Isogeometric free vibration of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials

Journal Pre-proofs Isogeometric free vibration of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials Yantao Zhang, Guoyong Jin, Mingfei Chen, Tiangui Ye, Zhigang Liu PII: DOI: Reference:

S2211-3797(19)33023-2 https://doi.org/10.1016/j.rinp.2019.102889 RINP 102889

To appear in:

Results in Physics

Received Date: Revised Date: Accepted Date:

11 October 2019 14 December 2019 16 December 2019

Please cite this article as: Zhang, Y., Jin, G., Chen, M., Ye, T., Liu, Z., Isogeometric free vibration of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials, Results in Physics (2019), doi: https://doi.org/10.1016/j.rinp.2019.102889

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Isogeometric free vibration of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials Yantao Zhang, Guoyong Jin*, Mingfei Chen, Tiangui Ye, Zhigang Liu College of Power and Energy Engineering, Harbin Engineering University, Harbin, 150001, P. R. China

Abstract An isogeometric numerical procedure based on non–uniform rational B–splines is applied to solve the free vibration of carbon nanotubes reinforced and functionally graded material sector cylindrical shells. A typical isogeometric analysis cylindrical shell element is introduced to transform curvilinear domain to parametric domain. The effects of the shear deformation, rotary inertia and axis extensibility are taken into consideration. The material properties are estimated by a mixture rule and effective carbon nanotube parameters. The weak form is derived in detail by applying virtual principle. The novelty of this paper is to use the isogeometric finite element method to analyze the dynamic behaviors of sector cylindrical shells with functionally graded material and four types of carbon nanotubes distributed material. By numerical examples, the convergence and accuracy of the current method are validated. Then, a series of natural frequencies and mode shapes are presented to serve as benchmark solutions in future researches. Moreover, the effects of boundary conditions, geometric proprieties as well as material parameters on the frequencies of the carbon nanotubes reinforced and functionally graded material sector cylindrical shells are examined.

Keywords: Free vibration; Carbon nanotubes; functionally graded material; Sector cylindrical shells; Isogeometric analysis

1. Introduction *

Corresponding author, Tel: +86 451-82589199, E-mail address: [email protected] (G, Jin)

1

In recent years, as one of novel excellent promising structural materials, the carbon nanotubes reinforced (CNTR) materials have gradually captured the attention of numerous material engineers and scientists. Due to the remarkable thermal, electrical and mechanical properties [1], CNTR material is usually regarded as a candidate for reinforcement of engineering structures. Simulations and experimental studies show that CNTR materials have exceptional properties over carbon fibers [2] . Additionally, compared with metallic and polymeric materials, CNTR materials possess higher elasticity modulus, resulting in the stiffness of structure enhanced. The static and dynamic behaviours of CNTR beams, plates and shells have been extensively studied by many researchers with different solutions, such as experimental, analytical and numerical investigations [3-7]. The mechanical behaviours of CNTR structures have drawn increasing attention. Heshmati and Yas [8] solved the vibration of CNTR nanocomposite beams by adopting Eshelby-Mori-Tanaka method. Lin and Xiang [9, 10] used a Ritz method to study the vibration of CNTR beam. In their work, the first-order as well as third order beam theories are considered. Ke et al.[11] used the Ritz method to solve the nonlinear vibration of CNTR beam. Similarily, the CNTR plates and shells were investigated by many methods, for instance, Zhang et al. [12-16] used an element-free IMLS-Ritz approach to investigate the buckling and free vibrational characteristic for CNTR skew, rectangular and triangular plates. Shahrbabaki and Alibeigloo [17] used a kp-Ritz method to solve three-dimensional (3D) vibration problem of CNTR plates. Malekzadeh et al. adopted a differential quadrature method (DQM) to analyse the natural vibration of plates with edges restrained elastically[18], the large amplitude vibration of tapered plates [19], orthotropic skew plates [20] and linear vibration of laminated thick plates [21]. Then Malekzadeh et al. applied the DQM to analyse free vibration characteristic of the 3D CNTR composite laminated plates [22] 2

and quadrilateral plates [23]. Alibeigloo et al. also used the DQM to analyse vibration of CNTR beams [24] and plates[25], and then they studied 3D thermo-elasticity solution of CNTR plates with piezoelectric layers [26]. Zhu et al. [27] applied the finite element method (FEM) to investigate the vibration of the CNTR plates with the first-order shear deformation theory (FSDT). Mehar et al. [28] also used the FEM to solve the nonlinear thermoelastic vibration analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels. Shen and Zhang [29, 30] studied the thermal buckling mechanical characteristic for CNTR plates and shells with perturbation method. Qin et al. [31] used a Chebyshev–Ritz formulation to solve the natural vibration of rotating CNTR cylindrical shells with general boundary conditions. Nguyen and his cooperators studied the nonlinear dynamic behaviors of CNTR truncated conical shells in thermal environment [32] and nanocomposite elliptical cylindrical shells resting on elastic foundations [33]. To meet the demands of higher thermal and mechanical specifications, functionally graded materials (FGMs) have been employed for many engineering structures, such as FGM beams, plates and shells. Li et al. [34, 35] used a semi-analytical method for the natural vibration of the functionally graded porous (FGP) and stepped FGM curved shells with elastic boundary conditions. Qin et al. [36, 37] used a Rayleigh-Ritz approach to solve the vibration of cylindrical shells and nanocomposite sandwich plate under thermo-mechanical loads, and then they analysed traveling wave of rotating functionally graded graphene platelet reinforced nanocomposite cylindrical shells [38]. Jin et al. [39, 40] analysed the vibration of FGM and laminated doubly-curved shells by the modified Fourier series method. Later, they also used this method to solve the thermo-elastic vibration of FGM beams based on higher-order shear deformation theory (HSDT) [41]. Duc and his co-authors developed various theoretical and numerical approaches to carry out the nonlinear 3

dynamic and stability analysis of FGM composites structures [42-48]. They also studied the nonlinear buckling of eccentrically oblique stiffened sandwich FGM double curved shells [49] and discussed the effect of cracks on the stability of the FGM plates with variable-thickness [50]. Some researchers have moved their attention into the mechanical analysis of FGM/CNTR structures. Baltacioglu and Civalek [51] calculated the vibration frequencies of the curved shells with CNT reinforced and FGM composites. By compared with other numerical methods, the isogeometric analysis (IGA) based on non– uniform rational B–splines (NURBS) [52] has many advantages, such as, geometry exactness, robust mesh refinement and higher order continuity. In addition to frictional contact [53], blood flow [54] and design optimization [55], the IGA has been widely applied in the static and dynamic analysis of various structures [56-63]. Nguyen et al. [64, 65] also analyzed the postbuckling and vibration of CNTR shells. To make the local refinement feasible, the adaptive extended isogeometric analysis (XIGA) based on the polynomial splines over hierarchical T-meshes (PHT-splines) [66], adaptive XIGA based on locally refined (LR) B-splines [67], and adaptive multi-patch isogeometric analysis based on LR B-splines [68] were proposed. However, to the best of authors’ knowledge, the isogeometric approach has not been applied to analyze free vibration of the FGM/CNTR sector cylindrical shells yet. In this present work, the isogeometric formulation is applied to investigate the free vibration of FGM/CNTR sector cylindrical shells with FSDT. Four CNT distribution types and functionally graded materials along the thickness direction of shells are approximated by effective CNT parameters with the mixture rule. By several numerical examples, the accuracy and convergence of the current solution are demonstrated. The effects of geometrical properties, boundary conditions 4

and material properties are investigated in detail.

2. FGM/CNTR cylindrical shell theory 2.1

FGM/ CNTR cylindrical shell The considered FGM/CNRT cylindrical shell is shown in Fig.1. The Cartesian and curvilinear

coordinate systems are used in this analysis.

X , Y, Z

and x, y, z

represent Cartesian and

curvilinear coordinate axes, respectively. Symbols r,  , h and L represent respectively the radius, angle, thickness, and length of the shell. The symbols u , v , w and x ,  y displacements and rotation components, respectively. Edge 2

Edge 3

w

L

u,  x

Edge 1

v,  y x

Edge 4 r

y



h

Z Y

z

X

Fig. 1. The coordinates and model of the FGM/CNTR sector cylindrical shell.

