Khorshidi, K., Fallah, A. (2017). Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories. Mechanics of Advanced Composite Structures, 4(2), 127-137. doi: 10.22075/macs.2017.1800.1094
Korosh Khorshidi; Abolfazl Fallah. "Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories". Mechanics of Advanced Composite Structures, 4, 2, 2017, 127-137. doi: 10.22075/macs.2017.1800.1094
Khorshidi, K., Fallah, A. (2017). 'Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories', Mechanics of Advanced Composite Structures, 4(2), pp. 127-137. doi: 10.22075/macs.2017.1800.1094
Khorshidi, K., Fallah, A. Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories. Mechanics of Advanced Composite Structures, 2017; 4(2): 127-137. doi: 10.22075/macs.2017.1800.1094
Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories
In the present study, a vibration analysis of functionally graded rectangular nano-/microplates was considered based on modified nonlinear coupled stress exponential and trigonometric shear deformation plate theories. Modified coupled stress theory is a non-classical continuum mechanics theory. In this theory, a material-length scale parameter is applied to account for the effect of nanostructure size that earlier classical plate theories are not able to explain. The material properties of the plate were assumed to vary according to a power-law form in the thickness direction. The governing equation of the motion of functionally graded, rectangular nano-/microplates with different boundary conditions were obtained based on the Rayleigh-Ritz method using complete algebraic polynomial displacement and rotation functions. The advantage of the present Rayleigh-Ritz method is that it can easily handle the different conditions at the boundaries of moderately thick rectangular plates (e.g., clamped, simply supported, and free). A comparison of the results with those available in the literature has been made. Finally, the effect of various parameters, such as the power-law index, thickness-to-length scale parameter ratio h/l, and aspect ratio a/b, on the natural frequency of nano/micro-plates are presented and discussed in detail.
Free Vibration Analysis of Size-Dependent, Functionally Graded, Rectangular Nano/Micro-plates based on Modified Nonlinear Couple Stress Shear Deformation Plate Theories
K. Khorshidi a,b*, A. Fallah a
a Department of Mechanical Engineering, Arak University, Arak, Iran
b Institute of Nanosciences & Nanotechnolgy, Arak University, Arak, Iran
Paper INFO
ABSTRACT
Paper history:
Received 2016-12-17
Revised 2017-02-07
Accepted 2017-03-01
In the present study, a vibration analysis of functionally graded rectangular nano-/microplates was considered based on modified nonlinear coupled stress exponential and trigonometric shear deformation plate theories. Modified coupled stress theory is a non-classical continuum mechanics theory. In this theory, a material-length scale parameter is applied to account for the effect of nanostructure size that earlier classical plate theories are not able to explain. The material properties of the plate were assumed to vary according to a power-law form in the thickness direction. The governing equation of the motion of functionally graded, rectangular nano-/microplates with different boundary conditions were obtained based on the Rayleigh-Ritz method using complete algebraic polynomial displacement and rotation functions. The advantage of the present Rayleigh-Ritz method is that it can easily handle the different conditions at the boundaries of moderately thick rectangular plates (e.g., clamped, simply supported, and free). A comparison of the results with those available in the literature has been made. Finally, the effect of various parameters, such as the power-law index, thickness-to-length scale parameter ratio h/l, and aspect ratio a/b, on the natural frequency of nano/micro-plates are presented and discussed in detail.
The potential and applications of nano-/micromaterials in the development of technologies such as electronics, energy, environmental remediation, nano-/microsystem, medical and health, future transportation, etc. are important factors that encourage scientists to choose it for future projects. Today, scientists and engineers can reduce production costs, energy consumption, and maintenance with the aid of nanotechnology and its integration with other technologies. Also, using nano-/microtechnology has increased the durability of engineering structures.
Generally, size-dependent material models can be developed based on size-dependent continuum theories like classical couple stress theory [1], nonlocal elasticity theory [2], and strain gradient theory [3]. Couple stress theory is one of the higher-order continuum theories that contains material-length scale parameters and can cover the size effects of nano-/microstructures.
Functionally graded materials (FGMs) are heterogeneous composite materials in which the material properties vary continuously from one surface to the other surface. This is achieved by gradually varying the volume fraction of mixture materials. The merit of using these materials is that they can survive high thermal gradient environments. FGMs were first used as thermal barrier materials for aerospace structural applications and fusion reactors. Recently, they have been developed for general application as structural components in high-temperature environments [4]. Typically, an FGM is a mixture of ceramic and metal for the purpose of thermal protection against large temperature gradients. The ceramic material provides high-temperature resistance due to its low thermal conductivity, while the ductile metal prevents fracture due to its greater toughness. Because of the wide use of nano-/microplates in engineering applications, the study of functionally graded (FG), rectangular nano-/microplates has received considerable attention in recent years.
