Khorshidi, K., Asgari, T., Fallah, A. (2015). Free Vibrations Analysis of Functionally Graded Rectangular Nanoplates based on Nonlocal Exponential Shear Deformation Theory. Mechanics of Advanced Composite Structures, 2(2), 7993. doi: 10.22075/macs.2015.395
Korosh Khorshidi; Tahmoores Asgari; Abolfazl Fallah. "Free Vibrations Analysis of Functionally Graded Rectangular Nanoplates based on Nonlocal Exponential Shear Deformation Theory". Mechanics of Advanced Composite Structures, 2, 2, 2015, 7993. doi: 10.22075/macs.2015.395
Khorshidi, K., Asgari, T., Fallah, A. (2015). 'Free Vibrations Analysis of Functionally Graded Rectangular Nanoplates based on Nonlocal Exponential Shear Deformation Theory', Mechanics of Advanced Composite Structures, 2(2), pp. 7993. doi: 10.22075/macs.2015.395
Khorshidi, K., Asgari, T., Fallah, A. Free Vibrations Analysis of Functionally Graded Rectangular Nanoplates based on Nonlocal Exponential Shear Deformation Theory. Mechanics of Advanced Composite Structures, 2015; 2(2): 7993. doi: 10.22075/macs.2015.395
Free Vibrations Analysis of Functionally Graded Rectangular Nanoplates based on Nonlocal Exponential Shear Deformation Theory
^{}Department of Mechanical Engineering, Faculty of Engineering, Arak, Iran
Abstract
In the present study the free vibration analysis of the functionally graded rectangular nanoplates is investigated. The nonlocal elasticity theory based on the exponential shear deformation theory has been used to obtain the natural frequencies of the nanoplate. In exponential shear deformation theory an exponential functions are used in terms of thickness coordinate to include the effect of transverse shear deformation and rotary inertia. The nonlocal elasticity theory is employed to investigate the effect of the small scale on the natural frequency of the functionally graded rectangular nanoplate. The govering equations and the corresponding boundary conditions are derived by implementing Hamilton’s principle. To show the accuracy of the formulations, the present results in specific cases are compared with available results in the literature and a good agreement is seen. Finally, the effect of the various parameters such as the nonlocal parameter, the power law indexes, the aspect ratio , and the thickness to lenghth ratioon the natural frequencies of the rectangular FG nanoplates is presented and discussed in detail.
Free Vibrations Analysis of Functionally Graded Rectangular Nanoplates based on Nonlocal Exponential Shear Deformation Theory
K. Khorshidi^{*}, T. Asgari, A. Fallah^{ }
Department of Mechanical Engineering, Faculty of Engineering, Arak, Iran
Paper INFO
ABSTRACT
Paper history:
Received 24 September 2015
Received in revised form 6 November 2015
Accepted 22 November 2015
In the present study the free vibration analysis of the functionally graded rectangular nanoplates is investigated. The nonlocal elasticity theory based on the exponential shear deformation theory has been used to obtain the natural frequencies of the nanoplate. In exponential shear deformation theory an exponential functions are used in terms of thickness coordinate to include the effect of transverse shear deformation and rotary inertia. The nonlocal elasticity theory is employed to investigate the effect of the small scale on the natural frequency of the functionally graded rectangular nanoplate. The govering equations and the corresponding boundary conditions are derived by implementing Hamilton’s principle. To show the accuracy of the formulations, the present results in specific cases are compared with available results in the literature and a good agreement is seen. Finally, the effect of the various parameters such as the nonlocal parameter, the power law indexes, the aspect ratio , and the thickness to lenghth ratioon the natural frequencies of the rectangular FG nanoplates is presented and discussed in detail.
The invention of carbon nanotubes (CNTs) initiated a new era in the nano world [1]. Since then, many studies have been performed in the field of the mechanical, electrical, physical and chemical behaviors of the nanostructures. The primary studies show that the mechanical properties of the nanostructures are different from other wellknown materials [2]. The superior properties of these structures have led to their applications in many fields such as nanodevices, nanooscillators, nanobearings, hydrogen storage, and electrical batteries. The platelike nanostructures such as nanoplates or nanoscale sheets are very important types of the nanostructures with twodimensional shapes. They possess extraordinary mechanical properties[16] and these unique properties make them ideal candidates for multifarious field of nanotechnology industry including energy storage[7], nano electromechanical systems [8], strain, mass and pressure sensors [9, 10], solar cells [11], photocatalytic degradation of organic dye [12], composite materials [13] and ect. The continuum modeling of the nanomaterials has received a great deal of attention of the scientific community because the controlled experiments in nanoscale are difficult and molecular dynamic simulations are highly expensive computationally. There are various size dependent continuum theories such as couple stress theory [14], strain gradient elasticity theory [15], modified couple stress theory [16] and nonlocal elasticity theory [17]. Among these theories, the nonlocal elasticity theory has been widely applied [1824]. To overcome the shortcomings of the classical elasticity theory, Eringen and Edelen[17] introduced the nonlocal elasticity theory in 1972. They modified the classical continuum mechanics for taking into account the small scale effects. According to the nonlocal elasticity theory, the stress tensor at an arbitrary point in the domain of nanomaterial depends not only on the strain tensor at that point but also on the strain tensor at all other points in the domain. Both the atomistic simulation results and the experimental observations on phonon dispersion have shown the accuracy of this observation [25, 26]. The Functionally Graded Materials (FGMs) are the new generation of novel composite materials in the family of engineering composites, whose properties are varied smoothly in the spatial direction microscopically to improve the overall structural performance. These materials offer a great promise in high temperature environments, for example, wearresistant linings for handling large heavy abrasive ore particles, rocket heat shields, heat exchanger tubes, thermoelectric generators, heat heat engine