Document Type: Research Paper
Authors
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Abstract
Keywords

Mechanics of Advanced Composite Structures 3 (2016) 99112 

Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
M. Mohammadimehr^{*}, H. Mohammadi Hooyeh, H. Afshari, M.R. Salarkia
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Paper INFO 

ABSTRACT 
Paper history: Received: 20160405 Revised: 20160519 Accepted: 20160625 
In this paper, sizedependent effects on the vibration behavior of Timoshenko microbeams under prestress loading embedded in an elastic foundation, using modified strain gradient theory (MSGT) and surface stress effects, were studied. To consider the surface stress effects, the Gurtin–Murdoch continuum mechanical approach was employed. Using Hamilton’s principle, the governing equations of motion and boundary conditions were obtained and solved numerically using the differential quadrature method (DQM). The effects of prestress loading, surface residual stress, surface mass density, Young’s modulus applied to the surface layer, three material length scale parameters, and the elastic foundation coefficients were investigated. For higher aspect ratios, this study found that the effect of the prestress loading was negligible for higher modes. Considering sizedependent effects led to increase the stiffness of the matrix and enhance the dimensionless natural frequencies of the Timoshenko microbeam. The MSGT results were higher than those found using other theories. In addition, this research discovered that there were negligible surface stress effects with each of the three material length scale parameters.




Keywords: Size dependent effect Prestress loading DQM Vibration behavior of Timoshenko microbeam



