Effect of scaling effect parameters on the vibration characteristics of nanoplates

Abstract

The Rayleigh–Ritz method has been applied to solve governing differential equations of the free vibration of nonhomogeneous rectangular nanoplates. Nonhomogeneity is assumed by taking a variety of combinations of linear as well as quadratic variations in the Young's modulus and the density of the material. Boundary characteristic orthogonal polynomials in two variables are used in the Rayleigh–Ritz method and have been generated with the help of the Gram–Schmidt process. The generalized eigenvalue problem has been solved by employing MATLAB code. Both graphical and tabular results are given to show the effects of nonhomogeneous parameters, the nonlocal parameter, boundary conditions, aspect ratio and the length of the nanoplate on the frequency parameters. Results are compared in special cases and are found to be in good agreement.

Keywords

Boundary characteristic orthogonal polynomial nonlocal elasticity theory nonhomogeneous nanoplate vibration mode shapes

1. Introduction

The production of nanostructures has become one of the challenging areas in the field of nanotechnology. Graphene sheets are one type of nanomaterial which can be used in the design of new sensors, gas detection and composite materials. This is due to their superior mechanical, electronic and thermal properties (Ruud et al., 1994). Therefore, analysis of graphene sheets has become an important topic for researchers. Generally, three approaches have been developed in the analysis of graphene sheets. These are (a) atomistic modeling, (b) hybrid atomistic–continuum mechanics and (c) continuum mechanics. Continuum mechanics includes both the classical (local) continuum approach and the nonclassical continuum approach. Since atomistic modeling and hybrid atomistic–continuum mechanics are computationally expensive and are not suitable for large-scale systems, continuum mechanics came into existence. The classical continuum approach is computationally less expensive but the application of this approach at small scale is doubtful. At nanoscale size, the lattice spacing between individual atoms becomes prominent and cannot be neglected. In other words, the material properties of nanostructures are size-dependent. Hence, for better prediction of their dynamic behavior, the small length scale effect should be considered. Among various nonclassical continuum theories, the nonlocal elasticity theory developed by Eringen (1972, 1983) has attracted much attention among researchers. According to this theory, the stress at a specific point depends on the strain tensors of the entire body. A nonlocal parameter is included in this theory which is determined either by experiment or by molecular dynamics (Duan et al., 2007; Wang et al., 2008; Huang et al., 2012; Liang and Han, 2012). Graphene sheets have been modeled as two-dimensional nanoplates by using nonlocal continuum plate models such as classical plate theory and first-order shear deformation plate theory. Few investigations have been done for the vibration of nanoplates. A finite difference method has been developed by Ravari and Shahidi (2013) to study the axisymmetric buckling behavior of circular annular nanoplates and solid disks under uniform compression. Ritz solutions have been presented by Anjomshoa (2013) for the buckling analysis of embedded orthotropic circular and elliptical micro- or nanoplates based on nonlocal elasticity theory. Malekzadeh and Shojaee (2013) used the differential quadrature (DQ) method to investigate the free vibration of nanoplates based on nonlocal two-variable refined plate theory. Mechanisms of nonlocal effects on the vibration of nanoplates have also been analyzed by Wang et al. (2011). Wang and Wang (2011) studied the vibration behavior of simply supported Kirchhoff and Mindlin nanoscale plates by using Navier's approach. Free vibration of orthotropic straight-sided quadrilateral nanoplates based on first-order shear deformation theory has been studied by Malekzadeh et al. (2011). The effect of small scale on the free in-plane vibration of nanoplates has been investigated by Murmu and Pradhan (2009a). Adali (2012) proposed a semi-inverse method to derive a variational principle for a nonlocal continuum model of orthotropic graphene sheets embedded in an elastic medium. An analytical solution for the vibration of nanoplates has been presented by Pradhan and Phadikar (2009) based on classical plate theory and first-order shear deformation plate theory.