Matsunaga [5] analyzed the natural frequencies and buckling stresses of plates made of FG materials by taking into account the effects of transverse shear, normal deformations, and rotatory inertia. By expanding the power series of displacement components, a set of FG plates was derived using Hamilton’s principle. Salehipour et al. [6] have developed a model for static and vibrating FG nano-/microplates based on the modified couple stress and three-dimensional elasticity theories. Ansari et al. [7] investigated the size-dependent vibrational behavior of FG, rectangular, Mindlin microplates, including geometrical nonlinearity. In their work, the FG Mindlin microplate was considered to be made of a mixture of metal and ceramic according to a power-law distribution. Kim and Reddy [8] have presented analytical solutions of a general third-order plate theory that accounts for the power-law distribution of two materials through thickness- and microstructure-dependent size effects. Thai and Vo [9] proposed a size-dependent model for the bending and free vibration of an FG plate based on the modified couple stress theory and sinusoidal shear deformation theory. Shaat et al. [10] developed a new Kirchhoff plate model using a modified couple stress theory to study the bending behavior of nanosized plates, including surface energy and microstructure effects. Lou and He [11] studied the nonlinear bending and free vibration responses of a simply supported, FG microplate lying on an elastic foundation within the framework of the modified couple stress theory, the Kirchhoff/Mindlin plate theory, and von Karman’s geometric nonlinearity. He et al. [12] developed a new, size-dependent model for FG microplates by using the modified couple stress theory. Based on the strain gradient elasticity theory and a refined shear deformation theory, Zhang et al. [13] developed an efficient, size-dependent plate model to analyze the bending, buckling, and free vibration problems of FG microplates resting on an elastic foundation. Lou et al. [14] proposed a unified higher-order plate theory for FG microplates by adopting the modified couple stress theory to capture size effects and using a generalized shape function to characterize the transverse shear deformation. Thai and Kim [15] developed a size-dependent model of the bending and free vibration of an FG Reddy plate. Gupta et al. [16] presented an analytical model for the vibration analysis of partially cracked isotropic and FG, rectangular plates based on a modified couple stress theory. Li and Pan [17] developed a size-dependent, FG, piezoelectric microplate model based on the modified couple stress and sinusoidal plate theories. Nguyen et al. [18] studied the size-dependent behaviours of FG microplates using a novel quasi-3D shear deformation theory based on modified couple stress theory. Lei et al. [19] presented a size-dependent FG microplate model based on a modified couple stress theory requiring only one material-length scale parameter. Jandaghian and Rahmani [20] investigated the free vibration analysis of FG, piezoelectric-material, nanoscale plates based on Eringen's nonlocal Kirchhoff plate theory under simply supported–edge conditions. Şimşeka and Aydınc [21] considered the static bending and forced vibration of an imperfect FG microplate carrying a moving load based on Mindlin plate theory and the modified couple stress theory. Thai and Choi [22] presented an analytical solution for size-dependent models for the bending, buckling, and vibration of FG Kirchhoff and Mindlin plates based on modified couple stress theory. Khorshidi et al. [23], investigated the free vibrations of size-dependent, FG, rectangular plates with simply supported–boundary conditions based on nonlocal, exponential shear deformation theory using a Navier-type solution. Khorshidi and Fallah [24] analyzed the buckling response of FG, rectangular nanoplates with all edges simply supported based on nonlocal, exponential shear deformation theory according to Navier-type solutions. Khorshidi and Khodadadi [25] used a new, refined trigonometric shear deformation plate theory to study the out-of-plane vibration of rectangular, isotropic plates with different boundary conditions. Reddy and Kim [26] adopted a higher-order shear deformation theory to develop a size-dependent model for FG microplates. Simsek and Reddy [27] examined the bending and free vibration of microbeams based on various higher-order beam theories. Using first-order plate theory, Jung et al. [28, 29] investigated the buckling, static deformation, and free vibration of sigmoid, FG-material nano-/microplates embedded in a Pasternak elastic foundation.
The free vibration problem of plates can be solved using either the energy functional or the governing partial differential equations. Both can be taken by using standard analytical and numerical techniques. Among the techniques available are the finite element method [30], the boundary element method [31], the finite difference method [32], the differential quadrature method [33], the collocation method [34], the Galerkin method [35], and the Ritz method [36–39]. In this article, a modified couple stress theory according to the nonlinear exponential and trigonometric shear deformation theories was applied to analyze the free vibration of FG, rectangular nano-/microplates. The natural frequencies of the FG nano-/microplates were calculated using the Rayleigh-Ritz method based on minimizing the Rayleigh quotient. The novelty of the present paper is that the analytical solution was developed for size-dependent, FG, rectangular nano-/microplates using the modified nonlinear couple stress shear deformation theories for a combination of different boundary conditions (i.e., simply supported [S], clamped [C], and free [F]), as follows: SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SCCC, CFCF, SSFF, CFSF, CFFF, SFCS, CFCC, SFCC, FFCC, CFCS, CSFF, SFFF, and FFFF. A comparison of the results with those available in the literature has been made. Finally, the effect of various parameters such as the power-law index, thickness-to-length scale parameter ratio (h/l), and aspect ratio (a/b) on the natural frequencies of nano-/microplates are presented and discussed in detail.
