Axisymmetric Free Vibration Analysis of Annular and Circular Mindlin Plates Using the Nonlocal Continuum Theory

This study aims to investigate and analyze the axisymmetric free vibration of non-local annular and circular Mindlin plates at the micro/nano scale which are modeled using Eringen's nonlocal elasticity theory, taking into consideration the small scale effect. The governing equations are derived using the nonlocal differential constitutive relations of Eringen. For this purpose, the resulted eigenvalue problem is solved numerically by applying the Chebyshev collocation method. The effects of the inner to outer radius ratio, the thickness to outer radius ratio, the nonlocal scale effect and the boundary conditions on the natural frequencies are studied.


INTRODUCTION
Due to their significant role in different engineering and modern technology fields such as aerospace, communications, composites, electronics, micro electro mechanical systems and nano electromechanical systems, micro and nano structures have gained appreciated consideration.These structures have more superior mechanical, electrical and thermal properties comparing with other structures at the normal length scale as they are high sensitive and high frequency devices for different applications (Murmu and Pardhan, 2009a).
In order to design a realistic model of a micro or a nanostructure, optimize and improve their performance and to well understand it, the small-scale effects and the atomic forces must be taken into consideration.In objects at the micro and nano scales, the dimensions, wavelengths and sizes of these structures are no longer considered much larger than the characteristic dimensions of the microstructure.In these cases, the internal length scales of the material are comparable with the structure size.Moreover, the particles affect each other by long range cohesive forces in addition to the contact forces and heat diffusion.Consequently, the internal length scale should be considered as a material parameter called nonlocal parameter in the constitutive and governing equations and relations.
Although the experimental and atomistic simulations and models are both capable to show the effects of the small-scale on the mechanical properties of the micro/nanostructures, these methods are expensive and restricted by computational capacity.It is well known that the local continuum theories for beams (Euler and Timoshenko) and plates (Kirchhoff and Mindlin) are scale free.Therefore, they are not able to capture the small scale effect on the mechanical, electrical and thermal properties for very small beam and plate like structures, which makes them inadequate in describing the dynamical behavior for these structures (Wang et al., 2007).In order to apply the continuum mechanics approach in the analysis of the micro and nanostructures, logical and reasonable modifications that take into consideration the scale effect should be proposed.For this purpose, several theoretical models have been suggested, such as the strain gradient theory, the modified coupled stress theory and the nonlocal elasticity theory which will be utilized in this study to analyze the free vibration problem of nonlocal annular and circular Mindlin plates.
The nonlocal elasticity theory was introduced by Eringen (1983) accounts for the small-scale effects arising at the nano scale level by assuming that the stress at a point is a function of the strains at all points in the domain.Many researchers applied the nonlocal elasticity theory to study the free vibration, buckling, deflection and dynamic problems of micro and nanostructures.For example, Reddy (2007) obtained analytical solutions for the bending, buckling and vibration problems for simply supported Euler, Timoshenko, Reddy and Levinson beams using Eringen's nonlocal theory.Murmu and Adhikari (2010a) studied the nonlocal transverse in and out-ofphase vibrations of double nanobeam systems, in which explicit closed-form expressions for natural frequencies were derived.Shakouri et al. (2011) applied the Galerkin approach to study the free vibration problem of nonlocal Kirchhoff plates with different boundary conditions; this study showed that the nonlocal parameter and Poisson's ratio have significant effects on the vibration.Wang et al. (2007) applied the Hamilton's principle, Eringen's nonlocal elasticity theory and Timoshenko beam theory to analyze the free vibration problem of the micro/nanobeams; it was concluded that the effects of small scale, rotary inertia and transverse shear deformation are important on the vibration behavior of short, stubby micro/nanobeams.Moreover, Murmu and Adhikari (2010b) applied the differential quadrature method and the nonlocal elasticity theory to study the free vibration of a rotating carbon nanotube modeled as an Euler-Bernoulli beam.It was concluded that the vibration is significantly influenced by the angular velocity, preload and the nonlocal parameter.Lu et al. (2006) derived the dispersion relation for a harmonic flexural wave propagation in an Euler-Bernoulli beam, as well as the frequency equations and modal shape functions for beam with different boundary conditions based on Eringen's nonlocal elasticity theory.Murmu and Pradhan (2009b) implemented the nonlocal elasticity theory to study the vibration response of single graphene sheets embedded in an elastic medium modeled as Winkler and Pasternak foundations, where the differential quadrature method was employed to numerically solve for the fundamental natural frequencies of plates with clamped and simply supported edges.In a similar manner, Murmu and Pradhan (2009b) applied the nonlocal elasticity theory to investigate the free vibration problem of nanoplates under uniaxially pre stressed conditions by utilizing the differential quadrature method to obtain the fundamental natural frequencies for simply supported and clamped nanoplates.In this study, it was observed that buckling occurs at smaller critical compressive load compared to the classical plate theory.Gürses et al. (2012) studied the free vibration analysis of thin nano-sized annular sector plates utilizing Eringen's nonlocal elasticity theory to formulate the equation of motion.The discrete singular convolution method was applied after transforming the irregular physical domain into a rectangular domain by using geometric coordinate transformation.This study showed that the effects of the nonlocal parameter is significant in the vibration analysis.Hashemi et al. (2013a) applied an exact analytical approach along with Eringen's theory to study the free vibration problem of thick circular and annular functionally graded Mindlin nanoplates with different combinations of boundary conditions.The effects of the plate radius, material properties which vary through the material according to a power-law distribution and the nonlocal parameter on the natural frequencies were examined.In another study, Hashemi et al. (2013b) introduced potential functions and used the separation of variables method to obtain closed-form solutions for non-local rectangular Mindlin plates with Levy-type boundary conditions in which the effects of the nonlocal parameter, thickness to length ratio and aspect ratio on the natural frequencies were investigated.
Furthermore, Ansari et al. (2010) applied the generalized differential quadrature method, Eringen's nonlocal elasticity theory and the molecular dynamics simulations to carry out the vibration analysis of single layered graphene sheets modeled as rectangular Mindlin plates.Additionally, the appropriate values of the nonlocal parameter suitable for each boundary condition were evaluated.Duan and Wang (2007) obtained exact solutions for the axisymmetric bending of micro and nano circular plates under general loading using a nonlocal plate theory.It was concluded that nonlocal parameter has a significant effect on the deflections, moments and bending stiffness.
The main objective of this article is to study the axisymmetric free vibration problem of non-local annular and circular Mindlin plates.For this purpose, Erigen's nonlocal elasticity theory along with Hamilton's principle will be applied and the Chebyshev collocation method (as a numerical technique) will be used to discretize the problem and to obtain the algebraic eigenvalue problem and hence, solving for the natural frequencies.It is worthwhile to mention that the Chebyshev collocation method was successfully employed to carry out the free vibration analysis of local continuous systems with different shapes and geometries in previous studies (Sari et al., 2011;Sari andButcher, 2011a, b, 2012).