Since conducting experiments at nanoscale size is quite difficult to handle, development of mathematical models and finding efficient solutions is a challenge in the field of nanotechnology. The literature reveals that most of the studies have been carried out using analytical methods such as Levy-type or Navier-type ones (Aksencer and Aydogdu, 2011), and numerical methods such as the DQ method (Murmu and Pradhan, 2009b) and the finite element method (Phadikar and Pradhan, 2010) for simple boundary conditions, that is, simply supported. One may note that the Levy-type solution is applied only for plates with two opposite edges that are simply supported and with the remaining ones under any arbitrary boundary condition. Similarly, the Navier-type solution is applied only for simply supported plates. One may also note that basis functions taken in the finite element method should be valid for the whole domain. As a result, for a deflection curve of smooth shape, we need to take a very large number of elements which is quite difficult to handle and takes more time. Moreover, the above-mentioned computational techniques may not be sufficient to analyze free vibration of nanoplates for all types of boundary condition with ease. Therefore, it is necessary to develop a reliable and computationally efficient numerical procedure to study the free vibration of nanoplates. In this regard, the Rayleigh–Ritz method is an approximate numerical method for handling any set of classical boundary conditions at the edges with ease. The Rayleigh–Ritz method has been extensively used in the vibration of plates but not much work has been done in the case of nanoplates. Authors have used boundary characteristic orthogonal polynomials (Bhat, 1985; Dickinson and Di Blasio, 1986; Bhat, 1991; Singh and Chakraverty, 1994(a,b,c); Rajalingham et al., 1996; Rizk and Ashour, 2001; Chakraverty et al., 2007; Behera and Chakraverty, 2014; Bhat, 2014; Chakraverty and Behera, 2014) in the Rayleigh–Ritz method to investigate the vibration of plates. They have generated boundary characteristic orthogonal polynomials with the help of the Gram–Schmidt process. The advantage of boundary characteristic orthogonal polynomials is that some of the entries of stiffness and mass matrices of generalized eigenvalue problems become either zero or one due to the orthonormality of the assumed shape functions.

As such, the present study is based on the application of boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method. In this article, the novelty of the method is in handling any set of boundary conditions. To the best of the authors' knowledge, this article gives results for the first time for boundary conditions which have not been found previously. In this article, the effect of nonhomogeneity on the vibration of nanoplates based on classical plate theory is also considered for the first time. The effects of nonhomogeneity, aspect ratio, boundary conditions and length of nanoplates on the free vibration of isotropic rectangular nanoplates have been investigated. A detailed analysis has been given which may help researchers of nanotechnology in the production of nanomaterials. The proposed numerical method easily handles all types of classical boundary conditions since some of the entries of the stiffness and mass matrices of the generalized eigenvalue problem become either zero or one due to the orthonormality of the assumed shape functions. The proposed method may be applied in any engineering problems for handling any set of classical boundary conditions at the edges.

2. Theoretical formulation of nanoplates based on classical plate theory

Let us consider an isotropic rectangular nanoplate with the domain a ≤ x ≤ b, a ≤ y ≤ b in the xy-plane where a and b are the length and breadth of the nanoplate respectively. The x- and y-axes are taken along the edges of nanoplate and the z-axis is perpendicular to the xy-plane. The origin is assumed to be at one of the corners of the nanoplate (Figure 1) and the middle surface is taken along z = 0.

Figure 1.

Geometry of the nanoplate.

In the case of isotropic nanoplates, the maximum potential energy is given by (Adali, 2012; Anjomshoa, 2013)

U = 12 D \int_{0}^{a} \int_{0}^{b} {(\partial 2 W \partial x 2) 2 + 2 ν (\partial 2 W \partial x 2 \partial 2 W \partial y 2) + (\partial 2 W \partial y 2) 2 + 2 (1 - ν) (\partial 2 W \partial x \partial y) 2} d x d y

(1)

where W is the deflection function, ν is Poisson's ratio and D is the flexural rigidity which is defined by Chakraverty (2009), Jomehzadeh and Saidi (2012), Anjomshoa (2013) and Ravari and Shahidi (2013) as follows:

D = Eh 312 (1 - ν 2)

where h is the height of the nanoplate and E is Young's modulus. Maximum kinetic energy may be written as (Adali, 2012; Anjomshoa, 2013)

T = 12 m_{0} ω 2 \int_{0}^{a} \int_{0}^{b} {W 2 + μ ((\partial W \partial x) 2 + (\partial W \partial y) 2)} d x d y

(2)

where ω is the natural frequency of the nanoplate, and μ = (e₀ l_int)² is the nonlocal parameter where e₀ is a constant and l_int is the internal characteristic length of the system (lattice parameter, granular distance, distance between C–C bonds). In equation (2) m₀ is the mass per unit area of the nanoplate and is defined as (Anjomshoa, 2013)

m_{0} = \int_{- h / 2}^{h / 2} ρ d z

. Here ρ denotes the mass per unit volume of the nanoplate.