(b)

(a)

5

denote the

(c)

(d)

Fig. 2. CNT Configurations in the section. (a) type-UD; (b) type -V; (c) type -O; (d) type -X. Assume that the sector cylindrical shell is made up of top material and bottom material. Meanwhile, the FGM sector cylindrical shell is also reinforced by CNT. Based on the Voigt’s rule, the effective FGM properties can be calculated as  E FG  ( Et  Eb )Vt  Eb  FG    ( t  b )Vt  b  FG    ( t  b )Vt  b

(2)

where E ,  and  denote respectively the Young modulus, mass density and Poisson ratio. The superscript FG denotes the effective FGM properties. Subscripts b and t denote the bottom and top material, respectively. Vt is the top material volume fraction, which can be defined as 1 z Vt     2 h

pn

(3)

where superscript pn denotes the volume fraction index. The CNT distributions: Type-UD, Type-V, Type-O and Type-X are given in Fig. 2. Assume they are defined as follows: * VCNT ( z )  VCNT  V ( z )  (1  2 z )V * CNT  CNT h  2z *  )VCNT VCNT ( z )  2(1  h  4z *  VCNT VCNT ( z )  h 

where

6

(Type-UD) (Type-V) (Type-O) (Type-X)

,

(4)

* VCNT 

wCNT   

CNT

wCNT   wCNT    CNT  m  m

(5)

in which  is the density, and superscripts CNT and FG denote carbon nanotube and FG material.

wCNT is carbon nanotube mass fraction. Furthermore, the relation between FG material volume fraction VFG and CNT’s volume fraction VCNT is the following form

VFG  VCNT  1

(6)

The effective material properties E11 , E22 and G12 are written by employing a mixture rule [69, 70] with CNT effective parameters

E11  1VCNT E11CNT  Vm E FG

2 E22

3 G12

(7)



VCNT VFG  FG CNT E E22

(8)



VCNT VFG  G12CNT G FG

(9)

where G and E denote shear modulus and Young modulus. Meanwhile, the CNT effective parameters i (i  1, 2, 3) are introduced for scale-dependent material properties. Similarly, effective Poisson ratio 12 and density  can be written as * 12  11CNT VCNT   FGVFG

(10)

   CNT VCNT   FGVFG

(11)

where Poisson ratio 12 is regarded as constant value along the thickness direction. 2.2

Kinematic relations According to the assumptions of FSDT, the displacements of the FGM/ CNTR cylindrical shell

are expressed:

w( x, y, z , t )  w( x, y, t )

(12)

u ( x, y, z , t )  u ( x, y, t )  z x ( x, y, t )

(13)

7

v ( x, y, z , t )  v( x, y, t )  z y ( x, y, t ) ,

(14)

where w , v and u represent respectively the displacements of mid-surface in the z, y and x directions.  y and  x are the rotations with respect to y and x axes:

u v w v u ,  y0   ,  xy0   x y R x y

(15)

w w v   x ,  yz0    y , x y R

(16)

  y  x  x .  ,  y  y ,  xy  x y x y

(17)

 x0 

 xz0  x 

And the linear strain-displacement relations as

 x   x0  z  x ,  y   y0  z  y ,  xy   xy0  z  xy

 xz   xz0 ,  yx   yz0 .

(18) (19)

The strain-stress relationship can be expressed as

 x   Q11 ( z ) Q12 ( z ) 0 0 0   x       0 0 0   y   y  Q12 ( z ) Q22 ( z )     0 0 0   xy  , Q66 ( z )  xy    0     0 0 0 0   xz  Q66 ( z )  xz     0 0 0 Q66 ( z )   yz   yz   0    

(20)

where the Qij ( z ) (i, j = 1, 2, 6) denote the elastic constants, and they are expressed as

E11 ( z )   1   ( z )  ( z )  12 21  Q11 ( z )   Q ( z )   21 ( z ) E11 ( z )   12       1  12 ( z ) 21 ( z )  . Q22 ( z )    E22 ( z ) Q66 ( z )    1  12 ( z ) 21 ( z )    G12 ( z )   The stresses and moments can be given as

N , N x

y

, N xy , M x , M y , M xy   

h /2

 h /2

 8

x

,  y , xy , z x , z y , z xy  dz

(21)

(22)

Q , Q    x

where M x , M y , M xy

and

y

kn  xz , yz  dz ,

h /2

 h /2

(23)

denote respectively the moments and in-plane force

N x , N y , N xy

resultants. Qx and Qy denote the transverse shear force resultants. The shear correction factor kn in Eq. (23) is set as 5/6 in this paper. Through substituting Eqs. (18)-(21) into the stresses and moments in Eqs. (22) and (23), the following matrix can be obtained Nx   A  N   11  y   A12  N xy   0     M x   B11    M y   B12 M   0  xy   Qx   0    0 Qy  

where

Aij , Bij and Dij

A12 A22

0 0 A66

0 B12 B22

0 0 B66

0 0 0

B11 B12

B12 B22

0 D11 D12

0 D12 D22

0 0 0

0 0 0

0 0

0 0 B11 0 0 D66 0 0

0 0 0 0 0 0 kn A66 0

0    x0    0    y0  0    xy0    0   x   , 0   y   0    xy    0    xz   kn A66    yz 

(24)

( i, j  1, 2, 6 ) represent extensional, coupling, and bending elastic

stiffness coefficients, respectively, written as

A , B , D    ij

ij

ij

h /2

 h /2

Qij ( z ) 1, z , z 2  dz .

(25)

The strain energy ( U st ) and kinetic energy ( Tst ) of the FGM/CNTR sector cylindrical shell are expressed as

U st 

1  x x   yz yz   y y   xz xz   xy xy  dzdydx , 2 V

 u 2  v 2  w 2  1 Tst             dxdydz V 2  t   t   t  

(26) (27)

where V represents the integral volume. The strain and kinetic energy can be expressed as

U st 

1  N x x0  M x  xz  N y y0  M y  y  N xy xy0  Qx xz  Qy yz dxdy , 2 S

9

(28)

2 2 2      I 0  u    v    w      t   t   t       u  x v  y   1  Tst   2 I1    dxdy ,  2 S t t   t t    2 2    I   x     y     2  t   t        where S represents the integral surface. I 2 , I1 and I 0 are the inertia terms and given by

I 2 , I1 , I 0    h /2  ( z )  z 2 , z,1 dz h /2

(29)

(30)

Thus, the total energy  of a FGM/CNTR sector cylindrical shell can be expressed as

  Tst  U st

(31)

By applying the Hamilton’s principle, the equilibrium equations of the FGM/CNTR sector cylindrical shell are derived as N x N xy  2  2u   I 0 2  I1 2x x y t t

N y y



N xy x



Qy R

 y  2v  I1 2 2 t t 2

 I0

(33)

Qx Qy N y 2w    I0 2 x y R t

(34)

M x M xy  2 x  2u   I1 2  I 2 2 x y t t

(35)

  2v   I1 2  I 2 2 y . y x t t

M y

2.3

(32)

M xy

2

(36)

NURBS basis functions The B-spline basis functions can be defined by a number of ways [71], such as by a recurrence

formula, by blossoming, and by divided differences of truncated power functions. As one of the useful ways for computer implementation, the recurrence formula is used in this paper. Let E = { 1 ,

2 , 3 ⋯  n  p ,  n  p 1 } be a non-decreasing knot vector i.e., i 1  i ,1  i  n  p  1 . The ith B-spline 10

function Bi ,0 ( ) is given as

1 if i    i 1 Bi ,0 ( )   . 0 otherwise

(37)

For the Bi , p ( ) (p>0)

Bi , p ( ) 

i 1     i ( ) , Bi , p 1 ( )   p  B i  p  i i  p 1  i 1 i 1, p 1

(38)

To depict the free-form curves and surfaces, the NURBS functions can be derived by introducing weights N ip, j,q ( , ) 

i , j Bi , p ( ) M j ,q ( )

  m

j 1

n

 B ( ) M j ,q ( ) i 1 i , j i , p

,

(39)

where Bi , p ( ) and M j ,q ( ) are the B-spline basis functions. p, q and n, m represent the polynomial orders and numbers of basis functions, respectively. i , j is the associated weight. The displacement field U of the FGM/CNTR cylindrical shell is approximated by NURBS, given as nm

U( , )   N a ( , )u a .

(40)

a 1

For each element: Nel

U e ( , )   N a ( , )u ea ,

(41)

a 1

where N a ( , ) denotes the NURBS function, u a  (ua , va , wa ,  xa ,  y a ) represents unknown variables. Nel denotes the number of control points in each element. 2.4

Isogeometric formulation As described above, the NURBS function possesses many advantages of the B-splines.

Therefore, the NURBS function can be used as interpolating function in IGA. As the basis functions are constructed based on parameter vector, there is a parameter domain for NURBS functions in 11

 denote the curvilinear, parametric and parent ˆ and  IGA, seen in Fig. 3 [62]. Symbols  , 

ˆ and f :  ˆ   denote the mapping functions. domains, respectively. The S :    Curvilinear domain

e

y

S

Z x

ξi+1

 j 1  ˆe

X

1

j

ξi Parametric domain

Y

ξ



-1

 

1

-1 Parent domain

f

Fig. 3. NURBS shell elements for the mapping from curvilinear domain to parent domain.

