On dynamic instability of a pressurized functionally graded carbon nanotube reinforced truncated conical shell subjected to yawed supersonic airflow

Composite Structures 153 (2016) 938–951

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On dynamic instability of a pressurized functionally graded carbon nanotube reinforced truncated conical shell subjected to yawed supersonic airflow M. Mehri a, H. Asadi b,c,⇑, Q. Wang d a

Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran d Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong b c

a r t i c l e

i n f o

Article history: Received 10 March 2016 Revised 11 May 2016 Accepted 2 July 2016 Available online 5 July 2016 Keywords: Aeroelastic flutter Functionally graded carbon nanotube Truncated conical shell Novozhilov nonlinear shell theory Yawed supersonic airflow Hydrostatic pressure

a b s t r a c t The aeroelastic flutter characteristics of a functionally graded carbon nanotube reinforced composite (FGCNTRC) truncated conical shell under simultaneous actions of a hydrostatic pressure and yawed supersonic airflow are scrutinized. The nonlinearity in geometry of the conical shell is considered in Green– Lagrange sense and the model is derived according to the Novozhilov nonlinear shell theory. The aerodynamic pressure is modeled based on the quasi-steady Krumhaar’s modified supersonic piston theory by considering the effect of the panel curvature and flow yaw angle. Parametric studies are conducted to investigate the effects of boundary conditions, semi-vertex angle, distribution and volume fraction of CNT, Mach number and airflow yaw angle on the stability boundaries and flutter characteristics. The results show that the semi-vertex angle and CNT distribution may alter the stability boundaries. It is also found that the aeroelastic flutter responses of the structure can be significantly improved through a functionally graded distribution of CNT in a polymer matrix. Moreover, the aeroelastic characteristics of the FG-CNTRC truncated conical shell are found to be very sensitive to geometrical parameters and the airflow yaw angle. The results of this study shed a light into developing and using ultra-high-strength and low-weight composites reinforced with CNT for aerospace applications. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Thin-walled structures have been widely used in various engineering structures, especially aircraft structures. External skins of high-speed aircrafts are mainly subjected to an external pressure, which results in the possible buckling instability. Essentially, the buckling instability does not indicate a structural failure, however, it may result in aerodynamic shape changes leading a reduction of the flight performance. On the other side, flutter is a kind of dynamic instability phenomenon, which is resulted from an interaction between the inertial, elastic and aerodynamic loads owing to the supersonic airflow. Predictions of the aeroelastic behaviors of lifting and control surfaces at high flow velocities are foremost from the perspective of both design optimization as well as safe testing of designs [1,2]. The onset of the flutter may be very sudden

⇑ Corresponding author at: Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada. E-mail address: [email protected] (H. Asadi). http://dx.doi.org/10.1016/j.compstruct.2016.07.009 0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.

and hence may lead to a rapid destruction of the lifting surface, and thus it is perilous to the structural stability and safety. Therefore, investigations of the aeroelastic flutter instability of high-speed structures are significant steps in designing the external skins of high-speed structures. There are a large number of researches in the literature on the buckling, vibration and flutter of thin-walled structures [3–26]. For instance, Shin et al. [18] conducted a numerical study based on the layerwise theory to investigate the aerothermoelastic responses of an aerothermally buckled cylindrical composite shell with different damping treatments. The aeroelastic stability of a geometrically imperfect cylindrical shell under a supersonic gas flow was addressed by Amabili and Pellicano [25] by employing Galerkin approach and Donnell shallow-shell theory. In their study, effects of viscous structural damping, asymmetric and axisymmetric imperfections of cylindrical shells were taken into consideration. It was shown that the onset of flutter is very sensible to initial imperfections of the shells. Recently, Asadi et al. [26] studied the nonlinear aerothermal flutter instability and investigated a

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possibility of enhancing the aerothermal stability boundaries and decreasing aerothermal postbuckling deflections of laminated composite beams using shape memory alloys fibers. Recent years have witnessed a strong interest and intensive research activities in the design and analysis of advanced materials and structures, which have been widely used in engineering fields, particularly, in the aviation and aerospace sectors. Among several kinds of advanced materials, the extraordinary stiffness and specific tensile strength of carbon nanotubes (CNTs) make them well-suited as reinforcing components in next generation of high-performance multifunctional composites. In fact, their exceptional properties along with their high aspect ratio, low density have attracted enormous researches on developments of the advanced composites using CNTs. Various studies assessed the influence of CNT on the dynamic and static responses of CNTRC structures [27–60]. An overview of the researches is presented hereafter. Wuite and Adali [28] performed a multiscale analysis of the stresses and bending deflection of the CNTRC beam. The influence of CNT volume fraction and nanotube diameter on the transverse deflection was studied, and comparisons were made with carbon fiber reinforced composites. Yas and Samadi [30] analyzed free vibration and buckling responses of a nanocomposite beam reinforced with SWCNTs resting on Pasternak elastic foundation by means of generalized differential quadrature (GDQ) method. Their results indicated that embedding CNTs in a polymer matrix could result in an increase in the buckling strength. Lei et al. [34,35] investigated large deflection and buckling of FG-CNTRC plates under various in-plane loads by employing the element-free kpRitz method. In these works, the effective material properties were calculated in accordance with either the Eshelby–Mori–Tanaka or the extended rule of mixture. Effects of CNT and elastic foundation on the onset of the buckling of FG-CNTRC skew plate were examined by Liew and his colleagues [37–39] by means of elementfree IMLS-Ritz method. The first shear deformation theory (FSDT) was used for formulation of energy functional to incorporate the influence of the transverse shear deformation and rotary inertia. Liew et al. [40,41] presented a numerical study using the element-free kp-Ritz approach to study dynamic stability and postbuckling characteristics of FG-CNTRC cylindrical panels under axial compressive load. According to kernel particle approximations for the field variables, the Ritz method was used to develop the discretized governing equations and the bending stiffness was evaluated by a stabilized conforming nodal integration scheme. Linear thermal and mechanical buckling of FG-CNTRC conical shells were studied by Kiani and his colleagues [42,43] using Donnell shell theory in conjunction with the first-order shear deformation theory (FSDT). Sankar et al. [56] studied dynamic instability behavior of sandwich panels with CNT reinforced facesheets under in-plane periodic load by employing a shear flexible QUAD-8 serendipity element. They found that HSDT predicts a narrow instability region and the onset of primary instability zone happens at lower forcing frequency compared to the FSDT. The flutter characteristics of sandwich plates with CNT reinforced facesheets was investigated by Sankar et al. [57] using QUAD-8 shear flexible element in conjunction with HSDT. Their results showed that the occurrence of type of flexural/extensional modes in the thickness direction depends on the position of the structures. Recently, Asadi et al. [59] investigated buckling and free vibration responses of FG-CNTRC truncated conical shells according to nonlinear Novozhilov shell theory. Their results indicated that the circumferential mode number associated with the fundamental natural frequency is independent of the CNT volume fraction and distribution as opposed to stability characteristics in which volume fraction and distribution of CNT may change the buckling configuration. Most recently, Zhang et al. [60] presented a comprehensive

