Vibration in an Electrically Affected Hygro-magneto-thermo-flexo Electric Nanobeam Embedded in Winkler-Pasternak Foundation

1. Selvamani^a*, J. Rexy^b, F. Ebrahimi^c

^a Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore-641114, Tamilnadu, India

^b Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, 641114, India

^c Department of Mechanical Engineering, Imam Khomeini International University Qazvin, Iran

1.     Introduction

Structural monitoring of electrical nanobeams, nanoplates, and nanomembranes is a recent novel field for many researchers due to their quality properties. The classical continuum theory is applied practically in the mechanical behavior of macroscopic structures. Still, it is improperly for the size effect on the mechanical treatments on micro- or nanoscale structure. Nevertheless, the classical continuum theory needs to be extended to factor in the nanoscale results. The prime magneto-electro-elastic (MEE) was used in the 1970s, and the MEE composite consisting of the piezoelectric and piezo magnetic phases was discovered this year. Van Den Boomgard et al. [1] the MEE nanomaterials (BiFeO3, BiTiO3-CoFe2O4, NiFe2O4-PZT) and their nanostructures became a significant role in researches (Zheng et al. [2], Martin et al. [3], Wang et al. [4], Prashanthi et al. [5]). For this reason, nanostructure's major potential for amplification many applications, their mechanical behavior should be investigated and well-identified before new designs can be proposed. The classical mechanic continuum theories demonstrate that to predict the response of structures up to a minimum size sub, they fail to provide accurate predictions. The nonlocal theories add a size parameter in the modeling of the continuum. This paper studied models that developed according to the greatly used nonlocal elasticity theory (Eringen [6], Eringen [7], Eringen [8], Eringen [9]). Timoshenko beam theory nonlocal elasticity was investigated in their study. So based on an elastic medium, the stability response of SWCNT is described. Winkler and Pasternak parameters, the aspect ratio of the SWCNT, and the nonlocal parameter were studied. Nonlinear free vibration of SWCNTs based on Eringen's nonlocal elasticity theory was developed by Yang et al. [10]. Ehyaei and Akbarizadeh [11] discussed vibration analysis of the micro composite thin beam based on modified couple stress theory elaborately.

Later, Ebrahimi and Barati [12] developed a unified formulation to model inhomogeneous nonlocal beams. Free vibration analysis of chiral double-walled carbon nanotube embedded in an elastic medium using nonlocal elasticity theory and Dihaj et al. [13] have investigated Euler- Bernoulli beam model. The vibration and buckling of piezoelectric and piezomagnetic nanobeams are verified based on the third-order beam model by Ebrahimi and Barati [14, 15, 16, 17]. The vibration, buckling, and bending of Timoshenko nanobeams based on a meshless method was proposed by Roque et al. [18]. Embedded in the nonlocal component relevance of Eringen, major articles were published searching to enlarge nonlocal beam models for nanostructures Peddieson et al. [19]. A novel method was proposed via nonlocal Euler–Bernoulli and Timoshenko beam theory, accepted by many studies to verify bending of the beam by (Civalek and Demir [20], Wang [21], Wang et al. [22]). During the years of research the small-size agents in SWCNTs, several studies were devoted by (Murmu and Pradhan [23], Karami et al. [24]) for the wave propagation of functionally graded anisotropic nanoplates resting on the Winkler-Pasternak foundation. The propagation of elastic waves in thermally affected embedded carbon-nanotube-reinforced composite beams via various shear deformation plate theories was analyzed by Ebrahimi and Rostami [25].

Hajnayeb and Foruzande [26] verify free vibration analysis of a piezoelectric nanobeam using nonlocal elasticity theory with its numerical computation. Thermo-magneto-electro vibrations of FG nanobeam and plates are studied for stability analysis via various physical variants by Ebrahimi et al. [27], Ebrahimi et al. [28], Ebrahimi et al. [29]. Yazdi has discussed the large amplitude forced vibration of the functionally graded nanocomposite plate with piezoelectric layers resting on a nonlinear elastic foundation [30]. Thermo-magneto-electro-elastic analysis of a functionally graded nanobeam integrated with functionally graded piezomagnetic layers was proposed in composite structures by Arefi and Zenkour [31]. An investigation has been conducted on the wave propagation analysis of smart strain gradient piezo-magneto-elastic nonlocal beams by Ebrahimi and Barati [32]. The static analysis of laminated piezo-magnetic size-dependent curved beam was concentrated on the modified couple stress theory by Arefi [33]. Studies were conducted on the analytical solutions to magneto-electro-elastic beams by Aimin and Haojiang [34].