© 2016 Published by Semnan University Press. All rights reserved. 
Nano technology is one of the most powerful technologies that can produce many materials and devices across a range of applications, such as electronics, biomaterials, medicine, and energy production [13]. Vibration analysis of composite beams has been a research topic in many engineering fields because vibration plays an important role in the design of turbine blades, helicopter blades, propeller blades, drill bits, and fluted cutters. In practice, these structures are typically modeled as either Euler or Timoshenko beams. The design of micro and nanoelectromechanical systems (MEMS/NEMS) requires widespread use of microrods and microbeams with different complex behaviors. Currently, microcomposite beams are employed in microturbo machines, ultrasonic piezoelectric micromotor designs, and medical micro devices.
Recently, many researchers have investigated the mechanical behaviors of micro and nanoscale materials using beam, plate, and shell theories. Ghorbanpour Arana et al. [4] analyzed the pulsating fluidinduced dynamic instability of doublewalled carbon nanotubes (DWCNTs), based on a sinusoidal strain gradient theory using the differential quadrature method (DQM) and the Bolotin method. Their results depicted that the imposed magnetic field was an effective controlling parameter for dynamic instability of viscoDWCNTs. In another work, Ghorbanpour Arani et al. [5] presented the nonlinear vibration of coupled nano and microstructures conveying fluid flow based on a Timoshenko beam model under a twodimensional magnetic field. They expressed that the magnetic field played an important role in the stability of the carbon nanotubes (CNTs) and controls the stability of the nanosystem.
Simsek [6] studied the free vibration analysis of nanobeams with various boundary conditions, based on the nonlocal elasticity theory for large amplitude. In that research, the effect of nonlocal parameters on the nonlinear frequency ratio was examined. Their results showed that the nonlocal effects should be considered in the analysis of the mechanical behavior of nanostructures. Sahmani and Bahrami [7] analyzed the dynamic stability of microbeams subjected to piezoelectric voltage, using the strain gradient theory (SGT). In their results, for a special value of applied piezoelectric voltage, increasing the dimensionless length scale parameter decreased the difference between stability responses predicted by the classical and nonclassical beam models. In addition, Mohammadimehr and Golzari [8] investigated the elliptic phenomenon effect of crosssections on the torsional buckling of a nanocomposite beam reinforced by a singlewalled carbon nanotube (SWCNT). With an increase in the matrix thickness, the tangential and longitudinal strains of SWCNT decreased, and the opposite effect occurred for the interface stress and the dimensionless stress of the outer surface.
Alternately, Mohammadimehr and Rahmati [9] considered the smallscale effects on electrothermomechanical vibration analysis of a singlewalled boron nitride nanorod under electric excitation. They represented that the natural frequency decreased with an increase in the smallscale effects or aspect ratios. On the other hand, the smallscale effects were significant for lower aspect ratios and higher natural frequencies. Atabakhshian et al. [10] employed vibration of a smart coupled electrothermal nanobeam system with an internal flow, based on nonlocal elasticity theory, while Ansari et al. [11] derived free vibration analysis from the evaluation of single and doublewalled carbon nanotubes based on nonlocal elastic shell models. They concluded that the smallscale effects in the nonlocal model made nanotubes more flexible. Akgoz and Civalek [12] studied higherorder shear deformation in microbeam models, based on the strain gradient elasticity theory. Their results showed that microbeams derived from the nonclassical theories, specifically modified strain gradient theory (MSGT), were stiffer than those based on the classical theory (CT).
Asgharifard Sharabiani and Haeri Yazdi [13] illustrated the nonlinear free vibrations for functionally graded (FG) nanobeams, including their surface effects. The results showed that the surface effects at higher volume fraction indices were either less or more dominant, in small and large amplitude ratios, respectively. Ke et al. [14] investigated the nonlinear vibrations of piezoelectric nanobeams based on the nonlocal and the Timoshenko beam theories. Their results demonstrated that a change in the external electric voltage from a positive value to a negative value led to a decrease in the nonlinear frequency ratio. Ansari et al. [15] analyzed the bending, buckling, and free vibration responses of FG Timoshenko microbeams, and they observed that the critical buckling loads and natural frequencies predicted by the beam models, based on MSGT and CT, provided the maximum and minimum values, respectively. Tounsi et al. [16] illustrated sizedependent bending and vibration analysis of FG microbeams, based on MCST and neutral surface positions. They represented that the inclusion of the couple stress effect makes a microbeam stiffer and decreased the vertical displacement and increased the natural frequency.
Alternately, Nazemnezhad et al. [17] employed an analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects. They observed that the effect of the surface density on the variation of the natural frequency of the nanobeam versus the thickness ratio decreases consistently with the increase of the mode number. Nejat Pishkenari et al. [18] examined the surface elasticity and size effects on the vibrational behavior of silicon nanoresonators. They developed a continuum model for nanobeam vibrations that was capable of predicting the results of molecular dynamics (MD) simulations with considerably lower computational effort. Yue et al. [19] proposed a microscale Timoshenko beam model for piezoelectricity using flexoelectricity and surface effects. Their results observed that the change of surface properties not only directly affected the static bending but also significantly changed the natural frequency of the beam. Preethi et al. [20] presented surface and nonlocal effects of the nonlinear analysis of Timoshenko beams using Eringen’s nonlocal theory and the GurtinMurdoch approach, where the nonlocal parameters and the positive surface parameters’ values decreased the stiffness of the beam and resulted in larger deflections and lower frequencies.
In this research, a Timoshenko microbeam model, based on the modified strain gradient theory (MSGT) and surface stress effects subjected to prestress loading, is presented. The MSGT and surface stress effects were considered together in this study because both of them affect the structure at the microscale. Despite the fact that the surface and small scale effects have been investigated individually in some papers, the novelty of this study lies in the evaluation of sizedependent effects, including three material length scale parameters, and the surface residual stress based on strain gradient, and the surface stress elasticity effects on the dimensionless natural frequency of Timoshenko microbeams, subjected to prestress loading and considered simultaneously at a microscale. Moreover, the sizedependent effects increased the dimensionless natural frequency due to increasing flexural rigidity, which then enhanced the stability of the microstructures. The governing equations of motion were obtained using Hamilton’s principle and energy method. The equations were solved using the differential quadrature method (DQM).
A schematic view of a straight Timoshenko microbeam model based on surface layers, an elastic medium, and prestress load is shown in Figure 1. The displacement fields for this model can be stated as [21]
(1) 

(2) 

(3) 
where and are axial and transverse displacements for the neutral axis, respectively, and is the rotational transverse normal angle about the xaxis.
The components of normal ( ) and shear ( ) strains, using Eqs. (1), (2), and (3), are considered as follows:
, 
(4) 
. 
(5) 
Figure 1. A schematic view of a Timoshenko microbeam model with a surface layer, elastic medium, and prestress load.
The strain energy for the linear isotropic elastic material, based on MSGT, is considered as follows [21,22]:
(6) 
where and are the Cauchy stress tensor and the strain tensor, respectively. The expressions and denote the dilatation gradient tensor, the deviatoric stretch gradient tensor, and the symmetric rotation gradient tensor, respectively, which are defined as the following forms [23,24]
(7) 