research on an active control aerothermoelastic flutter of FGCNTRC plates using piezoelectric actuators and sensors. In their work, the optimal area and location of piezoelectric patches were also determined using the genetic algorithm. It is found that the aeroelastic flutter stability may be enhanced where the CNTs are concentrated near the neutral plane of the FG-CNTRC plates. The above literature review clearly indicates that despite various attempts to study effects of CNT on the structural responses of flat and curved structures, a rigorous investigation for examining effects of CNT on the aeroelastic flutter response of CNTRC flat and curved structures seems to be absent. It is noteworthy that the incorporation of CNT may significantly enhance the strength and stiffness of the composites with a minimal increase in weight. Since Young modulus of CNTs are superior to all carbon fibers with value greater than 1 ðTPaÞ and their density may be only 1300 ðKg=m3 Þ. This motivates us to conduct the present research. The main purpose of the present research is to fill this important gap in the literature. The present work scrutinizes the aeroelastic flutter characteristics of a FG-CNTRC truncated conical shell subjected to hydrostatic pressure and exposed to yawed supersonic external airflow. Nonlinear equations of motion of the FG-CNTRC truncated conical shell are derived according to the Novozhilov nonlinear shell theory and Green–Lagrange geometrical nonlinearity via Hamilton principle. The Krumhaar’s modified supersonic piston model is adopted to evaluate the aerodynamic pressure. A detailed parametric study is carried out to investigate the effect of all geometrical parameters, CNT distribution and volume fraction on the supersonic aeroelastic flutter behavior of the FG-CNTRC truncated conical shell. The main contribution of the present work is to reveal an efficient application of CNT to enhance the aeroelastic flutter characteristics of supersonic curved structures. 2. Basic formulation 2.1. Modeling of FG-CNTRC As illustrated in Fig. 1A, a FG-CNTRC truncated conical shell is considered in the cylindrical coordinate system ðx; h; zÞ. The truncated conical shell is shown with the thickness h, length L, semi-vertex angle a and end radii R1 < R2 . It is assumed that the truncated conical shell is subjected to a hydrostatic pressure ðqH Þ and yawed supersonic airflow (see Fig. 1B).

(A)

(B)

Fig. 1. Schematic and geometric characteristics of the FG-CNTRC conical shell.

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The SWCNT reinforcement is assumed to be distributed either uniformly (referred to as UD) or functionally graded through the thickness (referred to as FG). In the present work, the extended rule of mixture is used to evaluate the effective materials properties of the two-phase nanocomposite, mixture of the CNT as a fiber and an isotropic polymer as a matrix. The effective material properties could be obtained as follows [54] m E11 ¼ g1 V CN ECN 11 þ V m E g2 V CN V m ¼ þ m E22 ECN E 22

g3 G12

¼

V CN GCN 12

þ

ð1Þ

Vm Gm

q ¼ qCN V CN þ qm V m CN CN where ECN 11 ; E22 , and G12 are the Young’s modulus and shear modulus of the SWCNTs, respectively. In addition, Em , and Gm represent the properties of the isotropic polymer matrix. Also, V CN and V m stand for the volume fraction of CNT and polymer matrix, which satisfies the condition

V CN þ V m ¼ 1

ð2Þ

in Eq. (1), the coefficients g1 ; g2 , and g3 are introduced to account for the scale dependent material properties. It is noteworthy that these constants are estimated by matching the effective properties of CNTRC obtained from molecular dynamic (MD) simulations with those from the rule of mixtures [55]. Five different kinds of CNT distributions are taken into account as shown in Fig. 2. The volume fraction of CNT in each case may be obtained as

8  V CN > >
 > > > > 2V CN 1  2 jzjh > > <
 V CN ðzÞ ¼ 4V  jzj CN h > > >   > > > V CN 1 þ 2 hz > >  :   V CN 1  2 hz

UD


 ^ -2 / ¼ ^h 1 þ ^f2  w
 ^ 1 þ ^f1  ^h-1 w¼w

ð6Þ

Furthermore

ð3Þ

FG-V FG-K

The effective Poisson’s ratio is not sensitive to position of the CNT and may be obtained as [55] m m12 ¼ V CN mCN 12 þ V m m

where

v ffi ^f1 þ ^f2 þ ^f1^f2  -1 -2

FG-O FG-X

Fig. 2. Distribution types of CNTs of FG-CNTRC conical shell [42].

ð4Þ

2.2. Displacement field and strains The Novozhilov nonlinear shell theory is used here to investigate the aeroelastic flutter behavior of the FG-CNTRC truncated conical shell. Based on the Novozhilov nonlinear shell theory, displacements u1 ; u2 and u3 of points at distance z from the midplane may be written in terms of the mid-plane displacements ðu0 ; v 0 ; w0 Þ as [61]

^f1 ¼ 1 @u0 þ 1 @A1 v 0 þ w0 A1 @x A1 A2 @h R1 1 @v 0 1 @A1 -1 ¼  u0 A1 @x A1 A2 @h ^h ¼  1 @w0 þ u0 A1 @x R1 1 @ v 1 @A2 w0 0 ^f2 ¼ þ u0 þ A2 @h A1 A2 @x R2 1 @u0 1 @A2  -2 ¼ v0 A2 @h A1 A2 @x ^ ¼  1 @w0 þ v 0 w A2 @h R2

ð7Þ

where R1 and R2 are the first and second principal radius of curvature. For the truncated conical shell, we have

A1 ¼ 1;

A2 ¼ RðxÞ;

RðxÞ ¼ R1 þ x sin a;

R1 ¼ 1;

u1 ¼ u0 þ z/

u2 ¼ v 0 þ zw

u3 ¼ w0 þ zv

ð5Þ

R2 ¼

RðxÞ cos a ð8Þ

By substituting Eqs. (6)–(8) into Eq. (5), the displacement fields of the truncated conical shell may be expressed as

! @w0 1 @ v 0 @w0 sin a @w0 cos a @w0 1 @w0 @u0 sin a @w0 cos a @u0 sin 2a 2   u0  w0 þ 2  2 v0  2 v0 þ 2 v0 RðxÞ @h @x @x RðxÞ @x RðxÞ @x @h @h 2R ðxÞ R ðxÞ @h @h R ðxÞ R ðxÞ    1 @w0 1 @u0 @w0 cos a @u0 1 @u0 @w0 sin a @w0 þ v0 1 þ v0 u2 ¼ v 0 þ z   þ  RðxÞ @h RðxÞ @x @h RðxÞ @h @x RðxÞ @x RðxÞ @x  @u0 1 @ v 0 sin a cos a 1 @ v 0 @u0 sin a @u0 cos a @u0 1 @u0 @ v 0 sin a @ v 0 u3 ¼ w0 þ z v0 þ þ u0 þ w0 þ þ u0 þ w0  þ RðxÞ @h @x @x RðxÞ @h RðxÞ RðxÞ RðxÞ @x RðxÞ @x RðxÞ @h @x RðxÞ @x u1 ¼ u0 þ z 