components, plasma facings for fusion reactors, and electrically insulating metal/ceramic joints, and also these are widely used in many structural applications such as mechanics, civil engineering, optical, electronic, chemical, biomedical, energy sources, nuclear, automotive fields, and ship building industries to minimize the thermomechanical mismatch in metalceramic bonding. Most structures, irrespective of their use, are subjected to dynamic loads during their operational life. Increased use of nanoFGMs in various structures such as thermonanoactuators, thermonanosensors, thermaopressure sensors andetcapplications necessitates the development of accurate theoretical models to predict their response.In past decades, the free vibration of functionally graded materials has been studied extensively. Malekzadeh and Heydarpour [27] investigated the free vibration analysis of rotating functionally graded cylindrical shells subjected to thermal environment based on the Firstorder Shear Deformation Theory (FSDT) of shells. The formulation included the centrifugal and coriolis forces due to the rotation of the shell. The differential quadrature method was adopted to discretise the thermoelastic equilibrium equations and the equations of motion. Ungbhakorn and wattanasakulpong[28]presented thermoelastic vibration response of functionally graded plates carring distributed patch mass based on thirdorder shear deformation theory. The solutions were obtained using the energy method. In addition, the forced vibration analysis with external dynamic load acting on the subdomain of the patch mass was also discussed. Kumar and Lal [29] predicted the first three natural frequencies of the free axisymmetric vibration of the twodirectional functionally graded annulr plates resting upon winkler foundation using differential quadrature method and Chabyshev collocation technique. Frequency equations for a plate clamped at both the edges and another plate simply supported at both the edges were obtained. Based on the threedimensional theory of elasticity and assuming that the mechanical properties of the materials varied continuously in the thickness direction and had the same exponentlaw variations, the threedimensional free and forced vibration analysis of functionally graded circular plate with various boundary conditions was achieved by Nie and Zhong[30]. Hung et al [31] investigated the free vibrations of rectangular FGM plates through internal cracks using the Ritz method. Threedimensional elasticity theory was employed, and new sets of admissible functions for the displacement fields were proposed to enhance the effectiveness of the Ritz method in modelling the behaviors of the cracked plates. Matsunaga [32] analysed the natural frequencies and buckling stresses of plates made of functionally graded materials by taking into account the effects of transverse shear and normal deformations and rotatory inertia. By using the method of power series expansion of displacement components, a set of functionally graded (FG) plates was derived using Hamilton’s principle. Malekzadeh and Alibeygi[33] presented the free vibration of functionally graded arbitrary straightsided quadrilateral plates under the thermal environment and based on the firstorder shear deformation theory. The differential quadrature method was adopted to discretise the equilibrium equations. The free vibration of functionally graded micro/nano plates was also considered in recent years.Ke et al [34] developed a nonclassical microplate model for the axisymmetric nonlinear free vibration analysis of annular microplates made of functionally graded materials based on the modified couple stress theory, Mindlin plate theory and vonKarman geometric nonlinearity theory. The nonclassical model was capable of shear deformation and rotary inertia.Ke et al [35] also studied the bendig, buckling and free vibration of annular microplates made of functionally graded materials based on the modified couple stress theory and Mindlin plate theory. The material properties of the FGM microplates were assumed to vary in the thickness direction and were estimated using the MoriTanaka homogenization technique. Asghari and Taati[36] presented a sizedependent formulation for mechanical analyses of inhomogeneous microplates based on the modified couple stress theory. The govering differential equations of the motion were derived for functionally graded plates with arbitrary shapes utilizing a variational approach. Utilizing the derived formulation, the freevibration behaviour as well as the static response of a rectangular FG microplate was proposed. Nataraja et al [37] investigated the sizedependent linear free flexural vibration behaviour of functionally graded nanoplates using the isogeometric based finite element method. The field variables were approximated by nonuniform rational Bsplines. The nonlocal constitutive relation was based on the Eringen’s differential form of the nonlocal elasticity theory.
In the present study, as a first endeavor, the free vibration of functionally graded nanoplates is investigated based on the exponential shear deformation theory and using the nonlocal elasticity theory. The Navier solution is used to study the free vibration of the functionally graded nanoplates. It is assumed that the material properties are varying through the thickness according to the power law distribution. The results of the present work may be used as bench marks for future studies.
2. Theoretical formulations
Consider a rectangular nanoplate of length , width , and total thickness and composed of functionally graded materials through the thickness as shown in Figure 1. The properties of the nanoplate are assumed to vary through the thickness of the nanoplate with a powerlaw distribution of the volume fractions of the two materials between the two surfaces. In fact, the top surface ( ) of the nanoplate is ceramicrich whereas the bottom surface ( ) is metalrich.
Young’s modulus and mass density are assumed to vary continuously through the thickness as what follows [38]
(1)
(2)
where the subscripts and c represent the metallic and ceramic constituents, respectively, and is the volume fraction that may be given by the following equation:
(3)
where is the powerlaw index and takes only posetive values. Poisson’s ratio is the same for all the metal/ceramic materials that are used here, so it is assumed to be constant and is taken to be 0.3 throughout the analysis[39]. The typical values for metals and ceramics used in the FG nanoplate are listed in Table 1.
