV at room temperature T = 300 K. In Table 3, the extended Voigt model (rule of mixture) is adopted and the CNT efficiency parameters are taken to be η₁ = 0.149, η₂ = η₃ = 0.934 for the case of = 0.11, and η₁ = 0.150, η₂ = η₃ = 0.941 for the case of VCN∗= 0.14, and η₁ = 0.149, η₂ = η₃ = 1.381 for the case of VCN∗= 0.17. It can be seen that for the UD, FG-∧, FG-V and FG-X cases the results of Pouresmaeeli and Fazelzadeh [67] are lower than the present solutions whereas for the FG-O case the results of Pouresmaeeli and Fazelzadeh [67] are higher than the present solutions. As a second example, the dimensionless fundamental frequencies for Al/ZrO₂ doubly curved panels, which have ceramic-rich outer surface and metal-rich inner surface, are calculated and compared in Table 4 with the analytical hybrid Laplace–Fourier transformation results of Kiani et al. [11] and the isoparametric finite element approach results of Kar and Panda [13]. The conventional Voigt model was adopted by Kiani et al. [11] and Kar and Panda [13]. The value of N in Table 4 is the index of volume fraction. The material properties of the panels do not include the effect of temperature with the properties of Aluminum being E_m = 70 GPa, ν_m = 0.3 and _m = 2702 kg/m³ and Zirconia being E_c = 151 GPa, ν_c = 0.3, and _c = 3000 kg/m³. The third comparison study is presented in Figure 2 on the nonlinear-to-linear frequency ratios ω_NL/ω_L for a (0/90/0) laminated square spherical panel from the present method and from the FEM results of Singh and Panda [4] based on a higher order shear deformation theory. The panel is of movable in-plane boundary condition. The geometric parameters and the material properties used in the computation study are: a/b = 1, b/h = 100, R₁/a = 5, R₂/b = 5, E₁₁ = 40E₂₂, G₂₃ = 0.5E₂₂, G₁₂ = G₁₃ = 0.6E₂₂, ν₁₂ = 0.25. Only the vibration mode of (m, n) = (1, 1) is considered. The three comparison studies have shown that good agreement is achieved between the results from the present solution method and from existing research work in the open literature.

Figure 2

Comparisons of frequency-amplitude curves of a (0/90/0) laminated square spherical panel

Tables 5-7 and Figures 3-7 present the numerical results for the large amplitude vibration of doubly curved GRC laminated panels with h = 2 mm, a/b = 1.0, b/h = 10 and 20. Note that GRCs may contain the volume fraction of graphene reinforcement by up to 21%[68] and in this study the maximum graphene volume fraction is 11%. Four FG and one UD GRC laminated doubly curved panels are considered. The UD GRC panel consists of 10 GRC layers with the graphene volume fraction for each layer being identical, i.e.V_G = 0.07. The four FG-GRC panels consist of 10 GRC layers with piece-wise graphene volume fractions, i.e. the FG-∧ type with [(0.03)₂/(0.05)₂/(0.07)₂/(0.09)₂/(0.11)₂], the FG-V with [(0.11)₂/(0.09)₂/(0.07)₂/(0.05)₂/(0.03)₂], the FG-O type with [0.03/0.05/0.07/0.09/0.11]_S and the FG-X type with [0.11/0.09/0.07/0.05/0.03]_S. Note that the total graphene volume fraction for all considered GRC laminated panels are the same. The non-dimensional frequency parameter Ω~=Ωb2/hρ0/E0is used in this study, where ρ₀ and E₀ are the reference values of ^m and E^m at T = 300 K. The panels are simply supported with immovable in-plane condition, unless stated otherwise.