ð9Þ

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The Novozhilov nonlinear shell theory incorporated with the Green–Lagrange geometrical nonlinearity, when applied to the truncated conical shell, yield the strain fields and may be written as

xx ¼  hh ¼ 

0 x þz x 0 h þz h 0 xh þ z xh

cxh ¼ c

j j j

ð10Þ

After some mathematical manipulations by retaining only the main nonlinear terms, the strain–displacement relations can be written as " 2  2  2 # @u0 1 @u0 @v 0 @w0 0 x ¼ þ þ þ @x 2 @x @x @x

1 @ v 0 sin a cos a  ¼ þ u0 þ w0 RðxÞ @h RðxÞ RðxÞ " 2  2 1 1 @u0 sin a 1 @ v 0 sin a cos a þ v0 þ  þ u0 þ w0 2 RðxÞ @h RðxÞ @h RðxÞ RðxÞ RðxÞ  2 # 1 @w0 cos a þ v0  RðxÞ @h RðxÞ   @v 1 @u0 sin a @u c0xh ¼ 0 þ v0 1 þ 0  RðxÞ @h @x RðxÞ @x  @v 0 1 @ v 0 sin a cos a þ þ u0 þ w0 @x RðxÞ @h RðxÞ RðxÞ  @w0 1 @w0 cos a v0  þ @x RðxÞ @h RðxÞ 0 h

jx jh

@ 2 w0 ¼ @x2  cos a @u0 1 @v 0 sin a @w0 1 @ 2 w0 ¼ þ   2 RðxÞ @x RðxÞ @h RðxÞ @x R ðxÞ @h2

jxh ¼ 

In addition, stress resultants in terms of mid-plane characteristics are calculated by substitution of Eqs. (10)–(13) into Eq. (14)

8 Nxx > > > > > Nhh > > > > > < Nxh

9 > > > > > > > > > > =

A11 6 6 A12 6 6 6 A16 ¼6 6 > M xx > > > 6 B11 > > > > 6 > > > > 6 > > M hh > > 4 B12 > > > > : ; M xh B16



 Aij ; Bij ; Dij ¼

Z

t2

> :

rhh ¼ 6 4 Q 21 Q 22 > sxh ; Q 61 Q 62

ð12Þ

where ½Q ij  represents the transformed reduced stiffness matrix and its coefficients are obtained as

E11 ; 1  m12 m21 E22 ¼ ; 1  m12 m21

m12 E11 ; 1  m12 m21

Q 12 ¼

Q 22

Q 66 ¼ G12

ð13Þ

Furthermore, the stress resultants N ij and M ij can be calculated as

8 9 8 9 > < Nxx > = Z 2h > < rxx > = Nhh ¼ rhh dz > > > h > : ; 2 : Nxh sxh ; 8 8 9 9 > < M xx > < rxx > = Z 2h > = M hh ¼ z rhh dz > > > > h : : ; 2 M xh sxh ;

B12

B22

A66

B16

B26

B12

B16

D11

D12

B22

B26

D12

D22

B26

B66

D16

D26

ð15Þ

Z

h 2



h 2

 Q ij ; zQ ij ; z2 Q ij dz

ð16Þ

ð17Þ

where dU; dV and dT, are the virtual strain energy, virtual work done by external forces and virtual kinetic energy, respectively. The total virtual strain energy of the FG-CNTRC truncated conical shell is given by L

Z 2p Z 0

h 2

h 2

 z RðxÞdzdhdx ðrxx dxx þ rhh dhh þ sxh dcxh Þ 1 þ R2

The virtual work done by external forces is provided as

0

Q 11 ¼

A26

A26

ðdU þ dV  dT Þdt ¼ 0

dV ¼

9 38 Q 16 > < xx > = 7 Q 26 5 hh > > : cxh ; Q 66

A22

9 38 B16 > 0x > > > > 7> > > B26 7> > 0h > > > > > 7> > > > 7> B66 7< c0xh = 7 7 > D16 7> > > jx > > > 7> > > > > 7 > D26 5> > jh > > > > > : ; D66 jxh

ð18Þ

The stress–strain relation of the FG-CNTRC truncated conical shell in a plane state of stress could be expressed as

Q 12

B12

t1

Z

Q 11

B11

The equations of motion of the FG-CNTRC truncated conical shell are developed using the Hamilton principle according to the Novozhilov nonlinear shell theory. Based on the Hamilton principle, an equilibrium position in structures occurs when the following condition is met

0

2.3. Constitutive equations

2

A16

3. Equations of motion

Z

ð11Þ

8 9 > < rxx > =

A12

where, stretching, stretching-bending coupling and bending stiffness parameters Aij ; Bij , and Dij are calculated as

dU ¼

cos a @u0 cos a @ v 0 2 sin a @w0 2 @ 2 w0 þ þ 2 þ 2 RðxÞ @h RðxÞ @x @h @x@h R ðxÞ R ðxÞ

2

L

Z 2p h i qH z RðxÞdhdx du0  qH dw0 þ DPdw0 1 þ R2 2 0

ð19Þ

where DP is the aerodynamic load. Krumhaar [62] proposed a piston theory curvature correction term for cylindrical shells, which may be modified for the truncated conical shells by letting ðR ¼ R1 þ x sin a ¼ RðxÞÞ and the corrected piston theory can be written as follows [63] ! c P1 M2 @w0 1 @w0 1 M2  2 _0 w DP ¼  paffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos u þ sin u þ RðxÞ @h Ma1 M2  1 M2  1 @x ! 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w0  ð20Þ 2RðxÞ M2  1 where a1 is the free stream speed of sound, P1 is the free stream static pressure, M is the Mach number, ca is the adiabatic exponent, and u is the flow yaw angle (See Fig. 1B). Also, the total virtual kinetic energy can be computed as:  Z L Z 2p Z h 2 z dT ¼ qðzÞðu_ 0 du_ 0 þ v_ 0 dv_ 0 þ w_ 0 dw_ 0 Þ 1 þ RðxÞdzdhdx h R2 0 0 2

ð21Þ

ð14Þ

After performing some mathematical manipulations and performing Green-Gauss theorem to relieve the virtual displacement gradients, the following equations of motion are obtained to be

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M. Mehri et al. / Composite Structures 153 (2016) 938–951

du0 :

N xx;x þ þ 

1 R2 ðxÞ sin a R2 ðxÞ

sin a sin a sin a N xx þ N xx u0;x þ ðNxx u0;x Þ;x  Nhh RðxÞ RðxÞ RðxÞ

ðN hh u0;h Þ;h  Nhh v 0;h 

sin a R2 ðxÞ

sin 2a 2R2 ðxÞ

ðNhh v 0 Þ;h 

Nhh w0 þ

sin

2

a

R2 ðxÞ

4. Solution procedures To find an alternate numerical technique using fewer grid points for obtaining results with satisfactory accuracy, the differential quadrature method (DQM) was presented by Bellman et al. [64,65]. The DQM has some merits compared to finite element method (FEM). First of all, the FEM uses lower-order polynomials to approximate a function on a local element. While, the DQM approximates a function on the global domain using higher-order polynomials. Furthermore, the DQM directly approximates the derivatives of a function at a point, while the FEM approximates a function over a local element and derivatives may then obtained from approximate function. Owing to the higher-order polynomial approximation in the DQM, it usually requires less grid points compared to the FEM. Thus, very little computational efforts and virtual storage are needed when DQM is employed. In the present work, the harmonic differential quadrature method (HDQM) is applied to solve the governing equations in the spatial domain. With regard to the HDQ method, the nth-order derivation of the function wðxÞ at any discrete point could be calculated roughly as