Equating maximum kinetic and potential energies, one may obtain the Rayleigh-quotient for λ² as

λ 2 = \int_{0}^{a} \int_{0}^{b} [(\partial 2 W \partial x 2) 2 + 2 ν (\partial 2 W \partial x 2 \partial 2 W \partial y 2) + (\partial 2 W \partial y 2) 2 + 2 (1 - ν) (\partial 2 W \partial x \partial y) 2] d x d y \int_{0}^{a} \int_{0}^{b} [W 2 + μ ((\partial W \partial x) 2 + (\partial W \partial y) 2)] d x d y

(3)

where m₀ is taken as ρh and λ² is defined as follows (Pradhan and Phadikar, 2009; Anjomshoa, 2013):

λ 2 = ρ h ω 2 D

We have introduced the nondimensional variables as X = x/a and Y = y/b. Also, we have assumed that Young's modulus and the density of the nanoplates vary in the space coordinates by the following functional relations:

E = E_{0} (1 + α X + β x 2)

ρ = ρ_{0} (1 + γ X + δ X 2)

The nondimensional form of equation (3) is given as follows:

λ 2 = \int_{0}^{1} \int_{0}^{1} A [(\partial 2 W \partial X 2) 2 + 2 ν R 2 (\partial 2 W \partial X 2 \partial 2 W \partial Y 2) + R 4 (\partial 2 W \partial Y 2) 2 + 2 (1 - ν) R 2 (\partial 2 W \partial X \partial Y) 2] d X d Y \int_{0}^{a} \int_{0}^{b} B [W 2 + μ a 2 ((\partial W \partial X) 2 + (\partial W \partial Y) 2)] d X d Y

(4)

where R = a/b, A = (1 + αX + βX²) and B = (1 + γX + δX²).

Here the nondimensional frequency parameter is given by (Pradhan and Phadikar, 2009; Anjomshoa, 2013)

λ 2 = ρ_{0} h a 4 ω 2 D_{0}

3. Solution methodology

Let us assume the deflection function

W (X, Y) = \sum_{i = 1}^{N} c_{i} \overset{\land}{φ_{i}} (X, Y)

(5)

where N is the order of approximation and c_i are the unknowns. Here

\overset{\land}{φ_{i}}

are orthonormal polynomials generated with the help of the Gram–Schmidt process which is discussed below.

Orthogonal polynomials have been generated over the region 0 ≤ X ≤ 1 and 0 ≤ Y ≤ 1 with the help of the linearly independent set of functions η_i = χθ_i, i = 1,2,3,…,N with χ = X^p(1 − X)^qY^r(1 − Y)^s, θ_i = {1,X, Y,X²,XY,Y²,X³,X²Y,XY²,Y³, …}. Here p = 0, 1 or 2 when the edge X = 0 is free, simply supported or clamped. The same justification can be given for q, r and s for the edges X = 1, Y = 0 and Y = 1 respectively. Orthogonal polynomials are generated by the following steps:

φ_{1} = η_{1}, φ_{k} = η_{k} - \sum_{j = 1}^{i - 1} τ_{kj} φ_{j}

(6)

Here

τ_{kj} = < η_{k}, φ_{j} >< φ_{j}, φ_{j} >

where <,> denotes the inner product of two functions.