Figure 1. The geometry of a FGM plate
Table 1. The material properties of the used FG plate
Material
Properties
E (GPa)
Aluminum (Al)
70
0.3
2702
Alumina ( )
380
0.3
3800
Zirconia ( )
200
0.3
5700
2.1. The nonlocal elasticity theory
In nonlocal theory stress field at each body point body is a function of the strain field. So stress plays a major role in the theory which is defined as what follows [40]:
(4)
where is a point on the body that the stress tensor on its efficacy, can be any point else in the body, is the volume of a region of the body that integral is taken on it, is the local stress tensor, is the nonlocal kernel function related to the internal characteristic length. With respect to properties of nonlocal kernel function that are discussed by Eringen[41], taking in a Greens function of a linear differential operator, , can be defined as following,
,
(5)
Substituting Eq. (5) into Eq. (4), the primary relation (1) form of the following differential equation is obtained as
,
(6)
For the nonlocal linear elastic solids, the equations of motion have the following form [42]:
,
(7)
Where is the mass density, body forces and is the displacement vector. Substituting Eq. (7) into Eq. (6) yields to what follows:
,
(8)
The nonlocal model with the linear differential operator for the twodimensional case is defined by the following equation [9]:
,
(9)
where is the Laplace operator, which in cartesian coordinates is and , is the internal characteristic length and is the material constant which is specified by the experiment. The value of the smallscale parameter is dependent on the boundary condition, the chirality, the mode shapes, the number of walls, and the nature of motions [43]. There is no accurate way to calculate this factor, but it is suggested that the coefficient be determined by conducting a comparison of dispersion curves from nonlocal continuum mechanics and lattice dynamics of nanomaterial crystal structure [43].
2.2. The assumptions made in the proposed theory
1. The displacement components and are the inplane displacements of the middle surface in and directions respectively, and is the deflection of the middle surface in direction. The magnitude of the deflection is not of the same order as the thickness of the plate and is small with respect to the plate thickness.
2. The inplane displacements, and , include three parts:
a) A displacement component equivalent to the displacement in the classical plate theory.
b) A displacement component owing to the shear deformation which is assumed to be harmonic in nature with respect to the thickness coordinate.
c) The shear strains in direction are zero in the top and bottom surfaces of the plates.
3. The deflection in z direction is assumed to be a function of and coordinates.
4. The plate is subjected to the transverse load only.
The displacement field of the exponential shear deformation theory is given as below [44]:
,
(10a)
,
(10b)
,
(10c)
Where and , and are displacements in the , and directions respectively, and and are the midplane displacements and are the rotation functions. With the linear assumption of vonKarman strain, the displacement strain field will be as what follows:
i, j=x,y,z
(11)
Considering Hooke's Law for stress field, the normal stress is assumed to be negligible in comparison with the plane stresses and .Thus stressstrain relationship will be as what follows:
,
(12a)
,
(12b)
,
(12c)
,
(12d)
,
(12e)
Substuting Eqs. (11) into Eq. (12), the displacement stress field will be as what follows:
(13)
Using Eq. (6) the stressdisplacement constitutive relation of a nonlocal FG plate can be written as:
(14)
where is the Young's modulus, is the Poisson's ratio, and is the shear modulus of the plate.
The Hamilton’s principle is employed to extract the equation of motion. The Hamilton’s principle in case of local form is obtained as what follows [44]:
,
(15)
where is the variation operator, is the strain energy, is the work done by external forces, and is the kinetic energy.
(16)
where the dottop index contract indicates the differentiation with respect to the time variable. Exertion of variation operator on Eq. 16 should be as follows:
(17a)
(17b)
(17c)
Using Eqs. (17a)(17c) and substituting Eq. (10) and (11) into Eq. (14), the following equations are obtained as follows,
(18)
Using integration by parts and lemma of calculus of variations can be derived the equations of motion and boundary conditions from Eq. (16).
The relations between stress resultants in local and nonlocal theories can be find in Eq. (18), by calculating the coefficients of , , , , , in Eq. (16), the nonlocal equations of motion may be expressed as Eqs. (19a)–(19e).
(19a)
(19b)
(19c)
(19d)
(19e)
where
(20a)
,
(20b)
(20c)
,
(20d)
,
(20e)
,
(20f)
,
(20g)
N and R are the force and M is the moment that is acting on the body. The following sets of boundary conditions at the edges of the plate are obtained as a result of the application of the Hamilton’s principle:
Either or prescribed at
Either or prescribed at
(21a)
Either or prescribed at Either or prescribed at
(21b)
Either or prescribed at
Either or prescribed at
(21c)
Either or prescribed at
Either or prescribed at
(21d)
Either or at
Either or at
(21e)
Introducing the nonlocal stress resultants:
(22a)
(22b)
(22c)
,
(22d)
where
,
(23a)
,
(23b)
,
(23c)
Substituting Eqs. (22a)–(22d) into Eqs. (19a)–(19e), yields to the following equations:
(24a)
(24b)
(24c)
(24d)
(24e)
It can be seen that Eqs. (24a)(24e) are coupled functions of displacements. The permissible displacement and rotation functions that can be satisfied with the simply supported boundary conditions at all edges of the plate are trigonometric series. Using Navier's solution, the explanation of the displacement and rotations is as what follows:
(25a)
(25b)
(25c)
(25d)
(25e)
Where and are the numbers of half wave correlation to x and y directions. , and are the amplitudes of translation and and are the amplitudes of rotations. is the frequency of the linear free vibration. is the width of the edge and is the length of the edge. Substituting Eqs. (25a)(25e) into Eqs. (24a)(24e)yields to what follows:
,
(26)
The requisite for answering equation (26) except the obvious answer is that the determinant of coefficients matrix must be zero. Using this principle can be derived the characteristic equation.
3. Numerical example
Since the results of nanoplate made of FGM are not available in the open literature, to validate the results, in this paper have used two separate parts;in the first part, have been validated Isotropic rectangular nanoplate, and in the second part, it does for FGM one.
3.1. Isotropic rectangular nanoplate
Only homogeneous plate ( ) is used herein for the verification. Tables 24 list the first three nondimensional frequency and Frequency Ratios (FR) for simply supported boundary condition with various values of aspect ratio ( = ), specified values of nondimensional nonlocal parameter ( ) and the thickness to lenghth ratio on rectangular nanoplates. The natural frequency parameters expressed in dimensionless form , are the flexural rigidity. The nanoplate is made of the following material properties: , and ( ). The calculated frequencies based on the nonlocal exponential shear deformation theory are compared with those reported by HosseiniHashemi et al. [40] based on Mindlin Plate Theory (MPT). Also, the Frequency Ratio (FR) relation between the nonlocal and local dimensionless frequencies is expressed as what follows:
(27)
where is the nondimensianal nonlocal frequency parameter, and is the nondimensional local frequency parameter.
