Document Type : Research Article
Authors
^{1} Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore641114, Tamilnadu, India
^{2} Department of Mathematics, Karunya Institute of Technology and Sciences , Coimbatore, 641114, India
^{3} Department of Mechanical Engineering, Imam Khomeini International University Qazvin, Iran
Abstract
Keywords
Vibration in an Electrically Affected Hygromagnetothermoflexo Electric Nanobeam Embedded in WinklerPasternak Foundation
^{a }Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore641114, 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
Keywords 

ABSTRACT 
Applied voltage Hygro thermomagnetic effect Flexoelectric nanobeam Nonlocal elasticity theory Refined beam theory 
This paper presents applied electric voltage performance in hydrothermal magneto flexo electric nanobeams embedded in the WinklerPasternak foundation based on nonlocal elasticity theory. Higherorder refined beam theory via Hamilton's principle is utilized to arrive at the governing equations of nonlocal nanobeams and solved by implementing an analytical solution. A parametric study is presented to analyze the effect of the applied electric voltage on dimensionless deflection via nonlocal parameters, slenderness, moisture constant, critical temperature, and foundation constants. It is found that physical variants and beam geometrical parameters significantly affect the dimensionless deflection of nanoscale beams. The accuracy and efficiency of the presented model are verified by comparing the results with that of published researches. A good agreement has arrived. The numerical examples are presented to explain how each variant can affect the structure's stability endurance. This type of model and its physical output show the great potential of hygromagnetothermoflexo electric combination in the design of intelligent composite structures and use in structural health scanners. Recent advances in the application of nanotechnology have resulted in the manufacture of nanoelectromechanical devices. The attractiveness of them is due to their excellent and distinctive mechanical and electrical properties. 
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 magnetoelectroelastic (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, BiTiO3CoFe2O4, NiFe2O4PZT) 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 wellidentified 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 doublewalled 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 thirdorder 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 smallsize 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 WinklerPasternak foundation. The propagation of elastic waves in thermally affected embedded carbonnanotubereinforced 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. Thermomagnetoelectro 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]. Thermomagnetoelectroelastic 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 piezomagnetoelastic nonlocal beams by Ebrahimi and Barati [32]. The static analysis of laminated piezomagnetic sizedependent curved beam was concentrated on the modified couple stress theory by Arefi [33]. Studies were conducted on the analytical solutions to magnetoelectroelastic beams by Aimin and Haojiang [34].
Furthermore, many researchers have presented the static and dynamic characteristics of beams and plates exposed to hygrothermal 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 hygrothermal loadings have been developed using several beam theories by Alzahrani et al. [37]. They extended the nonlocal constitutive relations of Eringen to contain the hygrothermal effects. Also, Sobhy [38] studies show the frequency response of simplysupported shear deformable orthotropic graphene sheets exposed to hygrothermal loading.
Ghorbanpour Arani and Zamani [39] examined the bending of electromechanical 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 threeunknown 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 powerlaw exponent has a wide influence on the responses of FG nanobeam. Free vibration of sizedependent magnetoelectroelastic 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 thermomagnetoelectroelastic of nanoplates. Arefi and Zenkour [48] studied the sizedependent electroelastic analysis of a sandwich microbeam based on higherorder sinusoidal shear deformation theory and strain gradient theory. Arefi and Rabczuk [49] analyzed the nonlocal higherorder shear deformation theory for electroelastic 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 hygrothermal environments based on nonlocal strain gradient theory. They found that the dynamic deflection increased as the temperature/moisture increased. Bending vibration and electromagnetoelastic 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 thermoelectromechanical 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 viscoPasternak foundations have been studied by Sobhy and Zenkour [54] and Zenkour and ElShahrany [55].
Barati and Zenkour [56] analyzed the forced vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygrothermal 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 electromechanical energy absorption, resonance frequency, and lowvelocity 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 nonpolynomial viscoelastic composite opentype shell under residual stresses. Dynamic simulation of the ultrafastrotating sandwich cantilever disk via finite element is modeled by Wu and Habibi [60]. AlFurjan et al. [61] introduced the vibrational characteristics of a higherorder laminated composite viscoelastic annular microplate via modified couple stress theory. AlFurjan et al. [62] proposed the threedimensional frequency response of the CNTCarbonFiber reinforced laminated circular/annular plates under initial stresses. Nonpolynomial framework for stress and strain response of the FGGPLRC disk using threedimensional refined higherorder theory was derived by AlFurjan et al. [63]. Dai et al. [64] investigated the frequency characteristics and sensitivity analysis of a sizedependent laminated nanoshell.
Wang et al. [65] developed the frequency and buckling responses of a highspeed 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 statespace method and 3Delasticity theory. Ma et al. [70] developed the chaotic behavior of graphenereinforced annular systems under harmonic excitation. Jiao et al. [71] proposed the coupled particle swarm optimization method with a genetic algorithm for the staticdynamic performance of the magnetoelectroelastic 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 nonuniform microtube.
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 WinklerPasternak 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, WinklerPasternak parameters effect on deflection characteristics of a nanobeam are investigated.
The component of displacement via refined shear deformable beam can be expressed by:
(1)
(2)
where is axial midplane 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
Nonzero strains of the suggested beam model can be expressed as follows:
(4)
(5)
where g(z)=1df(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 f_{13} 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)
The nonlocal theory can be extended for the piezo magnetic nanobeams as:
(22)
(23)
(24)
(25)
where corresponds with the components of the fourthorder elasticity tensor, also, is the relative dielectric susceptibility and is the flexoelectric coefficient. Also, is a nonlocal parameter which is introduced to describe the sizedependency 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. (2631) over the crosssection 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)
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]

