nano-spherical shell based on the modified couple stress theory

Author’s Accepted Manuscript On the free vibrations of size-dependent closed micro/nano spherical shell based on the modified couple stress theory Shahrokh Hosseini-Hashemi, Farzad Sharifpour, Mohammad Reza Ilkhani www.elsevier.com/locate/ijmecsci

PII: DOI: Reference:

S0020-7403(16)30120-5 http://dx.doi.org/10.1016/j.ijmecsci.2016.07.007 MS3344

To appear in: International Journal of Mechanical Sciences Received date: 18 February 2016 Revised date: 17 June 2016 Accepted date: 6 July 2016 Cite this article as: Shahrokh Hosseini-Hashemi, Farzad Sharifpour and Mohammad Reza Ilkhani, On the free vibrations of size-dependent closed micro/nano spherical shell based on the modified couple stress theory, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.07.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. 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.

On the free vibrations of size-dependent closed micro/nano spherical shell based on the modified couple stress theory Shahrokh Hosseini-Hashemia,b, Farzad Sharifpoura, Mohammad Reza Ilkhania*1 a

Impact Research Laboratory, Department of Mechanical Engineering, Iran University of, Science and Technology, Narmak, Tehran 16848-13114, Iran.

b

Center of Excellence in Railway Transportation, Iran University of Science and Technology, Narmak, 16842-13114 Tehran, Iran. [email protected] [email protected] [email protected] *

Corresponding Author: Mohammad Reza Ilkhani. Tel: +98 21 77240540; fax: +98 21 77240540;

ABSTRACT Since classical theory is incapable to justify size-dependency of small scale systems such as micro-electro-mechanics (MEMs) or nano-electro-mechanics (NEMs) systems, Modified Couple Stress Theory (MCST) has been developed in order to capture the size effect in the small size investigation. A novel study on the free vibrations of the micro/nano scale spherical shell based on First-order Shear Deformation Theory (FSDT) and MCST is done. Fullerene ( C 60 ) is an appropriate example of spherical micro scale structures. The governing equations of the modified couple stress spherical shell are derived by using Hamilton’s principle. Obtained equations are solved using Generalized Differential Quadrature (GDQ) method. The influences of changing geometrical parameter and scale parameter on the natural frequency are investigated. It is shown that the scale parameter is extremely effective on the natural frequency of the micro/nano sphere. This issue is bolder in the thick spherical shell. Finally, proper scale parameters are proposed for different nano scale spheres by comparing numerical results with experimental results. Keywords: Modified Couple Stress Theory, Size-dependency, Fullerene, FSDT, and Spherical Shell. 1

Postal Address: Iran University of Science and Technology, Narmak, Tehran 16848-13114, Iran.

1

1.

Introduction

Contemporary technological applications need materials that can employ in the micro or nanostructures. In this field, graphene base materials due to their unique mechanical, electrical, and thermal properties attract researcher’s attention [1-3]. By introducing Fullerene in 1985, as a carbon base nano-sphere, different kinds of application have been growing, widely. It has a wide range of application in biomedical including intelligent drug delivery, curing cancer and HIV [4], high performance MRI contrast agent and Photodynamic therapy as well as other application in nano machines [5] and electronics [6, 7]. Fullerene would be a strong representative of a spherical material in nano/micro scale. In all of these researches, analyzing dynamic behavior of structures are presented as one of the main steps of designed structures. There are different methods for dynamic analysis of such structure such as experiment, molecular dynamics simulations, and continuum base methods. Molecular dynamic simulations (MD) consider interatomic and inter-molecular interactions in each step of their analysis which it creates high computational cost, especially when number of atoms increasing. Among modeling methods, continuum based ones are more faster respect to the others and they provide good predictions of general behavior for micro/nano structures, too. But, experiments show that classical continuum theories are not able to predict material behavior properly. So, size dependent theories have been presented which all of them are based on a real: “deformations of a point of structure are dependent to the deformations of all other points of structures and length scale parameters play major role in behavior of micro/nano materials”. Therefore, many types of non-classical theories have been developed to overcome this problem. Among these higher order theories, couple stress theory [8-11], strain gradient theory [12], non-local elasticity [13-15] and surface elasticity [16] are more popular among researchers. It should be noted that couple stress theory is a special case of strain gradient theory. Recently, the couple stress theory has been improved by Yang et al. [17] by adding the equilibrium relation of moment of couples to the conventional force equilibrium and moment equilibrium. Consequently, linear isotropic materials only need one new additional parameter-material length scale parameter- beside the classical parameters to consider the size-dependency in micro/nano scale investigations.

2

In recent years, both modified strain gradient theories [18] and modified couple stress theory have been extremely utilized in micro/nano systems to study vibration, bending and buckling of structures such as beams, plates and shells. Park and Gao [19] scrutinized the modified couple stress theory in Bernoulli-Euler beam in order to analyze static deformation micro cantilever subjected to the point load. In similar way, Kong et al. [20] studied free vibration of a Bernoulli-Euler beam. Ma et al. [21] utilized named theory to investigate static bending and free vibration a Timoshenko beam. They found out that the deflection (similarly rotation) and natural frequency are predicted smaller and higher than classical model results, respectively. Xia et al. [22] established a non-classical Bernoulli-Euler beam based on modified couple stress to investigate free vibration, static deformation and buckling of micro beams. Also, Free vibration of an embedded micro beam that endures the moving load has been investigated by Simsek [23] as a Bernoulli-Euler beam. In this study, results has been compared with the classical beam theory and conform to the previous consequences. Fu and Zhang [24] considered size-dependency for micro tubules with the Timoshenko beam model. Asghari et al. [25] used the modified couple stress theory for static and dynamic analyze in functionally graded micro-beams. Kahrobayian et al. [26] investigated the behavior of cantilever atomic force microscopes (AFMs) based on the modified couple stress theory. Akgoz and Civalek [27] studied the static deformation and free vibration of higher-order beam based on the strain gradient theory. All the results are compared with other beam theories. Results indicate in smaller slenderness ratio beam, the shear deformation effect is undeniable. Also, it is shown that non-classical theory recognize beam stiffer than corresponding classical beam. Also, researchers such as Reddy [28], Reddy and Arbind [29], Ke et al. [30], Kahrobayian et al. [31], Simsek and Reddy [32], Arbind et al. [33] applied modified couple stress and modified strain gradient theories on functionally graded micro beams. Extensive interest of using non-classical elasticity theory has not been only restricted on the beam theories. There are many attempts which done for on the static and dynamic behavior of the low scale plates. Firstly, Tsiatas [34] applied the modified couple stress theory on the Kirchhoff plate model. The proposed model is applicable for micro-plates with complex geometry and different boundary conditions. It is shown that nonlinearity decrease with the increase of material length parameter. Also, decreasing of deflection is concluded with the increasing Poisson’s ratio. Yin et al. [35] investigated vibration of micro-plate with nonclassical Kirchhoff theory based on the modified couple stress theory. Ma et al. [36] utilized 3

