nano shell based on modified couple stress theory with considering thickness stretching effect

Journal Pre-proof Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect Maryam Lori Dehsaraji, Mohammad Arefi, Abbas Loghman PII:

S2214-9147(19)31252-8

DOI:

https://doi.org/10.1016/j.dt.2020.01.001

Reference:

DT 585

To appear in:

Defence Technology

Received Date: 7 December 2019 Revised Date:

26 December 2019

Accepted Date: 2 January 2020

Please cite this article as: Dehsaraji ML, Arefi M, Loghman A, Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect, Defence Technology (2020), doi: https://doi.org/10.1016/ j.dt.2020.01.001. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Production and hosting by Elsevier B.V. on behalf of China Ordnance Society.

Size Dependent Free Vibration Analysis of Functionally Graded Piezoelectric micro/nano Shell Based on Modified Couple Stress Theory with Considering Thickness Stretching Effect Maryam Lori Dehsaraji1, Mohammad Arefi*1 and Abbas Loghman11 1

Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran.

Abstract: Higher-order shear and normal deformation theory is used in this paper to account thickness stretching effect for free vibration analysis of the cylindrical micro/nano shell subjected to an applied voltage and uniform temperature rising. Size dependency is included in governing equations based on the modified couple stress theory. Hamilton’s principle is used to derive governing equations of the cylindrical micro/nano shell. Solution procedure is developed using Navier technique for simply-supported boundary conditions. The numerical results are presented to investigate the effect of significant parameters such as some dimensionless geometric parameters, material properties, applied voltages and temperature rising on the free vibration responses. Keywords: Thickness stretching effect, shear and normal deformation theory, vibration analysis, length scale parameter, modified couple stress theory.

1

Corresponding author, PhD E-mail: [email protected]

Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect Abstract: Higher-order shear and normal deformation theory is used in this paper to account thickness stretching effect for free vibration analysis of the cylindrical micro/nano shell subjected to an applied voltage and uniform temperature rising. Size dependency is included in governing equations based on the modified couple stress theory. Hamilton’s principle is used to derive governing equations of the cylindrical micro/nano shell. Solution procedure is developed using Navier technique for simplysupported boundary conditions. The numerical results are presented to investigate the effect of significant parameters such as some dimensionless geometric parameters, material properties, applied voltages and temperature rising on the free vibration responses. Keywords: Thickness stretching effect, shear and normal deformation theory, vibration analysis, length scale parameter, modified couple stress theory. 1. Introduction The responses of materials and structures subjected to various types of loadings have been defined by constitutive relations. These relations are presented in dimensionless forms. The behaviors of structures in small scales are defined using some nonclassical theories that account size dependency [1, 2]. Nano structures are extensively used in various situations as nano generator, nano resonator or chemical nano sensor [3-6]. There are some important works on the experimental analysis and atomistic simulations of nano-structures [7, 8] to measure size effects of small scale structures. To overcome the incompleteness of the classical continuum theory in precise prediction of small scale structures, some size-dependent theories have been developed to predict the size effects. These size-dependent theories are included the strain gradient theory (SGT), modified couple stress theory (MCST), micro-polar theory, and nonlocal elasticity theory. There are some small scale parameters in the aformentioned theories for more accurate analysis of size-dependent analysis [9- 11]. Yang, et al [12] used the modified couple stress theory (MCST) including only one small scale parameter. Based on the modified couple stress theory, the strain energy density function is depending on the symmetric part of the curvature tensor that leads to more easier formulation than the classical couple stress theory. There are some important works on the application of modified couple stress theory for analysis of micro or nanosized structures such as micro actuators, carbon nanotubes, micro cantilevers, micro beams and microfilms[13-17]. Lim and He [18] studied the size-dependent nonlinear response of thin elastic films with nano-scale thickness based on a continuum approach and by applying Kirchhoff’s

hypotheses. Park and Gao [19] investigated bending of a Euler-Bernoulli beam by using modified couple stress theory. Lu et al. [20] used a general thin plate-like theory with considering surface effects for analysis of thin film structures. Kong et al. [21] analyzed dynamic behavior of Euler–Bernoulli beams with simply supported boundary conditions based on the modified couple stress theory. Ma et al [22] studied vibration and bending analysis of beams according to Timoshenko beam model. Lam et al. [23] used the modified strain gradient elasticity theory for analysis of small scale structures as reported in References [24–30]. Xia et al. [31] studied nonlinear static bending, free vibration and postbuckling analysis of micro-beams based on the MCST by using the Euler–Bernoulli beam theory. Xia and Wang [32] studied vibration and stability analysis of micro pipes transmitter fluid using Timoshenko beam model. Free vibration of single-layered graphene sheet resting on Pasternak foundation with simply supported boundary condition based on the modified couple stress theory was studied by Akgöz and Civalek [33]. Jomehzadeh et al [34] investigated vibration analysis of micro-plates based on modified couple stress theory. The free vibration responses were presented in terms of different length scale parameters, different dimensionless aspect ratios and various boundary conditions. Analysis of the nano and micro shells has attracted various researchers due to various applications in technical applications [35-38]. Tadi Beni et al [39] derived equations of motion of functionally graded cylindrical shell based on modified couple stress theory and first order shear deformation shell theory. Khalili et al [40] studied dynamic behavior of functionally graded (FG) beams under moving loads by using Euler–Bernoulli beam theory to investigate the effect of power index parameter and moving load on the dynamic behavior. Li et al. [41] studied bending of simply-supported micro plate under external load based on the strain gradient elasticity theory. Ashoori et al. [42] obtained governing equations of the Kirchhoff micro plate model by using the modified strain gradient elasticity theory. The accuracy and trueness of the present theory was justified using comparison with previous works. Dynamic behavior of nanotubes based on nonlocal theory and using the three dimensional theory of elasticity was studied by Alibeigloo et al [43] to study the effect of small scale parameter on the natural frequency. Zeighampour and tadibeni [44] used modified couple stress theory for dynamic analysis of transporter fluid carbon nanotube. The obtained results have been presented in terms of small scale parameter and fluid velocity based on the classical theory and the modified couple stress theory. Dynamic stability of FGM micro cylindrical shells based on MCST theory was studied by Sahmani et al. [45]. They showed that by decreasing the size parameter, the domain of the instability district of system increases. The piezoelectric cylindrical shells because their direct and reverse piezoelectric effects, are extensively used as useful structures in the design of spatial transporters [46-51] in various modes such as vibration, buckling and bending behaviors subjected to the electric and thermal loads. Mohammadimehr et al [52] presented the thermo- electro-mechanical buckling and vibration analysis of a double-bonded nano composite piezoelectric plate reinforced by a boron nitride nanotube by using the Mori-Tanaka method and modified couple stress theory. Mosallaie Barzoki et al. [53] analyzed the thermo-electromechanical buckling analysis of a piezoelectric polymeric cylindrical shell reinforced by

double-walled boron nitride nanotubes. Liu et al. [54] studied nonlocal thermo-electromechanical vibration analysis of the piezoelectric nanoplate. The vibration responses have been presented in terms of the nonlocal parameter, axial force, temperature change and external electric voltage. Hadi et al [55] employed couple-stress theory for vibration analysis of Euler-Bernoulli nano-beams made of fgms with various boundary conditions using generalized differential quadrature method. Zeighampour and Tadi Beni [56] used the couple stress theory for vibration analysis of a shear deformable cylindrical shell model. Also free vibration analysis of FGM cylindrical nanoshells was studied by Tadi Beni et al. [57]. Zenkour [58] used a new theory including four unknown shear and normal deformations for bending of FG plate. Linear and nonlinear vibration of carbon nanotubes were investigated using perturbation technique and representative volume element method [59, 60]. Modal analysis of carbon nanotubes reinforced by polymer based on finite element method was presented by Fereidoon et al [61]. Some works presented a computational design methodology for topology optimization of piezoelectric and flexoelectric materials [62, 63]. Also different methods are developed for analyzing cracks and fractures in structures [64-66]. Rabczuk et al [67] treated fluid and fracturing structure by using a meshfree method under impulsive loads. Nanthakumar et al [68] solved inverse problem of detecting inclusion interfaces in a piezoelectric structure. Anitescu et al [69] developed a model for solving the partioal differential equations using rtificial neural networks and an adaptive collocation strategy. Guo et al [70] proposed a deep collocation method for bending analysis of a thin plate based on Kirchhoff theory. Rabczuk et al [71] proposed a new novel nonlocal parameter for solution of partial differential equations based on variational principle. Literature review was completed using summarizing some important works on the analysis of cylindrical micro/nano shells, higher order shear deformation theory and size dependent analysis. Based on the best author’s knowledge, there is no published works on the free vibration analysis of functionally graded cylindrical piezoelectric nano/micro shell with accounting thickness stretching effect. The thickness stretching effect is accounted based on shear and normal deformation theory. Size dependency is included in this analysis based on modified couple stress theory. The governing equations of motion are derived based on Hamilton’s principle. The accuracy and trueness of present numerical results are justified using comparison with existing numerical results of literature. The effect of significant parameters such as small scale parameters, length to radius ratio, radius to thickness ratio, applied voltage and temperature change on the natural frequencies of micro/nano shell. 2. Governing Equations of Motion The FGP cylindrical micro/nano shell with length L, radius R and thickness h under the external electric voltage V0 and uniform temperature change ∆T, in thickness direction is shown in Fig. 1. FGPM is usually made of the combination of two materials with electric properties in which the material properties varies continuously and uniformly from PZT_5H properties at the internal surface of the cylindrical micro/nano shell to the properties of the PZT_4 at the external surface. Volume fractions of PZT_4 and PZT_5H according to power law distribution are expressed as:

3

z 1 VPZT_4 = ( + ) N h 2 VPZT_5H = 1 − VPZT_ 4

(1)

In which N is power index. The displacement field based on higher order shear and normal deformation theory to account thickness stretching is expressed as: ∂w ∂φ U ( x, θ , z ) = u 0 ( x , θ ) − z ( x, θ ) − ψ 1 ( x, θ ) ∂x ∂x (2) z ∂w 1 ∂φ V ( x , θ , z ) = v0 ( x , θ ) − ( x, θ ) − ψ 1 ( x, θ ) r ∂θ r ∂θ W ( x , θ , z ) = w( x , θ ) + φ ( x , θ ) + ψ 2 χ ( x , θ ) In which, u 0 ( x , θ ) and v0 ( x , θ ) are displacements of middle surface, w ( x , θ ) and φ ( x , θ ) are the bending and shear components of the radial displacement, and χ ( x , θ ) is

an extra function of x and θ to shows variation of radial displacement along the radial direction. The shape functions appeared in Eq. 2 are suggested as [58]: ψ1 = z − ψ2

h πz sin ( ) π h

(3)

πz = cos ( ) h

Fig. 1. The schematic figure of a cylindrical micro/nano shell subjected to applied voltage.

