Static and free vibration analysis of functionally graded conical shells reinforced by carbon nanotubes

Accepted Manuscript

Static and Free Vibration Analysis of Functionally Graded Conical Shells Reinforced by Carbon Nanotubes M. Nejati , A. Asanjarani , R. Dimitri , F. Tornabene PII: DOI: Reference:

S0020-7403(17)30574-X 10.1016/j.ijmecsci.2017.06.024 MS 3753

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

7 March 2017 15 June 2017 15 June 2017

Please cite this article as: M. Nejati , A. Asanjarani , R. Dimitri , F. Tornabene , Static and Free Vibration Analysis of Functionally Graded Conical Shells Reinforced by Carbon Nanotubes, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.06.024

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Highlights Static and Free Vibration Analysis of CNT-Reinforced Composite Conical shells is performed The effect of volume fraction, agglomeration and geometry of CNTs is analysed The Generalized Differential Quadrature Method is applied for numerical analyses The sensitivity of the response to some geometry parameters of the cone is shown

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Static and Free Vibration Analysis of Functionally Graded Conical Shells Reinforced by Carbon Nanotubes

M. Nejati1, A. Asanjarani1, R. Dimitri2, F. Tornabene3 Young Researchers and Elite Club, Arak Branch, Islamic Azad University, Arak, Iran 2 Department of Innovation Engineering, Università del Salento, Lecce, Italy 3 DICAM Department, University of Bologna, Bologna, Italy

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Abstract This study investigates the static and free vibration behavior of rotating functionally graded (FG) truncated conical shells reinforced by carbon nanotubes (CNTs) with a gradual distribution of the volume fraction through the thickness. CNTs are here selected as

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reinforcement, because of their noteworthy physical and chemical properties, together with their ability to enhance the mechanical properties of the whole composite structure. A twoparameter agglomeration model is considered to describe the micromechanics of such particles, which tend to agglomerate into spherical regions when scattered in a polymer matrix. From the macro-mechanical point of view, the conical structures are characterized by

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a gradual variation of their mechanical properties along the thickness direction, since different distributions are explored to describe the volume fraction of the reinforcing phase.

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The governing equations of motion for the rotating truncated composite conical shells are derived and solved numerically by means of the Generalized Differential Quadrature (GDQ)

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method combined with the third-order shear deformation theory (TSDT) in small deformations. The GDQ approach has recently emerged as a very promising numerical tool to

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solve complex problems without passing through any variational formulation, but solving directly the equations of motion in a strong form. In this paper, a parametric study based on

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the GDQ is systematically performed to exploit the effect of some geometry parameters, i.e. the length, the radius, the thickness and the semi-vertex angle of the cone, as well as the different distribution of CNTs along the thickness, on the frequency at different circumferential wave numbers and rotating speeds. A convergence study of the numerical results is also made in terms of deflection and stress distributions of the structure, which proves the efficiency of the GDQ approach, also for coarse mesh discretizations in the meridional direction. Keywords: Carbon nanotube; Conical shell; GDQ method; Nanocomposite structure; TSDT. 2

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1. Introduction Over the past few decades, fiber reinforced composite materials have found many engineering applications due to their special privileges compared with conventional steel or alloy materials. More specifically, structures reinforced by carbon nanotubes (CNTs) have been widely employed in car bodies forming, cooling towers, wind turbine blades, etc. Due to the rapid development of the nano-materials technology, the nano-reinforced composite

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materials has attracted the attention of most designers as an appropriate alternative to the conventional materials. Indeed, it is well known from the literature that the mechanical properties of composite materials with a low volume fraction of nanotubes, can improve significantly. The nanotechnology products, including CNTs, have usually a high stiffness and strength besides their low weight. A considerable research has been increasingly

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performed in the last decades on functionally graded (FG) materials or composite structures such as beams, shells or plates reinforced by CNTs, as briefly reviewed in the following. Among the most interesting works from the literature, Liew and Zhao [1] investigated the free vibration of thin conical shells reinforced with CNTs under different boundary conditions. The analysis was based on the classical thin-shell theory, and was carried out

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using the element-free kp  Ritz method. Wu et al. [2] studied the mechanical properties of single-walled CNTs by using an energy-equivalent model. An equivalent Young's modulus

continuum mechanics.

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and shear modulus was derived by combining the methods of molecular mechanics and

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Kalamkarov et al. [3] presented two different approaches for modeling the behavior of CNTs. The first approach considered CNTs as an inhomogeneous cylindrical network shell based on

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an asymptotic homogenization, whereas the second approach was based on a finite element (FE) formulation, as also developed by Giannopoulos et al. [4]. According to the last

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formulation, the nodes were located at the atoms, whereas the atomic interactions were properly simulated by spring-type elements interconnecting these nodes. Rossi and Meo [5] also proposed a FE modeling of single-walled CNTs based on a molecular mechanics theory in order to evaluate the mechanical properties in terms of Young’s modulus, ultimate strength and ultimate strain. The novelty of the model relied on the use of non-linear and torsional spring elements to evaluate the mechanical properties and tensile failure of single-walled CNTs. Shen [6] analysed the nonlinear bending of FG nanocomposite plates reinforced by singlewalled CNTs, subjected to a transverse uniform load or a sinusoidal load in thermal 3

ACCEPTED MANUSCRIPT environments. The material properties of single-walled CNTs were assumed to be temperature-dependent and graded along the thickness direction. Similarly, Ke et al. [7] investigated the nonlinear free vibration of FG nanocomposite beams reinforced by singlewalled CNTs, based on a Timoshenko beam theory and von Kármán geometric nonlinearity. The Ritz method was employed by the authors to derive the eigenvalue equation governing the problem. Wang and Shen [8] studied the large amplitude vibration of nanocomposite plates reinforced by single-walled CNTs in thermal environments, resting on an elastic

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foundation with varying properties along the thickness direction.

A semi-analytical approach was proposed by Shadmehri et al. [9] to obtain the linear buckling response of conical composite shells under an axial compression loading using the first order shear deformation theory (FSDT). Hedayati and Sobhani Aragh [10] presented a 3D elasticity solution for a free vibration analysis of continuously graded CNT-reinforced

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annular sectorial plates resting on a Pasternak elastic foundation. A generalized differential quadrature method (GDQM) was applied by the authors to discretize the equations of motion and to implement different boundary conditions.

A dynamic stability analysis of FG nanocomposite beams reinforced by single-walled CNTs

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was provided by Ke et al. [11], based on the application of the Timoshenko beam theory. The material properties of the FG-CNTs were assumed to vary within the thickness and were

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estimated with the simple application of the rule of mixture. Moreover, Yas and et al. [12] studied the vibrational behavior of FG nanocomposite cylindrical panels reinforced by CNTs based on the three-dimensional theory of elasticity. The CNT-reinforced cylindrical panel

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featured a varying fraction of CNT in the radial direction, with material properties determined through an extended rule of mixture. The dynamic response of FG multi-walled MWCNT-

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polystyrene nanocomposite beams subjected to multi-moving loads was presented by Heshmati and Yas [13] while applying the Timoshenko beam theory. Lei et al. [14]

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investigated the free vibration of FG nanocomposite plates reinforced with single-walled CNTs, by using the element-free kp-Ritz method. The governing equations were based on the FSDT plate theory and the 2D displacement field was approximated by mesh-free kernel particle functions. A detailed analysis of the flexural strength and free vibration of CNT-reinforced composite cylindrical panels was carried out by Zhang et al. [15], by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach, in order to estimate their material properties. Lin and Xiang [16] investigated the linear free vibration of nanocomposite beams reinforced by single-walled CNTs, under the double assumption of 4

ACCEPTED MANUSCRIPT uniform and FG reinforcement distributions along the beams. Zhang et al. [17] determined the free vibration frequencies and modal shapes for FG-CNT skew plates with a moderate thickness, while employing the FSDT–based approach. The main free vibration features of FG nanocomposite triangular plates reinforced by single-walled CNTs was additionally presented by Zhang et al. [18]. In this case, the FSDT was employed to account for the effect of the transverse shear deformation of plates, whereas the element-free IMLS-Ritz method was adopted for numerical computations. Asanjarani et al. [19,20] studied the free vibration

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behavior of one- and two-dimensional FG truncated conical shells resting on a Winkler– Pasternak foundation, for an isotropic and inhomogeneous material along the length and thickness directions, while implementing the FSDT-based approach. Pourasghar and Kamarian [21] studied the dynamic behavior of a non-uniform column reinforced by singlewalled CNTs resting on an elastic foundation and subjected to follower forces. The problem

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was solved by applying the GDQ method. Additional works from the literature treat the buckling problem and the free vibration of FG cantilever beams or shells reinforced with CNTs (see i.e. [22-27] among others), while considering the agglomeration effect of CNTs within the structure.