Governing Equation of Motion and Solution Procedure
Consider a size-dependent, rectangular nano-/microplate with uniform thickness h, length a, and width b made up of FG material as shown in Fig. 1. The properties of the nano-/microplate are assumed to vary through the thickness of the nanoplate according to a power-law distribution of the volume fractions of two materials between the two surfaces. The top surface ( ) of the size-dependent plate is fully ceramic, whereas the bottom surface ( ) is fully metal. The plate regions are given by Eq. (1) as follows:
, ,
(1)
where , and are Cartesian coordinates. Poisson’s ratio of the plate ϑ is assumed to be constant for ceramic and metal throughout the analysis.
Young’s modulus and mass density are assumed to vary continuously through the plate thickness direction as
,
(2)
,
(3)
,
(4)
where the subscripts m and c represent the metallic and ceramic constituents, respectively; is the plate density per unit area of the FG plate; is the Young's modulus of the FG plate; is the volume fraction; and g is the power-law index and takes only positive values.
Figure 1. Rectangular plate geometry, dimensions, and coordinate system
According to Eqs. (2) and (3), when the power-law index g approaches zero or infinity, the plate is fully ceramic or metal, respectively. According to the following assumptions, the displacement field of the proposed plate theory is given as follows:
The displacement components u and v are the in-plane displacements in the x and y directions, respectively, and w is the transverse displacement in the z-direction. These displacements are small in comparison with the plate thickness.
The in-plane displacement u in the x direction and v in the y direction each consist of two parts.
(a) A displacement component similar to displacement in classical plate theory.
(b) A displacement component due to shear deformation, which is assumed to be exponential in exponential shear deformation theory and trigonometric in trigonometric shear deformation theory with respect to the thickness coordinate.
The transverse displacement w in the z direction is assumed to be a function of the x and y coordinates.
Based on the assumptions mentioned above, the displacement field can be described as
,
(5)
,
(6)
,
(7)
where, for exponential shear deformation plate theory, , and for trigonometric shear deformation plate theory, [23–25]. Also , , and are the displacement in the , , and directions, respectively; and are the mid-plane displacements; and and are the rotation functions. With the assumed linear von Karman strain, the displacement-strain field will be as follows [22]:
,
(8)
,
(9)
,
(10)
,
(11)
.
(12)
In the Eqs. (8–12), are normal strains and are shear strains. Considering Hooke's law for stress fields, the normal stress is assumed to be negligible in comparison within plane stresses and . Thus, the stress-strain relationship will be as follows:
,
(13)
,
(14)
,
(15)
,
(16)
(17)
where is the shear modulus of the plate.
In the modified couple stress theory, the strain energy of a linearly elastic continuum body on volume ∀ is defined by a function of both strain tensor and curvature tensor as
,
(18)
where , and are the components of the stress, normal strains, and shear strain tensors, respectively [1]. Also, are the components of the deviatoric part of the symmetric couple stress tensor, and are the components of the symmetric curvature tensor defined by
,
(19)
,
(20)
where is the length scale parameter, and are the components of the rotation vector related to the displacement field. These are defined as follows:
,
(21)
,
(22)
.
(23)
The kinetic energy of the FG nano-/microplate is defined as follows:
,
(24)
where the dot-top index contract indicates the differentiation with respect to the time variable.
In this section, the Rayleigh-Ritz method is employed to analyze the free vibration of the FG, rectangular nano-/microplates using coupled stess theory. In the Rayleigh-Ritz method, the admissible trial displacement and rotation functions can be introduced as follows [34]:
,
(25)
,
(26)
,
(27)
,
(28)
,
(29)
where , , , and are the generalized constant coefficients of the admissible trial functions; is the natural frequency of the plate; is the imaginary number; is the order of approximation; and are the fundamental functions. The fundamental functions of the moderately thick, FG, rectangular nano-/microplates that satisfy the geometric boundary conditions are introduced as
.
(30)
The fundamental functions for different boundary conditions of the moderately thick, FG, rectangular nano-/microplates that were considered in the present study are listed in Table 1. In the Rayleigh-Ritz approach, the Lagrangian function of the system is given as
.
(31)
With the application of the Rayleigh-Ritz minimization method, the eigenvalue equation can be derived from Eq. (32).
,
(32)
where is the vector of generalized coordinates and contains an unknown, undetermined coefficient. Eq. (32) can be written in matrix form as below:
,
(33)
where
, ,
(34)
is the stiffness matrix, and is the mass matrix. This eigenvalue problem is solved to obtain the natural frequency parameters and vibration modal shapes of the FG, rectangular nano-/microplate.
Table 1. Fundamental functions of the admissible trial displacement and rotation functions for different combinations of boundary conditions
Boundary Conditions
Fundamental Functions
SSSS
SCSS
SCSC
SSSF
SFSF
SCSF
CCCC
SSCC
SCCC
CFCF
SSFF
CFSF
CFFF
SFCS
CFCC
SFCC
FFCC
CFCS
CSFF
SFFF
FFFF
1
1
1
1
1
Numerical Results and Disscusions
In this section, the natural frequency parameters are obtained from the Rayleigh-Ritz method, presented here, and expressed in dimensionless form as . Numerical calculations have been performed for different combinations of boundary conditions (SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SCCC, CFCF, SSFF, CFSF, CFFF, SFCS, CFCC, SFCC, FFCC, CFCS, CSFF, SFFF, FFFF). In the numerical calculations, Poisson’s ratio has been used. The FG nano-/microplate is made up of the following material properties: , , , and the small scale parameter is .