Based on the modified couple stress theory, the strain energy is expressed as [52]:

δ U = ∫∫∫ {σ ijδε ij + mijδχ ij − Diδ Ei }( R + z )dzdθ dx

(4)

Where σ ij , ε ij , Di and m ij are the components of the Cauchy stress tensor, strain tensor, electric displacement vector and the higher order stress tensor. In addition, E i and χ ij are the components of electric field vector and the symmetrical part of rotation gradient tensor. The constitutive relations are defined as:

σ ij = Cijkl ( z ) ε kl − emij ( z ) Em − β ij ∆T Di = eikl ε kl + λij E j + Pi ∆T mij = 2 l 2 c66 χ ij 1 2

χ ij = (∇θ + ∇θ T )

(5)

→ 1 2 Ei = −∇ϕ In which C ijkl , e mij , D i , λij , Pi and β ij are the stiffness coefficients, piezoelectric coefficients,

θ = (curl u )

electric displacement vector, dielectric constants, pyroelectric constants and thermal moduli respectively. Also ϕ , Em , θ and ∆ T are electric potential, electric field vector, rotation vector and temperature change, respectively. Based on the displacement field relation (2), the components of strain are expressed as: ψ 1 ∂ 2φ 1 ∂V W 1 ∂V0 z ∂2w w εθ = + = − − + + 2 2 2 2 r ∂θ r R + z ∂θ ( R + z ) ∂θ R+ z ( R + z ) ∂θ

φ R+ z

+

ψ2

R+ z

χ (6)

∂u0 ∂U ∂ w ∂φ = −z − ψ1 2 ∂x ∂x ∂x ∂x 2 2

εx =

2

∂W ∂ψ 2 χ = ∂z ∂z ψ 2ψ 1 1 1 ∂W ∂V V 1  2z ∂w ∂φ ψ 2 ∂χ = ( + − )=  +( 2 + ) + 2 2 2 r ∂θ ∂z r 2  ( R + z ) ∂θ R + z ( R + z ) ∂θ R + z ∂θ

εz =

ε zθ

−

 1 v0  (R + z) 

1 ∂V 1 ∂U 1  ∂v 2 z ∂ 2 w 2ψ 1 ∂ 2φ 1 ∂u0  )=  0 − − + εθ x = ( +  2 ∂x r ∂θ 2  ∂x R + z ∂θ∂x R + z ∂θ∂x R + z ∂θ  1 ∂U ∂W 1  ∂φ ∂χ  + ) = ψ 2 +ψ 2 ε xz = ( 2 ∂z ∂x 2 ∂x ∂x  Moreover, the non-classical strain components are obtained as: ∂u0 1 1 χ zz = [ ] 2 2 ( R + z ) ∂θ

χ xx =

∂2w ∂ψ ∂ 2φ ∂v0 ∂2 χ 1 [2 + (1 + 1 ) − +ψ 2 ] 2( R + z ) ∂x ∂θ ∂z ∂x ∂θ ∂x ∂x ∂θ

χθθ =

∂ψ ∂v 1 ∂2w ∂ 2φ ∂2 χ 1 ∂u0 [ −2 − (1 + 1 ) + 0 −ψ 2 − ] 2( R + z ) ∂x ∂θ ∂z ∂x ∂θ ∂x ∂x ∂θ ( R + z ) ∂θ

(7)

(8)

5

∂w ∂ψ 1 ∂ 2ψ 1 ∂φ ∂ 2 v0 1 1 ∂ 2 u0 + + + − R + z + − [− 2 (1 ( ) ) 4( R + z ) ( R + z ) ∂θ 2 ∂x ∂z ∂z 2 ∂x ∂x ∂θ ∂ψ 2 ∂χ (( R + z ) −ψ 2 ) ] ∂z ∂x 2 ∂ψ 1 ∂ 2φ 1 ∂ w ∂2χ 2 ∂2w 1 =  −2 2 − (1 + + ) 2 −ψ 2 2 + [(1 2 2 ∂z ∂x ∂x 4  ∂x ( R + z ) ∂θ ( R + z)2

χθ z =

χ xθ

∂ψ 1 ∂ 2φ ∂2χ  ) 2 − v0 + ψ 2 2 ] ∂z ∂θ ∂θ  ∂ 2u0 ∂ 2 v0 1 1 2 ∂w 1 ∂ψ 1 χ xz =  − + 2 − − − [(1 + 2 2 4  ( R + z ) ∂x ∂θ ∂x ∂z ( R + z ) ∂θ ( R + z ) +

∂ 2ψ 1 ∂φ ∂ψ 2 ∂χ  (R + z) ) − v0 − ( R + z ) + ψ 2 ) ] 2 ∂z ∂θ ∂z ∂θ 

The constitutive relationships for piezoelectric cylindrical shell in general state are given by [53]: 0 0 C C12 C13 0  ε σ xx   11   xx  0 0 σ  C12 C11 C13 0  εθθ   θθ  C   C13 C33 0 0 0 σ zz   13   ε zz   = 0 0 0 C55 0 0  γ xz  σ xz     0 0 0 C55 0 σ θ z   0  γ θ z       1 0 0 0 0 ( C11 − C12 ) γ xθ  σ xθ   0 (9) 2   0 0 0 0  ∆T   0 0 e31   β11 0 0 0 e  0 β 0 0 0 0  ∆T  31  11   E    0 0 e33   x   0 0 β33 0 0 0  ∆T  −   Eθ  −    0 0 0 0 0  0   0 e15 0   E   0 e15 0 0   z   0 0 0 0 0 0  0       0 0 0 0 0  0  0 0 0 0

0  Dx   0 0    0  Dθ  =  0 0  D  e  z   31 e31 e33

0 e15

e15 0

0

0

ε xx  ε  0   θθ   ε  0   zz  + γ 0   xz  γ θ z    γ xθ 

(10)

0   Ex   P1  λ11 0 0 λ   E  +  P  ∆T 0 11   θ   1  0 0 λ33   Ez   P3 

According to Ref. [53], the electric potential as a combination of a cosine and linear variation, that satisfied the Maxwall equation, is assumed as follows [72-74]: 2 zV0 (11) ϕ ( x,θ , z ) = − cos ( β z ) Q( x, θ ) + h In which, β = h , Q ( x , θ ) and V0 represent variation of the electric potential in the midπ

plane and the external electric voltage, respectively. The electric field components are defined as [72-74]: ∂Q Ex = cos( β z) ∂x 1 ∂Q (12) Eθ = cos( β z) R + z ∂θ 2V Ez = −β sin( β z )Q − e h By substitution of strain components, we obtain the stress and electric displacement components as:  2 1 ∂v0 z ∂2w ψ 1 ∂ 2φ 2 σ xx =  c11 ( ∂u0 − z ∂ w2 −ψ 1 ∂ φ2 )+ c12 ( − − ∂x ∂x ∂x  R + z ∂θ ( R + z ) 2 ∂θ 2 ( R + z ) 2 ∂θ 2  ψ w φ + + + 2 χ ) + c13 ( ∂ψ 2 χ )  − e31 Ez − β11∆T ∂z  R+z R+z R+z  2 2 ∂ v 2 ψ 1 ∂2φ + σ θθ =  c12 ( ∂u0 − z ∂ w2 −ψ 1 ∂ φ2 )+ c11 ( R1+ z 0 − z 2 ∂ w2 − ∂θ ( R + z ) ∂θ ( R + z )2 ∂θ 2 ∂x ∂x ∂x  (13) w + φ + ψ 2 χ ) + c ( ∂ψ 2 χ )  − e E − β ∆T 31 z 11 13  R+ z R+ z R+ z ∂z   ∂ 2φ 1 2 σ zz =  c33 ( ∂ψ 2 χ )+c13 ( ∂u0 − z ∂ w2 −ψ 1 2 +

∂z



ψ1

∂x

∂x

∂x

∂v0 z ∂2w − − R + z ∂θ ( R + z ) 2 ∂θ 2

 ∂φ w + + φ + ψ 2 χ )  − e33 Ez − β 33 ∆T 2 ( R + z ) ∂θ R + z R+ z R+ z  2

2

7

 2 z ∂w ψ 2ψ 1 ∂φ ψ 2 ∂χ 1 +( 2 + + − ) v0  − e15 Ex . 2 2 R + z ( R + z ) ∂θ R + z ∂θ ( R + z )   ( R + z ) ∂θ ∂χ   ∂φ τ zx = c55 ψ 2 +ψ 2 − e15 Eθ ∂x ∂x   1 2 z ∂ 2 w 2ψ 1 ∂ 2φ 1 ∂u0   ∂v τ xθ = (c11 − c12 )  0 − − +  2  ∂x R + z ∂θ∂x R + z ∂θ∂x R + z ∂θ  

τ zθ = c55 

 2z  ψ 2ψ 1 ∂w ∂φ ψ 2 ∂χ 1 Dx = e15  +( 2 + ) + − v0  2 2 R + z ( R + z ) ∂θ R + z ∂θ ( R + z )   ( R + z ) ∂θ ∂Q + β11 cos( β z ) + P1∆T ∂x ∂χ  1 ∂Q  ∂φ Dθ = e15 ψ 2 +ψ 2 + β11 cos( β z ) + P1∆T  ∂x ∂x  R + z ∂θ 