Mindlin plate theory as a special case of first-order shear deformation theory. Both static deformation and natural frequency of the plate have been investigated in this article. Akgoz and Civalek [37] analyzed single layered Graphene sheet by Kirchhoff thin plate model resting on elastic foundation with the same approach. Afterwards, Ke et al. [38] explored on free vibration of Mindlin plate with two different boundary conditions. Shaat et al. [39] modeled a nano-plate, including surface effects based on the modified couple stress theory. Chen and Li [40] continued previous research with Kirchhoff plate for anisotropic elasticity. Also, Salehipour et al. [41] found closed-form answer for free vibration of functionally graded micro/nano plate. Using a new approach, Wang et al. [42] investigated static deformation of the circular Kirchhoff plate by the MCST. It is noticeable that all the noted researches have been investigated on Cartesian and rectangular coordinate. Hence, there is no need for changing the formulas of the modified couple stress theory (strain gradient theory). However, changing the coordinate for using MCST in shell investigation is inevitable. Therefore, some probes have been done on curvilinear coordinate. Zhao and Pedroso [43], Guzev and Qi [44], Ashoori and Mahmoodi [45] found modified strain gradient theory (MCST) formulation in general curvilinear coordinates. Due to complexity of shell theory few researches have been done via MCST. Recently, Zhou and Wang [46] surveyed on the vibration of micro-scale cylindrical shell that conveys fluid based on MCST. One of the most interesting results of this research is shifting the natural frequency by changing the fluid velocity. Zeighamipour and Tadi Beni [47-49] applied MSCT on cylindrical shell to analyze vibrations and investigating the effect of a length parameter. FSDT has been used to introduce displacement field. Additionally, Zeighamipour et al. [50] studied conical shell using FSDT and MCST. Diversions of results by changing the length parameter in the different apex angle have been investigated in this article. Lately, Tadi Beni et al. [51] utilized MCST in a cylindrical FGM shell that used FSDT for defining the displacement field. The obtained results show that natural frequencies in MCST are higher than classical theory results. By comparing literature, it can be easily seen that although MCST and MSGT are trusted methods in micro scale structure, but lack of investigation of the spherical shell is clearly evident. In this research, free vibrations of spherical shell by using non-classical the modified couple stress theory are studied. First-order shear deformation theory –Sanders 4

type is used to define displacement relations. The governing equations of the modified couple stress spherical shell are derived by using Hamilton’s principle. Equations are solved analytically and characteristics, Young’s module and Poisson’s ratio, of different nanospheres are used in this path. The shape functions and natural frequencies are exactly assessed. Results are compared with classical theory results, finite element results and experimental results to validate the present method. The size-dependency of shell and change of scale parameter on the natural frequency are investigated. The necessity of using higher-order non-classical theory in small and thick spherical shell has been shown. Behavior of nano-sphere by changing geometrical properties beside the scale parameter is studied and finally, proper values of scale parameter for different nano-spheres are proposed.

2. Mathematical formulations 2.1 Modified Couple Stress Theory The modified couple stress introduced by Yang et al [17], expressed strain energy as a function of strain tensor and the gradient of rotation tensor by adding the length parameter to the two classical parameters –Young’s module and Poisson’s ratio-. It should be noted that length scale parameter shall be obtained from experimental results. Therefore, strain energy of a three- dimensional body that occupies v volume in rectangular coordinate is given U  udv    ij ij  mij ij  dv, V

i, j  1, 2,3 (1)

V

 ij 

1 ui, j  u j ,i  (2) 2

ij 

1 θi, j  θ j ,i  (3) 2

 ij  ij kk  2 ij (4) mij  2l 2 ij (5)

Where  ij ,  ij ,  ij and mij are strain tensor, symmetric part of rotation gradient tensor, classical stress (Cauchy stress) tensor and higher-order stress tensor. Also, λ and  are lame’ constants and  ij and l are Kronecker delta and the material length scale parameter,

5

respectively. Moreover, ui and  i stand for displacement vector and rotation vector and defined

1 i  eijk uk , j (6) 2 That eijk is a permutation symbol. Due to symmetry of  ij higher-order stress tensor is symmetric, also. According to Yang et al [17], in the modified couple stress theory, only symmetric part of strain tensor and rotation gradient tensor contribute to the deformation energy that is the main contrary to couple stress theory. 2.2

Geometrical configuration and material properties

Figure 1 depicts the geometrical configuration of thick spherical shell with uniform thickness h and mean radius R and the neutral axis settled on the mid surface while the body is under the free boundary condition without any external loads. It is assumed that the material is isotropic. Then, Young’s module and Poisson ratio remain constant. Orthogonal spherical coordinate (  , , z ) is assumed in order to define deformation and geometry. Although, using both spherical orthotropic and rectangular orthotropic are practical for shallow shells. 2.3

Displacement field and Strain based on the FSDT assumptions

A major issue for studying micro/nano structure is using appropriate strain-displacement relation. Leissa [52] has made a good survey on different strain-displacement relations and comparing them. Also, there are many useful literatures on shell theory [53-55]. A review on these books and related research papers [56-58] approves that Sanders first order shear deformation shell relations are a reliable and accurate choice. Also, reviews done by researchers show that FSDT has enough accuracy to analysis nano-shells [59, 60]. Therefore, using S-FSDT due to doubly-curved shell’s complicated relations seems logical. So, in the first step, a first order displacement field is considered. By assuming normal vector of midsurface remains straight after deformation, the displacement field consistent with FSDT can be assumed as [53]: u  , , z, t   u0  , , t   z   , , t  v  , , z, t   v0  , , t   z   , , t  (7)

w  , , z, t   w0  , , t 

6

Where u0 , v0 , w0 are displacement of mid-surface and   and   denote the rotation of midsurface about  and  axis (Figure 2), respectively. By using defined displacement field of Eqs. (7), strain at any point of shell can be define in terms of mid-surface and also changes of curvature as [54, 55]:

 

1   zk  1  z / R  0 

 

1   zk  1  z / R  0 θ

 

1   zk  1  z / R  0 

 

1   zk  (8) 1  z / R  0 

z 

1   z   / R  1  z / R  0 z

 θz 

1   z   / R   1  z / R  0 z





Here  , R , z , and k is contribution factor (that changes according to different references and assumes one in this study), radius of curvature, distance from mid-surface, and curvature changes, respectively. Therefore, by substituting Lame’ parameters normal, in-plane and shear strain for thick spherical shell can be easily achieved [55]:

 0 

 1  u0  w0   R   

 0 

 1  1 v0  u0 cot    w0    R  sin    

 0 

1  v0    (9) R   

 0 

 1  1 u0  v0 cot      R  sin    

7

 0 z 

 1  w0  u0     R   

 0 z 

 1  1 w0  v0      R  sin    

Also the curvature changes of thick spherical shell are [55]

 

1   , R 

 

 1  1     cot    , (10)   R  sin    

 

1   , R 

 

 1  1     cot      R  sin    

It should be noted that the normal transverse strain is assumed zero.