We define the inner product of two functions say, φ_i and φ_k, as

< φ_{i}, φ_{k} > = \int_{0}^{1} \int_{0}^{1} φ_{i} (X, Y) φ_{k} (X, Y) d X d Y

(7)

Normalized functions $\overset{\land}{φ_{i}}$ may be obtained by dividing each φ_i by its norm, that is,

\overset{\land}{φ_{i}} = φ_{i} ‖ φ_{i} ‖

where ‖φ_i‖ is the inner product of φ_i and φ_i, that is,

‖ φ_{i} ‖ = < φ_{i}, φ_{i} > = \int_{0}^{1} \int_{0}^{1} φ_{i} (X, Y) φ_{i} (X, Y) d X d Y ‖ φ_{i} ‖ = \int_{0}^{1} \int_{0}^{1} φ_{i}^{2} (X, Y) d X d Y

(8)

Substituting equation (5) into equation (4), we get a generalized eigenvalue problem

[K] {Y} = λ 2 [M] {Y}

(9)

where matrices K and M are given in the Appendix. It may be noted that the elements of the mass matrix are the product of φ_i and φ_j along with other terms. When φ_i′s are orthogonal, the inner product of φ_i and φ_j will be constant for i = j and 0 for i ≠ j. But if these are orthonormal then the inner product of φ_i and φ_j will be 1 for i = j and 0 for i ≠ j. Accordingly, orthonormal functions will make the computations efficient.

4. Numerical results

Frequency parameters λ have been obtained numerically by solving the generalized eigenvalue problem using a computer programme developed by the authors in MATLAB. The first three lowest eigenvalues corresponding to the first three frequency parameters have been reported here. The letters C, S and F refer to clamped, simply supported and free edge conditions respectively. Edge conditions are taken in an anticlockwise direction starting at the edge X = 0 (Figure 2) and are obtained by taking the various values for p, q, r and s as 0, 1 or 2 for free, simply supported or clamped edge conditions respectively. Various possible cases are taken to investigate the effect of nonhomogeneous parameters on the frequency parameters taking Poisson's ratio (ν) = 0.3. Numerical results are given for the following cases of variations of Young's modulus and density:

E varies linearly and ρ varies quadratically;

E varies quadratically and ρ varies linearly;

E and ρ vary quadratically;

E and ρ vary linearly.

Table 1 shows the convergency study of SCSC and SCSF nanoplates (α = 0.1, β = 0.2, γ = 0.3, δ = 0.4) with aspect ratio R = 1, nonlocal parameter μ = 2 nm² and length a = 5 nm. From this table, we may observe that increase in the number of terms approaches the solution. Variation of the first two frequency parameter ratios with length a is given in Figure 3 to show the comparison of proposed method with the analytical solution in a special case (α = β = γ = δ = 0) taking boundary conditions (BC) as SSSS and R as 2. It may be noted that a close agreement of the results with Aksencer and Aydogdu (2011) is seen. The frequency parameter ratio is calculated as the ratio of the frequency parameter calculated using nonlocal theory and the frequency parameter calculated using local theory. It may be seen that frequency parameter ratios decrease with increase in the nonlocal parameter. It is also observed that the ratio decreases with length. The same observations may also be seen in the available results (Aksencer and Aydogdu, 2011).

Table 1.

Convergence of first three frequency parameters of SCSC and SCSF nanoplates.

		SCSC				SCSF
n	First	Second	Third	First	Second	Third
5	16.2881	25.6801	28.4824	8.2811	17.2944	22.4682
10	16.2403	22.6655	27.5842	8.1454	14.8992	18.4923
15	16.2393	22.6655	27.5842	8.1424	14.8648	18.4496
20	16.2393	22.6200	27.5818	8.1424	14.8643	18.3621
25	16.2393	22.6200	27.5680	8.1424	14.8642	18.3611
30	16.2393	22.6198	27.5680	8.1424	14.8641	18.3608
35	16.2393	22.6198	27.5680	8.1424	14.8641	18.3600
36	16.2393	22.6198	27.5679	8.1424	14.8641	18.3600

Figure 3.

Variation of first and second frequency parameter ratios with the length of nanoplates (SSSS), and comparison with Aksencer and Aydogdu (2011).

Figure 2.

Boundary conditions.

The first three frequency parameters are given in Table 2 for nanoplates whose density varies quadratically and Young's modulus varies linearly taking aspect ratios as 1, 2, 3. Frequency parameters are shown taking α = 0.1 and δ = 0.2, 0.4. Results are given for nanoplates subjected to three different boundary conditions (SCSC, SFSC, CFCS) having a = 5 nm and μ = 1 nm². One may see that frequency parameters increase with aspect ratio for a particular set of parameters. Figures 4 and 5 are the pictorial representations to show the effect of the first four frequency parameters with δ taking R = 1 and with α taking R = 2 respectively. In this figure, results are shown for SCSC nanoplates having μ = 1 nm² and a = 5 nm. It is observed that frequency parameters decrease with δ and increase with α. This is due to the fact that frequency parameter λ² is directly proportional to α and inversely proportional to δ in equation (4). Hence frequency parameters decrease with δ and increase with α.