(8) 

(9) 

(10) 

(11) 
where and are the dilatation strain and the displacement vector, respectively, according to Akgöz and Civalek [25]. The Knocker symbol is , and the permutation symbol is :
(12) 
The constitutive equations for linear, elastic, and isotropic materials are given by the following forms [25]
(13) 

(14) 

(15) 

(16) 
where is the deviatoric strain, which can be written as follows [25]:
(17) 
where , , and denote three additional independent material length scale parameters associated with the dilatation gradient tensor, deviatoric stretch gradient tensor, and symmetric rotation gradient tensor, respectively. In addition, the parameters λ and μ are the Lame coefficients which are given as [26,27]
, 
(18) 
where E and denote Young’s modulus and Poisson’s ratio, respectively.
Using Eqs. (3), (4), and (5), the following equations are given by
(19) 

(20) 
Using Eqs. (3), (4), and (5), the nonzero components of the deviatoric stretch gradient tensor can be derived as follows:
(21) 
Substituting Eqs. (19)–(21) into Eq. (11) yields the following form:
(22) 
Using Eqs. (15) and (21), the higher order stresses for MSGT are obtained as the following forms:
(23) 
Using Eqs. (14), (16), (19), (20) and (22), we have
(24) 
The nonzero stresses are obtained as follows:
(25) 
In Eq. (25), denotes the shear correction factor which depends on the shape of microbeam crosssection.
Because of a high surfacetovolume ratio, the surface stress effect plays an important role with micro and nanoscale materials. For this purpose, the constitutive equation of the Gurtin–Murdoch continuum mechanics approach is considered as follows [28]:
(26) 
where is the residual surface stress under unstrained condition, and and are the surface Lame constants. The components of normal and shear surface stress can be written as follows:
(27) 
The classical beam theory does not satisfy our model. For solving this problem, it is assumed that the stress component varies linearly through the beam thickness and satisfies the balance conditions on the surfaces. Therefore, can be written as [28]
(28) 

Using Eq. (27), can be written as
(29) 
The normal and shear components, considering bulk and surface effects and using Eq. (29), can be written as
(30) 
is the total potential energy that includes the strain energy, kinetic energy, and work done by the external loads, which can written as [4]
(31) 
where
(32) 
and where , , and are the strain and kinematic energies for bulk and surface effects, respectively. Moreover, and are the work done by the external forces, including the prestress load and the elastic foundation, respectively.
Using Hamilton’s principle and a variational method for Timoshenko microbeam model, based on strain gradient theory and the surface stress effects embedded in an elastic medium subjected to prestress loading , yields the following equation [4]:
(33) 
The surface strain energy is obtained as
(34) 
where

Using the presented equations, the strain energy for bulk and surface effects are explained in Appendix A with details. Using Eqs. (14)–(16), the kinetic energy of the Timoshenko microbeam model for bulk and surface effects can be written as
(35) 
The work done by external forces, including prestress load and elastic foundation, can be written as
(36) 
where is the prestress load, and and are Winkler’s spring and Pasternak’s shear modulli of elastic foundation, respectively.
By substituting Eqs. (A1), (A2), (35) and (36) into Eq. (33), one can obtain the governing equations of motion and boundary conditions as follows:
(37) 
For boundary conditions
(38) 
where
(39) 
The dimensionless geometric, mechanical, and surface residual stress, surface mass density, Young’s modulus of surface layer, and three material length scale parameters can be defined as follows [28]:
(40) 
To use the differential quadrature (DQ) method, first we should convert Eqs. (37) and (38) into dimensionless equations. Thus, substituting Eq. (40) into Eqs. (37) and (38) yields the Eqs. (41a) and (41b).
(41a) 
(41b) 
The dimensionless simply supported (SS) boundary conditions for the microbeam model are considered as follows:
(42) 
3. Using the DQ method to solve the Timoshenko microbeam model
The (DQ) method was used to solve Eq. (41) and the associated boundary conditions in Eq. (42) to determine the free vibration frequencies of the beam. The basic concept of the DQM was defined as the derivative of a function at a given point that can be approximated as a linear sum of a weighted function at all sample points [29,30]. Using this approximation, the differential equations are then reduced to a set of algebraic equations. This approach is convenient for solving problems governed by fourth or higherorder differential equations.
According to this method, the m^{th} order derivative of the function f(x) with respect to x at a grid point x_{i}, is approximated by a linear sum of all the functional values in the whole domain as follows [22]:

