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.

References

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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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