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A thorough analysis on the statics and dynamics of FG rotating truncated conical shell reinforced by CNTs could be complex to be performed analytically. Thus, a similar problem

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is here explored numerically with the use of the GDQ method, and the application of the third-order shear deformation theory (TSDT) in small deformations. The efficiency of the GDQ method is verified by means of a convergence study in terms of displacement field and

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stress distribution within the structure. We also check the ability of the method to capture the sensitivity of the structural response to the CNT patterns along the thickness, as well as to

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some geometry parameters, i.e. the length, the radius, the thickness and the semi-vertex angle of the cone.

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The organization of the paper is given in the following. The problem formulation of the rotating FG conical shells reinforced with CNTs is introduced in Section 2, while giving the mechanical properties of the materials in Section 3. Section 4 is concerned with a brief numerical overview of the GDQ method and its numerical application to the rotating problem of the composite shell. This falls within a wide parametric investigation aimed at investigating the sensitivity of the response to different geometry parameters. Finally, conclusions are drawn in Section 5.

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ACCEPTED MANUSCRIPT 2. Problem formulation In this section, we describe the geometry and the governing equations of motion for a rotating FG conical shell reinforced with CNTs. The truncated conical shell has a constant thickness h as shown in Fig.1, and is referred to the ( x, , z ) coordinate system. This is located at the middle surface of the shell in the reference configuration. The x-coordinate is measured along the cone generator with the origin located at the semi-vertex, the angle  refers to the

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circumferential coordinate, and z is the coordinate along the thickness. Let R1 and R2 denote the radii of the truncated cone at the top and bottom edges, respectively. As also visible in Fig.1,  is the semi-vertex angle of the cone and L is its length along the x-

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direction.

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Figure 1. Conical shell: geometrical scheme.

If R denotes the general radius of the truncated cone at any point along the x-direction, then it

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is

R  x sin   R1 for 0  x  L

(1)

Based on the assumptions of TSDT shell theory, the displacement field can be written as (for more details, see [28])

u ( x, , z, t )  u0 ( x,  , t )  z x ( x,  , t )  z 2x ( x,  , t )  z 3 x ( x,  , t ) v( x, , z, t )  v0 ( x, , t )  z  ( x, , t )  z 2 ( x, , t )  z 3 ( x, , t ) w( x, , z, t )  w0 ( x, , t )

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(2)

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where u0 and v0 denote the in-plane displacements of the mid-plane, w0 is the transverse displacement,  x ,   are the slopes of the normal unit vector of the mid-surface in the   z plane, and z  x plane, respectively. In Eq. (2), i , i ( i  x , ) define the higher-order displacement parameters at the mid-higher plane, and t refers to the general time step of the motion. The strain-displacement relations for the shell are expressed as

 x

u 1  v  ,    w cos   ,  u sin   x x sin    

1  u  v   v sin    ,  x sin     x

 z

 zx 

u w  , z x

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x 

v 1  w     v cos    z x sin    

(3)

The mechanical constitutive relation between stresses and strains in our coordinate system can be defined as

0 0 0 0 C55 0

0   x    0    0   z    0   z  0   zx    C66   x 

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C13 0 C23 0 C33 0 0 C44 0 0 0 0

(4)

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 x   C11 C12       C12 C22  z  C13 C23   0  z   0    0 0  zx   0  x   0

Based on the TSDT shell theory [28], the transverse normal strain component  z is zero,

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considering the displacement field (2), whereas the transverse normal stress component  z can be neglected, due to the small dimension of the thickness compared to the in-plane

invoked.

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dimensions. Thus, under these assumptions, a case of both plane strain and plane stress is

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The equation of motion together with its boundary conditions can be derived by applying the

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Hamilton’s principle, whose dynamic version reads t2

  K   P  W  dt  0

(5)

t1

where K is the kinetic energy, P is the potential energy and W is the total work related to the external forces. These functions are defined, respectively, as

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ACCEPTED MANUSCRIPT 2 2 2 1 h /2  l2   u   v   w   K                R( x)dxd dz 2  h /2 0 l1   t   t   t  

1 h /2  l2  x x       z z   z  z   zx zx   x  x  R( x)dxd dz 2  h /2 0 l1 1 h /2  l2 W      ( z ) 2  u sin   w cos   R( x)dxd dz 2  h /2 0 l1 P

(6)

The third term of Eq. (6) represents a first approximation of the external forces related to the

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rotation of the cone around its axis of revolution for the static inflection. The terms related to the slope  x and to the higher order displacements  x and  x have been neglected due to the fact that we consider that the thickness is small compared to the in-plane dimensions. Under these assumptions, their contributions are small and thus they can be neglected without losing accuracy.

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Upon substitution of the virtual variation of these quantities in the Hamilton’s principle, combined with the application of the boundary conditions, the governing equations of the problem become as follows

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N x N 1 N x N x     I1u0  I 2 x  I 3x  I 4 x  I1 R 2 sin  x x x sin   x N x N x 1 N cos   V  2   I1v0  I 2   I 3  I 4 x sin   x sin  x x Vx cos  1 V Vx  N    I1w0  I1R 2 cos  x x sin  x sin   x M x M  M M 1 x  Vx    x  I 2u0  I 3 x  I 4x  I 5 x x x x sin   x M M x 1 M  cos   V  F  2 x   I 2 v0  I 3   I 4  I 5 x sin   x sin  x x Qx Q 1 Qx Qx  2 Fx      I 3u0  I 4 x  I 5x  I 6 x x x x sin   x Q Q 1 Q cos   2 F  R  2 x  x  I 3v0  I 4   I 5  I 6 x sin   x sin  x x Px P 1 Px Px  3Rx      I 4u0  I 5 x  I 6x  I 7 x x x x sin   x P P 1 P cos   3R  S  2 x  x  I 4v0  I 5   I 6  I 7 x sin   x sin  x x

with

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(7)

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( N i , M i , Qi , Pi ) 



 i (1, z , z 2 , z 3 )dz

i  x, 

 h /2 h /2

( N x , M x , Qx , Px ) 



 x (1, z, z 2 , z 3 )dz

 h /2 h /2

(Vi , Fi , Ri ) 



 zi (1, z , z 2 )dz

i  x, 

(8)

 h /2



 z z 3dz

 h /2 h /2

( I1 , I 2 , I 3 , I 4 , I 5 , I 6 , I 7 ) 



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h /2

S 

 (1, z , z 2 , z 3 , z 4 , z 5 , z 6 )dz

 h /2

By combining Eqs. (4) and (8), the elastic coefficients can be written as L11 L22 L33 L44 L55 L66 L77

Q11 Q22 Q33 Q44 Q55 Q66 Q77

K11 K 22 K 33 K 44 K 55 K 66 K 77

1  T11    T22  z  2 T33  h2  z     T44     z 3  C11 C12   T55   h2  z 4    z5  T66    6 T77   z 

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B11 B22 B33 B44 B55 B66 B77

C22

C44

C55

C66  dz

(9)

