Computationally efficient model for flow-induced instability of CNT reinforced functionally graded truncated conical curved panels subjected to axial compression

Accepted Manuscript Computationally efficient model for flow-induced instability of CNT reinforced functionally graded conical curved panels subjected to axial compression M. Mehri, H. Asadi, M.A. Kouchakzadeh PII: DOI: Reference:

S0045-7825(16)31606-1 http://dx.doi.org/10.1016/j.cma.2017.02.020 CMA 11347

To appear in:

Comput. Methods Appl. Mech. Engrg.

Received date: 17 November 2016 Revised date: 10 January 2017 Accepted date: 15 February 2017 Please cite this article as: M. Mehri, H. Asadi, M.A. Kouchakzadeh, Computationally efficient model for flow-induced instability of CNT reinforced functionally graded conical curved panels subjected to axial compression, Comput. Methods Appl. Mech. Engrg. (2017), http://dx.doi.org/10.1016/j.cma.2017.02.020 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 proof before it is published in its final 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.

Computationally Efficient Model for Flow-Induced Instability of CNT Reinforced Functionally Graded Conical Curved Panels Subjected to Axial Compression

M. Mehri*, H. Asadi†,§, M. A. Kouchakzadeh‡

*

Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran

†

Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada

‡

Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran

§ Corresponding author, Email address: [email protected]

1

Abstract As a first endeavor, the aeroelastic responses of functionally graded carbon nanotube reinforced composite (FG-CNTRC) truncated conical curved panels subjected to aerodynamic load and axial compression are investigated. The nonlinear dynamic equations of FG-CNTRC conical curved panels are derived according to Green’s strains and the Novozhilov nonlinear shell theory. The aerodynamic load is estimated in accordance with the quasi-steady Krumhaar’s modified supersonic piston theory by taking into account the effect of the panel curvature. Matrix transform method along with the harmonic differential quadrature method (HDQM) are employed to solve the nonlinear equations of motion of the FG-CNTRC truncated conical curved panel. The advantage of the matrix transform method is that we only need to discretize the meridional direction. Effects of semi-vertex angle of the cone, subtended angle of the panel, boundary conditions, geometrical parameters, volume fraction and distribution of CNT, and Mach number on the aeroelastic characteristics of the FG-CNTRC conical curved panel are put into evidence via a set of parametric studies and pertinent conclusions are outlined. The results prove that the panels with different FG distributions have different critical dynamic pressure. It is found that the semi-vertex and subtended angles play a pivotal role in changing the critical circumferential mode number of the flutter instability. Besides, the research shows that the superb efficiency of proposed method with few grid points, which requires less CPU time, are attributed to the matrix transform method and the higher-order harmonic approximation function in the HDQM.

Keywords: Flow-Induced instability; Functionally graded carbon nanotube; Truncated conical curved panel; Novozhilov nonlinear shell theory; HDQM.

2

1. Introduction Fluid-structure interaction (FSI) is a multiphysics coupling between laws, which describe fluid dynamics and structural mechanics. This phenomenon is characterized by interactions between inertia, elastic and aerodynamic loads of flight structures. The topic of the fluid-structure interactions owing to supersonic airflow is a matter of interest because of its significance in launch vehicle design and supersonic aircraft. In the flutter condition, vibration amplitudes of the structure become dynamically unstable and increase exponentially with time, affecting substantially the accumulated fatigue life, reliability, cost and safety of existing aeronautical components. This type of instability has received much attention and the related phenomena are reasonably well understood for structures made of conventional materials [1-9]. The demand for lighter, stronger materials with tailored properties, which provide additional engineering functionality has spawned a new type of materials called multifunctional materials. Fiber reinforced polymer composites (FRPCs) for advanced aerospace structures usually require the most demanding multifunctional properties. The introduction of nanotechnology in the field of composite materials with nanoscales fillers, including carbon nanotubes (CNTs) and carbon nanofibers (CNFs), has offered new opportunities to improve multifunctional properties of FRPCs. Due to their extraordinary stiffness and strength, extremely high aspect ratio, and large surface area, these nanofillers inspire researchers to use these newly developed materials as reinforcement phases in a matrix medium to build up advanced composite materials. As background for understanding static and dynamic responses of CNT reinforced beams [1016], plates [17-27] and curved panels [28-41], a brief review of relevant research works is provided hereinafter. Buckling and free vibration responses of CNTRC beams resting on elastic foundation were investigated by Yas and Samadi [13] using differential quadrature method (DQM). Results of 3

their study showed that CNTs play a pivotal role in increasing the buckling strength. Large deflection and bifurcation behavior of FG-CNTRC plates subjected to various in-plane loads were studied by Lei et al. [18-19] by means of the element-free kp-Ritz approach. In their works, the effective material properties were evaluated through Eshelby-Mori-Tanaka method. Liew and his co-workers [21-23] investigated influence of elastic foundation and CNTs on the bifurcation characteristics of FGCNTRC skew plate by utilizing element-free IMLS-Ritz approach. In order to take into account the effect of the transverse shear deformation and rotary inertia, the first-order shear deformation theory (FSDT) were employed to formulate the potential energy. Liew et al. [28-29] initiated benchmark researches on stability characteristics of FG-CNTRC cylindrical panels under axial compressive load using element-free method. To get rid of shear locking for a thin cylindrical panel, the bending stiffness was determined by means of a stabilized conforming nodal integration scheme. Asadi et al. [39] conducted a numerical study using harmonic differential quadrature method to evaluate the bifurcation and vibration responses of FG-CNTRC truncated conical shells. Their results showed that fundamental frequency is not generally associated with the circumferential mode number one, and geometrical parameters play an essential role in circumferential mode number associated with the fundamental natural frequency of FG-CNTRC conical shells. Despite various attempts to expatiate on influence of CNTs on the static and dynamic characteristics of flat and curved structures, there is a research gap in investigating the influence of CNTs on the aeroelastic flutter instability of CNTRC flat and curved structures. This research aims to bridge this gap in the literature. The present research is primarily focused on the aeroelastic flutter characteristics of an FGCNTRC truncated conical curved panel under axial compression and exposed to supersonic airflow. Nonlinear dynamic equations of the FG-CNTRC truncated conical curved panel are obtained through the Novozhilov shell theory along with Green’s strains and Hamilton principle. To determine the aerodynamic pressure, the Krumhaar’s modified supersonic piston theory is used. A comprehensive 4

parametric study is conducted to shed light on the effect of CNT volume fraction and distribution, geometrical parameters, Mach number on the stability boundaries and flutter characteristics of the FG-CNTRC truncated conical curved panels.

