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A Levy Type Solution for Free Vibration Analysis of a Nano-Plate Considering the Small Scale Effect

Written By

E. Jomehzadeh and A. R. Saidi

Submitted: December 14th, 2010 Published: September 9th, 2011

DOI: 10.5772/24828

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1. Introduction

Experimental results show that as length scales of a material are reduced, the influences of long-range interatomic and intermolecular cohesive forces on the mechanical properties become prominent and cannot be neglected. It is well known that surfaces and interfaces in nano structures behave differently from their bulk counterparts. For nanostructures with size less than 100nm, the surface to volume ratio is significant and the effective properties are altered by surface and nonlocal effects. Therefore, at nanolength scales, size effects often become prominent, the causes of which need to be explicitly addressed especially with an increasing interest in the general area of nanotechnology (Sharma et al., 2003).

Due to the vast computational expenses of nano-structures analyses when using atomic lattice dynamics and molecular dynamic simulations, there is a great interest in applying continuum mechanics for analysis of nano-structures. Classical continuum elasticity, which is a scale free theory, cannot predict the size effects. Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with microstructures or nanostructures. It has been showed that it is possible to represent the integral constitutive relations of nano-structures in an equivalent differential form (Eringen, 1983). Eringen presented a nonlocal elasticity theory to account for the small scale effect by specifying the stress at a reference point is a functional of the strain field at every point in the body. Since then, many studies have been carried out nonlocal theory of elasticity for bending, buckling and vibration analyses of nano-structures.

Small scale effect on static deformation of micro- and nano-rods or tubes is revealed through nonlocal Euler–Bernoulli and Timoshenko beam theories by Wang and Liew (2007). Li and Wang (2009) investigated a theoretical treatment of Timoshenko beams, in which the influences of shear deformation, rotary inertia, and scale coefficient are taken into account.

Murmu and Pradhan (2009a) studied vibration response of nano cantilever considering non-uniformity in the cross sections using nonlocal elasticity theory.

Although graphite sheet has many superior properties, such as low electrical and thermal conductivities normal to the sheet but high electrical and thermal conductivities in the plane of the sheet, relatively little research have been reported in the literature for mechanical analyses of graphene sheets.

Kitipornchai et al. (2005) used the continuum plate model for mechanical analysis of graphene sheets. He et al. (2005) investigated vibration analysis of multi-layered graphene sheets in which the van der Waals interaction between layers is described by an explicit formula. Behfar and Naghdabadi (2005) studied nano scale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium based on the classical plate theory. Lu et al. (2007) derived the basic equations of nonlocal Kirchhoff and Mindlin plate theories for simply supported nano-plates. Axisymmetric bending of micro/nanoscale circular plates was studied using a nonlocal plate theory by Duan and Wang (2007). Pradhan and Phadikar (2009a) presented classical and first order shear deformation plate theories for vibration of nano-plate. Their approach is based on the Navier solution and for a nano-plate with all edges simply supported. Pradhan and Phadikar (2009b) carried out vibration analysis of multilayered graphene sheets embedded in polymer matrix employing nonlocal continuum mechanics.

In-plane vibration of nano-plates was investigated by Murmu and Pradhan (2009b) employing nonlocal continuum mechanics and considering small scale effect.

Aghababaei and Reddy (2009) developed a higher order plate theory for buckling and vibration analyses of a simply supported plate accounting the small scale effect. A nonlocal plate model was developed to study the vibrational characteristics of multi-layered graphene sheets with different boundary conditions embedded in an elastic medium using finite element method (Ansari et al., 2010). Pradhan and Kumar (2010) investigated the small scale effect on the vibration analysis of orthotropic single layered graphene sheets embedded in an elastic medium. Jomehzadeh and Saidi (2011a) investigated the nonlocal three dimensional elastodynamics theory to study the vibration of nano-plates. Recently, they (2011b & 2011c) studied the nonlinear vibration of graphene sheets using classical plate theory.

In this chapter, the vibration analysis of a nano-plate is presented by considering the small scale effect. The three coupled governing equations of motion are obtained based on the nonlocal continuum theory and are decoupled into two new equations. Solving these two decoupled partial differential equations, the natural frequencies of the nano-plate with arbitrary boundary conditions are determined. Finally, a detailed study is carried out to understand the effects of boundary condition, nonlocal parameter, thickness to length and aspect ratios on the vibration characteristics of nano-plates.

Results for natural frequencies of nano-plates with arbitrary boundary conditions are given for the first time and these can serve as reference values for other numerical analysis.


2. Constitutive relations

According to nonlocal elasticity theory, the stress at a reference point X is considered to be a function of the strain field at every point X' in the body. The nonlocal stress tensor σnl at point X can be expressed as (Eringen, 1983)


where σl is the classical stress tensor and K(|X'X|) is the Kernel function represents the nonlocal modulus. While the constitutive equations of classical elasticity is an algebraic relation between stress and strain tensors, that of nonlocal elasticity involves spatial integrals which represent weighted averages of contributions of the strain of all points in the body to the stress at the given point. Eringen showed that it is possible to represent the integral constitutive relation in an equivalent differential form as


where μ=(e0a)2 is nonlocal parameter, aan internal characteristic length and e0 a constant. Also, 2is the Laplacian operator.