Boundary conditions 
The functions 

At x=0, a 

SS 





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) = q_{0} is the uniform load density, and x_{0} 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.
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 nondimensional frequencies of the supported higherorder magnetoelectroelastic 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 magnetoelectroelastic 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 nondimensional frequencies of hygro magnetoelectroelastic 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 BiTiO3CoFe2O4 composite material
Properties BiTiO3CoFe2O4 
Elastic (GPa) c11 = 226, c12 = 125, c13 = 124, c33 = 216, c44 = 44.2, c66 = 50.5 Piezoelectric/(C・m^{−2}) e31 = −2.2, e33 = 9.3, e15 = 5.8 Dielectric/(10^{−9}C・V^{−1}・m^{−1}) k11 = 5.64, k33 = 6.35 Piezomagnetic/(N・A^{−1}・m−1) q15 = 275, q31 = 290.1, q33 = 349.9 Magnetoelectric/(10^{−12}Ns・V^{−1}・C^{−1}) s11 = 5.367, s33 = 2 737.5  297 Mass density (Kg/m3) ρ=5.55 Hygrothermal(/K) 
Table 3. Comparison of dimensionless deflections of nanobeam for electric voltage and magnetic potential.
L/h µ L/h µ (nm^{2} ) (nm^{2} ) 


Arefi Present and Zenkour (2016) 

Arefi Present and Zenkour (2016) 
10 
1 3.68 3.5781 2 3.71 3.6482 3 3.77 3.7302 4 3.84 3.79011 5 3.94 3.8952 

0 3.68 3.59892 10 1 1.3333 .66921 2 1.3645 3.74018 3 1.3958 3.80234 4 1.4270 3.92011 
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).
µ 