ˆ   is expressed as: The relationship of mapping f :  1 1  (   )  (i 1  i )     2 i 1 i  2   , 1 1      (   )  (   )  j j 1 j  2 j 1  2

(42)

then the associated Jacobian determinant is expressed as:

1 | J | (i 1  i )( j 1   j ) . 4 The relationship of the curvilinear coordinate y with nature coordinate can be written as: y ( ) 

 0 



 0 0

(

dY 2 dZ )  ( 0 ) 2 d 0 , x( )  X ( ) , 0 d d

(43)

(44)

and the radius of curvature is 3

 dY 2 dZ 2  2  ( d )  ( d )   . R( )   dY d 2 Z dZ d 2Y  d d 2 d d 2

(45)

For the FGM/CNRT cylindrical shell with FSDT, the first-order partial derivatives are given as 12

follows:     x            x        

y            x     J  x  , y          y     y 

(46)

where J is the associated Jacobian matrix. It can be seen easily that y /   0 and x /   0 from Fig. 3 and Eqs. (44)-(46). Therefore, J is expressed as:  x   J    0 

 0  . y   

(47)

The following partial derivatives can be obtained from Eq. (45)        x     J 1   .         y   

(48)

where J 1 is the inverse of the Jacobian matrix J . Substituting Eqs. (40) and (41) into Eq. (31), the vibration governing equations of the FGM/CNTR cylindrical shell are obtained as 2 K  k M  d k  = 0 ,

(49)

where K and M represent respectively global stiffness matrix and mass matrix. k and d k represent respectively the kth eigenvalue and associated eigenvector. The d k  u, v, w,  x ,  y 

T

are expressed as

w  {w1 , w2  wnm }, u  {u1 , u2  unm }, v  {v1 , v2  vnm }

(50)

 y  { y ,1 ,  y ,2  y ,nm },  x  { x ,1 ,  x ,2  x ,nm } .

(51)

In the FEM, the global matrices can be derived by assembling operator. The element matrix

K e can be easily expressed as

13

K e   e B kI  Ck  B kI  dxdy ,  T

(52)

with

 A11 A  12  0  B Ck    B11 12   0  0   0

A12 A22 0 B12 B22 0 0 0

 RI , x  0   RI , y  0 k B I     0   0  0   0

0 0 A66 0 0 B66 0 0

B11 B12

B12 B22

0 D11 D12

0 D12 D22

0 0 0

0 0 0

0 0 B11 0 0 D66 0 0

0 0 0 0 0 0 kn A66 0

0 0 0 RI , y RI / R 0 RI , x 0 0 0 0 RI , x 0 0 0 0 0 RI , y 0 RI , x RI 0  RI / R RI , y

0  0  0   0  , 0   0  0   kn A66 

0  0  0   0  , RI , y   RI , x  0   RI 

(53)

(54)

where RI  {N1 , N 2 , N Nel } is the NURBS vectors, RI , x and RI , y represent the first-order partial derivatives of RI . The M e can be expressed as

M e   e B mI  Cm  B mI  dxdy ,  T

(55)

with  I0 0  Cm    0   I1  0

0 I0 0 0 I1

0 0 I0 0 0

I1 0 0 I2 0

 RI 0  B mI    0  0  0

0 RI 0 0 0

0 0 RI 0 0

0 0 0 RI 0

0 I1  0,  0 I 2  0 0  0 .  0 RI 

(56)

(57)

By using the transformation formulas in Eqs. (41)-(47), the integral expressions in Eqs. (52) 14

and (55) can be expressed



e

f ( x, y )dxdy   ˆ e f ( , ) | J | d  d    e f (, ) | J |  | J | d d . 



(58)

By using finite element assembling operator, the global matrices in Eq. (48) can be derived easily. After solving the eigenvalue problems in Eq. (49) with Arnoldi method, the mode shapes of the FGM/CNTR sector cylindrical shell are calculated by the natural eigenvectors and associated NURBS basis functions. The integral results can be calculated by numerical integration with the Gauss quadrature. The physical region of each element is transformed to parent region for numerical integration. More integral details can be seen in Ref. [73]. In this paper, Fig. 1 shows the edges of the FGM/CNTR shell named along the anticlockwise directions. Symbols F, S and C represent that associated edges are free, simply supported and clamped, respectively. The classical boundary conditions of the FGM/CNTR shell for edges 1 and 3 can be defined as

w  u  v  x   y  0   w  u  v  0, M y  M xy  0   N y  N xy  Qy  M y  M xy  0

(C) (S)

(59)

(F)

For convenience, a four-letter string is utilized to represent the restraint conditions of the FGM/CNTR shell, for instance, CCCC and SSSS mean that all edges are clamped and simply supported, respectively, and CSCF means that edges 1 and 3 are clamped, but edge 2 is simply supported and edge 4 is free.

2.5

Refinement algorithms The order of NURBS functions and control points can affect the accuracy of IGA. There are

many kinds of refinement strategies, seen in Ref. [72]. In this paper, three refinements (p–, h– and 15

k–refinements) are discussed briefly. According to the knot vector E = {0 0 0 0.5 1 1 1}, Fig. 4 (a), Fig. 5 (a) and Fig. 6 (a) give the unrefined curves and associated NURBS basis functions. Fig. 4(b) and Fig. 4(c) mean the first two h–refinements. New knots are inserted in non–zero knot span in the h–refinement. Fig. 5(b) and Fig. 5(c) present respectively the first and second p–refinements. The p–refinement can increase the orders of NURBS functions. The k-refinement can firstly elevate the order of original curve and then insert new knots. Fig. 6 (b) and Fig. 6 (c) present the first two k-refinements.

1

1

1

0.5

0.5

0.5

0 0

0.5 (a)

1

0 0

0.5 (b)

1

0 0

0.5 (c)

1

Fig. 4. Curves and relevant B-spline functions: (a) initial curve; (b) the first h-refined curve; (c) the second h-refined curve.

1

1

1

0.5

0.5

0.5

0 0

0.5 (a)

1

0 0

0.5 (b)

1

0 0

0.5 (c)

1

Fig. 5. Curves and relevant B-spline functions: (a) initial curve; (b) the first p-refined curve; (c) the second p-refined curve.

16

1

1

1

0.5

0.5

0.5

0 0

0.5 (a)

0 0

1

0.5 (b)

1

0 0

0.5 (c)

1

Fig. 6. Curves and relevant B-spline functions: (a) initial curve; (b) the first k-refined curve; (c) the second k-refined curve.

3. Numerical results 3.1

Convergence study Table 1 gives material properties of Aluminum (Al), Zirconium oxide (ZrO2) and CNT fiber.

The geometrical parameters: r=1m,   90 , L=2m, h=0.1m and the material properties in table 1 are used in the rest analysis unless otherwise noted. Table 2 provides the effective parameters * i (i  1, 2,3) of the various volume fraction VCNT .

Table 1. Material properties.

CNT Al ZrO2

E11 (Gpa)

E22 (Gpa)

G12 (Gpa)

12

  kg/m3 

5646.6 70 168

7080 70 168

1944.5 – –

0.175 0.3 0.3

1400 2707 5700

Table 2. Effective parameters of the selected CNT. * VCNT

1

2

3

0 0.11 0.14 0.17 0.28

0 0.149 0.150 0.149 0.141

1 0.934 0.941 1.381 1.582

1 0.934 0.941 1.381 1.582

17

In consideration of capacity limitation and calculation speed, the mesh of the CNTR/FGM shell should be refined appropriately. Therefore, it is necessary to conduct a convergence study by using this method. Table 3 gives the first six natural frequencies of the FGM sector cylindrical shell * reinforced by UD-Type CNT with CSCS boundary. The volume fraction VCNT =0.11 and the FGM

power-law index pn  1 are considered in this table. The mesh of the CNTR/FGM shell is refined by IGA with orders p=q=2, p=q=3, p=q=4 and elements 4  4, 8  8, 16  16, 32  32. As seen from Table 3, the IGA method with higher-order function has faster convergence and the IGA method with the order p=q=4 and element 16  16 can be regarded as convergent. Therefore, the IGA with order p=q=4 and element 16  16 are used in the following discussions. Table 3. The first six frequencies f   / (2 ) (Hz) of the FGM/CNTR sector cylindrical shell * with CSCS boundary condition (Type-UD, pn  1 , VCNT =0.11).

Order

Frequency f (Hz)

Frequency f (Hz)

1400 1200 1000 1st 2nd 3rd

800 600 400 0.1

Type-V

1600

0.2

0.3 *

Volume fraction VCNT

1400 1200 1000 1st 2nd 3rd

800 600 400 0.1

4 1076.27 911.65 907.07 906.73 929.92 907.48 906.77 906.70 911.37 906.93 906.71 906.70

0.2

1200 1000

*

1st 2nd 3rd

800 600 0.2

0.3 *

Volume fraction VCNT

Volume fraction VCNT

18

6 1495.71 1255.70 1234.46 1233.56 1303.72 1237.56 1233.57 1233.52 1258.21 1234.03 1233.52 1233.51 Type-X

1600

1400

400 0.1

0.3

5 1306.93 1146.08 1139.21 1138.92 1188.64 1139.64 1138.94 1138.91 1149.12 1139.04 1138.91 1138.91

Type-O

1600

Frequency f (Hz)

Type-UD

1600

3 918.31 884.75 882.80 882.67 892.21 882.95 882.68 882.66 884.49 882.74 882.66 882.66

Frequency f (Hz)

Element Mode No. 1 2 p=q=2 4  4 809.00 849.62 8 8 661.37 764.92 16  16 656.81 761.31 32  32 656.59 761.18 p=q=3 4  4 681.22 781.42 8 8 657.04 761.50 16  16 656.60 761.18 32  32 656.57 761.17 p=q=4 4  4 661.16 765.59 8 8 656.67 761.22 16  16 656.58 761.17 32  32 656.57 761.17