2.4

Rotation gradient tensor

It is obvious changing coordinate alters rotation gradient tensor relations. Therefore, Eq. (3) is used in impractical in spherical coordinate. Thus, a new set of equations need to satisfy this condition. By performing this procedure, classical and non-classical components of strain can be obtained (see Appendix A). Substituting Eqs. (7) into (A-1) symmetric rotation gradient tensor component are as follows:

 

 

    u cos   v0  z     0  z  2   z  2 R sin      1   R   1

1

  w0  2 w0    R sin   cot             

   v   R cos     R   sin    0  z   2    z  2 R sin      1    R 1

1

(11)

8

 w0  2 w0   cot          

 z   z

     u w0  cos    0  z   sin           2 2 1 1   sin     u0  z      sin  2   v0  z       z  2 R 2 sin 2      2      1     R      2 v0  2    sin   z  cos  2  v0  z       2 2        

 w0    v  2sin    u0  z    cot    0  z  2sin       1 1   z   z  2       2 v0  1  z  4 R 2 sin     2   1   2u0   z          sin     2  R  2       

   u0 z    cos   v0  z    R cos       1 1    z  z  2 R 2 sin       v      R   sin    0  z    R sin    1     R       

2.5

          

      

      

Strain, potential and kinetic energy

If obtaining a set of equations based on Young’s module and Poisson ratio is required, then

 

E

1  1  2 

,

E (12) 2 1  

By rewriting Eq. (4-5), stress-strain relations is as follow: 1         0        E    2 0    1      z  0      z   0 



0 0

0 0

0 0

0

1  2

0

0

0

0

1  2

0

0

0

0

1  2

0

0

0

0

1

9

0  0       0         0    (13)      0   z     z  1    2 

Now if we consider strain energy in spherical coordinate, then

U Classical Strain 

 h /2          1       2     2   1  z / R  R sin   dzd d   2    h /2        z  z    z  z  

U non Classical Strain 

 h /2 m   m   m   m      1   z z    2  2   1  z / R  R sin   dzd d (14)    m   2 m   2 m  2    h /2  z z z z     

Where

m  m11 ,

m  m22 ,

mz  m33

m  m12 ,

m  m21 ,

m z  m13 (15)

mz  m31 ,

m z  m23 ,

mz  m32

Hence, strain energy of the system is calculated by substituting Eq. (8-12) and (13) into (14). In the same manner, by integrating over thickness kinetic energy has been obtained:

 ˙2 ˙2 ˙ 2   I3   ˙ 2 ˙ 2   I  I2 I u  v  w I         I 3  4  52    0 0 0  1 2    2R R   2 R R  2 1    T     R sin   d d (16) 2   I3 I4     u v I     0  0    2 2 R R 2   Where dot symbols represent derivation respect to time and I i are inertia terms

 I1 , I 2 , I3 , I 4 , I5  

 h /2

  1, z, z

 h /2

2

, z 3 , z 4  dz (17)

 is a mass density of shell per unit area of midsurface. Hence, Lagrangian functional can be expressed as V  T  UClassical Strain  U non Classical Strain  (18)

2.6

Equations of motion

In order to find the equations of motion, energy functional should be computed. Therefore, by substituting Eqs. (14) and (15,16) into (18) energy functional is calculable. Hence, governing equations can be derived using Hamilton’s principle as follows 10

t

 Vdt  0 (19) 0

Where  represent variation symbol. It should be noted that according to the geometrical condition of problem there is compatibility condition in equations. Now by taking variation respect to the time from Eq. (19) five equations of motion can be obtained as follows:

u :  F151 F

1 21

F

1 25

 F331

F u  ,  , t   F 1 11 0

1 12

u0  , , t   v0  , , t  

  2

 3v0  , , t 

F

  2

1 22

 3

F

1 26

 3 w0  , , t  

t 2  3u0  , , t 

 F161

 3v0  , , t 

F

1 42

    ,  , t 

F

1 51

v:  F152 F

2 23

 F272 F

2 32

t 2    , , t   2 11

F

 2 v0  , , t   2  4 v0  , , t    2

2

 F542

F

1 52

F

 F282 F

2 33

    ,  , t      , , t  

 2

 2 v0  , , t 



 3

 3 v0  , , t   

3

w0  , , t  

F



 3v0  , , t   2

 F292



 2

F

2 25

 2 w0  , , t 

 2   , , t 

 3

1 45

 2 u 0  ,  , t 

 F212 v0  , , t   F222

(20)

 2   , , t   2



2 34

4

    ,  , t  t

0

11

 3

 2 v0  ,  , t   2

 F312

w0  , , t  

 F412

2

2

 4u0  , , t 

t 2

F

 3 w0  , , t   2

 F

2 14

 2 v0  , , t 

2 26

 4 v0  , , t 

F

 F512   , , t   F522





2 13

F

 3v0  , , t 

 4v0  , , t 

 F411   , , t 

3

   , , t 

F

 2  2

 F321

 3 w0  , , t 

1 44

 3u0  , , t 

 4 u0  ,  , t 

F

 4  4u0  , , t 

0



F

2 24

 3



F

1 24

2

 4u0  , , t 

1 14

 F181

 2 w0  , , t 

 F351

    ,  , t 

2 12

 F162

 3 w0  , , t  2

 F422

F

1 43



 2 



F

2

2

 2v0  , , t 

2

u0  , , t 

 3u 0   ,  , t 



1 31

 2 w0  , , t  

 2  2u0  , , t 

F

 

2

F

1 23

3

 2 u 0  ,  , t 

1 13

 F171

 4 v0  , , t 

 F341

2

 2 u0   ,  , t 

   , , t  

    ,  , t  2

 F523

 2

(21)

F113u0  ,  , t   F123

w:  F153 F

3 23

 F333  F373

 2 u0   ,  , t   2 v0  , , t   2 w0  , , t 

F

 2  2 w0  , , t 

3 42

 2   , , t   2  2   , , t  

 F154  F324

3 54

 2 u0   ,  , t   2 w0  , , t  

 F334

 F    ,  , t   F 4 41

F

4 45

F

4 51

 F554

 2 

4 42

   , , t     , , t  

t 2

F F

 3   , , t   2 

 2   , , t  4 46



 2

4 52

 F564

F

4 43

 3   , , t   3  3

 2 w0  , , t  t 2  3 w0  , , t   3 w0  , , t   4

 3   , , t 

F



 2

3  F310

3 44

   , , t 

F

 2 

 F523

 2 u 0  ,  , t  

2

 2v0  , , t  

 3   , , t   3

 2  2

F

4 53

F

 2  2   , , t 

0

12

4 44

u0  , , t  

 2 w0  , , t   2

 F354

 2   , , t 



 F144

 F314

 2 w0  , , t 

 2   , , t 

F

 4   , , t 

 3

(22)

4 47

 2 

 3

4 13

 F344

 3   , , t 

 3v0  , , t 

 F363

 3 w0  , , t 



 F224

 3 w0  , , t 



 2

0

2



w0  , , t 

 F513

 2 u0   ,  , t 

v0  , , t 

 F214

F

 3   , , t 

t

 F223 3 32

   , , t 

3 43

 3

4 12



 F393

 3   , , t 

F u  ,  , t   F 4 11 0

v0  , , t 

 F353

 2  2

 F463



 3u0  , , t 

 F143

 F w0  ,  , t   F

 4 w0  , , t 

 2

u0  , , t 

3 31

 4

 2   , , t 

F

 F213

 4 w0  , , t 

 F383

 2

  :