2. Governing Equations 2.1 Modeling of FG-CNTRC Consider an FG-CNTRC truncated conical curved panel subjected to supersonic airflow and axial compression, simultaneously. The semi-vertex angle, subtending angle, thickness and length of the panel are denoted by  ,  , h and L , respectively. R1 and R2 are the radii of the conical panel at its small and larger edges, respectively. The truncated conical curved panel is referred to a coordinate system  x , , z  , as shown in Fig. 1. The CNT reinforcement is assumed to be distributed either uniformly referred to as UD or functionally graded through the thickness referred to as FG. The modified rule of mixture is adopted to model the effect of CNTs on overall properties of the composite conical curved panel. Based on this model, the effective material properties could be determined as

E11  1VCN E11CN  Vm E m

2 E22



VCN Vm  m CN E22 E

3

V V  CN  mm CN G12 G12 G

(1)

  CNVCN   mVm where, the super/sub scripts ‘CN’ and ‘m’ stand for the CNT and polymer matrix, respectively. In addition, Young’s modulus, shear modulus and material density are donated by E, G and  , 5

respectively. Furthermore, the coefficients 1 , 2 , and 3 are introduced to account for the scale dependent material properties. It should be noted that these constants are assessed by matching the effective properties of CNTRC determined from molecular dynamic simulations (MDS) with those from the rule of mixtures [14]. Besides, VCN and Vm are the volume fraction of CNTs and polymer matrix, which fulfils the relationship of Vm  VCN  1. Three types of distributions for the CNTs in the FG-CNTRC truncated conical curved panel are considered in which these distributions are illustrated in Fig. 2. In this figure, the uniform distribution and the other two CNT distributions are denoted by UD, FG-X and FG-O, respectively. The mathematical expression of the CNT volume fraction as a function of the z direction, in each case is expressed as

  * VCN   z  *  VCN  z   2VCN 1  2  h    *  z   4VCN    h

UD FG  O

(2)

FG  X

* in which, V CN represents the volume fraction of CNTs.

The effective Poisson’s ratio is not sensitive to location of the CNTs and could be evaluated as * 12  12CNVCN   mVm

(3)

The previous researches by Bakis et al. [42] showed that the damping properties of CNT reinforced structures may be enhanced owing to the stick-slip frictional motion of CNTs. However, the micro-mechanical nanotube/resin interaction model is overlooked in this study, and as most of

6

the literatures [11-12]. The main focus of this study lies on the macro-mechanical characteristics of CNT reinforced curved structures.

2.2

Structural model

The conical curved panel is modeled according to the Novozhilov nonlinear shell theory and Green’s strains. Accordingly, the displacements  u1 , u2 , u3  of a generic point are related to the middle surface displacements  u0 , v0 , w0  , can be taken in the form [50] u1  u0  z u2  v0  z

(4)

u3  w0  z 

where

   1   2    2    1  1    1   1   2  1 2   1 2 in addition

7

(5)

1 

w 1 u0 1 A1  v0  0 A1 x A1 A2  1

1 

1 v0 1 A1  u0 A1 x A1 A2 

 

1 w0 u0  A1 x 1

(6)

w 1 v0 1 A2 2   u0  0 A2  A1 A2 x 2

2 

1 u0 1 A2  v0 A2  A1 A2 x

 

1 w0 v0  A2  2

where 1 , and 2 are the first and second principal radius of curvature and for the truncated conical curved panel can be written as

A1  1,

A2  R  x  ,

R  x   R1  x sin  ,

1  ,

2 

R  x cos 

(7)

Substituting Eqs. (5-7) into Eq. (4), the displacement fields of the conical curved panel can be reformulated as [50]

8

 w w 1 v0 w0 sin  w0 cos  u1  u0  z   0   u0  w0 0 x R  x  x  x R  x   x R  x  

w0 u0 sin  w u cos  sin 2 2   2 v0 0  2 v0 0  2 v0  R  x    R  x   R  x   2 R  x   1

2

 1 w0 1 u0 w0 cos   u0  1 u0 w0 sin  w0  u2  v0  z     v0 1   v0   x  R  x   x R  x  x   R  x   R  x  x  R  x    u 1 v0 sin  cos  1 u0 v0 u3  w0  z  0   u0  w0  R  x R  x  x   x R  x   R  x  

(8)

u sin  u0 cos  1 u0 v0 sin  v0  u0  w0 0   v0  R  x  x R  x  x R  x   x R  x  x 

With regard to the Green’s strains and the Novozhilov nonlinear shell theory, the strain fields can be written as

 xx   x0  z x   0  z

(9)

 x    z x 0 x

By retaining the main nonlinear terms and performing some mathematical manipulations, the strain-displacement relations may be expressed as

9

2 2 2 u0 1  u0   v0   w0             x 2  x   x   x   0 x

2 1 v0 sin  cos  1  1 u0 sin      u0  w0    v0  R  x   R  x  R  x 2  R  x   R  x    0

2

 1 v0 sin  cos    1 w0 cos      u0  w0     v0  R x   R x R x R x   R x              

 x0  

2

   

v0  1 u0 sin    u0  v0  1 u0   v0  1      R  x   R  x    x  x  R  x   sin  cos   w0  1 w0 cos   u0  w0    v0   R  x R  x   x  R  x   R  x  

 2 w0 x 2 cos   u0 1 v0  sin  w0 1  2 w0        R  x   x R  x    R  x  x R 2  x   2 cos  u0 cos  v0 2sin  w0 2  2 w0  x   2    R  x   R  x  x R 2  x   R  x  x

x  

(10)

The elastic constitutive equations of FG-CNTRC truncated conical curved panel in a plane state of stress may be written as  xx   Q11 Q12        Q21 Q22   Q  x   61 Q62

Q16   xx    Q26     Q66    x 

(11)

where Qij  denotes the transformed reduced stiffness matrix and its coefficients are determined as follows Q11 

E11

1  1221

,

Q12 

12 E11 , 1  1221

Q22 

10

E22

1  1221

,

Q66  G12

(12)

In accordance with Eq. (9), the stress resultants N ij , and M ij can be expressed in terms of the mid-plane displacements as    A  N      M    B 

 B    0   D   

(13)

The stiffness matrices in Eq. (13) are defined as h2

 A ,  B ,  D   Q 1, z, z dz 2

(14)

h 2

3. Dynamic Equations The equations of motion for the FG-CNTRC truncated conical curved panel may be obtained from the Hamilton’s principle. Based on this principle, an equilibrium position occurs in a structure when the following condition is fulfilled t2

 U  W   K  dt  0

(15)

t1

where the virtual strain energy  U , the virtual kinetic energy  K , and the virtual work done  W owing to the external loads can be determined as follows L  h 2

U   

 



xx

 xx        x   x   1  

0 0 h 2 L  h 2

K  



z    z u  u v v w w  1  

0 0 h 2 L 



z   R  x  dzd  dx 2 

  R  x  dzd  dx 2 

  F z   u 0  P w 0  1   R  x  d  dx   2  0 0   R x 

W    

11

(16)

where, F and P denote the axial compressive force and the aerodynamic load, respectively. To estimate the aerodynamic load, the quasi-steady Krumhaar’s modified supersonic piston theory [43] is used. It should be emphasized that this theory was originally proposed for cylindrical shells, which can be modified for truncated conical shells [44], by letting R  R  x   R1  x sin  , thus the modified piston theory may be expressed as

P  

 a P M 2  w 0

 1 M 2 2 1    w  w   0 0 Ma  M 2  1  M 2  1  x 2R  x  M 2  1 

(17)

where, a is the free stream speed of sound, P is the free stream static pressure, M is the Mach number and  a is the adiabatic exponent. Recalling Eqs. (9-17) and integrating the above expressions by parts to relieve the virtual displacements, and performing some mathematical manipulations lead to the dynamic equations of the FG-CNTRC truncated conical curved panel in which can be written as

 u0 :

N xx , x 

sin  sin  sin  1 N xx  N xxu0, x   N xxu0, x , x  N  2  N u0, , R  x R  x R  x R  x



sin  sin 2  sin  sin 2 1 N v  N u0  2 N v0,  2 N w0  N x ,    0 , 2 2 R  x R  x R  x 2R  x  R  x



1 1 sin  sin  cos  N x u0, x ,  N x u0, , x  N x v0, x  M  , x  N x v0 , x    R  x R  x R  x R  x R  x



cos  M x ,  I 0u0  I1w0, x R2  x 

12

 v0 :

sin  1 sin 2  sin  N xx v0, x   N xx v0, x , x  N ,  2 N v0  2 N u0, R  x R  x R  x R  x 1