Table 2. The variations of the nondimensional frequency ( ) and the frequency ratio (FR) for the nonlocal plate(m=1, n=1)
Method
=0
=0.2
=0.4
=0.6
=0.8
FR
FR
FR
FR
FR
Present
35.015
1.0000
0.6335
0.3789
0.2633
0.2005
Exact [40]
35.0643
1.0000
0.6335
0.3789
0.2633
0.2005
Present
24.2084
1.0000
0.7051
0.4451
0.1346
0.2412
Exact [40]
24.2330
1.0000
0.7050
0.4451
0.3146
0.2412
Present
19.0684
1.0000
0.7475
0.4904
0.3512
0.2708
Exact [40]
19.0840
1.0000
0.7475
0.4904
0.3512
0.2708
Table 3. The variations of the nondimensional frequency ( ) and the frequency ratio (FR) for the nonlocal plate(m=2, n=1)
Method
=0
=0.2
=0.4
=0.6
=0.8
FR
FR
FR
FR
FR
Present
60.1556
1.0000
0.5216
0.2923
0.1997
0.1511
Exact [40]
60.2869
1.0000
0.5216
0.2923
0.1997
0.1511
Present
50.2147
1.0000
0.5594
0.3197
0.2194
0.1663
Exact [40]
50.3100
1.0000
0.5594
0.3197
0.2194
0.1664
Present
45.5048
1.0000
0.5799
0.3353
0.2308
0.1752
Exact [40]
45.5845
1.0000
0.5799
0.3353
0.2308
0.1752
Table 4. The variations of the nondimensional frequency ( ) and the frequency ratio (FR) for the nonlocal plate(m=2, n=2)
Method
=0
=0.2
=0.4
=0.6
=0.8
FR
FR
FR
FR
FR
Present
121.356
1.0000
0.3789
0.2005
0.1352
0.1018
Exact [40]
121.7700
1.0000
0.3789
0.2006
0.1352
0.1018
Present
86.9898
1.0000
0.4451
0.2412
0.1635
0.1233
Exact [40]
87.2357
1.0000
0.4451
0.2412
0.1635
0.1233
Present
69.8517
1.0000
0.4904
0.2708
0.1843
0.1393
Exact [40]
70.0219
1.0000
0.4904
0.2708
0.1844
0.1393
According to the Eqs. (1) and (2),when the power law index approaches zero or infinity,the plate is isotropic composed of fully ceramic or metal, respectively.Three fundamental frequency parameters of SSSS square plate ( ) are presented in Table 5 for and 0.2.The results are compared with those obtained by Shufrin and Eisenberger[45] based on the HSDT. It is found that when gradient index approaches zero or infinity the frequency parameters of FG plate converge to relevant isotropic one. The excellent agreement among the reults confirms the high accuracy of the current analytical approach.
3.2. FGM square plate
Table 6 shows a comparison of the frequency parameters for square moderately thick plates with those obtained by HosseiniHashemi et al. [46], Zhao et al. [47] and Masunaga [32] when , 0.5, 1, 4 and 10. In addition, the corresponding mode shapes m and n, denoting the number of halfwaves in the x and y directions, respectively, are present for any of the frequency parameters . Also, in Table 7,a comparison of the frequency parameters for a simply supported square plates with those of twodimensional higherorder theory [32], threedimensional theory by employing the power series method [48], finite element HSDT method [49], finite element FSDT method[49] and an analytical FSDT solution [46] is shown. From Tables 6 and 7, it is evident that there is a very good agreement among the results confirming the high accuracy of the current analytical approach.
Table 5. The comparison of the fundamental frequency parameters ( ) for square plates ( )
Material
Method
g
0.1
0.2
Fully ceramic
Present
5.7681
5.2826
10
5.7701
5.2843
10
5.7703
5.2845
10
5.7703
5.2845
HSDT[45]
0
5.7694
5.2813
Fully metallic
Present
10
2.8235
2.8235
10
2.7051
2.7051
10
2.9389
2.6913
10
2.9372
2.6900
HSDT[45]
2.9376
2.6891
As it is seen, the present solution reports a good agreement with those obtained by the HSDT [32]for the thicker FG square plates ( ) particularly at the higher modes of vibration. The difference between the present natural frequencies from those obtained by the 3D method [48] may be due to the estimation of the material properties at a point where expressed by the local volume fractions and the material properties of the phases using two methods: MoriTanaka [50, 51]and the selfconsistent scheme [52], whereas, in the present analysis, Material properties of the FGM layer are assumed to vary in the thickness direction according to a power law distribution. The difference between the present solutions from those obtained by the analytical FSDT solutions [46] is caused by vanishing of the inplane displacement components of FG plate in Ref [46]. In fact, as the present procedure provided, the inplane displacement components u and v should be taken into account and are coupled with the transverse displacement components w, , and m and n are wave numbers in directions x and y, respectively.
4. Results and discussion
The vibration modes of FG plate may be divided into two main categories: the outofplane (transverese) modes and the inplane modes. For the inplane modes, the magnitude of the transverse displacement is very smaller than the magnitude of the inplane displacements, u and v. There is a main difference between inplane modes of isotropic plates and FG ones. When an isotropic plate has inplane mode, there is no transverse displacement and the plate can only move along the inplane directions but due to the existing coupling between inplane and outofplane vibration in FG plate, inplane mode includes two kinds of motion whereas inplane vibration is dominant. InTabel 8, based on the present navier solutions and finite element method, the numerical results have been performed for square plates ( ) when p=1. The lenghth of square plates is 1 m. Three different thicknesses 0.05 m (corresponding to thin plates), 0.1 m and 0.2 m (corresponding to moderately thick plate) and 0.3 m (corresponding to thick plates) have been used. All of the calculations are obtained for the first four natural frequencies.
