Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory

Accepted Manuscript Fr ee vibr ation analysis of functionally gr aded piezoelectr ic cylindr ical nanoshell based on consistent couple str ess theor y Hamed Razavi, Asghar Faramarzi Babadi, Yaghoub Tadi Beni PII: DOI: Reference:

S0263-8223(16)31102-3 http://dx.doi.org/10.1016/j.compstruct.2016.10.056 COST 7882

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

2 July 2016 23 August 2016 18 October 2016

Please cite this article as: Razavi, H., Faramarzi Babadi, A., Tadi Beni, Y., Fr ee vibr ation analysis of functionally gr aded piezoelectr ic cylindr ical nanoshell based on consistent couple str ess theor y, Composite Structures (2016), doi: http://dx.doi.org/10.1016/j.compstruct.2016.10.056

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Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory Hamed Razavi1*, Asghar Faramarzi Babadi1, Yaghoub Tadi Beni2 1

Department of Mechanical Engineering, Golpayegan University of Technology, Golpayegan, Iran 2 Faculty of Engineering, Shahrekord University, Shahrekord, Iran * [email protected]

Abstract In this study, the governing equations of electromechanical vibration of cylindrical nanoshell made of functionally graded piezoelectric material (FGPM) are developed using the consistent couple stress theory and the cylindrical shell model. The new consistent couple stress has only one material length scale parameter and is able to consider size effects relating to the nanoshell structure in the nanoscale. The governing equations and boundary conditions are determined using the energy method and Hamilton’s principle. Afterwards, using Navier and Galerkin methods, the free vibration of a special model of FGPM nanoshell under different boundary conditions are investigated. Finally, the effect of different parameters such as dimensionless length scale parameter, variation of index, length-to-radius ratio, and radiusto-thickness ratio are investigated. It is demonstrated that the length-to-radius ratio, radius-tothickness ratio, and dimensionless length scale parameter play a significant role in the vibration behavior of the FGPM cylindrical nanoshell based on the size-dependent piezoelectric theory. Keywords: size-dependent piezoelectric, nanotube, thin-walled cylindrical shell, free vibration functionally graded piezoelectric material, Galerkin method 1. Introduction In mechanical systems, unwanted vibrations cause cracks in parts, loose joints, structural failure, malfunction of electronic devices, and many other problems. By measuring vibration features of the system such as its natural frequencies, it is possible to select speeds remote from resonance and to prevent many adverse effects of vibration. To solve these problems, piezoelectric materials were introduced to industry. Applying mechanical load to such materials generates an electric current in them (direct effect), and exposing them to an electric field creates mechanical strain in them (inverse effect). This two-way property has made piezoelectric materials ideal for making sensors (using direct effect) and actuators (using inverse effect) [1, 2]. Besides, the two-way action of turning mechanical energy to electric energy and vice versa has made piezoelectric materials useful in a wide range of applications such as air and vehicle transport systems (e.g. large space antenna control, active or non-active vibration control, etc.), and miniature positioning devices (e.g. micro-robots, medical devices, and micro-pumps). The pioneering work by Zheng [3] on piezoelectric nanomaterials (e.g. ZnO, ZnS, PZT, GaN, and BaTiO3) and their nanostructures (e.g. 1

nanowires, nanorings, nanobelts, nanosprings, etc.) attracted the attention of many researchers to this area. Due to their intrinsic piezoelectric properties at the nanoscale, piezoelectric nanostructures are considered the next generation of piezoelectric materials. Besides, FG piezoelectric materials were made to eliminate the deficiencies of homogeneous piezoelectric materials such as major bending displacements, high stress concentration, creeping in high temperature, and low resistance to external loads [4]. Fang et al. [5] examined the effect of increased piezoelectricity; electric, mechanical, and chemical properties; and the relation between piezoelectricity and semiconductors. These distinctive features make piezoelectric materials suitable for potential application in many nanodevices such as nanoresonators [6], light emitting diodes [7], chemical sensors [8], and nanogenerators [9]. Piezoelectric structures have different dimensions ranging from a few nanometers to hundreds of nanometers. At this scale, size effects become very important. Size-dependent properties of piezoelectric nanostructures has been observed in both empirical experiments and molecular dynamic simulation. By using strain gradient theory and incorporating rotation gradient effect, Wang et al. [10] developed a new ‘size-dependent piezoelectric theory’ to express size effects in piezoelectric solids. Studies based on the sizedependent piezoelectric theory can be effective in identifying size-dependent electromechanical properties of piezoelectric nanowires, nanofilms, and nanoshells. These properties are crucial for designing piezoelectric nanodevices. Voigt suggested the presence of couple stress in materials [11], and the Cosserat brothers introduced the first mathematical model to analysis materials with couple stress afterwards [12]. The couple stress theory was later developed for elastic materials by Toupin, Mindlin, and Tiersten, Koiter et al. [13, 14, 15]. In these theories, the gradient of the rotation vector is used as a curvature tensor. Due to the presence of body couple in constitutive equations and the consequent difficulty of use of the above theories, these inconsistent theories are called indefinite couple stress theories [16, 17, 18]. Other types of couple stress theories which were developed later with definite boundary conditions and energy conjugacy are in fact not consistent [19, 20]. The consistent couple stress theory has recently been developed by Hadjesfandiari and Dargush [16]. Based on previous research, Wang et al. developed the size-dependent piezoelectric theory where the rotation gradient effect is expressed based on couple stress theory [21]. It could be argued that the aforementioned size dependent piezoelectric theory is the dielectric polarization which is dependent on curvature strain and in fact, shows that there exists flexoelectric effect for dielectric materials in the isotropic case [22, 23]. In this respect, other theories have also been developed [24, 25], all of which suffer from lack on using of consistent second order gradients of deformation and negligence of couple stress effect [22]. Based on his previous work [16], Hadjesfandiari has recently developed the consistent size-dependent piezoelectric theory which is based on the electromechanical formulation and expresses the behavior of continuous materials in the small scale [22]. Another theory, being used by Ke and Wang, and Ke et al. is the nonlocal theory which has been utilized to model the properties of piezoelectric materials too [26, 27]. Liang et al. [28] investigated the nonlinear vibration of a piezoelectric nanobeam based on the nonlocal theory and Timoshenko beam model, demonstrating the effect of the nonlocal 2