N hh u0

1 N xh;h RðxÞ

þ

1 1 sin a ðNxh u0;x Þ;h þ ðN xh u0;h Þ;x  ðN xh v 0 Þ;x RðxÞ RðxÞ RðxÞ



sin a cos a cos a N xh v 0;x þ Mhh;x  2 Mxh;h RðxÞ RðxÞ R ðxÞ

€ 0 þ I1 w € 0;x ¼ I0 u 2

sin a 1 sin a Nhh v 0 N hh;h  2 N xx v 0;x þ ðN xx v 0;x Þ;x þ RðxÞ RðxÞ R ðxÞ

dv 0 : þ þ

sin a R2 ðxÞ cos a 2

R ðxÞ

Nhh u0;h þ

1 R2 ðxÞ

ðN hh w0 Þ;h 

ðNhh v 0;h Þ;h þ

cos2 a R2 ðxÞ

N hh v 0 þ

sin a R2 ðxÞ

cos a R2 ðxÞ

ðN hh u0 Þ;h

N hh w0;h

n

d wðxi Þ

2 sin a sin a 1 þ ðN xh v 0;x Þ;h N xh þ N xh;x þ Nxh u0;x þ RðxÞ RðxÞ RðxÞ þ ¼

dx

 

cos a R2 ðxÞ cos a R2 ðxÞ

Nhh v 0;h 

sin 2a 2R2 ðxÞ

ðN hh v 0 Þ;h 

Nhh u0 þ

1 R2 ðxÞ

K X

@ijðnÞ wðxj Þ

ð25Þ

j¼1

where xi is a discrete point in the solution domain, wðxj Þ is the func-

sin a cos a cos a cos a ðN xh u0 Þ;x þ ðNxh w0 Þ;x þ 2 Mhh;h þ Mxh;x RðxÞ RðxÞ RðxÞ R ðxÞ ! ! 2 cos a cos2 a 1 cos a € 0;h I2 v€ 0  I 1 þ 2 I2 w I0 þ I1 þ 2 RðxÞ RðxÞ R ðxÞ R ðxÞ

ðnÞ

tion value at point xj and @ij is the weighting coefficients for the

sin a cos a cos2 a N hh w0 N xx w0;x þ ðN xx w0;x Þ;x  N hh  2 RðxÞ RðxÞ R ðxÞ

dw0 :



n

ðN hh w0;h Þ;h

cos a 1 1 N xh;x w0;h þ ðNxh w0;x Þ;h Nxh þ RðxÞ RðxÞ RðxÞ

1 cos a 2 sin a N xh w0;xh  þ ðN xh;x v 0 Þ;x þ Mxx;xx þ Mxx;x RðxÞ RðxÞ RðxÞ sin a 1 2 sin a 2 Mxh;h  Mxh;xh þ DP þ qH Mhh;x þ 2 Mhh;hh þ 2 RðxÞ RðxÞ R ðxÞ R ðxÞ ! sin a 1 cos a sin a € 0 þ I1 u € 0;x € 0;x þ I1 u w þ I2 2 ¼ I1 v€ 0;h  I2 RðxÞ RðxÞ RðxÞ R ðxÞ

nth-order derivation of the function in the x direction. The weighting coefficients could be calculated through the functional approximations in the x direction. Utilizing harmonic functions as the approximating test functions in the quadrature method to handle with periodic problems, results in explicit terms for the weighting coefficients for HDQ. Furthermore, using the harmonic functions instead of polynomial as the test function results in circumventing the limitation for the number of grid point in the conventional DQM on the basis of polynomial test functions. The harmonic test function  hk ðxÞ employed in this work is [66]





 ðxxkþ1 Þp sin ðxx20 Þp    sin ðxx2k1 Þp sin    sin ðxx2K Þp 2



 hk ðxÞ ¼ ðx x Þp 0 Þp    sin ðxk x2k1 Þp sin k 2kþ1    sin ðxk x2 K Þp sin ðxk x 2



€ 0;xx  I2  I2 w

1 2

R ðxÞ

ð26Þ In accordance with the HDQ concept, the weighting coefficients ð1Þ

of the first-order derivatives @ij for i – j could be calculated as p Cðx Þ

@ð1Þ ij ¼

€0 € 0;hh þ I0 w w ð22Þ

2

Cðxj Þ sin

Z Ii ¼

h 2

h 2

qðzÞzi dz

i ¼ 0; 1; 2

ð23Þ

In the present work, two types of movable boundary conditions are taken into account for each end of the truncated conical shell. Each end of the conical shell may be either clamped (C) or simply supported (S). The mathematical expressions for each boundary condition are

v 10 ¼ w10 ¼ w10;x ¼ ðRðxÞNxx þ RðxÞNxx u0;x þ Nxh u0;h Nxh sin av 0 þ M hh cos aÞ ¼ 0 S : v 10 ¼ w10 ¼ M 1xx ¼ ðRðxÞNxx þ RðxÞNxx u0;x þ Nxh u0;h Nxh sin av 0 þ M hh cos aÞ ¼ 0

C:

K Y

Cðxi Þ ¼

sin

j¼1; j–i

ð27Þ

 ðxi  xj Þp 2

ð28Þ ð2Þ

The weighting coefficients of the second-order derivatives @ij for i – j can be computed as



ð1Þ ð1Þ @ð2Þ 2@ij  p cot ij ¼ @ij



 ðxi  xj Þp ; 2

i; j ¼ 1; . . . ; K

ð29Þ

The weighting coefficients of the first and second-order derivaðrÞ

tives @ii for i ¼ j can be calculated as

@ðrÞ ii ¼ ð24Þ



ðxi xj Þp 2

where

.

where in the above equations, ( ), ð Þ;x and ðÞ;h stand for the derivatives with respect to time t, the x and h directions, respectively. Also, Ii are the inertia terms and obtained as follows


i

K X

@ðrÞ ij

r ¼ 1 or 2 and for i ¼ 1; . . . ; K

ð30Þ

j¼1; j–i ð1Þ

ð2Þ

With regard to the @ij and @ij , the weighting coefficients of the third and fourth-order derivatives may be obtained as

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M. Mehri et al. / Composite Structures 153 (2016) 938–951

@ð3Þ ij ¼

K X

ð2Þ @ð1Þ ij @ij

k¼1 ð4Þ ij

@

¼

K X

ð31Þ ð2Þ ij

where N 0xx ; N 0hh and N0xh are the prebuckling load resultants of the truncated conical shell and can be evaluated as

@ @

k¼1

The accuracy of HDQM results depends on the number of grid points, and the relative spacing of the discrete points. Selecting equally spaced discrete points may be an easy choice while nonuniformly spaced discrete points may result in predictions with higher level of accuracy. It is found that one of the optimal selections of the discrete points in vibration analysis is the zeros of the well-known Chebyshev polynomials and can be obtained as [66]