(14)

∂u0 ψ 1 ∂ 2φ ∂2w ∂ 2φ z ∂2w 1 ∂v0 − z 2 −ψ 1 2 + − − ∂x ∂x ∂x R + z ∂θ ( R + z )2 ∂θ 2 ( R + z ) 2 ∂θ 2 ∂ψ 2V ψ w φ + + + 2 χ ) + e33 ( 2 χ ) − β33 β sin( β z )Q − e + P3 ∆T R+z R+z R+z ∂z h

Dz = e31 (

Substitution of strain components into strain energy gives: 1 ∂T 1 ∂ 2T 1 ∂ 2Txz0 ∂Tθθ1  ∂M 1,0 ∂N ∂N ∂M 1xθ δ U = ∫∫ [δ u0 − xx − xθ − ( zz + θ2z + )  + δ v0 − θ − − ∂x ∂θ 2 ∂θ ∂θ ∂x ∂θ ∂x ∂θ ∂θ 

{

{

∂Tθθ0 ∂Tx1θ ∂Txx0  1 ∂ 2T 0 ∂ 2Txz−1 ∂N 0 1 − Nθ z + ( θ z + + T − + + )  + δχ { Q1zz + N 20θθ − 2θ z − xz 2 ∂θ ∂x ∂x ∂θ ∂x  2 ∂x ∂θ ∂N21xz 1 ∂Q21θ z ∂T20θ z ∂T2−xz1 ∂Q20xz ∂ 2T20θθ ∂ 2T21xθ ∂ 2T2−x1θ ∂T20xx  + − + − − − + + ) ∂x ∂x 2 ∂x ∂θ ∂θ ∂x ∂θ ∂x 2 ∂θ 2 ∂x ∂θ  +δ w{− +

1,1 −1,1 0 1 2 0 2 −1 ∂ 2 Mθθ−1,1 ∂ 2 M xx ∂ 2 M x0,1 θ − 2∂M θ z − ∂Tθ z + ∂Txz − ∂ Tθθ − ∂ Txθ + N − − 2 θ ∂θ 2 ∂x2 ∂x∂θ ∂θ ∂x ∂θ ∂x ∂θ ∂x2

∂Txx0  ∂ 2Tx1θ ∂2 N −1 ∂2 N11xx ∂2 N10xθ ∂N20θ z ∂N1−θ1z ∂N21xz + 2 ) +δφ{− 12θθ + Nθ − −2 − −2 − − 2 2 ∂x∂θ ∂θ ∂θ ∂x ∂θ ∂x ∂x ∂θ  ∂θ

1 ∂Tθ0z ∂P20θ z ∂P31θ z ∂Txz1 ∂P2−xz1 ∂P30xz ∂ 2Tθθ0 ∂ 2 P20θθ ∂ 2Tx−θ1 ∂ 2 P21xθ ∂ 2Tx1θ − + + + − − − − − 2 + 2 2 ∂x ∂x ∂x ∂θ ∂θ ∂θ ∂x ∂θ ∂x ∂θ ∂x2 ∂x ∂θ +

h /2 0 ∂P20xx  ∂ 2 P2−x1θ ∂Txx  ∂Dx cos(β z )( R+ z )−∂Dθ cos(β z )− + + + δ Q  ∫ ∂θ 2 ∂x ∂θ ∂x ∂θ  ∂θ − h/2 ∂x

Dz β sin(β z )( R+ z )dz}]dθ dx

(15)

In which the resultant components are defined in Appendix A. The variation of kinetic energy is defined as: t2 t2 h /2 ∂U ∂δ U ∂V ∂δ V ∂W ∂δ W ∫t δ Tdt = ∫t1 ∫−h/2 ∫∫A ρ ( ∂t ∂t + ∂t ∂t + ∂t ∂t ) ( R + z)dA dz dt 1

(16)

Substitution of components of displacement field from relation (2) into leads to: t ∂ 2u0 ∂3 w ∂3φ  δ δ T u R I I R I I R J J = − ( + ) + ( + ) + ( + ) [ ∫ 0 00 10 10 20 00 10 2 2 2 2

∂t

t1

+ δ v0 [ −( R I 00 + I10 ) − ( R L00 + L10 )

∂x∂t

∂x∂t 

∂2v0 ∂3 w ∂3φ  ∂2 w δχ I J R L L + + + − ( + ) [ 10 00 00 10 ∂t 2 ∂θ∂t 2 ∂θ∂t 2  ∂t 2

∂ 2φ ∂2 χ  ∂2 w δ R K K w R I I − + + − + − ( ) ( ) [ 00 10 00 10  ∂t 2 ∂t 2  ∂t 2

∂3u0 ∂2φ ∂2 χ ∂4 w − ( R L + L ) − ( R I + I ) + ( R I + I ) 00 10 10 20 20 30 ∂t 2 ∂t 2 ∂x∂t 2 ∂x2∂t 2 ∂3v0 ∂ 4φ ∂4 w ∂ 4φ  + (R J10 + J 20 2 2 − I10 + + I J 21 11  + δφ [ − ( R I00 ∂x ∂t ∂θ∂t 2 ∂θ 2∂t 2 ∂θ 2∂t 2 

( R I00 + I10 )

(17)

∂3u0 ∂2 w ∂2φ ∂2 χ − + − + − + + ( R I I ) ( R L L ) ( R J J ) 00 10 00 10 00 10 ∂x∂t 2 ∂t 2 ∂t 2 ∂t 2 ∂3v0 ∂4 w ∂ 4φ ∂4 w ∂ 4φ  + + ( R J10 + J 20 ) 2 2 + ( R S00 + S10 2 2 − J 00 J S 11 01 ∂x ∂t ∂x ∂t ∂θ∂t 2 ∂θ 2∂t 2 ∂θ 2∂t 2  + I10 )

The unknown coefficients in Eq. 17 are defined in Appendix B. The work done by external loadings such as thermal and electrical loads are expressed as:

δ W = ∫∫ [( NTx + N Ex ) A

∂ 2W ∂ 2W + N + N R]dxdθ , ( ) Tθ Eθ ∂x 2 ∂θ 2

δ W = ∫∫ {δχ [ ( R + z ) ψ 2 ( NTx + N Ex )(ψ 2 A

(ψ 2

∂ 2 χ ∂ 2 w ∂ 2φ ψ + 2 + 2 ) + 2 ( NTθ + N Eθ ) 2 ∂x ∂x ∂x R+ z

∂ 2 χ ∂ 2 w ∂ 2φ  ∂ 2 χ ∂ 2 w ∂ 2φ + + ) + w ( N + N ) ( + + )+ δ ψ [ 2 Tx Ex  ∂θ 2 ∂θ 2 ∂θ 2  ∂x 2 ∂x 2 ∂x 2

(18)

1 ∂ 2 χ ∂ 2 w ∂ 2φ  ( NTθ + N Eθ )(ψ 2 2 + 2 + 2 )  + δφ [ ( R + z )( NTx + N Ex ) R+ z ∂θ ∂θ ∂θ  ∂ 2 χ ∂ 2 w ∂ 2φ 1 ∂ 2 χ ∂ 2 w ∂ 2φ  (ψ 2 2 + 2 + 2 ) + ( NTθ + N Eθ ) (ψ 2 2 + 2 + 2 )  dθ dx} ∂x ∂x ∂x R+z ∂θ ∂θ ∂θ  In which ( NTx , NTθ ) and ( NEx , N pθ ) are thermal and electrical loads in x and θ directions respectively. Hamilton principle is used to derive governing equations of motion as: ∫t1 δ T − δ U + δ W dt = 0 t2

(19)

9

By substituting Eqs. (15), (17) and (18) into Eq. (19), the six governing equations of motion are derived as: ∂2u0 ∂2u0 ∂2v0 ∂χ ∂w ∂3 w ∂3 w ∂φ δ u0 : a1 2 + a2 2 + a3 + a4 + a5 + a6 3 + a7 + a8 + 2 ∂x ∂θ ∂x ∂θ ∂x ∂x ∂x ∂x ∂θ ∂x a9

∂4u0 ∂4v0 ∂4u0 ∂3φ ∂3φ ∂Q ∂3 χ + + + + + + a a a a a a 10 11 12 13 14 15 ∂x3 ∂x ∂θ 2 ∂x ∂θ 4 ∂x ∂θ 3 ∂x ∂θ 2 ∂x2 ∂θ 2

(20-a)

∂4v0 ∂2u0 ∂3 w ∂3φ ( R I I ) ( R I I ) ( R J J ) = − + + + + + 00 10 10 20 00 10 ∂θ ∂x3 ∂t 2 ∂x∂t 2 ∂x∂t 2 3 3 ∂2u ∂2v ∂2v ∂χ δ v0 : b1 ∂x ∂θ0 + b2 v0 + b3 20 + b4 20 + b5 ∂θ + b6 ∂∂w + b7 ∂ w3 + b8 ∂ w 2 + θ ∂θ ∂x ∂θ ∂θ ∂x + a16

b9

∂Q ∂4u0 ∂4v0 ∂4u0 ∂φ ∂3φ ∂3φ + b10 3 + b11 + b + b + b + b ∂θ ∂θ ∂θ ∂x2 12 ∂θ 13 ∂x ∂θ 3 14 ∂x2 ∂θ 2 15 ∂x3 ∂θ

+ b16

(20-b)

∂v ∂4v0 ∂w ∂φ ∂3χ ∂3χ + b17 + b18 3 = −( R I 00 + I10 ) 20 + I10 + J 00 4 2 2 ∂x ∂θ ∂x ∂θ ∂t ∂θ∂t ∂θ∂t 2 2

