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 2 (2015) 113126


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://macs.journals.semnan.ac.ir 
Nonlocal Buckling and Vibration Analysis of TripleWalled ZnO Piezoelectric Timoshenko Nanobeam Subjected to MagnetoElectroThermoMechanical Loadings
M. Mohammadimehr*, S.A.M. Managheb, S. Alimirzaei
Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 30 October 2015 Received in revised form 7 January 2016 Accepted 31 January 2016 
In this study, using the nonlocal elasticity theory, the buckling and vibration analysis of triple walled ZnO piezoelectric Timoshenko beam on elastic Pasternak foundation is analytically investigated under magnetoelectrothermomechanical loadings. Using the Timoshenko beam free body diagram, the equilibrium equation of Timoshenko nanobeam model is obtained and solved by Navier’s method for a simplysupported nanobeam. The surrounding elastic mediums are simulated by Winkler and Pasternak models and interlayer forces are considered by LenardJones potential. The effects of various parameters including the elastic medium, small scale, length, thickness, van der Waals force on the critical buckling load and nondimensional natural frequency of triple walled ZnO nanobeam are investigated. The results of this study show that the critical buckling load reduces with increasing the temperature change, direct electric field, magnetic field, and length of nanotube, and vice versa for the thickness of nanotubes, and two parameters elastic foundations. Also, the nonlocal critical buckling load and nondimensional natural frequency of Timoshenko nanobeam are smaller than the local critical buckling load and nondimensional natural frequency. The results can be useful for designing the smart nanobeam for MEMS and NEMS. 



Keywords: Nonlocal buckling and vibration analysis Triplewalled ZnO piezoelectric Timoshenko nanobeam Elastic foundation Magnetoelectrothermomechanical loadings. 