  L ði  1Þp xi ¼ 1  cos 2 ðK  1Þ

for i ¼ 1; 2; . . . ; K

ð32Þ

Before applying the HDQ, the stability equations of the FGCNTRC truncated conical shell may be derived through an application of the adjacent equilibrium criterion [61]. To this end, a perturbed equilibrium position from a prebuckling state is considered. An equilibrium position in prebuckling state is defined   with components u00 ; v 00 ; w00 . With an arbitrary perturbation  1 1 1 u0 ; v 0 ; w0 , the truncated conical shell experiences a new equilibrium configuration adjacent to the primary one described with dis  placement components u00 þ u10 ; v 00 þ v 10 ; w00 þ w10 . Dropping the inertia terms from the Eq. (22) and applying the adjacent criterion leads to stability equations expressed as


 sin a 1 sin a 0 1 sin a 1 N þ N u þ N0xx u10;x  N ;x RðxÞ xx RðxÞ xx 0;x RðxÞ hh 2 1
0 1  sin a
0 1  sin a 0 1 þ 2 Nhh u0;h  2 Nhh v 0  2 Nhh u0 ;h ;h R ðxÞ R ðxÞ R ðxÞ N1xx;x þ

  dv 0 :

sin a R2 ðxÞ cos a R2 ðxÞ

N0hh

v 10;h 

sin 2a 2R2 ðxÞ

N0hh w10

1 cos a 1 þ N1 þ M RðxÞ xh;h RðxÞ hh;x

N0xh

ð34Þ

Considering Eq. (34) and the periodicity conditions of the truncated conical shell in the circumferential direction, the following separation of variables satisfies the circumferential constraints of the truncated conical shell

u10 ðx; hÞ ¼ UðxÞ sinðmhÞ

v 10 ðx; hÞ ¼ VðxÞ cosðmhÞ

ð35Þ

where m is the full-wave length in circumferential direction. Substituting Eq. (35) into Eq. (33) and employing the HDQ discretization for the x-dependent functions lead to the following system of equations 9 8 9 0 UU 2 UU UW 3 UW 18 ½K G  ½K UV ½K E  ½K UV > E  ½K E  G  ½K G  < fU g > = > < f0g > = B VU 6 VU VW 7 VV VW C  V ¼ 0 f g f g @ ½K E  ½K VV 5 4 A  ½K   ½K  ½K  ½K E E G G G > > > : ; > : ; fW g f0g ½K WU  ½K WV  ½K WW  ½K WU  ½K WV  ½K WW  E

E

E

G

G

G

ð36Þ where ½K E  is the elastic stiffness matrix, which contains the unknown circumferential full-wave length number m. Also, ½K G  is the geometric matrix, which contains both mechanical and aerodynamics load parameters. The critical buckling load of the FG-CNTRC truncated conical shell can be determined by solving the standard eigenvalue problem (Eq. (36)). In this work, the solution procedure by means of the HDQM is implemented in a MATHEMATICA code. 4.2. Aeroelastic panel flutter The modal shapes for a vibrating truncated conical shell are introduced as a product of unknown functions in the meridional direction and known trigonometric functions in the circumferential direction. The following separation of variables fulfills the boundary conditions and is also compatible with the equations of motion specified in Eq. (22), is considered

M1xh;h ¼ 0


 sin a 0 1 1 sin a 0 1 N1  Nxx v 0;x þ N0xx v 10;x þ N v ;x RðxÞ hh;h R2 ðxÞ hh 0 RðxÞ sin a 1
0 1  sin a
0 1  þ 2 N0hh u10;h þ 2 Nhh v 0;h þ 2 Nhh u0 ;h ;h R ðxÞ R ðxÞ R ðxÞ
 2 cos a 0 1 cos a 0 1 cos a 0 1 þ 2 Nhh w0  2 Nhh v 0 þ 2 Nhh w0;h ;h R ðxÞ R ðxÞ R ðxÞ 2

u0 ðx; h; tÞ ¼ ent UðxÞ sinðmhÞ

v 0 ðx; h; tÞ ¼ ent VðxÞ cosðmhÞ

ð37Þ

w0 ðx; h; tÞ ¼ ent WðxÞ sinðmhÞ After substituting Eq. (37) into Eq. (22) and applying the HDQ discretization for the x-dependent functions, the following system of equations is established

2 sin a 1 cos a cos a 1 þ N þ N1xh;x þ 2 M 1hh;h þ M ¼0 RðxÞ xh RðxÞ xh;x R ðxÞ
 sin a 0 1 cos a 1 cos2 a 0 1 dw0 : Nxx w0;x þ N0xx w10;x  Nhh  2 Nhh w0 ;x RðxÞ RðxÞ R ðxÞ cos a sin 2a 1
0 1   2 N0hh v 10;h  2 N0hh u10 þ 2 N hh w0;h ;h R ðxÞ R ðxÞ 2R ðxÞ cos a
0 1  cos a 1 2 sin a N þ M 1xx;xx þ M1xx;x  2 Nhh v 0  ;h RðxÞ xh RðxÞ R ðxÞ

20

UU ½K UU E þ KG 

UV ½K UV E þ KG 

6B VU 4@ ½K E þ K VU G 

VV ½K VV E þ KG 

½K UW þ K UW E G 

3

7 ½K VW þ K VW E G  5

þ K WU ½K WV þ K WV ½K WW þ K WW ½K WU E G  E G  E G  2 UU UV UW 3 ½C  ½C  ½C  6 7 þn4 ½C VU  ½C VV  ½C VW  5 WU

WV

ð38Þ

WW

½C  ½C  ½C  9 8 9 2 UU 3 18 > ½M  ½M UV  ½MUW  < fU g > = > < f0g > = 6 7C ¼ f0g þn2 4 ½M VU  ½M VV  ½MVW  5A fV g > > > : ; > : ; fW g f0g ½M WU  ½M WV  ½MWW 

sin a 1 1 2 sin a 1  M þ M1 þ 2 Mxh;h RðxÞ hh;x R2 ðxÞ hh;hh R ðxÞ 

N0hh

w10 ðx; hÞ ¼ WðxÞ sinðmhÞ

4.1. Aeroelastic buckling

du0 :

qH RðxÞ 2 cos a q RðxÞ ¼ H cos a ¼0

N0xx ¼ 

ð2Þ ij

2 M 1 þ DP ¼ 0 RðxÞ xh;xh ð33Þ

where ½M is the mass matrix and ½C is the aerodynamic damping matrix. The eigenvalues n are in general complex numbers and

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M. Mehri et al. / Composite Structures 153 (2016) 938–951

given by n ¼ C þ ix, where C is the measurement of the damping, and x is the eigenfrequencies. It is worth mentioning that the aerodynamic damping term in Eq. (20) is small and always stabilizes the flutter boundary [8–10]. Thus it is advantageous to ignore it when analyzing the behavior in the frequency domain [8–10]. Once the complex eigenvalues of the system are solved, the natural frequencies ðxj Þ and damping ratio ð1j Þ of the system can be determined as

xj ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðImðnj ÞÞ2

C

ð39Þ

j 1j ¼  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2j þ C2j

It is worth recalling that when flutter occurs, two eigenmodes of structures merge at a certain flight velocity, which is called flutter velocity. The flutter leads to dynamic instability owing to aerodynamic load. To improve the aeroelastic flutter characteristics of an aircraft structure, one would manage to postpone efficiently the coalescent point of two flutter modes.