parameter and the variation of temperature and external electric voltage on the size-dependent nonlinear vibration of piezoelectric nanobeams. Ghorbanpour et al. [29] developed a semianalytical method to analyze time-dependent thermo-electromechanical creep for radial polarization of piezoelectric cylinders. Ghorbanpour et al. [30] studied the heterogeneity effect of the material on the thermo-electromechanical behavior of an FG piezoelectric circular shaft rotating around its axis at a constant angular velocity. The results of that study demonstrated that heterogonous materials exert a considerable impact on the thermoelectromechanical behavior of the FG piezoelectric shaft rotating around its axis; hence, this must be considered in the design of the shafts. Yan et al. [31] investigated the flexoelectricity effect on the static bending and free vibration of a simply-supported piezoelectric nanobeam based on extended linear piezoelectric theory and Timoshenko beam model, demonstrating that flexoelectricity has a considerable effect on curved beam deflection. Furthermore, the simulation results demonstrated that the flexoelectricity effect on the vibration behavior of piezoelectric nanobeam is stronger in beams with less thickness. Ghorbanpour et al. [32] investigated the effect of time-dependent creep on the thermo-electromechanical behavior of a piezoelectric sphere using Mendelson’s method. Hosseini et al. [33] studied the free vibration of piezoelectric nanobeams by using the nonlocal theory and applying the effect of surface parameters and surface density. In a number of studies, Tadi et al. [34- 38] examined the vibration and buckling of carbon nanotubes based on modified couple stress theory and strain gradient theory under cylindrical shell and conical shell model. They investigated the effects of nanotube length, diameter, and thickness and length scale parameter on the natural frequencies and buckling loads of the nanotubes. Zhang et al. [39] introduced a fourunknown shear deformation theory and then further developed it for the shear deformable cylindrical microshell model using strain gradient elasticity theory. The results of their investigation demonstrated that the size effect is considerable when microshell thickness approaches the length scale parameter. Akgoz and Civalek [40] studied the bending response of an inhomogeneous microbeam discretized on a medium Pasternak foundation based on modified strain gradient elasticity theory in relation to different beam theories. Based on Murdock- Gartyn theory, Rouhi et al. developed a size-dependent continuum model to investigate the free vibration of a cylindrical shell in the nanoscale [41]. Akgoz and Civalek investigated the static bending, buckling, and free vibration of a sine microplate based on the shear deformation plate model and shear deformation plate theory. Their results indicated that when the microplate thickness approaches size scale parameter, microplate size dependency increases [42]. Akgoz and Civalek [43] developed shear beam models together with a new shear correction factor based on modified strain gradient elasticity theory and investigated bending and buckling parameters of a functionally graded microbeam subjected to fixed supported boundary conditions. Zeighampour et al. [44] studied the vibration of an axial FG nanobeam using strain gradient theory and Euler-Bernoulli beam model. Sahmani et al. [45] investigated the nonlinear buckling and postbuckling of a piezoelectric cylindrical nanoshell subjected to a compressive mechanical load in the presence of free surface energy effects. Using the modified strain gradient elasticity theory, Akgoz and Civalek developed a new microstructure-dependent sinusoidal beam model for buckling of microbeams using modified 3

strain gradient theory [46]. Fakhrabadi investigated the mechanical behavior of an active electrostatic carbon nanotube based on consistent couple stress theory and Euler-Bernoulli beam model. Comparison between the results of the classical theory and those of the couple stress theory demonstrated that the use of the consistent couple stress theory provides a model of carbon nanotubes with higher stiffness [47]. Akgoz and Civalek developed a new sizedependent beam model on the basis of hyperbolic shear deformation beam and modified strain gradient theory [48]. Shojaeian et al. [49] investigated the electromechanical buckling of beam-type nanoelectromechanical systems based on modified strain gradient theory. Shojaeian and Tadi [50] attempted to investigate the electromechanical buckling of beamtype nanoelectric systems by considering nonlinear geometric effect and intermolecular force (Casimir force and van der Walls force) based on modified couple stress theory. Tadi et al. [51] investigated the effect of the Casimir attraction on the torsion/bending coupled pull-in instability of an electrostatic nanoactuator for the first time. Tadi et al. [52] used modified couple stress theory to study the effect of intermolecular van der Waals force on the size dependent pull-in of nanobridges and nanocantilevers. Sedighi et al. [53] investigated the dynamic instability of functionally graded nano-bridges considering Casimir attraction and electric filed actuation. Mohammadi Dashtaki et al. [54] investigated the effects of heat and size on the buckling behavior of a nanobeam positioned symmetrically between two electrodes subjected to the influence of the nonlinear external forces including electrostatic and Casimir attractions based on modified couple stress theory and Euler-Bernoulli beam model. Akgoz and civalek [55] presented a new microstructure-dependent sinusoidal beam model for buckling of microbeams using modified strain gradient theory. In particular, Arani et al. conducted comprehensive research on the size-dependent behavior of boron nitride nanotubes using the nonlocal piezoelectric Timoshenko beam theory [56] and the shell theory [57]. Ke et al. [58] studied the size-dependent thermoelectromechanical vibrations of a cylindrical piezoelectric nanoshell under different boundary conditions using the nonlocal theory. They investigated the effect of increased temperature, external electric voltage, radius-to-thickness ratio, and length-to-external voltage ratio on the natural frequency of a piezoelectric nanoshell. Using the two-dimensional elastic theory and airy stress function Shi and Chen [59] developed a closed-form solution for an FG piezoelectric clamped beam with linear variation in mass density and piezoelectric constant. Lu et al. [60] proposed a precise solution for a simply supported FG piezoelectric plate based on Stroh-like formulation. Using finite element method under a set of mechanical, electric, and thermal loads, Behjat et al. [61] conducted comprehensive studies on the statics, dynamics, and natural frequency responses of FG piezoelectric panels. In the present paper, the free vibration of a piezoelectric nanoshell is presented for the first time based on Hajesfandiari’s size-dependent piezoelectric theory and Love’s thin-shell theory under two different boundary conditions, and the results are compared with those of the classical theory. Furthermore, the effects of FG piezoelectric properties, thickness, length, radius, and size scale parameter on the natural frequency of a piezoelectric nanoshell are investigated for the first time based on the aforementioned theory. Hamilton’s principle is used to determine and verify the governing equations and boundary conditions. Analytical solution is carried out using the Navier and Galerkin methods for simply supported and clamped-clamped piezoelectric nanoshells.