 2 

 F343

 F    ,  , t   F

3 53

 3v0  , , t 

F



3 41

 F453

 3

3 24

 F133

 2

 3u0  , , t 

 F163

 2

 2 u0   ,  , t 

 3 w0  , , t   3

 4   , , t   4

F

4 48

F

4 54

 4   , , t   2  2  4   , , t   3

(23)

  : F

5 11

 F235

u0  , , t  

 2 v0  , , t  

2

F

5 12

 F245

 w0  , , t 

 F425 F

5 46



F

5 54

 F585

 3   , , t   3  4   , , t     , , t    3   , , t   3



 2   , , t  

 F    ,  , t   F

F

 F595

 2 

w0  , , t  

 4

 3 t 2

5 56

 F415

 4   , , t 

 2   , , t 

F

 4   , , t 

 2

 F445 5 52

 3   , , t 

 F315

 w0  , , t 

 F345

5 51

5 55

2

t 2

3



 F435

5 22

 2 v0  , , t 

 F255

 w0  , , t 

 F335

 2v0  , , t 

 F v  ,  , t   F 5 21 0

2

 3

 3



v0  , , t 

3

 F325

 2 u 0  ,  , t 

 2



 F455

F

 2   , , t 

   , , t 

5 53

 3   , , t   2

(24)

 2   , , t   2

F

5 57

 4   , , t   2  2

0

Where superscript, first subscript, second subscript represent equation number, function number and dummy counter, respectively and Fijk are represented in appendix B. Eqs. (20-24) show equations of motion of spherical shell based on the modified couple stress theory by considering shear deformation and rotation inertia for an isotropic homogenous material. It should be noted that these equations can be reduced to classical FSDT spherical shell theory by assuming the length parameter scale ( l  0 ) .

2.7

Boundary Conditions

Due to geometry of problem, kinematical and physical compatibility shall be satisfied at the

  0 , 2 .the kinematic and physical compatibility conditions refer to continuity of displacement and continuous conditions for stress resultants Kinematical Compatibility Condition: u 0  0, , t   u 0  2 , , t  ,

   0, , t      2 , , t  ,

v 0  0, , t   v 0  2 , , t  ,

w 0  0, , t   w 0  2 , , t  ,

   0, , t      2 , , t  (25)

Physical Compatibility Condition:

13

N   0, , t   N   2 , , t  ,

N   0, , t   N   2 , , t  ,

M   0, , t   M   2 , , t  ,

M   0, , t   M   2 , , t  , Q  0, , t   Q  2 , , t 

Where N , Q and M stands for normal force, shear force and bending moment stress resultant, respectively. Stress resultants can be defined as follows N 

Eh  0   0  1  2

Eh  0   0  2 1   

N 

Eh3 M  12 1   2



M  

Q 

k



  k  (26)

Eh3  k  k  24 1   

 Eh  0 z 2 1   

Where 2.8



in most reliable references proposed the value of 5 / 6 .

Semi-Analytical Solution for Free Vibration of Spherical Shell

In order to analyze the free vibrations of the spherical shell based on the modified couple stress theory, governing equation of motion should be solved. If harmonic vibrations are assumed, displacement field can be expressed as follows: u0  , , t   u   cos  n  sin t  v0  , , t   v   sin  n  sin t  w0  , , t   w   cos  n  sin t  (27)

   , , t   Ψ   cos  n  sin t  14

   , , t   Ψ   sin  n  sin t  u  φ   u   , v   , w   ,Ψ   ,Ψ    (28) T

Where n and  depict mode number and natural frequency, respectively. Also, new vector can be defined as follows: By applying assumed solutions (27) in equations of motion (20-24) five partial differential equations are reduced into ordinary differential equations. Due to complexity of obtained equations, finding closed-form solution is not practical. So, using some semi-analytical method seems more applicable. The Generalized Differential Quadrature (GDQ) [61] method is one of the popular techniques for computation of numerical solution due to its simplicity in implementation and compact form. The GDQ method is used to discretize the derivatives in governing equations. Among different types of grid distribution (see [62]) Chebyshev-Gauss-Lobatto polynomials has better accuracy and convergence speed. By utilizing this method, 4th order differential equations (20-24) change to linear algebraic system of equations. Therefore, natural frequent of the system can easily found by solving equations. Here, the GDQ method has been descripted, briefly. The GDQ method is used to discretize the derivatives and boundary condition. This method by computing weighting coefficient alters the equations to linear algebraic system of equation. Weighting coefficient can be obtained easily by calculating first-order derivative coefficient (while the first-order derivative coefficient can be obtained by a simple algebraic formulation). Let approximate continuous function of f  x  as follows

f  x   p j  x  f  x j  , N

j  1, 2,, N (29)

j 1

Where N indicates number of grid points and p j  x  are the Lagrange interpolated polynomials [62] that can be defined as follows p j  x 

L  x

 x  x  L   x  1

j

,

j  1, 2,, N (30)

j

Where

15

N

L  x     x  xi  ,

1

L

x    x N

j

i 1

j

i 1,i  j

 xi  (31)

Now by taking derivative eq. (27) respect to x , first-derivative at a specified point can be obtained

f 1  xi    pj1  xi  f  x j    ij1 f  x j  , N

N

j 1

j 1

i  1, 2,, N (32)

That  ij1 defines GDQ weighting coefficient of first derivative and described as follows 1

pj

 xi    ij

1

L1  xi 



 x  x  L   x  1

i

j

,

i, j  1, 2,, N , i  j (33)

j

By some mathematical calculation [62]  ii1 can be obtained according to definition in below N



1 ij

N

1

 0   ii  

j 1

  , 1 ij

i, j  1, 2,, N (34)

j 1, j  i

Therefore, by considering eqs. (33) and (34) weighting coefficient  ij1 is calculable. In same manner, nth-order derivative function can be approximated by the following definition d n f  x dx n

N

x  xi

  ij n  f ( x j ) ,

i  1, 2,, N (35)

j 1

It should be noted that  ij n  depicts nth-order derivative weighting coefficient. By same approach, second- and higher-order derivatives can be obtained by a recurrence relation. Weighting coefficient can be determined by following formulas  n

 ij N

  n 1 1  ij n 1  n   ii  ij   xi  x j 

n n  ij   0   ii    j 1

 ,  

N

  , n ij

i  j, n  2,3,, N 1, i, j  1, 2,..., N

n  2,3,, N 1, i, j  1, 2,..., N (36)

j 1, j  i

By utilizing different grid nodes that have been distributed by the definition of ChebyshevGauss-Lobatto polynomials differential equations of motion change to the linear algebraic equation. It is worth to note that Tornabene and Viola [62] investigated the effect of node number and node distribution in the results. They found out for achieving accurate results, 16

using at least 21 grid nodes for higher frequencies is necessary. Moreover, ChebyshevGauss-Lobatto was found as the best grid distribution among other non-uniform typical grid distribution. Now by using these relations interior point equations can be easily discretized. Afterward equations are combined with compatibility condition. Therefore, by using eqs. (910) and (25-26) utilizing the GDQ method for closed geometry are applicable. Now, by combining eqs. (20-24) and eq. (27) and setting GDQ method on 5 sets of equations for interior grid points and using eqs. (37) for boundaries, the whole system of equations can be discretized and summarized into the linear algebraic matrix as follows [62]  Kbb K  db