5. Simulation results In this section, the aeroelastic flutter instability of the FGCNTRC truncated conical shell is investigated to show the effectiveness of the CNT in enhancing the aeroelastic flutter characteristics. Firstly, a convergence study is performed to obtain the sufficient grid points of the HDQM. Then, to ensure the accuracy and the reliability of the present results, verifications of the model are conducted. As a benchmark study, a rigorous parametric study is carried out to elaborate on the effect of CNT volume fraction and distributions, geometrical parameters and direction of the airflow on the aeroelastic flutter characteristics of the supersonic truncated conical shell. In this work, Poly (methy1 methacrylate), referred to as PMMA, is considered for the matrix. The material properties and scale dependent parameters of CNT gi ði ¼ 1; 2; 3Þ are introduced as follows [47]. ECN 11 ¼ 5:6466 TPa, GCN 13 m

CN GCN 12 ¼ G13 ¼ 1:9445 TPa,

ECN 22 ¼ 7:0822 TPa,

¼ 2:3334 TPa, q ¼ 1400 kg=m , m q ¼ 1150 kg=m3 , mm ¼ 0:34 CN

3

CN 12

¼ 0:175; Em ¼ 2:5 GPa,

8 > < g1 ¼ 0:137; g2 ¼ 1:022; g3 ¼ 0:715; if g1 ¼ 0:142; g2 ¼ 1:626; g3 ¼ 1:138; if > : g1 ¼ 0:141; g2 ¼ 1:585; g3 ¼ 1:109; if

V CN ¼ 0:12 V CN ¼ 0:17 V CN ¼ 0:28

5.1. Convergence study To determine the sufficient grid points of the HDQM, the critical freestream static pressure ðP Cr 1 Þ of the FG-CNTRC truncated conical shell with different CNT distributions is addressed in Fig. 3. It is concluded that the results converge well as the number of grid points increases. It is evident that, when K ¼ 20 the desired accuracy is reached, because both K ¼ 20 and K ¼ 18 obtain the same results. Thus, K ¼ 20 is adopted for computation of the results. 5.2. Comparison studies

Fig. 3. Convergence study of the critical freestream static pressure of the FG-X CNTRC truncated conical shell.

Table 1 A comparison on the non-dimensional natural
 Kg L sin a ¼ 0:25; Rh2 ¼ 100; h ¼ 4 mm; m ¼ 0:3; E ¼ 70 GPa; q ¼ 2710 m 3 . R2

að Þ

Due to the lack of any results on this research in the literature, for the sake of validation, it is assumed that aerodynamic pressure is absent and natural frequencies and critical buckling pressure of truncated conical shell are addressed. To validate the proposed model and solution procedure, the first eight natural frequencies of an isotropic truncated conical shell are calculated and listed in Table 1. From comparisons of the natural frequencies obtained by the present method with those by [67– 69], it can be found that they are almost the same values, which

 frequencies

X ¼ xR2

qffiffiffiffiffiffiffiffiffiffiffiffiffi qð1m2 Þ E

of

isotropic

truncated

conical

Modes 2

3

4

5

6

7

8

a ¼ 30

Present study Li [67] Lam [68] Irie [69]

0.7909 0.8431 0.8429 0.7910

0.7285 0.7416 0.7376 0.7284

0.6356 0.6419 0.6362 0.6352

0.5537 0.5590 0.5528 0.5531

0.4959 0.5008 0.4950 0.4949

0.46667 0.4701 0.4661 0.4653

0.4660 0.4687 0.4660 0.4645

a ¼ 45

Present study Li [67] Lam [68] Irie [69]

0.6878 0.7642 0.7655 0.6879

0.6974 0.7211 0.7212 0.6973

0.6668 0.6747 0.6739 0.6664

0.6312 0.6336 0.6323 0.6304

0.6045 0.6049 0.6035 0.6032

0.5936 0.5928 0.5921 0.5918

0.6015 0.6005 0.6001 0.5992

a ¼ 60

Present study Li [67] Lam [68] Irie [69]

0.5721 0.6342 0.6348 0.5722

0.6004 0.6236 0.6238 0.6001

0.6060 0.6146 0.6145 0.6054

0.6088 0.6113 0.6111 0.6077

0.6175 0.6172 0.6171 0.6159

0.6365 0.6347 0.6350 0.6343

0.6677 0.6653 0.6660 0.6650

shell

M. Mehri et al. / Composite Structures 153 (2016) 938–951 Table 2 A comparison on the critical buckling pressure (qCr ðKPaÞ) of CNTRC truncated conical
 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi shell Rh1 ¼ 100; L ¼ ð300R1 hÞ; h ¼ 1 mm; V CN ¼ 0:12 .

að Þ

CNT distributions

945

theory in conjunction with the first-order shear deformation theory. While, results of this study are calculated on the basis of the Novozhilov nonlinear shell theory in which shear deformation effect is neglected. This comparison study clearly proves the efficiency and accuracy of the Novozhilov shell theory for moderately thick curved structures without considering the influence of transverse shear deformation.

FG-X

UD

FG-O

Present Study Jam and Kiani [42]

19.97 (8) 20.34 (8)

18.38 (8) 18.75 (8)

17.16 (8) 17.51 (8)

a ¼ 30

Present study Jam and Kiani [42]

12.42 (9) 12.64 (10)

11.27 (9) 11.47 (10)

10.31 (10) 10.50 (10)

5.3. Parametric studies



Present study Jam and Kiani [42]

9.42 (10) 9.58 (10)

8.46 (10) 8.61 (10)

7.62 (10) 7.78 (10)

In this subsection, the aeroelastic flutter characteristics including, stability boundaries, damping ratios and critical freestream static pressure will be studied in details. To determine the flutter boundary (dynamically unstable zone), the freestream static pressure is increased at a certain hydrostatic pressure, till the coalescence of two eigenmodes (i.e. flutter occurs). In addition, to determine the buckling boundary (statically unstable zone), the critical buckling pressure is obtained with the presence of the freestream static pressure as a known parameter. In the first place, let us scrutinize the flutter characteristics of the FG-X truncated conical shell. As illustrated in Fig. 4A, the mode coalescence between either ð3; 2Þf and ð4; 2Þf or ð6; 2Þ1c and ð6; 2Þ2c

a ¼ 10

a ¼ 40



effectively verifies the validity of the proposed solution procedure. The maximum discrepancy of less than 1% is observed between the present results and those published in the literature. Furthermore, in Table 2, the critical buckling pressures of the FG-CNTRC truncated conical shell under lateral pressure are compared with those obtained by [42]. A brief review of this table confirms the efficiency and accuracy of the present model and the maximum difference is about 3%. It should be noted that results of the reference [42] were obtained according to the Donnell shell

  2 . (A) Mode coalescence, (B) normalized deflection in the meridian direction. Fig. 4. Aeroelastic characteristics of the FG-X truncated conical shell R ¼ R1 þR 2

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mode shapes results in a self-excited oscillation, namely panel flutter. The corresponding critical freestream static pressure are about 425 ðKPaÞ and 650 ðKPaÞ, respectively. As a result, a simple way to enhance the flutter characteristics is to suppress the mode coalescence or to increase the frequency interval of flutter modes. To better understand the deformation pattern of the FG-X CNTRC truncated conical shell, the normalized deflection of each mode shapes discussed in Fig. 4A, in the meridional direction are plotted in Fig. 4B and C. In addition the 3D flutter mode shapes and normalized deflection of the FG-X truncated conical shell discussed in Fig. 4A, are also depicted in Fig. 5A and B, for three regions including flat and stable, flutter point and dynamically unstable zone.