R2  x 

 N v0, , 

sin  cos  cos 2  N u  N w    0 , 2   0 , 2 N v0 R2  x  R  x R  x



cos  2sin  sin  1 N w0,  N x  N x , x  N x u0, x   N x v0, x , 2 R  x R  x R  x R  x



sin  cos  cos  cos  M x , x  N x u0 , x   N x w0 , x  2 M  ,  R  x R  x R  x R  x

  1 2 cos  cos 2   cos     I 0  I1  2 I 2  v0   I1  2 I 2  w0, R  x R  x  R  x     R  x

 w0 :

sin  cos  cos 2  cos  N xx w0, x   N xx w0, x , x  N  2 N w0  2 N v0, R  x R  x R  x R  x 

sin 2 1 cos  cos  N u0  2 N w0, ,  2 N x  N v0 ,   2 2R  x  R  x R  x R  x



1 1 1 cos  N x , x w0,  N x w0, x ,  N x w0, x    N x , xv0 , x R  x R  x R  x R  x

 M xx , xx 

2sin  sin  1 2sin  M xx , x  M  , x  2 M  ,  2 M x , R  x R  x R  x R  x

(18)

 2 sin  1 cos   M x , x  P  I1 u0  I1u0, x   I1  I2 2  v0, R  x R  x R  x    R  x sin  1 I2 w0, x  I 2 w0, xx  I 2 2 w0,  I 0 w0 R  x R  x



in which, the comma subscript notation stands for the partial derivatives with respect to the x and  directions and    indicates the differentiation with respect to time. Furthermore, I i are the inertia terms and determined as h 2

Ii 



z     z  dz

i  0,1, 2

i

h 2

13

(19)

Each end of panel could be clamped (C) or simply supported (S). Mathematical expressions for these types of edge supports may be written as x  0, L

   R  x  N 1  u   N u

 cos    0

C:

1 v 01  w 01  w 0,1 x  R  x  N xx 1  u 0,1 x   N x  u 0,1  v 01 sin    M  cos   0

S:

1 v 01  w 01  M xx

xx

1 0, x

x

1 0,

1 v 01 sin    M 

  0,   N  M x1    u 0,1  v 01   N x  u 0,1 x  1  cos    0 R x   R x  

C:

u 01  w 01  w 0,1 

S:

 N  M x1 1 u 01  w 01  M     u 0,1  v 01   N x  u 0,1 x  1  cos    0 R x   R x  

(20)

4. Methodology With recent advances of computer technology, simulations of scientific and engineering problems have become increasingly sophisticated and complicated. From computational point of view, simulations of such complex models not only need more virtual storage, but also are an extremely time consuming process. To fill the technological gap, an efficient numerical method can be used to investigate in detail the interaction between fluids and solids, which is considered as a multidisciplinary problem. In this regard, finite element method (FEM) and finite difference method (FDM) have been the two well-known methods for engineering problems during the last five decades. The method of differential quadrature (DQM) is recently proposed by Bellman et al. [45-46]. This method is a highly efficient numerical technique for mathematical physics and engineering problems. The DQM has several advantages compared to the FEM and FDM [47-49]. For instance, the DQM approximates a function on the global computational domain by means of higher-order polynomials; whereas, the FEM employs lower-order polynomials to approximate a function on a local element. As a result of using higher-order polynomial approximation in the DQM, this method requires less grid

14

points compared to the FEM and FDM, therefore, very little computational costs, virtual storage and less CPU time are needed by implementing the DQM. Unlike the classic DQM in which the Lagrangian interpolation shape functions are used as test functions, the newly proposed DQM chooses other interpolation shape functions as test functions (i.e. harmonic functions, Hermite interpolation shape functions). These new test functions have some merits compared to the Lagrangian interpolation shape functions. For example, employing harmonic functions as the approximating test functions results not only in explicit terms for the weighting coefficients, but also in circumventing the limitation for the number of grid points in the classic DQM. In this work, the harmonic differential quadrature method (HDQM) is implemented. The capability of HDQM to handle the nonlinear problems is demonstrated by [47-49]. With regard to the concept of HDQM, the  th-order derivation of the function w  x  at any discrete point may be approximated as:

d w  x i  K    C ij w  x j  dx  j 1

(21)

where x i is a discrete point in the solution domain; w  x j  is the function value at point x j ; and  C ij  is the weighting coefficients for the

 th-order derivation of the function in the x direction. The

weighting coefficients may be determined by the functional approximations in the x direction. The harmonic test function

k

x 

employed in this technique may be written as:

15

k

  x  x 0     x  x k 1   sin   ...sin  2 2    x     x  x 0     x k  x k 1   sin  k  ...sin  2 2   

   x  x k 1     x  x K    sin   ...sin   2 2         x k  x k 1     x k  x K    sin   ...sin   2 2      (22)

In the HDQ, the weighting coefficients of the first-order derivatives C ij1 for i  j may be computed as:

C ij(1)

  x i  2    x i  x j      x j  sin    2  

(23)

where

x i  

  x i  x j  sin   2 j 1, j i  K



   

(24)

The weighting coefficients of second-order derivatives C ij 2 for i  j could be estimated by the following formula:

  x i  x j π C ij(2)  C ij(1)  2C ij(1)  πctg   2  

  ,  

i , j  1, 2,..., K

(25)

 The weighting coefficients of the first and second order derivatives C ii

for i  j may be

r

calculated as:

C ii r   

K



j 1, j i

C ij r  ,

r  1, 2,

16

j  1, 2,..., K

(26)

With regard to the C ij1 and C ij 2 , the weighting coefficients of the third and fourth-order derivatives could be estimated as K

C ij(3)  C ik(1)C kj(2) k 1

(27)

K

C

(4) ij

 C C k 1

(2) ik

(2) kj

Since the Chebyshev-Gauss-Lobatto grid point distribution has the most convergence and stability [47-49], the grid points may be generated as

xi 

  i  1  L 1  cos  2    K  1

   , i  1, 2,..., K 

(28)

4.1 Aeroelastic buckling analysis Before implementing the HDQ, the stability equations of the FG-CNTRC truncated conical curved panels may be obtained in accordance with an application of the adjacent equilibrium criterion [52]. To this end, a perturbed equilibrium position from a prebuckling state is considered. An equilibrium configuration at bifurcation point is defined with components  u00 , v00 , w00  . Then, the equilibrium configuration is perturbed by the tentative and arbitrary perturbation  u01 , v01 , w01  , thus the conical curved panel experiences a new equilibrium configuration adjacent to the primary one described with displacements  u00  u01 , v00  v10 , w00  w01  . By dropping the inertia terms from the Eq. (18), and applying the adjacent criterion, the stability equations can be written as

17

 u0 :

N 1xx , x 

sin  1 sin  0 1 sin  1 1 N xx  N xxu0, x   N xx0 u0,1 x   N  2  N0 u0,1  , , x R  x R  x R  x R  x



sin  sin 2  0 1 sin  0 1 sin 2 0 1 1 0 1 N v  N u0  2 N v0,  2 N w0  N 1x ,    0 , 2 2 R  x R  x R  x 2R  x  R  x



cos  1 cos  M  , x  2 M x ,  0 R  x R  x

sin  0 1 1 sin 2  0 1 sin  0 1 1 N xx v0, x   N xx0 v0,1 x   N  N v0  2 N u0, , ,x R  x R  x R2  x  R  x

 v0 :