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 A11 A  22  A33   A44  A55   A66 A  77

Finally, the unknown coefficients can be determined through the enforcement of the

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boundary conditions (BC), i.e.

u0  v0  w0   x      x    x    0 for Clamped-clamped BCs

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v0  w0  N x  M x  Qx  Px  M x  Qx  Px  0 for Simply supported-simply supported BCs

(10)

For a modal analysis, the following solutions may be assumed to define the displacement

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field [29, 30]

U 0 ( x,  , t )  u0 ( x) cos(m )eit V0 ( x,  , t )  v0 ( x) sin( m )eit W0 ( x,  , t )  w0 ( x) cos(m )eit  x ( x,  , t )   x ( x) cos( m )eit  ( x,  , t )   ( x) sin( m )eit  x ( x,  , t )   x ( x) cos( m )eit  ( x,  , t )    ( x) sin( m )eit  x ( x,  , t )   x ( x) cos(m )eit  ( x,  , t )   ( x) sin( m )eit

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(11)

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3. Mechanical properties of the FG CNT truncated conical shell For a composite with a uniaxial reinforcement and subjected to constant strain conditions, the dependence of the elastic modulus with the CNT volume fraction can be estimated by means of the rule of the mixture as follows. More specifically, based on the recent work by Omidi et

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al. [31], the Young’s modulus Ec of a CNT-reinforced polymer composite can be estimated as follows

Ec  (kl ko kw ECNT  Em )VCNT e VCNT  Em

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with kl  1 

2l d



2 Em ECNT (1  vm ) ln VCNT

(14)

(15)

Ec  Em (kl ko kw ECNT  Em )VCNT

(16)

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

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 

(13)

ln(  ) VCNT

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

tanh 

(12)

In the previous relations E CNT and E m refer to the longitudinal elastic modulus for the CNT

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and the pure polymer, respectively. V CNT is the CNT volume fraction; kl is the CNT length efficiency parameter, ko is the CNT orientation efficiency factor, k w is the CNT waviness

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parameter, l is the length of the CNT, d is the diameter of the CNT, and v m is the Poisson’s ratio of the polymer. The hat symbol in Eq. (16) is applied to unknowns determined experimentally via some tensile tests and curve fitting procedures for higher percentages in weight of CNTs. It should be noted that expressions (12)-(16) represent a slightly different approach for the evaluation of the mechanical properties of a composite compared to the classical rule of mixture, as illustrated by Mehri et al. [32]. In this last work, indeed, the relationship between the Young’s modulus of the composite and the volume fraction of the nanoparticles is assumed to be linear. Nevertheless, Omidi et al. [31], demonstrated that the 10

ACCEPTED MANUSCRIPT relation at issue cannot be linear. This aspect becomes even more evident for increasing volume fraction of CNTs. As a consequence, an exponential term is added in Eq. (12). This aspect is justified by the experimental data available in the literature [31]. On the other hand, the mass density  and the Poisson’s ratio of the MWCNT/ Polystyrene (PS) composite can be computed by means of the classic rule of mixture as follows

  VCNT CNT  Vm m v  VCNT vCNT  Vmvm

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(17)

in accordance with the linear rule of mixtures, where CNT and  m refer to the densities of the CNT and the polymer, respectively.

In the present work, we study a PS matrix reinforced by MWCNTs with volume fractions

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related by

VCNT  Vm  1

(18)

We also consider the experimental data as provided by Andrews et al. [33] for the Young’s modulus of MWCNT/ PS composites with different volume fractions of MWCNTs, as here

parameters provided in Tab.1.

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used to fit the above mentioned rule of mixture. The best fitting is achieved with the model

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Table 1. Material properties for the PS matrix and the MWCNTs. Mechanical properties of PS

Mechanical properties of CNT

m  1050( Kg / m )  m  0.34

CNT  2100(kg / m3 )  CNT  0.28

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Em  1.9(GPa)

ECNT  900(GPa)

Characteristics of CNT

k0  0.2

kw  0.1

Ec  3.8(GPa)

VCNT  0.15%

d  25(nm)

l  60(  m)

It is important to underline that the current approach is valid when the reinforcing phase made of CNTs are straight and aligned inside the matrix. This hypothesis is extremely common in the literature, even if more complex models are available. These advanced approaches, in fact, could take into account also the effect of CNT agglomeration and waviness on the mechanical properties of CNT reinforced composites. Further details about these aspects can be found in the work by Shi et al. [34]. 11

ACCEPTED MANUSCRIPT Different kinds of material profiles are here assumed along the thickness of the shell in order to examine the effect of the CNT distribution on the static and free vibration behavior of the FG-CNT conical shell. We also consider a typical linear profile of CNTs within the composite with different distributions: namely, uniform ( U ), asymmetrical ( V ) or two

* VNT (U )   1  2 z  V * (V )  NT   h   VCNT ( z )   4 | z | * VNT (X )  h   1 | z | * 4  2  h  VNT (O)   

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symmetrical distributions ( X and O ), defined by the following relations

(19)

* VNT 

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* where h is the thickness of the shell, VNT is the volume fraction of the nanotube defined as

wNT

  wNT  CNT  CNT wNT m m

(20)

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while wNT is the mass fraction of the nanotube, CNT and  m are the densities of the CNT and

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the matrix, respectively. Fig. 2 shows the variation of the CNT volume fraction VCNT  %  through the thickness of the composite shell, here represented in a dimensionless way, i.e.

  z / h . As clearly visible from Fig. 2, a symmetrical X -distribution of CNTs within the

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shell leads to the highest volume fraction at the external and internal surfaces   0.5 ,

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while reaching a null value at the mid-surface (i.e. VCNT  0 for   0 ). On the contrary, for a symmetrical O -distribution of CNTs, the volume fraction of CNTs reaches the maximum

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value at the mid-surface and vanishes at the outer surfaces. As expected, the asymmetrical V distribution of CNTs increases linearly from zero (at the internal surface) up to the maximum value (at the external surface), whereas the volume fraction of CNTs maintains constant along the whole thickness for a uniform distribution U .

12

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Figure 2. Variation of the CNT volume fractions through the thickness for different distributions of CNTs. The different distribution of CNTs within the shell can affect significantly the mechanical

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behavior of the composite. This is visible, for example, in Fig. 3 in terms of Young’s modulus E , whose variation along the dimensionless thickness clearly changes for different CNTs distributions, and could be repeated for other mechanical properties. Finally, it should be mentioned that the assumption of a perfect bonding between the reinforcing phase and the

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matrix has been assumed by hypothesis (as also considered in the studies [34-36]). The present paper, indeed, is not focused on the physical phenomenon at the nanoscale level,

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but it only investigates the free vibrations of CNT reinforced structured at the macroscale level. More complicated models, however, could be used for a further investigation, by including some physically consistent cohesive zone models, as increasingly applied in the

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CE

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literature for the study of composite materials and interfaces (see e.g. [37-43]).

Figure 3. Variation of the Young’s modulus through the thickness for different distributions of CNT.

13

ACCEPTED MANUSCRIPT 4. The numerical applications 4.1 Introduction In the present work, we employ the GDQ technique to solve the governing equations of the problem in a strong form, as increasingly used in the last years in the literature due to its great features of accuracy and reliability [44-55]. The main aspects of the method are illustrated in

domain (for more details see Tornabene et al. [56]).