Fig. 6 illustrates the effect of semi-vertex angle on the stability boundaries of the FG-X truncated conical shell subjected to hydrostatic pressure and aerodynamic force simultaneously. As can be seen, the semi-vertex angle plays a major role in aeroelastic flutter characteristics and may alter the stability boundaries. The critical freestream static pressure of the FG-X shell can be reduced by increasing the semi-vertex angle. In fact, in the truncated conical shell, the dramatic drop of the stability boundaries of the FGCNTRC truncated conical shell subjected to hydrostatic pressure and aerodynamic load are observed as the semi-vertex angle increases. Moreover, for the semi-vertex angle 0 and 30 , the flutter instability occurs by the coalescence between ð4; 1Þ and ð5; 1Þ modes at the critical freestream static pressure, while for the

  2 . Fig. 5. Normalized deflection in the meridian direction and 3D flutter mode shapes of the FG-X CNTRC truncated conical shell R ¼ R1 þR 2

M. Mehri et al. / Composite Structures 153 (2016) 938–951

Fig. 6. The effect of the semi-vertex angle on the stability boundaries of the FG-X   2 CNTRC trundated conical shell R ¼ R1 þR . 2

semi-vertex angle 45 , the mode coalescence between ð3; 1Þ and ð4; 1Þ leads to a panel flutter. Of possible further interest is that the presence of the supersonic airflow results in a noticeable decrease in the critical buckling pressure of the truncated conical shell. The reason for this observation would be related to both types of mechanical loading and non-symmetric geometry of the truncated conical shell. For instance, at a constant hydrostatic pressure parameter qqCr ¼ 0:8, increasing the semi-vertex angle from 0

to 45 leads to 77% decrease in the critical freestream static pressure. To investigate the effect of Mach number on aeroelastic flutter characteristics of the FG-X truncated conical shell, the stability boundaries and damping ratio are depicted in Fig. 7A and B, respectively. It is assumed that the truncated conical shell is subjected to

947

hydrostatic pressure and a yaw angle of 0 . It is evident that increasing Mach number leads to a drastic decrease in the critical freestream static pressure. It is noteworthy that Mach number may not affect the mode coalescence and for all cases the mode coalescence between ð3; 1Þ and ð4; 1Þ leads to a panel flutter. Moreover, a decrease in Mach number results in a wide flat and stable zone in which the FG-CNTRC truncated conical shell is stable and neither flutter nor buckling instability occurs. For instance, decreasing Mach number from 5 to 2 results in 134% increase in the critical freestream static pressure and 55% decrease in the aerodynamic load. By comparing Fig. 7A and B, it can be concluded that the flutter characteristics determined by computing the natural frequencies are the same as those by calculating the damping ratios. Moreover, another study is conducted to elaborate the effect of Mach number on the critical freestream static pressure of the FG-X truncated conical shell. To this end, the variation of the critical freestream static pressure versus Mach number is plotted in Fig. 8 for four mean radius-to-thickness ratios. As expected, increasing the mean radius-to-thickness ratio leads to a reduction in the critical freestream static pressure owing to the decrease of the flexural rigidity of the truncated conical shell. In other words, increasing the flexural rigidity of the FG-CNTRC truncated conical shell leads to a retardation of the coalescence between ð3; 1Þ and ð4; 1Þ modes. For example, at a constant Mach number of 2, increasing the mean radius-to-thickness ratio from 100 to 175 results in 58% decrease in the critical freestream static pressure. To expatiate on the effect of airflow direction on the aeroelastic flutter behavior of the FG-CNTRC truncated conical shell, the stability boundaries of the FG-X CNTRC truncated conical shell are demonstrated in Fig. 9 for different yaw angles. It can be concluded that an increase in the airflow yaw angle increases the frequency interval of flutter modes and the critical freestream static pressure owing to the retardation of coalescence between ð3; 1Þ and ð4; 1Þ modes, almost regardless of all other parameters. In other words, the onset of the flutter is prompted and the flutter instabilities of the truncated conical shell are more likely, when the truncated conical shell is subjected to axial supersonic airflow ðu ¼ 0 Þ. To elaborate on the effect of the yaw angle on the aeroelastic responses, the variation of the critical freestream static pressure

  2 . (A) stability boundaries, (B) damping ratio. Fig. 7. the effect of Mach number on aeroelastic flutter characteristics of the FG-X truncated conical shell R ¼ R1 þR 2

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Fig. 8. Influence of Mach number on the onset of flutter instability of the FG-X   2 truncated conical shell R ¼ R1 þR . 2

Fig. 10. The effect of airflow direction on the onset of flutter instability of the FG-X   2 . truncated conical shell R ¼ R1 þR 2

ratios of the FG-X CNTRC truncated conical shell are depicted in Fig. 11A and B for three different CNT volume fractions. It is evident that increasing the CNT volume fraction results in a drastic enhancement of the stability boundaries and also the critical freestream static pressure dramatically increases by increasing the volume fraction of CNT. It is understandable that an efficient way to improve the aeroelastic characteristic of structures is to increase the structural stiffness and decrease the mass ratio. As a result, CNT could be selected as a promising candidate of reinforcement in high-performance multifunctional composites, since the incorporation of CNT may greatly increases the strength and stiffness of composites with minimal increase in weight. For example, at a constant hydrostatic pressure parameter qqCr ¼ 0:5, increasing the volume fraction of CNT from 0:12 to 0:28 leads to about 120% increase in the onset of the flutter instability. Figs. 12 and 13 are presented to examine the influence of CNT distributions on the stability boundaries and the critical freestream static pressure of the FG-CNTRC truncated conical shell. As can be seen from Fig. 12, the distribution of CNT has a complex effect on the stability boundaries of the FG-CNTRC. The major reason of such complex effect is the inertia terms (i.e. I1 ; I2 ) and bending stiffness B2