 

N  x

1 R2

1  v0,   0

,

sin  cos  cos 2  0 1 0 1 0 1 N u  N w  N v0      0 ,  0 , R2  x  R2  x  R2  x 

cos  0 1 2sin  1 cos  1 cos  1 N w0,  N x  N 1x , x  2 M  ,  M x , x  0 2 R  x R  x R  x R  x

sin  0 1 cos  1 cos 2  0 1 cos  0 1 N xx w0, x   N xx0 w0,1 x   N  2 N w0  2 N v0, ,x R  x R  x R  x R  x

 w0 :



sin 2 0 1 1 cos  cos  1 N u0  2 N0 w0,1    2 N0 v01   N x   2 ,  ,  2R  x  R  x R  x R  x

M 

1 xx , xx

2sin  1 sin  1 1 2sin  1 1  M xx , x  M  , x  2 M  M x , ,  R  x R  x R  x R2  x 

(29)

2 M 1x , x  P  0 R  x

where, N xx0 , N0 and N x0 are the prebuckling load resultants of the conical curved panel and can be determined as N xx0  

P  R  x  cos 

N x0  0,

N0  0

where,  and P represent the subtending angle and axial force, respectively.

18

(30)

In accordance with the periodicity conditions of the conical curved panel in the circumferential direction and in the absence of the shear prebuckling force N x0 , the following solution is proposed, which satisfies the circumferential restraints of the conical panel

 m  u01  x,    U  x  sin       m  v01  x,    V  x  cos       m  w01  x,   W  x  sin     

(31)

where, m is the wave number through the circumferential direction. Substituting Eq. (31) into Eq. (29) and implementing the HDQ discretization for the x  dependent functions result in following system of equations

   K EUU   K EUV   K EUW     KGUU   KGUV   KGUW     U   0                      VU VV VW VU VV VW    K   K   K      K   K   K    V    0   E   E   E   G   G   G        K WU   K WV   K WW     K WU   K WV   K WW    W  0 E E E G G G                

(32)

where  K E  is the elastic stiffness matrix, which contains the aerodynamic load parameter and the unknown circumferential wave number m. In addition,  KG  is the geometric matrix, which contains unknown mechanical loads. The critical buckling load of the FG-CNTRC truncated conical curved panel may be evaluated by solving the standard eigenvalue problem.

19

4.2

Aeroelastic flutter analysis

The modal shapes for a vibrating FG-CNTRC conical curved panel are proposed as a product of known trigonometric functions in the circumferential direction and unknown functions in the meridional direction. Following displacement field, which satisfies the boundary conditions is considered [39]

 m   i t u 01  x ,  , t   U  x  sin  e     m   i t v 01  x ,  , t  V  x  cos  e     m   i t w 01  x ,  , t  W  x  sin  e   

(33)

Eq. (33) is substituted into Eq. (18) and by performing the HDQ discretization for the x  dependent function, the following system of equations is established    K EUU  KGUU   K EUV  K GUV   K EUW  K GUW            K VU  K VU   K VV  K VV   K VW  K VW   G  G  G   E  E  E  WU WU WV WV WW    K  K   K  K   K  K WW   G  G  G   E  E   E  C UU  C UV  C UW     M UU   M UV   M UW    U   0                     C VU  C VV  C VW    2 2   M VU   M VV   M VW     V    0          C WU  C WV  C WW     M WU   M WV   M WW    W  0             

(34)

where C  and  M  stand for the aerodynamic damping matrix and the mass matrix, respectively. The eigenvalues  are in general complex numbers and given by     i , where  is the measurement of the damping and  is the frequency. It should be emphasized that the aerodynamic damping term in Eq. (33) is fairly small and always stabilizes the flutter boundary [8-9]. Consequently, it is advantageous to ignore it when analyzing the behavior in the frequency domain [8-9]. Once the 20

complex eigenvalues of the system are determined, the natural frequencies  j  and damping ratio

  of the system can be obtained as j

j 

 Im   , 2

j

j 

j

 2j   2j

(35)

It should be pointed out that when flutter occurs, two eigen modes of structures merge at a certain aerodynamic pressure, which is called flutter pressure.

5. Numerical illustrations and discussions The aeroelastic characteristics of FG-CNTRC truncated conical curved panels subjected to supersonic airflow are investigated to reveal the efficiency of the CNTs in enhancing the aeroelastic responses of the conical curved panels. Moreover, the accuracy, efficiency and computational costs of the proposed solution procedure, which is based on the Novozhilov nonlinear shell theory along with the HDQM are evaluated and discussed in details in this section. At first, a convergence study is conducted to obtain the sufficient grid points of the HDQM. Afterwards, verifications of the free vibration and buckling analysis of curved panels in the absence of the aerodynamic pressure are performed to ensure the proficiency of proposed method. As a benchmark investigation, a rigorous parametric study is directed to examine the influence of the CNT volume fraction and distribution, geometrical parameters and boundary conditions on the aeroelastic responses of the supersonic truncated conical curved panel.

21

The polymer matrix is assumed to be Poly methy1 methacrylate referred to as PMMA. For the SWCNTs with chiral index (10,10), the CNTs efficiency parameters are given in Table 1. Besides, the material properties of PMMA and CNTs are considered as [38]

E m  2.5  GPa  ,

 m  1150  kg/m3  ,

E 11CN  5.6466  TPa  ,

 m  0.34,

CN E 22  7.080  TPa  ,

G12CN  1.9445  TPa  ,

 CN  1400  kg/m3  ,

 CN  0.175 In the rest of this paper, the following convention is established for boundary conditions. For example, an CSCS truncated conical shell, indicates a conical shell which is clamped at x  0 and L , simply supported at   0 and  .

5.1 Convergence study To determine the sufficient grid points of the HDQM, the critical freestream static pressure

P  Cr 

of the simply supported FG-CNTRC truncated conical curved panel with different CNT

distributions are demonstrated in Figure 3. It can be inferred that the results converge well as the number of grid points increases. As noticed, the fast rate of convergence of the proposed method is evident from this figure and it is revealed that after adoption of 19 grid points for each variable, the critical freestream static pressure is determined with an accuracy up to six digits.

5.2

Verifications

As stated earlier, no work has been conducted on the aeroelastic responses of the FG-CNTRC truncated conical curved panels. Thus, for the sake of comparison, the case of the FG-CNTRC cylindrical panels in the absence of the aerodynamic pressure is addressed. To this end, the first five natural frequencies and the critical buckling load of cylindrical panels are listed in Table 2 and 3, respectively. 22

In Table 2, the first five natural frequencies of the FG-CNTRC cylindrical panel are compared with those obtained by Zhang et al. [38] based on the Ritz method. It should be noted that Zhang et al. [38] used an equivalent continuum model based on the Eshelby-Mori-Tanaka method to estimate the material properties of the nanocomposite panel. An excellent agreement with a maximum discrepancy of less than 1% can be observed in the predictions. As part of the verification of the present method, the buckling for simply supported cross-ply laminated composite cylindrical panels under axial compression are compared in Table 3 with a GDQ method results of Abediokhchi et. al. [51] using Donnell shell theory. As can be seen, the present results are in good agreement with those obtained in [51].