(43) 
where x_{i} is the location of ith sample point in the domain; N is the number of sampling points; is the functional value at point x_{i}, is the weighting coefficient of the m^{th} order differentiation attached to these functional values. To avoid illconditioning, the Lagrange interpolation basis functions are used as the following form [22]:
(44) 
To determine the unequallyspaced positions of the grid points, the Chebyshev–Gauss–Lobatto polynomials were employed as follows [22]:
(45) 
The first order weighting matrix can be obtained completely from Eq. (44). Higherorder coefficient matrices can be obtained from the firstorder weighting matrix as follows [22]:

(46) 
Then, substituting Eqs. (44) and (46) into Eqs. (41a) and (42) obtained the following equations of motion using the DQ method

(47) 
For (SS) boundary conditions, we have:
(48) 
The general solutions of motion equations are considered as
(49) 
where is the dimensionless natural frequency. is the fundamental natural frequency, and denotes the density of microbeam.
The stiffness and mass matrices for the Timoshenko microbeam, using strain gradient theory and surface stress effects under prestress loading, can be written as
(50)

where are the stiffness and mass matrices and the subscripts b and d stand for the boundary and domain points, respectively. By solving Eq. (50), the dimensionless natural frequencies for and their associated vibration mode shapes can be extracted.
4. Numerical Results and Discussion
The mechanical and geometric properties of a Timoshenko microbeam is considered as [28,31]
(51) 
The material length scale parameter is crucial for the successful application of the MSGT. DehrouyehSemnani and NikkhahBahrami [32] presented an indepth discussion on how to determine this parameter numerically, and they compared the numerical results obtained to those obtained by experimental testing [32]. They showed the bending rigidity of an epoxy microcantilever versus thickness for a modified couple stress model (MCST), using , and the experimental data reported by Lam et al. and found that the results based on the constitutive beam model validated the experimental data, while the EulerBernoulli beam model overestimated the bending rigidity of the microcantilever. In addition, they depicted that the material length scale parameter of epoxybased materials on the EulerBernoulli beam model equals to . According to the results of DehrouyehSemnani and NikkhahBahrami, the EulerBernoulli beam model validated the experimental data very well, but the constitutive beam model underestimated the bending rigidity of the epoxy microcantilever. Therefore, in this work, we used the material length scale parameter equal to .
Table 1 gives the dimensionless natural frequencies for the Timoshenko microbeam under various boundary conditions. An excellent agreement was found between the present results and the analytical solutions.
The results, obtained by the present work, are compared with the reported results by Ansari et al. [33] in Figure 2, where they demonstrate good agreement each other. In addition, the trend of the results was the same. On the other hand, increasing the aspect ratio (L/h) reduced the dimensionless natural frequency. Moreover, the stiffness of the Timoshenko microbeam decreased with increasing the aspect ratio.
Table 1. Comparison of dimensionless natural frequencies with various thicknesses for different boundary conditions. 

Thickness (nm) 
SS 
SC 
CC 

Ansari et al. [28] 
h=1 
0.1830 
0.2148 
0.2524 
Present work 
0.1863 
0.2169 
0.2553 

Ansari et al. [28] 
h=5 
0.1255 
0.1643 
0.2117 
Present work 
0.1258 
0.1652 
0.2120 
Figure 2. The dimensionless natural frequency versus aspect ratio.
Table 2 shows the first three dimensionless natural frequencies of the Timoshenko microbeam model for the different values of aspect ratio ( ), and surface residual stress ( ). As shown in Table 2, by increasing the aspect ratio, the value of the first three dimensionless natural frequencies decreases, and the opposite occurs for the surface residual stress.
The latter subject has been illustrated for dimensionless fundamental natural frequencies in Figure 3. Considering the surface residual stress, the Timoshenko beam at a microscale becomes stiffer, but the effect of this parameter on the dimensionless natural frequency is not noticeable. Therefore, surface residual stress can be ignored in the results.
Table 2. First, second, and third dimensionless natural frequencies of a Timoshenko microbeam model for the different values of and for . 