Furthermore, many researchers have presented the static and dynamic characteristics of beams and plates exposed to hygro-thermal environments because of the considerable effects on the structure's behavior. An analytical method to determine Gayen and Roy [35] presented the stress distributions in circular tapered laminated composite beams under hygro and thermal loadings in detail. Kurtinaitiene et al. [36] introduced the effect of additives on the hydrothermal synthesis of manganese ferrite nanoparticles. The size effects on static behavior of nanoplates resting on elastic foundation subjected to hygro-thermal loadings have been developed using several beam theories by Alzahrani et al. [37]. They extended the nonlocal constitutive relations of Eringen to contain the hygro-thermal effects. Also, Sobhy [38] studies show the frequency response of simply-supported shear deformable orthotropic graphene sheets exposed to hygro-thermal loading.

Ghorbanpour Arani and Zamani [39] examined the bending of electro-mechanical sandwich nanoplate based on silica aerogel foundation with physical variables. They described that the influence of parameters on nanostructures such as applied voltage, porosity index, foundation characteristics, parameter, plate aspect ratio, and thickness ratio was studied on bending response of sandwich nanoplate. It was shown that the three-unknown shear and normal deformations nonlocal beam theory for the bending analysis by Simsek et al. [40]. He researched the bending and buckling of the FG based on the nonlocal Timoshenko and Euler–Bernoulli beam theory. Also, he was described that the power-law exponent has a wide influence on the responses of FG nanobeam. Free vibration of size-dependent magneto-electro-elastic nanoplates was derived by authors (Ke et al. [41], Ramirez et al. [42]).

Authors, Zur et al. [43], Arefi et al. [44], Arefi and Zenkour [45], Arefi and Zenkour [46], Arefi et al. [47] have investigated free vibration and bending analyses of thermo-magneto-electro-elastic of nanoplates. Arefi and Zenkour [48] studied the size-dependent electro-elastic analysis of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory. Arefi and Rabczuk [49] analyzed the nonlocal higher-order shear deformation theory for electro-elastic analysis of a piezoelectric doubly curved nanoshell. They concluded that with an increase of nonlocal parameters, the stiffness of the nanoshell is decreased. Consequently, the displacement components, rotation component, and maximum electric potential are increased significantly.  Barati et al. [50] reported the dynamic response of nanobeams subjected to moving nanoparticles and hygro-thermal environments based on nonlocal strain gradient theory. They found that the dynamic deflection increased as the temperature/moisture increased. Bending vibration and electro-magneto-elastic responses of piezomagnetic curved nanobeams have been developed by authors Arefi and Zenkour [51, 52]). Zenkour and Alghanmi [53] constructed the analytical model for the static response of sandwich plates with FG core and piezoelectric faces under thermo-electro-mechanical loads and resting on elastic foundations. They have shown that the inclusion of Pasternak's foundation reduces deflections. The modified coupled stress theory and vibration analysis of nanobeams resting on visco-Pasternak foundations have been studied by Sobhy and Zenkour [54] and Zenkour and El-Shahrany [55].

Barati and Zenkour [56] analyzed the forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments. They also found that increase of Kerr foundation parameters leads to postponement in resonance frequencies of FG nanobeams. Daik and Zenkour [57] reported the bending of Functionally Graded Sandwich Nanoplates Resting on Pasternak Foundation under Different Boundary Conditions. The electro-mechanical energy absorption, resonance frequency, and low-velocity impact analysis of the piezoelectric doubly curved system were derived by Guo et al. [58]. Dai et al. [59] developed on the vibrations of the non-polynomial viscoelastic composite open-type shell under residual stresses. Dynamic simulation of the ultra-fast-rotating sandwich cantilever disk via finite element is modeled by Wu and Habibi [60]. Al-Furjan et al. [61] introduced the vibrational characteristics of a higher-order laminated composite viscoelastic annular microplate via modified couple stress theory. Al-Furjan et al. [62] proposed the three-dimensional frequency response of the CNT-Carbon-Fiber reinforced laminated circular/annular plates under initial stresses. Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory was derived by Al-Furjan et al. [63]. Dai et al. [64] investigated the frequency characteristics and sensitivity analysis of a size-dependent laminated nanoshell.

Wang et al. [65] developed the frequency and buckling responses of a high-speed rotating fiber metal laminated cantilevered microdisk. Ghabussi et al. [66] constructed the seismic performance assessment of a novel ductile steel braced frame equipped with Steel curved dampers. Ghabussi et al. [67] proposed the frequency characteristics of a viscoelastic graphene nanoplatelet–reinforced composite circular microplate. Ghabussi et al. [68] studied the improving seismic performance of portal frame structures with steel curved dampers. They observed that there were significant improvements in the seismic performance of both types of portal frames by utilizing proposed steel curved dampers. Zhao et al. [69] presented the bending and stress responses of the hybrid axisymmetric system via the state-space method and 3D-elasticity theory. Ma et al. [70] developed the chaotic behavior of graphene-reinforced annular systems under harmonic excitation. Jiao et al. [71] proposed the coupled particle swarm optimization method with a genetic algorithm for the static-dynamic performance of the magneto-electro-elastic nanosystem. Huang et al. [72] have discussed the computer simulation via a couple of homotopy perturbation methods and the generalized differential quadrature method for nonlinear vibration of functionally graded non-uniform micro-tube.