4

2. Preliminaries 2.1. Material properties of FG piezoelectric shell The material properties of the FG piezoelectric are a blend of two-manufactured substance, and it is assumed that variation in the joined substances is expressed through shell thickness according to its volume fraction. According to the power law, the volume fraction function is as follows [62]: zˆ V o = ( )N h

(1)

Vi =1−Vo

Where parameter N represents the power law index which is 0 ≤  ≤ ∞. N=0 represents the entire material 1, and  = ∞ represents the entire material 2. ̂ stands for the distance of an arbitrary surface of the shell from the neutral axis of the shell. The volume variation of the metal through thickness for different powers is demonstrated in Fig. (1). the properties of the FG piezoelectric cylindrical shell made up of two parts are as follows [62]: E (zˆ ) = E i + Eo i V o ( zˆ ), Eoi = Eo − E i v (zˆ ) = v i + v o i V o (zˆ ), v o i = v o − v i

ρ ( zˆ ) = ρi + ρo i Vo (zˆ ), ρoi = ρo − ρi

(2)

According to Eq. (2) and when ̂ is equal to zero, (̂ ) equals zero; hence (̂ ) =  , (  ̂ ) =   and (̂ ) =  . Similarly, when ̂ = ℎ, (̂ ) is equal to 1, and (̂ ) = ,  (̂ ) =  and (̂ ) =  , in which  ,   and  , and ,  and  stand for Young’s modulus, Poisson’s ratio, and density for the internal and external surfaces of the cylindrical shell. 2.2. Size-dependent piezoelectric theory

In order to explain the size effect on piezoelectric solids and linear electromechanical coupling in isotropic materials, Hajesfandiari developed the consistent size-dependent piezoelectric theory. This theory is based on couple stress theory [16] in which the antisymmetric part of the microscopic rotation gradient tensor is regarded as a deformation size. According to this theory, strain energy is expressed as follows [22]:

∫∀ (σ ji δ eij + µ ji δ kij − Di δ Ei ) dV = ∫s ti δ ui ds + ∫s mi + ∫ Fi δ u i dV + ∫ d δΦ d s − ∫ ρ e δΦ d V V s V (n)

()

(n)

δ Wi d s (3)

()

where  stands for force traction vector,  represents momentum traction vector, and  is force per unit volume.

Also,  ,  ,  ,  ,  , and  represent the Cauchy stress tensor, the couple stress tensor, the antisymmetric part of rotation gradient tensor, electric displacement field, quasistatic electric field, and electric charge density, respectively, and they are defined as:

σ ji = λ ekk δij + 2µ eij + 2µl 2∇2ω ji

(4)

µji = εi jk µk

(5)

5

k ij =

1 eijk ωk 2

(6)

Where λ and µ are Lame constants, l is the size scale parameter, and e , ω , and ∇" are the components of strain tensor, rotation gradient tensor, and Laplacian actuator, respectively, and they are defined as follows: 1 (( ∇ u ) + ( ∇ u )T ) 2 1 ω ij = eijk u k 2

ε ij =

(7) (8)

∇2ωij = ∇× (∇× ωij )

(9)

Also, #$ is the permutation tensor, %$ is the displacement vector, &$ is the rotation tensor, ' is the electric potential, ( is induced polarization of dielectric materials, and )* is the vacuum permittivity factor. The structural equations in linear elastic materials according to the size-dependent piezoelectric theory are expressed as follows:

σij = Cijkl εkl

(10)

µij = εijk µk , µi = −8µ (zˆ )l 2k i + 2fEi

(11)

In Eq. (11), + is the size scale parameter, (̂ ) is the FG shear modulus, and f is the flexoelectric coefficient. Using plane stress assumption, the stress for FG piezoelectric cylindrical thin shell is expressed as follow [62]:

(̂)

(̂)5(̂ ) 2 " " -11 − 5(̂ ) 1 − 5(̂ )

(̂ ) ,.. / = 1 (̂ )5(̂ ) 1 -. " " 11 − 5(̂ ) 1 − 5(̂ ) 0 0 0

0 8 7 #-7 , #.. / 0 7 2# -. 7 (̂ ) 6

(12)

It should be noted that since in the nanoscale microscopic stress and strains are not fixed and depend on the decrease in the length scale of the nanostructure, the deformation gradient is very intense [63]. It is also worth noting that determining the material length scale parameter is still a challenging issue and there are some ambiguities in this connection [64]. However, the length scale parameter of polymer materials is calculated at a few nanometers by molecular dynamic estimation and at more than a few millimeters by experimental measurement [65]. Besides, the experiment procedure can have a strong influence on the values measured. In general, different methods may be observed in the tension experiment, as compared with the bending or torsion experiments. Yet length scale parameters can be determined through molecular dynamics (MD) simulation [66] or empirical experiments [67]. Previous studies have attempted to determine the size effect parameters using MD simulation [65, 68, 69]. A detailed discussion on the methods of measuring the length scale is presented in Ref. [70]. 2.3. Displacement field for the cylindrical shell

6

Consider a cylindrical shell of radius R, length L, and thickness h in the coordinates shown in Fig. 1. The displacement components are demonstrated by u, v, and w in coordinates x, y and z according to Love’s thin shell theory and they can be written as [71]: ∂W ( x , θ , t ) ∂x z ∂W (x , θ , t ) v ( x ,θ , z , t ) = V ( x ,θ , t ) − ( −V ( x ,θ , t )) R ∂θ w ( x , θ , z , t ) =W ( x , θ , t ) u ( x , θ , z , t ) = U (x , θ , t ) − z

(13)

In Eq. (13), U(x, θ, t), V(x, θ, t), and W(x, θ, t) represent neutral plane displacements, the position of which is determined by considering the equilibrium equation along x axis under bending as follows [72]:

∫A

σ xx dA = ∫

E ( zˆ ) ∂2 w ( ) dA = 0 z A 1 − v2 ( zˆ ) ∂ x2

(14)

where (15)

z = zˆ − zˆc Substituting Eq. (15) into Eq. (14) yields the following results: E ( zˆ )

zˆc =

ˆ ∫A 1 − v2 ( zˆ) zdA

(16)

E ( zˆ )

∫A 1 − v2 ( zˆ ) dA

3. Governing equations of FG piezoelectric cylindrical shell

The equations of motion and classical and nonclassical boundary conditions are developed using displacement components in Eq. (13). To develop these equations for the shell, the components of the strain tensor and the antisymmetric part of the gradient tensor for the cylindrical shell which are presented in Appendix A must be determined. The nonzero components of strain which are developed by substituting Eq. (13) into Eq. (A.1) are as follows [34]:

ε xx =

∂V ∂2W −z ∂x ∂x 2

εθθ =

1 ∂V W z ∂2W + − 2 R ∂θ R R ∂θ 2 1 ∂ V 1 ∂V 2z ∂ 2 W ) + − 2 ∂x R ∂θ R ∂θ ∂x

(17)

εθx = ε xθ = (

The nonzero components k determined by substituting Eq. (13) into Eq. (A.2) and by applying Eq. (8) are as follows:

7

1 1 ∂V 1 ∂ 2W ∂ 2W ) kθ x = − ( 2 − 2 − 2 R ∂θ R ∂θ 2 ∂x 2 1 1 ∂ 2V ∂ 2V k zx = ( − ) 4 R ∂θ ∂x ∂x 2 1 1 ∂ 2V 1 ∂ 2V 1 ∂W ) k z θ = − (− 2 + + 2 R ∂θ ∂x R ∂x 4 R ∂θ

(18)

The strain energy of the FG piezoelectric cylindrical shell is determined by using the sizedependent piezoelectric theory and Love’s thin shell theory, by applying the simplifiable ": : : assumption (1 − ; ) ≃ 1, =1 + ;? ≃ 1and (;)" ≃ 0, and by substituting the classical and higher order strains and stresses into Eq. (3) as follows: U =∫