Kbd   b  0   b  0  2        (37) K dd   d  0 M dd   d 

Where subscripts b and d are symbols to show boundary and domain, respectively. Therefore by some computation natural frequencies of system can be determined

K



 K db  Kbb  Kbd   2 M dd  0 (38) 1

dd

3. Results and discussion In this section, numerical results which are found from previous calculations are presented in two groups. At first, results are compared with literatures in different tables and validity of the results are approved. Also, for the first time proper MCST’s scale parameters are presented for natural frequencies of different nano-sphere such as Fullerene. Secondly, behavior of Fullerene under variations of geometrical and scale parameters are studied in the different figures. For these purposes, different geometry and material properties are used which they are defined in the table 1 in detail. These properties are found from literature as they are indicated in Refs [63-67]. Before any numerical calculations, convergence of results should be checked. In other word, the convergence of GDQ method that has been applied on this special problem should be investigated. For this purpose, convergence of the results is investigated in figure 3. The convergence of two parameters are studied. Firstly, variations of the non-dimensional fundamental natural frequency of a Fullerene (1) by increasing the grid numbers are investigated for two different scale parameter as l=0 and l=1.57h, in figure (3-a) and (3-b), 17

respectively. Figures (3-a) and (3-b) show that the results of both conditions are converged after 25 grids. In addition to convergence of results for grid numbers, convergence of compatibility conditions should be checked. Compatibility conditions are written in the edge of θ=0, 2π. But our investigations show that using exact value of 2π produces divergence. So, in order to find the proper closer values to the 0 and 2π, convergence analysis are presented in figure (3-c) and (3-d). In these figures, variations of non-dimensional natural frequency of a Fullerene by changing μ is presented. μ is logarithmic non-dimensional parameter as

   log(  ) which

are considered as small deviations from edges as θ=0±μ,

2π±μ. In addition to convergence of results, figures (3-c, 3-d) show that the proper value for this analysis is about  / 109 . As a result, the first set of results only provides the values that are appropriate for utilizing GDQ method. In fact, the required grid numbers, approaching to merge of boundary for more accuracy in different condition (scale parameter) and geometry and also dependency to different condition are studied. As the first comparison, ten fundamental natural frequencies of a spherical steel are compared with literature [65] and finite element results in table 2. In fact, result compared in order to validate the FSDT theory in macro scale. For this purpose, sphere is modeled in the traditional finite element software using 3D brick elements with sufficient numbers of elements. The percent of error for both of the comparison are presented in the table. Table 2 approves validity and accuracy of the present method in the macro scale. It is noticeable to know l=0 considered for macro scale comparison. In modified couple stress theory, natural frequencies are dependent to scale parameter l. So, to find the frequency, scale parameter should be stated. In all of the size dependent theories such as nonlocal elasticity or modified couple stress, scale parameter is determined by comparing results of those theories with experimental results or molecular dynamics simulations. Considering an optimum tool, such as least square, a scale parameter is found which approaches frequencies of modified couple stress to experiments and MD results. In the next comparison, natural frequencies of different nano sphere are presented in the table 3. Results are presented for different materials which their specifications are presented in table 1. Calculated frequencies are compared with experimental results which are presented in the literature [68-70]. Table 3 shows that by setting scale parameter l=0 natural frequencies are lower than the experimental results. In other word, the MCST would approach to the proper result from lower value of that frequency or increase the natural 18

frequency by increasing scale parameter. So, proper scale parameters are considered for each specification and these values are presented in the third column. Table 3 shows that modified couple stress is a proper method to find natural frequencies of nano-sphere when the first order shear deformation shell theory is applied. Because in opposite of nonlocal elasticity which decreases natural frequencies of sphere [71, 72], modified couple stress increase frequencies. However, results of first order shear deformation shell theory are lower than experimental results and the modified couple stress theory can increase them as we want. Also, it is shown that scale parameter of each nano-sphere is dependent to the geometry of the nano-sphere and of course its thickness or its radius. In the next table, table 4, six fundamental natural frequencies of a Fullerene are calculated according to the properties presented in the table 1. In a similar manner with table 3, natural frequencies of Fullerene without scale parameter are lower than experimental results. In order to find proper scale parameter for Fullerene, results are compared with experimental results of literature [73]. Hence, results are calculated for scale parameters l=0 and l=1.57h. Therefore, natural frequencies are approached to the experimental results by increasing scale parameter. So, Table 3 and 4 shows that using first order shear deformation shell theory beside the modified couple stress theory is an appropriate combination for analyzing nanospheres. By the confirmation of present method, some graphical analyses are done to show behavior of modified couple stress theory for Fullerene by changing effective parameters. As the first one, Figure 4 shows effect of increasing thickness to radius ratio on different modes of fullerene. Non-dimensional frequencies ( Ω   R  / E ) are plotted in two figures with constant meridian (m) and circumferential (n) modes number. Figure 4 shows that increasing thickness to radius ratio increases natural frequencies in all mode numbers. This increment has more effect on the higher mode numbers in both figures. As it is seen in figure 4 that increasing thickness has same effect on both of the meridian and circumferential modes so, in the next study, effect of MCST scale parameter is added to assumptions of figure 4, only for circumferential modes and results are presented in figure 5. Variations of non-dimensional frequencies with different circumferential modes of fullerene are plotted for different thickness to radius ratios in figure 5, when MCST scale parameter is considered as l=1.57h. Figure 5 shows that increasing scale parameter increases natural frequencies when scale parameter is zero or not. Also, increasing scale parameter has more 19

effect on geometries with higher thickness. Additionally, figure 5 shows that scale parameter has more effect on frequencies with higher mode numbers. In the next study, figure 6 shows variations of non-dimensional fundamental natural frequency of nano-sphere by changing thickness to radius ratio (h/R) for different scale parameters. Material properties of a Fullerene are considered for analysis. Figure 6 shows that increasing scale parameter has same effect on nano-spheres with different thickness. Generally, for all of the scale parameters frequencies increase more than 50 percent.

4. Conclusion Free vibrations of nano-sphere have been analyzed by applying modified couple stress on first order shear deformation shell theory. Five coupled partial differential equations have been found to define the equations of motion which have been solved by using generalized differential quadrature method. A computer code developed and natural frequencies of macro/nano spheres have been calculated. Validity and accuracy of the present analysis has been approved by comparing results with experimental results in nano scale. It has been approved that setting scale parameter to zero gives lower natural frequencies respect to experimental results. So, applying modified couple stress which increases natural frequency is a good choice for modeling nano-spheres such as Fullerene. Additionally, proper values of scale parameters have been presented for nano-spheres with different material properties such as Fullerene. The graphical investigations show that the modified couple stress theory has more effect on higher modes of nano-spheres. Also, it is shown that increasing or decreasing the scale parameter has similar effect that increasing or decreasing the thickness has. Therefore, combinations of modified couple stress and first order shear deformation theory is a good choice for vibrational analysis of nano-sphere.

