Nonlinear Vibration Analysis of Functionally Graded Porous Plates Reinforced by Graphene Platelets on Nonlinear Elastic Foundations

This paper presents a nonlinear vibration analysis of functionally graded graphene platelet (GPL) reinforced plates on nonlinear elastic foundations. Uniformly or non-uniformly distributed internal pores were present in the plates. Based on the modified Halpin-Tsai micromechanics model and the extended rule of mixture, the material properties were evaluated. The governing equations, coupled with the effect of nonlinear foundations, were derived by using the higher-order plate theory and general von Kármán-type equations. A two-step perturbation technique was employed to obtain the nonlinear frequency and transient response. After the present method was verified, the effects of pores, GPLs, and elastic foundations were investigated in detail. A new finding is that the influence of the porosity coefficient on the natural frequency and dynamic response is relevant to foundation parameters. Moreover, the influence of the nonlinear foundation parameter can be negligible. effects of pores and nonlinear elastic foundations are drawn.


INTRODUCTION
Due to the excellent load-carrying capacity with stronger bonding between the matrix and carbonaceous nanofillers [1] and [2], nano graphene platelets (GPL) are now increasingly used in many engineering fields, including aerospace, automobile, and civil engineering. It is necessary to study the dynamic characteristics of GPL-reinforced composite structures.
In recent years, few results about the dynamic behaviour for functionally graded (FG) plates reinforced with GPL have been reported [3]. Song et al. [4] employed the first-order shear deformation plate theory to study the free and forced vibrations of FG multilayer GPL/polymer plates. Their results illustrated that a small amount of GPL can lead to higher natural frequency and lower dynamic response. By using the finite element method, Zhao et al. [5] studied investigated the bending and free vibration behaviours of FG trapezoidal plates. Their results also illustrated that the transient deflection was decreased by using a small number of graphene platelets. Gholami and Ansari [6] investigated the nonlinear harmonically excited vibration of FG graphenereinforced composite plates. They found that the nonlinear buckling was increased with the rise of GPLs weight fraction. Moreover, Wu et al. [7] studied the dynamic stability of FG nanocomposite plates subjected to thermal and mechanical loads.
Recently, a novel kind of graphene-reinforced nanocomposite was made by dispersing GPL into metal foams [8] to [10]. Due to the excellent mechanical and thermal properties, the novel composite materials have attracted some attention. Kittipornchai et al. [11] proposed a micromechanical model to estimate the typical properties of FG graphene reinforced nanocomposite, in which both GPL and internal pores are uniformly dispersed within each layer. Later, the model was employed to study the dynamic behaviour of graphene-reinforced nano-composite plates [12] and [13]. In these papers, the effect of internal pores has been discussed in detail, and the results showed that both pore volume fraction and the distribution of pores can affect the dynamic characteristics of porous structures.
In practical engineering, plates are sometimes rested on elastic foundations. Thus, a suitable model is needed to evaluate foundation interaction. The simplest is the Winkler model, in which a series of springs are used to calculate the tensile and compressed forces of elastic foundations. This model was later developed into the Pasternak model, in which a shear spring is added to estimate the shear forces of the Winkler foundations. Thereafter, the Pasternak model has been widely used to study the static and dynamic behaviours of plates on elastic foundations [14] to [16]. To estimate the nonlinear interaction between foundations and plates, some researchers recently presented a nonlinear foundation model. For example, Nath et al. [17], Civalek [18] and Najafi et al. [19] studied the nonlinear dynamic behaviour of composite plates by using a nonlinear foundation model. Their results revealed that the nonlinear foundation parameter had a distinct influence on the dynamic characteristics.
As mentioned above, the number of studies focused on the dynamic behaviour of FG porous plates reinforced with GPL is still rather scarce. According to the authors' knowledge, no previous work has been done to study the nonlinear dynamic behaviour of FG graphene platelets reinforced porous plates on nonlinear elastic foundations. Hence, this paper attempts to study the nonlinear vibration of the plates. A modified material properties model is proposed, and the effects of internal pores and a nonlinear elastic foundation are discussed.

A POROUS NANO-COMPOSITE PLATE
As depicted in Fig. 1, a functionally graded graphenereinforced porous plate (length a, width b, thickness h) on a nonlinear elastic foundation is taken into account. The origin of the coordinate system (X, Y, Z) is located at one corner of the middle plane of the plate. The Z-axis is perpendicular to the X-Y plane and points upwards. Three types of porosity distributions are considered (Fig. 2). "P-1" indicates that the largest-size pores are distributed in the middle. In contrast, "P-2" indicates that the largest-size pores are on the bottom and top surfaces. The symbol of even distribution is "P-3".
Unlike the material properties model presented by Kittipornchai et al. [11], the present model is based on the volume fraction of pores, which are assumed to be: (1) In the above equation, e 0 , e 0 * (0 ≤ e 0 (e 0 * ) < 1 and α denote the porosity coefficients for P -1, P -2, and P -3 distributions, respectively.
Yang's elastic modulus E(Z) and shear elastic modulus G(Z) of the porous plate can be expressed by using the rule of mixture: where E 0 and G 0 are the corresponding variations of the graphene-reinforced nanocomposites without internal pores. Because the mass density ρ(Z) cannot be calculated by using the rule of mixture [11], it is assumed to be where E m and V GPL are Young's elastic modulus of matrix and GPL volume fraction, respectively. The coefficients ξ L , ξ W , η L and η W are defined by in which E GPL , a GPL , b GPL and h GPL are Young's modulus, average length, width, and thickness of graphene platelets, respectively.
Based on the rule of mixture, Poisson's ratio ν, and mass density ρ 0 can be calculated as where ρ GPL is the mass density of GPL, and ρ m and ν m are the mass density and Poisson's ratio of the matrix. The shear modulus G 0 can be obtained by As shown in Fig. 3, three types of GPL dispersion where ν i1 , ν i2 and ν i3 are the maximum value of V GPL , i (i = 1, 2, 3) indicate the three types of porosity distributions. ν i1 , ν i2 and ν i3 can be reckoned as follows: in which V GPL T and W GPL are the total volume and weight fractions of GPL, respectively.