3. Governing equations of motion

The first order shear deformation plate theory assumes that the plane sections originally perpendicular to the longitudinal plane of the plate remain plane, but not necessarily perpendicular to the longitudinal plane. This theory accounts for shear strains in the thickness direction of the plate and is based on the displacement field


where u0 and v0 are displacement components of the midplane, wis transverse displacement, tis time, ψxand ψy are the rotation functions of the midplane normal to x and y directions, respectively. Using the Hamilton’s principle, the nonlocal bending governing equations of motion for a single layered nano-plate are obtained as follows (Pradhan and Phadikar, 2009a)


In above equations, dot above each parameter denotes derivative with respect to time, Gis the shear modulus, D=Eh3/12(1ν2)denotes the bending rigidity of the plate, Eand ν Young modulus and Poisson’s ratio, respectively and κ2 the shear correction factor. Also, qis the transverse loading in z direction. Mass moments of inertia, I1andI2, are defined as


in which ρ is the density of the plate. It can be seen that the governing equations (Eq. 4) are generally a system of six-order coupled partial differential equations in terms of the transverse displacement and rotation functions.


4. Solution

In order to solve the governing equations of motion (Eq. 4) for various boundary conditions, it is reasonable to find a method to decouple these equations. Let us introduce two new functions ς and φ as


Using relations (Eq. 6), the governing equations (4) can be rewritten as


Doing some algebraic operations on Eq. (7), the three coupled partial differential equations Eq. (4) can be replaced by the following two uncoupled equations


where C denotesD(1ν)/2. It can be seen that the above equations are converted to the classical equations of the Mindlin plate theory whenμ=0. Like the classical elasticity (Reissner, 1985), Eq. (8) and (Eq. 9) are called edge-zone (boundary layer) and interior equations, respectively. Also, the rotation functions ψx and ψy can be defined in terms of w and φ as


By obtaining transverse displacement and rotation functions (w, ψxandψy), the stress components of the nano-plate can be computed by using the nonlocal constitutive relations in the following forms


Here, a rectangular plate (a×b) with two opposite simply supported edges at x=0 and x=a and arbitrary boundary conditions at two other edges is considered. For free harmonic vibration of the plate, the transverse loading q is put equal to zero and the transverse deflection w and boundary layer function φ are assumed as


which exactly satisfy the simply supported boundary conditions at x=0 andx=a. In these relations, ωnis the natural frequency of the nano-plate and βn denotesnπ/a. Substituting the proposed series solutions (Eq. 11) into decoupled Eq. (8), yields


where the constant coefficients λi(i=1,..,5) are material constants. The above equations are two ordinary differential equations with total order of six. The solutions of Eq. (12) can be expressed as


where Ci(i=1,..,6) are constants of integration and parametersη1, η2and η3 are defined as


Six independent linear equations must be written among the integration constants to solve the free vibration problem. Applying arbitrary boundary conditions along the edges of the plate at y=0 andy=b, leads to six algebraic equations. Here, three types of boundary conditions along the edges of the nano-plate in y direction are considered as

Simply supported (S) w=Myy=ψx=0E30
Clamped (C)   w=ψx=ψy=0E31
Free (F)Myy=Mxy=Qy=0E32

where the resultant moments Myy and Mxy and resultant force Qy are expressed as


In order to find the natural frequencies of the nano-plate, the various boundary conditions at y=0 and y=b should be imposed. Applying these conditions and setting the determinant of the six order coefficient matrix equal to zero, the natural frequencies of the nano-plate are evaluated.


5. Numerical results and discussion

For numerical results, the following material properties are used throughout the investigation


In order to verify the accuracy of the present formulations, a comparison has been carried out with the results given by Pradhan and Phadikar (2009a) for an all edges simply supported nano-plate. To this end, a four edges simply supported nano-plate is considered. The non-dimensional natural frequency parameter Ω=ωa2π4I1/D is listed in Table 1 for some nonlocal parameters. From this table, it can be found that the present results are in good agreement with the results in literature when the rotary inertia terms have been neglected. It can be also seen that the rotary inertia terms have considerable effects especially in second mode of vibration and cause the natural frequency decreases. Hereafter, the rotary inertia terms are considered in numerical results.

To study the effects of boundary condition, the nonlocal parameter (μ) and thickness to length ratio (h/a) on the vibrational behavior of the nano-plate, the first two non-dimensional frequencies are obtained for a single layered nano-plate. The results are tabulated in Tables 2-6 for five possible boundary conditions at y=0 and y=b as clamped- clamped (C-C), clamped-simply (C-S), clamped-free (C-F), simply-free (S-F) and free-free (F-F).