Appendix A To calculate the strain energy in orthogonal curvilinear coordinate, the definition of classical strain and symmetric rotation gradient tensor are used and expressed as follows [44-46]:

 ij 

1 2 gii g jj



gii ul

  ,j

g jj u j



,i

 2 g kk uk Γijk

20



(A1)



 

 







n n   g u  g nn un Γ ml  g nn un Γim  g mm um Γlin   mm m ,li ,i ,l ,n g    jk    eklm p n p n     Γln Γim  Γ mn Γli g pp u p  4 g gii g jj   n  g ki g mm um  g nn un Γ ml  g nn un Γ njm  g nn un Γljn  ,lj ,j ,l ,n  

ij s



Where

,

and





 

 









are permutation symbols, fundamental covariant tensor and

Christoffel symbols, respectively. Covariant tensor in spherical coordinate can be expressed as [46, 53]:

g

  z    R 1    R   

g

2

 z     R sin   1     R  

2

(A2)

g zz  1 and gij  0  i  j By defining Christoffel symbols as [47]: Γijk 

Also,

1 il  g jl g kl g jk  g    l  (A3) 2  x k x j x 

is contravariant tensor. Therefore, in spherical coordinate Christoffel symbols can

be obtained as follows:

z z 1   z z Γ   R 1   ,     Γ   R 1   sin 2   ,     Γz  , z   R  R R 1    R  Γ   sin   cos   ,     Γz 

1 z  R 1    R

,     Γ  cot   (A4)

Appendix B Here coefficient of governing equation can be expressed as follows:

F  1 11









Eh csc   2l 2  1    l 2  1    2 R 2 1   cos  2   2 R 2 3   4 sin2  



8R 1  2

21

2





 h   12R  h   3

1 12

F



F 

F161 

F  1 17

F221 

F  1 25

F261  F311 

8R 2 1  



8R 1  2

2





Ehl 2 cot   csc  ] 8R 2 1  





Eh 8R 2  l 2  1   sin  



8R 1  2

2



Ehl 2 csc   8R 2 1  







Eh cot   4 R 2  3    l 2  1    4 R 2  3    l 2  1    cos  2 

16  1    R sin   2

2



2

Ehl 2 cot  

8  8   R 2 sin2   

F231   F241 

Ehl 2csc3  



F 

F 

8R 1  

Eh 8R 2  l 2  1   cos  

1 18

1 21



2

F141  

F 

12 R 2

Eh l 2  2 R 2  2 R 2cos  2  csc3  

1 13

1 15

2



Eh csc   2 R 2  l 2  1    2 R 2  2 R 2  1   cos  2   8 R 2 sin2  

Ehl 2 8  8  R 2 sin2  







8 1  2 R 2 sin  





Ehl 2 cot  

 4  4  R sin   Ehl 2 R 2  8  8 

 

Ehl 2 cot   csc  

F  1 32

4 R 2 1  







8R 1   2

Ehl csc   2

F331  



Eh csc   6 R 2  2l 2  1     2 R 2  2 R 2  3     l 2  1    cos  2   8 R 2 sin 2  

8R 2 1  

22

2





F  1 34

F  1 35

F  1 41

F451 

Ehl 2 sin   8R 2 1  



F  F441 

8R 2 1  





Eh l 2Cos   Cot    l 2  4 R 2 sin  

F421   1 43

Ehl 2 cos  

8R 1  



h3  48R

Ehl 2 csc  

4 R  4 R Ehl 2 cos  

8R  8R Ehl 2 sin  

8R  8R Ehl 2 cos   F511  8R  8R  sin   F521 

F  2 11

3Ehl 2 8R 1  



F 

F  2 13

F  2 21

3Ehl 2 cot  



8 1   R 2 sin 2  

F







  1   sin  

8 R



2

2



Ehl 2 cot   R 2  4  4 

Ehl 2 R 2  8  8 

 









Eh 5l 2  2 R 2  8l 2 cos  2   l 2  2 R 2 cos  4  csc3   32 R 1   2

 h   12R  h   3

2 22

2

Eh csc   2 R 2  3l 2  1    2 R 2  2 R 2  1   cos  2   8 R 2 sin2  

Ehl 2 8  8  R 2 sin2  

F152  

F162 



8R 1  2

2 12

F142 



Eh cot   l 2  1   cot 2    2 2 R 2  3    l 2  1    2l 2  1   csc 2  

2

12 R 2

23





F  2 23

F  2 24

F  2 25

F  2 26





Eh 4 R 2 cos    3l 2 cot   csc  



8R 1  





2

Ehl 2 cot   csc  



8R 2 1  





Eh 3l 2 cos   cot    4 l 2  R 2 sin   8R 1  



2

F  2 28

F  2 29

F 



16 R 2 1  2

F272  

2 31



Ehl 2 csc   8R 2 1  

Ehl 2 cos   4 R 2 1  

Ehl 2 sin   8R 2 1  



Eh 2 R 2  3    l 2  1    4 R 2



4 R 2 1  2

F322  





Ehl 2csc 2  

8  8  R 2 Ehl 2 cot   2 F33   8  8  R 2 Ehl 2 F  2 8R 1   2 34

F412  

F422  

F  2 51

F522  

F532 



Eh 8R 2  5l 2  1    8R 2  l 2  1   cos  2  csc3  

Ehl 2 cot  

8  8  R

3Ehl 2 8R 1  









Eh 3l 2  2 R 2  l 2  2 R 2 cot  2  csc   8R 1  

h3  48R

Ehl 2 csc  

8R  8R Ehl 2 cos   F542   4 R  4 R 24

F552   F  3 11

F123  F  3 13

F  3 14

F153 

F163 

F  3 21

F223 

Ehl 2 sin   4 R  4 R







8R 2 1  2 8R 2 1  







16 R 1   2

Ehl 2 cos   4 R 2 1  

Ehl 2 sin   8R 2 1  



  8R  1  

Eh 2 2 R 2  3    l 2  1    csc 2   l 2  1    8 R 2 sin2   2

2

Ehl 2 8  8  R 2 sin2   Ehl 2 cot  

8  8  R 2

2 Eh sin  1  

 1   2

 h   12R  h   3

F  3 33

F343  

F 



4 R 2 1  

Ehl 2 F  2 8R 1  

3 35

2

Ehl 2 csc  

3 24

F



EhCsc   12 R 2  5l 2  1     4 R 2  4 R 2  3     3l 2  1    Cos  2   16 R 2 sin 2  

F 

3 32





Ehl 2 cot   csc  

3 23

F313 



Eh cot   csc   2 R 2  3    2 R 2  3    l 2  1   cos  2   8 R 2 sin2  

2



12 R 2







Eh l 2  R 2  l 2  R 2 cos  2  csc3   4 R 1   2

Ehl 2csc3   8R 2 1  





Eh 2 l 2  2 R 2 cos    l 2 cot   csc   8R 2 1  



25





F  3 36

F  3 37

Ehl 2 cot   csc   4 R 2 1  



4 R 2 1  

Ehl 2 cos   4 R 2 1  

3 F310 

F  3 41

F 

F  3 44

F  3 45

F463 

F  3 51

8R 1  



Ehl 2 csc  

F393  

3 43



2

F383  

F423 



Eh l 2Cos   Cot    4 l 2  R 2 sin  

Ehl 2 sin   8R 2 1  

 