The percentage difference given in Table 8 is defined as what follows:
Table 6. The comparison of the natural frequency parameter ( ) for square plates ( )
(m,n)
Method
G
0
0.5
1
4
10
0.05
(1,1)
Present
0.0148
0.0125
0.0113
0.0098
0.0094
FSDT[46]
0.0148
0.0128
0.0115
0.0101
0.0096
FSDT[47]
0.0146
0.0124
0.0112
0.0097
0.0093
0.1
(1,1)
Present
0.0577
0.0490
0.0442
0.0381
0.0364
HSDT[32]
0.0577
0.0492
0.0443
0.0381
0.0364
FSDT[46]
0.0577
0.0492
0.0445
0.0383
0.0363
FSDT[47]
0.0568
0.0482
0.0435
0.0376
0.3592
(1,2)
Present
0.1377
0.1174
0.1059
0.0902
0.0856
HSDT[32]
0.1381
0.1180
0.1063
0.0904
0.0859
FSDT[47]
0.1354
0.1154
0.1042

0.085
(2,2)
Present
0.2114
0.1808
0.1632
0.1377
0.1300
HSDT[32]
0.2121
0.1819
0.1640
0.1383
0.1306
FSDT[47]
0.2063
0.1764
0.1594

0.1289
0.2
(1,1)
Present
0.2114
0.1808
0.1632
0.1377
0.1300
HSDT[32]
0.2121
0.1819
0.1640
0.1383
0.1306
FSDT[46]
0.2112
0.1806
0.1650
0.1371
0.1304
FSDT[47]
0.2055
0.1757
0.1587
0.1356
0.1284
(1,2)
Present
0.4629
0.3993
0.3611
0.2976
0.2772
HSDT[32]
0.4658
0.4040
0.3644
0.3000
0.2790
(2,2)
Present
0.6691
0.5807
0.5254
0.4280
0.3947
HSDT[32]
0.6753
0.5891
0.5444
0.4362
0.3981
Table 7. The comparison of the fundamental frequency parameter ( ) for square plates ( )
Method
g=0
g=1
g=2
g=3
g=5
Present
0.4629
0.0577
0.0158
0.0619
0.2278
0.2288
0.2301
0.2327
HSDT[32]
0.4658
0.0577
0.0158
0.0619
0.2285
0.2264
0.2270
0.2281
3D[48]
0.4658
0.0577
0.0153
0.0596
0.2192
0.2197
0.2211
0.2225
HSDT[49]
0.4658
0.0578
0.0157
0.0613
0.2257
0.2237
0.2243
0.2253
FSDT[49]
0.4619
0.0577
0.0162
0.0633
0.2323
0.2325
0.2334
0.2334
FSDT[46]
0.4618
0.0576
0.0158
0.0611
0.2270
0.2249
0.2254
0.2265
An excellent agreement is observed between the present Navier solution and the FEM. The frequencies rise with an increase in the thickness of the plate due to an increase in the stiffiness of the plate. This phenomenon originates from increasing the rigidity of the plate. Although it is observed that the present solution can be predicts the inplane modes of the plate. From the results are presented in this study one can be find that the natural frequencies are increased by increasing the thickness of the plate. This feature is due to the fact that the strain energy of the plate has significant sensibility with respect to the thickness. It is to be reminded that the outofplane modes depend on the bending energy, directly. Then, the number of the outofplane modes is increased. The influence of the aspect ratio on the frequency parameters of a rectangular plate ( , g=1) is shown in Table 9. From Table 9, it can be inferred that with a decrease in the aspect ratio, the frequency parameter increases, whereas the plate considering here is assumed to be simply supported in all edges, with decrease in length in a constant width, the degree of freedom (DOF) of the plate decreases, and it causes to increase the stiffness and the frequency parameter. In Table 10, the effects of different parameters on the nondimensional frequencies of the rectangular FG nanoplate are shown.From this Table, it is found that by increasing the nonlocal parameter, the rate of variation of nondimensional frequencies decreases, because by increasing the nonlocal parameter, the strain energy decreases, and it causes a decrease in the plates rigidity. In Fig. 3, the effects of the aspect ratio and the nonlocal parameter on the nondimensional frequancy of the rectangular nanoplates are shown. It is shown that with an increase in the aspect ratio, the nondimensional frequancy increases.
Table 8. The first four natural frequency (Hz) for square plates ( , g=1)
Method
Mode
1
2
3
4
0.05
Present
359.92
889.33
1406.9
1745.7
FEM[53]
357.37
883.58
1398.6
1736.2
Diff(%)
0.7130
0.6500
0.5930
0.5470
0.1
Present
703.44
1686.0
2597.5
2597.5
FEM[53]
699.30
1679.7
2578.7
2592.5
Diff(%)
0.5920
0.3750
0.7290
0.1930
0.2
Present
1298.8
2575.5
2873.5
3635.2
FEM[53]
1296.3
2574.6
2883.0
3633.3
Diff(%)
0.1930
0.0350
0.3230
0.0520
0.3
Present
1755.2
2569.7
3618.8
3607.4
FEM[53]
1759.8
2567.7
3613.3
3613.3
Diff(%)
0.2610
0.0780
0.1520
0.1630
Table 9. The frequency parameter ( ) for square plates ( , g=1)
2
1.5
1
2/3
0.5
Mode
3.1198
3.3720
4.9325
6.9551
9.9853
Table 10. The effect of the nondimensional nonlocal parameterand the power law index g on the nondimensional frequencies of the rectangular FG nanoplate
Power low index
0
5
10
0.0
0.5
0.2
0.2114
0.1357
0.0856
0.1
0.0365
0.0239
0.0231
1.0
0.2
0.2310
0.1356
0.1300
0.1
0.0577
0.0377
0.0364
0.1
0.5
0.2
0.1299
0.1239
0.0808
0.1
0.0345
0.0226
0.0218
1.0
0.2
0.1932
0.1239
0.1188
0.1
0.0527
0.0344
0.0332
0.2
0.5
0.2
0.1127
0.0728
0.0700
0.1
0.0299
0.0196
0.0189
1.0
0.2
0.1580
0.1014
0.0972
0.1
0.0431
0.0282
0.0272
0.3
0.5
0.2
0.0948
0.0613
0.0589
0.1
0.0251
0.0165
0.0159
1.0
0.2
0.1269
0.0814
0.0780
0.1
0.0346
0.0226
0.0218
0.4
0.5
0.2
0.0798
0.0516
0.0496
0.1
0.0212
0.0139
0.0134
1.0
0.2
0.1037
0.0665
0.0638
0.1
0.0283
0.0185
0.0178
It is illustrated that for the lower aspect ratios, the influence of the nonlocal parameters decreases. Also in Fig. 4, the effects of the aspect ratio and the nonlocal parameter on the frequancy ratio of the rectangular nanoplates are shown for different modes of vibration. From this figure, it seems that the frequancy ratios for the lower modes are more than those for the upper modes.