© 2015 Published by Semnan University Press. All rights reserved. 
The piezoelectric thin films have been widely used in transducers, actuators, and resonators. The piezoelectric transducers generate electrical signals in response to the mechanical vibrations and produce mechanical energy in response to the electrical signals. In addition, the piezoelectric thin films have been used in sensors, MicroElectroMechanical Systems (MEMSs), and NanoElectroMechanical Systems (NEMSs). The piezoelectric ZnO layer has large electromechanical coupling coefficients; furthermore, it can be epitaxial growth on the high acoustic velocity substrates at a relatively low temperature, which makes it a promising candidate for the multilayers thin film acoustic devices. Nano science is developing rapidly and this development is due to the interest of many researchers in the various structures, such as nanobeam, graphene sheet, and nanoshell that the small scale effect plays an important role at nano scale.
The nonlocal elasticity theory was presented first by Eringen [1] in 1983. So many researches in the field of Piezoelectricity analysis are done by researchers [2]. The wave propagation in SingleWalled Carbon NanoTube (SWCNT) is investigated by Narendar et al. [3] under the longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory. They investigated the effect of longitudinal magnetic field on wave dispersion characteristics of an equivalent continuum structure of the SWCNT embedded in an elastic medium. Their obtained results show that the velocity of flexural waves in the SWCNTs increases with an increasing in the longitudinal magnetic field exerted on it in the frequency ranges such as 0–20 THz.
Kiani [4] studied magnetothermoelastic fields caused by an unsteady longitudinal magnetic field in a conducting nanowire accounting for eddycurrent loss. His results reveal that the eddycurrent loss plays a crucial role in dynamic elastic fields within the nanowire, particularly for low initial duration of the applied magnetic field as well as low levels of the insulation of the nanowire’s surface. Ghorbanpour Arani et al. [5] investigated electrothermotorsional buckling of an embedded armchair DoubleWalled Boron Nitride NanoTube (DWBNNT) using nonlocal shear deformation shell model. Their numerical results depict that the critical buckling load decreases with considering the piezoelectric effect. In the other work, they [6] illustrated the effect of Carbon NanoTube (CNT) volume fraction on the magnetothermoelectromechanical behavior of smart nanocomposite cylinder. Their results show that increasing the CNT volume fraction enhances the strength of the nanocomposite cylinder. Thermomagnetodynamic stresses and perturbation of magnetic field vector in a nonhomogeneous hollow cylinder are presented by Kong et al. [7]. Their numerical results described the effects of the nonhomogeneous exponent of material for the hollow cylinders on the response amplitude and the phase of thermomagnetostresses and magnetic field vector perturbation.
Murmu et al. [8] presented the axial vibration of the embedded nanorods under transverse magnetic field effects via nonlocal elastic continuum theory. They investigated the effects of an external transverse magnetic field on the axial vibration of a nanorod such as a carbon nanotube with and without the elastic medium. Wang and Li [9] investigated the nonlinear primary resonance of a nano beam with an axial initial load using the nonlocal continuum theory. They show that the resonant amplitude decreases with increasing Winkler foundation modulus and decreasing the ratio of the length to the diameter. Mohammadimehr and Rahmati [10] considered the electrothermomechanical nonlocal axial vibration analysis of SingleWalled BoronNitride Nano Rods (SWBNRs) under the electric excitation. They obtained the constitutive equation for the nano rods under electrothermomechanical loadings, then they discussed about the effects of the aspect ratio, small scale parameter, clampedclamped and clampedfree boundary conditions on the natural frequency.
A piezoelectric ZnOCNT nanotube under the axial strain and electrical voltage is illustrated by Zhang et al. [11]. Their numerical results show that the critical buckling axial decreases as the length of ZnOCNT and the applied voltage increase Akbari Alashti et al. [12] studied the thermoelastic analysis of a functionally graded spherical shell with the piezoelectric layers Using Differential Quadrature Method (DQM). They obtained the effects of the power index of material properties; temperature change and thickness of piezoelectric layers on stress and displacement were presented. Nonlinear softening and hardening nonlocal bending stiffness of an initially curved monolayer graphene sheet are illustrated by Jomehzadeh et al. [13]. They find out that the bending stiffness of graphene strongly depends on the initial configuration, showing no obvious maxima and minima, and suggesting the possibility of a smart tuning.
Lei et al. [14] presented the asymptotic frequencies of various damped nonlocal beams and plates. They investigated the asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. Mohammadimehr et al. [15] investigated biaxial buckling and bending of smart nanocomposite plate reinforced by carbon nanotube under electromagnetomechanical loadings based on the extended mixture rule approach. Their results indicate that the nonlocal deflection of the smart nanocomposite plate decreases with an increase in the magnetic field intensity. Ansari et al. [16] studied the free vibration of sizedependent functionally graded microbeams based on using the strain gradient Reddy beam theory. They observe that the critical buckling loads and natural frequencies predicted by the beam models based on the MSGT and CT are the maximum and minimum values, respectively. Also, it is shown that increasing the material property gradient index leads to lower nondimensional natural frequencies.
Najar et al. [17] developed the nonlinear nonlocal analysis of the electrostatic nanoactuators. Their simulated results show that the effect of the initial deflection seems to beproducing frequency veering. Reddy and ElBorgi [18] obtained the Eringen’s nonlocal theories of beams accounting for moderate rotations. Their numerical results for bending response are presented to illustrate the parametric effect of boundary conditions and the influence of the nonlocal parameter. It is shown that for all beams, except for those beams for which both the transverse displacement and slope are not specified at a boundary point, the nonlocal parameter has the effect of increasing the deflections.