2π

L

∫0

0

−2 −

(

Yθ x ∂ 2 W Yzx ∂ 2 V Nθθ Y ∂W ∂2 W M W + zθ − (Mxx + Yθ x ) 2 − ( θθ2 + 2 ) + R R ∂x 2 ∂x2 ∂x R R ∂θ 2

Mθ x ∂ 2 W N Y ∂V ∂V ∂U Nθ x ∂U Yzθ ∂ 2 V + Nθ x + ( θθ + θ x2 ) + Nxx + + R ∂θ ∂x ∂x R R ∂θ ∂x R ∂θ 2 R ∂θ ∂x

h − zˆc Yzθ ∂ 2 U Yzx ∂ 2 U ∂Φ ∂Φ − + ∫ (−ε + 4 fkz ) dz 2 2 ˆ − z c 2 R ∂θ ∂x ∂z ∂z 2 R ∂θ

+∫

h − zˆc 1 −ε ∂Φ ∂Φ ∂Φ ∂Φ ( + 4 fkθ ) dz + ∫ (−ε + 4 fkx ) dz ) Rdθ dx − zˆc R R ∂θ ∂θ ∂x ∂x

h − zˆc

− zˆc

(19)

The classical and nonclassical forces and momentums are expressed as:

Nij = ∫

h− zˆc

σ ij dz

(20)

σ ij zdz

(21)

µij dz

(22)

− zˆc

M ij = ∫ Y ij = ∫

h− zˆc

− zˆc h − zˆc

− zˆc

The kinetic energy of the cylindrical shell is determined as follows [34]:

T =

1 L 2π h − zˆc ∂U ∂ 2W 2 ∂V z ∂ 2W 2 ∂W 2 ˆ z z ( ρ ( )( − ) +( − ) +( ) ) R dzdθ dx 2 ∫0 ∫0 ∫− zˆc ∂t ∂x ∂t ∂t R ∂θ ∂t ∂t

(23)

In Eq. (23),  (̂ ) represents the density of the FG cylindrical shell. The external load done on the cylindrical shell is expressed as follows [22, 34]: Wd =

∫x ∫θ (f x U

+ f θ V + f z W ) R dx d θ

(24)

∂U ∂V ∂W   W b = ∫ N xuU + Pxuh + N xv V + Pxvh + N xw + M xw  R dθ θ ∂x ∂x ∂x  ∂U ∂V ∂W  θ0  + ∫ N θuU + Pθuh + N θv V + Pxvh + N θwW + M θw dx x ∂θ ∂θ ∂θ  0

8

L 0

(25)

Welc. = ∫ d.Φ ds − ∫ ρe .Φ dV s

(26)

V

A-B ,  A-C ,  A-D ,  A.B ,  A.C ,  A.D , (E.BF , and (E.CF are the in which @- , @. , and @: are volume forces,  classical and nonclassical forces and momentums acting upon the lines or edges of (G = HIJKLJ) and (M = HIJKLJ).

The following parameters for the FG cylindrical shell are introduced as follows [73]:

( D1,i ) = ∫

h− zˆc

− zˆc

(D3,i ) = ∫

ˆ E(z) ( zi ) dz 2 1 − v ( zˆ) ˆ v( zˆ) i E (z) ( z ) dz 1 − v2 ( zˆ)

h− zˆc

− zˆc

(D 5,i ) = ∫

h −zˆc

− zˆc

(I 1,i ) = ∫

h − zˆc

− zˆc

µ (zˆ )(z i ) dz

i = 0,1,2

(27)

i = 0,1, 2

(28)

i = 0,1, 2

ˆ z i ) dz ρ (z)(

(29)

i = 0,1, 2

(30)

Hamilton’s principle is presented in the following form: t2

∫t

(δ T − δ U + δ W ) dt = 0

(31)

1

Afterwards, by substituting Eqs. (19), (23) and (24) into Eq. (31), and according to FG parameters (Eq. 27 – 30), and by performing lengthy calculations, the equations of motion and classical and nonclassical boundary conditions are developed for the FG cylindrical shell using the size-depended piezoelectric theory.

δW : −

D1,0 R2

W + (2

D 3,1 R

+

4 D 5,0 l 2 ∂ 2 W D1,1 ∂2W D1,2 4 D 5,0 l 2 ∂4W ) + 2 − ( + ) R2 ∂x 2 R 3 ∂θ 2 R4 R4 ∂θ 4

D 3,2 D 5,2 D 5,1 l 2 D 5,0 l 2 ∂ 4W D1,0 ∂V ∂ 4W −(D1,2 + 4 D 5,0 l ) − (2 + 4 − 4 + 8 ) − ∂x 4 R2 R2 R3 R 2 ∂x 2∂θ 2 R 2 ∂θ 2

D 5,0 l 2 ∂3V D 3,1 D 5,1 D 5,0 l 2 ∂3V D 5,1 l 2 ∂ 5V +( 3 + 4 ) +( +2 +6 ) +2 R R R ∂θ ∂x 4 R R 4 ∂θ 3 R 2 ∂θ ∂x 2 D1,1

D 5,1 l 2 ∂ 5V D 3,0 ∂U D 3,1 D 5,1 l 2 D 5,0 l 2 ∂ 3U ∂ 3U +2 − + D + ( + 2 − 2 ) 1,1 R ∂x R 3 ∂θ 3∂x 2 ∂x 3 R2 R2 R 3 ∂θ 2∂x D 5,1 l 2 ∂ 5U D 5,1 l 2 ∂ 5U f h − zˆc ∂3Φ ∂4W −2 − 2 + 2 dz + I 1,2 R 2 ∂θ 2∂x 3 R 4 ∂θ 4∂x R 2 ∫− zˆc ∂z ∂θ 2 ∂x 2∂t 2 +

f R

h −zˆc

∫−zˆ

c

∂2Φ dz + 2f ∂x 2

h − zˆc

∫−zˆ

c

I1,2 ∂ 4W I1,1 ∂ 3U ∂3Φ ∂ 2W + − − + fz = 0 dz I 1,0 ∂z ∂x 2 R 2 ∂θ 2∂t 2 R ∂θ ∂t 2 ∂t 2

9



(32)

D 1,0 ∂W D 1,1 4D 5,0 l 2 ∂3W D 3,1 D 5,1 D 5,0l 2 ∂3W δV : 2 −( 3 + ) −( +2 +4 ) R R R ∂θ R R4 ∂θ 3 R 2 ∂θ ∂x 2 4D 5,0l 2 ∂2V D 3,0 D 5,0 ∂2U ∂2V +( 2 + ) + D + ( + ) 5,0 R R ∂θ ∂x R R4 ∂θ 2 ∂x 2 D 1,0