4.2 The GDQ method

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what follows for a 1D domain  x0 , x1  for simplicity, and can be easily extended to a 2D

at a general point xi   x0 , x1  as follows d n f  x dx

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Based on the GDQ approach, the n th derivative of a smooth function f  x  can be evaluated

  cij( n ) f  x j  N

n x  xi

j 1

for i = 1, 2,..., N

(21)

where N defines the total number of discrete points defined along the domain, and cij( n ) refers

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to the weighting coefficients of the summation, evaluated recursively as detailed in [56]. A proper grid distribution must be chosen to discretize the domain. In a 2D setting, a discrete

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grid has to be applied along the two principal directions. In the simplest case of 1D domain, the discrete node can be placed according to the following expression (22)

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  i  1    x1  x0  xi  1  cos     x0 for i  1, 2,..., N 2  N 1   

which defines a Chebyshev-Gauss-Lobatto (Che-Gau-Lob) grid. This distribution is

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employed to solve different kinds of structural problems due its accuracy. However, different grid distributions can be selected due to the general features of the present numerical

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approach, as briefly listed in the work [56]. After performing some mathematical manipulations, one finally obtains a system of linear algebraic eigenvalue equations. In order to reduce the computational efforts, related to the specified boundary and domain degrees of freedom (DOF) one can break the major coefficient matrix into the number of sub-matrices. If one separates the system DOF into domain, d  , and boundary, b  , then the GDQ discretized form of the equations of motion and the related boundary conditions can be obtained in the matrix form, respectively, as

14

ACCEPTED MANUSCRIPT K dbb  K dd d m2 Md  0

(23)

K bbb  K bd d  0

(24)

The elements of the stiffness matrices K di (i  b, d ) and the mass matrix M are obtained from the separation between the boundary points and the domain points. The elimination of the boundary degrees of freedom from Eq. (23) and Eq. (24), yields to the

K   M d  0 2 m

whose solution can be found in terms of natural frequencies.

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following system of eigenvalue problem (25)

Using the discrete transformation and the GDQ rules, the equations of motions and the related boundary conditions can be redefined for numerical computations. More details about

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the discretized equations of motion and boundary conditions in the x-direction at each grid point of the domain are given in Appendix.

4.3 Numerical implementation and results

The efficiency of the GDQ approach for the numerical solution of the problem is now

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investigated by means of a convergence study. The accuracy of the GDQ-based results is verified with respect to those ones available from the literature (see, e.g. [1, 57-59]), and the

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numerical predictions based on the classical finite element method (FEM). In the last case, we have adopted the Abaqus commercial code for numerical computations. Some

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comparative results are listed in Tab. 2 in terms of the first eight natural frequencies

   R2  (1  v 2 ) E , for three different semi-vertex angles  of the conical shell (i.e.

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  30, 45, 60 respectively). The conical shell is clamped-clamped along the two opposite

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sides in the axial direction, and is defined by the geometry ratios R2 / h  20 and L / R2  0.5 .

Table 2. Comparative evaluation of the results for the first eight natural frequencies.   300

Present Ref. [1]

m=0 0.9925 0.9930

m=1 0.8775 0.8776

m=2 0.6420 0.6422

m=3 0.4799 0.4803

m=4 0.3808 0.3816

m=5 0.3299 0.3311

m=6 0.3200 0.3216

m=7 0.3429 0.3450

  450

Present Ref. [1] Ref. [57]

0.8726 0.8731 0.8732

0.8117 0.8120 0.8120

0.6694 0.6696 0.6696

0.5426 0.5430 0.5428

0.4563 0.4570 0.4565

0.4085 0.4095 0.4088

0.3956 0.3970 0.3961

0.4133 0.4151 0.4141

  600

Present Ref. [1]

0.6682 0.6685

0.6313 0.6316

0.5520 0.5523

0.4780 0.4785

0.4290 0.4298

0.4083 0.4093

0.4145 0.4159

0.4448 0.4466

15

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As visible from Tab. 2, the results based on the proposed formulation resemble very well those presented in [1, 57], which demonstrate the correct solution of the problem based on a GDQ approach. Furthermore, the natural frequencies of FGM conical shells for various FG power low index (p) are compared with predictions from the literature [58, 59] in Tab. 3. Also in this case, it is worth noticing the excellent agreement between results.

1 2 3 4 5 6 7 8 9 10

p=0 Present Ref. study [58] 209.81 209.9 209.81 209.9 231.79 231.9 231.79 231.9 287.36 287.5 287.36 287.5 321.93 322.5 321.93 322.5 356.32 356.9 356.32 356.9

Ref. [59] 210.0 210.0 232.0 232.0 287.5 287.5 322.6 322.6 357.0 357.0

p=1 Present Ref. study [58] 204.36 204.4 204.36 204.4 223.89 224.0 223.89 224.0 275.78 276.2 275.78 276.2 315.69 315.7 315.69 315.7 346.52 346.8 346.52 346.8

Ref. [59] 204.9 207.9 224.4 224.4 276.7 276.7 316.3 316.3 347.7 347.7

p=5 Present Ref. study [58] 203.46 203.5 203.46 203.5 226.92 227.3 226.92 227.3 283.71 283.8 283.71 283.8 308.94 309.0 308.94 309.0 345.19 346.3 345.19 346.3

Ref. [59] 203.9 203.9 227.7 227.7 284.3 284.3 304.8 304.8 341.5 341.5

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Mode No.

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Table 3. Comparative evaluation of the natural frequencies of a FGM conical shell with l cos   2m,  =40°, R1 =0.5m, h=0.1m. p=20 Present Ref. study [58] 200.54 200.6 200.54 200.6 223.82 224.1 223.82 224.1 279.75 279.9 279.75 279.9 304.68 304.8 307.68 304.8 341.74 341.9 341.74 341.9

Ref. [59] 200.8 200.8 224.3 224.3 280.2 280.2 305.0 305.0 341.9 341.9

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We also analyze comparatively the wall deflection W * of a rotating conical shell with L / R2  0.5, R2 / h  20,   30 as given by the present GDQ-based formulation, and the

AC

CE

PT

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ABAQUS code (see Fig. 4).

Figure 4. Wall deflection of the rotating FGM conical shell with a V  distribution. L / R2  0.5, R2 / h  20,   30,   300 rad/s . An angular speed of   300rad/s is here assumed for the dynamic analysis, while the wall deflection is computed in a dimensionless form as 16

ACCEPTED MANUSCRIPT W *  105 w0 D /  R22 L4 2

(26)

where D  Eh3 / 12(1  v2 ) is the stiffness of the shell, and w0 is the initial transverse deflection of the shell. As visible from the plots of Fig. 4, a perfect agreement between the GDQ and FEM approaches is obtained, which proves once again the capability of the proposed GDQ method to describe correctly the problem. The efficiency of the method is also evaluated trough a convergence study of the natural

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frequencies for an increasing number of collocation points N from 5 up to 20 for each element. The main results for the first four natural frequencies are listed in Tab. 4 for the same geometry of conical shell as considered before, and a V -distribution of CNTs within the composite shell. Based on this convergence study, a limited number of grid points N  11 is sufficient to obtain a stable solution, with a reduced computational effort despite the

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complexity of the numerical problem.