5.3

Computational efficiency

To examine the computational efficiency of the proposed methodology, the CPU time and CPU usage of the outlined method are compared with those of the FEM. A finite element analysis is conducted using ABAQUS 6.11 software. A simply supported isotropic truncated conical curved panel is modeled in ABAQUS software using the eight node doubly-curved rectangle elements (S8R) with six degrees of freedom (DOF) at each node. It should be pointed out that the predictions of the FEM are strongly dependent on mesh sizes. To preserve the accuracy of results, the mesh size may further have refined depending on geometrical parameters of the truncated conical curved panel. A modal dynamic analysis is performed using ABAQUS software. In Table 4, the CPU usage and time of the HDQM to determine the non-dimensional fundamental natural frequency of the isotropic truncated conical curved panel are compared with those of the FEM. A review of this table discloses that the computational costs and virtual storage of the outlined method are tremendously less than the FEM. It is evident that, each HDQ result takes 23

less than 10 (sec) computer running time; whereas, the minimum computer running time for the FEM is about 10 (min). Besides, this comparison shows very good agreement with a maximum discrepancy of less than 4.5%. In Table 5, the CPU usage and time of the HDQM to determine the critical buckling load of the isotropic truncated conical curved panel are compared with those of the FEM. A review of this table reveals that the computational costs and virtual storage of the proposed method are terrifically less than the FEM. It can be inferred from this table that each HDQ result takes less than 15 (sec) computer running time; whereas, the minimum computer running time for the FEM is about 11 (min). Besides, this comparison shows a close agreement with a maximum discrepancy of less than 2.4%. The main reason for this observation is that the accurate HDQ results may be determined using a noticeably fewer grid points, so the computational costs are minimal. Hence, the outlined method is significantly efficient from the computational point of view.

5.4

Parametric studies

To study the influence of the CNT volume fraction and distribution, geometrical parameters, the Mach number and boundary condition on the stability boundaries, onset of the aeroelastic buckling and flutter instabilities, a set of parametric study is given in the following subsections.

5.4.1

Stability boundaries

In Figure 4, the stability boundaries of the FG-CNTRC truncated conical curved panel for different distributions of CNTs are illustrated. It can be inferred that distribution of CNTs plays an essential role in improvement of the stability boundaries of the FG-CNTRC truncated conical curved panel. 24

For instance, the ratio of the flat and stable area of the FG-X CNTRC truncated conical curved panel to those of the FG-O CNTRC truncated conical curved panel is about 2.7. Besides, the ratio of the flat and stable area of the FG-X CNTRC truncated conical curved panel to those of the UD CNTRC truncated conical curved panel is about 1.65. This observation indicates that the increase of the stiffness near the surfaces of the FG-CNTRC truncated conical curved panel is more effective approach to improve the stability boundaries. The stability boundaries of the FG-X CNTRC truncated conical curved panel are exhibited in Figure 5, for different volume fractions of CNTs. It can be revealed from this figure that volume fraction of CNTs plays a major role in enhancement of the stability boundaries of the CNTRC truncated conical curved panel. For instance, the ratio of the flat and stable zone of the FG-X CNTRC * truncated conical curved panel with V CN  0.28 to those of the FG-X CNTRC truncated conical * curved panel with V CN  0.12 is about 2.5.

In Figure 6, the influence of the subtended angle on the stability boundaries of the FG-X CNTRC truncated conical curved panel is depicted. It is found that the subtended angle plays a pivotal role not only in changing the critical circumferential mode number of the aeroelastic flutter instability, but also in improving the stability boundaries of the FG-X CNTRC truncated conical curved panel. For example, for case of the    60  , the critical circumferential mode number is 2; whereas, the critical circumferential mode number is 1, for the case of the    180  . In addition, the ratio of the flat and stable area of the FG-X CNTRC truncated conical curved panel with    60  to those of the FG-X CNTRC truncated conical curved panel with    180  is about 1.45. In Figure 7, the stability boundaries of the FG-X CNTRC truncated conical curved panel are exhibited for different semi-vertex angles. It can be inferred that increasing the semi-vertex angle 25

results in noticeable decrease in the flat and stable area. Of possible further interest is that for all cases except   60  , increasing the axial compressive load results in an oscillation in the dynamically unstable zone (flutter boundary). The reason for such oscillation behavior is that both the semi-vertex angle and axial compression may change the critical circumferential flutter mode number. For instance, for the case of the   15  , the critical circumferential flutter mode number is 1 till the non-dimensional axial compressive load  F FCr  reaches to 0.8. Beyond this point, the critical circumferential flutter mode number changes from 1 to 3.

5.4.2

Aeroelastic buckling instability

Figure 8 illustrates variations of the critical buckling load of the FG-X CNTRC truncated conical curved panel versus the circumferential buckling mode number  m  for different subtended angles. It can be concluded that the critical buckling load is not generally associated with the circumferential buckling mode number one  m  1 , and the subtended angle plays a major role in circumferential mode number corresponding to the critical buckling load of the FG-X CNTRC truncated conical curved panel. For instance, for the case of the    60  , the circumferential buckling mode number 2 corresponds to the critical buckling load; whereas, for the case of the    180  , the circumferential mode number 5 corresponds to the critical buckling load. Figure 9 is depicted to investigate the effect of the subtended angle on the onset of the aeroelastic buckling instability of the FG-X CNTRC truncated conical curved panel. It can be inferred that increasing subtended angle not only increases the critical buckling load, but also, may change the buckling mode shape of the FG-X CNTRC FG-X CNTRC truncated conical curved panel. In fact, 26

variations of the critical buckling load versus the subtended angle exhibits oscillatory behavior as a result of the change in the circumferential buckling mode number of the FG-X CNTRC truncated conical curved panel. Of possible further interest is that the number of local extrema, which indicate alternation in buckling modes, increases with decreasing the semi-vertex angle. Besides, some buckled configurations of the FG-X CNTRC truncated conical curved panel are illustrated in Figure 10. Figure 11 is provided to expatiate on the effect of the aerodynamic pressure on the buckling characteristics of the FG-CNTRC truncated conical curved panel. It can be seen that increasing the aerodynamic pressure results in a noticeable increase in the critical buckling load of the FG-CNTRC truncated conical curved panel. The reason for this observation is the stabilizing effect of the aerodynamic pressure. Of possible further interest is that the aerodynamic pressure not only plays a crucial role in the onset of the aeroelastic buckling instability, but also may alter the buckling configuration of the FG-CNTRC truncated conical curved panel. For instance, for the case of the

P  10  kPa  , the circumferential buckling mode number is 6; while, for the case of the P  80  kPa  , the circumferential buckling mode number is 4. 5.4.3

Aeroelastic flutter instability

Figures 12(a) and 12(b) are presented to evaluate the effect of CNT distribution and volume fraction on the aeroelastic flutter instability of the FG-CNTRC truncated conical curved panel. It can be inferred from figure 12(a) that the maximum and minimum critical freestream static pressure correspond to the FG-X and FG-O distributions of CNTs, respectively. Of possible further interest is that the difference between these two distributions decreases as the subtended angle increases. For example, increasing the subtended angle from 60 to 180 , decreases the difference between these two distributions from 78% to 50%. Besides, it is evident from figure 12(b) that increasing the volume fraction of CNTs leads to a significant enhancement of the aeroelastic flutter characteristics of the 27