3

3

∂u0 ∂v ∂2 χ ∂2 χ ∂4 χ ∂4 χ ∂4 χ + C2 0 + C3 χ + C4 2 + C5 2 + C6 4 + C7 4 + C8 2 2 ∂x ∂θ ∂x ∂θ ∂x ∂θ ∂x ∂θ 2 2 4 4 4 + C9 w + C10 ∂ w2 + C11 ∂ w2 + C12 ∂ w4 + C13 ∂ w4 + C14 ∂2 w 2 + C15φ ∂θ ∂x ∂x ∂θ ∂x ∂θ ∂2Q ∂2φ ∂2φ ∂4φ ∂4φ ∂4φ + C16 2 + C17 2 + C18 4 + C19 4 + C20 2 2 + C21Q + C22 2 + ∂θ ∂x ∂x ∂θ ∂x ∂θ ∂θ

δχ : C1

∂u ∂ v0 ∂v ∂ w ∂Q + C24 2 0 + C25 + C26 30 = −( R L00 + L10 ) 2 2 ∂x2 ∂θ ∂x ∂θ ∂x ∂θ ∂t 2

C23

−( R L00 + L10 )

3

3

3

(20-c)

2

∂ 2φ ∂2 χ ( R K K ) − + 00 10 ∂t 2 ∂t 2

∂u ∂3u0 ∂3u ∂v ∂3v ∂3v0 ∂2 χ δ w : d1 ∂x0 + d 2 + d3 30 + d 4 0 + d5 30 + d6 + d 7 χ + d8 2 + 2 2 ∂θ ∂x ∂θ ∂x ∂θ ∂θ ∂x ∂θ 2 2 ∂2 χ ∂4 χ ∂4 χ ∂4 χ d9 2 + d10 4 + d11 4 + d12 2 2 + d13 w + d14 ∂ w2 + d15 ∂ w2 ∂x ∂x ∂θ ∂x ∂θ ∂θ ∂x 2 2 4 4 4 4 ∂4 w +d17 ∂ w4 +d18 ∂2 w 2 +d19φ +d20 ∂ φ2 +d21 ∂ φ2 +d22 ∂ φ4 +d23 ∂ φ4 4 ∂θ ∂x ∂θ ∂θ ∂x ∂x ∂θ ∂x 2 2 2 2 4 ∂w ∂φ +d24 ∂2 φ 2 +d25 ∂ Q2 +d26 ∂ Q2 =−( R I00 + I10 ) 2 − ( R I00 + I10 ) 2 ∂x ∂θ ∂x ∂θ ∂t ∂t

+d16

∂ 3u0 ∂2 χ ∂4w − ( R I + I ) + ( R I + I ) +( R J10 10 20 20 30 ∂t 2 ∂x∂t 2 ∂x 2∂t 2 ∂ 3v0 ∂ 4φ ∂4w ∂ 4φ + J 20 2 2 − I10 + I + J 21 11 ∂x ∂t ∂θ∂t 2 ∂θ 2 ∂t 2 ∂θ 2 ∂t 2 −( R L00 + L10 )

(20-d)

δφ : E1

∂ 3v0 ∂u0 ∂3u0 ∂3u0 ∂v0 ∂3v E E E + E2 + + + + E6 30 + E7 χ 3 4 5 2 3 2 ∂x ∂θ ∂x ∂θ ∂x ∂θ ∂θ ∂x

∂2 χ ∂2 χ ∂4 χ ∂4 χ ∂4 χ + E9 2 + E10 4 + E11 4 + E12 2 2 + E13 w + 2 ∂θ ∂x ∂x ∂θ ∂x ∂θ 2 2 4 4 4 ∂2φ E14 ∂ w2 + E15 ∂ w2 + E16 ∂ w4 + E17 ∂ w4 + E18 ∂2 w 2 + E19φ + E20 2 ∂θ ∂x ∂x ∂θ ∂x ∂θ ∂θ 2 2 ∂Q ∂Q ∂2φ ∂4φ ∂4φ ∂4φ + E21 2 + E22 4 + E23 4 + E24 2 2 + E25 2 + E26 2 = ∂x ∂x ∂θ ∂x ∂θ ∂x ∂θ

+ E8

(20-e)

∂2w ∂ 2φ ∂2 χ R I I R L L − ( + ) − ( + ) − ( R J 00 + J10 ) 00 10 00 10 ∂t 2 ∂t 2 ∂t 2 ∂ 3u0 ∂ 3v0 ∂4w ∂ 4φ ∂4w ( R J J ) ( R S S J J + + + + − + 10 20 00 10 00 11 ∂x∂t 2 ∂x 2 ∂t 2 ∂x 2 ∂t 2 ∂θ∂t 2 ∂θ 2 ∂t 2 ∂ 4φ + S01 2 2 . ∂θ ∂t ∂2χ ∂2χ ∂2w ∂2 w ∂u0 ∂v δ Q : f1 + f 2 0 + f3 χ + f 4 2 + f5 2 + f 6 2 + f7 2 + ∂x ∂θ ∂θ ∂x ∂x ∂θ (20-f) 2Q 2Q 2φ 2φ ∂ ∂ ∂ ∂ + f 8 w+ f 9φ + f10 2 + f11 2 + f12Q+ f13 2 + f14 2 =0 ∂x ∂θ ∂x ∂θ The coefficients a1 ... a16 , b1 ... b18 , C1 ... C 27 , d1 ... d 26 , E1 ... E 26 and f1 ... f14 are expressed in Appendix B. − ( R I 00 + I10 )

3. Solution Procedure The thermal and electrical resultant components are defined as [53]: h/ 2 h/2 2e31Ve N Tx = N Tθ = − ∫ β11∆Tdz , N Ex = N Eθ = − ∫ dz (21) h −h/ 2 −h/2 In this section, the solution of Eqs. (20) for the simply supported FGP cylindrical nano/micro shell subjected to temperature change ∆T and external electric voltage V0 , is investigated. The unknown functions based on the Navier solution are expressed as: U 1 = U 1 mn ( t ) cos( V1 = V1mn ( t ) sin(

mπx ) cos( nθ ) L

mπx ) sin( nθ ) L

W1 = W1mn ( t ) sin(

mπx ) cos( nθ ) L

(22)

mπx φ1 = φ1 mn ( t ) sin( ) cos( nθ ) L

χ1 = χ1 mn ( t ) sin(

mπx ) cos( nθ ) L

Q1 = Q1 mn ( t ) sin(

mπx ) cos( nθ ) L

11

In which U1mn , V1mn , W1mn , χ1mn , U φ1mn and Q1mn are amplitudes of unknown functions, and m, n are axial and circumferential wave numbers respectively. Thus, by substituting approximate solutions from Eq. (22) into Eqs. (20), the governing equations are obtained

{} ..

as known format [ K ]6×6 {d}6×1 + [ M ]6×6 d

6×1

= 0 , in which [ K ] and [ M ] is stiffness and

mass matrix respectively. The harmonic unknown functions are assumed as:

{d } = [U1mn

V1mn χ1mn W1mn φ1mn Q1mn ] eiωt in which ω is natural frequency. To investigate the shell frequency, the non-trivial solutions are obtained by solving the determinant of the coefficients matrix. T

4. Numerical Results and Discussion In this section, the free vibration of an FG piezoelectric micro/nano shell with simply supported boundary conditions is investigated based on the modified couple stress theory. The obtained results are validated using comparison with previous valid works. After validation, the numerical results are presented in terms of significant parameters such as dimensionless size parameter l/h, power index N, length-to-radius ratio L/R, temperature change and external electric voltage. 4-1 Validation of results The present section investigates the verification of present numerical results using comparison with some valid references. The verification is performed with the relevant references (Alibeigloo and Shaban [43], Tadi Beni et al [57], Zeighampour and Tadi Beni [56]). The material properties for the present comparison are assumed as follows. R = 2nm, E = 1.06 TPa, ν = 0.3, L = R, m = 1 Tables 1, 2 list comparison of dimensionless natural frequency for different thickness to radius ratios h/R and various circumferential wave numbers n for m = 1. The dimensionless natural frequency is assumed as Ω = ω R ρ / E . One can conclude that the present results with considering thickness stretching effect leads to significant improvements of numerical results respect to previous results without thickness stretching effect. Table 1. Comparison of dimensionless natural frequency for different thickness to radius ratios and various circumferential wave numbers (m=1). (L/R=1, R=2 [43] [57] Present [57] [56] Present N nm)H/R (l=0) (FSDT)(l=0) study(l=0) (FSDT)(l=h) (l=h) study(l=h) 1 0.913 0.933 0.9784 1,126 1.122 1.1785 0.1 2 0.762 0.776 0.8197 1.0688 1.046 1.1438 3 0.699 0.713 0.7464 1.207 1.136 1.3075 1 0.993 1.048 1.0846 1.537 1.528 1.6450 0.2 2 0.936 0.971 1.0092 1.590 1.553 1.7711 3 0.999 1.052 1.0903 1.928 1.837 2.2005 1 1.112 1.181 1.2109 1.878 1.860 2.0570 0.3 2 1.116 1.162 1.2057 1.974 1.931 2.2415 3 1.245 1.330 1.3880 2.415 2.338 2.7850

Table 2. Comparison of dimensionless natural frequency for different thickness to radius ratios and various circumferential wave numbers. (L/R=1,R=2 nm) [56] Present N H/R (l=2h) study(l=2h) 1 1.495 1.6157 2 1.517 1.7358 0.1 3 1.794 2.0662 1 2.101 2.3790 2 2.214 2.6220 0.2 3 2.736 3.2767 1 2.520 2.8928 2 2.749 3.2555 0.3 3 3.478 4.1216

4-2 Effect of length scale parameter on dimensionless natural frequency The complete numerical results are presented in this section in terms of significant parameters of the cylindrical micro/nano shell. The material properties of cylindrical micro/nano shell are expressed in Table 3. The dimensionless natural frequency is

assumed as: Ω = ω R ( ρ / c11 )PZT4 . The effect of the dimensionless length scale parameter and the various power index N on natural frequencies according to the classical continuum theory (l=0) and size dependent modified couple stress theory (l=3 h) are presented in Fig. 2. Fig. 2 shows that increase in the length scale parameter, leads to increase in dimensionless natural frequency, also the highest natural frequency is related to PZT4 and the lowest is for PZT5H. When the material length scale parameter changes from l/h=0 to l/h=5, the dimensionless natural frequency changes from 0.2539 to 0.3751 for N=0 in shells made from PZT4, and from 0.2360 to 0.34812 for N=∞ in shells made from PZT-5H. Table 3: Material properties of the PZT-4 and the PZT-5H. [53] Elastic moduli/GPa

PZT4 PZT5H

c11

c12

c13

c 22

c 23

c33

c 44

c55

c 66

139

77.8

74

139

74

115

25.6

25.6

30.6

126

79.1

83.9

126

83.9 117

23

23

23.5

Piezoelectric moduli /(C·m-2) PZT-4 PZT-5H

Dielectric moduli/(10-9 F·m-1)

e15

e 24

e31

e32

e33

λ11

λ 22

λ33

12.7 17

12.7 17

-5.2 -6.5

-5.2 -6.5

15.1 23.3

6.46 15.05

6.46 15.05

5.62 13.02

Thermal moduli/(105 N·m-2·k-1) PZT-4 PZT-5H

Pyroelectric constants/(10-5 C·m-2·k-1)

β 11

β 22

β 33

P1

P2

P3

5.19 5.19

5.19 5.19

4.8 4.8

5.4 5.4

5.4 5.4

5 5

13

Fig. 2. Variation of dimensionless natural frequencies in terms of dimensionless small scale parameter l/h for various gradient index N.