1400 1200 1000 1st 2nd 3rd

800 600 400 0.1

0.2

0.3 *

Volume fraction VCNT

(a)

1000 800 1st 2nd 3rd

400 200 0.1

0.2

1200 1000 800 400 200 0.1

0.3

1st 2nd 3rd

600

*

0.2

1000 800 400 200 0.1

0.3

1st 2nd 3rd

600

0.2

1200 1000 800 400 200 0.1

0.3

1st 2nd 3rd

600

*

Volume fraction VCNT

Type-X

1400

1200

*

Volume fraction VCNT

Type-O

1400

Frequency f (Hz)

Frequency f (Hz)

Frequency f (Hz)

1200

600

Type-V

1400

Frequency f (Hz)

Type-UD

1400

0.2

0.3 *

Volume fraction VCNT

Volume fraction VCNT

(b)

800 1st 2nd 3rd

400 0.1

0.2

Volume fraction

0.3 * VCNT

1000 800 1st 2nd 3rd

600 400 0.1

0.2

Volume fraction

Type-O

1200 1000 800

400 0.1

0.3 * VCNT

1st 2nd 3rd

600 0.2

Volume fraction

Type-X

1400

Frequency f (Hz)

Frequency f (Hz)

Frequency f (Hz)

1000

600

Type-V

1200

Frequency f (Hz)

Type-UD

1200

0.3 * VCNT

1200 1000 800

1st 2nd 3rd

600 400 0.1

0.2

Volume fraction

0.3 * VCNT

(c) Fig. 7. Variation of natural frequencies versus the CNT volume fractions of sector cylindrical shell: (a) CCCC boundary condition; (b) SSSS boundary condition; (c) CSCF boundary condition. 3.2

Validation of results The rectangular plate can be thought as one of the sector cylindrical shells when all the weights

are equal to 1. Consequently, the validation of natural vibrations of the CNTR rectangular plates and FGM sector cylindrical shells should be presented. Tables 4 shows the comparison of the non-dimensional frequency parameters    ( L2 / h )  t / Et

for a CNTR square plate under

CCCC boundary condition. Et  Eb = 2.1Gpa , t  b  0.34, t  b  1150kg/m3 and the CNTR materials properties listed in Table 1 are used in this analysis. The compared results in Table 4 are obtained from Zhu et al. [27]. To further validate the accuracy of the developed solution, Table 5 gives the comparison of the fundamental frequencies f   / (2 ) (Hz) of a CCCC FGM sector cylindrical shell. The top and bottom materials are set as ZrO2 and Al (seen in Table 1), 19

respectively. The geometrical parameters are given as: R=1m, L/R=2, h/R=0.1. The referential results are calculated by modified Fourier series method [74]. The difference between the current result and referential result is calculated by (diff .)  rc  rf rf , where rc and rf

mean

respectively the current result and referential result. As seen from Table 4, the (diff .)  3% and a well agreement for vibration results of the CNTR square plate is achieved. For the results tabulated in Table 5, the (diff .) is smaller than 0.03% and the current vibration results of FGM cylindrical shell do agree well with the results obtained by Su et al. [74]. Consequently, the accuracy of vibration results of cylindrical shell with FGM and CNTR materials are verified. Table 4. Comparison of the first three non-dimensional frequencies    ( L2 / h)  m / Em

* VCNT

0.11

0.14

0.17

CNTR square plate with CCCC boundary condition. Mode Type-UD Type-V L/h No. Current Ref. [27] Current Ref. [27]  10 1 17.966 17.625 1.94% 17.511 17.211 2 23.227 23.041 0.81% 22.968 22.818 3 33.332 33.592 0.77% 33.263 33.070 20 1 28.302 28.400 0.34% 26.529 26.304 2 32.906 33.114 0.63% 31.568 31.496 3 43.658 44.559 2.02% 42.936 43.589 10 1 18.494 18.127 2.02% 18.123 17.791 2 23.780 23.572 0.88% 23.589 23.413 3 34.003 34.252 0.73% 33.999 34.101 20 1 30.339 29.911 1.43% 28.234 27.926 2 34.797 34.516 0.82% 33.129 32.976 3 45.451 45.898 0.97% 44.404 44.989 10 1 22.416 22.011 1.84% 21.896 21.544 2 29.013 28.801 0.74% 28.778 28.613 3 41.666 42.015 0.83% 41.730 41.431 20 1 35.707 35.316 1.11% 32.932 32.686 2 41.462 41.253 0.51% 39.332 39.279 3 54.938 55.627 1.24% 53.700 54.560

20

 1.74% 0.66% 0.58% 0.85% 0.23% 1.50% 1.86% 0.75% 0.30% 1.10% 0.46% 1.30% 1.63% 0.58% 0.72% 0.75% 0.13% 1.58%

of a

Type-O Current Ref. [27] 16.968 16.707 22.343 22.253 32.476 32.378 24.634 24.486 29.788 29.795 41.170 41.895 17.606 17.311 22.899 22.782 33.041 33.411 26.352 26.127 31.254 31.186 42.376 43.034 21.138 20.833 27.745 27.651 40.279 40.501 30.480 30.325 36.814 36.848 50.835 51.757

 1.56% 0.41% 0.30% 0.60% 0.02% 1.73% 1.70% 0.51% 1.11% 0.86% 0.22% 1.53% 1.46% 0.34% 0.55% 0.51% 0.09% 1.78%

Table 5. Comparison of the fundamental frequencies f   / (2 ) (Hz) of a CCCC FGM sector cylindrical shell ( Eb  70Gpa , ρ b  2707 kg / m3 , b  0.3 , Et  168Gpa , t  5700kg/m3 , t  0.3 R=1m, L/R=2, h/R=0.1) . 60  pn Curren Ref.  t [74] 0. 0.004 897.79 897.83 5 % 0.002 1 889.68 889.70 % 0.002 5 886.60 886.62 % 0.002 20 876.24 876.26 % 3.3

Curren t

120  Ref. [74]

493.83

493.87

489.18

489.20

493.74

493.76

488.41

488.43

 0.008 % 0.005 % 0.005 % 0.004 %

Curren t

240  Ref. [74]

396.56

396.64

0.021%

392.86

392.93

0.017%

396.15

396.22

0.017%

391.77

391.84

0.019%



Effects of parameters In this part, the vibration behaviors of the FGM/CNTR cylindrical shells are discussed in a few

examples. Then, the influence of boundary condition, geometry and material parameters is investigated and some mode shapes of FGM/CNRT cylindrical shells are given. Tables 6-8 give the influence of CNT volume fraction on the first six vibration frequencies

f   / (2 ) (Hz) of the FGM/CNTR sector cylindrical shells with various boundary conditions, including CCCC, SSSS and CSCF. The natural vibration frequencies

f   / (2 ) (Hz) of the

*  0.28 under CCCC boundary restraint is the Type-X FGM/CNT cylindrical shell with VCNT

largest in these tables. Through comparing Table 6 with Table 7, it is easily seen that the stronger boundary conditions can result in the larger natural vibration frequencies. The variations of the natural vibration frequencies versus CNT volume fractions with CCCC, SSSS and CSCF boundary restraints are presented in Fig. 7. For different CNT distribution types, the first three frequency parameters increase with the increasing volume fraction. As another important factor for analyzing

21

the vibration characteristics, the mode shapes are also investigated in this paper. Figs. 8-10 present some mode shapes of the FGM/CNTR shell with different CNT distribution types, volume fractions * VCNT as well as boundary conditions, respectively. The associated frequencies can be found in

Tables 6-8. These figures show that the mode shapes of the FGM/CNTR sector cylindrical shell are * independent of volume fraction VCNT and CNT distribution type, but can be affected by the types

of boundary condition directly. Table 6. Effects of the CNT volume fraction on the first six natural frequencies f   / (2 ) (Hz) of the FGM/CNTR sector cylindrical shell with CCCC boundary condition. * VCNT

Type-UD

Type-V

Type-O

Type-X

0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28

Mode No.

1 683.28 705.26 873.98 1042.69 690.16 714.27 890.00 1082.80 670.60 688.40 851.33 999.50 699.82 728.39 909.67 1137.70

2 780.56 805.37 1003.39 1198.80 788.94 817.58 1025.99 1266.51 774.76 798.28 996.44 1196.76 791.01 819.98 1025.84 1261.30

3 958.12 995.16 1206.12 1439.19 953.40 991.90 1210.99 1487.60 935.04 966.67 1175.79 1405.85 984.39 1029.12 1250.08 1532.36

4 984.01 1020.01 1238.92 1474.55 987.40 1024.93 1253.20 1519.95 956.85 985.33 1196.71 1407.08 1014.33 1060.13 1294.58 1602.27

5 1244.70 1283.86 1569.20 1869.41 1243.05 1292.55 1538.95 1873.27 1215.86 1254.25 1490.72 1764.15 1272.42 1322.36 1651.07 2025.86

6 1282.87 1340.63 1612.13 1925.51 1266.75 1315.66 1639.60 2005.06 1236.44 1275.53 1576.03 1858.68 1337.39 1409.19 1678.39 2094.65

Table 7. Effects of the CNT volume fraction on the first six natural frequencies f   /  2  (Hz) of the FGM/CNTR sector cylindrical shell with SSSS boundary condition. * VCNT

Type-UD

0.11 0.14 0.17 0.28

Mode No.