Table 4. Convergence study of the results for the three four natural frequencies, as obtained * by the proposed formulation for L / R2  0.5, R2 / h  20,   30, VCN  0.3 and V distribution. N 7

N 9

2.0149 1.9833 1.9250 1.8937

2.0151 1.9835 1.9251 1.8937

N  11

N  13

N  15

N  17

N  19

2.0151 1.9835 1.9251 1.8937

2.0151 1.9835 1.9251 1.8938

2.0151 1.9835 1.9252 1.8938

2.0151 1.9835 1.9252 1.8938

2.0151 1.9835 1.9252 1.8938

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N 5 2.0268 1.9931 1.9299 1.8924

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m=0 m=1 m=2 m=3

Furthermore, we evaluate the influence of the R2 / h ratio, and of the distribution of CNTs

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within the thickness, on the free vibration of the shell. The main results related to the first four natural frequencies of the structure are listed in Tabs. 5-7 and represented in Figs. 5a-c

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for the only first natural frequency, and for the semi-vertex angles   30, 45, 60 , respectively. As visible from the lists of Tabs. 5-7, as well as from the plots in Figs. 5a-c, a

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decrease of the frequency  is observed with increasing R2 / h ratios. Moreover, the curve obtained for a symmetrical O  distribution of CNTs lies always below those ones based on a V  or U  distribution. These last two curves in turn are lower than the one based on a

symmetrical X  distribution. This means that, for a given R2 / h ratio, the O  distribution of CNTs always provides the lowest value of frequency, whereas the X  distribution of CNTs yields to the highest estimation of the frequency for the same structure. From a design point of view, this means that the highest stiffness for a truncated conical structure can be physically reached with an X  distribution of CNTs, independently of the semi-vertex angle. 17

ACCEPTED MANUSCRIPT This would be of extreme interest for engineers working with space fight, rocket, aviation, or submarine technology. Another systematic investigation is performed on the sensitivity of the response to the L / R2

geometry ratio, for fixed angles   30, 45, 60 , and for a fixed R2 / h

dimensionless ratio. Tabs. 8-10 report the first four natural frequencies of the shell for varying CNTs distributions, and for semi-vertex angles   30, 45, 60 , respectively. As it

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is reasonable to expect, an increase in natural frequencies is obtained for fixed L / R2 ratios and increasing semi-vertex angles, whereas the increasing values of L / R2 yield a meaningful decrease of the frequency for all the CNTS configurations. Based on a comparative evaluation of Tabs. 8-10, it can be concluded that FG conical shells with an X-distribution of CNTs are always the most rigid ones, among different possibilities of distribution, while

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reaching the highest stiffness for increasing angles and decreasing L / R2 ratios

(more

specifically, for   60 and L / R2  0.2 ).

R2 / h

CE

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10

AC

15

20

O

V

U

X

3.9423 3.9181 3.9159 3.9981 2.8726 2.8525 2.8321 2.8624 2.2836 2.2572 2.2144 2.2070 1.9586 1.9257 1.8640 1.8275

3.9615 3.9389 3.9409 4.0298 2.9417 2.9223 2.9041 2.9386 2.3517 2.3262 2.2865 2.2839 2.0151 1.9835 1.9252 1.8938

4.1807 4.1525 4.1502 4.2429 3.1556 3.1347 3.1161 3.1550 2.5282 2.5024 2.4637 2.4660 2.1601 2.1282 2.0709 2.0439

5.2553 5.3286 5.5422 5.8787 3.4701 3.5269 3.6929 3.9562 2.5925 2.6345 2.7574 2.9536 2.1167 2.1478 2.2390 2.3858

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5

m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3

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Table 5. First four natural frequencies for different ratios R2 / h and different distributions of * CNTs. L / R2  0.5,   30, VCN  0.3.

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ACCEPTED MANUSCRIPT Table 6. First four natural frequencies for different ratios R2 / h and different distributions of * CNTs. L / R2  0.5,   45, VCN  0.3.

10

15

20

V

U

X

3.8841 3.8721 3.8962 4.0149 2.7970 2.7855 2.7867 2.8469 2.1888 2.1701 2.1478 2.1687 1.8473 1.8218 1.7797 1.7706

3.9020 3.8918 3.9206 4.0464 2.8663 2.8555 2.8590 2.9236 2.2583 2.2407 2.2215 2.2472 1.9060 1.8818 1.8432 1.8393

4.1197 4.1052 4.1320 4.2650 3.0797 3.0680 3.0726 3.1436 2.4348 2.4172 2.4002 2.4324 2.0505 2.0265 1.9901 1.9920

5.2192 5.3046 5.5520 5.9389 3.4241 3.4902 3.6821 3.9843 2.5331 2.5826 2.7267 2.9547 2.0446 2.0819 2.1910 2.3648

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5

m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3

O

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R2 / h

Table 7. First four natural frequencies for different ratios R2 / h and different distributions of * CNTs. L / R2  0.5,   60, VCN  0.3.

PT

10

AC

CE

15

20

3.8048 3.8112 3.8707 4.0314 2.6914 2.6943 2.7273 2.8249 2.0532 2.0491 2.0601 2.1193 1.6846 1.6741 1.6669 1.6980

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5

m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3

V

U

X

3.8203 3.8286 3.8928 4.0609 2.7605 2.7641 2.7994 2.9013 2.1248 2.1216 2.1353 2.1990 1.7467 1.7373 1.7332 1.7690

4.0364 4.0426 4.1086 4.2881 2.9738 2.9775 3.0157 3.1262 2.3018 2.2994 2.3166 2.3884 1.8916 1.8831 1.8825 1.9255

5.1708 5.2686 5.5512 5.9907 3.3605 3.4361 3.6549 3.9971 2.4492 2.5071 2.6745 2.9369 1.9409 1.9858 2.1161 2.3211

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O

R2 / h

19

  30

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(a)

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ACCEPTED MANUSCRIPT

  45

AC

CE

PT

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(b)

  60 Figure 5. Variation of the first natural frequency  vs. the R2 / h ratios for different * distributions of CNTs. L / R 2  0.5, V CN  0.3, m  1. (c)

20

ACCEPTED MANUSCRIPT Table 8. First four natural frequencies for different L / R2 ratios, and different distributions * of CNTs. R2 / h  10,   30, VCN  0.3.

0.4

0.6

0.8

V

U

X

3.9111 3.9072 3.9106 3.9469 1.7999 1.6590 1.5176 1.5310 1.6417 1.1184 0.8603 0.9930 1.6409 1.0257 0.7836 0.9798

3.9848 3.9811 3.9859 4.0247 1.8340 1.6949 1.5608 1.5849 1.6549 1.1270 0.8815 1.0358 1.6526 1.0318 0.8045 1.0231

4.2654 4.2607 4.2644 4.3051 1.9443 1.8000 1.6672 1.7042 1.7326 1.1773 0.9329 1.1173 1.7255 1.0762 0.8525 1.1045

4.9681 5.0180 5.1651 5.4024 1.8303 1.8848 2.0390 2.2808 1.1098 1.3095 1.1953 1.3284 0.8077 1.1037 0.9381 1.2519

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0.2

m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3

O

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L / R2

Table 9. First four natural frequencies for different L / R2 ratios, and different distributions * of CNTs. R2 / h  10,   45, VCN  0.3. O

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0.4

CE

0.6

U

X

6.3945 6.4010 6.4240 6.4710 2.4342 2.4091 2.3971 2.4660 1.5903 1.3466 1.2030 1.3725 1.5652 1.2385 1.0751 1.3158

6.7886 6.7946 6.8170 6.8648 2.6143 2.5882 2.5780 2.6557 1.6754 1.4220 1.2848 1.4832 1.6442 1.3017 1.1468 1.4236

8.5639 8.6048 8.7266 8.9259 2.8107 2.8819 3.0867 3.4050 1.5110 1.5690 1.6856 1.9483 1.4612 1.4797 1.4595 1.7626

AC

0.8

6.3478 6.3539 6.3757 6.4208 2.3717 2.3456 2.3301 2.3932 1.5686 1.3229 1.1672 1.3178 1.5480 1.2212 1.0430 1.2610

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0.2

m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3

V

M

L / R2

21

ACCEPTED MANUSCRIPT Table 10. First four natural frequencies for different L / R2 ratios, and different distributions * of CNTs. R2 / h  10,   60, VCN  0.3.

