1 Introduction

Plates having a sandwich construction with low strength core and high strength laminated face sheets provide unique solutions in many engineering structures,where weight minimization is a major challenge. Structural components made with composite materials,e.g.,beam,plates, and shells,are used in many engineering applications because of their high stiffness to weight ratios and high modulus to weight ratios. Understanding their true dynamic and static behavior is of increasing importance.

The problem of buckling and vibration of thick plates has attracted considerable attention in recent years. Akhavan et al.^{[1, 2]} introduced exact solutions for the buckling analysis of rectangular Mindlin plates subjected to uniformly and linearly distributed in-plane loading on two opposite edges simply supported resting on elastic foundation. Flexural stability of a homogeneous plate compressed in its plane and lying on an elastic foundation was studied by Morozov and Tovstik^[3]. The governing equations of elasticity theory for natural vibration and buckling of anisotropic plate were derived by Lu and Li^[4] from Hellinger-Reissner’s variational principle with nonlinear strain-displacement relations.

None of the above researchers have considered laminated structures. Reddy^[5] studied the effect of transverse shear deformation on deflection and stresses of laminated composite plates subjected to a uniformly distributed load using finite element analysis. A method based on newly presented state space formulations was developed by Ding et al.^[6] for analyzing the bending,vibration,and stability of laminated transversely isotropic rectangular plates with simply supported edges. Analysis of composite plates using a higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method was presented by Ferreira et al.^[7]. Swaminathan and Ragounadin^[8] applied an analytical solution for static analyzing of antisymmetric angle-ply composite and sandwich plates. An investigation on the nonlinear bending of simply supported,functionally graded nanocomposite plates reinforced by single walled carbon nanotubes (SWCNTs) subjected to a transverse uniform or sinusoidal load in thermal environments was investigated by Shen^[9]. Heydari et al.^[10] presented an exact solution for transverse bending analysis of embedded laminated Mindlin plate.

None of the above studies have considered size effects in mechanical analysis of structures. In micro scale,small size effects are important. The nonlocal elasticity theory was initiated in the papers of Eringen^[11]. He regarded the stress state at a given point as a function of the strain states of all points in the body,while the local continuum mechanics assumes that the stress state at a given point depends uniquely on the strain state at the same point. Thermal buckling characteristics of rectangular flexural microplates subjected to a uniform temperature were investigated by Farahmand et al.^[12] using a higher continuity p-version finite element framework. Ahmadi et al.^[13] investigated elastic buckling of rectangular flexural micro plates using a higher continuity p-version finite-element framework based on the Galerkin formulation. Electro-thermo nonlinear vibration of a piezo-polymeric rectangular micro plate made from polyvinylidene fluoride (PVDF) reinforced by zigzag double walled boron nitride nanotubes (DWBNNTs) was studied by Ghorbanpour et al.^[14]. Nonlinear free vibration of micro-plates was studied by Ramezani^[15] based on a strain gradient elasticity theory. Nonlinear vibration and instability of a boron nitride micro-tube (BNMT) conveying ferrofluid under the combined magnetic and electric fields were investigated by Ghorbanpour et al.^[16]. In another work by Ghorbanpour et al.^[17],nonlinear vibration and instability analysis of a bonded double-smart composite microplate system (DSCMPS) conveying microflow based on nonlocal piezoelasticity theory were presented.

In the present study,the nonlocal nonlinear vibration behavior of laminated micro/nanoplates resting on orthotropic Pasternak medium is investigated. The nonlinear governing equations are obtained based on the Hamilton principal along with the orthotropic Mindlin plate theory. The differential quadrature method (DQM) is applied for nonlinear frequency of the laminated micro-plate. The effects of the elastic medium,aspect ratio,vibrational modes,and boundary conditions on the frequency of the system are discussed in detail.

2 Nonlinear Mindlin plate theory

As shown in Fig. 1,a laminated plate with length L,width b,and thickness h is considered. The laminated plate is surrounded by an orthotropic Pasternak medium,which is simulated by K_w G_ξ ,and G_η corresponding to the Winkler foundation parameter and shear foundation parameters in the ξ- and η-directions,respectively.