μh/bMode 1Mode 2
0.2026 a
Pradhan (2009a)0.13320.2026
0.1236 a
0.1730 a
Pradhan (2009a)0.12360.1730
0.0942 a
0.1432 a
Pradhan (2009a)0.09420.1432
0.0874 a
0.1224 a
Pradhan (2009a)0.08740.1224
0.0769 a
0.1170 a
Pradhan (2009a)0.07690.1170
0.0714 a
0.0999 a
Pradhan (2009a)0.07140.0999
0.0666 a
0.1013 a
Pradhan (2009a)0.06660.1013
0.0618 a
0.0865 a
Pradhan (2009a)0.06180.0865

Table 1.

Comparison of non-dimensional frequency parameter Ω=ωa2π4I1/D of a nano-plate with all edges simply supported (a Neglecting the rotary inertia terms)

Based on the results in these tables, it can be concluded that for constanth/a, the frequency parameter decreases for all modes as the nonlocal parameter μ increases. The reason is that with increasing the nonlocal parameter, the stiffness of the nano-plate decreases. i.e. small scale effect makes the nano-plate more flexible as the nonlocal model may be viewed as atoms linked by elastic springs while the local continuum model assumes the spring constant to take on an infinite value. In sum, the nonlocal plate theory should be used if one needs accurate predictions of natural frequencies of nano-plates.

μh/bMode 1Mode 2

Table 2.

First two non-dimensional frequency parameters Ω=ωa2π4I1/D of a C-C nano-plate

μh/bMode 1Mode 2

Table 3.

First two non-dimensional frequency parameters Ω=ωa2π4I1/D of a C-S nano-plate

μh/bMode 1Mode 2

Table 4.

First two non-dimensional frequency parameters Ω=ωa2π4I1/D of a C-F nano-plate

The influence of thickness-length ratio on the frequency parameter can also be examined by keeping the nonlocal parameter constant while varying the thickness to length ratio. It can be easily observed that as h/a increases, the frequency parameter decreases. The decrease in the frequency parameter is due to effects of the shear deformation, rotary inertia and use of term a2/h in the definition of the non-dimensional frequencyΩ. These effects are more considerable in the second mode than in the first modes.

μh/bMode 1Mode 2

Table 5.

First two non-dimensional frequency parameters Ω=ωa2π4I1/D of a S-F nano-plate

μh/bMode 1Mode 2

Table 6.

First two non-dimensional frequency parameters Ω=ωa2π4I1/D of a F-F nano-plate

To study the effect of the boundary conditions on the vibration characteristic of the nano-plate, the frequency parameters listed in a specific row of tables 1-6 may be selected from each table. It can be seen that the lowest and highest values of frequency parameters correspond to F-F and C-C edges, respectively. Thus like the classical plate, more constrains at the edges increases the stiffness of the nano-plate which results in increasing the frequency.

The effect of variation of aspect ratio (b/a) on the natural frequency of a C-S nano-plate is shown in Fig. 1 for various nonlocal parameters. It can be seen with increasing the aspect ratio, the natural frequency of the nano-plate decreases because of decreasing of stiffness.

In Fig. 2, the relation between natural frequency and nonlocal parameter of a square C-C nano-plate is depicted for different thickness to length ratios. It can be seen that nonlocal theories predict smaller values of natural frequencies than local theories especially for higher thickness to length ratios. Thus the local theories, in which the small length scale effect between the individual carbon atoms is neglected, overestimate the natural frequencies. The effect of boundary conditions on the natural frequency of a nano-plate is shown in Fig. 3. It can be concluded that the boundary condition has significant effect on the vibrational characteristic of the nano-plates.

Figure 1.

Variation of natural frequency with respect to aspect ratio for a C-S nano-plate

Figure 2.

Variation of natural frequency with nonlocal parameter for a C-C nano-plate

Figure 3.

Variation of natural frequency with nonlocal parameter for nano-plates with different boundary conditions at two edges


6. Conclusion

Presented herein is a variational derivation of the governing equations and boundary conditions for the free vibration of nano-plates based on Eringen’s nonlocal elasticity and first order shear deformation plate theory. This nonlocal plate theory accounts for small scale effect, transverse shear deformation and rotary inertia which become significant when dealing with nano-plates. Coupled partial differential equations have been reformulated and the generalized Levy type solution has been presented for free vibration analysis of a nano-plate considering the small scale effect. The accurate natural frequencies of nano-plates have been tabulated for various nonlocal parameters, some thickness to length ratios and different boundary conditions. The effects of boundary conditions, variation of nonlocal parameter, thickness to length and aspect ratios on the frequency values of a nano-plate have been examined and discussed.



The authors wish to thank Iran Nanotechnology Initiative Council for its financial support.


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Written By

E. Jomehzadeh and A. R. Saidi

Submitted: December 14th, 2010 Published: September 9th, 2011