MATERIALS AND METHODS
Chebyshev spectral collocation: Gauss-Chebyshev-Lobatto or Chebyshev extreme points are the points in the interval (-1, 1) defined by: Chebyshev points are the projections on (-1, 1) of equally spaced points on the upper half of the unit circle and they are numbered from right to left as shown in Fig. 1 (Trefethen, 2000).For the set of N+1 Chebyshev points we have an (N+1) X (N+1) Chebyshev differentiation matrix D N .The Chebyshev differentiation matrix is obtained by interpolating a Lagrange polynomial of degree N at each Chebyshev point, differentiating the polynomial and then finding the derivative of the polynomial at each Chebyshev point.The entries of this matrix are: Nonlocal theory: In local elasticity theory, the stress at a reference point in a body depends on the strain at the same point.On the other hand, In the non-local elasticity theory pioneered by Eringen (1983), the stress at a point in a linear, homogeneous, isotropic, elastic domain is related to the stress field at all points in the domain.Eringen's theory is based on the atomic theory of lattice dynamics and experimental results on phonon scattering and dispersion (Eringen, 1983;Gürses et al., 2012).For non-local linear elastic solids, the stress tensor t ij is defined by: in which e 0 is a material constant estimated by experiments or other models and theories, such that the non-local theory relations could result in approximate solutions to those obtained by atomic theory.The value of e 0 was taken to be 0.39 in Eringen's analysis.Moreover, the constant l represents the internal characteristic length which is of the same order of the external length.The Laplace operator 2  is defined in the polar coordinates as: Since the axisymmetric vibration of the annular non-local Mindlin plates will be studied, all derivatives with respect to the θ axis will be zero and hence, the Laplace operator is simplified as: According to Eringen, the integral constitutive relation Eq. ( 1) could be simplified and rewritten as: (5) Due to its simple form, Eq. ( 3) has been extensively employed by many researchers in applying the non-local theory to study and analyze the vibration and mechanics of micro and nanostructures.
From Eq. ( 3) and utilizing Hook's law, the constitutive relations can be expressed as: where,  r and ε r : The normal stress and normal strain in the radial direction r  θ and ε θ : The normal stress and normal strain in the circumferential direction θ E : Young's modulus v : The Poisson's ratio In the case of axisymmetric vibration, the normal strains ε r and ε θ are defined as: where, the z axis is shown in Fig. 2 and ψ is the slope rotation in the r-z plane at z = 0. Accordingly, the resultant moments and shear are expressed as: The resultant moments and shear are defined in terms of the normal stresses  r and  θ as: Based on Hamilton's principle, the first order shear deformation theory FSDT and Eringen's non-local constitutive relations, the following equilibrium equations of the free vibration are obtained for an annular axisymmetric isotropic non-local Mindlin plate with a uniform thickness shown in Fig. 2: where, t = The time ρ = The density of plate material Substituting Eq. ( 9)-( 12) into Eq.( 13) yields: Note that the governing Eq. ( 14) reduce to that of the local Annular Mindlin plate model when the characteristic length l is set to zero (Han and Liew, 1999).For convenience, the following non-dimensional parameters are introduced: And assuming harmonic solutions in time as: The left hand side of each equation is written in terms of Chebyshev collocation matrices and the Kronecker product operator as: The right hand side of each equation is written as: where, diag(R) and diag(R 2 ) are diagonal M X M matrices with values of R i and R i 2 on the main diagonals, respectively and M = N+1 and i = 2, 3,…, N. The dimensions of GE1, GE2 RH1 and RH2 are M X 2M.Equation ( 16) is written as: It is worthwhile to mention that for the annular plates, the Chebyshev points are shifted to the range [b/a, 1], while for the circular plates these points are shifted to the range [0,1] as the solid part of the annulus begins at r = b or R = b/a.This different range of the points in both cases leads to different Chebyshev differentiation matrices since the entries of these matrices depend on the distribution of the points.