Table 2 shows a comparison study of the nondimensional natural frequency parameters ( ) for a simply supported, FG, square nano-/microplate with those reported based on Mindlin plate theory by Thai and Choi [22]. The effect of the length scale parameter and length-to-thickness ratio a/h on the first two nondimensional natural frequency parameters for simply supported, FG, rectangular nano-/microplates with a/b = 1 and different power-law indices are shown in Table 2.
From the results shown in Table 2, it can be observed that the present results, which were obtained by the Rayleigh-Ritz method, have greater values than those reported by Thai and Choi [22]. This is because in the Rayleigh-Ritz method, the admissible trial displacement and rotation functions that can satisfy the different boundary conditions at all edges of the plate are in the form of a finite polynomial series. Reducing the number of series terms decreases the degree of freedom of the plate and increases the stiffness and frequency parameter, in contrast with what was reported in Thai and Choi’s work based on the Navier method (exact solution) [22]. Moreover, the different distribution of shear stress and rotary inertia in the thickness direction led to differences in the gained results, which are explained by the exponential, trigonometric, and first-order shear deformation plate theories. The results in Table 2 show that there is a good agreement between the present results and those of Thai and Choi [22].
Tables 3 and 4 show the effect of different boundary conditions, power-law index ( , 1, and 10) and aspect ratios (a/b = 0.2, 0.5, and 1) on the dimensionless natural frequency ( ) of FG, rectangular nano-/microplates using the exponential and trigonometric shear deformation plate theories. From the results presented in Tables 3 and 4, it can be observed that an increasing aspect ratio (a/b) leads to an increase in the dimensionless natural frequency parameters because decreasing the width of a plate with a constant length decreases the degrees of freedom of the plate and increases the stiffness.
Table 2. Comparison of nondimensional natural frequency of an FG nano-/microplate with all edges simply supported
TSDT
ESDT
Ref. [22]
TSDT
ESDT
Ref. [22]
TSDT
ESDT
Ref. [22]
5
0
5.48130
5.48140
5.38710
4.99490
4.99510
4.87440
5.66210
5.66230
5.58180
12.1209
12.1214
11.6717
10.9028
10.9035
10.7905
12.2316
12.2316
11.9931
11.2690
11.2692
11.1311
11.2779
11.2784
4.0451
11.2364
11.2364
11.1666
23.9416
23.9418
23.7023
23.8609
23.8615
23.6723
23.8985
23.8987
23.7146
10
0
6.21220
6.21220
5.93010
5.39620
5.39630
5.26970
5.13090
5.13120
5.09030
14.2254
14.2256
14.0893
12.7138
12.7139
12.6460
14.6621
14.6625
14.6464
12.9139
12.9143
12.6360
12.6693
12.6693
12.4128
12.7405
12.7409
12.7302
29.6572
29.6576
29.4588
29.1949
29.1953
29.1174
30.0109
30.0113
29.6008
20
0
6.35950
6.35960
6.09970
5.55870
5.5590
5.38800
6.56880
6.56910
6.38370
15.1465
15.1466
15.0319
13.4209
13.4210
13.3192
15.8744
15.8748
15.7108
13.4285
13.4287
13.1786
13.2017
13.2020
12.8871
13.3998
13.3999
13.3030
32.5074
32.4947
32.4952
32.2374
31.6689
31.6689
31.6012
32.6132
32.6133
Table 3. Comparison of the fundamental nondimensional natural frequency parameter for SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SFCC, and FFCC, FG, square nano-/microplates for different aspect ratios and power-law index values
ESDT
TSDT
a/b
B.Cs.