Adhikari et al. [19] reported the nonlocal normal modes in the nanoscale dynamical systems. Their theoretical results are applied to three representative problems, namely (a) an axial vibration of a singlewalled carbon nanotube, (b) a bending vibration of a doublewalled carbon nanotube, and (c) a transverse vibration of a singlelayer graphene sheet. The DQM for a nonlinear nonlocal buckling analysis of a Double Layer Graphene Sheet (DLGS) integrated with the ZnO piezoelectric layers are studied by Ghorbanpour Arani et al. [20]. Their results present that the intensifying magnetic field makes the system more stable. Furthermore, increasing the thickness of both piezoelectric and graphene layers makes the system stiffer, and consequently the critical buckling load becomes more. The thermal postbuckling behavior of the sizedependent Functionally Graded (FG) Timoshenko microbeams is studied by Ansari et al. [21]. They illustrated influences of the material gradient index, length scale parameter, and boundary conditions on the thermal postbuckling behavior of FG microbeams are comprehensively investigated and also, considered the effect of geometrical imperfection on the buckling deformation of microbeams in prebuckled and postbuckled states.
Karlicic et al. [22] analyzed the dynamics of multiple viscoelastic carbon nanotube based on nanocomposites with an axial magnetic field. Their results show that the influence of the longitudinal magnetic field on the free vibration response of viscoelastically coupled multinanobeam system (MNBS) and discussed in details. An enhanced performance of a ZnO nanowirebased selfpowered glucose sensor by piezotronic effect is reported by Yu et al. [23]. Their results show that the performance of the glucose sensor is generally enhanced by the piezotronic effect when applying a –0.79% compressive strain on the device, and magnitude of the output signal is increased by more than 200%; the sensing resolution and sensitivity of sensors are improved by more than 200% and 300%, respectively. Liew et al. [24] presented the postbuckling of the carbon nanotubereinforced functionally graded cylindrical panels under axial compression using a meshless approach. They used several numerical cases to study the effect of various parameters including the carbon nanotube volume fraction, lengthtothickness ratio and radius on the postbuckling behaviour of CNTRFG cylindrical panels.
Mohammadimehr et al. [25] investigated the free vibration of viscoelastic doublebonded polymeric nanocomposite plates reinforced by Functionally Graded Carbon Nanotube reinforced composites (FGSWCNTs) using a Modified Strain Gradient Theory (MSGT), sinusoidal shear deformation theory and meshless method. Their results show that the elastic foundation, van der Waals (vdW) interaction and magnetic field increase the dimensionless natural frequency of the doublebonded nanocomposite plates for Classical Theory (CT), Modified Couple Stress Theory (MCST) and MSGT. Also, the material length scale parameter effects on the nondimensional natural frequency of the double bonded nanocomposite plates are negligible at for CT, and MCST and MSGT. In the other study, Mohammadimehr et al. [26] investigated the size dependent effect on the buckling and vibration analysis of doublebonded nanocomposite piezoelectric plate reinforced by Boron Nitride NanoTube (BNNT) based on MCST. The results of their research show that the critical buckling load decreases with an increase in the dimensionless material length scale parameter.
The vibration of nonlinear graduation of nanoTimoshenko beam considering the neutral axis position is studied by Eltaher et al. [27]. Their obtained numerical results reflected the significant effect of neutral axis position, material distribution profile, and the nonlocality parameter on the fundamental frequencies of the nanoTimoshenko beams. The exact buckling solution for the twolayer Timoshenko beams with interlayer slip is presented by Le Grognec et al. [28]. It is shown that their results are in much better agreement with the numerical values than the obtained solutions with the simplified kinematic assumptions. Aydogdu [29] analyzed the propagation of longitudinal waves in the multiwalled carbon nanotubes using nonlocal theory. He studied the effects of van der Waals force, scale parameter and radius on the wave propagation. Ghorbanpour Arani et al. [30] studied viscosurfacenonlocal piezoelectricity effects on the nonlinear dynamic stability of the graphene sheets integrated with ZnO sensors and actuators using a refined zigzag theory. The results depict that the magnetic field and external voltage are effective controlling parameters for dynamic instability region of system.
Ghannadpour et al. [31] examined the stress analysis, buckling and vibrations of EulerBernoulli beam. They developed analytical formulas to find buckling stiffness matrix and mass matrix, and then, they obtained the critical buckling load, natural frequency and deflection using Ritz method and desired boundary conditions. Wang and Adhikari [32] studied vibration analysis of the composite nanotubes synthesized by coating CNTs with piezoelectric ZnO. Their results show that the half wave number related to deformation energy, does not have observable influence on the vibration. Also the vdW interaction can play a predominant role in the mechanics of the nanostructures. Rahmati and Mohammadimehr [33] investigated the vibration analysis of the nonuniform and nonhomogeneous Boron Nitride NanoRod (BNNR) embedded in an elastic medium under combined loadings using DQM. They conclude that the nondimensional frequency ratio of nonhomogeneous BNNR decreases with the presence of electrothermal loadings, and their effect on the nondimensional frequency ratio is higher in short nanorods and high nonlocal parameter.
In this study, the nonlocal buckling and vibration analysis of triplewalled ZnO piezoelectric Timoshenko nanobeam is subjected to magnetoelectrothermomechanical loadings. The equilibrium equation using Timoshenko beam free body diagram and writing equilibrium equation of force and momentum are achieved and solved by Navier's method for a simplysupported nanobeam. Also, the effects of various factors including the elastic medium, small scale, length, thickness, interlayer van der Waals force on the critical buckling load and natural frequency of triplewalled ZnO nanotube are investigated.
2. TripleWalled Timoshenko Beam Model
In this article, the triplewalled ZnO piezoelectric nanobeam model is investigated based on the Timoshenko nanobeam theory. As an improvement of the multiEuler beam model, the multiTimoshenko beam model is proposed. In this model, each nanotube is simulated by a Timoshenko beam that allows for the effects of transverse shear deformation and rotary inertia. The deflections of the adjacent tubes are coupled through the van der Waals force and elastic foundation which are determined by the interlayer spacing. A schematic configuration of the ZnO piezoelectric Timoshenko nanobeam has been illustrated in Fig. 1.
The displacement fields for the Timoshenko beam model can be considered as follows [34]:
(1) 
where w is displacement component at the midsurface of the beam, is the section normal vector rotations about the y, and t denotes time.
Figure 1. A schematic configuration of the ZnO piezoelectric Timoshenko nanobeam 
Consider the element of the Timoshenko beam shown in Figure 2. The angle denotes the shear deformation of the element so we have what follows, from Figure 2 [35].
(2)