4.3. Effect of dimensionless length to radius ratio L/R on dimensionless natural frequencies Fig. 3 presents variation of dimensionless natural frequencies of micro/nano shell in terms of various length to radius ratio L/R based on the size-dependent theory (l=3h) and classical continuum theory for different materials. The numerical results indicate that the dimensionless natural frequencies are decreased with increase of length to radius ratio L/R. In addition, the natural frequencies for modified couple stress theory is significantly more than the results of classical continuum theory. It is concluded that with length to ratio more than 10, L/R>10, the natural frequencies are closed to an assymptotic value.

Fig. 3. Variation of dimensionless natural frequencies in terms of dimensionless length to radius ratio L/R based on classical and modified couple stress theory for various gradient index N.

4.4. Effects of axial and circumferential wave numbers on dimensionless natural frequencies Figs. 4-6 show the variation of dimensionless natural frequencies of cylindrical micro/nano shell based on calssical and modified couple stress theories for different circumferential and axial wave numbers n, m. It is observed that increasing in axial and circumferential wave numbers increases the dimensionless natural frequency for both

theories. This increasing for modified couple stress theory is more significant than classic theory.

Fig. 4. Variation of dimensionless natural frequencies in terms of axial and circumferential wave numbers based on classical and modified couple stress theory.

Fig. 5. Variation of dimensionless natural frequencies in terms of circumferential wave numbers for various length to radius ratio L/R based on classical and modified couple stress theory.

Fig. 6. Variation of dimensionless natural frequency in terms of circumferential and axial wave numbers based on modified couple stress theory (l=3h).

4.5. Effect of thickness-to-radius ratio on dimensionless natural frequency

15

Fig 7 shows the effect of thickness to radius ratio h/R of micro/nano shell as well as circumferential wave numbers on the dimensionless natural frequency. It is observed that the natural frequencies are increased with increase of thickness to radius ratio h/R for both theories of classic and modified couple stress. In addition, increase in circumferential wave number leads to increase in dimensionless natural frequency.

Fig. 7. Variation of dimensionless natural frequencies in terms of circumferential wave numbers for various thickness to radius ratio h/R based on classical and modified couple stress theory.

4.6. Effect of uniform temperature changes on dimensionless natural frequency Fig. 8 shows the effect of the uniform temperature changes ∆T on the dimensionless frequency of micro/nano shells based on the modified couple stress theory for different applied voltages. The numerical results indicate that with increasing the temperature rise, stiffness of micro/nano shell decreases that causes the fundamental dimensionless frequency decreases. It is also observed that increasing the applied voltage reduces the natural frequency.

Fig. 8. Variation of dimensionless natural frequency in terms temperature rising and external applied voltage based on modified couple stress theory (l=3h).

4.7 Effects of external electric voltage on dimensionless natural frequency

The effect of external electric voltage V0 on the dimensionless natural frequency of micro/nano shells based on the modified coupling stress theory is illustrated in Fig. 9. The numerical results indicate that increasing in electric voltage applied leads to decrease in fundamental dimensionless frequency. It is concluded that increase of external electric voltage in a piezoelectric cylindrical shell leads to decrease of stiffness.

Fig. 9. Variation of dimensionless natural frequency in terms temperature rising and external applied voltage based on modified couple stress theory (l=3h).

4.6 Effect of thickness stretching As said in Abstract, the main novelty of the present paper is conclusion on the effect of thickness stretching. To conclude the effect of thickness stretching effect in improvement of numerical results respect to the previous cases, the natural frequencies are depicted with and without thickness stretching effect. Shown in Fig. 10 is comparison between natural frequencies of cylindrical micro/nano shell with and without thickness stretching effect in terms of thickness to radius ratio h/R. It is concluded that accounting thickness stretching leads to increase of the natural frequencies rather than the cases that ignore thickness stretching. This improvement becomes very important for modified couple stress theory.

Fig. 10. Comparison between natural frequencies of cylindrical micro/nano shell with and without thickness stretching effect in terms of thickness to radius ratio h/R

17

Table 4 lists variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of dimensionless small scale parameter l/h for various power index N. It is observed that the dimensionless natural frequencies are increased with increase of dimensionless small scale parameter l/h while increase of power index N leads to decrease of natural frequencies. Listed in Table 6 is variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of various temperature rising and external electric voltage. It is concluded that the natural frequencies are decreased with increase of temperature rising and external electric voltage.

Table 4. Variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of dimensionless small scale parameter l/h for various power index N. (L/R=5,R=2µm,∆T=V0=0, l /h h/R=0.1) Ω N 0 0.2539 1 0.3238 0 (PZT4) 2 0.34223 3 0.3527 0 0.2450 1 0.3120 1 2 0.3302 3 0.3400 0 0.2423 1 0.3086 2 2 0.3265 3 0.3361 0 0.2394 1 0.3052 5 2 0.3231 3 0.3326 0 0.23596 1 0.3012 2 0.3198 ∞ (PZT5H) 3 0.3293

Table 5. Variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of various temperature rising and external electric voltage. (h/R=0.1,R=2 µm, N=0,l=3h, L/R=5) Ve (v) Ω

∆T/K -50

-100 0 50

0.3856 0.3529 0.3345

0

+50

+100

100 -100 0 50

0.3145 0.3855 0.3527 0.3343

100

0.3143

-100 0 50 100 -100 0 50 100

0.3853 0.3525 0.3341 0.3141 0.3852 0.3523 0.3339 0.3139

5. Conclusion Size dependent free vibration analysis of a functionally graded cylindrical micro/nano shell was studied in this paper based on a new higher-order shear and normal deformation theory and modified couple stress theory. The cylindrical micro/nano shell was subjected to thermal and electrical loads. Hamilton’s principle was used to derive governing equations of motion. The analytical solution was proposed for the cylindrical shell with simply-supported boundary conditions. The main numerical results of the present paper are summarized as: Comparison between numerical results with and without thickness stretching effect show that accounting thickness stretching effect leads to significant improvement of natural frequencies specially for modified couple stress theory. Size dependency was included in constitutive relations based on the modified couple stress theory. The numerical results show that consideing size-dependency based on the modified couple stress thoery and increase of small scale parameter leads to increase of stiffness of material and increase of natural frequencies. Investigation on the effect of applied electric voltage and uniform temperature rising indicates that the natural frequencies are decreased with increase of these parameters. The numerical results have been presented in terms of dimensionless geometric parameters. The numerical results indicate that the natural frequencies are decreased with increase of length to radius ratio L/R and decrease of thickness to radius ratio h/R.

6. REFERENCES

[1] Wang ZL. Zno nanowire and nanobelt platform for nanotechnology. Materials Science and Engineering: R. 2009;64:33–71. Https://doi.org/10.1016/j.mser.2009.02.001 [2] Xu S, Wang ZL. One-dimensional zno nanostructures: solution growth and functional properties. Nano Research, 2011;4:1013–1098 . Https://doi.org/10.1007/s12274-0110160-7

19

[3] Wang Q, Li QH, Chen YJ, Wang TH, He XL, et al. Fabrication and ethanol sensing characteristics of zno nanowire gas sensors. Applied Physics Letters 2004;9:3435–3439. Https://doi.org/10.1063/[email protected] [4] Wang ZL, Song JH. Piezoelectric nanogenerators based on zinc oxide nanowire arrays. Science, 2006;312:242–246. DOI: 10.1126/science.1124005 [5] Tanner SM, Gray JM, Rogers CT, Bertness KA, Sanford NA. High-Q gan nanowire resonators and oscillators. Applied Physics Letters, 2007;91:203117. Https://doi.org/10.1063/1.2815747 [6] He JH, Hsin CL, Liu J, Chen LJ, Wang ZL. Piezoelectric gated diode of a single zno nanowire. Advance Materials .2007;19:781–784. Https://doi.org/10.1002/adma.200601908. [7] Chen CQ, Shi Y, Zhang YS, Zhu J, Yan YJ. Size dependence of Young’s modulus in zno nanowires. Physical Review Letters. 2006;96:075505. DOI:https://doi.org/10.1103/physrevlett.96.075505. [8] Stan G, Ciobanu CV, Parthangal PM, Cook RF. Diameter-dependent radial and tangential elastic moduli of zno nanowires. Nano Letters. 2007;7:3691–3697. Https://doi.org/10.1021/nl071986e. [9] Toupin, R. A. Elastic materials with couple stresses. Archives of Rational for Mechanical and Analysis, 1962;11:385–414. [10] Mindlin, R. D., & Tiersten, H. F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 1962;11:415–448. [11] Mindlin, R. D. Influence of couple-stresses on stress concentrations. Experimental Mechanics, 1963;3:1–7. [12] Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. Couple stress based strain gradient theory for elasticity. Solids and Structures, 2002;39:2731–2743. Https://doi.org/10.1016/S0020-7683(02)00152-X [13] Asghari, M., Ahmadian, M. T., Kahrobaiyan, M. H., & Rahaeifard, M. On the sizedependent behavior of functionally graded micro-beams. Materials and Design, 2010;31:2324–2329. Https://doi.org/10.1016/j.matdes.2009.12.006. [14] Kong, S., Zhou, S., Nie, Z., & Wang, K. Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Engineering Science, 2009;47:487–498. Https://doi.org/10.1016/j.ijengsci.2008.08.008. [15] Ma, H. M., Gao, X. L., & Reddy, J. N. A nonclassical Reddy–Levinson beam model based on a modified couple stress theory. Multiscale Computational Engineering, 2010;8:167–180.