Figure 4.

Variation of frequency parameter with δ.

Figure 5.

Variation of frequency parameter with α.

Table 2.

First four frequency parameters of nanoplates with linear variation in E and quadratic variation in ρ.

		δ = 0.2			δ = 0.4
R	B.C.	λ₁	λ₂	λ₃	λ₁	λ₂	λ₃
1	SCSC	20.9436	31.1152	38.1769	20.3183	30.1862	37.0371
	SFSC	10.1029	20.1410	25.0047	9.8012	19.5397	24.2582
	CFCS	18.1921	22.1450	32.1444	17.6489	21.4838	31.1847
2	SCSC	51.3607	53.9392	60.9858	49.8272	52.3287	59.1649
	SFSC	15.5798	27.9621	39.4661	15.1147	27.1273	38.2877
	CFCS	18.6271	33.0469	36.2263	18.0709	32.0602	35.1447
3	SCSC	84.1980	84.4329	86.3172	81.6841	81.9119	83.7400
	SFSC	20.7049	32.6712	45.9368	22.0270	31.6957	44.5652
	CFCS	19.0528	36.7839	46.7481	18.4840	35.6856	45.3523

Frequency parameters for nanoplates whose density varies linearly and Young's modulus varies quadratically are shown in Table 3 taking different aspect ratio values. Frequency parameters have been shown for nanoplates having μ = 2 nm² and a = 10 nm taking γ = 0.3 and β = 0.2, 0.4. Here also, one may observe that frequency parameters increase with aspect ratio for a particular set of parameters. Variation of the first four frequency parameters of SSCF nanoplates with β and γ is shown in Figures 6 and 7 taking μ = 2 nm², R = 1 and a = 10 nm. It is seen that frequency parameters increase with β and decrease with γ. The cause of this behavior is that in equation (4), frequency parameter λ² is directly proportional to β and inversely proportional to γ.

Figure 6.

Variation of frequency parameter with β.

Figure 7.

Variation of frequency parameter with γ.

Table 3.

First four frequency parameters of nanoplates with quadratic variation in E and linear variation in ρ.

		β = 0.2			β = 0.4
R	B.C.	λ₁	λ₂	λ₃	λ₁	λ₂	λ₃
1	CCSF	14.7789	25.5460	35.8605	15.2338	26.3323	36.9642
	SSCF	14.2479	22.6691	35.6680	14.6864	23.3668	36.7658
	FFSC	6.2081	17.2213	19.9383	4.9897	14.8200	18.8952
2	CCSF	20.1545	38.8919	50.4910	20.7748	40.0888	52.0449
	SSCF	16.3862	37.3403	40.5120	16.8905	38.4895	41.7588
	FFSC	13.0427	22.4668	36.6413	13.4441	23.1583	37.7690
3	CCSF	28.7000	44.3000	64.2000	29.6000	45.7000	66.2000
	SSCF	18.6081	39.5660	60.3492	19.1808	40.7837	62.2066
	FFSC	23.5000	32.1000	44.8000	24.2000	33.1000	46.2000

Results for nanoplates when E and ρ vary quadratically are given in Table 4 taking β constant (0.2), with δ varying (0.1, 0.5), and Table 5 taking δ constant (0.3), with β varying (0.2, 0.4). Results are given for different boundary conditions taking μ = 1 nm² and a = 5 nm. It is noticed that frequency parameters decrease with δ and increase with β. This behavior is due to the fact that in equation (4), frequency parameter λ² is directly proportional to β and inversely proportional to δ. Frequency parameters of nanoplates whose E and ρ vary linearly are given in Tables 6 (α constant (0.1), γ varies (0.3, 0.5)) and 7 (γ constant (0.3), α varies (0.2, 0.4)). It is observed that frequency parameters decrease with γ and increase with α. It may be explained as follows. In equation (4), frequency parameter λ² is directly proportional to α and inversely proportional to γ.

Table 4.