The bending moment M and the shear force V are written as follows:
(3)

Where w and are the transverse displacement, the slope of the beam due to bending deformation alone, respectively, x, I, A, E, G, and denote the axial coordinate, the area moment of inertia, the crosssectional area, the Young’s modulus, the shear modulus and the shear correction factor (which are dependent on the shape of the cross section), respectively. Since the triplewalled ZnO piezoelectric Timoshenko nanobeam is modelled as a single beam with hollow annular cross section area, the dependence of the shear correction factor on its crosssectional shape is considered, and is determined by the following formula [36]:
(4a)

where is the ratio of the innermost and the outermost diameters of the tube; therefore, k equals what follows:
(4b)

Figure 2. A schematic view of the Timoshenko beam element [35] 
2.1. Governing equation of motion
Based on the Eringen's elasticity theory, the nonlocal constitutive formulation can be expressed in the following form:
(5)

Where and represent the nonlocal and local stress, respectively, is the Laplacian operator; e_{0} and a are a constant associated with each material and the internal characteristic length, respectively.
So, the governing equations of motion for ZnO piezoelectric Timoshenko nanobeam can be obtained by equations that are mentioned in Appendix A. [35, 37]:
(6)

Finally, for the triplewalled ZnO piezoelectric Timoshenko nanobeam, the governing equations of motion are written as follows using Eq. (6):
(7a) 

(7b)


(7c)


Where p and q(x) are axial and distributed transverse loads, respectively that are obtained by the following form.
(7d)

2.2. External force
The external forces can be divided into the following parts:
• Winkler and Pasternak foundations.
• One dimensional magnetic field applied to the nanobeam.
• Van der Waals interaction forces.
2.2.1. Elastic medium
The Timoshenko nanobeam is resting on an elastic foundation whose supporting action is described by Pasternaktype relationship. The Pasternak foundation force can be expressed as follows [34]:
(8)

where w is the transverse deflection of the beam, K_{w} and K_{g} are spring and shear coefficients of the elastic foundation, respectively.
2.2.2. Magnetic field
For a nanobeam subjected to a magnetic field, H, the exerted body force can be calculated as what follows [3, 20]:
(9)

For simplifying the analysis, a longitudinal magnetic field vector is considered as H= (H_{x}, 0, 0).
The current density (J) and the Maxwell equations are given by the following equations:
(10)

where is the magnetic field permeability.
Therefore, the components of Lorentz forces along the x, y and z directions are as follows:
(11)

2.2.3. Van der Waals force
As shown in Figure 1, the triplewalled ZnO piezoelectric Timoshenko nanobeam can be assumed as a set of concentric nanobeam with vdW interaction forces between the inner and the outer layers which are equal in the magnitude and opposite in the sign. Therefore, the vdW interaction forces between adjacent layers can be expressed as follows [5]:
(12)

where is vdW interaction coefficient.
So, the forces between adjacent layers can be expressed as what follows:
(13a)

Substituting Eq. (12) into Eq. (13a) yields to the following equations:
(13b)

The triplewalled ZnO piezoelectric Timoshenko nanobeam under the electrothermomechanical loading can be considered as what follows:
(14a)

Where superscript M, T and E indicate the mechanical, thermal and electrical components of load as what follows [38]:
(14b)