1 519.47 535.84 666.75 796.13

2 731.15 753.27 943.19 1126.74

3 830.05 858.64 1052.58 1253.48 22

4 878.41 909.13 1118.26 1335.11

5 1060.62 1092.69 1369.78 1637.45

6 1138.94 1188.25 1414.15 1685.38

Type-V

Type-O

Type-X

0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28

526.25 545.30 682.64 839.73 512.74 526.94 654.57 774.43 529.24 549.80 689.16 859.61

735.77 759.48 954.78 1158.22 725.03 745.10 932.48 1108.17 741.83 768.60 968.44 1198.82

841.76 874.85 1081.07 1329.98 813.06 836.93 1026.70 1214.91 850.67 886.57 1091.99 1350.65

879.82 912.76 1130.17 1383.76 864.79 892.09 1100.28 1315.38 896.37 933.28 1151.39 1415.48

1081.06 1122.51 1391.94 1708.56 1049.91 1078.88 1347.34 1605.66 1078.60 1118.62 1412.94 1762.72

1122.85 1171.44 1431.50 1787.75 1097.37 1136.59 1362.20 1620.48 1182.49 1243.51 1481.93 1814.43

Table 8. Effects of the CNT volume fraction on the first six natural frequencies f   / (2 ) (Hz) of the FGM/CNTR sector cylindrical shell with CSCF boundary condition. * VCNT

Type-UD

Type-V

Type-O

Type-X

0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28 0.11 0.14 0.17 0.28

Mode No.

1 568.99 586.25 734.63 878.44 573.58 592.46 746.14 909.58 559.31 572.99 715.34 837.13 582.53 605.67 766.32 966.89

Type-V

2 717.32 739.86 923.75 1105.83 727.82 754.45 948.59 1173.23 715.06 737.16 921.09 1100.40 724.08 749.82 940.81 1158.57

3 745.23 768.86 955.28 1140.53 753.60 780.05 973.56 1187.02 733.69 753.49 934.02 1108.45 761.12 791.37 991.05 1238.87

4 790.00 814.76 1016.16 1214.19 801.13 830.48 1042.74 1288.55 785.94 809.91 1011.52 1216.05 798.87 827.44 1036.67 1274.54

Type -O

1st mode

23

5 942.82 978.04 1192.27 1423.63 942.05 979.62 1203.84 1483.28 924.58 955.49 1169.12 1401.17 965.18 1007.53 1230.73 1510.43

6 1020.51 1056.28 1290.94 1537.37 1029.18 1067.77 1313.34 1596.53 996.63 1025.52 1253.06 1475.40 1048.72 1094.35 1344.87 1667.74

Type -X

2nd mode

3rd mode

Fig. 8. Mode shapes of the FGM/CNTR sector cylindrical shell versus various CNT types under * CCCC boundary condition ( VCNT =0.11).

* VCNT =0.11

* VCNT =014

* VCNT =0.17

1st mode

2nd mode

3rd mode

Fig. 9. Mode shapes of the sector FGM/CNTR sector cylindrical shell versus various volume fractions under CCCC boundary condition (Type-V). CCCC

SSSS

1st mode

2nd mode

24

CSCF

3rd mode

Fig. 10. Mode shapes of the sector FGM/CNTR sector cylindrical shell under various boundary * conditions. ( VCNT =0.11, Type-V). The influence of the length on natural vibration frequencies

f   / (2 ) (Hz) of the

FGM/CNTR sector cylindrical shell with CCCC boundary restraint is presented in Table 9. In this analysis, the length of the cylindrical shell is selected as 1.0m, 1.4m, 1.8m, respectively; the * power-law index pn =1 and CNT volume fraction VCNT =0.11 are selected in this calculation. The

variations of natural vibration frequencies versus length of the FGM/CNTR cylindrical shell are presented in Fig.11. The length L increases from 1m to 1.8m by the step 0.2m in this figure. From the table and figure, the effects of the length on the vibration frequencies can be seen easily. The flexibility of the FGM/CNTR sector cylindrical shell becomes bigger as the length L increases, so the natural vibration frequencies of the shells will decrease. Therefore, when the length L increases, the natural vibration frequency f   / (2 ) (Hz) will decrease. Table 9. Effects of the length on the first six natural frequencies f   /  2  (Hz) of the * FGM/CNTR sector cylindrical shell with CCCC boundary condition ( VCNT =0.11, pn =1).

L(m) Type-UD Type-V Type-O Type-X

1.0 1.4 1.8 1.0 1.4 1.8 1.0 1.4 1.8 1.0 1.4 1.8

1 1067.78 826.37 716.18 1056.13 831.30 722.95 1031.78 806.24 702.09 1104.81 849.86 734.07

2 1082.65 866.60 797.96 1078.88 869.04 805.24 1044.80 851.96 790.51 1121.67 885.16 809.96

Mode No. 3 4 1495.51 1929.86 1289.03 1323.20 1029.09 1074.74 1498.96 1871.19 1260.33 1332.61 1018.74 1074.96 1457.45 1822.13 1231.61 1298.95 998.26 1041.22 1539.33 2025.36 1344.53 1355.12 1062.16 1110.76

25

5 2017.62 1372.87 1262.54 2001.93 1360.86 1273.39 1938.49 1317.23 1242.83 2066.26 1427.97 1290.28

6 2038.42 1731.95 1437.11 2032.75 1730.86 1398.05 1978.72 1678.95 1365.26 2129.36 1788.96 1505.21

900 800 700

1

1.2 1.4 1.6 1.8 Length L (m)

Mode 3

Type-UD Type-V Type-O Type-X

1100 1000 900 800 700

1

1.2 1.4 1.6 1.8 Length L (m)

Type-UD Type-V Type-O Type-X

1600 1400 1200 1000 1

1.2 1.4 1.6 1.8 Length L (m)

Mode 4

2200

Frequency f (Hz)

1000

Frequency f (Hz)

Frequency f (Hz)

1100

Mode 2

1200

Type-UD Type-V Type-O Type-X

Frequency f (Hz)

Mode 1

1200

Type-UD Type-V Type-O Type-X

2000 1800 1600 1400 1200 1000

1

1.2 1.4 1.6 1.8 Length L (m)

Fig. 11. Variation of the natural frequencies of the FGM/CNTR sector cylindrical shell versus the length under CCCC boundary condition. The radius r is another important geometrical parameter, and the effect of radius on the vibration of the FGM/CNTR sector cylindrical shell is a particularly relevant message in engineering designs. Table 10 gives the first six natural vibration frequencies f   / (2 ) (Hz) of the FGM/CNTR sector cylindrical shell under CCCC boundary condition with different radiuses. * The radius r  0.4m, 0.8m and 1.2m, the power-law index pn =1 and the volume fraction VCNT

=0.11 are used in this analysis. The variations of natural vibration frequencies f   /  2  (Hz) of the FGM/CNTR sector cylindrical shell versus different radiuses are depicted in Fig. 12. From Table 10 and Fig. 12, it is easy to see that the first four natural vibration frequencies decrease with the radius r increases, and the rate of decline decreases gradually with the increasing of radius r. Table 10. Effects of the radius on the first six natural frequencies

f   / (2 ) (Hz) of the

* FGM/CNTR sector cylindrical shell with CCCC boundary condition ( VCNT =0.11, pn =1).