7.03770
6.70340
6.74270
7.03750
6.70330
6.74260
0.2
SSSS
9.77880
7.85750
7.92220
9.77850
7.85740
7.92210
0.5
12.9404
12.8693
12.9143
12.9402
12.8691
12.9141
1
7.05570
6.72190
6.76450
7.05530
6.72160
6.76440
0.2
SCSS
9.38490
5.76980
8.46440
9.38480
5.76960
8.46420
0.5
14.7465
14.0618
14.3214
14.7463
14.0617
14.3211
1
7.07760
6.74540
6.79230
7.07760
6.74500
6.79180
0.2
SCSC
15.6274
14.8164
15.6684
15.6274
14.8162
15.6683
0.5
18.1387
17.3800
17.7942
18.1385
17.3798
17.7941
1
1.12700
1.09390
1.10290
1.12500
1.09340
1.10260
0.2
SSSF
8.72160
8.34450
8.24750
8.72150
8.34450
8.24740
0.5
8.72160
8.24750
8.34450
8.72150
8.24750
8.34410
1
0.27100
0.24870
0.25730
0.27090
0.24870
0.25680
0.2
SFSF
1.78370
1.53100
1.70780
1.78350
1.53900
1.70760
0.5
7.51420
7.19320
7.28030
7.51360
7.19290
7.28010
1
2.79750
2.71570
2.77240
2.79740
2.71550
2.77230
0.2
SCSF
4.40560
4.25520
4.32510
4.40540
4.25510
4.32500
0.5
9.21430
9.04810
9.17290
9.21430
9.04800
9.17270
1
15.0082
14.4513
14.8819
15.0081
14.4513
14.8817
0.2
CCCC
15.9753
15.3828
15.7562
15.9753
15.3826
15.7562
0.5
23.9918
2.35750
23.4679
23.9911
2.35710
23.4676
1
10.5243
10.1382
10.3519
10.5238
10.1382
10.3518
0.2
SSCC
11.4301
10.9700
11.2156
11.4295
10.9700
11.2156
0.5
17.3995
16.8890
17.2510
17.3994
16.8890
17.2500
1
2.85220
2.76970
2.82770
2.85210
2.76970
2.82760
0.2
SFCC
5.04160
4.88120
4.98250
5.04160
4.88110
4.98240
0.5
13.1621
12.7617
13.0365
13.1615
12.7611
13.0364
1
2.50770
2.42680
2.48490
2.50750
2.42670
2.48450
0.2
FFCC
3.14900
3.05610
3.12160
3.14700
3.05580
3.12170
0.5
5.29150
5.18080
5.25770
5.29150
5.18070
5.25760
1
Table 4. Comparison of the fundamental nondimensional natural frequency parameter for CFCS, CFCF, SSFF, CFSF, CFFF, SCCC, CFCC, SFCS, CSFF, SFFF, and FFFF, FG, square nano-/microplates for different aspect ratios and power-law index values.
ESDT
TSDT
a/b
B.C
*
1.35310
1.31750
1.33920
1.35290
1.31710
1.33890
0.2
CFCS
5.05720
4.89360
4.99820
5.05700
4.89340
4.99810
0.5
17.8132
17.3941
17.7298
17.8126
17.3940
17.7295
1
0.65900
0.69230
0.65680
0.65800
0.69230
0.65660
0.2
CFCF
4.32290
4.18700
4.28060
4.32290
4.18690
4.27970
0.5
19.8915
16.8847
19.7786
19.8914
16.8847
19.7784
1
0.20740
0.54510
0.55020
0.20700
0.54510
0.54950
0.2
SSFF
1.40520
1.36300
1.37260
1.40500
1.36100
1.37250
0.5
2.95410
2.78790
2.83330
2.95380
2.78790
2.83340
1
0.44400
0.42000
0.43520
0.44380
0.41960
0.43440
0.2
CFSF
2.89690
2.78050
2.85190
2.89660
2.78050
2.85180
0.5
11.8647
11.4962
11.7224
11.8645
11.4958
11.7223
1
0.19600
0.09900
0.10260
0.19550
0.09840
0.10220
0.2
CFFF
0.67300
0.64800
0.66570
0.67290
0.64780
0.66530
0.5
2.84280
2.67270
2.73150
2.84260
2.67270
2.73140
1
14.9707
14.4491
14.7614
14.9706
14.4490
14.7612
0.2
SCCC
15.6274
15.0347
15.3964
15.6270
15.0343
15.3964
0.5
20.3359
19.7427
20.2216
20.3358
19.7427
2.22150
1
2.91680
2.83320
2.89300
2.91670
2.83310
2.89300
0.2
CFCC
5.98950
5.81000
5.93590
5.98900
5.89910
5.93560
0.5
18.1564
17.7509
18.1155
18.1563
17.7508
18.1151
1
1.23070
1.19640
1.21090
1.22990
1.19590
1.21040
0.2
SFCS
3.90480
3.75220
3.82210
3.90450
3.75200
3.82210
0.5
12.5125
12.2400
12.4766
12.5124
12.2398
12.4761
1
0.60808
0.59630
0.60450
0.60870
0.59610
0.60400
0.2
CSFF
1.78730
1.73100
1.74190
1.78730
1.73000
1.74180
0.5
4.44620
4.35090
4.38030
4.44600
4.35090
4.37990
1
0.08840
0.08440
0.08780
0.08800
0.08430
0.08740
0.2
SFFF
0.58860
0.55230
0.57920
0.58840
0.55210
0.57890
0.5
2.81410
2.04150
2.17190
2.81390
2.04140
2.17170
1
0.03810
0.03430
0.03750
0.03790
0.03400
0.03750
0.2
FFFF
0.34860
0.32080
0.33190
0.34800
0.32070
0.33190
0.5
1.22530
1.17160
1.21460
1.22510
1.17150
1.21410
1
As the results show in Tables 2–4, the effect of the power-law index on dimensionless natural frequencies is very interesting. It is observed that increasing the power-law index value initially decreases, reaches a minimum, and then increases the frequency. This is because decreasing or increasing the dimensionless natural frequency depends on the kind of material researchers choose to study. For example, Matsunga [5] presented the free vibration and stability of FG plates according to a 2D, higher-order deformation theory in which the frequency parameter decreases with an increase in the power-law index. On the other hand, Thai and Choi [22] analyzed size-dependent, FG, Kirchhoff and Mindlin plate models based on a modified couple stress theory. Their results show that the frequency parameter decreases first and then rises. This phenomenon could be due to the fact that the frequency parameter of FG materials are dependent on both Young’s modulus (Young’s modulus plate rigidity) and density (density plate softening). In the presented material research, and similar to the results reported by Thai and Choi [22], with an increase in the power-law index, the dimensionless natural frequency decreases first and then rises because an increase in the power-law index in this research’s material caused Young’s modulus and density to decrease. A reduction in the Young’s modulus, consequently, caused the plates rigidity and frequency parameter to decrease. However, a decrease in the density leads to an increase in the frequency parameter. So, first, the effect of the Young’s modulus is greater than the effect of density on the frequency parameter; consequently, the dimensionless natural frequency first decreases. But after reaching a minimum, the effect of the density becomes greater than the effect of Young’s modulus, and it causes the dimensionless natural frequency to increase. By comparing the obtained dimensionless natural frequencies of the different boundary conditions that are shown in Tables 2–6, it was found that the dimensionless natural frequencies increase as the degrees of freedom of the plate decrease (increasing the geometric constraints on the edges of the plate). Because of the decreased degree of freedom at each edge of the rectangular plate, the plate gets stiffer, leading to increased dimensionless natural frequencies.