Finally, substituting Eq. (14b) into Eq. (14a) yields to the following expressions:
(14c)

3. Navier’s Type Approach
The developed governing differential equations of section 2 have been solved by Navier’s approach for simply supported boundary conditions. The simply supported boundary condition for nanobeam is considered as follows:
x=0 and x=L:
The following expressions of various generalized displacements have been assumed to be as what follows:
(15)

where .
By substituting Eq. (15) into Eq. (7), three linear algebraic equations can be obtained that are shown in Appendix B. The matrix form of Eqs. (15a) (15c) is derived as follows:
(16)

where the coefficients in Eq. (16) are shown in Appendix C.
The natural frequencies and critical buckling load are obtained from the following expression:
(17)

4. Numerical Results and Discussions
In this paper, using the nonlocal elasticity theory, the buckling and free vibration analysis of triple walled Zno piezoelectric Timoshenko nanobeam on elastic Pasternak foundation under magnetoelectrothermomechanical loadings using Navier’s type approach is investigated. The physical, geometrical and mechanical parameters of Zno piezoelectric Timoshenko nanobeam are considered in Table1. The physical, geometrical and mechanical parameters of the ZnO piezoelectric Timoshenko nanobeam are considered in Table1. In Table 2, to validate this study with the obtained results by Wu and Lai, the dimensionless natural frequencies of singlewalled carbon nanotube are calculated using Navier’s method, and Timoshenko beam theory for various small scale effects is employed. The results of this research are compared with the obtained results by Wu and Lai [39] for the following mechanical properties:
(18) 
From Table 2, it can be observed that the results of current study are in good agreement with the obtained results by Wu and Lai [39] for various nonlocal parameters.
The effect of various loadings such as all fields (magnetoelectrothermomechanical fields) (cases a and b), magnetothermomechanical fields (case c), and electrothermomechanical fields (case d) and length of triplewalled ZnO piezoelectric Timoshenko nanobeam on the critical buckling load is investigated in Table 3. It is shown that the the critical buckling load for all states occurs in m=2. Also, the buckling load under the combined loadings (all fields) is less than the other states, and decreases with an increase in the length of nanobeam.
This study presents the buckling and vibration behaviors of simply supported triplewalled ZnO piezoelectric Timoshenko nanobeam. The effect of various loadings on the critical buckling load of the triplewalled ZnO piezoelectric Timoshenko nanobeam is investigated in Figure 3.
Table 1. The physical, geometrical and mechanical parameters of the ZnO piezoelectric Timoshenko nanobeam [3, 5]
parameter 
value 
parameter 
value 
h 
0.066 nm 
L 
90nm 
E 
5.5 TPa 
0.19 

K 
0.5 
K_{g} 

K_{w} 

1.09nm 

0.256nm 

0.056nm 

T 
200 

It can be observed that employing the thermal, electrical and magnetic fields in longitudinal direction of the triplewalled ZnO piezoelectric Timoshenko nanobeam decreases the critical buckling load. Also, it can be observed in this figure that the difference between various loadings is more obvious in m=2 and the critical buckling load for all states occurs in m=2. In addition, it can be seen from this figure that the maximum buckling load is related to the mechanical loading, and the buckling load under combined loadings (all fields) is less than the other states.
Figures 4a and 4b demonstrate the effect of Van der Waals force on the critical buckling load and nondimensional natural frequency of the triplewalled ZnO piezoelectric Timoshenko nanobeam under combined loadings versus longitudinal wave number. This figure illustrates that, because of the presence of Van der Waals force, the system becomes stiffer. On the other hand, the system is a more stable. Also, it is significant in these results that with considering the first and third layers of Van der Waals force, the buckling load diagram nearly doubles, but the change of critical buckling load is negligible. The effect of the external electric voltage of the ZnO piezoelectric nanobeam on the critical buckling load is demonstrated in Figure 5.
Table 2. The comparison between the results of the current study and the obtained results by Wu and Lai [39]
Vibration mode=1 
Present work 
Ref. [39] 