I 2f h − zˆc ∂2Φ ∂3W ∂2V + 2∫ dz + 1,1 − I + fθ = 0 1,0 R ∂θ ∂t 2 R − zˆc ∂θ ∂z ∂t 2

(33)

D 3,0 ∂W D 3,1 D 5,1 D 5,0l 2 ∂3W ∂3W δU : − D 1,1 3 − ( 2 + 2 2 − 4 ) R ∂x ∂x R R R3 ∂x ∂θ 2 +(

+

D 3,0 R

D 5,0 ∂2V 2D 5,0l 2 ∂4V D 5,0l 2 ∂4V ∂2 U + ) + + 2 + D 1,0 R ∂x ∂θ R ∂θ ∂x 3 R 3 ∂θ 3∂x ∂x 2

D 5,0l 2 ∂4 U D 5,0l 2 ∂4 U ∂3W ∂2 u − 2 − 2 + I − I +fx =0 1,1 1,0 R 2 ∂θ 2 R 4 ∂θ 4 R 2 ∂θ 2 ∂x 2 ∂x ∂t 2 ∂t 2

D 5,0 ∂2U

δΦ = ∫

h −zˆc

−zˆc

(ε (

∂2Φ

1 ∂2Φ ∂2Φ ∂2W + + ) − )dz = 0 f ∂z 2 R ∂θ 2 ∂x 2 ∂x 2

(34)

(35)

Boundary conditions for an edge with (x=constant) are applied as follows:

∫θ

(

D 3,0 R

+ D 1,0

∫θ

W − D 1,1

D 3,1 ∂2W D 3,0 ∂V D 5,0 l 2 ∂3V D 5,0 l 2 ∂3V − + + − R ∂θ R ∂x 2 R 2 ∂θ 2 ∂θ ∂x 2 R 2 ∂θ 2 ∂x

∂2W

h − zˆc f ∂2Φ ∂U −∫ dz − N U x ) d θ x = 0, L = 0 or δU x = 0, L = 0 2 2 ˆ − z ∂x c R ∂θ

(−2

D 5,1 ∂2W ∂V D 5,0 ∂U + D 5,0 + − N Vx ) dθ x =0,L = 0 or δ V x =0,L = 0 R ∂θ ∂x R ∂x ∂θ

2 D 5,0l 2 ∂2U 2∂ V (D 5,0l − 2 θ R

∫

or

∂x

∂θ ∂x

(36)

(37)

h − zˆc f ∂φ dz − PxV h ) d θ x = 0,L = 0 − zˆc R ∂θ

−∫

∂δ V =0 ∂x x = 0,L

(38)

10

3 D 5,0 l 2 ∂W D 3,2 D 5,0 l 2 D 5,2 2 ∂ W (( + 4 ) − ( D + 4 D l ) − ( + 4 + 4 1,2 5,0 ∫θ R ∂x R2 ∂x3 R2 R2 R2

D 3,1

D 5,1 l 2

−4

R3

∂θ 2 ∂x

+(

D 3,1 R

D 5,1 l 2 ∂ 4 V

+2

∂4 U

R2

∂θ 2 ∂x2

W + D 1,1

∂φ

∫θ ( ∂z

D 5,1 ∂ 2 V D 5,1l 2 ∂ 4 V +2 ) +2 R ∂θ ∂x R ∂θ ∂x3

∂U ∂x

D 5,1

h − zˆc 2 f ∂φ ∂ 2φ ( +2f ) dz − N xw ) x = 0, L = 0 or − zˆc R ∂x ∂z ∂x

+∫

((−D 1,2 + 4D 5,0l 2 )

R

R2

∂ 2U

D 5,1 l 2

D 3,1

+6

D 5,0 l 2

D 5,0 l 2 ∂ 2U D 5,1l 2 ∂ 4U + D1,1 + (2 −2 ) −2 ∂θ 3 ∂x ∂x 2 R2 R3 ∂θ 2 R 4 ∂θ 4

R3

−2

∫θ

)

∂3 W

∂ 2W ∂x 2

−∫

h − zˆc

− zˆc

+ (−

2f

D 3,2 R2

+4

x = 0, L

=0

(39)

D 5,0l 2 ∂ 2W D 3,1 D 5,0l 2 ∂V ) +( −4 ) R R2 ∂θ 2 R2 ∂θ

∂φ dz − M W x ) dθ x = 0, L = 0 ∂z

1 ∂ φ ∂φ h − zˆ + ) d θ x = 0, L − zˆ c = 0 or c R ∂θ ∂ x

+

δW

δφ

or

∂δ W =0 ∂x x = 0, L

h − zˆc x = 0, L − zˆc = 0

(40)

(41)

Boundary conditions for an edge with (θ=constant) are applied as follows:

∫x

((−2

D 5,1 R

+4

3 D 5,0 l 2 ∂2W D 5,0 l 2 ∂3V ∂V 2∂ V ) + D + 2 D l + 2 5,0 5,0 ∂θ ∂x ∂x R2 ∂x 3 R 2 ∂θ 2∂x

D 5,0 ∂U D 5,0 l 2 ∂3U D 5,0 l 2 ∂3U θ θ + −2 −2 − N θU ) dx 0 0 = 0 or δ U 0 0 = 0 3 3 2 R ∂θ R R ∂θ ∂θ ∂x

(42)

D 5,0 l 2 ∂W D 5,0 l 2 ∂ 2V D 5,0 l 2 ∂ 2U h − zˆc f ∂φ θ Uh ∫x (2 R 2 ∂x + R 2 ∂θ ∂x − R 3 ∂θ 2 + ∫− zˆc R ∂θ + Pθ ) dx 00 = 0 θ

or

∂δ U 0 =0 ∂θ 0

(43)

D 5,0 l 2 ∂2W D 1,1 D 5,0 l 2 ∂2W ∫x ( R W − (D 3,1 + 6 R ) ∂x 2 − ( R 2 + 4 R 3 ) ∂θ 2 D 1,0

+(

D 1,0 R

+4

2 D 5,0 l 2 ∂V ∂U D 5,0 l ∂3U ) + D 3,0 + ∂θ ∂x R3 R2 ∂θ 2 ∂x

h − zˆc 2f ∂φ ∂2φ θ ( −f ) dz − N θV dx 00 = 0 or 2 − zˆc R ∂z

+∫

∂x

11

δV

θ0 0

=0

(44)

D 5,0 l 2 ∂3W D 3,2 D 5,2 D 5,1 l 2 ∫x ( R 2 ∂θ − ( R 3 + 4 R 3 ) ∂θ 3 − ( R + 4 R − 4 R 2 D 1,1 ∂W