Literature review reveals the lack of an analytical investigation concerning bending of hygro thermo magneto flexo electric nanobeams based on nonlocal elasticity theory. Thus, the authors aimed to construct an analytical model of bending hygro thermo magneto flexo electric nanobeams based on nonlocal elasticity theory.         

This paper studied the bending of hygro thermo magneto flexo electric nanobeams (Fig. 1) based on the nonlocal elasticity theory. Governing equations of a nonlocal nanobeam on Winkler-Pasternak substrate are derived via Hamilton's principle. Galerkin method is implemented to solve the governing equations. Effects of different factors such as nonlocal parameter, slenderness, moisture constant, critical temperature, applied voltage and magnet potential, Winkler-Pasternak parameters effect on deflection characteristics of a nanobeam are investigated.

2.     Formulation of the Problem

The component of displacement via refined shear deformable beam can be expressed by:

                                 (1)

 (2)

where is axial mid-plane displacement and  denote the bending and shear components of transverse displacement, respectively. Also, f(z) is the shape function representing the shear stress/strain distribution through the beam thickness, which for the present study has a trigonometric essence; thus, a shear correction factor is not required.

                                    (3)

Fig. 1. The geometry of nanobeam resting on elastic foundation

Non-zero strains of the suggested beam model can be expressed as follows:

                                           (4)

                                                                       (5)

where g(z)=1-df(z)/dz.                                       

According to Maxwell's equation, the relation between electric field (E_x, E_z) and electric potential and magnet field (Q_x, Q_z) and magnet potential can be obtained as Ke et al. [41]

(6)

                                       (7)

Through extended Hamilton's principle, the governing equations can be derived as follows:

                                                             (8)

where  is the total strain energy is the work done by externally applied forces. The first variation of strain energy  can be calculated as:

          (9)

Substituting Eqs.(1) - (5) into Eq.(9) yields:

 (10)

in which the forces and moments expressed in the above equation are defined as follows:

     (11)

The first variation of the work done by applied forces can be written in the form:

     (12)

where , and  are linear, shear coefficients of the medium,   electric, and magnetic loading, respectively.

                                                       (13)

                                                       (14)

K_w, k_p, and f₁₃ are Winkler, Pasternak, and damping constants and are applied forces due to variation of temperature and moisture as

                                                (15)

                                               (16)

where and  are thermal and moisture expansion coefficients, respectively. T and H denote the temperature and moisture variation, respectively.

The following Euler–Lagrange equations are obtained by inserting Eqs. (10), (12) in Eq. (8) when the coefficients of  are equal to zero:

                                                                                   (17)

                         (18)

                               (19)

                        (20)

            (21)

3.     Nonlocal Elasticity Theory  

The nonlocal theory can be extended for the piezo magnetic nanobeams as:

                                                                       (22)

       (23)

          (24)

                           (25)

where  corresponds with the components of the fourth-order elasticity tensor, also,  is the relative dielectric susceptibility and  is the flexoelectric coefficient. Also,  is a nonlocal parameter which is introduced to describe the size-dependency of nanostructures.

where   is the Laplacian operator. The stress relations can be expressed by:

                                                              (26)

(27)

(28)

(29)

(30)

(31)

where  and  are nonlocal and length scale parameters.

Integrating Eq. (26-31) over the cross-section area of nanobeam provides the following nonlocal relations for a refined beam model as also, normal forces and moments due to the electrical field can be defined by:

                                      (32)

                       (33)

                      (34)

                      (35)

                      (36)

in which:

                                                                      (37)

                   (38)

    (39)

   (40)                                 

                      (41)                               (42)

The governing equations of nonlocal strain gradient nanoplate under electrical field in terms of the displacement can be derived by substituting Eqs. (32) -(36), into Eqs. (17) - (21) as follows:

(43)

(44)

     (45)

                                                      (46)

                                                      (47)

4.     Solution Procedure

The displacement quantities are presented in the following form to satisfy the boundary conditions (given in Table 1) as :

                                                        (48)

                                                        (49)

                                                        (50)

                                                             (51)

                                                           (52)

where (  , , , ) are the unknown coefficients and for different boundary conditions ( , ) :

                                    (53)

where [K], and [F] are the stiffness, loading matrixes for nanobeam, respectively.

      (54)  



 

in which:

           (55)

Table 1. The admissible functions  Sobhy [38]

The uniform load is supposed that lead to bending and is expressed by the following form:  

                         (56)

                                                 (57) 

Qn is the Fourier coefficient, q(x) = q₀ is the uniform load density, and x₀ is the centroid coordinate. Also, in the case of concentrated point load, the following expression for the harmonic load intensity can be written: 

                                                   (58)

                                                             (59)

in which  is the Dirac delta.