Fig. 9. Effect of yaw angle on the aeroelastic stability boundaries of the FG-X   2 . truncated conical shell R ¼ R1 þR 2

versus the airflow yaw angle is addressed in Fig. 10 for four different Mach numbers. It is visible that an increase in the airflow yaw angle results in a decrease in the aerodynamic load and also an increase in the flutter boundary by inducing the retardation of coalescence of flutter modes. An interesting phenomenon also appears when the truncated conical shell is subjected to low Mach number airflow; it is seen that for Mach number is 2, the circumferential flutter modes number is 1 when the airflow yaw angle is less than approximately 40 . When the airflow yaw angle is higher than 40 , the circumferential flutter modes number is 2. In other words, the coalescence point of flutter modes is changed and the aeroelastic characteristics of the FG-X truncated conical shell alter as a result of this phenomenon. To examine the influence of the CNT volume fraction on the aeroelastic flutter instability, the stability boundaries and damping

), which are totally dependent on the distriparameter (i.e. D11  A11 11 bution of CNT through the thickness. It is worth recalling that the inertia coupling (i.e. I1 ) and bending-extension coupling (i.e. ½Bij ) only exist in the FG-CNTRC truncated conical shell with nonsymmetrically distributed CNT. Of possible further interest is that the distribution of CNT may affect the mode coalescence and for all cases, except in the FG-O, the mode coalescence between ð3; 1Þ and ð4; 1Þ leads to the flutter instability. While for the FGO, the mode coalescence between ð3; 3Þ and ð4; 3Þ results in the flutter instability. In addition, it can be concluded from Fig. 13 that the FG-X CNTRC truncated conical shell has the most critical freestream static pressure, while the FG-O CNTRC truncated conical shell has the least critical freestream static pressure. As a result, the structural stiffness may dramatically increase when the outer surface of the truncated conical shell is CNTs rich. Thus, inserting CNTs in the outer surface is more effective than in the inner surface for a possible delay of the onset of the flutter instability. For instance, at a   constant mean radius-to-thickness ratio Rh ¼ 75 , changing the

M. Mehri et al. / Composite Structures 153 (2016) 938–951

949

  2 . (A) stability boundaries, (B) damping ratio. Fig. 11. the effect of CNTs volume fraction on aeroelastic flutter characteristics of the FG-X truncated conical shell R ¼ R1 þR 2

Fig. 12. The effect of CNTs distribution on aeroelastic stability boundaries of the FG  2 X truncated conical shell R ¼ R1 þR . 2

distribution of CNT from the FG-X to the UD, results in 28% decrease in the critical freestream static pressure. On the other hand, changing the CNT distribution from the FG-X to the FG-O, results in 55% decrease in the critical freestream static pressure. Furthermore, it is evident that geometrical parameters such as mean radius-to-thickness ratio also paly an essential role in aeroelastic flutter characteristics of the truncated conical shell. For instance, in the case FG-X CNTRC truncated conical shell, decreasing the mean radius-to-thickness ratio from 95 to 75 leads to 90% increase in the critical freestream static pressure. Finally, to elaborate on the effect of boundary condition on the onset of flutter instability, variations of the critical freestream static pressure versus the Mach number are addressed in Fig. 14, for two different edge supports with two length-to-mean radius ratios. As expected, the FG-X CNTRC truncated conical shell with clamped (CC) boundary conditions has the higher critical freestream static

Fig. 13. Influence of CNTs distribution on the onset of flutter instability of the FG-X   2 truncated conical shell R ¼ R1 þR . 2

pressure compared to the FG-X CNTRC truncated conical shell with simply supported (SS) edge supports, due to the higher flexural rigidity of the clamping condition. Moreover, it can be clearly revealed from the figure that the length-to-mean radius ratio plays a pivotal role in the aeroelastic flutter behavior of the truncated conical shell. For instance, in the case FG-X CNTRC truncated conical shell with clamped edge supports, increasing the length-tomean radius ratio from 1.5 to 3 results in 46% decrease in the critical freestream static pressure. Of possible further interest is that length-to-mean radius ratio may not only change the circumferential flutter modes number but also change the axial flutter modes number. For instance, the FG-X CNTRC truncated conical shell with clamped edge supports, when RL is 1.5, mode coalescence between ð3; 1Þ and ð4; 1Þ leads to a panel flutter, while RL is 3, mode coalescence between ð4; 2Þ and ð5; 2Þ results in a panel flutter. In addition, for the simply supported FG-X CNTRC truncated conical

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M. Mehri et al. / Composite Structures 153 (2016) 938–951

The length-to-mean radius ratio and mean radius-to-thickness ratio play an essential role not only in the onset of flutter instability but also in changing the coalescence point and flutter modes. The direction of airflow also plays a major role in aeroelastic stability boundaries and the onset of flutter instability. Increasing the airflow yaw angle substantially enhances the aeroelastic characteristics of the FG-CNTRC truncated conical shell. The aeroelastic flutter behavior of the truncated conical shell may significantly improve by inserting CNT as reinforcement into a polymer matrix. Increasing the CNT volume fraction leads to a considerable increase of the frequency interval of flutter modes. The aeroelastic flutter characteristics of the truncated conical shell may effectively improve through functionally graded distribution of CNT in a polymer matrix by retarding the coalescence of flutter modes and compensating the hydrostatic pressure.

References Fig. 14. The effect of boundary condition on the critical freestream static pressure   2 of the FG-X truncated conical shell R ¼ R1 þR . 2

shell, with the ratio of RL being 1.5, the mode coalescence between ð3; 1Þ and ð4; 1Þ leads to a panel flutter. When RL is 3, the mode coalescence between ð4; 3Þ and ð5; 3Þ leads to a panel flutter. Apparently, the circumferential flutter mode shapes are highly sensitive to the flexural characteristics of the support. 6. Concluding remarks In the current research, aeroelastic characteristics of the FGCNTRC truncated conical shell under simultaneous actions of the hydrostatic pressure and yawed supersonic airflow are investigated. The equations of motion are established according to the Hamilton principle, the Green–Lagrange type nonlinear kinematics within the frame work of Novozhilov nonlinear shell theory. The aerodynamic pressure is modeled according to the quasi-steady Krumhaar’s modified piston theory. A semi-analytical method based on the trigonometric function expansion and HDQ approach is presented. The accuracy of the proposed solution procedures are confirmed by some comparison and convergence studies. Extensive parametric studies are conducted to examine the effects of various influential parameters including, the semi-vertex angle, the mean radius-to-thickness ratio, the length-to-mean radius ratio, direction of airflow, distribution and volume fraction of CNT. The numerical results elaborate on the fact that designers need to be careful about following points: It is well known that presence of supersonic airflow postpones the critical buckling load in the flat and cylindrical shells. However, present results indicate that on the contrary, the critical buckling pressure of the truncated conical shell decreases in the presence of supersonic airflow. It is owing to types of mechanical load (i.e. hydrostatic pressure) and non-symmetric geometry of the truncated conical shells. The semi-vertex angle plays a pivotal role in the aeroelastic stability boundaries of the truncated conical shells. A significant drop of the flutter boundaries of the FG-CNTRC truncated conical shell is observed as the semi-vertex angle increases. With increasing Mach number, the aerodynamic load noticeably increases and the critical freestream static pressure considerably decreases. It is owing to the decrease of the frequency interval of flutter modes with increasing Mach number.

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