First four frequency parameters of nanoplates with quadratic variations in E and ρ taking β constant.

		δ = 0.1			δ = 0.5
R	B.C.	λ₁	λ₂	λ₃	λ₁	λ₂	λ₃
1	CCCS	23.2290	36.0616	39.8315	21.8614	33.9385	37.4864
	CFFF	3.2376	6.8421	13.7721	3.0470	6.4393	12.9612
	SSSF	9.6455	17.9938	25.3976	9.0777	16.9344	23.9023
2	CCCS	40.9262	51.0768	64.5923	38.5166	48.0696	60.7893
	CFFF	3.2092	8.6499	14.3243	3.0202	8.1406	13.4809
	SSSF	11.9743	27.0780	32.0263	11.2693	25.4838	30.1407

Table 5.

First four frequency parameters of nanoplates with quadratic variations in E and ρ taking δ constant.

		β = 0.2			β = 0.4
R	B.C.	λ₁	λ₂	λ₃	λ₁	λ₂	λ₃
1	CCCS	22.5141	34.9518	38.6056	23.2070	36.0275	39.7938
	CFFF	3.1379	6.6315	13.3482	3.2345	6.8356	13.7590
	SSSF	9.3487	17.4400	24.6159	9.6364	17.9767	25.3735
2	CCCS	39.666	49.5048	62.6043	40.8874	51.0284	64.5311
	CFFF	3.1104	8.3837	13.8834	3.2061	8.6417	14.3107
	SSSF	11.6057	26.2447	31.0406	11.9629	27.0524	31.9959

Table 6.

First four frequency parameters of nanoplates with linear variations in E and ρ taking α constant.

		γ = 0.3			γ = 0.5
R	B.C.	λ₁	λ₂	λ₃	λ₁	λ₂	λ₃
1	SFFF	5.0584	9.7153	14.2582	4.8518	9.3186	13.6760
	FSSS	9.0715	16.9299	23.8860	8.7011	16.2318	22.9107
	CCCF	17.7065	23.1258	34.5544	16.9834	22.1815	33.1434
2	SFFF	7.1728	9.7544	20.9040	6.8799	9.3561	20.0504
	FSSS	23.8857	29.7554	38.9516	22.9104	28.5404	37.3611
	CCCF	19.9440	35.5227	39.2148	19.1296	34.0722	37.6135

Table 7.

First four frequency parameters of nanoplates with linear variations in E and ρ taking γ constant.

		α = 0.2			α = 0.4
R	B.C.	λ₁	λ₂	λ₃	λ₁	λ₂	λ₃
1	SFFF	5.1774	9.9439	14.5938	5.4076	10.3861	15.2427
	FSSS	9.2850	17.3211	24.4481	9.6978	18.0913	25.5353
	CCCF	18.1231	23.6700	35.3676	18.9290	24.7225	36.9402
2	SFFF	7.3416	9.9840	21.3959	7.6681	10.4279	22.3473
	FSSS	24.4478	30.4556	39.8682	25.5349	31.8099	41.6410
	CCCF	20.4134	37.9754	41.9223	21.3211	37.9754	41.9223

Variation of the fundamental frequency parameter with length is given in Figure 8 for CCCC nanoplates with R = 2, α = 0.1, β = 0.2, γ = 0.3 and δ = 0.4. Results are given for different nonlocal parameters (0 nm², 1 nm², 2 nm², 3 nm²). In this graph, one may see that frequency parameters are highest in the case μ = 0. This shows that frequency parameters are over-predicted in the case of local elasticity theory and hence nonlocal elasticity theory should be considered for the free vibration of nanoplates. This fact may also be seen in terms of relative error percentage (REP) which is defined below.

Figure 8.

Variation of frequency parameter with length.

Let us define the REP as

REP = | Local Result - Nonlocal Result | | Local Result | \times 100

Neglecting nonlocal effect, REPs of the fundamental frequency parameter for a = 5 nm and a = 30 nm with μ = 1 nm² are 46.2839% and 3.1625% respectively. Hence nonlocal theory should be taken into account for vibration analysis of small enough nanoplates. From the above graph, we may also observe that frequency parameters increase with length for each value of the nonlocal parameter. This is due to the size dependency in nonlocal elasticity theory. In other words, size effect is determined by the scaling effect parameter which is e₀l_int divided by a. Therefore, assuming l_int to be constant, the scaling effect parameter is inversely proportional to a. This means that if we increase the length of the nanoplates, then the small scale effect will decrease. As a result, decreasing the small scale effect will increase the frequency parameter. It is also noted that for a particular length, the frequency parameters decrease with increase in the nonlocal parameter.