Ω=0.05 
Ω=0 
Ω=+0.05 







ΔH =0.2 
ΔH =1 
ΔH =5 
ΔH =0.2 
ΔH =1 
ΔH =5 
ΔH =0.2 
ΔH =1 
ΔH =5 











0 
V=5 
35.6083 
32.9563 
31.4249 
36.477 
33.5046 
31.6119 
37.3256 
34.044 
31.7979 

V=0 
35.4747 
32.5345 
30.7053 
36.3466 
33.0897 
30.8967 
37.1982 
33.6358 
31.0870 

V=+5 
35.3406 
32.1071 
29.9685 
36.2158 
32.6696 
30.1646 
37.0703 
33.2226 
30.3594 











1 
V=5 
29.9003 
27.8605 
26.7829 
30.9298 
28.5069 
27.0022 
31.9261 
29.1390 
27.2197 

V=0 
29.7411 
27.3602 
25.9349 
30.7759 
28.0182 
26.1613 
31.7771 
28.6610 
26.3857 

V=+5 
29.5810 
26.8506 
25.0582 
30.6212 
27.5207 
25.2924 
31.6273 
28.1749 
25.5245 











2 
V=5 
26.1741 
24.5567 
23.7980 
27.3442 
25.2877 
24.0444 
28.4663 
25.9981 
24.2884 

V=0 
25.9920 
23.9876 
22.8394 
27.1700 
24.7354 
23.0961 
28.2990 
25.4613 
23.3500 

V=+5 
25.8087 
23.4046 
21.8388 
26.9947 
24.1705 
22.1071 
28.1307 
24.9128 
22.3722 











3 
V=5 
23.4875 
22.1912 
21.6780 
24.7848 
22.9976 
21.9483 
26.0176 
23.7765 
22.2153 

V=0 
23.2845 
21.5598 
20.6211 
24.5925 
22.3889 
20.9050 
25.8344 
23.1883 
21.1852 

V=+5 
23.0796 
20.9093 
19.5070 
24.3987 
21.7631 
19.8069 
25.6500 
22.5847 
20.1024 
Table 5. Dispersion of dimensionless frequency of nanobeam for the various nonlocal parameter, magnetic potentials, and electric voltages ( =30).
µ 