R(m) Type-UD Type-V Type-O Type-X

0.4 0.8 1.2 0.4 0.8 1.2 0.4 0.8 1.2 0.4 0.8 1.2

Mode No. 1 2230.54 931.91 549.09 2258.34 940.88 554.46 2223.28 916.24 538.09 2252.82 953.23 562.91

2 2320.89 997.28 641.07 2341.28 1009.17 647.07 2305.53 992.55 633.94 2350.56 1008.34 651.72

3 2509.29 1147.44 843.83 2515.08 1147.80 835.29 2475.39 1127.62 817.86 2555.42 1172.34 871.67 26

4 2666.85 1210.25 857.38 2688.79 1219.18 856.04 2642.17 1183.53 828.99 2707.35 1242.14 887.63

5 2810.73 1439.00 946.49 2795.27 1416.76 955.10 2747.72 1388.25 932.50 2877.21 1490.93 966.72

6 2833.05 1594.44 1138.11 2859.38 1594.01 1140.81 2804.50 1544.41 1108.31 2881.56 1647.42 1172.63

1000 500 0 0.4 0.6 0.8 1 1.2 1.4 Radius r (m)

2000

Mode 2 Type-UD Type-V Type-O Type-X

1500 1000

3000

500 0.4 0.6 0.8 1 1.2 1.4 Radius r (m)

Mode 3 Type-UD Type-V Type-O Type-X

2500 2000 1500 1000

3000

Frequency f (Hz)

1500

Type-UD Type-V Type-O Type-X

2500

Frequency f (Hz)

2000

Mode 1

Frequency f (Hz)

Frequency f (Hz)

2500

500 0.4 0.6 0.8 1 1.2 1.4 Radius r (m)

2500 2000

Mode 4 Type-UD Type-V Type-O Type-X

1500 1000 500 0.4 0.6 0.8 1 1.2 1.4 Radius r (m)

Fig. 12. Variation of the natural frequencies of the FGM/CNTR sector cylindrical shell versus the radius under CCCC boundary condition. Table 11 presents the first six frequencies

f   / (2 ) (Hz) of the FGM/CNTR sector

* cylindrical shell under CCCC boundary condition with various angles. The volume fraction VCNT

=0.11, the angles  =30  , 60  and 90  are taken into account. Fig. 13 depicts the variation of the first four vibration frequencies versus angles of the FGM/CNTR sector cylindrical shell with CCCC boundary condition. The angles  =36  , 60  , 90  , 120  , 150  and 180  are used in the figure. From Table 11 and Fig. 13, it is easily seen that the natural frequencies decrease and its rate of fall becomes slower with the deceasing of angle  . The reason as follows: the increase of angle  results in bigger flexibility in the y direction, then the frequency will decrease. Table 11. Effects of the angles on the first six natural frequencies

f   / (2 ) (Hz) of the

* FGM/CNTR sector cylindrical shell with CCCC boundary condition ( VCNT =0.11).

 Type-UD

Type-V

Type-O

Type-X

30  60  90  30  60  90  30  60  90  30 

Mode No. 1 1909.27 935.69 683.28 1931.42 945.98 690.16 1886.64 928.72 670.60 1944.00

2 2020.59 1086.92 780.56 2036.93 1084.72 788.94 1989.41 1063.24 774.76 2063.11

3 2237.83 1280.96 958.12 2241.59 1292.81 953.40 2187.61 1260.00 935.04 2297.07 27

4 2574.36 1387.04 984.01 2559.88 1363.05 987.40 2494.96 1331.54 956.85 2658.20

5 3023.67 1489.79 1244.70 2988.55 1501.31 1243.05 2908.93 1458.98 1215.86 3136.18

6 3567.18 1820.79 1282.87 3512.07 1775.39 1266.75 3415.44 1729.05 1236.44 3708.59

Type-UD Type-V Type-O Type-X

1000 500 30 60 90 120 150 180 Angle  ( )

1115.17 791.01

1309.99 984.39

Mode 2 2000 1500

Type-UD Type-V Type-O Type-X

1000 500 30 60 90 120 150 180 Angle  ( )

2500

1442.87 1014.33

1528.02 1272.42

Mode 3 Type-UD Type-V Type-O Type-X

2000 1500 1000

500 30 60 90 120 150 180 Angle  ( )

3000

Frequency f (Hz)

1500

Mode 1

Frequency f (Hz)

Frequency f (Hz)

2000

948.75 699.82

Frequency f (Hz)

60  90 

2500 2000

1877.47 1337.39 Mode 4 Type-UD Type-V Type-O Type-X

1500 1000 500 30 60 90 120 150 180 Angle  ( )

Fig. 13. Variation of the natural frequencies of the FGM/CNTR sector cylindrical shell versus the angles under CCCC boundary condition. The effects of the thickness on the first six natural frequencies f   / (2 ) (Hz) of the CCCC * FGM/CNTR sector cylindrical shell are presented in Table 12, where the volume fraction VCNT

=0.11, radius r=1m and h=0.05m, 0.10m, 0.15m are used. Fig. 14 shows the variation of the first four frequencies versus the thickness of the CCCC FGM/CNTR sector cylindrical shell. In the figure, the thickness h is set from 0.02m to 0.25m by the step of 0.05m. The material and other geometrical parameters for calculation are the same as those used in Table 12. It is easily seen that the first four natural frequencies increase as the thickness increases. The reason may be that the flexural rigidity increases faster than the mass as the thickness increases. Table 12. Effects of the thickness on the first six natural frequencies f   / (2 ) (Hz) of the * FGM/CNTR sector cylindrical shell with CCCC boundary condition ( VCNT =0.11).

h(m) Type-UD

Type-V

Type-O

0.05 0.10 0.15 0.05 0.10 0.15 0.05 0.10 0.15

Mode No. 1 434.10 683.28 865.39 440.19 690.16 871.79 428.03 670.60 857.08

2 564.15 780.56 905.55 570.91 788.94 913.43 554.93 774.76 889.73

3 664.26 958.12 1120.96 670.74 953.40 1108.38 651.54 935.04 1086.51 28

4 713.88 984.01 1264.29 720.42 987.40 1268.02 699.07 956.85 1230.59

5 894.77 1244.70 1566.69 904.65 1243.05 1529.99 877.68 1215.86 1494.99

6 906.19 1282.87 1622.54 906.21 1266.75 1635.36 889.55 1236.44 1598.92

442.61 699.82 878.59

1000 800 Type-UD Type-V Type-O Type-X

600

400 0.05 0.1 0.15 0.2 0.25 Thickness h (m)

679.94 984.39 1156.80

Mode 2

Frequency f (Hz)

Frequency f (Hz)

Mode 1

576.84 791.01 926.20

1000

600

Type-UD Type-V Type-O Type-X

904.69 1272.42 1630.38

Mode 3

1200

800

731.91 1014.33 1300.97

400 0.05 0.1 0.15 0.2 0.25 Thickness h (m)

1400 1200 1000 800 600

Type-UD Type-V Type-O Type-X

937.22 1337.39 1658.26 Mode 4

Frequency f (Hz)

0.05 0.10 0.15

Frequency f (Hz)

Type-X

400 0.05 0.1 0.15 0.2 0.25 Thickness h (m)

1600 1400 1200 1000 800 600

Type-UD Type-V Type-O Type-X

0.05 0.1 0.15 0.2 0.25 Thickness h (m)

Fig. 14. Variation of the natural frequencies of the FGM/CNTR sector cylindrical shell versus the thickness under CCCC boundary condition. The influence of power-law index pn on the first six natural frequencies f   / (2 ) (Hz) of the FGM/CNTR sector cylindrical shell is presented in Table 13 and Fig.15. In this analysis, the * volume fraction VCNT =0.11, clamped boundary condition and different pn are used. The FGM of

the shell is composed of Al and ZrO2. From the table and figure, it is easily seen that the natural frequency decreases for low indices pn ( pn <1) and then shows an increasing-decreasing trend as the power-law index increases. Fig.16 presents the effects of power-law index pn on vibration frequencies of CCCC FGM/CNTR sector cylindrical shell. In this analysis, the components of the FGM are Al and Al2O3, and the other material properties and geometrical parameters are same as those in Fig.15. It is easily seen that the free frequencies decrease monotonously with the power-law index increasing. Therefore, we can believe that the effects of power-law index on the vibration frequency are dependent on the components of FGM. Table 13. Effects of the power-law index on the first six natural frequencies f   / (2 ) (Hz) of * the FGM/CNTR sector cylindrical shell under CCCC boundary condition ( VCNT =0.11).

pn Type-UD

0.5

Mode No. 1 691.52

2 788.76

3 961.66 29

4 991.33

5 1257.69

6 1281.41

Frequency f (Hz)

740

Mode 1

720 700 680 660 640

Type-UD Type-V Type-O Type-X

Mode 2

820

620 -2 -1 0 1 2 10 10 10 10 10 Power-law index Pn

958.12 952.03 957.61 953.40 939.26 942.13 935.04 916.49 984.48 984.39 990.22

800 780 760

Type-UD Type-V Type-O Type-X

740 720 10

-2

-1

0

1

10 10 10 10 Power-law index Pn

2

984.01 997.88 994.73 987.40 998.00 966.76 956.85 958.95 1019.17 1014.33 1039.56

1244.70 1246.40 1246.93 1243.05 1247.38 1221.33 1215.86 1212.35 1286.08 1272.42 1282.69

Mode 4

950 Type-UD Type-V Type-O Type-X

900 10

1282.87 1303.18 1277.44 1266.75 1270.64 1250.30 1236.44 1231.35 1327.96 1337.39 1377.65

Mode 3

1000

Frequency f (Hz)

Type-X

780.56 759.06 795.00 788.94 761.66 784.41 774.76 748.45 797.55 791.01 775.18

Frequency f (Hz)

Type-O

683.28 690.21 697.09 690.16 695.26 679.52 670.60 672.34 707.23 699.82 712.59

Frequency f (Hz)

Type-V

1 20 0.5 1 20 0.5 1 20 0.5 1 20

-2

-1

0

1

10 10 10 10 Power-law index Pn

2

1050 1000 950

Type-UD Type-V Type-O Type-X

900 10

-2

-1

0

1

10 10 10 10 Power-law index Pn

2

Fig. 15. Variation of the natural frequencies of the CNTR Al/ZrO2 sector cylindrical shell versus the power-law index under CCCC boundary condition.