+4

+2

D 5,0 l 2 R

D 1,2

4 D 5,0 l 2 ∂2V 2∂ V ) + 2 D + ( + 4 ) + 2 D l 5,1 5,1 ∂θ ∂x 2 ∂x 2 R2 R3 ∂θ 2 ∂x 4

∂3W

D 5,1 l 2

∂2V

D 1,1

D 3,1 ∂2 U D 5,1 l 2 ∂4 U D 5,1 l 2 ∂4 U + −2 −2 R ∂θ ∂x R R 3 ∂θ 3 ∂x ∂θ 2∂x 2 ∂θ ∂x 3 ∂4V

R2

h − zˆc 2f ∂2φ θ0 dz − N W θ ) dx 0 = 0 or − zˆc R ∂z ∂θ

+∫

∫x +(

(

δW

θ0 0

=0

(45)

D 3,2 D 5,0 l 2 ∂ 2W D 1,2 D 5,0 l 2 ∂2W W − ( + 4 ) − ( + 4 ) R R R2 ∂x 2 R3 R3 ∂θ 2

D 1,1

D 1,1

+4 R2

D 5,0 l 2 ∂V D 3,1 ∂U h − zˆc 2f ∂φ θ0 ) + + dz − M W θ ) dx 0 = 0 ∫ 3 ˆ − z ∂θ R ∂x R ∂z c R

θ

∂δ W 0 or =0 ∂θ 0 ∂φ

∫x ( ∂z

+

1 ∂φ ∂φ θ h − zˆ − ) dx 00 − zˆ c = 0 or c R ∂θ ∂x

(46)

θ

h − zˆ

∂ φ 00 − zˆ c = 0 c

(47)

4. Analyzing the vibration of FG piezoelectric cylindrical shell with simply-supported and clamped support 4.1. Boundary conditions of FG piezoelectric cylindrical shell with simply-supported and clamped-clamped support

In this section, according to the newly-developed equations, the free vibration of an FG piezoelectric cylindrical shell with simple and clamped support is investigated as a special case. The boundary conditions of the simply-supported cylindrical shell are set at the two ends. The boundary conditions exist according to Eqs. (42-47), and the boundary conditions in Eqs. (36-41) are satisfied. Hence, the essential and natural boundary conditions, without neutral momentum and higher order stresses in the classical and nonclassical cases, respectively, corresponding to the simply-supported cylindrical shell according to Eqs. (36-41) are as follows: W x = 0, L = 0

(48)

V x = 0,L = 0

(49)

Φ x = 0, L = 0

(50)

12

D 3,0 ∂V D 5,0 l 2 ∂3V D 5,0l 2 ∂3V ( W − D 1,1 − + + − R R ∂θ R ∂θ ∂x 2 ∂x 2 R 2 ∂θ 2 R 2 ∂θ 2 ∂x ∂ 2W

D 3,0

+ D 1,0

D 3,1 ∂ 2W

h − zˆc f ∂2φ ∂U −∫ dz − N U =0 x 2 2 ˆ − z ∂x x = 0, L c R ∂θ

2 D 5,0 l 2 ∂2U h −zˆc f ∂φ 2∂V (D 5,0 l − − =0 dz − PxVh 2 ˆ − z x =0, L R ∂ ∂ x R ∂ θ θ c ∂x

∫

{(

R

+2

−2

D 5,1 l 2 R3

)

∂3W ∂θ 2 ∂x

+(

D 3,1 R

D 5,1l 2 ∂4V D 5,1l 2 2

+6

D 5,0 l 2 R2

D 5,1 ∂2 V D 5,1l 2 ∂4V +2 ) +2 R ∂θ ∂x R ∂θ ∂x 3

D 5,0 l 2 ∂2U D 5,1l 2 ∂4U + D 1,1 + (2 −2 ) −2 ∂θ 3 ∂x ∂x 2 R2 R3 ∂θ 2 R 4 ∂θ 4

R3

R

(52)

3 D 5,0 l 2 ∂W D 3,2 D 5,0 l 2 D 5,2 2 ∂W +4 ) − (D 1,2 + 4D 5,0 l ) −( +4 +4 ∂x R2 R2 R2 R2 ∂x 3

D 3,1

−4

(51)

∂2U

∂4U 2

∂θ ∂x

2

+∫

h − zˆc

−zˆc

(

D 5,1

2f ∂φ ∂2φ + 2f ) dz − N Vx } =0 x = 0, L R ∂x ∂z ∂x

(53)

2 D 3,2 D 5,0 l 2 ∂ 2W 2 ∂ W ( W + ( − D 1, 2 + 4 D 5,0 l ) + (− +4 ) R ∂x 2 R2 R2 ∂θ 2

D 3,1

+(

D 3,1 R

−4

D 5,0 l 2 ∂V h − zˆc ∂U ∂φ ) + D 1,1 −∫ 2f =0 dz − M W x 2 ˆ − z θ ∂ ∂ x ∂ z x = 0, L c R

(54)

And the boundary conditions for the clamped-clamped support conditions are as follows: ∂W W x = 0, L = V x = 0, L = U x = 0, L = = Φ x = 0, L = 0 ∂x x = 0, L

(55)

Therefore, the governing equations of motions and boundary conditions for the FG piezoelectric cylindrical shell with simple and clamped-clamped support conditions are presented based on the size-dependent piezoelectric theory in Eqs. (32-35) and Eqs. (48-54). by setting + to zero (+ = 0), the equations of cylindrical thin shell in the classical theory can be developed. 4.2. Solution method

Free vibration analysis was carried out using the Navier procedure for the FG piezoelectric cylindrical shell with simple support and using the Galerkin method for the one with clamped-clamped support.

13

4.2.1. Navier procedure

In Navier procedure, the displacement and potential functions as a result of known and unknown functions and trigonometric functions for satisfying simply-supported boundary conditions are expressed as: mπ x ) cos ( nθ ) L mπ x V (x , θ , t ) = ∑ m ∑ nV mn (t ) sin ( ) sin (nθ ) L mπ x W (x ,θ , t ) = ∑ m ∑ nW mn (t ) sin ( ) cos (nθ ) L mπ x Φ (x ,θ , t ) = ∑ m ∑ n Φ mn (t ) sin ( ) cos ( nθ ) L

U (x ,θ , t ) = ∑ m ∑ n U mn (t ) cos (

(56)

In the above equations, m and n are axial and circumferential wavenumbers, and OP (), P (), QP (), and RP are the unknown functions of t. Hence, the displacement equations in the Navier procedure satisfy the essential and natural boundary conditions presented in Eqs. (48-54). 4.2.2. Galerkin solution

In the Galerkin method, like in the Navier method, the displacement and potential functions as a result of known and unknown functions and trigonometric functions for satisfying clamped-clamped boundary conditions are expressed as follows [74, 75]:

U( x,θ ,t ) = U mn ( t )( − sin(

ε L

ε L

( − x )))cos(n θ ) R 2 R 2 ε L ε L V ( x,θ ,t ) = Vmn ( t )( − cos( ( − x )) + k cosh( ( − x )))sin(n θ ) R 2 R 2 ε L ε L W( x,θ ,t ) = Wmn ( t )( − cos( ( − x )) + k cosh( ( − x )))cos(n θ ) R 2 R 2 ε L ε L Φ ( x,θ ,t ) = Φmn ( t )( − cos( ( − x )) + k cosh( ( − x )))cos(n θ ) R 2 R 2 (

− x )) + k sinh(

(57)

in which k is determined through the following equation: sin (

k=

ε .L

) 2R ε .L sin h( ) 2R

(58)

and # is determined by solving the following two equations: tan (

ε .L 2R

)+− tanh (

ε .L 2R

)=0

(59)

By substituting Eqs. (56) and (57) for simply-supported and clamped-clamped boundary conditions into Eqs. (32-35), the equations of motion are expressed as: ..