Tables 5 and 6 show the effect of the different boundary conditions and length-to-thickness ratios (a/h = 5, 8, 15, and 20) on the 3 first dimensionless natural frequencies ( ) of homogeneous, rectangular nano-/microplates using the exponential and trigonometric shear deformation plate theories. From the results in Tables 5 and 6, it can be found that, with an increase in the length-to-thickness ratio (constant length and thickness decreases), the dimensionless natural frequency increases. From these results, it can be seen that if the thickness increases, the effective stiffness and effective mass of the plate increase, but the growth of the effective stiffness is greater than the effective mass, so the natural frequency of the nano-/microplate increases.
As shown in Table 2, it can be found that the dimensionless natural frequency of nano-/microplates according to couple stress theory is greater than the dimensionless natural frequency of the plate, due to classical linearly elastic continuum mechanics ( ). This is because the potential energy of linearly elastic continuum mechanics is only defined by a function of the strain tensor in the classical exponential and trigonometric shear deformation plate theories.
Table 5. Comparison of the three first nondimensional natural frequency parameters for SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SCCC, CFCC, SSCC, and SFCC, homogeneous, square nano-/microplates for different length-to-thickness ratios (g = 0, a/b = 1, )
ESDT
TSDT
a/h
B.C
Third mode
Second mode
First mode
Third mode
Second mode
First mode
19.0885
17.3734
7.51870
19.0881
17.3733
7.51870
5
SSSS
27.7967
2404702
10.0574
27.7967
24.4701
10.0573
8
48.3788
41.5901
16.5255
48.3785
41.5901
16.5254
15
63.4543
54.2770
21.4145
63.4540
54.2730
21.4144
20
19.7053
19.0885
8.78180
19.7049
19.0883
8.78180
5
SCSS
28.9468
28.6739
12.0817
28.9464
28.6735
12.0816
8
50.3357
49.9104
20.1517
50.3354
49.9971
20.1517
15
65.9542
65.4489
26.1877
65.9540
65.4489
26.1871
20
19.0952
19.0816
10.4078
19.0949
19.0814
10.4071
5
SCSC
30.4398
30.0424
14.8677
30.4398
30.0424
14.8677
8
57.2944
53.0401
24.8475
57.2942
53.0401
24.8474
15
76.2987
68.8502
32.9207
76.2986
68.8501
32.9207
20
12.3624
12.1604
5.61230
12.3623
12.1597
5.61190
5
SSSF
19.4566
17.8130
7.75900
19.4560
17.8060
7.75880
8
36.4812
30.6762
12.8755
36.4799
30.6762
12.8753
15
48.6416
401171
16.7025
48.6415
40.1169
16.7024
20
8.97430
7.07110
4.50870
8.97390
7.07060
4.50840
5
SFSF
14.3589
9.86410
6.15520
14.3589
9.86400
6.15490
8
26.9229
16.5391
16.2031
26.9227
16.5390
16.2031
15
35.8972
21.4781
13.2281
35.8966
21.4775
13.2273
20
12.3628
12.1609
5.61270
12.3625
12.1599
5.61260
5
SCSF
19.4569
17.8170
7.73590
19.4568
17.8168
7.73590
8
36.4815
30.6764
12.8759
36.4815
30.6763
12.8754
15
48.6419
40.1173
16.7029
48.6419
40.1171
16.7027
20
50.7356
36.5427
33.6441
50.7359
36.5426
33.6435
5
CCCC
27.0417
19.6007
19.2698
27.0412
19.5999
19.2698
8
50.7359
36.5427
33.6441
50.7354
36.5425
33.6437
15
67.4061
48.5932
44.6753
67.4057
48.5931
44.6753
20
19.0906
19.0848
11.6717
19.0905
19.0884
11.6715
5
SCCC
30.5488
30.5417
16.9560
30.5487
30.5411
16.9500
8
57.2544
57.1773
28.6683
57.2543
57.1772
28.6675
15
82.0035
76.3459
37.7139
82.0034
76.3458
37.7139
20
15.0734
12.1604
10.1567
15.0731
12.1604
10.1566
5
CFCC
22.2911
19.4566
14.9637
22.2904
19.4554
14.9629
8
38.8576
36.4812
26.0918
38.8571
36.4809
26.0903
15
50.9237
48.6416
34.2048
50.9229
48.6405
34.2044
20
20.5762
19.0885
10.2617
20.5756
19.0878
10.2615
5
SSCC
30.5416
30.1799
14.4481
30.5415
30.1797
14.1480
8
57.2654
52.4612
24.4775
57.2644
52.4607
24.4789
15
76.3539
68.7305
31.9135
76.3535
68.7304
31.9134
20
16.9181
12.1604
7.63680
16.9177
12.1697
7.63620
5
SFCC
19.6768
19.4566
10.8800
19.6766
19.4565
10.8778
8
36.4812
34.0332
18.5534
36.4805