FG-CNTRC truncated conical curved panel. It should be pointed out that increasing the CNT volume fraction results in an increase in the frequency interval of flutter modes. For instance, for the case of the    60  , increasing the CNT volume fraction from 0.12 to 0.28 leads to a noticeable increase about 130% in the critical freestream static pressure. OF possible further interest is that for all cases, the critical circumferential flutter mode number is 1 and it is independent of the CNT volume fraction and distribution. Figure 13 is provided to expatiate on the effect of the Mach number on the onset of aeroelastic flutter instability of FG-CNTRC truncated conical curved panel. It can be concluded that the Mach number plays a major role in the onset of the aeroelastic flutter instability of the FG-CNTRC truncated conical curved panel. For example, for the case of the    60  , increasing the Mach number from 2 to 5 results in a tremendous decrease about 54% in the critical freestream static pressure of the FGCNTRC truncated conical curved panel. The interesting point is that the critical circumferential flutter mode number is 1 for all cases, and it is independent of the Mach number. Of possible further interest is that for the case of the  M  2  , increasing the subtended angle from 60 to 180 leads to a substantial decrease about 42% in the critical freestream static pressure of the FG-CNTRC truncated conical curved panel. Figure 14 is presented to elaborate on the influence of the boundary conditions of the truncated conical curved panel on the aeroelastic characteristics of the FG-CNTRC conical curved panel. As expected, the minimum critical freestream static pressure corresponds to the fully simply supported boundary conditions as it has the lower flexural rigidity compared to the clamped boundary condition. Due to non-symmetric geometry of the truncated conical curved panel, the trend of the critical freestream static pressure of the conical curved panel with SSCC boundary conditions is always opposite to that of the conical curved panel with CCSS boundary conditions. It should be noted that 28

for the case of the    60  , the difference between the critical freestream static pressure of the conical curved panel with CCSS and SSCC boundary conditions is about 26%. Figure 15 is provided to elucidate the effects of radius-to-thickness ratio on the aeroelastic flutter responses of the FG-CNTRC truncated conical curved panels. It can be inferred from figure 15 that by increasing the radius-to-thickness ratio, the critical freestream static pressure decreases significantly. It is due to the lower flexural rigidity of the thin FG-CNTRC conical curved panel. For instance, at a constant subtended angle    60  , increasing the radius-to-thickness ratio from 100 to 175 results in a considerable decrease about 80% in the onset of the aeroelastic flutter instability. Apparently, the critical circumferential flutter mode number is independent of the radius-to-thickness ratio and for all cases, the critical circumferential mode number is equal to one. Finally, Figure 16 is depicted to elaborate the effect of length-to-radius ratio on the aeroelastic flutter responses of the FG-CNTRC truncated conical curved panels. It can be seen from figure 16 that by increasing the length-to-radius ratio, the critical freestream static pressure declines remarkably. It is due to the lower flexural rigidity of the long FG-CNTRC conical curved panel. For instance, at a given subtended angle

   120 ,

increasing the length-to-radius ratio from 1 to 1.5 leads to

noticeable decrease about 52% in the onset of the aeroelastic flutter instability. Apparently, the critical circumferential flutter mode number is independent of the length-to-radius ratio.

6. Concluding remarks The present research deals with the aeroelastic characteristics of FG-CNTRC truncated conical curved panels subjected to axial compression and aerodynamic load, simultaneously. According to the modified rule of mixture, equivalent material properties of the FG-CNTRC conical curved panel are 29

estimated. The equations of motion are established within the framework of Novozhilov nonlinear shell theory in conjunction with the Green’s strains. The aerodynamic pressure is estimated with regard to the quasi-steady Krumhaar’s modified piston theory. A semi-analytical solution based on the matrix transform method and HDQM is proposed. Fast rate of convergence of the proposed method is demonstrated and its high accuracy and efficiency with less computational costs and CPU time is illustrated by comparing the results with those reported in the literature. Various parametric studies are accomplished to examine the effect of CNT volume fraction and distribution, boundary conditions, geometrical parameters and the Mach number on the aeroelastic responses of the FGCNTRC truncated conical curved panel. Following conclusions may be drawn from the numerical simulations: 

The semi-vertex and subtended angles play an essential role not only in increasing the critical buckling load of the FG-CNTRC truncated conical curved panel, but also in changing the circumferential buckling mode number.



The aerodynamic pressure not only increases the critical buckling load of the FGCNTRC truncated conical curved panel, but also may alter the buckling configuration of the conical curved panel.



The critical freestream static pressure noticeably decreases as the Mach number increases. It is due to the decrease in the frequency interval of flutter modes with increasing the Mach number.



Volume fraction and distribution of CNTs play a pivotal role in enhancement of the aeroelastic buckling and flutter instabilities of the FG-CNTRC truncated conical curved panel. It is found that the aeroelastic behaviors of the FG-CNTRC truncated conical curved panel may be tremendously enhanced by the FG-X distribution of CNTs; while,

30

the aeroelastic characteristics of the FG-CNTRC truncated conical curved panel may be worsened by the FG-O distribution of CNTs. 

It is inferred that the critical circumferential flutter mode number is independent of the volume faction and distribution of CNTs.



It is found that the critical circumferential flutter mode number may be changed by increasing the applied compressive load. As a result of changes in the critical circumferential flutter mode number, an oscillatory behavior is observed in the dynamically unstable zone (flutter boundary).



Results of this research indicate that the outlined solution procedure is effective in reducing computational costs and virtual storage requirements.



It is expected that the capability of the HDQM to handle complex flows and large structural deformations will be discovered.

Reference [1] E. H. Dowell, H. M. Voss. Theoretical and experimental panel flutter studies in the Mach number range 1.0 to 5.05. AIAA Journal. 1965, 3, 2292-2304. [2] E. H. Dowell. Panel flutter: a review of the aeroelastic stability of plates and shells. AIAA Journal. 1970, 8, 385-399. [3] R. L. Bisplinghoff, H. Ashley. Principles of aeroelasticity. New York, Wiley. 1962. [4] F. M. Li, M. G. Song. Aeroelastic flutter analysis for 2D Kirchhoff and Mindlin panels with different boundary conditions in supersonic airflow. Acta Mechanica. 2014, 225, 3339-3351. [5] D. Y. Xue, C. Mei. Finite element nonlinear panel flutter with arbitrary temperatures in supersonic flow. AIAA Journal. 1993, 31, 154-162. 31

[6] M. Amabili, F. Pellicano. Multimode approach to nonlinear supersonic flutter of imperfect circular cylindrical shells. Journal of Applied Mechanics. 2012, 69, 117-129. [7] M. A. Kouchakzadeh, M. Rasekh, H. Haddadpour. Panel flutter analysis of general laminated composite plates. Composite Structures. 2010, 92, 2906-2915. [8] Z. G. Song, F. M. Li. Aeroelastic analysis and active flutter control of nonlinear lattice sandwich beams. Nonlinear Dynamics. 2014, 76, 57-68. [9] M. Samadpour, H. Asadi, Q. Wang. Nonlinear aero-thermal flutter postponement of supersonic laminated composite beams with shape memory alloys. European Journal of Mechanics-A/Solids. 2016, 57, 18-28. [10] K. M. Liew, Z. X. Lei, L. W. Zhang. Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review. Composite Structures. 2015, 120, 90-97. [11] J. Wuite, S. Adali. Deflection and stress behavior of nanocomposite reinforced beams using a multiscale analysis. Composite Structures. 2005, 71, 388-396. [12] M. Rafiee, J. Yang, S. Kitipornchai. Thermal bifurcation buckling of piezoelectric carbon nanotube reinforced composite beams. Journal of Computers & Mathematics with Applications. 2013, 66, 1147-1160. [13] M. H. Yas, N. Samadi. Free vibrations and buckling analysis of carbon nanotube reinforced composite Timoshenko beams on elastic foundation. International Journal of Pressure Vessels and Piping. 2012, 98, 119-128. [14] H. S. Shen, Y. Xiang. Nonlinear analysis of nanotube reinforced composite beams resting on elastic foundations in thermal environments. Engineering Structures. 2013, 56, 698-708. [15] G. Formica, W. Lacarbonara, R. Alessi. Vibrations of carbon nanotube-reinforced composites. Journal of Sound and Vibration. 2010, 329, 1875-1889.