0 
9.7297 
9.7254 

2 
8.9202 
8.9163 

4 
8.2841 
8.2804 
Table 3. The effect of various loadings and length of triplewalled ZnO piezoelectric Timoshenko nanobeam


m=1 
107.513 
114.029 
107.905 
107.693 
m =2 
67.637 
68.979 
67.954 
67.783 
m =3 
94.508 
76.795 
94.824 
94.6527 
m =4 
147.821 
109.883 
148.137 
147.966 
m =5 
220.746 
158.685 
221.062 
220.891 
m=6 
311.557 
220.746 
311.873 
311.702 
m=7 
419.660 
295.211 
419.9767 
419.805 
Figure 3. The effect of various loadings on the critical buckling load of the triplewalled ZnO piezoelectric Timoshenko nanobeam 
(a)
(b) Figure 4. The effect of van der Waals (vdW) force on the critical buckling load and nondimentional natural frequency of the triplewalled ZnO piezoelectric Timoshenko nanobeam under the combined loadings versus m 
It shows applying positive voltage shifts of system to lower the critical buckling load. This is due to the fact that the imposed positive and negative voltages generate the axial compressive and tensile forces in the top of ZnO piezoelectric nanobeam, respectively. Hence, the imposed external voltage is an effective controlling parameter for the dynamic stability of system.
Figure 6 shows the effect of an induced upward and downward magnetic field of the Zno piezoelectric nanobeam on the critical buckling load. The influence of an induced upward magnetic field is more than an induced downward magnetic field. With the increasing number of longitudinal wave for m>10, the difference of critical buckling load between two cases increases and it cannot be ignored, while for m<10, this result is not noticeable.
Figure 5. The effect of the external electric voltage of the Zno piezoelectric nanobeam on the critical buckling load 
Figure 6. The effect of an induced upward and downward magnetic field of the ZnO piezoelectric nanobeam on the critical buckling load 
Figs. 7a and 7b present the effect of spring constant on the critical buckling load and nondimensional natural frequency of ZnO piezoelectric nanobeam, respectively. It is observed from the results that the critical buckling loads and the dimensionless natural frequency with increasing the spring constant increases. On the other hand, the triplewalled ZnO piezoelectric nanobeam becomes stiffer, then the stability of system enhances. Also, in this ﬁgure the slope of curves is not constant. These increases in the nondimensional natural frequency versus the wave number are found to be almost dependent on the change of the value of the elastic layer stiffness especially at higher values. At lower values of the elastic layer stiffness, this effect is not signiﬁcant.
Figs. 8a and 8b depict the local and nonlocal critical buckling and the dimensionless natural frequency loads under the magnetoelectrothermomechanical loading versus the axial half wave number. At nano scale, the small scale effect (e_{0} a) is considered in formulation that this effect at macro scale is negligible, while at nano scale this parameter must be considered and this effect cannot be ignored. This effect has been shown in Figures 8a and 8b. On the other hand, in classical or local theory of continuum mechanics, the stress at a point is only proportional to the strain at that point. This theory is valid for a large scale. In a small scale, the stress at a reference point x is a function of the strain at all the other points of the body. This phenomenon is known as smallscale effect which is shown in the constitutive equations by the parameter e_{0} a and its theory is identiﬁed as the smallscale or nonlocal theory. For a structure in the nanoscale, it is not reasonable to ignore the smallscale effect (e_{0} a). Ignoring this term (e_{0} a = 0), the nonlocal theory reduces to a local or classical theory which has no desired accuracy for the analysis of CNTs. It is shown that with considering the small scale effect, the critical buckling load and nondimensional natural frequency decrease. Moreover, the difference between the local and nonlocal critical buckling load increases with increasing the axial half wave number. It is due to the fact that the nonlocal theory introduces a more flexible model. Also, for lower values of (e_{0}a) the critical buckling loads are higher while the critical buckling loads are lower for highscale coefficients (e_{0}a). It is due to the fact that the nonlocal theory introduces a more flexible model wherein the atoms are joined by the elastic springs.
Figs. 9a and 9b illustrate the effects of the length of the triplewalled ZnO piezoelectric Timoshenko nanobeam on the critical buckling load and nondimensional natural frequency, respectively.
It is shown that with an increase in the length of the nanobeam, the instability of the system increases. Therefore, the lower length should be taken into account for ZnO in optimum design of nano/micro devices.
In Figures 10a and 10b, the effect of different values of ZnO piezoelectric nanobeam radius on the critical buckling load is investigated. According to this figure, increasing the radius of nanotubes and the number of longitudinal wave, the critical buckling load increases, and increasing the longitudinal wave number and radius, the difference between local and nonlocal theories increases.
(a)
(b) Figure 7. The effect of spring constant on the critical buckling load and nondimentional natural frequency of the ZnO piezoelectric nanobeam 
(a)
(b) Figure 8. The effect of nonlocal parameter on the critical buckling load and nondimentional natural frequency of the triplewalled ZnO piezoelectric nanobeam 
5. Conclusion
In this study, based on the Timoshenko beam theory, the buckling and free vibration analysis of the triple walled ZnO piezoelectric Timoshenko nanobeam on the elastic Pasternak foundation under magnetoelectro thermomechanical loadings is analytically investigated.
The effects of various parameters including the elastic medium, small scale parameter, length, interlayer van der Waals force on the critical buckling load and nondimensional natural frequency of the triplewalled ZnO piezoelectric Timoshenko nanobeam were presented.
The result of this research can be listed as follows:
1 The effects of the surrounding elastic medium, such as the spring constant of the Winklertype and the shear constant of the Pasternaktype, as well as the vdW forces between the inner and the outer nanotubes were taken into account. Generally, the critical bucking load and nondimensional natural frequency increasein the presence of the surrounding elastic medium, including the Winkler and Pasternak foundations. Also, the presence of van der Waals force has a positive role on the stability of the system.
2 Applying positive voltage shifts to the buckling of system to lower frequency zone and vice versa. It is also seen that the difference between various loadings is more obvious, and the critical buckling load for all states occurs in m=2.
(a)
(b) Figure 9. The effect of the length of the triplewalled Zno piezoelectric Timoshenko nanobeam on the critical buckling load and nondimentional natural frequency for 
(a)
(b) Figure 10. The effect of different values of ZnO piezoelectric nanobeam radius on the critical buckling load and nondimentional natural frequency