34.0324
18.5530
15
48.6416
44.5388
24.2071
48.6414
44.5387
24.2066
20
Table 6. Comparison of the three first nondimensional natural frequency parameter for SSCC, SFCC, FFCC, CFCS, SFCS, CSFF, CFCF, SSFF, CFSF, CFFF, SFFF, and FFFF, homogeneous, square nano-/microplates for different length-to-thickness ratios (g = 0, a/b = 1, )
ESDT
TSDT
a/h
B.C
Third mode
Second mode
First mode
Third mode
Second mode
First mode
20.5762
19.0885
10.2617
20.5756
19.0878
10.2615
5
SSCC
30.5416
30.1799
14.4481
30.5415
30.1797
14.1480
8
57.2654
52.4612
24.4775
57.2644
52.4607
24.4789
15
76.3539
68.7305
31.9135
76.3535
68.7304
31.9134
20
16.9181
12.1604
7.63680
16.9177
12.1697
7.63620
5
SFCC
19.6768
19.4566
10.8800
19.6766
19.4565
10.8778
8
36.4812
34.0332
18.5534
36.4805
34.0324
18.5530
15
48.6416
44.5388
24.2071
48.6414
44.5387
24.2066
20
10.1397
7.74920
3.09180
10.1395
7.74880
3.09160
5
FFCC
13.6711
12.3987
4.38680
13.6710
12.3987
4.38650
8
23.2475
22.8323
7.59510
23.2474
22.8321
7.59480
15
30.9967
29.6265
9.96270
30.9967
29.6263
9.96240
20
16.9181
12.1604
9.92720
16.9177
12.1604
9.27710
5
CFCS
20.0742
19.4566
14.6355
20.0741
19.4550
14.6350
8
36.4812
34.6875
25.5669
36.4810
34.6873
25.5664
15
48.6416
45.3792
33.5370
48.6414
45.3788
33.5368
20
16.9181
12.0543
7.33010
16.9175
12.0537
7.32940
5
SFCS
19.4566
17.1289
10.4140
19.4562
17.1287
10.4110
8
36.2810
29.1811
17.7667
36.2807
29.1804
17.7661
15
48.6416
38.0750
23.1912
48.6409
38.0744
23.1911
20
10.1397
7.74920
2.51770
10.1395
7.74920
2.51760
5
CSFF
12.3987
10.8310
3.60230
12.3984
10.8308
3.60220
8
23.2475
17.9114
6.29800
23.2474
17.9114
6.29700
15
30.9967
23.2027
8.28150
30.9965
23.2027
8.28140
20
16.5349
11.0593
8.97390
16.5346
11.0581
8.97350
5
CFCF
16.3444
14.3596
14.3243
16.3444
14.3596
14.3237
8
28.4434
26.9226
26.0234
28.4434
26.9226
26.0234
15
37.2508
35.8962
32.7453
37.2508
35.8962
32.7453
20
8.1157
7.32850
1.60520
8.11560
7.32850
1.60500
5
SSFF
11.7012
9.77340
2.27190
11.7010
9.77330
2.27180
8
20.2377
15.8028
3.91690
20.2374
15.8020
3.91660
15
26.4931
20.3743
5.12480
26.4930
20.3741
5.12450
20
15.5240
8.83300
6.83580
15.5240
8.83300
6.83580
5
CFSF
14.3589
12.6404
9.76200
14.3587
12.6404
9.76190
8
26.9229
21.5324
16.7452
26.9226
21.5320
16.7449
15
38.8972
28.0669
21.8875
35.8968
28.0662
21.8871
20
7.96570
3.34000
1.65540
7.96540
3.33970
1.65510
5
CFFF
5.71810
5.34400
2.28760
5.71790
5.34350
2.28730
8
10.0615
10.0199
3.88530
10.0614
10.0199
3.88520
15
13.3599
13.2335
5.07200
13.3599
13.2334
5.07170
20
3.54470
1.48690
0.76020
3.54460
1.48690
0.75970
5
SFFF
3.16780
2.93870
1.27670
3.16750
2.93840
1.27650
8
5.57400
5.55500
2.15720
0.68850
0.53760
0.06170
15
7.41210
7.32740
5.07200
1.34340
0.56070
0.10610
20
1.44240
0.60490
0.45110
1.44230
0.60490
0.45090
5
FFFF
1.28630
1.19490
0.51920
1.28640
1.19480
0.51910
8
2.26600
2.26020
0.87710
2.26100
2.27960
0.87680
15
4.39310
4.34430
2.06430
4.39290
4.34430
2.06410
20
Conclusion
The free vibration of size-dependent, rectangular, FG, nano-/microplates was analyzed based on nonlinear shear deformation plate theories using modified couple stress theory. The modified couple stress theory contains one material-length scale parameter, and it can also be degenerated to the classical FG, rectangular plate by setting the material-length scale parameter equal to zero. Equations of motion for free vibration can be found through an implementation of the Rayleigh-Ritz method, which may satisfy any combination of boundary conditions, including: SSSS, SCSS, SCSC, SSSF, SFSF, SCSF, CCCC, SSCC, SCCC, CFCF, SSFF, CFSF, CFFF, SFCS, CFCC, SFCC, FFCC, CFCS, CSFF, SFFF, and FFFF. Material properties were assumed to change continuously through the thickness according to a power-law distribution. A comparison of the present results with those reported in the literature for size-dependent, rectangular, FG nano-/microplates illustrated the high accuracy of the present study. This research shows the effects of variations of the length scale parameter, length-to-thickness ratio, power-law index, and the aspect ratio as well as different boundary conditions on the free vibration of a size-dependent, rectangular, FG nano-/microplate.