10 
3.6763 
7.3545 
11.0340 

15 
3.6387 
7.2933 
10.9432 

20 
3.6208 
7.2618 
10.8893 

10 
3.6762 
7.3543 
11.0336 

15 
3.6386 
7.2931 
10.9429 

20 
3.6207 
7.2616 
10.8891 

10 
3.6761 
7.3540 
11.0333 

15 
3.6385 
7.2929 
10.9426 

20 
3.6207 
7.2614 
10.8888 
Figure 3. The dimensionless fundamental natural frequency versus aspect ratio for different values of
Tables 3 and 4 depict the first three dimensionless natural frequencies of the Timoshenko microbeam model for the values of the aspect ratio ( ), surface mass density ( ), and Young’s modulus of surface layer ( ), respectively. By increasing of the , the value of the first three dimensionless natural frequencies increases and vice versa for surface mass density. A change in and led to increase stiffness and mass of the micro structure, respectively. Moreover, the results, shown in Figures 4 and 5, are similar to those shown in Tables 3 and 4. Furthermore, Figures 4 and 5 demonstrate that the effect of on the dimensionless natural frequency is more than . However, the effect of on the dimensionless natural frequency is not noticeable, and it can be ignored in the results.
Table 3. First, second, and third dimensionless natural frequencies of the Timoshenko microbeam model for the different values of and for 



10 
3.6764 
7.3546 
11.0342 

15 
3.6388 
7.2935 
10.9435 

20 
3.6209 
7.2620 
10.8896 

10 
3.6760 
7.3538 
11.0329 

15 
3.6383 
7.2927 
10.9422 

20 
3.6205 
7.2611 
10.8884 

10 
3.6722 
7.3462 
11.0213 

15 
3.6346 
7.2852 
10.9309 

20 
3.6169 
7.2538 
10.8772 
Figure 4. The dimensionless fundamental natural frequency versus aspect ratio for different values of .
Figure 5. The dimensionless fundamental natural frequency versus aspect ratio for different values of
Table 4. First, second, and third dimensionless natural frequencies of a Timoshenko microbeam model for the different values of and for 