From Fig. 5, it is found that for lowerpower low index, the rate of The nondimensional frequencies are higher,and The effect of nondimensional nonlocal parameter on The nondimensional frequencies are significant, and diminishes with increase in that.Because by increase in the power low index, property of plate approaches to metal, and it’s stiffness decreases, thus itcauses to decrease nondimentionalfrequencuy.
From this figure, it is shown that in investigating the FG nanoplates, the effects of nonlocal parameter cannot be ignored so the theories for macro plates arenot suitable for nanoplates.
Figure 3. The effects of the aspect ratio and the nonlocal parameter on the nondimensional frequancy
Figure 4. The effects of the aspect ratio and the nonlocal parameter on the nondimensional frequancy
Figure 5.The effects of the nonlocal parameter and the power law index on the nondimensional frequency of the square nanoplate (mode (1,1), )
5. Conclusion
A Navier method was applied to the free vibration of the functionally graded rectangular nanoplates. The formulations were based on the exponential shear deformation theory using the nonlocal elasticity theory, and Hamilton’s principle was used to derive the equations of motion and associated boundary conditions. Comparing the cases with those reported in the literature for simply supported rectangular FG nanoplates demonstrates a high stability and accuracy of the present solution. What presented herein shows the effects of the variations of the nonlocal parameter, the ratio of the thickness to the length, the power law indexes and the aspect ratio on the frequency values of a FG nanoplate. It’s shown that the frequency ratio decreases with increasing the mode number and the value of the nonlocal parameter, and also increasing the power law index causes the nondimensional frequencies to decrease. All analytical results presented here can provide other research groups with a reliable source to check out their analytical and numerical solutions.
References
[1] Iijima S. Helical microtubules of graphitic carbon. nature 1991; 354: 5658.
[2] Miller RE, Shenoy VB. Sizedependent elastic properties of nanosized structural elements. Nanotechnology 2000; 11: 139.
[3] Shen HS. Zhang CL. Torsional buckling and postbuckling of doublewalled carbon nanotubes by nonlocal shear deformable shell model. Compos Struct 2010; 92: 10731084.
[4] Aydogdu M. Axial vibration of the nanorods with the nonlocal continuum rod model. Phys E 2009; 41: 861864.
[5] Wang CM, Duan W. Free vibration of nanorings/arches based on nonlocal elasticity. Appl Phys 2008; 104: 014303.
[6] Pradhan S, Phadikar J. Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys Lett A 2009; 373: 10621069.
[7] Ma M, et al. Electrochemical performance of ZnO nanoplates as anode materials for Ni/Zn secondary batteries. J Power Sources 2008; 179: 395400.
[9] SakhaeePour A, Ahmadian M, Vafai A. Applications of singlelayered graphene sheets as mass sensors and atomistic dust detectors. Solid State Commun 2008; 145: 168172.
[10] SakhaeePour A, Ahmadian M, Vafai A. Potential application of singlelayered graphene sheet as strain sensor. Solid State Commun 2008; 147: 336340.
[11] Aagesen M. Sorensen C. Nanoplates and their suitability for use as solar cells. Proceedings of Clean Technol 2008; 109112.
[12] Ye C, et al. Thicknessdependent photocatalytic performance of ZnO nanoplatelets. J Phys Chem B 2006; 110: 1514615151.
[13] Rafiee MA, et al. Fracture and fatigue in graphene nanocomposites. small 2010; 6: 179183.
[14] Kong S, et al. The sizedependent natural frequency of Bernoulli–Euler microbeams. Int J Eng Sci 2008; 46: 427437.
[15] Fleck N, Hutchinson J. Strain gradient plasticity. Adv Appl Mech 1997;33 :296361.
[16] Yang F, et al. Couple stress based strain gradient theory for elasticity. Int J Sol Struct 2002; 39: 27312743.
[17] Eringen AC, Edelen D. On nonlocal elasticity. Int J Eng Sci 1972; 10: 233248.
[18] Peddieson J, Buchanan GR, McNitt RP. Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 2003; 41: 305312.
[19] Reddy J. Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 2007; 45: 288307.
[20] Reddy J. Pang S. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J App Phys 2008; 103: 023511.
[21] Heireche H, et al. Sound wave propagation in singlewalled carbon nanotubes using nonlocal elasticity. Phys E 2008; 40: 27912799.
[22] Murmu T, Pradhan S. Buckling analysis of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Phys E 2009; 41: 12321239.
[23] Murmu T, Pradhan S. Thermomechanical vibration of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Comput Mater Sci 2009; 46: 854859.
[24] Wang L. Dynamical behaviors of doublewalled carbon nanotubes conveying fluid accounting for the role of small length scale. Comput Mater Sci 2009; 45: 584588.
[25] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J appl phys 1983; 54: 47034710.