Ω=0.05 
Ω=0 
Ω=+0.05 







ΔH =0.2 
ΔH =1 
ΔH =5 
ΔH =0.2 
ΔH =1 
ΔH =5 
ΔH =0.2 
ΔH =1 
ΔH =5 











0 
V=5 
35.5080 
32.1523 
30.4349 
36.376 
33.11246 
31.534 
37.2222 
33.144 
30.7779 

V=0 
34.4347 
32.345 
30.234 
36.1466 
32.0997 
30.5967 
36.0982 
33.445 
30.8870 

V=+5 
35.3406 
32.0061 
29.0685 
36.0158 
32.6096 
30.0646 
36.0703 
33.0226 
29.3294 











1 
V=5 
29.789 
27.456 
26.657 
30.567 
28.2349 
26.4022 
31.0261 
29.0290 
27.123 

V=0 
29.5411 
27.3002 
25.7249 
30.6559 
27.8182 
26.0613 
31.6771 
28.5610 
26.4857 

V=+5 
29.423 
26.6506 
24.7582 
30.3212 
27.0007 
25.134 
31.5273 
28.0549 
25.235 











2 
V=5 
26.0241 
24.327 
23.5680 
27.1242 
25.0877 
23.8444 
28.0663 
25.7781 
24.145 

V=0 
25.754 
23.4876 
22.344 
27.0500 
24.524 
22.5961 
28.0990 
25.0613 
23.0500 

V=+5 
25.567 
23.3046 
21.728 
26.337 
24.0705 
22.0071 
28.0307 
24.7128 
22.0722 











3 
V=5 
23.0875 
22.0912 
21.210 
24.6248 
22.3476 
21.6483 
25.0176 
23.5765 
22.1153 

V=0 
23.0845 
21.2398 
20.5211 
24.4325 
22.2889 
20.6050 
25.6344 
23.0883 
21.0852 

V=+5 
22.0796 
20.723 
19.3070 
24.2787 
21.7001 
19.4069 
25.5500 
22.4847 
20.0024 
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, , ).
External electric voltages on the deflection of Hygromagnetoelectroelastic (HMEE) embedded nanobeams are studied in this article. The governing equations of nonlocal nanobeams based on higherorder 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.
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. Multiferroic BaTiO3CoFe2O4 nanostructures. Science, 303, pp.661–663.
[3] Martin, L.W., Crane, S.P. and Chu, Y.H., 2008. Multiferroics and magnetoelectrics: Thin films and nanostructures. Journal of Condensed Matter Physics, 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 SolidiRapid Research, 6, pp.244–246.
[6] Eringen, A., 1968. Mechanics of micromorphic continua, in E. Kroner (Ed.), Mechanics of Generalized Continua. SpringerVerlag, 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 theory, In Continuum Physics. Eringen A.C. (Ed.) Academic Press, New York, 106.
[9] Eringen, A., 2002. Nonlocal Continuum Field Theories. Springer, New York, 105.
[10] Yang, J., Ke, L.L. and Kitipornchai, S., 2010. Nonlinear free vibration of singlewalled carbon nanotubes using nonlocal Timoshenko beam theory. Physica E: Lowdimensional Systems and Nanostructures, 42, pp.17271735.
[11] Ehyaei, J. and Akbarizadeh, M.R., 2017. Vibrationanalysis of micro composite thin beam based on modified couple stress. Structural Engineering and Mechanics, 64, pp.793802.
[12] Ebrahimi, F. and Barati, M.R., 2016. Buckling analysis of nonlocal thirdorder shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39, pp.937952.
[13] Dihaj, A., Zidour, M., Meradjah, M., Rakrak, K., Heireche, H. and Chemi, A., 2018. Free vibration analysis of chiral doublewalled carbon nanotube embedded in an elastic medium using nonlocal elasticity theory and Euler Bernoulli beam model. Structural Engineering and Mechanics, 65(3), pp.335342.
[14] Ebrahimi, F. and Barati, M.R., 2016a. Dynamic modeling of a thermo–piezoelectrically actuated nanosize beam subjected to a magnetic field. Applied Physics, 122(4), pp.118.
[15] Ebrahimi, F. and Barati, M.R., 2016b. Electromechanical buckling behavior of smart piezoelectrically actuated higherorder sizedependent graded nanoscale beams in thermal environment. International Journal of Smart and Nano Materials, 7(2), pp.6970.
[16] Ebrahimi, F. and Barati, M.R., 2016c. An exact solution for buckling analysis of embedded piezoelectromagnetically actuated nanoscale beams. Advance Nano Research, 4, pp.6584.
[17] Ebrahimi, F. and Barati, M.R., 2016d. Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magnetoelectrical 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. International Journal of Engineering Science, 49, pp.976984.
[19] Peddieson, J., Buchanan, G.R. and Mcnitt, R.P., 2003. Application of nonlocal continuum models to nanotechnology. 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 nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 98, pp.124301.
[22] Wang, C.M., Kitipornchai, S., Lim, C.W. and Eisenberger, M., 2008. Beam bending solutions based on nonlocal Timoshenko beam theory. Journal of Engineering Mechanics, 134, pp.475–481.
[23] Murmu, T. and Pradhan, S.C., 2009. Buckling analysis of a singlewalled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM. Physica E: Lowdimensional Systems and Nanostructures, 41, pp.12321239.
[24] Karami, B., Janghorban, M. and Tounsi, A., 2017. Effects of triaxial magnetic field on the anisotropic nanoplates. Steel and Composite Structure, 25, pp.361374.
[25] Ebrahimi, F. and Rostami, P., 2018. Propagation of elastic waves in thermally affected embedded carbonnanotubereinforced composite beams via various shear deformation plate theories. Structural Engineering and Mechanics, 66, pp.495504.