0.4

0.6

0.8

V

U

X

8.3374 8.3452 8.3700 8.4150 3.0364 3.0423 3.0770 3.1668 1.4698 1.3927 1.3852 1.5902 1.3521 1.2219 1.2021 1.4811

8.3478 8.3559 8.3817 8.4285 3.1077 3.1141 3.1506 3.2439 1.5071 1.4330 1.4346 1.6541 1.3798 1.2531 1.2463 1.5445

8.8112 8.8194 8.8456 8.8942 3.3439 3.3505 3.3892 3.4897 1.6106 1.5357 1.5477 1.7939 1.4650 1.3356 1.3434 1.6774

11.5756 11.6112 11.7172 11.8919 3.8424 3.9128 4.1176 4.4410 1.5999 1.7058 1.9837 2.3995 1.4005 1.5063 1.7472 2.1835

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0.2

m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3 m=0 m=1 m=2 m=3

O

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L / R2

Some results related to the first natural frequency, are also plotted in Figs. 6-8 for the three semi-vertex angles, which confirm the decreasing trend of the frequency  for increasing values of S  L sin  / R2 . The lowest values of  are reached with an O  distribution of

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CNTs, whereas the highest values of  are provided always by the X  distribution of CNTs. The estimations provided by a U  and a V  distribution fall always within them.

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This means that an O  distribution or an X  distribution of CNTs can be selected as the best choices for practical applications, depending on whether a flexible or rigid structure is Similar results can be also concluded regarding the variation of the first natural

PT

required.

frequency  vs. the semi-vertex angle  for all the distributions of CNTs (see Fig. 9).

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Accordingly, predictions based on a V  or U  distribution stay always within the range of predictions as provided by a O  or X  distribution (see Fig. 9). For completeness, we check

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the variation of the first seven natural frequencies  vs. the semi-vertex angle  for each fixed distribution of CNTs.

22

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ACCEPTED MANUSCRIPT

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M

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Figure 6. Variation of the first natural frequency  vs. the S ratio for different distributions * of CNTs through the thickness of the shell. R2 / h  10,   30, VCN  0.3, m  1.

AC

CE

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Figure 7. Variation of the first natural frequency  vs. S ratio, for different distributions of * CNTs through the thickness of the shell. R2 / h  10,   45, VCN  0.3, m  1.

Figure 8. Variation of the first natural frequency  vs. the S ratio, for different distributions * of CNTs through the thickness of the shell. R2 / h  10,   60,VCN  0.3, m  1.

23

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ACCEPTED MANUSCRIPT

Figure 9. Variation of the first natural frequency  vs. the angle  , for different distributions of CNTs through the thickness of the shell. * L / R2  0.5, R2 / H  20, VCN  0.3, m  1.

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As visible from Tabs. 11-14, the first five natural frequencies decrease monotonically for all the distributions herein assumed, whereas the last two natural frequencies here analyzed increase monotonically with increasing angles  . This agrees with some recent findings obtained in the recent literature by Dung et al. [60].

1.9862 1.9060

m=2 1.9425

m=3 1.9055

m=4 1.9183

m=5 1.9878

m=6 2.1144

1.9561

1.9025

1.8785

1.9085

1.9990

2.1501

1.8818

1.8432

1.8393

1.8920

2.0074

2.1857

1.7373

1.7332

1.7690

1.8615

2.0166

2.2355

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1.7467

m=1 2.0037

ED

m=0 2.0360

and a V-distribution.

1.5932

1.6007

1.6334

1.7077

1.8350

2.0223

2.2721

1.5344

1.5489

1.5963

1.6852

1.8250

2.0234

2.2834

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  25   35   45   60   75   85

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Table 11. Seven natural frequencies for different angles  * L / R2  0.5, R2 / h  20, VCN  0.3.

Table 12. Seven natural frequencies for different angles  and an X-distribution.

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* L / R2  0.5, R2 / h  20, VCN  0.3.

  25   35   45   60   75   85

m=0 2.1306

m=1 2.1599

m=2 2.2463

m=3 2.3856

m=4 2.5732

m=5 2.8050

m=6 3.0787

2.0976

2.1305

2.2272

2.3823

2.5898

2.8451

3.1455

2.0446

2.0819

2.1910

2.3648

2.5956

2.8777

3.2079

1.9409

1.9858

2.1161

2.3211

2.5898

2.9143

3.2901

1.8440

1.8958

2.0451

2.2774

2.5783

2.9376

3.3493

1.8082

1.8626

2.0189

2.2612

2.5736

2.9452

3.3693

24

ACCEPTED MANUSCRIPT Table 13. Seven natural frequencies for different angles  and an O-distribution. * L / R2  0.5, R2 / h  20, VCN  0.3.

m=0 1.9798

m=1 1.9464

m=2 1.8818

m=3 1.8399

m=4 1.8465

m=5 1.9086

m=6 2.0267

1.9291

1.8978

1.8407

1.8115

1.8351

1.9178

2.0599

1.8473

1.8218

1.7797

1.7706

1.8168

1.9243

2.0930

1.6846

1.6741

1.6669

1.6980

1.7842

1.9312

2.1401

1.5280

1.5348

1.5652

1.6357

1.7572

1.9364

2.1759

1.4690

1.4830

1.5286

1.6140

1.7482

1.9386

2.1881

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  25   35   45   60   75   85

Table 14. Seven natural frequencies for different angles 

  25   35   45   60   75   85

m=0 2.1811

m=1 2.1484

m=2 2.0880

m=3 2.0551

2.1310

2.1008

2.0485

2.0294

2.0505

2.0265

1.9901

1.8916

1.8831

1.7403 1.6836

and a U-distribution.

m=4 2.0758

m=5 2.1571

m=6 2.2996

2.0685

2.1722

2.3410

1.9920

2.0550

2.1852

2.3831

1.8825

1.9255

2.0299

2.2019

2.4430

1.7492

1.7861

1.8686

2.0089

2.2146

2.4884

1.6995

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* L / R2  0.5, R2 / h  20, VCN  0.3.

1.7513

1.8486

2.0018

2.2191

2.5037

M

Finally, we expose how the composite shell structure with a fixed geometry of L / R2  1 ,

ED

R2 / h  10 and   30 can withstand vibrations, i.e. which is the structural response in

terms of displacements (see Fig. 10) and internal forces for varying distributions of CNTs

PT

(see Figs.11a-d). All the results are reported as a function of the dimensionless length x / L in the longitudinal direction for all the distributions of CNTs. From Fig. 10, it is worth noticing that the lowest amplitude of transverse displacement is reached with an O  distribution of

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CNTs within the thickness (at least for the selected geometry of conical shell), whereas the worst response would be obtained for a V  distribution of CNTs. This is also confirmed by

AC

the results obtained in terms of transverse force or moment force distributions, as plotted in Fig. 11. For the numerical analyses, it is assumed a rotating speed of the conical shell of





* 4 2   300 rad / s , and we compute the transversal displacement as W  10 Dw m R2  ,

D  Em h  2(1  vm )  being the stiffness of the shell, as well as the internal stresses M x*  100R2 M x D ,

Qx*  100R2Qx D , Px*  100R2 Px D , N x*  100R2 N x D , which define,

respectively, the bending moment, transverse force, higher-order transverse force, and axial stress. 25

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ACCEPTED MANUSCRIPT

(b) Transverse force Qx*

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PT

ED

(a) Bending Moment M x*

M

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Figure 10. Transverse displacement W * vs. the dimensionless length x / L for different distributions of CNTs through the thickness of the shell. L / R2  1, R2 / h  10,   30.

(c) Higher-order transverse force Px*

AC

(d) Axial force N x*

Figure 11. Internal forces vs. the dimensionless length x / L , for different distributions of CNTs through the thickness of the shell. L / R2  1, R2 / h  10,   30. For the most conservative distribution of CNTs (i.e. the O  distribution), we finally analyse the variation of the amplitude of transversal displacements and internal forces along the longitudinal length x / L for varying semi-vertex angles. As shown in Fig. 12, the amplitude of transverse displacements reduces for  increasing from 30° to 60°, in accordance with the

26

ACCEPTED MANUSCRIPT internal bending moment M x* (Fig. 13a), transverse force Qx* (Fig. 13b), and higher-order transverse force Px* (Fig. 13c). Differently, the magnitude of the axial force N x* seems to increase in the longitudinal direction for increasing semi-vertex angles. A similar investigation could be repeated for the other configurations of CNTs within the structure, but

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it is here not reported for the sake of brevity.