5.      Numerical Results and Discussions

The bending of hygro magneto thermos piezoelectric nanobeam is analyzed in this section. The material properties are shown in Table 2 (Ramirez et al. [42]). The validity of the present study is proved by comparing the bending of this model with those of Arefi and Zenkour [31]. Arefi and Zenkour [31] for various nonlocal parameters as presented in Table 3. The role of multiple parameters like magnetic potential ( ), electric voltage ( ), moisture constant (ΔH), and nonlocal parameter ( ) on the non-dimensional frequencies of the supported higher-order magneto-electro-elastic nanobeams at L/h=20 and L/h=30 are exposed in Tables 4 and Table 5. Here, it is noticeable that with the rise of nonlocal parameters, the natural frequencies of hygro magneto-electro-elastic nanobeam reduce for all magnetic potentials and external voltages due to the existence of nonlocality weakens the beam. Also, it is referred that when the moisture constant arose, the non-dimensional frequencies of hygro magneto-electro-elastic nanobeam decreased, especially for lower moisture constant. Moreover, it is concluded that negative values of magnetic potential and external electric voltage produce lower/higher frequencies than positive ones.

The length of the nanobeam is considered to be L = 10 nm.

Also, the dimensionless deflection is adopted as

                                                                     (60)

Figures 2 and 3 are investigated for the effect of nonlocal parameters on a dimensionless deflection through various magnetic potentials. The increasing value of nonlocal parameter caused an increase of dimensionless deflection, but positive electric voltage leads to a reduction in deflection and negative potential rise the dimensionless deflection. We understand from this subject that magnetic potential has a significant role under dimensionless deflection and dimensionless deflection null effect during zero electric voltage. It must be mentioned that the nonlocal parameter weakens the nanobeam structure. The impact of humid is observed in Fig.3 through the rise in dimensionless deflection.

Dimensionless deflection of the nanobeam concerning slenderness ratio through various electric voltages are presented in Figs. 4 and 5. It is found that the external electric voltage caused that softening deflection of nanobeam for positive values and external voltage for negative values of nanobeam demonstrated a hardening effect. From this, the axial tensile and compressive forces are exposed in the nanobeams via the constructed positive and negative voltages, respectively. In addition, it is lightly observed that the dimensionless deflection is approximately independent of the slenderness ratio for zero electric voltages ( ). The increase in magnetic potential hardens the deflection.

Table 2.  Material properties of BiTiO3-CoFe2O4 composite material

Table 3. Comparison of dimensionless deflections of nanobeam for electric voltage and magnetic potential.

Fig. 2. Effect of nonlocal parameters on dimensionless deflection via  (L/h=10,,K_w= K_p=20,ΔH=1.5).

Fig. 3.  Effect of nonlocal parameters on dimensionless deflection via  (L/h=10,,K_w= K_p=20,ΔH=1.5).

 Fig. 4.  Effect of slenderness ratio on dimensionless deflection via  (L/h=10,,K_w= K_p=20,ΔH=1.5).

Fig. 5.  Effect of slenderness ratio on dimensionless deflection via  (L/h=10,,K_w= K_p=20,ΔH=1.5).

Table 4. Variation of dimensionless frequency of FG nanobeam for the various nonlocal parameter, magnetic potentials, and electric voltages ( =20).

Table 5. Dispersion of dimensionless frequency of nanobeam for the various nonlocal parameter, magnetic potentials, and electric voltages ( =30).

In Figs. 6 and 7, the demonstration for the variation of dimensionless deflection of nanobeam with the moisture constant with different electric voltage via various magnetic potentials is given. From this, it is obtained that the rise in moisture constants weakens the value of deflection. Also, it is referred that the positive voltage values increase the stiffness than negative voltage. The obtained results of these figures indicate that the maximum deflection increases with increasing the magnetic potential of the nanobeam. Figures 8 and 9 expose the effect of critical temperature on the dimensionless deflection via moisture coefficient rise with different electric voltage values. It can be noticed that the increase in critical temperature drives to reduce the dimensionless deflection. Also, the moisture coefficient rise exposes the less magnitude rise in dimensionless deflection while increasing temperature values. The results reveal the truth that the moisture coefficient variations soften the variant values via critical temperature.

The variations of the dimensionless deflection of nanobeams versus the Winkler and Pasternak parameters for various electric voltages are shown in Fig.10 and 11, respectively. It is found from this figure that regardless of the sign and magnitude of electric voltage, the dimensionless deflection increases with the increase of Winkler and Pasternak parameters, So the increment in stiffens of the nanobeam. At a constant electric voltage, the growth of dimensionless defection with Pasternak parameter measurement has a higher rate than the Winkler parameter.