[26] Chen Y, Lee JD, Eskandarian A. Atomistic viewpoint of the applicability of microcontinuum theories. Int j sol struct 2004; 41: 20852097.
[27] Malekzadeh P, Heydarpour Y. Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment. Compos Struct 2012; 94: 29712981.
[28] Ungbhakorn V, Wattanasakulpong N. Thermoelastic vibration analysis of thirdorder shear deformable functionally graded plates with distributed patch mass under thermal environment. Appl Acoust 2013; 74: 10451059.
[29] Kumar Y, Lal R. Prediction of frequencies of free axisymmetric vibration of twodirectional functionally graded annular plates on Winkler foundation. Eur J Mech ASolid 2013; 42: 219228.
[30] Nie G, Zhong Z. Semianalytical solution for threedimensional vibration of functionally graded circular plates. Comput MethodAppl M 2007; 196: 49014910.
[31] Huang C, Yang P, Chang M. Threedimensional vibration analyses of functionally graded material rectangular plates with through internal cracks. Compos Struct 2012; 94: 27642776.
[32] Matsunaga H. Free vibration and stability of functionally graded plates according to a 2D higherorder deformation theory. Compos struct 2008; 82: 499512.
[34] Ke L, et al. Axisymmetric nonlinear free vibration of sizedependent functionally graded annular microplates. Compos Part B:Engineering 2013; 53: 207217.
[35] Ke LL, et al. Bending, buckling and vibration of sizedependent functionally graded annular microplates. Compos struct 2012; 94: 32503257.
[36] Asghari M, Taati E. A sizedependent model for functionally graded microplates for mechanical analyses. J Vib Cont 2013; 19: 16141632.
[37] Natarajan S, et al. Sizedependent free flexural vibration behavior of functionally graded nanoplates. Comput Mater Sci 2012; 65: 7480.
[38] Thai HT, Choi DH. Sizedependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory. Compos Struct 2013; 95: 142153.
[39] Reddy J. Microstructuredependent couple stress theories of functionally graded beams. J Mech Phys of Solids 2011; 59: 23822399.
[40] HosseiniHashemi S, Zare M, Nazemnezhad R. An exact analytical approach for free vibration of Mindlin rectangular nanoplates via nonlocal elasticity. Compos Struct 2013; 100: 290299.
[41] Sahoo R, Singh B. A new trigonometric zigzag theory for static analysis of laminated composite and sandwich plates. Aerosp sci technol 2014; 35: 1528.
[42] Narendar S. Buckling analysis of micro/nanoscale plates based on twovariable refined plate theory incorporating nonlocal scale effects. Compos Struct 2011; 93: 30933103.
[43] HosseiniHashemi S, Bedroud M. Nazemnezhad R. An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity. Compos Struct 2013; 103: 108118.
[44] Sayyad AS, Ghugal YM. Bending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory. Appl Comput mech 2012; 6: 112.
[45] Shufrin I, Eisenberger M. Stability and vibration of shear deformable plates––first order and higher order analyses. Int j sol struc 2005; 42: 12251251.
[46] HosseiniHashemi S, et al. Free vibration of functionally graded rectangular plates using firstorder shear deformation plate theory. Appl Math Model 2010; 34: 12761291.
[47] Zhao X, Lee Y, Liew KM. Free vibration analysis of functionally graded plates using the elementfree kpRitz method. J sound Vibration 2009; 319: 918939.
[48] Vel SS, Batra R. Threedimensional exact solution for the vibration of functionally graded rectangular plates. J Sound Vibration 2004; 272: 703730.
[49] Pradyumna S, Bandyopadhyay J. Free vibration analysis of functionally graded curved panels using a higherorder finite element formulation. J Sound Vibration 2008; 318: 176192.
[50] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta metallurgica 1973; 21: 571574.
[51] Benveniste Y. A new approach to the application of MoriTanaka's theory in composite materials. Mech mater 1987; 6: 147157.
[52] Hill R. A selfconsistent mechanics of composite materials. J Mech Phys Sol 1965; 13: 213222.
[53] HosseiniHashemi S, Fadaee M, Atashipour SR. Study on the free vibration of thick functionally graded rectangular plates according to a new exact closedform procedure. Compos Struc 2011; 93: 722735.
References
[1] Iijima S. Helical microtubules of graphitic carbon. nature 1991; 354: 5658.
[2] Miller RE, Shenoy VB. Sizedependent elastic properties of nanosized structural elements. Nanotechnology 2000; 11: 139.
[3] Shen HS. Zhang CL. Torsional buckling and postbuckling of doublewalled carbon nanotubes by nonlocal shear deformable shell model. Compos Struct 2010; 92: 10731084.
[4] Aydogdu M. Axial vibration of the nanorods with the nonlocal continuum rod model. Phys E 2009; 41: 861864.
[5] Wang CM, Duan W. Free vibration of nanorings/arches based on nonlocal elasticity. Appl Phys 2008; 104: 014303.
[6] Pradhan S, Phadikar J. Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models. Phys Lett A 2009; 373: 10621069.
[7] Ma M, et al. Electrochemical performance of ZnO nanoplates as anode materials for Ni/Zn secondary batteries. J Power Sources 2008; 179: 395400.
[9] SakhaeePour A, Ahmadian M, Vafai A. Applications of singlelayered graphene sheets as mass sensors and atomistic dust detectors. Solid State Commun 2008; 145: 168172.
[10] SakhaeePour A, Ahmadian M, Vafai A. Potential application of singlelayered graphene sheet as strain sensor. Solid State Commun 2008; 147: 336340.
[11] Aagesen M. Sorensen C. Nanoplates and their suitability for use as solar cells. Proceedings of Clean Technol 2008; 109112.
[12] Ye C, et al. Thicknessdependent photocatalytic performance of ZnO nanoplatelets. J Phys Chem B 2006; 110: 1514615151.