(k ){d} + (M ){d} = 0

(60) 14

where {d } = {d0 }eiω t

(61)

If Eq. (61) is substituted into Eq. (62), then: (( k ) − ω 2 ( m ))(d0 ) = 0

(62)

In the above equations, & and {d*}V = {U* , V*, W*, Φ* }V are the frequency and unknown displacement amplitude vector, respectively. This small and significant solution is derived by setting the matrix coefficients to zero. 5. Results and discussion

In this section, the free vibration of an FG piezoelectric nanoshell with simple and clamped boundary conditions is investigated using the size-dependent piezoelectric theory. Afterwards, the equations derived as well as Navier and Galerkin solution procedures mentioned in Section 4 are used in this section. The results obtained by developing the new formulation are validated. Afterwards, the effect of dimensionless length scale parameter h/l, gradient index N, length-to-radius ratio L/R, and axial and circumferential wavenumbers n and m on the free vibration of the FG piezoelectric nanobeam are examined. 5.1.

Validation of results

Considering that so far no study has been done on FG piezoelectric nanoshells using the size-dependent piezoelectric theory and the shell model, the validation method is carried out for the structure of FGMs, which is equal to a piezoelectric isotropic homogeneous shell, by setting the power law index to zero. Therefore, in order to verify the results and their precision, the size-dependent free vibration of the piezoelectric cylindrical nanoshell [58] is reinvestigated. The properties of the material used in the comparison are displayed in Table 1 and the results are presented in Table 2. The dimensionless natural frequency is determined through the equation \ = ω. R. _(1 − υ" ). ρ/E . This can be seen in Table 2 in which the results are very close to one another in the classical case. Besides, in the engineering literature, the size-dependent piezoelectric theory has been used more frequently than the classical continuum theory to predict natural frequencies. According to what was mentioned previously, by setting + = 0, the governing equations and boundary conditions of the rotating cylindrical shell can be determined for the classical theory. 5.2. Effect of size parameter on natural frequencies

In this section, the material properties components of the FG piezoelectric nanoshell are displayed in Table 1 and then are evaluated. The effect of the dimensionless length scale parameter and the variation of coefficient N on natural frequencies according to the classical continuum theory and size-dependent piezoelectric theory for L=5R are presented in Fig.2. as illustrated in Fig.2 (a, b), in the classical continuum theory, the dimensionless length scale parameter has no effect on the vibration of the FG piezoelectric nanoshell for different N’s in simple and clamped support conditions. In contrast, considering the incorporation of size effects in the size-dependent piezoelectric theory, decrease in h/l, which is decrease in the length scale parameter, leads to increase in natural frequency (Fig.2-c,d). For instance, in 15

simple support conditions, when the material length scale parameter changes from += h (ℎ/ + =1) to + = 100h (h/ + = 0.05), the dimensionless natural frequency changes from 0.379 to 0.509 for  = ∞ in shells made entirely of Lithium -Niobate, and from 0.183 to 0.245 for N=0 in shells made entirely of Barium- Titanate. Similarly, in clamped boundary conditions, the dimensionless natural frequency changes from 0.509 to 0.901 for  = ∞, and from 0.257 to 0.456 for N=0. Besides, the classical continuum theory and the size-dependent piezoelectric theory generated similar results for ℎ/+ > 1. Therefore, as illustrated, increased slope index results in a wider gap between the dimensionless natural frequencies predicted by the size dependent piezoelectric theory and those predicted by the classical continuum theory. Besides, the results demonstrate that for all h/l ratios and N’s, the size-dependent piezoelectric theory yields higher dimensionless frequencies compared to the classical continuum theory. 5.3. Effect of dimensionless shell length on natural frequencies

Dimensionless frequency variation due to variation in shell length-to-radius L/R based on the size-dependent piezoelectric theory and classical continuum theory is presented in Fig.3 (a, b, c, d). As the length-to-radius ratio L/R increases, the dimensionless natural frequency decreases and instability increases. This is due to increased deflection in length increase. According to Fig.3 (a, b, c, d), this happens in both the size-dependent piezoelectric theory and the classical continuum theory and in both simple and clamped support conditions. For instance, in simply-supported condition, for N=1, the dimensionless natural frequency changes from 0.017 to 0.287 for a length increase from L=5R to L=50R based on the classical continuum theory, whereas it changes from 0.016 to 0.228 based on the sizedependent piezoelectric theory. In clamped-clamped boundary conditions, for N=1, the dimensionless natural frequency changes from 0.019 to 0.387 for a length increase from L=5R to L=50R based on the classical continuum theory, whereas it changes from 0.671 to 0.722 based on the size-dependent piezoelectric theory. 5.4. Effect of radius-to-thickness ratio of piezoelectric shell on dimensionless natural frequency

Fig.4 (a, b, c, d) illustrates the effect of radius-to-thickness ratio of piezoelectric shell on the dimensionless natural frequency. Based on both the classical continuum theory and the size-dependent piezoelectric theory, in simply-supported and clamped-clamped conditions, for all N’s, the dimensionless natural frequency decreases. This is because higher thicknessto-radius ratios indicate that the shell is thinner and less stiff. It should be noted that Love’s shell theory has been used in the present study to analyze the vibration behavior of the piezoelectric nanoshell. This theory is useful only for thin shells. If the thickness-to-radius ratio is equal to 10 (for isotropic cases) the shell can be considered a thin shell. Therefore, the ; radius-to-thickness 10 ≤ ≤ 100 has been used for this example. F