Fig. 6. Effect of moisture constant versus dimensionless deflection via , (L/h=10, K_p =K_w=20).

Fig. 7.  Effect of moisture constant versus dimensionless deflection via  (L/h=10, K_p =K_w=20).

Fig. 8. Effect of critical temperature versus dimensionless deflection via ΔH=0.5 (L/h=10, K_p =K_w=20, ).

 Fig. 9. Effect of critical temperature versus dimensionless deflection via ΔH=1.5 (L/h=10, K_p =K_w=20, ).

Fig. 10. Effect of the Pasternak foundation versus dimensionless deflection (L/h=10,   , ).

Fig. 11. Effect of the Winkler foundation versus dimensionless deflection (L/h=10,  , ).

6.     Conclusion

External electric voltages on the deflection of Hygro-magneto-electro-elastic (HMEE) embedded nanobeams are studied in this article. The governing equations of nonlocal nanobeams based on higher-order refined beam theory are obtained using Hamilton's principle and solved by an analytical solution. A parametric study is presented to observe the effect of the nonlocal parameter, slenderness, moisture constant, critical temperature, and the foundation constants on the deflection characteristics of nanobeam via different applied electric voltages. Some of the bolded highlights of this research are as follows.

1. The dimensionless deflection can be amplified using a higher nonlocal parameter.
2. The maximum dynamic response can be arrived at by choosing higher magnetic intensity.
3. The system's dimensionless deflection can be gradually amplified when the electric voltage is negative.
4. The moisture values soften the dimensionless deflection in the presence of magnetic potential.
5. The dimensionless deflection may be weakened by bigger values of critical temperature in a humid environment.
6. The Pasternak parameters propose a higher deflection than the Winkler parameter when the applied electric voltage is constant.
7. The compressive and tensile nature is proven to deflect nanobeams via positive and negative voltage generation.

References

[1]          Van Den Boomgard, J., Terrell, D.R. and Born, R.A.J., 1974. An in situ grown eutectic magnetoelectric composite material. Journal of Materials Science, 9, pp.1705–1709.

[2]          Zheng, H., Wang J. and Lofland S.E., 2004. Multifer-roic BaTiO3-CoFe2O4 nanostructures. Science, 303, pp.661–663.

[3]          Martin, L.W., Crane, S.P. and Chu, Y.H., 2008. Mul-tiferroics and magnetoelectrics: Thin films and nanostructures. Journal of Condensed Matter Phys-ics, 20, pp.434220.

[4]          Wang, Y., Hu, J.M. and Lin, Y.H., 2010. Multiferroic magnetoelectric composite nanostructures. NPG Asia Mater, 2, pp.61–68.

[5]          Prashanthi, K., Shaibani, P.M. and Sohrabi A., 2012.  Nanoscale magnetoelectric coupling in multiferroic BiFeO3 nanowires. Physica Status Solidi-Rapid Re-search, 6, pp.244–246.

[6]          Eringen, A., 1968. Mechanics of micromorphic con-tinua, in E. Kroner (Ed.), Mechanics of Generalized Continua. Springer-Verlag, 104, pp.18–35.

[7]          Eringen, A., 1972. Nonlocal polar elastic continua. International Journal of Engineering Science, 10, pp.1–16.

[8]          Eringen, A., 1976. Nonlocal micropolar field theo-ry, In Continuum Physics. Eringen A.C. (Ed.) Aca-demic Press, New York, 106.

[9]          Eringen, A., 2002. Nonlocal Continuum Field Theo-ries. Springer, New York, 105.

[10]        Yang, J., Ke, L.L. and Kitipornchai, S., 2010. Nonlin-ear free vibration of single-walled carbon nano-tubes using nonlocal Timoshenko beam theory. Physica E: Low-dimensional Systems and Nanostructures, 42, pp.1727-1735.

[11]        Ehyaei, J. and Akbarizadeh, M.R., 2017. Vibration-analysis of micro composite thin beam based on modified couple stress. Structural Engineering and Mechanics, 64, pp.793-802.

[12]        Ebrahimi, F. and Barati, M.R., 2016. Buckling anal-ysis of nonlocal third-order shear deformable func-tionally graded piezoelectric nanobeams embedded in elastic medium. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39, pp.937-952.  

[13]        Dihaj, A., Zidour, M., Meradjah, M., Rakrak, K., Heireche, H. and Chemi, A., 2018. Free vibration analysis of chiral double-walled carbon nanotube embedded in an elastic medium using non-local elasticity theory and Euler Bernoulli beam model. Structural Engineering and Mechanics, 65(3), pp.335-342.

[14]        Ebrahimi, F. and Barati, M.R., 2016a. Dynamic modeling of a thermo–piezo-electrically actuated nanosize beam subjected to a magnetic field. Ap-plied Physics, 122(4), pp.1-18.