 





Eh 2 l 2  R 2  l 2  2 R 2 cos  2  cot   csc   8R 1  

Ehl 2 cot   csc   R  8  8 



Eh l 2 cos   cot    4 R 2 sin   8R 1  



Ehl 2 csc   8R 1  

Ehl 2 cos   4 R 1  

Ehl 2 sin   8R 1  



 



Eh 2 l 2  R 2  l 2  2 R 2 cos  2  8 1   Rsin2  



Ehl 2 F  8  8  Rsin2   3 52

F533  

F543 

Ehl 2 cot  

8  8  R

Ehl 2 8R 1  

F  4 11

F124  









Eh 2 R 2  l 2  2 R 2 cos  2  csc   8R 1  

h3  48R 26

Ehl 2 csc  

F  4 13

F  4 14

F  4 15

F214 

F224  F314 

F  4 32

R  4  4 

Ehl 2 cos   R  8  8 

Ehl 2 sin   R  8  8 

Ehl 2 cot   R  8  8 

3Ehl 2 8R 1   Ehl 2 cot   csc   R  4  4 





F334  

F  4 34

F  4 35

8R 1  

R 2  8  8 

Ehl 2 cos   R  8  8 

Ehl 2 sin   8R 1  

   1     (12 R  l  2 R   1     h (l  1     2 R (1  2  1       )))Cos  2   2 h  l  1     R  3  2  1       1       4 sin    )  96 R  1    4

2

4



Ehl 2 csc  

Eh csc  2  ( 24 R

F41



Eh l 2 cos   cot    l 2  4 R 2 sin  

F  4 43

F444  

F  F464 

2

2

2

2

2

1 3 h 12





2

2

2

2

F424  

4 45

2

 

2

2





Eh 24l 2 R 2  h2 l 2  2 R 2  2 h 2  12l 2 R 2Cos  2  csc3   96 R 1   2

Eh3l 2csc3   96 R 2 1  

 





Eh h 2 8R 2  l 2  1    12l 2 R 2  1   cos  



96 R 2 1  2



Eh3l 2 cot   csc   96 R 2 1  

27

2

 1   

F  4 47

 





Eh h 2 8R 2  l 2  1    12l 2 R 2  1   sin  



96 R 2 1  2

F484  



Eh3l 2 csc   96 R 2 1  





Eh cot   (h 2 4 R 2  3    l 2  1     60l 2 R 2  1    

 h  4R 

 3    l 2  1     60l 2 R 2  1    cos  2 ) F 192  1  2   R 2 sin 2    Eh3l 2 cot   4 F52  96 1    R 2 sin2    2

4

51

2





Eh(18l 2 R 2  1    2 h 2  9l 2 R 2  1   cos  2   h (l  1    2 R  1    8 R 2 sin 2   2

F  4 53

F544  F554 

2

2

 



96 R 2 1  2 sin 2  

Eh3l 2 96 1   R 2 sin2  





Eh3l 2 cot  

 48  48  R 2

Eh3l 2 F  96 R 2 1   4 56

 

Ehl 2 cot  

F115  

F125 

F  5 21

8R  8R 

3Ehl 2 8R 1  



F225  

F  5 23



4 R 1  



 h3 48R

Ehl 2 csc   4 R 1  

F  5 24

F  5 25



Eh l 2Cos   Cot    2 l 2  R 2 sin  

Ehl 2 cos   4 R 1  

Ehl 2 sin   4 R  4 R

28



Eh l 2  2 R 2

F  5 31

4 1   R



Ehl 2 8R 1   sin2  

F325  

Ehl 2 cot  

F335  

8  8  R

Ehl 2 8  8  R

F345  





EhCot   (h 2 4 R 2  3    5l 2  1     60l 2 R 2  1    

 h  4R  3    l 2

F  5 41

F425  

Eh3l 2 cot  



32 1   R 2 sin2  



Eh 18l R

F43  5

 1     60l 2 R 2  1    cos  2 ) 192  1  2   R 2 sin 2   

2

2



 1     2  h

2

2

2

 9l

2

R

2

 1    cos  2   h

Eh3l 2 F  96 1   R 2 sin2  



F455  

F465 

2

2

 1     2 R  1     8 R sin      

 3l

  1    sin

96 R

5 44

2

2

F53 

48 1   R 2

 

h

l

2

2

R

2

2

2

2

2

4

2

 2     ) cos  2   12 R

2

2

l

2

2

 2R

2

2

2

  h l 2

2

2

l

 2R

2

2

 2R

2



 1  2      cos  4 )csc 2

3

 

384 R (1   ) 2

1 3 h 12

  8R

Eh h

F 

2

2

 5l

2

 1      12l

2

R

2

 1      h

 8R  l  1      12l R 192 R  1    2

2

2

F  5 54

5 55

2

Eh3l 2 96 R 2 1  

F525   5

2

Eh3l 2 cot  

2 2

F51 

2



Eh(5h l  2h R  60l R  72 R  12h R   6h R   8(6 R 5

2





 