[26] Hajnayeb, A.K.A. and Foruzande, H. 2017. Free vibration analysis of a piezoelectric nanobeam using nonlocal elasticity theory. Structural Engineering and Mechanics, 61, pp.617624.
[27] Ebrahimi, F., jafari, A. and Selvamani, R., 2020. Thermal buckling analysis of magneto electro elastic porous FG beam in thermal environment. Advances in Nano Research, 8, pp.8394.
[28] Ebrahimi, F., Hamed, S., Hosseini, S. and Selvamani, R., 2020. Thermoelectroelastic nonlinear stability analysis of viscoelastic doublepiezo nano plates under magnetic field. Structural engineering and mechanics, 73, pp.565584,
[29] Ebrahimi, F., Kokaba, M., Shaghaghi, G. and Selvamani, R., 2020. Dynamic characteristics of hygromagnetothermoelectrical nanobeam with non ideal boundary conditions. Advances in Nano Research, 8, pp.169182.
[30] Yazdi, A.A., 2018. Large amplitude forced vibration of functionally graded nanocomposite 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 magnetothermoelectric environment. Journal of Sandwich Structures and Materials, 18, pp.624651.
[32] Ebrahimi, F. and Barati, M.R., 2018. Wave propagation analysis of smart strain gradient piezomagnetoelastic nonlocal beams. Structural Engineering and Mechanics, 66, pp.237248.
[33] Arefi, M., 2019. Static analysis of laminated piezomagnetic sizedependent curved beam based on modified couple stress theory. Structural Engineering and Mechanics, 69, pp.145153.
[34] Aimin, J. and Haojiang, D., 2004. Analytical solutions to magnetoelectroelastic beams. Structural Engineering and Mechanics, 18, pp.195209.
[35] Gayen, D. and Roy, T., 2013. Hygrothermal effects on stress analysis of tapered laminated composite beam. International Journal of Composite Materials, 3, pp.46–55.
[36] Kurtinaitiene, M., Mazeika, K., Ramanavicius, S., Pakstas, V. and Jagminas, A., 2016. Effect of additives on the hydrothermal synthesis of manganese ferrite nanoparticles. Advances in Nano Research, 4, pp.114.
[37] Alzahrani, E.O., Zenkour, A.M. and Sobhy, M., 2013. Small scale effect on hygrothermomechanical bending of nanoplates embedded in an elastic medium. Composite Structures, 105, pp.163172.
[38] Sobhy, M., 2013. Buckling and free vibration of exponentially graded sandwich plates resting on elastic foundations under various boundary conditions. Composite structure, 99, pp.7687.
[39] Ghorbanpour Arani, A. and Zamani, M.H., 2017. Investigation of electric field effect on sizedependent bending analysis of functionally graded porous shear and normal deformable sandwich nanoplate on silica Aerogel foundation. Journal of Sandwich Structures and Materials, 21, 109963621772140.
[40] Şimşek, M. and Yurtcu, H.H., 2013. Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Composite Structures, 97, pp.378386.
[41] Ke, L.L, Wang, Y.S., Yang, J. and Kitipornchai, S., 2014. Free vibration of sizedependent magnetoelectroelastic nano plates based on the nonlocal theory. Acta Mechanica Sinica, 30, pp.516525.
[42] Ramirez, F., Heyliger, P.R. and Pan, E., 2006, Discrete layer solution to free vibrations of functionally graded magnetoelectroelastic plates, Mechanics of Advanced Materials and Structure, 13, pp.249266.
[43] Zur, K.K., Arefi, M., Kim, J. and Reddy, J.N., 2020. Free vibration and buckling analyses of magnetoelectroelastic FGM nanoplates based on nonlocal modified higherorder sinusoidal shear deformation theory. Composites Part B: Engineering 182, pp.107601.
[44] Arefi, M., Kiani, M. and Rabczuk, T., 2019. Application of nonlocal strain gradient theory to size dependent bending analysis of a sandwich porous nanoplate integrated with piezomagnetic facesheets. Composites Part B: Engineering, 168, pp.320333.
[45] Arefi, M. and Zenkour, A.M., 2017. Thermoelectromechanical bending behavior of sandwich nanoplate integrated with piezoelectric facesheets based on trigonometric plate theory. Composite Structures, 162, pp.108122.
[46] Arefi, M. and Zenkour, A.M., 2017. Nonlocal electrothermomechanical analysis of a sandwich nanoplate containing a Kelvin–Voigt viscoelastic nanoplate and two piezoelectric layers. Acta Mechanica, 228, pp.475–493.
[47] Arefi, M., MohammadRezaei Bidgoli, E., Dimitri, R. and Tornabene, F., 2018, Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets. Aerospace Science and Technology, 81, pp.108117.
[48] Arefi, M. and Zenkour, A.M., 2018. Sizedependent electroelastic analysis of a sandwich microbeam based on higherorder sinusoidal shear deformation theory and strain gradient theory. Journal of Intelligent Material Systems and Structures, 29(7), pp.13941406.
[49] Arefi, M. and Rabczuk, T., 2019. A nonlocal higher order shear deformation theory for electroelastic analysis of a piezoelectric doubly curved nano shell. Composites Part B: Engineering, 168, pp.496510.