[15]        Ebrahimi, F. and Barati, M.R., 2016b. Electro-mechanical buckling behavior of smart piezoelec-trically actuated higher-order size-dependent grad-ed nanoscale beams in thermal environment. Inter-national Journal of Smart and Nano Materials, 7(2), pp.69-70.

[16]        Ebrahimi, F. and Barati, M.R., 2016c. An exact solu-tion for buckling analysis of embedded piezoelec-tro-magnetically actuated nanoscale beams. Ad-vance Nano Research, 4, pp.65-84.

[17]        Ebrahimi, F. and Barati, M.R., 2016d. Vibration analysis of smart piezoelectrically actuated nano-beams subjected to magneto-electrical field in thermal environment. Journal of Vibration and Control, 1077546316646239.

[18]        Roque, C.M.C., Ferreira, A.J.M. and Reddy, J.N., 2011. Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method. Inter-national Journal of Engineering Science, 49, pp.976-984.

[19]        Peddieson, J., Buchanan, G.R. and Mcnitt, R.P., 2003. Application of nonlocal continuum models to nano-technology. International Journal of Engineering Science, 41, pp.305–312.

[20]        Civalek, O. and Demir, C., 2011. Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Applied Mathematical Sciences, 35, pp.2053–2067.

[21]        Wang Q., 2005. Wave propagation in carbon nano-tubes via nonlocal continuum mechanics.  Journal of Applied Physics, 98, pp.124-301.

[22]        Wang, C.M., Kitipornchai, S., Lim, C.W. and Eisen-berger, M., 2008. Beam bending solutions based on nonlocal Timoshenko beam theory. Journal of En-gineering Mechanics, 134, pp.475–481.

[23]        Murmu, T. and Pradhan, S.C., 2009. Buckling analy-sis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Physica E: Low-dimensional Systems and Nanostructures, 41, pp.1232-1239.

[24]        Karami, B., Janghorban, M. and Tounsi, A., 2017. Effects of triaxial magnetic field on the anisotropic nanoplates. Steel and Composite Structure, 25, pp.361-374.

[25]        Ebrahimi, F. and Rostami, P., 2018. Propagation of elastic waves in thermally affected embedded car-bon-nanotube-reinforced composite beams via var-ious shear deformation plate theories. Structural Engineering and Mechanics, 66, pp.495-504.

[26]        Hajnayeb, A.K.A. and Foruzande, H. 2017. Free vi-bration analysis of a piezoelectric nanobeam using nonlocal elasticity theory. Structural Engineering and Mechanics, 61, pp.617-624.

[27]        Ebrahimi, F., jafari, A. and Selvamani, R., 2020.  Thermal buckling analysis of magneto electro elas-tic porous FG beam in thermal environment. Ad-vances in Nano Research, 8, pp.83-94.

[28]        Ebrahimi, F., Hamed, S., Hosseini, S. and Selvamani, R., 2020. Thermo-electro-elastic nonlinear stability analysis of viscoelastic double-piezo nano plates under magnetic field. Structural engineering and mechanics, 73, pp.565-584,

[29]        Ebrahimi, F., Kokaba, M., Shaghaghi, G. and Selvamani, R., 2020. Dynamic characteristics of hygro-magneto-thermo-electrical nanobeam with non- ideal boundary conditions.  Advances in Nano Research, 8, pp.169-182.

[30]        Yazdi, A.A., 2018. Large amplitude forced vibration of functionally graded nano-composite plate with piezoelectric layers resting on nonlinear elastic foundation. Structural Engineering and Mechanics, 68, pp.203–213.

[31]        Arefi, M. and Zenkour, A.M., 2016. A simplified shear and normal deformations nonlocal theory for bending of functionally graded piezomagnetic sandwich nanobeams in magneto-thermo-electric environment. Journal of Sandwich Structures and Materials, 18, pp.624-651.‏

[32]        Ebrahimi, F. and Barati, M.R., 2018. Wave propaga-tion analysis of smart strain gradient piezo-magneto-elastic nonlocal beams. Structural Engi-neering and Mechanics, 66, pp.237-248.

[33]        Arefi, M., 2019. Static analysis of laminated piezo-magnetic size-dependent curved   beam based on modified couple stress theory. Structural Engineer-ing and Mechanics, 69, pp.145-153.

[34]        Aimin, J. and Haojiang, D., 2004. Analytical solu-tions to magneto-electro-elastic beams. Structural Engineering and Mechanics, 18, pp.195-209.

[35]        Gayen, D. and Roy, T., 2013. Hygro-thermal effects on stress analysis of tapered laminated composite beam. International Journal of Composite Materi-als, 3, pp.46–55.