Governing Equations
As noted by Civalek [18], the effect of nonlinear platefoundation interaction on the dynamic response of plates on elastic foundations must not be neglected. Therefore, the following three-parameter nonlinear foundation model is adopted: in which U , V , W , ψ 1 and ψ 2 are the displacements and rotations of a point (X, Y) on the mid-plane.
The von Karman strains associated with the displacement field in Eq. (14) can be stated as Based on Hook's law, the relationship between stresses and strains can be expressed as . (17) The in-plane forces N i , bending moment M i , higher-order bending moment P i , shear forces Q i , and higher-order shear forces R i are By using the Hamilton principle, the equations of motion for the plate can be derived as In Eq. (19), the constants I j and I j were given by Reddy [20]. The superposed dots indicate the differentiation with respect to time.
The strain compatibility equation is The in-plane forces N i can be expressed by stress function F X Y t ( , , ) : By substituting Eqs. (18) and (21) into Eqs. (19) and (20), the governing equations of nonlinear vibration for the plate can be derived as follows: where the constants I j ( j = 8, 9, 10), linear operators l ij and nonlinear operator l were given by Shen [21] and Huang and Zheng [22].

Solution Procedure
To solve the nonlinear equations, Eqs. (22) and (23), we first introduce the following dimensionless parameters: where the stiffness constants A ij , B ij , and D ij are defined in the standard way [20].
The dimensionless form of the equation (22) where the constants γ i (i = 1, 2, ..., 5), the dimensionless operators l ij and l can be seen in the previous work [22]. The dimensionless form of the boundary conditions in Eq. (23) are also rewritten as: The nonlinear ordinary equation, Eq. (29) can be solved by using the Runge-Kutta iteration Scheme [23]. For the free vibration problem (λ q (τ) = 0), the approximate nonlinear frequency can be derived as / .

RESULTS AND DISCUSSION
In the section, the several dimensionless parameters are used as follows:  [13]. The maximum error is about 2.3 %. This is because Gao et al. [13] employed the classic plate theory and differential quadrature method to calculate the fundamental frequencies, which is different from the present method. Example 2. The dynamic response of a FG GPL reinforced plate is investigated in this example. The dispersion pattern of GPL is G -2. The plate is subjected to the explore load:

Parametric Studies
In what follows, the effects of material properties and foundation parameters are investigated in detail. The material and geometric parameters are the same as those in Example 1. Unless specially stated, the weight fraction W GPL and pore coefficient e 0 are 5 % and 0.2. The dimensionless foundation parameters (k 1 , k 2 , k 3 ) are (50, 50, 50).
Tables 2 to 4 list the fundamental natural frequencies of the plate with different GPL dispersion pattern, porosity distribution, weight fractions W GPL , porosity coefficients e 0 and foundation parameters (k 1 , k 2 ). It can be seen that the natural frequency is increased with the rising weight fraction W GPL and foundation parameters (k 1 , k 2 ). The frequency for G -1 is higher than those for G -2 and G -3. This illustrates that G -1 can strengthen the plate more effectively than the other two patterns. Also, it can be seen that the frequency for P -1 is lower than those for P -2 and P -3. The fact demonstrated that P -1 can weaken the plate more seriously. In past studies [11] to [13], a conclusion was drawn that the natural frequency monotonously decreases with the increasing porosity coefficient. However, in the present case, the conclusion was correct only on P -2. On P -1 and P -3, the effect of the porosity coefficient     is relevant to the values of foundation parameters (k 1 , k 2 ). If foundation parameters are (50, 0) or (0, 0), the natural frequency is decreased. In contrast, the natural frequency for (50, 50) is increased. Figs. 5 to 8 reveal the influences of GPL dispersion pattern, porosity distribution, GPL weight fraction W GPL , and pore coefficient e 0 on the nonlinear to linear frequency ratio ω NL / ω L . As can be observed, the frequency ratio for G -2 / P -2 is higher than those for other GPL and pore distributions. The frequency ratio is decreased as pore coefficient e 0 rises but rose with the increase of GPL fraction W GPL . that the maximum dynamic deflection for G -2 / P -2 is the largest among all patterns and distributions. In contrast, the amplitude of the dynamic bending moment for G -2 / P -2 is the smallest. The effects of GPL weight fraction W GPL and pore coefficient e 0 on the transient responses is illustrated in Figs. 13 and 14. It is discerned that the rise of W GPL reduces the amplitude of transient deflection, but increases the amplitude of bending moment. The maximum deflection increases by about 8 % as the porosity coefficient e 0 rises from 0.0 to 0.4. Therefore, a conclusion may be made that the effect of the porosity coefficient on the dynamic response can be negligible. The effect of foundation parameters (k 1 , k 2 , k 3 ) on dynamic responses is presented in Fig. 15. As expected, Winkler and Pasternak elastic foundation parameters reduce the dynamic responses. The dynamic response for Pasternak elastic foundations (50, 50, 0) is very close to those for the nonlinear elastic foundations (50, 50, 50). Hence, a conclusion may be drawn that the effect of nonlinear foundation parameter k 3 on dynamic responses may be neglected, which is different from the statement mentioned by