[50] Barati, M.R., Faleh, N. and Zenkour, A.M., 2019. Dynamic response of nanobeams subjected to moving nanoparticles and hygrothermal environments based on nonlocal strain gradient theory. Mechanics of Advanced Materials and Structures, 26(19), pp.16611669.
[51] Arefi, M. and Zenkour, A.M., 2019. Influence of magnetoelectric environments on sizedependent bending results of threelayer piezomagnetic curved nanobeam based on sinusoidal shear deformation theory. Journal of Sandwich Structures and Materials, 21(8), pp.2751–2778.
[52] Arefi, M. and Zenkour, A.M., 2018. Sizedependent vibration and electromagnetoelastic bending responses of sandwich piezomagnetic curved nanobeams. Steel and Composite Structures, 29(5), pp.579–590.
[53] Zenkour, A.M. and Alghanmi, R., 2020. Static response of sandwich plates with FG core and piezoelectric faces under thermoelectromechanical loads and resting on elastic foundations. ThinWalled Structures, 157, pp.107025.
[54] Sobhy, M. and Zenkour, A., 2020. The modified couple stress model for bending of normal deformable viscoelastic nanobeams resting on viscoPasternak foundations. Mechanics of Advanced Materials and Structures, 27(7), pp.525–538.
[55] Zenkour, A.M. and ElShahrany, H.D., 2020. Vibration 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 vibration of sinusoidal FG nanobeams resting on hybrid Kerr foundation in hygrothermal environments. Mechanics of Advanced Materials and Structures, 25(8), pp.669–680.
[57] Daik, A. and Zenkour, A.M., 2020. Bending of Functionally 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. Electromechanical energy absorption, resonance frequency, and lowvelocity impact analysis of the piezoelectric 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 nonpolynomial viscoelastic composite opentype shell under residual stresses. Composite Structures, 263, pp.113599.
[60] Wu, J. and Habibi, M., 2021. Dynamic simulation of the ultrafastrotating sandwich cantilever disk via finite element and seminumerical methods. Engineering with Computers. https://doi.org/10.1007/s00366021013966
[61] AlFurjan, M.S.H., SamimiSohrforozani, E., Habibi, M., Jung, D.W. and Safarpour, H., 2021. Vibrational characteristics of a higherorder laminated composite viscoelastic annular microplate via modified couple stress theory. Composite Structures, 257, pp.113152.
[62] AlFurjan, M.S.H., Fereidouni, M., Sedghiyan, D., Habibi, M. and Jung, D.W., 2021. Threedimensional frequency response of the CNTCarbonFiber reinforced laminated circular/annular plates under initially stresses. Composite Structures, 257, pp.113146.
[63] AlFurjan, M.S.H., Habibi, M., Ghabussi, A., Safarpour, H., Safarpour, M. and Tounsi, A., 2021. Nonpolynomial framework for stress and strain response of the FGGPLRC disk using threedimensional refined higherorder theory. Engineering Structures, 228, pp.111496.
[64] Dai, Z., Jiang, Z., Zhang, L. and Habibi, M., 2021. Frequency characteristics and sensitivity analysis of a sizedependent laminated nanoshell. Advances in Nano Research, 10(2), pp.175189.
[65] Wang, Z., Yu, S., Xiao, Z. and Habibi, M., 2020. Frequency and buckling responses of a highspeed rotating fiber metal laminated cantilevered microdisk. Mechanics of Advanced Materials and Structures. https://doi.org/10.1080/15376494.2020.1824284
[66] Ghabussi, A., Asgari Marnani, J. and Sadegh Rohanimanesh, M., 2021. Seismic performance assessment of a novel ductile steel braced frame equipped with steel curved damper. Structures, 31, pp.8797.
[67] Ghabussi, A., Habibi, M., Shavalipour, A., Moayedi, H., Safarpour, H. and Noormohammadi O., 2020. Frequency characteristics of a viscoelastic graphene nanoplatelet–reinforced composite circular microplate. Journal of Vibration and Control, 27, 107754632092393.
[68] Ghabussi, A., Asgari Marnani, J. and Sadegh Rohanimanesh, M., 2020. Improving seismic performance of portal frame structures with steel curved dampers. Structures, 24, pp.2740.
[69] Zhao, Y., Moradi, Z., Davoudi, M. and Zhuang, J., 2021. Bending and stress responses of the hybrid axisymmetric system via statespace method and 3Delasticity theory. Engineering with Computers. https://doi.org/10.1007/s00366020012421
[70] Ma, L., Liu, X. and Moradi, Z., 2021. On the chaotic behavior of graphenereinforced annular systems under harmonic excitation. Engineering with Computers. https://doi.org/10.1007/s00366020012109
[71] Jiao, J., Ghoreishi, S.M., Moradi, Z. and Oslub, K., 2021. Coupled particle swarm optimization method with genetic algorithm for the static–dynamic performance of the magnetoelectroelastic nanosystem. Engineering with Computers. https://doi.org/10.1007/s0036602101391x
[72] Huang, X., Zhang, Y., Moradi, Z. and Shafiei, N., 2021. Computer simulation via a couple of homotopy perturbation methods and the generalized differential quadrature method for nonlinear vibration of functionally graded nonuniform microtube. Engineering with Computers. https://doi.org/10.1007/s00366021013957