3 As the system becomes stiffer, the critical buckling load becomes larger. Also it can be seen the maximum buckling load is related to the mechanical loading, and the buckling load under combined loadings (all fields) is less than the other states.
4 The influence of the small length scale on the buckling load was investigated. It is concluded that increasing the nonlocal parameter leads to a decrease in the system stability.
5 It is shown that the nonlocal critical buckling load decreases with increasing piezoelectric and magnetic field constant. On the other hand, considering the piezoelectric and magnetic fields leads to a decrease in the stability of system.
6 Increasing the radius of nanotubes and the number of longitudinal wave, the critical buckling load and dimensionless natural frequency increase.
7 With an increase in length of the beam, the instability of the system decreases.
8 The influence of an induced upward magnetic field is more than an induced downward magnetic field. It can be seen with the increasing number of longitudinal wave for m>10, the difference of critical buckling load between two cases increases and it cannot be ignored, while for m<10, this result is not noticeable.
9 The results of this research can be used for studying the electro thermomechanical buckling behavior of the smart piezoelectric nanotubes such as multiwalled BNNTs that are assumed to be surrounded by a bundle of CNTs.
Acknowledgments
The authors would like to thank the reviewers for their reports to improve the clarity of this article. Moreover, the authors are grateful to the University of Kashan for moral and financial supporting of this research by Grant No. 363452/14. They would also like to thank the Iranian Nanotechnology Development Committee for their financial support.
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Appendix A
To decouple the equations of motion, the following manner should be considered.
The shear force and moment for nonlocal Timoshenko nanobeam are considered as follows [40]:
(A1) 

(A2) 
The equations of motions are written as follows:
(A3) 

(A4) 
Substituting Eqs. (A1) and (A2) into Eqs. (A3) and (A4), one can obtain the following equations:
(A5)


(A6)

Differentiating Eq. (A1) with respect to the independent variable x three times, we have:
(A7)

Substituting Eq. (A3) into Eq. (A7), the following equation is derived:
(A8)




Also, using Eqs. (A1) and (A3), we have:
(A9)

Substituting Eqs. (A8) and (A9) into Eq. (A6) yields the governing equation of motion for ZnO piezoelectric Timoshenko nanobeam that is shown in Eq. (6).
Appendix B
Substituting Eq. (15) into Eq. (7), three linear algebraic equations for triplewalled ZnO piezoelectric Timoshenko nanobeam can be obtained that are shown in the following form:
(B1) 

(B2)


(B3)

Appendix C
The coefficients in Eq. (16) are written as follows:
(C1)


(C2)


(C3)