Based on the Mindlin plate theory,the displacement field can be expressed as follows^[10]:

where (u_x,u_x,u_z) denote the displacement components at an arbitrary point (x,y,z) in the plate,and (u,v,w) are the displacements of a material point at (x,y) on the mid-plane (i.e., z = 0) of the plate along the x-,y-,and z-directions,respectively; ψ_x(x,y) and ψ_y(x,y) are the rotations of the normal to the mid-plane about the x- and y-directions,respectively. The von K´arm´an strains associated with the above displacement field can be expressed in the following forms: where (ε_xx,ε_yy) are the normal strain components,and (γ_yz,γ_xz,γ_xy) are the shear strain components.

3 Orthotropic stress-strain relations

In Eringen’s nonlocal elasticity model,the stress state at a reference point in the body is regarded to be dependent not only on the strain state at this point but also on the strain states at all of the points throughout the body. The basic equations for homogeneous,isotropic,and nonlocal elastic solid with zero body forces are given by^[11]

where C_ijkl is the elastic module tensor of classical (local) isotropic elasticity; σ_ij and ε_ij are the stress and strain tensors,respectively,and u_i is the displacement vector. α(|x − x'| ,τ) is the nonlocal modulus. |x − x'| is the Euclidean distance,and τ =e₀a/l is defined that l is the external characteristic length,e₀ denotes a constant appropriate to each material,and a is an internal characteristic length of the material (e.g.,length of C-C bond,lattice spacing, granular distance). Consequently,e₀a is a constant parameter which is obtained with molecular dynamics,experimental results,experimental studies,and molecular structure mechanics. The constitutive equation of the nonlocal elasticity can be written as where the parameter e₀a denotes the small scale effect on the response of structures in nanosize, and ∇² is the Laplace operator in the above equation. The constitutive equation for stress σ and strain ε matrices in thermal environments may be written as follows: where ¯Q_ij denotes transformed elastic coefficients^[5]. 4 Energy method

The total potential energy V of the laminated plate is the sum of the strain energy U,the kinetic energy K,and the work done by the elastic medium W.

4.1 Potential energy

The strain energy can be written as

Combining of (2)-(6) and (10) yields

where the stress resultant-displacement relations can be written as where k' is the shear correction coefficient. 4.2 Kinetic energy

The kinetic energy can be written as

where ρ is the density of plate. 4.3 External work

The external work due to the orthotropic Pasternak medium can be written as

where P is related to orthotropic Pasternak medium which may be expressed as^[18] where the angle θ describes the local ξ direction of orthotropic foundation with respect to the global x-axis of the plate. 5 Governing equations

The governing equations can be derived by Hamilton’s principal as follows:

Substituting (11),(14),and (15) into (17) yields the following governing equations:

Substituting (9) and (2)-(7) into (12) and (13),the stress resultant-displacement relations can be obtained as follows: where where N is the total number of the laminated. Substituting (23) to (26) into (18) to (22) yields the governing equations as follows: δ_u : δ_v : δ_w : δψ_x : δψ_y : 6 DQM

In this method,the differential equations are changed into a first order algebraic equation by employing appropriate weighting coefficients. Because weighting coefficients do not relate to any special problem and only depend on the grid spacing. In other words,the partial derivatives of a function (say w here) are approximated with respect to specific variables (say x and y), at a discontinuous point in a defined domain (0 < x < Lx and 0 < y < Ly) as a set of linear weighting coefficients and the amount represented by the function itself at that point and other points throughout the domain. The approximation of the nth and mth derivative functions with respect to x and y,respectively,may be expressed in general form as follows^[10]

where Nx and Ny denote the numbers of points in the x- and y-directions,f(x,y) is the function, and Aik and Bjl are the weighting coefficients defined as where M and P are the Lagrangian operators defined as The weighting coefficients for the second,third and fourth derivatives are determined via matrix multiplication, Using the following rule,the distribution of grid points in domain is calculated as Finally,the governing equations in a matrix form can be expressed as where [KL] and [KNL] are,respectively,the linear and nonlinear coefficients which can be defined as follows:

The above nonlinear equation can now be solved using a direct iterative process as follows:

(i) First,nonlinearity is ignored by taking K_NL=0 to solve (37). This yields the linear deflection. The deflection is then scaled up.

(ii) Using linear deflection,[K_NL] could be evaluated. The problem is then solved by substituting [K_NL] into (37). This would give the nonlinear deflection.