2

2

2

 1     cos  2   csc  

2





Eh 24l 2 R 2  h 2 3l 2  2R 2  2 h 2  12l 2 R 2 cos  2  cot   csc   96 R 1   2

Eh3l 2 cot   csc   96 R 2 1  

29

3

F  5 56









E 3h3l 2 cos   cot    4h 12l 2 R 2  h 2 l 2  R 2 sin  

F  5 57

F  5 58

F595  

96 R 2 1  



Eh3l 2 csc   96 R 2 1  

Eh3l 2 cos   48R 2 1  

Eh3l 2 sin   96 R 2 1  

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[43] J. Zhao, D. Pedroso, Strain gradient theory in orthogonal curvilinear coordinates, Int. J. Solids Struct., 45 (2008) 3507-3520. [44] M.A. Guzev, C. Qi, Equations of the strain gradient theory in curvilinear coordinates, Far Eastern Mathematical Journal, 13 (2013) 35-42. [45] A. Ashoori, M. Mahmoodi, The modified version of strain gradient and couple stress theories in general curvilinear coordinates, European Journal of Mechanics-A/Solids, 49 (2015) 441-454. [46] X. Zhou, L. Wang, Vibration and stability of micro-scale cylindrical shells conveying fluid based on modified couple stress theory, Micro & Nano Letters, 7 (2012) 679-684. [47] H. Zeighampour, Y.T. Beni, Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory, Physica E, 61 (2014) 28-39. [48] H. Zeighampour, Y.T. Beni, Cylindrical thin-shell model based on modified strain gradient theory, Int. J. Eng. Sci., 78 (2014) 27-47. [49] H. Zeighampour, Y.T. Beni, A shear deformable cylindrical shell model based on couple stress theory, Archive of Applied Mechanics, 85 (2015) 539-553. [50] H. Zeighampour, Y.T. Beni, F. Mehralian, A shear deformable conical shell formulation in the framework of couple stress theory, Acta mech, (2015) 1-23. [51] Y.T. 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, Compos. Struct., 120 (2015) 65-78. [52] A.W. Leissa, Vibration of shells, Scientific and Technical Information Office, National Aeronautics and Space Administration Washington, DC, USA, 1973. [53] J.N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis, CRC press, 2004. [54] J.N. Reddy, Theory and analysis of elastic plates and shells, CRC press, 2006. [55] M.S. Qatu, Vibration of laminated shells and plates, Elsevier, 2004. [56] E. Efraim, M. Eisenberger, Dynamic stiffness vibration analysis of thick spherical shell segments with variable thickness, Journal of Mechanics of Materials and Structures, 5 (2010) 821-835. [57] S. Hosseini-Hashemi, M. Fadaee, On the free vibration of moderately thick spherical shell panel—a new exact closed-form procedure, J. Sound Vib., 330 (2011) 4352-4367. [58] S. Hosseini-Hashemi, M.R. Ilkhani, M. Fadaee, Identification of the validity range of Donnell and Sanders shell theories using an exact vibration analysis of functionally graded thick cylindrical shell panel, Acta mech, 223 (2012) 1101-1118. [59] B. Arash, Q. Wang, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Comput. Mater. Sci., 51 (2012) 303-313. [60] R. Rafiee, R.M. Moghadam, On the modeling of carbon nanotubes: A critical review, Composites Part B, 56 (2014) 435-449. [61] C. Shu, Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation, in, University of Glasgow, 1991. [62] F. Tornabene, E. Viola, Vibration analysis of spherical structural elements using the GDQ method, Computers & Mathematics with Applications, 53 (2007) 1538-1560. [63] E. Ghavanloo, S.A. Fazelzadeh, Nonlocal elasticity theory for radial vibration of nanoscale spherical shells, European Journal of Mechanics-A/Solids, 41 (2013) 37-42. [64] M.N. Sarvi, M. Ahmadian, Static and vibrational analysis of fullerene using a newly designed spherical super element, Scientia Iranica, 19 (2012) 1316-1323. [65] M. Tavakoli, R. Singh, Eigensolutions of joined/hermetic shell structures using the state space method, J. Sound Vib., 130 (1989) 97-123. [66] E. Ghavanloo, S. Fazelzadeh, Nonlocal shell model for predicting axisymmetric vibration of spherical shell-like nanostructures, Mech. Adv. Mater. Struct., 22 (2015) 597-603. [67] E. Ghavanloo, S.A. Fazelzadeh, Radial vibration of free anisotropic nanoparticles based on nonlocal continuum mechanics, Nanotechnology, 24 (2013) 075702. [68] H. Portales, L. Saviot, E. Duval, M. Fujii, S. Hayashi, N. Del Fatti, F. Vallée, Resonant Raman scattering by breathing modes of metal nanoparticles, J Chem Phys, 115 (2001) 3444-3447. [69] M.-Y. Ng, Y.-C. Chang, Laser-induced breathing modes in metallic nanoparticles: A symmetric molecular dynamics study, J Chem Phys, 134 (2011) 094116. 32

[70] V. Mankad, K. Mishra, S.K. Gupta, T. Ravindran, P.K. Jha, Low frequency Raman scattering from confined acoustic phonons in freestanding silver nanoparticles, Vib. Spectrosc., 61 (2012) 183187. [71] R. Zaera, J. Fernández-Sáez, J.A. Loya, Axisymmetric free vibration of closed thin spherical nano-shell, Compos. Struct., 104 (2013) 154-161. [72] J. Vila, R. Zaera, J. Fernández-Sáez, Axisymmetric free vibration of closed thin spherical nanoshells with bending effects, J. Vib. Control, (2015) 1077546314565808. [73] A. Vassallo, L. Pang, P. Cole-Clarke, M. Wilson, Emission FTIR study of C60 thermal stability and oxidation, J. Am. Chem. Soc., 113 (1991) 7820-7821.

. Figure 1. Notation in shell coordinates

Figure 2. Displacement notation in FSDT.

33

(a) l=0

(b) l=1.57h

(c) l=0

(d) l=1.57h

Figure 3. Convergence analysis for first natural frequency of fullerene: (a) and (b): grid number variations. (c) and (d): angle of compatibility variations.

(a)

34

(b)

Figure 4. Variations of different natural frequencies of fullerene by changing thickness to radius ratio for various mode numbers. a) n variations, b) m variations.

35

Figure 5. Variations of natural frequencies of fullerene against changing circumferential mode numbers for different thickness ratio considering effect of MCST.

Figure 6. Variations of non-dimensional fundamental frequency of fullerene under variations of h/R for different scale parameter.

Table 1. Material properties. 36

Material

E (TPa)



  kg / m3 

R (nm)

h (nm)

Fullerene [64]

1.05

0.159

1650

0.33

0.34

Steel [65]

207

0.3

7850

114.3

5.7

Silver (1) [63]

86.63

0.36

10500

1.3875

0.225

Silver (2) [63]

86.63

0.36

10500

1.8875

0.225

Silver (3) [63]

86.63

0.36

10500

4.7875

0.225

Gold (1) [63]

85.45

0.42

19280

2.7875

0.225

Gold (2) [63]

85.45

0.42

19280

5.6375

0.225

Gold (3) [63]

85.45

0.42

19280

9.9875

0.225

Table 2. First ten fundamental natural frequencies for a steel sphere. Frequency (Hz)

1 2 3 4 5 6 7 8 9 10

MCST ( l  0 ) 5178

FEM

Ref. [63]

Err. 1 (%)

Err. 2 (%)

5281.9

5281

1.97

1.95

6640

6311.1

6321

5.21

5.04

6950

6846.1

6883

1.52

0.98

7305

7336.4

-

0.43

-

8005

7946.2

-

0.74

-

8839

8755.4

-

0.95

-

9840

9801.8

-

0.39

-

11025

11096

-

0.64

-

12210

12125

-

0.70

-

12984

12632

-

0.03

-

Table 3. Fundamental natural frequency for various nano shells. Material Silver (1) Silver (2) Silver (3) Gold (1) Gold (2) Gold (3)

MCST ( l  0 ) (THz) 0.740 0.595 0.296 0.341 0.214 0.130

MCST considering l (THz)

Experimental Result (THz)

Reference

1.02 ( l  0.89h ) 0.834 ( l  0.85h ) 0.327 ( l  0.79h ) 0.557 ( l  0.87h ) 0.279 ( l  0.84h ) 0.158 ( l  0.84h )

1.03 0.828 0.330 0.552 0.281 0.161

Ref. [70] Ref. [70] Ref. [68] Ref. [69] Ref. [69] Ref. [69]

Table 4. First six fundamental natural frequencies for fullerene. 37

Frequency (Hz)

1 2 3 4 5 6

MCST ( l  0 ) THz 6.95

MCST ( l  1.57h ) THz

Ref. [73] (THz)

8.056

8.08

8.43

10.061

10.63

11.39

14.74

14.80

13.87

17.45

17.74

19.61

24.91

25.60

27.59

34.98

35.69

Highlights: 

Free vibrations of a micro/nano sphere is analyzed based on modified couple stress theory.



Accuracy and validity of the results are approved by comparing them with the experimental results.



New scale parameters are proposed for spherical nano-gold, nano-silver and fullerene.



Effects of different geometries and scale parameter on frequencies are investigated.



It is approved that increasing scale parameters has more effect on higher mode numbers.

38