1000 800 600

Type-UD Type-V Type-O Type-X

400 -2 -1 0 1 2 10 10 10 10 10 Power-law index Pn

Mode 3

1000 Type-UD Type-V Type-O Type-X

500 10

-2

-1

0

1

10 10 10 10 Power-law index Pn

2

1500

1000

Type-UD Type-V Type-O Type-X

500 -2 -1 0 1 2 10 10 10 10 10 Power-law index Pn

2000

Frequency f (Hz)

1200

Mode 2

1500

Frequency f (Hz)

Mode 1

Frequency f (Hz)

Frequency f (Hz)

1400

Mode 4

1500 1000

Type-UD Type-V Type-O Type-X

500 -2 -1 0 1 2 10 10 10 10 10 Power-law index Pn

Fig. 16. Variation of the natural frequencies of the CNTR Al/Al2O3 sector cylindrical shell versus the power-law index under CCCC boundary condition.

4. Conclusions An isogeometric method has been applied to analyze the free vibration behaviors of sector cylindrical shells with carbon nanotube-reinforced and functionally graded materials. The material 30

properties change continuously along the thickness direction, and the effective material properties are approximated by a mixture rule. The effects of boundary conditions, geometric properties as well as material parameters on the frequencies of the carbon nanotubes reinforced and functionally graded material sector cylindered shells are examined. Through this investigation, some remarks can be listed as follows:  The convergence study indicates that the IGA with higher-order basis functions can converge faster.  The current method can obtain accurate free vibration frequencies of FGM/CNRT shells.  The influence of the boundary conditions, CNT distribution types, CNT volume fractions, and geometrical parameters on the free vibration frequency parameters of FGM/CNTR sector cylindrical shells is significant. The free vibration frequency will increase with the increasing of CNT volume fractions. For the same order frequency, the Type-X FGM/ CNT plate has biggest value and the Type-O FGM/ CNT plate has the lowest value in the considered four types of FGM/ CNT plates. The effects of power-law index on the vibration frequency are dependent on the components of FGM.  Effects of boundary condition, CNT volume fraction and CNT distribution type on vibration mode shapes are investigated. The results show that CNT volume fraction and CNT distribution type have small influence on mode shapes, but the boundary condition has pronounced effect on mode shapes.

5. Conflicts of interest The authors declare that they have no conflicts of interest 31

6. Acknowledgment The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51822902, 51709066 and 51775125), Heilongjiang Provincial Natural Science Foundation (No. QC2018050), the Fundamental Research Funds for the Central Universities of China (No. HEUCF180305) and the Fund for Science and Technology on Reactor System Design Technology Laboratory.

References [1] Salvetat-Delmotte JP, Rubio A. Mechanical properties of carbon nanotubes: a fiber digest for beginners. Carbon 2002;40(10):1729-34. [2] Sun CH, Li F, Cheng HM, Lu GQ. Axial Young’s modulus prediction of single-walled carbon nanotube arrays with diameters from nanometer to meter scales. Appl Phys Lett 2005;87(19):1297. [3] Thostenson ET, Chou TW. On the elastic properties of carbon nanotube-based composites: modelling and characterization. J Phys D Appl Phys 2003;36(5):573. [4] Odegard GM, Gates TS, Wise KE, Park C, Siochi EJ. Constitutive modeling of nanotube– reinforced polymer composites. Compos Sci Technol 2001;63(11):1671-87. [5] Mokashi VV, Qian D, Liu Y. A study on the tensile response and fracture in carbon nanotube-based

composites

using

molecular

mechanics.

Compos

Sci

Technol

2007;67(3):530-40. [6] Griebel M, Hamaekers J. Molecular dynamics simulations of the elastic moduli of polymer– carbon nanotube composites. Comput Method Appl M 2004;193(17–20):1773-88. [7] Cadek M, Coleman JN, Barron V, Hedicke K, Blau WJ. Erratum: Morphological and mechanical properties of carbon-nanotube-reinforced semicrystalline and amorphous polymer composites. Appl Phys Lett 2002;81(27):5123-5. [8] Heshmati M, Yas MH. Free vibration analysis of functionally graded CNT-reinforced 32

nanocomposite

beam

using

Eshelby-Mori-Tanaka

approach.

J

Mech

Sci

Tech

2013;27(11):3403-8. [9] Lin F, Xiang Y. Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Appl Math Model 2014;38(15-16):3741-54. [10] Lin F, Xiang Y. Numerical analysis on nonlinear free vibration of carbon nanotube reinforced composite beams. Int J of Struct Stab Dy 2014;14(1).1350056. [11] Ke LL, Yang J, Kitipornchai S. Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos Struct 2010;92(3):676-83. [12] Zhang LW, Lei ZX, Liew KM. Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach. Compos Part B-eng 2015;75:36-46. [13] Zhang LW, Lei ZX, Liew KM. Vibration characteristic of moderately thick functionally graded carbon nanotube reinforced composite skew plates. Compos Struct 2015;122:172-83. [14] Zhang LW, Lei ZX, Liew KM. Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method. Compos Struct 2015;120:189-99. [15] Zhang LW, Cui WC, Liew KM. Vibration analysis of functionally graded carbon nanotube reinforced composite thick plates with elastically restrained edges. Int J Mech Sci 2015;103(24):9-21. [16] Lei ZX, Zhang LW, Liew KM. Free vibration analysis of laminated FG-CNT reinforced composite rectangular plates using the kp-Ritz method. Compos Struct 2015;127:245-59. [17]

Shahrbabaki

EA,

Alibeigloo

A.

Three-dimensional

free

vibration

of

carbon

nanotube-reinforced composite plates with various boundary conditions using Ritz method. Compos Struct 2014;111(11):362-70. [18] Malekzadeh P, Shahpari SA. Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM. Thin Wall Struct 2005;43(7):1037-50. [19] Malekzadeh P, Karami G. Large amplitude flexural vibration analysis of tapered plates with edges elastically restrained against rotation using DQM. Eng Struct 2008;30(10):2850-8. [20] Malekzadeh P, Fiouz AR. Large deformation analysis of orthotropic skew plates with nonlinear rotationally restrained edges using DQM. Compos Struct 2007;80(2):196-206. 33

[21] Karami G, Malekzadeh P, Mohebpour SR. DQM free vibration analysis of moderately thick symmetric

laminated

plates

with

elastically

restrained

edges.

Compos

Struct

2006;74(1):115-25. [22]

Malekzadeh

P,

Heydarpour

Y.

Mixed

Navier-layerwise

differential

quadrature

three-dimensional static and free vibration analysis of functionally graded carbon nanotube reinforced composite laminated plates. Meccanica 2015;50(1):143-67. [23] Malekzadeh P, Zarei AR. Free vibration of quadrilateral laminated plates with carbon nanotube reinforced composite layers. Thin Wall Struct 2014;82(82):221-32. [24] Alibeigloo A, Liew KM. Elasticity solution of free vibration and bending behavior of functionally graded carbon nanotube-reinforced composite beam with thin piezoelectric layers using differential quadrature method. Int J Appl Mech 2015;7(01):1550002. [25] Alibeigloo A, Emtehani A. Static and free vibration analyses of carbon nanotube-reinforced composite plate using differential quadrature method. Meccanica 2015;50(1):61-76. [26] Alibeigloo A. Three-dimensional thermoelasticity solution of functionally graded carbon nanotube reinforced composite plate embedded in piezoelectric sensor and actuator layers. Compos Struct 2014;118(1):482-95. [27] Zhu P, Lei ZX, Liew KM. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with First-order shear deformation plate theory. Compos Struct 2012;94(4):1450-60. [28] Mehar K, Panda SK , Bui TQ , Mahapatra TR. Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM. J Therm Stresses 2017; 40(7):1-18. [29] Shen HS, Zhang CL. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates. Mater Design 2010;31(7):3403-11. [30] Shen HS. Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells. Compos Part B-eng 2012;43(3):1030-8. [31] Qin ZY, Pang XJ, Safaei B, Chu FL. Free vibration analysis of rotating functionally graded CNT reinforced composite cylindrical shells with arbitrary boundary conditions. Compos Struct 2019; 220: 847-860. 34