32

[16] H. Asadi, Q. Wang. An investigation on the aeroelastic flutter characteristics of FG-CNTRC beams

in

the

supersonic

flow.

Composites

Part

B,

2016,

DOI:

10.1016/j.compositesb.2016.10.089, In Press. [17] A. Arani, S. Maghamikia, M. Mohammadimehr, A. Arefmanesh. Buckling analysis of laminated composite rectangular plates reinforced by SWNTs using analytical and finite element methods.Mechanical Science and Technology. 2011, 25, 809-820. [18] Z. X. Lei, K. M. Liew, J. I. Yu. Buckling analysis of functionally graded carbon nanotubereinforced composite plates using the element-free kp-Ritz method. Composite Structures. 2013, 98, 160-168. [19] Z. X. Lei, K. M. Liew, J. I. Yu. Large deflection analysis of functionally graded carbon nanotubereinforced composite plates using the element-free kp-Ritz method. Computer Methods in Applied Mechanics and Engineering. 2013, 256, 189-199. [20] M. Rafiee, X. Q. He ,K. M. Liew. Nonlinear dynamic stability of piezoelectric functionally graded carbon nanotube-reinforced composite plates with initial geometric imperfection. International Journal of Nonlinear Mechanics. 2014, 59, 37-51. [21] L.W. Zhang, Z.X. Lei, K.M. Liew. An element-free IMLS-Ritz framework for buckling analysis of FG-CNT reinforced composite thick plates resting on Winkler foundations. Engineering Analysis with Boundary Elements. 2015, 58, 7-17. [22] Z. X. Lei, L. W. Zhang, K. M. Liew. Buckling of FG-CNT reinforced composite thick skew plates resting on Pasternak foundations based on an element-free approach. Applied Mathematics and Computation. 2015, 266, 773-791. [23] L. W. Zhang, Z. X. Lei, K. M. Liew. Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach. Composites B. 2015, 75, 36-46. 33

[24] L. W. Zhang, Z. X. Lei, K. M. Liew. Vibration characteristic of moderately thick functionally graded carbon nanotube reinforced composite skew plates. Composite Structures. 2015, 122, 172183. [25] L. W. Zhang, Z. X. Lei, K. M. Liew. Free vibration analysis of functionally graded carbon nanotube reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method. Composite Structures. 2015, 120, 189-199. [26] M. Mohammadzadeh- Keleshteri, H. Asadi, M. M. Aghdam. Geometrical nonlinear free vibration responses of FG-CNT reinforced composite annular sector plates integrated with piezoelectric layers. Composite Structures. 2017. Accepted, In Press. [27] A. Sankar, S. Natarajan, M. Haboussi, K. Ramajeyathilagam, M. Ganapathi. Panel flutter characteristics of sandwich plates with CNT reinforced facesheets using an accurate higher-order theory. Journal of Fluids and Structures. 2014, 50, 376-391. [28] K. M. Liew, Z. X. Lei, J. L. Yu, L. W. Zhang. Postbuckling analysis of carbon nanotube reinforced functionally graded cylindrical panels under axial compression using a meshless. Computer Methods in Applied Mechanics and Engineering. 2014, 268, 1-17. [29] Z. X. Lei, L. W. Zhang, K. M. Liew, J. L. Yu. Dynamic stability analysis of carbon nanotube reinforced functionally graded cylindrical panels under axial compression using the element-free kp-Ritz method. Composite Structures. 2014, 113, 328-338. [30] M. Mehri, H. Asadi, Q. Wang. On dynamic instability of a pressurized functionally graded carbon nanotube reinforced truncated conical shell subjected to yawed supersonic airflow. Composite Structures. 2016, 153, 938-951. [31] H. S. Shen, Y. Xiang. Postbuckling of axially compressed nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Composites B. 2014, 67, 50-61. 34

[32] H. S. Shen, Y. Xiang. Thermal postbuckling of nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Composite Structures. 2015, 123, 383392. [33] H. S. Shen, Y. Xiang. Nonlinear response of nanotube-reinforced composite cylindrical panels subjected to combined loadings and resting on elastic foundations. Composite Structures. 2015, 131, 939-950. [34] H. S. Shen. Thermal buckling and postbuckling behavior of functionally graded carbon nanotubereinforced composite cylindrical shells. Composites B. 2012, 43, 1030-1038. [35] H. S. Shen. Torsional postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments. Composite Structures. 2014, 116, 477-488. [36] Y. Heydarpour, M. M. Aghdam, P. Malekzadeh. Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells. Composite Structures. 2014, 117, 187-200. [37] L. W. Zhang, Z. X. Lei, K. M. Liew, J. L. Yu. Large deflection geometrically nonlinear analysis of carbon nanotube reinforced functionally graded cylindrical panels. Computer Methods in Applied Mechanics and Engineering. 2014, 273, 1-18. [38] L. W. Zhang, Z. X. Lei, K. M. Liew, J. L. Yu. Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels. Composite Structures. 2014, 111, 205-212. [39] M. Mehri, H. Asadi, Q. Wang. Buckling and vibration analysis of a pressurized CNT reinforced functionally graded truncated conical shell under an axial compression using HDQ method. Computer Methods in Applied Mechanics and Engineering. 2016, 303, 75-100. [40] R. Ansari, J. Torabi. Numerical study on the buckling and vibration of functionally graded carbon nanotube-reinforced composite conical shells under axial loading. Composites B. 2016, 95, 196208. 35

[41] L. W. Zhang, Z. G. Song, K. M. Liew. Computation of aerothermoelastic properties and active flutter control of CNT reinforced functionally graded composite panels in supersonic airflow.Computer Methods in Applied Mechanics and Engineering. In Press. DOI: 10.1016/j.cma.2015.11.029. [42] X. Zhou, E. Shin, K. W. Wang, C. E. Bakis. Interfacial damping characteristics of carbon nanotube-based composites. Composite Science and Technology. 2004, 64(15), 2425-2437. [43] H. Krumhaar. The accuracy of linear piston theory when applied to cylindrical shells. AIAA. 1963, 1(6), 1448-1449. [44] S. C. Dixon, M. L. Hudson. Flutter, vibration, and buckling of truncated orthotropic conical shells with generalized elastic edge restraint. NASA, TN D-5759, 1970. [45] R. E. Bellman, B. G. Kashef, J. Casti. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics. 1972, 10, 40-52. [46] R. E. Bellman, J. Casti. Differential quadrature and long-term integration. Journal of Mathematical Analysis and Applications 1971, 34, 235-238 [47] H. Asadi, A. H. Akbarzadeh, Z. T. Chen, M. M. Aghdam. Enhanced thermal stability of functionally graded sandwich cylindrical shells by shape memory alloys. Smart Materials and Structures. 2015, 24, 045022. [48] H. Asadi, M. M. Aghdam, M. Shakeri. Vibration analysis of axially moving line supported functionally graded plates with temperature-dependent properties. Proc IMechE Part C: J Mech. Eng. Sci. 2014, 228(6), 953–969

36

[49] H. Asadi, Y. Kiani, M. Shakeri, M. M. Aghdam. Enhanced thermal buckling of laminated composite cylindrical shells with shape memory alloy. Journal of Composite Materials. 2016, 50(2), 243-256. [50] Amabili M. Nonlinear vibrations and stability of shells and plates. 1st ed. Cambridge University Press; 2008. [51] J. Abediokhchi, M. A. Kouchakzadeh, M. Shakouri. Buckling analysis of cross-ply laminated conical panels using GDQ method. Composites Part B. 2013, 55, 440-446. [52] Brush D. O. Prebuckling rotations and cylindrical shell analysis. ASCE, Journal of Engineering Mechanics. 1980, 106, 225-232.