6. Conclusion

In this paper, a new formulation was developed for FG piezoelectric cylindrical shells based on the size-dependent piezoelectric theory and the cylindrical shell model. Equations of motion and boundary conditions were developed based on Hamilton’s principle. This formulation which is developed for FG piezoelectric cylindrical shells can be reduced to the classical continuum theory in special cases. To demonstrate the results of the formulation, equations of motion and boundary conditions were developed for the free vibration of the 16

simply-supported and clamped-clamped boundary conditions FG piezoelectric cylindrical shell. In order to validate the results and their precision, the dimensionless natural frequency of a homogeneous piezoelectric nanoshell was reviewed based on the classical continuum theory. The convergence of results indicates the precision of the formulation. The effect of different parameters such as dimensionless length scale h/l, variation of index N, length-toradius ratio L/R, and radius-to-thickness ratio R/h were investigated for both the sizedependent piezoelectric theory and the classical continuum theory. It was demonstrated that the dimensionless length scale parameter h/l plays no significant role in the vibration behavior of the shell based on the classical continuum theory. However, due to the effect of size in the size-dependent piezoelectric theory, the dimensionless length scale parameter affects the vibration behavior of the FG piezoelectric nanoshell in this theory. The results also demonstrate that the length-to-radius ratio L/R and the radius-to-thickness ratio R/h play a significant role in the vibration behavior of the FG piezoelectric cylindrical nanoshell based on the size-dependent piezoelectric theory. Also, results of free vibrations indicate that the vibrational frequency increase with the increase in size parameter, and the results of the couple stress theory are higher than those of the classic theory. Finally, can be conclude that the frequency tuning is more efficient for piezoelectric nanoshells. Such a frequency tuning concept is expected to provide helpful guidelines for the design and application of piezoelectric nanoshells in nanodevices.

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Figures List

Fig. 1: Coordinates and displacements of a cylindrical shell Fig. 2-a: The effect of the dimensionless length scale parameter h/l on the dimensionless Frequencies of the FGPM Nano shell with different N, Simply support and Classical continuum theory (R=40h, L = 5R, m=1, n=1). Fig. 2-b: The effect of the dimensionless length scale parameter h/l on the dimensionless Frequencies of the FGPM Nano shell with different N, clamped-clamped and Classical continuum theory (R=40h, L = 5R, m=1, n=1). Fig. 2-c: The effect of the dimensionless length scale parameter h/l on the dimensionless Frequencies of the FGPM Nano shell with different N, Simply support and size depended piezoelectricity theory (R=40h, L = 5R, m=1, n=1). Fig. 2-d: The effect of the dimensionless length scale parameter h/l on the dimensionless Frequencies of the FGPM Nano shell with different N, clamped-clamped and size depended piezoelectricity theory (R=40h, L = 5R, m=1, n=1). Fig. 3-a: The effect of the length to radius ratio L/R on the dimensionless Frequencies of the FGPM Nano shell with different N, Simply support and Classical continuum theory (R=20h, m=1, n=1). Fig. 3-b: The effect of the length to radius ratio L/R on the dimensionless Frequencies of the FGPM Nano shell with different N, clamped-clamped and Classical continuum theory (R=20h, m=1, n=1). Fig. 3-c: The effect of the length to radius ratio L/R on the dimensionless Frequencies of the FGPM Nano shell with different N, Simply support and size depended piezoelectricity theory (R=20h, m=1, n=1). Fig. 3-d: The effect of the length to radius ratio L/R on the dimensionless Frequencies of the FGPM Nano shell with different N, clamped-clamped and size depended piezoelectricity theory (R=20h, m=1, n=1). Fig. 4-a: The effect of the radius-to-thickness ratio R/h on the dimensionless Frequencies of the FGPM Nano shell with different N, Simply support and Classical continuum theory (L=20R, m=1, n=2).

22

Fig. 4-b: The effect of the radius-to-thickness ratio R/h on the dimensionless Frequencies of the FGPM Nano shell with different N, clamped-clamped and Classical continuum theory (L=20R, m=1, n=2). Fig. 4-c: The effect of the radius-to-thickness ratio R/h on the dimensionless Frequencies of the FGPM Nano shell with different N, Simply support and size depended piezoelectricity theory (L=20R, m=1, n=2). Fig. 4-d: The effect of the radius-to-thickness ratio R/h on the dimensionless Frequencies of the FGPM Nano shell with different N, clamped-clamped and size depended piezoelectricity theory (L=20R, m=1, n=2).

Tables List

Table 1: Material properties of PZT-4, LiNbO3 and BiTiO3. ρ Table 2-a: Comparison of dimensionless frequencies (\ = &. e. f(1 − 5 ") g ) S-S

homogeneous macroscopic cylindrical shells with m = 1, L/R = 20, h/R = 0.01 and 5 = 0. 3.

ρ Table 2-b: Comparison of dimensionless frequencies (\ = &. e. f(1 − 5 " ) g ) C-C homogeneous macroscopic cylindrical shells with m = 1, L/R = 20, h/R = 0.01 and 5 = 0. 3.

23

Fig. 1: Coordinates and displacements of a cylindrical shell

24

(a)

(b)

25

(c)

(d) Fig. 2-a, b, c, d: Effect of the dimensionless length scale parameter h/l on the dimensionless Frequencies of the FGPM Nano shell with different N, S-S and C-C. a, b) Classical continuum theory; c, d) Size-dependent piezoelectricity theory (R=40h, L = 5R, m=1, n=1)

26

(a)

(b)

27

(c)

(d) Fig. 3-a, b, c, d: Effect of the length to radius ratio L/R on the dimensionless frequencies of the FGPM Nano shell with different N, S-S and C-C. a, b) Classical continuum theory; c, d) Sizedependent piezoelectricity theory (R=20h, m=1, n=1)

28

(a)

(b)

29

(c)

(d) Fig. 4-a, b, c, d: The effect of the radius-to-thickness ratio R/h on the dimensionless frequencies of FGPM cylindrical Nano shell with different N, S-S and C-C. a, b) Classical continuum theory; c, d) Size-dependent piezoelectricity theory (L=20R, m=1, n=2)

30

BiTiO3. [58, 76, 77] and Table 1: Material properties of PZT-4, LiNbO3 Mater ial

Density ijkglm n

Young's modulus (GPa)

Poisson's ratio

PZT4

7600

78

0.31

4640

170

0.25

6020

42.9

0.325

opq rpsp

Table2-a: Comparison of dimensionless frequencies (Ω = ω. R. _(1 − υ") ρ/E) S-S homogeneous macroscopic cylindrical shells with m = 1, L/R = 20, h/R = 0.01 and 5 = 0. 3. [58,78]

n

Loy et al.

Reddy et al.

present study

1

0.016101

0.01608

0.01612

2

0.009382

0.009381

0.011608

3

0.022105

0.022109

0.024826

4

0.042095

0.042099

0.044889

Table2-b: Comparison of dimensionless frequencies (Ω = ω. R. _(1 − υ" ) ρ/E) C-C homogeneous macroscopic cylindrical shells with m = 1, L/R = 20, h/R = 0.01 and 5 = 0. 3. [58, 78]

n

Loy et al.

Reddy et al.

present study

1

0.032885

0.032760

0.033844

2

0.013932

0.013893

0.015770

3

0.022672

0.022671

0.024826

4

0.042208

0.042213

0.045001

31