[32] Nguyen PD, Quang VD, Anh VTT, Duc ND. Nonlinear vibration of carbon nanotube reinforced composite truncated conical shells in thermal environment. Int J Struct Stab Dy 2019; http://dx.doi.org/10.1142/S021945541950158X. [33] Dat ND, Khoa ND, Nguyen PD, Duc ND. An analytical solution for and nonliear dynamic response and vibration of FG-CNT reinforced nanocomposite elliptical cylindrical shells resting on elastic foundations. ZAMM-Journal of Applied Mathematics and Mechanics 2019; http://dx.doi.org/10.1002/zamm.201800238. [34] Li HC, Pang FZ, Chen HL, Du Y. Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Compos Part B-eng 2019; 164: 249-264. [35] Pang FZ, Li HC, Jing FM, Du Y. Application of first-order shear deformation theory on vibration analysis of stepped functionally graded paraboloidal shell with general edge constraints. Materials 2018; 12(1): 69. [36] Qin ZY, Chu FL, Zu JA. Free vibrations of cylindrical shells with arbitrary boundary conditions: A comparison study. Int J Mech Sci 2017; 133: 91-99. [37] Safaei B, Moradi-Dastjerdi R, Qin ZY, Chu FL. Frequency-dependent forced vibration analysis of nanocomposite sandwich plate under thermo-mechanical loads. Compos Part B-eng 2019;161:44-54. [38] Qin ZY, Safaei B, Pang XJ, Chu FL. Traveling wave analysis of rotating functionally graded graphene platelet reinforced nanocomposite cylindrical shells with general boundary conditions. Results Phys, 2019, https://doi.org/10.1016/j.rinp.2019.10275. [39] Jin GY, Ye TG, Wang XR, Miao XH. A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions. Compos Part B-eng 2016; 89: 230-52. [40] Ye TG, Jin GY, Zhang YT. Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature. Compos Struct 2015; 133: 202-25. [41] Chen YK, Jin GY, Zhang CY, Ye TG, Xue YQ. Thermal vibration of FGM beams with general boundary conditions using a higher-order shear deformation theory. Compos Part 35

B-eng 2018; 153: 376-86. [42] Duc ND. Nonlinear static and dynamic stability of functionally graded plates and shells. Vietnam National University Press, Hanoi, 2014 [43] Duc ND, Tung HV. Nonlinear analysis of stability for functionally graded cylindrical panels under axial compression. Com Mater Sci 2010; 49 (4): 313-6. [44] Anh VTT, Bich DH, Duc ND. Nonlinear stability analysis of thin FGM annular spherical shells on elastic foundations under external pressure and thermal loads. Eur J Mech A-Solid 2015; 50: 28-38. [45] Duc ND, Nguyen PD, Khoa ND. Nonlinear dynamic analysis and vibration of eccentrically stiffened S-FGM elliptical cylindrical shells surrounded on elastic foundations in thermal environments. Thin Wall Struct 2017; 117: 178-89. [46] Duc ND, Lam PT, Quyen NV, Quang VD. Nonlinear dynamic response and vibration of 2d penta-graphene composite plates resting on elastic foundation in thermal environments. VNU Journal of Science: Mathematics-Physics 2019; 35 (3):13-29. [47] Duc ND, Anh VTT, Huong VT, Quang VD, Nguyen PD. Nonlinear dynamic response of nano-composite

sandwich

annular

spherical

shells.

VNU

Journal

of

Science:

Mathematics-Physics 2019; 35(3):1-12. [48] Kim SE, Duc ND, Nam VH, Sy NV. Nonlinear vibration and dynamic buckling of eccentrically oblique stiffened FGM plates resting on elastic foundations in thermal environment. Thin Wall Struct 2019; 142: 287-96. [49] Quan TQ, Cuong NH, Duc ND. Nonlinear buckling and post-buckling of eccentrically oblique stiffened sandwich functionally graded double curved shallow shells. Aerosp Sci Technol 2019; 90: 169-80. [50] Minh PP, Duc ND. The effect of cracks on the stability of the functionally graded plates with variable-thickness using HSDT and phase-field theory. Compos Part B-eng 2019; 175: 107086. [51] Baltacioglu AK, Civalek O.Vibration analysis of circular cylindrical panels with CNT reinforced and FGM composites. Compos Struct 2018; 202: 374-88. [52] Hughes TJR, Cottrell JA, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, 36

exact geometry and mesh refinement. Comput Method Appl M 2005;194(39–41):4135-95. [53] Kruse R, Nguyen-Thanh N, Lorenzis LD, Hughes TJR. Isogeometric collocation for large deformation elasticity and frictional contact problems. Comput Method Appl M 2015;296:73-112. [54] Zhang Y, Bazilevs Y, Goswami S, Bajaj CL, Hughes TJ. Patient-Specific Vascular NURBS Modeling for Isogeometric Analysis of Blood Flow. Comput Method Appl M 2007;196(29-30):2943. [55] Cho S, Ha SH. Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidiscip O 2009;38(1):53-70. [56] Thai CH, Nguyen-Xuan H, Nguyen-Thanh N, Le TH, Nguyen-Thoi T, Rabczuk T. Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach. Int J Numer Meth Eng 2012;91(6):571-603. [57] Liu S, Yu TT, Bui TQ, Bui TQ,Yin SH, Thai DK Tanaka S. Analysis of functionally graded plates by a simple locking-free quasi-3D hyperbolic plate isogeometric method. Compos Part B-eng 2017; 120:182-196. [58] Yin SH, Yu TT, Bui TQ, Zheng XJ, Yi G. Rotation-free isogeometric analysis of functionally graded thin plates considering in-plane material inhomogeneity. Thin Wall Struct 2017; 119:385-395. [59] Yu TT, Yin SH, Bui TQ, Chen L, Wattanasakulpong N. Buckling isogeometric analysis of functionally graded plates under combined thermal and mechanical loads. Compos Struct 2017; 162:54-69. [60] Xue YQ, Jin GY, Ma XL, Chen HL, Ye TG, Chen MF, Zhang YT. Free vibration analysis of porous plates with porosity distributions in the thickness and in-plane directions using isogeometric approach. Int J Mech Sci 2019; 152: 346–362. [61] Chen MF, Jin GY, Ye TG, Zhang YT. An isogeometric finite element method for the in-plane vibration analysis of orthotropic quadrilateral plates with general boundary restraints. Int J of Mech Sci 2017; 133:846–62. [62] Chen MF, Jin GY, Ma XL, Zhang YT, Ye TG, Liu ZG. Vibration analysis for sector cylindrical shells with bi-directional functionally graded materials and elastically restrained 37

edges. Compos Part B-eng 2018; 153: 346-63. [63] Chen MF, Jin GY, Zhang YT, Niu FL, Liu ZG. Three-dimensional vibration analysis of beams with axial functionally graded materials and variable thickness. Compos Struct 2019; 207: 304–22. [64] Nguyen TN, Thai CH, Luu AT, Nguyen-Xuan H, Lee J. NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells. Comput Method Appl M 2019; 347: 983-1003 [65] Nguyen TN, Thai CH, Nguyen-Xuan H, Lee J. NURBS-based analyses of functionally graded carbon nanotube-reinforced composite shells. Compos Struct 2018; 203:349-60. [66] Yang HS., Dong CY. Adaptive extended isogeometric analysis based on PHT-splines for thin cracked plates and shells with Kirchhoff–Love theory. Appl Math Model 2019; 76:759-99. [67] Gu JM, Yu TT, Lich LV, Nguyen TT, Yang Y, Bui TQ. Fracture modeling with the adaptive XIGA based on locally refined B-splines. Comput Method Appl M 2019; 354:527-67. [68] Gu JM, Yu TT, Lich LV, Nguyen TT, Bui TQ. Adaptive multi-patch isogeometric analysis based on locally refined B-splines. Comput Method Appl M 2018; 339: 704-38. [69] Esawi AMK, Farag MM. Carbon nanotube reinforced composites: potential and current challenges. Mater Des 2007;28:2394–401. [70] Fidelus JD, Wiesel E, Gojny FH, Schulte K, Wagner HD. Thermo-mechanicalproperties of randomly oriented carbon/epoxy nanocomposites. Compos Part A-Appl 2005;36:1555–61. [71] Piegl LA,Tiller W. The NURBS book. 2nd ed. Springer-Verlag;1995. [72] Hughes T J R. The finite element method : linear static and dynamic finite element analysis: Prentice-Hall, 2000. [73] Nguyen VP Anitescu C, Bordas SPA, Isogeometric analysis: an overview and computer implementation aspects. Math Comput Simul 2012; 117: 89-116. [74] Su Z, Jin G, Ye T. Free vibration analysis of moderately thick functionally graded open shells with general boundary conditions. Compos Struct 2014; 117:169-86.

38

Dear Editor, We the undersigned declare that this manuscript entitled “Isogeometric vibration analysis of sector cylindrical shells with carbon nanotubes reinforced and functionally graded materials” is original, has not been published before and is not currently being considered for publication elsewhere. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. CRediT author statement: Yantao ZHANG: Data curation, Software, Methodology,; Guoyong JIN: Conceptualization, Methodology, Writing- Reviewing and Editing; Mingfei CHEN: Methodology, Software, WritingOriginal draft preparation，Validation; Tiangui YE: Investigation, Writing- Reviewing and Editing, and Zhigang LIU: Supervision

We understand that the Corresponding Author is the sole contact for the editorial process. He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. Signed by all authors as follows: Yantao ZHANG, Guoyong JIN, Mingfei Chen, Tiangui YE, and Zhigang LIU

Dear Editor, 39

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “” Signed by all authors as follows: Yantao Zhang, Guoyong Jin, Mingfei Chen, Tiangui Ye, Zhigang Liu

Highlights:  Vibration analysis of CNTR /FGM sector cylindrical shells.  Exact geometry, refinement without geometry and higher continuity.  New results of the CNT/FGM sector cylindrical shells are presented.  The influence of materials, geometries and boundaries is investigated.

40