By looking into the present results, the following points may be concluded:
Increasing the aspect ratio (a/b) causes an increase in the dimensionless natural frequency.
With an increase in the power-law index, the dimensionless natural frequency decreases first and then increases.
The dimensionless natural frequencies of the size-dependent, rectangular nano-/microplate increase with a decreasing degree of freedom at the boundary conditions of the plate.
With an increasing length-to-thickness ratio (constant length and thickness decreases), the dimensionless natural frequency increases.
The transverse shear and rotary inertia have a dissimilar effect in the exponential and trigonometric shear deformation plate theories.
The dimensionless natural frequency of the FG, rectangular nano-/microplate based on the couple stress theory is more than the dimensionless natural frequency of the FG, calssical plate.
All analytical results presented here can be provided to other research groups of a reliable source to compare their analytical and numerical solutions.
Acknowledgments
The authors gratefully acknowledge the funding by Arak University, under Grant No. 95/8589.
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References
[1] Yang FA, Chong AC, Lam DC, Tong P. Couple stress based strain gradient theory for elasticity. Int J Solides Struct 2002; 39: 2731-2743.
[2] Eringen AC. Nonlocal polar elastic continua. Int J Eng Sci 1972; 10: 1-16.
[3] Nix WD, Gao H. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 1998; 46: 411-425.
[4] Lu C, Wu D, Chen W, Non-linear responses of nano-scale FGM films including the effects of surface energies. IEEE Trans Nanotechnol 2011; 10(6); 1321–1327.
[5] Matsunaga H. Free vibration and stability of functionally graded plates according to a 2- graded plates according to a 2-D higher-order deformation theory. Compos Struct 2008; 82; 499-512.
[6] Salehipour H, Nahvi H, Shahidi AR. Exact closed-form free vibration analysis for functionally graded micro/nano plates based on modified couple stress and three-dimensional elasticity theories. Compos Struct 2015; 124: 283-291.
[7] Ansari R, Faghih-Shojaei M, Mohammadi V, Gholami R, Darabi MA. Nonlinear vibrations of functionally graded Mindlin microplates based on the modified couple stress theory. Compos Struct 2014; 114: 124-134.
[8] Kim J, Reddy JN. Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory. Compos Struct 2013; 103: 86-98.
[9] Thai H-T, Vo TP. A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory. Compos Struct 2013; 96: 376-383.
[10] Shaat M, Mahmoud FF, Gao X-L, Faheem AF. Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects. Int J Mech Sci 2014; 79: 31-37.
[11] Lou J, He L. Closed-form solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory. Compos Struct 2015; 131: 810-820.
[12] He L, Lou J, Zhang E, Wang Y, Bai Y. A size-dependent four variable refined plate model for functionally graded microplates based on modified couple stress theory. Compos Struct 2015; 130: 107-115.
[13] Zhang B, He Y, Liu D, Shen L, Lei J. An efficient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation. Appl Math Model 2015; 39(13): 3814-3845.
[14] Lou J, He L, Du J. A unified higher order plate theory for functionally graded microplates based on the modified couple stress theory. Compos Struct 2015; 133: 1036-1047.
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