[16] Simsek, M. Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. Engineering Science, 2010;48:1721–1732. Https://doi.org/10.1016/j.ijengsci.2010.09.027. [17] Yang, F., Chong, A. C. M., Lam, D. C. C., &Wang, B., Zhao, J., & Zhou, S. A micro scale Timoshenko beam model based on strain gradient elasticity theory. Mechanics A/Solids, 2010;29:591–599. Https://doi.org/10.1016/j.euromechsol.2009.12.005. [18] C.W. Lim, L.H. He, Size-dependent nonlinear response of thin elastic films with nanoscale thickness, Mechanical Sciences. 2004;46:1715. Https://doi.org/10.1016/j.ijmecsci.2004.09.003. [19] Park, S. K., & Gao, X. L. Bernoulli–Euler beam model based on a modified couple stress theory. Micromechanics and Microengineering, 2006;16:2355–2359. Https://doi.org/10.1088/0960-1317/16/11/015 [20] Lu. P, He. L.H, Lee. HP, Lu. C, Thin plate theory including surface effects, Solids and Structures. 2006;4631. Https://doi.org/10.1016/j.ijsolstr.2005.07.036 [21] Kong, S., Zhou, S., Nie, Z., & Wang, K. The size-dependent natural frequency of Bernoulli–Euler micro-beams. Engineering Science, 2008;46:427–437. Https://doi.org/10.1016/j.ijengsci.2007.10.002 [22] Ma, H. M., Gao, X. L., & Reddy, J. N. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. The Mechanics and Physics of Solids, 2008;56: 3379–3391. Https://doi.org/10.1016/j.jmps.2008.09.007. [23] Lam DCC, Yang F, Chong ACM, Wang J, Tong P. Experiments and theory in strain gradient elasticity. The Mechanics and Physics of Solids 2003;51:1477–508. Https://doi.org/10.1016/S0022-5096(03)00053-X [24] Wang B, Zhou S, Zhao J, Chen X. A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. Mechanics - A/Solids.2011;30:517–24. Https://doi.org/10.1016/j.euromechsol.2011.04.001 [25] Tsiatas GC. A new Kirchhoff plate model based on a modified couple stress theory. Solids and Structures. 2009;46:2757–2764. Https://doi.org/10.1016/j.ijsolstr.2009.03.004 [26] Sh. Kong, Sh. Zhou, Z, Nie and K. Wang. Static and dynamic analysis of micro beams based on strain gradient elasticity theory, Engineering Science. 2008;47(4):487-498. Https://doi.org/10.1016/j.ijengsci.2008.08.008 [27] Civalek Ö, Demir C, Akgöz B. Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model. Math Comput Appl. 2010;15:289–98. Https://doi.org/10.3390/mca15020289 [28] Civalek Ö, Akgöz B. Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler–Bernoulli beam modeling, Scientia Iranica. Trans B – Mech Eng 2010;17:367–375.

21

[29] M. Hosseini, Bahaadini. R, Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory, Engineering Science,2016;101:1-13. Https://doi.org/10.1016/j.ijengsci.2015.12.012 [30] Shen HS. Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium. Physics Letters A. 2010;374:4030–9. Https://doi.org/10.1016/j.physleta.2010.08.006 [31] Xia W, Wang L, Yin L. Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Engineering Science. 2010;48:2044–2053. Https://doi.org/10.1016/j.ijengsci.2010.04.010 [32] Xia W, Wang L. Micro fluid-induced vibration and stability of structures modeled as microscale pipes conveying fluid based on non-classical Timoshenko beam theory. Microfluid Nanofluid. 2010;9:955–962. Https://doi.org/10.1007/s10404-0100618-z. [33] B. Akgöz and Ö Civalek, Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory. Materials & Design. 2012;42:164171. Https://doi.org/10.1016/j.matdes.2012.06.002. [34] Jomehzadeh E, Noori HR, Saidi AR. The size-dependent vibration analysis of microplates based on a modified couple stress theory. Physica E: Low-dimensional Systems and Nanostructures. 2011;43:877–83. Https://doi.org/10.1016/j.physe.2010.11.005 [35]. Kumar, C.S.S.R. Nanomaterials for Biosensors. Wiley, Weinheim .2007. [36]. Zabow, G., Dodd, S.J., Moreland, J., Koretsky, A.P. The fabrication of uniform cylindrical nanoshells and their use asspectrally tunable MRI contrast agents. Nanotechnology, 2009;20: 385301. Https://doi.org/10.1088/0957-4484/20/38/385301 [37]. Arefi, M. Rahimi, G.H. Three-dimensional multi-field equations of a functionally graded piezoelectric thick shell with variable thickness, curvature and arbitrary nonhomogeneity, Acta Mechanica 2012, 223 (1), 63-79. Doi:10.1007/s00707-011-05365. [38]. Zhu, J., Li, J.-J., Zhao, J.-W. Obtain quadruple intense plasmonic resonances from multilayered gold nanoshells by silvercoating: application in multiplex sensing. Plasmonics. 2013;8:1493. Https://doi.org/10.1007/s11468-013-9563-5. [39] Y.Tadi Beni, F. Mehralian, H. Razavi ,Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Composite Structures. 2014;09:065, https://doi.org/10.1016/j.compstruct.2014.09.065 [40] Khalili. S.M.R, Jafari. A.A, Eftekhari. S.A., A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads, Composite Structures, 2010; 92 (10): 2497-2511.https://doi.org/10.1016/j.compstruct.2010.02.012.

[41]. Li A, Zhou S, Zhou S, Wang B.A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory. Composite Structures. 2014;113:272-280. Doi: http://dx.doi.org/10.1016/j.compstruct.2014.03.028 [42]. Ashoori Movassagh A,Mahmoodi MJ. A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Mechanics - A/Solids. 2013;40:5059. Doi: http://dx.doi.org/10.1016/j.euromechsol.2012.12.008 [43]. Alibeigloo A, Shaban M. Free vibration analysis of carbon nanotubes by using threedimensional theory of elasticity. Acta Mechanics. 2013;224(7):1415-1427. Doi: http://dx.doi.org/10.1007/s00707-013-0817-2 [44]. Zeighampour H, tadibeni Y. Size-dependent vibration of fluid-conveying doublewalled carbon nanotubes using couple stress shell theory. Physica E: Low-dimensional Systems and Nanostructures. 2014;61:28-39. Doi: http://dx.doi.org/10.1016/j.physe.2014.03.011 [45] Sahmani S, Ansari R, Gholami R, Darvizeh A. Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory. Composites Part B: Engineering. 2013;51:44–53. Https://doi.org/10.1016/j.compositesb.2013.02.037 [46] Samaei AT, Gheshlaghi B, Wang GF. Frequency analysis of piezoelectric nanowires with surface effects. Current Applied Physics .2013;13: 2098-2102. Https://doi.org/10.1016/j.cap.2013.08.018. [47] Alijani F, Amabili M, Karagiozis K, Bakhtiari-Nejad F. Nonlinear vibration of functionally graded doubly curved shallow shells. Sound and Vibration. 2011;330:1432– 1454. Https://doi.org/10.1016/j.jsv.2010.10.003. [48] Chorfi SM, Houmat A. Nonlinear free vibration of a functionally graded doubly curved shallow shell of elliptical plan form. Composite Structures. 2012;92:2573–2581. Https://doi.org/10.1016/j.compstruct.2010.02.001 [49] Ansari R, Torabi J. Numerical study on the buckling and vibration of functionally graded carbon nanotube-reinforced composite conical shells under axial loading. Composites Part B: Engineering. 2016;95:196–208. Https://doi.org/10.1016/j.compositesb.2016.03.080 [50] Sofiyev. A.H, Aksogan. O. Non-linear free vibration analysis of laminated nonhomogeneous orthotropic cylindrical shells. Mechanical Engineers part K. 2003;217:293–300. Https://doi.org/10.1243/146441903322683040 [51] Sofiyev AH. Large-amplitude vibration of non-homogeneous orthotropic composite truncated conical shell. Composites Part B: Engineering. 2014;58:524–33. Https://doi.org/10.1016/j.compositesb.2013.06.040 [52] Mohammadimehr.M,Mohandes. M, Moradi. M, Size dependent effect on the buckling and vibration analysis of double-bonded nanocomposite piezoelectric plate reinforced by