Boundary conditions:
In order to obtain the natural frequencies of the circular annular non-local Mindlin plate, the boundary conditions at r = b and at r = a should be applied.In the present study, two types of boundary conditions, i.e., simply Supported (S) and Clamped (C) are taken into consideration.For a clamped edge, both displacements W and Ψ equal zero; whereas for a simply supported edge, the transverse displacement W and M r equal zero.In terms of Chebyshev collocation method, these conditions are applied at the inner edge (r = b) as: where, [1… 0] is a 1 X M vector with the first entry equals one and other entries are zero since it corresponds to the boundary r = b.The displacement vector [U] is defined as: Simply supported: At the outer edge (r = a), the conditions for the axisymmetric vibration are applied in a similar way as: Clamped: where, [0… 1] is a 1 X M vector with the last entry equals one and other entries are zero since it corresponds to the boundary r = a.Simply supported: where, D1 (1, :) is defined as: and D1 (end, :) is defined as: The boundary conditions considered here are the same for both local and nonlocal Mindlin plate models as the contribution of scale effect gets nullified at the ends as the displacements are zero.After applying the boundary conditions at R = β and R = 1, the system is written as: In this case, S BB is a 4 X 4 matrix, S BI is a 4 X (2M-4) matrix, S IB is a (2M-4) X 4 matrix, S II is a (2M-4) X (2M-4) matrix and U I and U B are the displacements vectors at the interior and boundary points, respectively.The frequency parameter λ is defined as: From the equations in matrix vector form, we obtain the final eigenvalue problem as in Eq. ( 22).Writing the coupled governing equations and the boundary conditions in matrix vector form with the use of the Kronecker product operator makes the procedure simpler in implementation.
For the case of axisymmetric vibration of circular non-local Mindlin plates, the governing equations are the same; whereas the Chebyshev points in this case are shifted to the range [0,1] unlike in the annular plate where these points are in the range [b/a, 1].Furthermore, the boundary conditions at R = 0 are: Ψ and Q R equal zero, while the edge R = 1 is considered to be either clamped or simply supported.It is worthwhile to mention that different range of the points in both cases leads to different Chebyshev differentiation matrices as the entries of these matrices depend on the distribution of the points.