37

Table 1. CNT efficiency parameters [14].

1

2

3

0.12

0.137

1.022

0.715

0.17

0.142

1.626

1.138

0.28

0.141

1.585

1.109

* V CN

 L2 Table 2. Comparison for the first five natural frequencies      h 

m 

 of the FG-CNTRC E m 

cylindrical panels with four edges simply supported boundary conditions.

UD Mode

Ref. [38]

FG-O Present Method

Ref. [38]

Present Method

FG-X Ref. [38]

Present Method

1

17.850

17.233

13.444

12.871

21.243

20.833

2

22.073

21.975

18.482

18.134

25.096

24.816

3

33.285

32.684

30.587

30.112

35.939

35.632

4

51.778

50.965

48.702

48.294

54.535

54.173

5

65.121

64.391

49.430

49.095

76.758

76.244

Table 3. Validation of the critical buckling load of symmetric cross-ply conical panels with two layers

E

11

E 22  40,G12 E 22  0.5, E 22  12.1 GPa  ,12  0.25, L  1(m), L R1  0.5 .

R h 𝛽(°)

80

𝛼 (°) Ref. [51]

30

60

100 Present Method

Ref. [51]

120 Present Method

Ref. [51]

Present Method

1

0.006485

0.006329

0.004529

0.004422

0.003412

0.003386

5

0.006431 0.006277

0.004487

0.004383

0.003395

0.003363

10

0.006266 0.006119

0.004361

0.004263

0.003338

0.003265

30

0.004732 0.004637

0.003200

0.003145

0.002397

0.002363

45

0.003147 0.003089

0.002035

0.002004

0.001474

0.001457

1

0.01297

0.012658

0.009057

0.008845

0.006117

0.006035

5

0.01286

0.012555

0.008975

0.008766

0.006077

0.005995

10

0.01253

0.012239

0.008722

0.008525

0.005952

0.005872

30

0.009465 0.009275

0.006401

0.006291

0.004777

0.004690

45

0.006293 0.006178

0.004070

0.004008

0.002948

0.002915

Table 4. A comparison on the computational efficiency of the proposed HDQ method with finite element method for free vibration analysis

 L2     h 

 , R h  200, L R  0.5, E  2.5GPa, =0.34  . E 



  45

Solution

  90

  60

  30

  60

  30

  60

  30

  60

Number of grid points

19

19

19

19

19

19

Present

DOF

57

57

57

57

57

57

Method

CPU time (sec)

5

5

8

8

10

10

(HDQM)

CPU usage (%)

25

25

25

25

25

25



19.4079

14.2453

18.1978

13.9007

18.0138

13.5267

Number of elements

5034

5772

5694

6004

5934

6592

DOF

30204

34632

34164

36024

35604

39552

CPU time (sec)

573

643

598

793

703

943

CPU usage (%)

80

80

81

83

81

86



20.04

14.87

18.93

14.45

18.56

14.12

Discrepancy (%)

3.15

4.2

3.85

3.8

2.95

4.2

method

Finite Element Method (ABAQUS)

Table 5. A comparison on the computational efficiency of the proposed HDQ method with finite element method for buckling analysis  R h  200, L R  0.5, E  2.5GPa, =0.34 .

  45

Solution

  90

  60

  30

  60

  30

  60

  30

  60

Number of grid points

19

19

19

19

19

19

Present

DOF

57

57

57

57

57

57

Method

CPU time (sec)

7

7

10

10

15

15

(HDQM)

CPU usage (%)

25

25

25

25

25

25

Buckling load (MN)

0.0451

0.0155

0.0602

0.0206

0.0902

0.0309

Number of elements

5275

6035

5863

6296

6304

6894

DOF

31650

36210

35178

37776

37824

41364

CPU time (sec)

641

683

663

712

724

748

CPU usage (%)

82

82

85

85

85

85

Buckling load (MN)

0.0456

0.0158

0.0607

0.0211

0.0908

0.0312

1.1

1.9

0.9

2.4

0.7

1

method

Finite Element Method (ABAQUS)

Discrepancy (%)

F

F

b Figure 1: Schematic and geometric characteristics of an FG-CNTRC truncated conical curved panel.

1

Figure 2: Three types of CNT distributions in the cross section of the FG-CNTRC truncated conical curved panel.

2

∞

α β

Figure 3: Convergence study of the critical freestream static pressure of the FG-CNTRC truncated conical curved panel.

3

β

∞

α

Figure 4: Influence of CNT distribution on the stability boundaries of the FG-CNTRC truncated conical curved panel.

4

β

∞

α

Figure 5: Effect of the CNT volume fraction on the stability boundaries of the FG-X CNTRC truncated conical curved panel.

5

∞

α

β β β

Figure 6: Effect of the subtended angle on the stability boundaries of the FG-CNTRC truncated conical curved panel.

6

∞

β

α α α α

Figure 7: Influence of the semi-vertex angle on the stability boundaries of the FG-CNTRC truncated conical curved panel.

7

α

β β β β

Figure 8: Variation of the critical buckling load of the FG-X CNTRC truncated conical curved panel versus the circumferential mode number m.

8

7 6

6

α α α α

5 5 4

5

4 3 3 2

4 3

4

2 2

1 1 1 1

3 2

β ° Figure 9: Effect of the subtended angle on the onset of the buckling instability of the FG-CNTRC truncated conical curved panel.

9

b=30 m=1

b=60 m=2

b=90 m=3

b=120 m=4

b=180 m=5

Figure 10: Effect of the subtended angle on the onset of the buckling instability of the FG-CNTRC truncated conical curved panel.

10

A) m=6

B) m=5

C) m=4

Figure 11: Influence of the aerodynamic pressure on the buckling mode shape of the FG-CNTRC truncated conical curved panel A) P∞ = 10(kP a), B) P∞ = 50(kP a), C) P∞ = 80(kP a).

11

(a)

∞

α

β °

(b)

∞

α

β °

Figure 12: Influence of the CNT distribution and volume fraction on the onset of the flutter instability of the FG-CNTRC truncated conical curved panel, a) CNT distribution, b) CNT volume fraction.

12

∞

α

β ° Figure 13: Influence of the Mach number on the onset of the flutter instability of the FG-CNTRC truncated conical curved panel.

13

∞

α

β ° Figure 14: Effect of boundary conditions on the onset of the flutter instability of the FG-CNTRC truncated conical curved panel.

14

∞

α

β ° Figure 15: Influence of the radius-to-thickness ratio on the onset of the flutter instability of the FG-CNTRC truncated conical curved panel.

15

°

∞

α

β

°

β

°

β

°

Figure 16: Influence of the length-to-radius ratio on the onset of the flutter instability of the FG-CNTRC truncated conical curved panel.

16

Highlights

1. Investigating application of CNTs in enhancing stability response of conical shell. 2. Investigating application of CNTs in improving dynamic response of conical shell. 3. Investigating the effect of CNTs on the buckling configuration of conical shell. 4. Investigating efficient CNTs distribution in improving structural response of shell. 5. The proposed solution procedure is effective in reducing computational costs and virtual storage requirements.