(iii) The new nonlinear deflection is scaled up again,and the above procedure is repeated iteratively until the difference between deflection values from the two subsequent iterations becomes less than 0.01%.

7 Numerical results and discussion

A computer program is prepared for the numerical solution of nonlinear vibration of laminated-plates resting on an orthotropic Psaternak foundation. The plate is simply supported, and the thickness to length ratio of each composite ply is h/a=1/50. The laminate configuration includes three layers of Graphite/Epoxy (Gp/Ep) with fiber orientations of [0/90/0]. The elas-tic properties of Graphite/Epoxy are^[19] E₁₁=132.38GPa,E₂₂=10.76GPa,E₁₃₃₁=10.76GPa, G₁₂=3.61GPa,G₁₃=5.65GPa,G₂₃=5.65GPa,ν₁₁=0.24,ν₂₃=0.24,ν₁₃=0.49,and ρ= 1 587 kg/m³. The laminated plate is considered with three kinds of boundary conditions,all edges simply supported (SSSS) or clamped (CCCC) and two opposite edges simply supported and the other two clamped (SCSC)^[20].

The effect of the grid point number in DQM on the dimensionless nonlinear frequency (i.e., of the laminated plate is demonstrated in Fig. 2. As can be seen,fast rates of convergence of the method are quite evident,and it is found that 15 DQ grid points can yield accurate results.

Figure 3 illustrates the effect of the different boundary conditions on the dimensionless nonlinear frequency of the laminated micro-plate versus nonlocal parameter. As can be seen increasing the nonlocal parameter decreases the dimensionless nonlinear frequency. This is due to the fact that the increase of nonlocal parameter decreases the interaction force between microplate atoms,and that leads to a softer structure. It is also found that the CCCC boundary condition has higher dimensionless nonlinear frequency. This is due to the fact that the CCCC boundary condition leads to a harder structure. Meanwhile,the effect of boundary conditions becomes more considerable at lower nonlocal parameters.

The dimensionless nonlinear frequency of the laminated micro-plate versus nonlocal parameter-is demonstrated in Fig. 4 for different elastic media. In this figure,four cases are considered as follows: Case 1: indicating without elastomeric medium. Case 2: indicating Winkler medium. Case 3: indicating orthotropic Pasternak medium. Case 4: indicating Pasternak medium. As can be seen,considering elastic medium increases frequency of the laminated micro-plate. It is due to the fact that considering elastic medium leads to stiffer structure. Furthermore,the effect of the Pasternak-type is higher than the Winkler-type on the nonlinear frequency of the laminated micro-plate. It is perhaps due to the fact that the Winkler-type is capable to describe just normal load of the elastic medium while the Pasternak-type describes both transverse shear and normal loads of the elastic medium.

The effect of the slenderness ratio on dimensionless nonlinear frequency of the laminated micro-plate versus nonlocal parameter is depicted in Fig. 5. As can be seen,the dimensionless nonlinear frequency decreases with increasing slenderness ratio. It is because that increasing-slenderness ratio leads softer structure. Meanwhile,the effect of slenderness ratio on the vibration of the laminated micro-plate becomes more prominent at lower nonlocal parameters.

Figure 6 shows the dimensionless nonlinear frequency of the laminated micro-plate versus nonlocal parameter for different vibrational modes. The same as other figures,increasing the nonlocal parameter decreases the dimensionless nonlinear frequency of the laminated micro-plate. It can be also found that the dimensionless nonlinear frequency of the laminated microplate increases with increasing vibrational modes.

8 Conclusions

Based on the orthotropic Mindlin plate and Eringen’s theories,nonlinear vibration analysis of an embedded laminated micro/nano-plate is studied in this paper. The system is surrounded in an orthotropic Pasternak medium. Using the strain-displacement relation,energy method, and Hamilton’s principle,the governing equations are derived. In order to obtain the nonlinear frequency of the system,DQM is performed. The effects of the elastic medium,aspect ratio, vibrational modes,boundary conditions,and nonlocal parameters are considered. Results show that considering elastic medium increases frequency of the laminated micro-plate. It is also concluded that 15 DQ grid points can yield accurate results. In addition,the dimensionless nonlinear frequency decreases with the increasing slenderness ratio. Furthermore,increasing the nonlocal parameter decreases the dimensionless nonlinear frequency.