RESULTS AND DISCUSSION
Dimensionless frequencies are calculated for annular and circular non-local Mindlin plates with clamped and simply supported boundary conditions, where N = 16 is used in the computations to attain convergence for the first six dimensionless frequencies.The convergence analysis for a SS non-local annular plates, with b/a = 0.3, h/a = 0.15 and μ = 0.8 are shown in Table 1, where the minimum number of points used in the computations is N = 6 and these points were increased till the results are converged up to four decimal places.For all cases considered in the present study, the shear correction factor is taken as κ = π 2 /12 and the Poisson's ratio as ν = 0.3.
The variation of the fundamental frequency parameter with the inner to outer radius b/a of a SS and CC annular Mindlin plates with h/a = 0.1 at different values of the dimensionless non-local parameter μ is presented in Fig. 3 and 4, respectively.It is shown that as the ratio b/a increases the frequency increases (Han and Liew, 1999).Further, it is observed that the natural frequency decreases by increasing the nonlocal parameter due to the decrease in the stiffness of the micro/nano-plate.In Eringen nonlocal elasticity theory, it may be viewed that atoms are bonded by elastic springs with finite value; while the classical local model assumes that the stiffness of springs have a value of infinity (Wang et al., 2007;Gürses et al., 2012).Moreover, it is noticed that as the non-local parameter μ increases, the frequency increases with a smaller rate as the ratio b/a increases.As an example, for the SS annular plates with μ = 0.1, the fundamental frequency λ 1 increases from 13.1342 at b/a = 0.1 to 31.6491 at b/a = 0.5 with an increasing rate of about 141%; while for μ = 0.8, the fundamental frequency λ 1 increases from 4.7564 at b/a = 0.1 to 7.3156 at b/a = 0.5 with an increasing rate of about 54%.For the CC annular plate       seen that as the frequency decreases as the ratio h/a increases (Han and Liew, 1999) and the decrease rate is higher at smaller values of the non-local scale effect.The variations of the first six frequency parameters with the non-local parameter for CC, CS and SS non-local annular Mindlin plates at b/a = 0.3, h/a = 0.15 are presented in Table 2 to 4. As in the annular plates, it is seen that as the non-local scale parameter increases, the frequency parameters decrease and this effect is more significant for higher modes.Moreover, it is observed that the frequency parameters are more affected by the nonlocal parameter when the annular plate has clamped boundary conditions as the structure becomes stiffer and more rigid.Additionally, at higher values of the non-local parameter, the frequency parameters get closer to each other and it is expected that the frequency parameters will no longer increase as the mode number increases at values of μ greater than 1.0.
The variations of the first six frequency parameters with the non-local parameter for clamped and simply supported non-local circular Mindlin plates with different values of the thickness to radius ratio h/a are presented in Table 5 and 6, respectively.It is observed that as the ratio h/a and the non-local parameter μ increase, the frequency parameters decrease.As in the non-local annular plates, the frequency parameters are more effected by the nonlocal parameter when the circular plate has clamped boundary conditions and at higher values of the non-local parameter, the frequency parameters get closer to each other.

CONCLUSION
The free vibration analysis of non-local annular and circular Mindlin plates was investigated and the nonlocal elasticity theory is employed to derive the governing equations of motion, where the Chebyshev spectral collocation method was utilized to solve for the frequency parameters.Convergence analysis was carried out to determine the sufficient number of grid points to be used in the computations.The effects of the nonlocal scale parameter, the inner to outer radius ratio (for annular plates), the thickness to the outer radius ratio and the boundary conditions on the frequency parameters are examined.It was found that the nonlocal scale parameter has a significant effect especially for higher frequency modes.Therefore, the scale parameter should be taken into account when modeling micro/nano annular and circular plates.Moreover, it was observed that the effect of nonlocal parameter on the frequencies of plates with clamped boundary conditions is more than that on plates with simply supported edges.In addition, it was found that the frequencies get closer to each other at higher values of the nonlocal scale parameter.The results presented herein may be useful for scientists working on and designing micro/nano annular and circular plates.For future work, it is suggested to study the free vibration analysis of nonlocal annular and circular Mindlin plates resting on elastic foundation, due to the fact that graphene sheets and circular nanotubes are found embedded in elastic medium, which will make the study more realistic as it has an industrial application.

Fig. 1 :
Fig. 1: Chebyshev points When solving ODEs or PDEs by the Chebyshev collocation method, the first derivative is represented by D1 = D N , the second derivative by D2 = (D1) 2 = (D N ) 2 and so on.
where, x is a reference point in the elastic domain, | ′ | is the non-local kernel attenuation function which introduces the nonlocal effects at the reference point x produced by the local stress at any point x' and | ′| is the distance in Euclidean form.In order to simplify Eq. (1), Eringen introduced a linear differential operator ς, defined by ς = deflection D = Eh 3 /12 (1-ν 2 ) : The flexural rigidity h : The thickness κ : The shear correction factor G : The shear modulus

Fig. 3 :
Fig. 3: Variation of the fundamental frequency parameter λ 1 with b/a of a SS non-local annular mindlin plate (h/a = 0.1)

Table 3 :
The first six frequency parameters of CS non-local annular mindlin plates with b/a = 0.3, h/a = 0.15 μ

Table 5 :
The first six frequency parameters of clamped non-local circular mindlin plates Axisymmetric vibration mode number -