PT

ED

M

Figure 12. Transverse displacement W * vs. the dimensionless length x / L for different angles  and an O-distribution. L / R2  1, R2 / h  5.

(b) Transverse force Qx*

AC

CE

(a) Bending moment M x*

(c) Higher-order transverse force Px*

(d) Axial force N x*

Figure 13. Internal forces vs. the dimensionless length x / L , for different angles  and an O-distribution. L / R2  1, R2 / h  5. 27

ACCEPTED MANUSCRIPT 5. Conclusions In this paper, an innovative technique known as Generalized Differential Quadrature (GDQ) method has been applied to study the static and free vibration behavior of rotating FG truncated conical shells reinforced with CNTs. The equations of equilibrium and motion for the rotating shells are derived by applying the TSDT in small deformations, whose numerical solutions are found in a strong form directly with their boundary conditions. The effect of the

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agglomeration of the reinforcing phase is investigated for a varying distribution of nanofibers scattered in the polymeric matrix, and a varying exponent of the through-the-thickness volume fraction distribution. Based on a systematic numerical investigation, we verify the sensitivity of the mechanical response of the rotating structure to the agglomeration of CNTs within the matrix and to some geometry parameters, i.e. the cone length, radius, thickness,

AN US

and angle. This is verified both in terms of deflection and stress distribution within the structure. Such an influence is verified to become even more significant for higher circumferential wave numbers. Moreover, a very good agreement between the numerical results and the reference ones available from the literature demonstrates the potentials and accuracy of the GDQ methodology for the vibrational analysis of the rotating conical shells

M

made of composite materials, also for coarse mesh discretizations. The reduced computational cost required by the GDQ approach makes it a useful and recommended

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numerical tool for the solution of challenging problems of rotating composite structures.

PT

Appendix. Equations of motion in the GDQ-based discrete form Herein we give additional details about the first nine equations of motion of the composite

CE

shell in a discrete form based on the GDQ approach. The 1st equation of motion reads

AC

L  mL L  mL L m 2T  m 2T  mT  L cos  m 2T  mT    112  2 112  u0i   2 11  2 11  v0i  112 w0i   222  2 222  xi   2 22  2 22   i   332  2 332   xi  xi sin  xi sin   xi sin    xi xi sin    xi sin  xi sin    xi  xi sin  xi sin    xi 2 N N  mL33 L  mL A  mT  mT  mT  m  2 33   i   442  2 442  xi   2 44  2 44  i    11 cij(1)  A11cij(2)  u0 j   2  cij(1) T11  B11  v0 j  xi sin   xi sin  j 1 j 1  xi  xi sin  xi sin    xi  xi sin  xi sin    N

c j 1

(1) ij

N A  B11 cos  m w0 j    22 cij(1)  A22cij(2)  xj  xi sin  xi sin  j 1  xi 

m xi sin 

j 1

(1) ij

A  m cij(1) T33  B33   j    44 cij(1)  A44cij(2)  xj   x x sin  j 1 j 1  i i  N

N

N A   B22   j    33 cij(1)  A33cij(2)   xj  j 1  xi 

N

 c T

22

N

 c T j 1

(1) ij

28

44

 B44  j   2  I1u0i  I 2 xi  I 3 xi  I 4 xi   I1 R 2 sin 

ACCEPTED MANUSCRIPT The 2nd equation of motion is defined as  mL   mL cos  mQ cos    mL mT  m2 L T Q cos 2   mT    2 11  2 11  u0i    2 11  112  112 2  v0i   2 11 2  2 11 2  w0i   2 22  2 22  xi  2 xi sin   xi sin    xi sin  xi sin    xi sin  xi  xi sin   xi sin  xi sin    m 2 L22  mL  m2 L mT  T T22 Q11 cos  Q22 cos 2   2Q cos  Q33 cos 2    2 2   i   2 33  2 33   xi    2 33  332  22  2 2   i   2 2  2  2 x sin  x x sin  x sin  x sin  x sin  x sin  x xi sin  xi sin   i i i i i  i   i   i 2 2 N N  mL44  mL T  3Q cos  Q44 cos   mT  T m  2 44  xi    2 44  442  33  2 2  i  cij(1) T11  B11  u0 j    11 cij(1)  T11cij(2)  v0 j   2  2 x sin  x sin  x sin  x x sin  x sin  x sin  x j 1 j 1  i i i i i i  i   i   N N N T  T  m  B22  xj    22 cij(1)  T22cij(2)   j  cij(1) T33  B33   xj    33 cij(1)  T33cij(2)   j   x sin  x j 1 j 1  xi j  1 j  1 i   i  N N  T44 (1) m (1) (2)  2 cij  T44cij  j    I1v0i  I 2  i  I 3 i  I 4 i    cij T44  B44  xj   xi sin  j 1 j 1  xi  N

m xi sin 

 c T (1) ij

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22

The 3rd equation of motion is expressed as 

 mL cos  mQ cos    L cos 2   K11 L22 cos   L11 cos  m 2Q  u0i   2 11 2  2 11 2  v0i   112 2  2 11  2  w0i    xi  2 2 xi sin  x sin  x sin  x sin  x sin  xi sin   i i  i   i   xi

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 mL22 cos  mQ22 cos   2 K 22 L33 cos    mL33 cos  mQ33 cos  2mQ22  mQ11    2   2 2  2 2   i     xi    2 2  2 2   i  x sin  x sin  x sin  x x sin  xi sin  xi sin   i i i i    i   xi sin   3K 33 L44 cos    mL44 cos  mQ44 cos  3mQ33  cos  N  2    xi    2 2  2 2  i   B11cij(1)u0 j  xi sin   xi sin  xi sin   xi sin  j 1  xi  xi sin  N N  K11 (1)    B33 cos   (1) B cos   (1) cij  K11cij(2)  w0 j    K11  22  cij  xj    2 K 22   cij  xj  xi sin   xi sin   j 1  xi j 1  j 1   N

 j 1



33



B44 cos   (1) 2 2  cij  xj   I1w0i  I1 R cos  xi sin  

The 4th equation of motion results

M



N

  3K

PT

ED

L  mL   mL L m 2T  mT  m 2T  mT  L cos    222  2 222  u0i   2 22  2 22  v0i  222 w0i   K11  332  2 332  xi   2 33  2 33   i  xi sin   xi sin  xi xi sin    xi  xi sin  xi sin     xi sin  xi sin     mL44   mL L m 2T  mT  L44 m 2T44  mT   2 44   i   3K33  552  2 552  xi   2 55  2 55  i   2 K 22  2  2 2   xi   2 xi xi sin   xi xi sin     xi sin  xi sin     xi sin  xi sin   N N N  A22 (1)   B cos   m cij  A22 cij(2)  u0 j  cij(1) T22  B22  v0 j   cij(1)  22  K11  w0 j     x x sin  x sin  j 1  i j 1 j 1 i   i  N N  A  m m cij(1)  A33cij(2)  xj  cij(1) T33  B33   j    44 cij(1)  A44cij(2)   xj   x sin  x x sin  j 1  i j 1 j 1  i i i   N N  A55 (1)  m cij  A55cij(2)  xj     c(1) T  B    2  I 2u0i  I3 xi  I 4 xi  I5 xi  x sin  j 1 ij 55 55  j j 1  xi i  N