The effect of aspect ratio on the fundamental frequency parameter is shown in Figure 9 for SSSS nanoplates with a = 10 nm, α = 0.1, β = 0.2, γ = 0.3 and δ = 0.4. From the above graph, one may notice that the nonlocal effect on the frequency parameter is more at higher values of the aspect ratio. This may be due to the fact that for a particular length, as we increase aspect ratio, nanoplates become smaller which will lead to increase in the small scale effect. It is also observed that frequency parameters increase with aspect ratio in all the nonlocal parameters. The following points may be considered in this context. Keeping b constant, aspect ratio R is directly proportional to length a. When aspect ratio R increases, the length will increase. Again, increasing length will increase the frequency parameter as mentioned above. One may notice that when we increase the nonlocal parameter, frequency parameters become smaller than those of its local counterpart. Neglecting nonlocal effect, the REPs for aspect ratios 1 and 3 with μ = 2 nm² are 15.2674% and 41.9565% respectively. This shows that nonlocal theory should be considered for the free vibration of nanoplates with high aspect ratios.

Figure 9.

Variation of frequency parameter with aspect ratio.

Figure 10 shows the variation of the first four frequency parameters with the nonlocal parameter for FFFF nanoplates with a = 5 nm and R = 1. In this figure, the nonlocal parameters are taken as 0 nm², 0.5 nm², 1 nm², 1.5 nm², 2 nm², 2.5 nm², 3 nm² and 4 nm². It is noticed that frequency parameters decrease with the nonlocal parameter. Neglecting the nonlocal effect, the REPs for the first and fourth mode numbers with μ = 1 nm² are found to be 27.5420% and 46.012% respectively. This shows that the nonlocal effect on the frequency parameters is more in higher modes.

Figure 10.

Variation of frequency parameter with nonlocal parameter.

Knowledge of higher modes is necessary before finalizing the design of engineering systems. So some higher mode shapes are given in Figures 11 to 18 for SCSC and CCCC nanoplates with μ = 3, a = 10 nm, R = 1, α = 0.1, β = 0.2, γ = 0.3 and δ = 0.4.

Figure 11.

First mode shape of SCSC nanoplates.

Figure 12.

Second mode shape of SCSC nanoplates.

Figure 13.

Third mode shape of SCSC nanoplates.

Figure 14.

Fourth mode shape of SCSC nanoplates.

Figure 15.

First mode shape of CCCC nanoplates.

Figure 16.

Second mode shape of CCCC nanoplates.

Figure 17.

Third mode shape of CCCC nanoplates.

Figure 18.

Fourth mode shape of CCCC nanoplates.

5. Conclusions

A computationally efficient numerical method has been developed to investigate the dependence of frequency parameters on Young's modulus and the density of nanoplates. Boundary characteristic orthogonal polynomials have been generated with the help of the Gram–Schmidt process. A convergence study is given for a particular set of parameters to observe the accuracy of the method. Results are compared with special cases and are found to be in good agreement. It depends upon the application where design engineers need such frequency parameters. The proposed method may more easily handle all types of boundary conditions at the edges and may be applied in some complicated problems. The present analysis has great significance for design engineers dealing with nanoplates in obtaining the desired frequency parameters. Moreover, the following conclusions may be drawn in this analysis:

Frequency parameters increase with mode number.

Frequency parameters decrease with δ, γ and increase with α, β.

The nonlocal effect on the frequency parameter is more in higher values of aspect ratio.

Frequency parameters increase with length for a particular set of parameters.

Frequency parameters decrease with the nonlocal parameter in all the modes of vibration.

Size-dependency plays a vital role in the vibration analysis of small enough nanoplates, the vibration analysis of nanoplates with high aspect ratios, and better prediction of the frequency parameters associated with higher modes.

Footnotes

Acknowledgment

The authors are thankful to the anonymous reviewers for their valuable suggestions to improve the paper.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Appendix

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