[13] Rafiee MA, et al. Fracture and fatigue in graphene nanocomposites. small 2010; 6: 179183.
[14] Kong S, et al. The sizedependent natural frequency of Bernoulli–Euler microbeams. Int J Eng Sci 2008; 46: 427437.
[15] Fleck N, Hutchinson J. Strain gradient plasticity. Adv Appl Mech 1997;33 :296361.
[16] Yang F, et al. Couple stress based strain gradient theory for elasticity. Int J Sol Struct 2002; 39: 27312743.
[17] Eringen AC, Edelen D. On nonlocal elasticity. Int J Eng Sci 1972; 10: 233248.
[18] Peddieson J, Buchanan GR, McNitt RP. Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 2003; 41: 305312.
[19] Reddy J. Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 2007; 45: 288307.
[20] Reddy J. Pang S. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J App Phys 2008; 103: 023511.
[21] Heireche H, et al. Sound wave propagation in singlewalled carbon nanotubes using nonlocal elasticity. Phys E 2008; 40: 27912799.
[22] Murmu T, Pradhan S. Buckling analysis of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Phys E 2009; 41: 12321239.
[23] Murmu T, Pradhan S. Thermomechanical vibration of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Comput Mater Sci 2009; 46: 854859.
[24] Wang L. Dynamical behaviors of doublewalled carbon nanotubes conveying fluid accounting for the role of small length scale. Comput Mater Sci 2009; 45: 584588.
[25] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J appl phys 1983; 54: 47034710.
[26] Chen Y, Lee JD, Eskandarian A. Atomistic viewpoint of the applicability of microcontinuum theories. Int j sol struct 2004; 41: 20852097.
[27] Malekzadeh P, Heydarpour Y. Free vibration analysis of rotating functionally graded cylindrical shells in thermal environment. Compos Struct 2012; 94: 29712981.
[28] Ungbhakorn V, Wattanasakulpong N. Thermoelastic vibration analysis of thirdorder shear deformable functionally graded plates with distributed patch mass under thermal environment. Appl Acoust 2013; 74: 10451059.
[29] Kumar Y, Lal R. Prediction of frequencies of free axisymmetric vibration of twodirectional functionally graded annular plates on Winkler foundation. Eur J Mech ASolid 2013; 42: 219228.
[30] Nie G, Zhong Z. Semianalytical solution for threedimensional vibration of functionally graded circular plates. Comput MethodAppl M 2007; 196: 49014910.
[31] Huang C, Yang P, Chang M. Threedimensional vibration analyses of functionally graded material rectangular plates with through internal cracks. Compos Struct 2012; 94: 27642776.
[32] Matsunaga H. Free vibration and stability of functionally graded plates according to a 2D higherorder deformation theory. Compos struct 2008; 82: 499512.
[34] Ke L, et al. Axisymmetric nonlinear free vibration of sizedependent functionally graded annular microplates. Compos Part B:Engineering 2013; 53: 207217.
[35] Ke LL, et al. Bending, buckling and vibration of sizedependent functionally graded annular microplates. Compos struct 2012; 94: 32503257.
[36] Asghari M, Taati E. A sizedependent model for functionally graded microplates for mechanical analyses. J Vib Cont 2013; 19: 16141632.
[37] Natarajan S, et al. Sizedependent free flexural vibration behavior of functionally graded nanoplates. Comput Mater Sci 2012; 65: 7480.
[38] Thai HT, Choi DH. Sizedependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory. Compos Struct 2013; 95: 142153.
[39] Reddy J. Microstructuredependent couple stress theories of functionally graded beams. J Mech Phys of Solids 2011; 59: 23822399.
[40] HosseiniHashemi S, Zare M, Nazemnezhad R. An exact analytical approach for free vibration of Mindlin rectangular nanoplates via nonlocal elasticity. Compos Struct 2013; 100: 290299.
[41] Sahoo R, Singh B. A new trigonometric zigzag theory for static analysis of laminated composite and sandwich plates. Aerosp sci technol 2014; 35: 1528.
[42] Narendar S. Buckling analysis of micro/nanoscale plates based on twovariable refined plate theory incorporating nonlocal scale effects. Compos Struct 2011; 93: 30933103.
[43] HosseiniHashemi S, Bedroud M. Nazemnezhad R. An exact analytical solution for free vibration of functionally graded circular/annular Mindlin nanoplates via nonlocal elasticity. Compos Struct 2013; 103: 108118.
[44] Sayyad AS, Ghugal YM. Bending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory. Appl Comput mech 2012; 6: 112.
[45] Shufrin I, Eisenberger M. Stability and vibration of shear deformable plates––first order and higher order analyses. Int j sol struc 2005; 42: 12251251.
[46] HosseiniHashemi S, et al. Free vibration of functionally graded rectangular plates using firstorder shear deformation plate theory. Appl Math Model 2010; 34: 12761291.
[47] Zhao X, Lee Y, Liew KM. Free vibration analysis of functionally graded plates using the elementfree kpRitz method. J sound Vibration 2009; 319: 918939.
[48] Vel SS, Batra R. Threedimensional exact solution for the vibration of functionally graded rectangular plates. J Sound Vibration 2004; 272: 703730.
[49] Pradyumna S, Bandyopadhyay J. Free vibration analysis of functionally graded curved panels using a higherorder finite element formulation. J Sound Vibration 2008; 318: 176192.
[50] Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta metallurgica 1973; 21: 571574.
[51] Benveniste Y. A new approach to the application of MoriTanaka's theory in composite materials. Mech mater 1987; 6: 147157.
[52] Hill R. A selfconsistent mechanics of composite materials. J Mech Phys Sol 1965; 13: 213222.
[53] HosseiniHashemi S, Fadaee M, Atashipour SR. Study on the free vibration of thick functionally graded rectangular plates according to a new exact closedform procedure. Compos Struc 2011; 93: 722735.