[36]        Kurtinaitiene, M., Mazeika, K., Ramanavicius, S., Pakstas, V. and Jagminas, A., 2016. Effect of addi-tives on the hydrothermal synthesis of manganese ferrite nanoparticles. Advances in Nano Research, 4, pp.1-14.

[37]        Alzahrani, E.O., Zenkour, A.M. and Sobhy, M., 2013. Small scale effect on hygro-thermo-mechanical bending of nanoplates embedded in an elastic me-dium. Composite Structures, 105, pp.163-172.

[38]        Sobhy, M., 2013. Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary condi-tions. Composite structure, 99, pp.76-87.‏

[39]        Ghorbanpour Arani, A. and Zamani, M.H., 2017. Investigation of electric field effect on size-dependent bending analysis of functionally graded porous shear and normal deformable sandwich na-noplate on silica Aerogel foundation. Journal of Sandwich Structures and Materials, 21, 109963621772140.

[40]        Şimşek, M. and Yurtcu, H.H., 2013. Analytical solu-tions for bending and buckling of functionally graded nanobeams based on the nonlocal Timo-shenko beam theory. Composite Structures, 97, pp.378-386.‏  

[41]        Ke, L.L, Wang, Y.S., Yang, J. and Kitipornchai, S., 2014. Free vibration of size-dependent magneto-electro-elastic nano plates based on the nonlocal theory. Acta Mechanica Sinica, 30, pp.516-525.

[42]        Ramirez, F., Heyliger, P.R. and Pan, E., 2006, Dis-crete layer solution to free vibrations of functional-ly graded magneto-electro-elastic plates, Mechanics of Advanced Materials and Structure, 13, pp.249-266.

[43]        Zur, K.K., Arefi, M., Kim, J. and Reddy, J.N., 2020. Free vibration and buckling analyses of magneto-electro-elastic FGM nanoplates based on nonlocal modified higher-order sinusoidal shear defor-mation theory. Composites Part B: Engineering  182, pp.107601.

[44]        Arefi, M., Kiani, M. and Rabczuk, T., 2019. Applica-tion of nonlocal strain gradient theory to size de-pendent bending analysis of a sandwich porous na-noplate integrated with piezomagnetic face-sheets. Composites Part B: Engineering, 168, pp.320-333.

[45]        Arefi, M. and Zenkour, A.M., 2017. Thermo-electro-mechanical bending behavior of sandwich nano-plate integrated with piezoelectric face-sheets based on trigonometric plate theory. Composite Structures, 162, pp.108-122.

[46]        Arefi, M. and Zenkour, A.M., 2017. Nonlocal elec-tro-thermo-mechanical analysis of a sandwich na-noplate containing a Kelvin–Voigt viscoelastic na-noplate and two piezoelectric layers.  Acta Mechan-ica, 228, pp.475–493.

[47]        Arefi, M., Mohammad-Rezaei Bidgoli, E., Dimitri, R. and Tornabene, F., 2018, Free vibrations of func-tionally graded polymer composite nanoplates re-inforced with graphene nanoplatelets. Aerospace Science and Technology, 81, pp.108-117.

[48]        Arefi, M. and Zenkour, A.M., 2018. Size-dependent electro-elastic analysis of a sandwich microbeam based on higher-order sinusoidal shear defor-mation theory and strain gradient theory. Journal of Intelligent Material Systems and Structures, 29(7), pp.1394-1406.

[49]        Arefi, M. and Rabczuk, T., 2019. A nonlocal higher order shear deformation theory for electro-elastic analysis of a piezoelectric doubly curved nano shell. Composites Part B: Engineering, 168, pp.496-510.

[50]        Barati, M.R., Faleh, N. and Zenkour, A.M., 2019. Dynamic response of nanobeams subjected to mov-ing nanoparticles and hygro-thermal environments based on nonlocal strain gradient theory. Mechan-ics of Advanced Materials and Structures, 26(19), pp.1661-1669.

[51]        Arefi, M. and Zenkour, A.M., 2019. Influence of magneto-electric environments on size-dependent bending results of three-layer piezomagnetic curved nanobeam based on sinusoidal shear de-formation theory. Journal of Sandwich Structures and Materials, 21(8), pp.2751–2778.

[52]        Arefi, M. and Zenkour, A.M., 2018. Size-dependent vibration and electro-magneto-elastic bending re-sponses of sandwich piezomagnetic curved nano-beams. Steel and Composite Structures, 29(5), pp.579–590.

[53]        Zenkour, A.M. and Alghanmi, R., 2020. Static re-sponse of sandwich plates with FG core and piezoe-lectric faces under thermo-electro-mechanical loads and resting on elastic foundations. Thin-Walled Structures, 157, pp.107025.

[54]        Sobhy, M. and Zenkour, A., 2020. The modified couple stress model for bending of normal deform-able viscoelastic nanobeams resting on visco-Pasternak foundations. Mechanics of Advanced Ma-terials and Structures, 27(7), pp.525–538.