 A33

N

 c T j 1

(1) ij

44

 B44   j 

CE

 x

The 5th equation of motion reads

AC

 mL  m2 L  mQ11  mL33 mT  mT  T Q cos  Q22 cos 2   mL cos  mQ22 cos     2 22  2 22  u0i    2 22  222  11  2 2  v0i    2 22 2  2 2  2 33  xi   w0i   2 2 x sin  x sin  x sin  x x sin  x sin  x sin  x sin  x sin  x sin  x i i i i i i i sin    i   i   i   i 2 2 2 2   mL  m L33 T33 Q22 cos  Q33 cos   3Q cos  Q44 cos   mT  m L T  2 2   i   2 44  2 44   xi   2Q22  2 44  44  33  2 2   i   Q11  2 2  2  xi sin  xi xi sin  xi sin   xi sin 2  xi2 xi sin  xi sin     xi sin  xi sin    N N  mL55  T  mT  m2 L T 4Q cos  Q55 cos 2   m  2 55  xi   3Q33  2 55  552  44  2 2  i  cij(1) T22  B22  u0 j    22 cij(1)  T22cij(2)  v0 j   2  2 x sin  x sin  x sin  x x sin  x sin  x sin  x j  1 j  1 i i i i i i i i       N N N N  T33 (1)  T44 (1) m m (1) (2)  (1) (2)  cij  T33cij   j  cij  T44cij   j     cij T33  B33  xj    cij T44  B44   xj   xi sin  j 1 xi sin  j 1 j 1  xi j 1  xi  

m xi sin 

N

 c T j 1

(1) ij

55

N T   B55  xj    55 cij(1)  T55cij(2)  j   2  I 2 v0i  I 3  i  I 4 i  I 5 i  j 1  xi 

29

ACCEPTED MANUSCRIPT

The 6th equation of motion is L  mL   mL m 2T  mT  L cos  L m 2T  mT    332  2 332  u0i   2 33  2 33  v0i  332 w0i   2 K 22  442  2 442  xi   2 44  2 44   i  xi sin   xi sin  xi xi sin    xi  xi sin  xi sin     xi sin  xi sin     mL55   mL L55 m 2T55  mT  L m 2T  mT   2 55   i   6 K33  662  2 662  xi   2 66  2 66  i   4 K33  2  2 2   xi   2 xi xi sin   xi xi sin     xi sin  xi sin     xi sin  xi sin   N N N  A33 (1)   B cos   m cij  A33cij(2)  u0 j  cij(1) T33  B33  v0 j   cij(1)  33  2 K 22  w0 j     x x sin  x sin  j 1  i j 1 j 1 i   i 

N

 A66

j 1



 x

i

 m cij(1)  A66 cij(2)  xj  x sin  i 

N

 c T j 1

(1) ij

44

N

 c T j 1

(1) ij

66

N A  m  B44   j    55 cij(1)  A55cij(2)   xj  x x sin  j 1  i i 

 B66  j   2  I 3u0i  I 4 xi  I 5 xi  I 6 xi 

The 7th equation of motion has the form

N

 c T j 1

(1) ij

55

 B55   j 

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 A44 (1)  m cij  A44 cij(2)  xj  x x sin  j 1  i i  N



AN US

 mL  m2 L  2mQ22 mL33 cos  mQ33 cos   mT  T 2Q cos  Q33 cos 2     2 33  2 33  u0i    2 33  332  22  2 2  v0i    2 2  2 2  w0i  2 xi sin  xi sin   xi sin  xi sin    xi sin  xi sin    xi sin  xi  xi sin   mL44   mL 3Q cos  Q44 cos 2   mT  mT  m2 L T  2 44  xi   2Q22  2 44  442  33  2 2   i   2 55  2 55   xi   2 2 x sin  x sin  x s in  x x sin  x sin  x sin  x i i i i i i sin    i     i

  mL  m 2 L55 T55 4Q44 cos  Q55 cos 2   mT  m2 L T 5Q cos  Q66 cos 2    2 2   i   2 66  2 66  xi   6Q44  2 66  662  55  2 2  i   4Q33  2 2  2  2 xi sin  xi xi sin  xi sin   xi sin  xi xi sin  xi sin     xi sin  xi sin    N N N N  T33 (1) T  m m (2)  cij(1) T44  B44  xj    44 cij(1)  T44cij(2)   j   cij  T33cij  v0 j   cij(1) T33  B33  u0 j    xi sin  j 1 x sin  x j 1  xi j  1 j  1 i   i  N T  m  B55   xj    55 cij(1)  T55cij(2)   j  xi sin  j 1 j 1  xi   2  I 3v0i  I 4  i  I 5 i  I 6 i  N

 c T (1) ij

55

N

 c T j 1

(1) ij

66

N T   B66  xj    66 cij(1)  T66cij(2)  j  j 1  xi 

M

m xi sin 

ED

The 8th equation of motion can be written

j 1 N

(1) ij

55

 c T j 1

N A  m  B55   j    66 cij(1)  A66cij(2)   xj  xi sin  j 1  xi 

(1) ij

77

N

 c T j 1

(1) ij

66

N A   B66   j    77 cij(1)  A77 cij(2)  xj  j 1  xi 

 B77  j   2  I 4u0i  I 5 xi  I 6 xi  I 7 xi 

AC

m xi sin 

N

 c T

CE

m xi sin 

PT

L  mL   mL L m 2T  mT  m 2T  mT  L cos    442  2 442  u0i   2 44  2 44  v0i  442 w0i   3K 33  552  2 552  xi   2 55  2 55   i  x x sin  x sin  x sin  x sin  x x sin  x sin  x i i i i i i sin    i   i     i 2 2   mL66   mL L66 m T66  mT  L mT  mT   2 66   i   9 K 55  772  2 772  xi   2 77  2 77  i   6 K 44  2  2 2   xi   2 x x sin  x sin  x sin  x x sin  x sin  x i i i i i i sin      i     i N N N N  A44 (1)   B cos   A  m cij  A44 cij(2)  u0 j  cij(1) T44  B44  v0 j   cij(1)  44  3K 33  w0 j    55 cij(1)  A55cij(2)  xj     x sin  x sin  x j 1  xi j  1 j  1 j  1 i   i   i 

The 9th equation of motion is  mL  m2 L  3mQ33 mL44 cos  mQ44 cos    mL55 3Q cos  Q44 cos 2   mT  mT  T   2 44  2 44  u0i    2 44  442  33  2 2  v0i    2 2  2 2  2 55  xi   w0i   2 2 x sin  x sin  x sin  x x sin  x sin  x sin  x sin  x sin  x sin  x i i i i i i i sin    i   i   i   i 2 2 2 2   mL  m L55 T55 4Q44 cos  Q55 cos   mT  m L T 5Q cos  Q66 cos    2 2   i   2 66  2 66   xi   6Q44  2 66  66  55  2 2   i   3Q33  2 2  2  xi sin  xi xi sin  xi sin   xi sin 2  xi2 xi sin  xi sin     xi sin  xi sin    N N  mL77  T  mT  m2 L T 6Q cos  Q77 cos 2   m  2 77  xi   9Q55  2 77  772  66  2 2  i  cij(1) T44  B44  u0 j    44 cij(1)  T44 cij(2)  v0 j   2  2 x sin  x sin  x sin  x x sin  x sin  x sin  x j  1 j  1 i i i i i i  i     i  N N N N  T55 (1)  T66 (1) m m (1) (2)  (1) (2)  cij  T55cij   j  cij  T66 cij   j     cij T55  B55  xj    cij T66  B66   xj   xi sin  j 1 xi sin  j 1 j 1  xi j 1  xi  

m xi sin 

N

 c T j 1

(1) ij

77

N T   B77  xj    77 cij(1)  T77 cij(2)  j   2  I 4v0i  I 5  i  I 6 i  I 7 i  j 1  xi 

30

ACCEPTED MANUSCRIPT where i  2,3,..., N 1 . For a clamped-clamped boundary condition, as assumed in this study, it is finally enforced the following relations

u0i  v0i  w0i   xi    i  xi   i  xi   i  0

i  1, N

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Graphical_Abstract

35