23

boron nitride nanotube based on modified couple stress theory. Vibration and Control. 2016; 22, (7): https://doi.org/10.1177/1077546314544513. [53] Mosallaie Barzoki A.A, Ghorbanpour Arani A, Kolahchi R and Mozdianfard M.R. Electro-thermo-mechanical torsional buckling of a piezoelectric polymeric cylindricalshell reinforced by dwbnnts with an elastic core. Applied Mathematical Modelling. 2012;36:2983–2995. Https://doi.org/10.1016/j.apm.2011.09.093 [54] Liu C, Ke LL, Wang YS, Yang J and Kitipornchai S. Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory. Composite Structures. 2013;106:167–174. Https://doi.org/10.1016/j.compstruct.2013.05.031. [55] Hadi A, Mohammad Z, Rastgoo, A; Hosseini, M. Buckling analysis of FGM EulerBernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory. Steel and Composite Structures. 2018; 26(6):663-672. Https://doi.org/10.12989/scs.2018.26.6.663 [56]. Zeighampour, H, Tadi Beni, Y. A shear deformable cylindrical shell model based on couple stress theory. Archive of Applied Mechanics. 2015;85:539–553. Https://doi.org/10.1007/s00419-014-0929-8 [57] Tadi Beni. Y, Mehralian. F, Razavi. H. Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Composite Structures. 2015;120:65–78. Https://doi.org/10.1016/j.compstruct.2014.09.065. [58] Zenkour, A.M, “Bending of FGM plates by a simplified four-unknown shear and normal deformations theory”. International Journal of Applied Mechanics. 2013;5:1–15, DOI: 10.1142/S1758825113500208. [59] Rafiee. R. Analysis of Nonlinear Vibrations of a Carbon Nanotube Using Perturbation Technique. Modares Mechanical Engineering. 2012; 12:60-67. [60] Fereidoon. A, Rafiee. R, Maleki Moghadam. R. A. Modal Analysis of CarbonNanotube Reinforced Polymer by Using A Multiscale Finite-Element. Mechanics of Composite Materials, 2013; 49(3):325-332. Https://doi.org/10.1007/s11029-013-9350-6 [61] Jamal Omidi. M, shayanmehr. M, Shokrollahi. S, Rafiee. R. A study on nonlinear vibration behavior of CNT-based representative volume element. Aerospace Science and Technology. 2016; 55:272-281. Http://dx.doi.org/10.1016/j.ast.2016.06.005. [62] Ghasemi. H, Park. H, Rabczuk. T. A level-set based IGA formulation for topology optimization of flexoelectric materials. Comput. Methods Appl. Mech. Engrg.2016; 312(C): 322-350. Http://dx.doi.org/10.1016/j.cma.2016.09.029. [63] Hamid Ghasemi. H, Park. H.S, Rabczuk. T. A multi-material level set-based topology optimization of flexoelectric composites. Comput. Methods Appl. Mech. Engrg. 2018; 332: 47-62. Https://doi.org/10.1016/j.cma.2017.12.005

[64] Nguyen-Thanh. N, Valizadeh. N, Nguyen. M.N et al. An extended isogeometric thin shell analysis based on Kirchhoff–Love theory. Comput. Methods Appl. Mech. Engrg. 2015;284:265–291. Http://dx.doi.org/10.1016/j.cma.2014.08.025. [65] Chau-Dinh. T, Zi. G, Lee. P.S, et al. Phantom-node method for shell models with arbitrary cracks. Computers and Structures. 2012;92-93:242–256. Doi: 10.1016/j.compstruc.2011.10.021. [66] Areias. P, Rabczuk. T, Msekh. M.A. Phase-field analysis of finite-strain plates and shells including element subdivision. Comput. Methods Appl. Mech. Engrg. 2016;312(C): 322-350. Http://dx.doi.org/10.1016/j.cma.2016.01.020 [67] Rabczuk. T, Gracie. R, Song. J.H, Belytschko. T. Immersed particle method for fluid– structure interaction. International Journal for Numerical Methods in Engineering, 2010; 81(1): 48-71. DOI: 10.1002/nme.2670. [68] Nanthakumar. S.S, Lahmer. T, Zhuang. X, Zi. G, Rabczuk. T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering. 2016; 24(1): 153 – 176. Http://dx.doi.org/10.1080/17415977.2015.1017485. [69] Anitescu, C. Atroshchenko, E. Alajlan, N. Rabczuk, T. Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems, Computers, Materials & Continua, 2019; 59(1): 345-359. Doi:10.32604/cmc.2019.06641. [70] Guo, H. Zhuang, X. Rabczuk, T. A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate, Computers, Materials & Continua, 2019, 59 (2): 433-456. Doi:10.32604/cmc.2019.06660. [71] Rabczuk, T. Ren, H. Zhuang, X. A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem, Computers, Materials & Continua, 2019, 59(1): 31-55. Doi:10.32604/cmc.2019.04567. [72] Arefi, M. Zenkour, AM. Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation, Physica B: Condensed Matter 2017, 521, 188-197. Https://doi.org/10.1016/j.physb.2017.06.066. [73] Arefi, M. Zenkour, AM. Transient sinusoidal shear deformation formulation of a sizedependent three-layer piezo-magnetic curved nanobeam, Acta Mechanica 2017, 228 (10), 3657-3674. Https://doi.org/10.1007/s00707-017-1892-6. [74] Arefi, M. Zenkour, AM. Influence of magneto-electric environments on sizedependent bending results of three-layer piezomagnetic curved nanobeam based on sinusoidal shear deformation theory, Journal of Sandwich Structures & Materials 2019, 21 (8), 2751-2778. Https://doi.org/10.1177/1099636217723186. Appendix A:

25

∫

h /2

∫

h /2

− h /2

− h /2

σ ij dz = Nij , ∫

h /2

− h /2

( R + z)k z lσ ij dz = M ijk ,l , i, j = x,θ , z...k , l = ... − 2, −1,0,1, 2,...

( R + z)k zlσ ij dz = M ijk ,l ,

1 mij dz = Tijk , − h /2 ( R + z ) k

∫

h /2

∫

h /2

− h /2

( R + z )k

∂ψ 2 σ ij dz = Qijk , ∂z

h /2 h /2 ∂ψ 2 mij dz = Q2kij , ∫ ( R + z )kψ 2σ ij dz = N2kij , ∫ ( R + z )kψ 2 mij dz = T2kij , − h /2 − h /2 ∂z h /2 h /2 h /2 k ∂ψ k k ∂ψ k k k ∫−h/2 (R + z) ∂z1 σ ij dz = Pij , ∫−h/2 (R + z) ∂z1 mij dz = P2ij , ∫−h/2 (R + z) ψ1σ ij dz = N1ij , 2 h /2 h /2 k k k ∂ ψ1 ( R + z ) m dz = T , ( R + z ) mij dz = P3kij . ψ 1 ij 1ij ∫ ∫−h/2 − h /2 ∂z 2

∫

h /2

− h /2

( R + z)k

Appendix B: 1. Unknown constants in governing equations i i h /2 h /2 I ij = ∫− h /2 ρ z j dz , J ij = ∫− h /2 ρ ψ 1 z j dz ( R+ z ) ( R+ z ) i 2 i Lij =∫−hh/2/2 ρ ψ 2 z j dz , Kij =∫−hh/2/2 ρ (ψ 2 ) zj dz ( R+ z ) ( R+ z ) 2 i h /2 Sij = ∫− h /2 ρ (ψ1 ) zj dz ( R+ z ) h /2

A = kl ij

∫

cij

− h /2 h /2

Bijkl =

∫

cij

− h /2

∫

kl

∧ kl

∫

( R + z )l

dz ,

kl ∂ψ 2 ψ 2 z k ∂ψ 2 2 z k dz , E c ( ij = ∫ ij ∂z ) ( R + z )l dz, ∂z ( R + z )l − h /2

cij

kl (ψ 2 ) 2 z k ψ 1ψ 2 z k dz , B c ij = ∫ ij ( R + z )l dz, ( R + z )l − h /2

h /2

h /2

− h /2

ψ 2 zk

cij

− h /2

C ij =

∫

− h /2

cij

h /2

h /2

∫

(R + z)

h /2

dz , Cijkl = l

∂ψ 1 z k ∂ψ 2 zk kl dz , E = c ij ∫ ij ∂z ( R + z )l dz , ∂z ( R + z )l − h /2

− h /2

C ij =

ψ 1z k

i, j , k , l = 1, 2, 3,...

cij

h /2

Dijkl =

zk dz , ( R + z )l

h /2

∧ kl

B ij =

h /2

∫

− kl ∂ψ 2 ψ 1 z k (ψ 1 ) 2 z k dz , B = c ij ∫ ij ( R + z)l dz, ∂z ( R + z )l − h /2 h /2

cij

− h /2 h /2

h/ 2

kl zk zk F = ∫ eij sin( β z ) dz , F ij = ∫ eij cos( β z ) dz , ( R + z )l ( R + z )l − h /2 − h/ 2 kl ij

− kl ∂ψ 2 ψ 2 zk zk F ij = ∫ eij sin( β z ) dz , F = e sin( z ) dz , β ij ∫ ij ∂z ( R + z )l ( R + z )l − h /2 − h /2 ∧ kl

h /2

− kl

h /2

C ij =

∫

h /2

eij cos( β z )

− h /2 kl

G ij =

h /2

∫

eij sin( β z )

− h /2 kl

h /2

∫

H ij =

ψ 2zk ( R + z)

ψ 1z k (R + z)

λij (sin( β z )) 2

− h/ 2

N xkl = ∫

h/2

− h/2

h /2

( NTx + N Ex )

dz , Gijkl = l

∫

eij cos( β z )

− h /2 − kl

dz , H ij = l

h /2

∫

λij sin( β z )

− h /2

ψ 1z k ( R + z )l

dz ,

zk dz , ( R + z )l

h /2

zk zk kl 2 dz , H = λ (cos( β z )) dz ij ij l ∫ ( R + z )l ( R + z ) − h /2

h/2 zk zk kl dz , N = ( N + N ) θ ∫− h /2 Tθ Eθ ( R + z )l dz. ( R + z )l

a1 = − RA1100 − A1110 , a2 = −( 1 ) ( A1101 − A1201 ) − l 2 A 03 66 , 2 1 00 00 10 a3 =−( 1 ) ( A1100 + A1200 ) + l 2 A02 66 , a4 =− C12 − RE13 − E13 , 2 4 00 a5 =− A1200 , a6 =( RA1110 + A1120 ),a7 = A1111 −l 2 A02 66 , a8 =− A12 , 1 02 10 a9 =( RB1100 + B1110 ),a10 = B1101 − l 2 (A02 66 + D66 ), a11 =−( β F31 ), 2 1 1 2 02 1 2 02 1 2 01 1 2 00 a12 = l 2 A03 66 , a13 = − l A66 , a14 =− l C66 , a15 = l A66 , a16 =− l A66 . 4 4 2 4 4