10 
3.6761 
7.3541 
11.0333 

15 
3.6385 
7.2930 
10.9427 

20 
3.6207 
7.2615 
10.8889 

10 
3.6762 
7.3543 
11.0337 

15 
3.6386 
7.2931 
10.9429 

20 
3.6207 
7.2616 
10.8891 

10 
3.6763 
7.3544 
11.0340 

15 
3.6386 
7.2932 
10.9431 

20 
3.6208 
7.2617 
10.8892 
Figures 6a and 6b show the influence of prestress load on the dimensionless first and third natural frequencies versus aspect ratio, respectively. These results demonstrated that the effect of prestress load on the greater mode is negligible for higher aspect ratios, and this effect was similar to the lower aspect ratios for all modes. Clearly, the stiffness of microbeam increased at lower aspect ratios. In this figure, the effect of the positive prestress load on the natural frequency was higher than that of the negative prestress load. Consequently, positive and negative prestress loads led to increase and decrease stiffness of the Timoshenko microbeam, respectively. These results are the same for dimensionless natural frequencies.
(a)
(b)
Figure 6. The influence of prestress load on the dimensionless first (a) and third (b) natural frequencies versus aspect ratios .
To consider the sizedependent effects (l denotes the material length scale parameter), the parameter at a microscale is taken into account, and it is nonzero for MSGT ( ) or MCST ( ), while at a macro scale, it is zero for CT ( ).
Figures 7a and 7b are plotted to illustrate the influence of various material length scale theories including modified strain gradient (MSGT) ( ), modified couple stress (MCST) ( ), and classical theories (CT) ( ) on the dimensionless first and third natural frequencies versus , respectively.
(a)
(b)
Figure 7. The influence of various material length scale theories on the dimensionless (a) first and (b) third natural frequencies versus h/l.
The effect of the material length scale parameters on the dimensionless natural frequencies for MSGT was higher than that of the other states, such as MCST and CT. This indicates that considering three material length scale parameters led to increase stiffness of the Timoshenko microbeam model, and therefore the dimensionless natural frequencies for MSGT enhanced.
Figures 8 and 9 present the influence of transverse and shear constants of the elastic foundation on the dimensionless fundamental natural frequencies with different values of aspect ratios. The dimensionless natural frequency increased with an increase in the transverse and shear constants of the elastic foundation, while the elastic foundation increased stiffness of the microstructure.
Figure 8. The influence of the transverse constant of the elastic foundation on the dimensionless fundamental natural frequencies with different values of aspect ratios , .
Figure 9. The influence of shear constant of elastic foundation on the dimensionless fundamental natural frequencies with different values of aspect ratios , .
Moreover, increasing the transverse and shear constants of the elastic foundation were directly related to the stiffness of the Timoshenko microbeam and the dimensionless natural frequency.
5. Conclusions
Sizedependent effects on the free vibration analysis of the Timoshenko microbeam model, based on MSGT and surface stress effects subjected to prestress loading embedded in an elastic medium, were investigated. The Gurtin–Murdoch continuum mechanical approach was considered, and the set of governing equations were derived using a variational method and solved using DQM. Effects of prestress load, surface residual stress, surface mass density, Young’s modulus of surface layer, material length scale parameters, and elastic foundation coefficients were studied.
The results of this article can be listed as follows:
· By increasing the aspect ratio, the values of natural frequencies decreased while the opposite occurred for surface residual stress. In addition, when increasing the value of , the value of the natural frequencies increased, while the surface mass density decreased. Variations in and led to increase stiffness and mass matrices for the micro structures, respectively. The numerical results showed that the effect of surface residual stress was more than the surface mass density or Young’s modulus of the surface layer.
· The effect of prestress loading in higher modes was negligible for higher aspect ratios, and this effect was similar to lower aspect ratios across all modes.
· The effect of the three material length scale parameters on the natural frequencies for MSGT was higher than that of the other theories. Application of each of the three material length scale parameters , increased the natural frequencies for MSGT, which was due to the increasing stiffness of the Timoshenko microbeam model.
Acknowledgments
The authors would like to thank the reviewers for their reports that improved the clarity of this article. Moreover, the authors are grateful to the Iranian Nanotechnology Development Committee for their financial support. We are also grateful to the University of Kashan for supporting this work through Grant no. 463855/1.
References
[1]Ramezani S. A micro scale geometrically nonlinear Timoshenko beam model based on strain gradient elasticity theory, Int J Nonlinear Mech 2012; 47: 86373.
[2] Rahaeifard M,Kahrobaiyan MH, Ahmadian MT. Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials; ASME, 2009.