[55]        Zenkour, A.M. and El-Shahrany, H.D., 2020. Vibra-tion suppression of magnetostrictive laminated beams resting on viscoelastic foundation. Applied Mathematics and Mechanics, 41(8), pp.1269–1286.

[56]        Barati, M.R. and Zenkour, A.M., 2018. Forced vibra-tion of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygro-thermal environments. Mechanics of Advanced Materials and Structures, 25(8), pp.669–680.

[57]        Daik, A. and Zenkour, A.M., 2020. Bending of Func-tionally Graded Sandwich Nanoplates Resting on Pasternak Foundation under Different Boundary Conditions. Journal of Applied and Computational Mechanics, 6, pp.1245–1259.

[58]        Guo, Y., Mi, H. and Habibi, M., 2021. Electro-mechanical energy absorption, resonance frequen-cy, and low-velocity impact analysis of the piezoe-lectric doubly curved system. Mechanical Systems and Signal Processing, 157, pp.107723.

[59]        Dai, Z., Zhang Z., Yasaman Bolandi, S. and Habibi, M., 2021. On the vibrations of the non-polynomial viscoelastic composite open-type shell under re-sidual stresses. Composite Structures, 263, pp.113599.

[60]        Wu, J. and Habibi, M., 2021. Dynamic simulation of the ultra-fast-rotating sandwich cantilever disk via finite element and semi-numerical methods. Engi-neering with Computers. https://doi.org/10.1007/s00366-021-01396-6

[61]        Al-Furjan, M.S.H., Samimi-Sohrforozani, E., Habibi, M., Jung, D.W. and Safarpour, H., 2021. Vibrational characteristics of a higher-order laminated compo-site viscoelastic annular microplate via modified couple stress theory. Composite Structures, 257, pp.113152.

[62]        Al-Furjan, M.S.H., Fereidouni, M., Sedghiyan, D., Habibi, M. and Jung, D.W., 2021. Three-dimensional frequency response of the CNT-Carbon-Fiber reinforced laminated circular/annular plates under initially stresses. Composite Struc-tures, 257, pp.113146.

[63]        Al-Furjan, M.S.H., Habibi, M., Ghabussi, A., Sa-farpour, H., Safarpour, M. and Tounsi, A., 2021. Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory. Engineer-ing Structures, 228, pp.111496.

[64]        Dai, Z., Jiang, Z., Zhang, L. and Habibi, M., 2021. Frequency characteristics and sensitivity analysis of a size-dependent laminated nanoshell. Advances in Nano Research, 10(2), pp.175-189.

[65]        Wang, Z., Yu, S., Xiao, Z. and Habibi, M., 2020. Fre-quency and buckling responses of a high-speed ro-tating fiber metal laminated cantilevered micro-disk. Mechanics of Advanced Materials and Struc-tures. https://doi.org/10.1080/15376494.2020.1824284

[66]        Ghabussi, A., Asgari Marnani, J. and Sadegh Ro-hanimanesh, M., 2021. Seismic performance as-sessment of a novel ductile steel braced frame equipped with steel curved damper. Structures, 31, pp.87-97.

[67]        Ghabussi, A., Habibi, M., Shavalipour, A., Moayedi, H., Safarpour, H. and Noormohammadi O., 2020.  Frequency characteristics of a viscoelastic gra-phene nanoplatelet–reinforced composite circular microplate. Journal of Vibration and Control, 27, 107754632092393.

[68]        Ghabussi, A., Asgari Marnani, J. and Sadegh Ro-hanimanesh, M., 2020. Improving seismic perfor-mance of portal frame structures with steel curved dampers. Structures, 24, pp.27-40.

[69]        Zhao, Y., Moradi, Z., Davoudi, M. and Zhuang, J., 2021. Bending and stress responses of the hybrid axisymmetric system via state-space method and 3D-elasticity theory. Engineering with Computers. https://doi.org/10.1007/s00366-020-01242-1

[70]        Ma, L., Liu, X. and Moradi, Z., 2021. On the chaotic behavior of graphene-reinforced annular systems under harmonic excitation. Engineering with Com-puters. https://doi.org/10.1007/s00366-020-01210-9

[71]        Jiao, J., Ghoreishi, S.M., Moradi, Z. and Oslub, K., 2021. Coupled particle swarm optimization method with genetic algorithm for the static–dynamic per-formance of the magneto-electro-elastic nanosys-tem. Engineering with Computers. https://doi.org/10.1007/s00366-021-01391-x

[72]        Huang, X., Zhang, Y., Moradi, Z. and Shafiei, N., 2021. Computer simulation via a couple of ho-motopy perturbation methods and the generalized differential quadrature method for nonlinear vibra-tion of functionally graded non-uniform micro-tube. Engineering with Computers. https://doi.org/10.1007/s00366-021-01395-7