27

1 1 1 2 03 01 b1 = − ( )( A1100 + A1200 ) + l 2 A 02 l A 66 , 66 , b2 = A55 + 2 4 4 1 1 1 2 01 00 10 00 10 b3 = − A1101 − l 2 A 03 l A 66 , 66 , b4 = − ( ) ( RA11 + A11 − RA12 − A12 ) − 4 2 2 1 02 1 2 03 12 − 1 l 2 A03 , b5 =−C1101 −C5501 − E1300 + l 2 E66 − l C66 ,, b6 =− A1101 − 2 A55 66 4 4 2 02

1 3 1 1 1 b7 = A1112 + l 2 A6603 ,, b8 = A1110 + l 2 A6601 ,b9 =− A1101 − 2 B5502 −C5501 + l 2 D66 − l 2 A6603 − l 2 D6603 , 2 2 4 4 4 01

1 1 3 3 1 b10 = B + l 2 A6603 + l 2 D6603 , b11 = B1100 + l 2 A6601 + l 2 D6601 , b12 = F 15 , b13 =− l 2 A6602 , 4 4 4 4 4 1 1 1 3 1 b14 = l 2 A6601 , b15 = − l 2 A6600 , b16 = l 2 A6600 , b17 = l 2C6601 , b18 = l 2C6603 , 4 4 4 4 4 02 11

1 1 C1 = ( RE1300 + E1310 + C1200 ), C2 = E1300 + C1101 + C5501 + l 2C6603 − l 2 E6602 , 4 4 00

00

10

∧ 01

∧ 00

∧ 10

00

C3 =2C13 + RE 33 + E33 +C11 ,C4 =− RC 55 −C 55 − R( N x )− 00

10

01

00

01

01

03

01

02

− ∧ − 12 1 ∧ l ( RE 66 + E 66 +C 66 −2C 66 ), C5 =−C 55 −( N θ ) − l 2 (C 66 + E 66 −2C 66 ), 4 4 00

10

03

01

∧ ∧ 1 1 ∧ 1 ∧ C6 = l 2 ( R C 66 + C 66 ),C7 = l 2 ( C 66 ), C8 = l 2 ( C 66 ), C9 = E1300 +C1101, 4 4 2 01

12 − N + 1 l 2 ( E 02 −C 03 ),C =− RE10 − E 20 −C10 − C10 =− E1311 −C1112 −2C55 θ 66 66 11 13 13 12 2 00

1 1 10 ), C = 1 l 2 (C 03 ), ( R N x )+ l 2 ( E6600 −C6601),C12 = l 2 ( R C6600 +C66 13 66 2 2 2 C14 =l 2 C6601 , C15 = E1300 +C1101,

∧ 01

02

02

∧ 01

01

C16 = − B13 − B11 − 2 B55 − C 55 − N θ − 02

03

01

02

00

10

00

∧ − ∧ ∧ 12 − l (− D66 +C6603 + D66 + A66 − E6602 − A66 ), C17 =− RB13 −B13 − B12 4 ∧ 10

∧ 00

00

00

01

00

− ∧ 1 −C 55 −RC55 −( RN x )− l 2 (− D66 +C6601 + D66 + R A66 + 4 10

00

00

10

∧ ∧ − 1 10 + RD + D ), A66 −E6600 − A66 ),,C18 = l 2 ( RC6600 +C66 66 66 4 03

01

00

10

00

∧ ∧ ∧ ∧ − 1 1 ,C19 = l 2 (C6603 + D66 ),C20 = l 2 (C6601+ D66 ),C21= Rβ F 33 + β F 33 +β F 31 , 4 2

−3 1 C22 =C15 ,C23 = RC15 +C15 ,C24 = l 2 (C6602 ), C25 = l 2 (C6601 ), 2 4 h /2 2V 1 C26 =− l 2 (C6603 ), C27 = ∫ (− β33∆T +ψ 2e31 e )dz. h 4 − h /2 − 01

− 00

− 10

d1 = A1200 , d 2 = − A1111 + l 2 A6602 , d3 = − RA1110 − A1120 , 1 1 12 d 4 = A1101 + 2 A55 + l 2 A6603 , d5 = − A1112 − l 2 A6603 , 2 2 3 d 6 = − A1110 − l 2 A6601 , d 7 = C1101 + E1300 , d8 = −C1112 − 2 01 1 12 2C55 − E1311 − N θ + l 2 ( E6602 − C6603 ), d9 = −C1210 − 2 00 1 E1320 − RE1310 − ( R N x ) + l 2 ( E6600 − C6601 ), 2 1 2 1 10 ), d11 = l 2 C6603 , d10 = l ( RC6600 + C66 2 2 2 01 01 d12 = l C66 , d13 = A11 , d14 = −2 A1112 − 4 A5523 − ( Nθ01 ) − l 2 A6603 , d15 = −2 A1210 − ( RN x00 ) − l 2 A6601 ,

29

10 d16 = RA1120 + A1130 + l 2 ( RA6600 + A66 ), d17 = A1123 + l 2 ( A6603 ),

d18 = 2 A1121 + 2l 2 ( A6601 ), d19 = A1101 , d 20 = − A1112 − B1102 − 02 1 13 12 4 B55 − 2C55 − Nθ01 + l 2 ( D 66 − A6603 − D6603 ), 2 00 1 d 21 = − B1200 − A1210 − ( RN x00 ) + l 2 ( D 66 − A6601 − D6601 ), 2 1 10 + RD00 + D10 ), d22 = RB1110 + B1120 + l 2 ( RA6600 + A66 66 66 2 1 d23 = B1113 + l 2 ( A6603 + D6603 ),d24 =2 B1111 +l 2 ( A6601 + D6601 ), 2 12

d25 =− Rβ F3110 −β F3120 , d26 =− β F3111 + 2 F 15 .

1 E1 = A1200 , E2 = − B1101 + l 2 ( A6602 + D6602 ), E3 = − RB1100 − B1110 , 2 02 1 E4 = A1101 + 2 B5502 + C5501 − l 2 ( D 66 − A6603 − D6603 ), E5 = − B1100 − 4 3 2 01 1 l ( A66 + D6601 ), E6 = − B1102 − l 2 ( A6603 + D6603 ), E7 = C1101 + E1300 , 4 4 02

∧ 01

∧ 01

01

00

∧ 10

E8 = − B11 − B13 − C 55 − 2 B5025 − N θ + C16 , E9 = − B12 − B13 − ∧ 00

∧ 00

∧ 10

00

R B13 − R C 55 − C 55 − ( R N x ) + C17 , ∧ 00 ∧ 10 1 10 E10 = d10 + l 2 ( RC6600 + C66 + R D 66 + D 66 ), 4 ∧ 03 ∧ 01 1 1 E11 = d11 + l 2 (C6603 + D 66 ), E12 = d12 + l 2 (C6601 + D 66 ), 4 2

12 13 E13 = A1101 , E14 = − B1102 − A1112 − 2C55 − 4 B55 − ( Nθ01 ) +

1 2 02 l ( D 66 − A6603 − D6603 ), E15 = − A1210 − B1200 − ( RN x00 ) − 2 00 1 2 01 1 10 l ( A66 + D6601 − D 66 ), E16 = RB1110 + B1120 + l 2 ( RA6600 + A66 + 2 2 1 10 RD6600 + D66 ), E17 = B1113 + l 2 ( A6603 + D6603 ), E18 = 2 B1111 + l 2 ( A6601 + D6601 ), 2 02

− 03

∧ 01

02

E19 = A1101 , E20 = −2 B1102 − 2 B55 − 4 B 55 − C 55 − 2 B 55 − Nθ01 − 01 ∧ 02 − 01 1 2 ∧ 01 l ( A66 + A6602 + E 66 + 2 D6602 − 2 D 66 − 2 E 66 ), 4 ∧

00

∧ ∧ ∧ 1 10 00 + A 10 + A00 + E 00 + E21 =−2 B1200 − RC 55 −C55 −( RN x00 + N x10 )− l 2 ( R A 66 66 66 66 4 00

10

− − − − 10 −2 RE 00 −2 E 10 ), E = RB + B + 1 l 2 ( RA 00 + − 2 D6600 −2 RD 00 2 D 11 11 66 66 66 66 22 66 4 00

10

03

03

∧ ∧ − 1 2 03 ∧ 00 10 03 A 10 66 + RE 66 + E 66 + 2 RD66 + 2 D 66 ), E23 = B11 + l ( A 66 + E 66 + 2 D66 ), 4 01

01

00

10

00

− ∧ − 1 01 + E + 2 D01 ), E =− R β G − β G + RC + E24 =2 B11 + l 2 ( A 66 66 31 31 15 66 25 2 − 10

01

− 01

C15 ,E26 =− β G31 +C15 +2G1502 , 01

f1 = R β F3100 + β F3110 , f 2 = β F3100 − F 15 , − 00

∧ 00

∧ 10

− 01

f3 = β F 31 + R β F 33 + β F 33 , f 4 = C 15 , − 00

− 10

f5 = R C 15 + C 15 , f 6 = − Rβ F3110 − β F3120 , 12

f 7 = − β F + 2 F 15 , f8 = β F3100 , f9 = β F3100 , 11 31

00

10

− 00

− 10

f10 = − β R G 31 − β G 31 + R C 15 + C 15 , 01

− 01

f11 = − β G 31 + 2G1500 + C 15 , 00

10

f12 = − β 2 R H 33 − β 2 H 33 , f13 = RH1100 + H1110 , f14 = H1101 , − 00 − 10 2Ve f15 = − β ( R H 33 + H 33 ) + h h /2

∫

P1∆T β sin( β z )( R + z )dz.

− h /2

31

Dear Professor Changgen FENG Editor in chief, Defence Technology I would like to express that there is no conflict of interest for this submission

Sincerely Yours M. Arefi Associate Professor of Mechanical Engineering Department of Mechanical Engineering Emails: [email protected], [email protected] Tel:+98-3155912405