[3] Simsek M, Yurtcu HH. Analytical solutions for bending and buckling of functionally gradednanobeams based on the nonlocal Timoshenko beam theory, Compos Struct 2013; 97: 37886.
[4] Ghorbanpour Arani A, Kolahchi R, Mosayebi M, Jamali M. Pulsating fluid induced dynamic instability of viscodoublewalled carbon nanotubes based on sinusoidal strain gradient theory using DQM and Bolotin method, Int J Mech Mater Des 2016; 12: 1738.
[5] Ghorbanpour Arani A, Dashti P, Amir S,Yousefi M. Nonlinear vibration of coupled nano and microstructures conveying fluid based on Timoshenko beam model under twodimensional magnetic field, Acta Mech 2015; 226: 272960.
[6]Simsek M. Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory, Compos Part B 2014; 56: 62128.
[7] Sahmani S, Bahrami M. Sizedependent dynamic stability analysis of microbeams actuated by piezoelectric voltage based on strain gradient elasticity theory, J Mech Sci Tech 2015; 29: 32533.
[8] Mohammadimehr M, Golzari E. The elliptic phenomenon effect of cross section on the torsional buckling of a nanocomposite beam reinforced by a singlewalled carbon nanotube, Proc Instit Mech Eng, Part N: J Nanoeng Nanosys 2016; 230: 5567.
[9] Mohammadimehr M, Rahmati AR. Small scale effect on electrothermomechanical vibration analysis of singlewalled boron nitride nanorods under electric excitation, Turkish J Eng, Env Sci 2013; 37: 115.
[10] Atabakhshian V, Shooshtari A, Karimi M. Electrothermal vibration of a smart coupled nanobeam system with an internal flow based on nonlocal elasticity theory, Phys B 2015; 456: 37582.
[11] Ansari R, Rouhi H, Sahmani S. Free vibration analysis of single and doublewalled carbon nanotubes based on nonlocal elastic shell models, J Vib Cont 2014; 20: 67078.
[12] Akgoz B, Civalek O. A sizedependent shear deformation beam model based on the strain gradient elasticity theory, Int J Eng Sci 2013; 70: 114.
[13]Asgharifard Sharabiani P, Haeri Yazdi MR. Nonlinear free vibrations of functionally graded nanobeams with surface effects, Compos Part B 2013; 45: 58186.
[14]Ke LL, Wang YS,Wang ZD. Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory, Compos Struct 2012; 94: 203847.
[15] Ansari R, Gholami R, Faghih Shojaei M, Mohammadi V, Sahmani S. Sizedependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory, Compos Struct 2013; 100: 38597.
[16] Tounsi A, AlBasyouni AS, Mahmoud SR. Size dependent bending and vibration analysis of functionally graded microbeams based on modified couple stress theory and neutral surface position, Compos Struct 2015; 125: 621630.
[17] Nazemnezhad R, Salimi M. Hosseini Hashemi SH, Asgharifard Sharabiani P. An analytical study on the nonlinear free vibration of nanoscale beams incorporating surface density effects. Compos Part B 2012; 43: 2893973.
[18] Nejat Pishkenari H, Afsharmanesh B, Akbari E. Surface Elasticity and Size Effect on the Vibrational Behavior of Silicon Nanoresonators, Current Appl Phys, 2015; 15: 13891396.
[19] Yue YM, Xu KY, Chen T. A micro scale Timoshenko beam model for piezoelectricity with flexoelectricity and surface effects, Compos Struct 2016; 136: 278286.
[20] Preethi K, Rajagopal A, Reddy JN. Surface and Nonlocal Effects for nonlinear analysis of Timoshenko beams, Int J Nonlinear Mech 2015; 76: 100111.
[21] Ke LL, Yang J, Kitipornchai S, Xiang Y. Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials, Mech Adv Mater Struct 2009; 16: 488502.
[22] Mohammadimehr M, Monajemi AA, Moradi M. Vibration analysis of viscoelastic tapered microrod based on strain gradient theory resting on viscopasternak foundation using DQM, J Mech Sci Tech 2015; 29: 2297305.
[23] Kahrobaiyan MH, Asghari M, Ahmadian MT. A strain gradient Timoshenko beam element: application to MEMS, Acta Mech 2015, 226: 50525.
[24] Allahbakhshi A, Allahbakhshi M. Vibration analysis of nanostructure multilayered graphenesheets using modified strain gradient theory, Front Mech Eng 2015; 10: 18797.
[25] Akgoz B, Civalek O. Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded microscaled beams, Int J Eng Sci 2011; 49: 126880.
[26] Mohammad Abadi M, Daneshmehr AR. An investigation of modified couple stress theory in buckling analysis of micro composite laminated Euler–Bernoulli and Timoshenko beams, Int J Eng Sci 2014; 75: 4053.
[27] Rajabi F, Ramezani S. A nonlinear microbeam model based on strain gradient elasticity theory, Acta Mech Solida Sinica 2013; 26: 2134.
[28] Ansari R, Mohammadi V, Faghih Shojaei M, Gholami R, Sahmani S. On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory, Compos. Part B: Eng 2014; 60: 15866.
[29] Ghorbanpour Arani A, Kolahchi R, Khoddami Maraghi Z. Nonlinear vibration and instability of embedded doublewalled boron nitride nanotubes based on nonlocal cylindrical shell theory, Appl Math Model 2013; 37: 76857707.
[30] Murmu T, Pradhan SC. 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: 123239.
[31] Zhang B, He Y, Liu D, Gan Z, Shen L. Nonclassical Timoshenko beam element based on the strain gradient elasticity theory, Finite Elem Anal Des 2014; 79: 2239.
[32] DehrouyehSemnani AM, NikkhahBahrami M. A discussion on evaluation of material length scale parameter based on microcantilever test, Compos Struct 2015; 122: 425429.
[33] Ansari R, Gholami R, Sahmani S. Free vibration analysis of sizedependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Compos Struct 2011; 94: 221228.
Appendix
The strain energies for bulk and surface effects are written as follows:
(A1) 

(A2) 