Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

Accepted Manuscript

Stability and free vibration analysis of double-bonded micro composite sandwich cylindrical shells conveying fluid flow M. Mohammadimehr , M. Mehrabi PII: DOI: Reference:

S0307-904X(17)30224-X 10.1016/j.apm.2017.03.054 APM 11695

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

26 December 2016 15 March 2017 22 March 2017

Please cite this article as: M. Mohammadimehr , M. Mehrabi , Stability and free vibration analysis of double-bonded micro composite sandwich cylindrical shells conveying fluid flow, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.03.054

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ACCEPTED MANUSCRIPT Highlights 

Development of stability and vibration of sandwich small-scale axisymmetric cylindrical shell model

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Presenting formulation based on the modified couple stress double-bonded Reddy shells theory Solving formulation using generalized differential quadrature method (GDQM)

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Study of the effects of hf / hc , u f , L / h and H z on the damping and natural

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frequency

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ACCEPTED MANUSCRIPT

Stability and free vibration analysis of double-bonded micro composite sandwich cylindrical shells conveying fluid flow M. Mohammadimehr1*, M. Mehrabi1 1*

Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran.

Abstract

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In this study, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow under magneto-thermo-mechanical loadings using modified couple stress theory are investigated. It is assumed that the cylindrical shells with foam core rested in an orthotropic elastic medium and the face sheets are made of composites with temperature-dependent material properties. Also, the Lorentz functions are applied to simulation of magnetic field in the

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thickness direction of each face sheets. Then, the governing equations of motions are obtained using Hamilton’s principle. Moreover, the generalized differential quadrature method is used to discretize the equations of motions and solve them. There are a good agreement between the obtained results from this method and the previous studies. Numerical results are presented to predict the effects of size-dependent length scale parameter, third order shear deformation theory, magnetic intensity,

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length-to-radius and thickness ratios, Knudsen number, orthotropic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of

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cylindrical shells. Also, it is demonstrated that the magnetic intensity, temperature changes and carbon nanotubes volume fraction have important effects on the behavior of micro composite sandwich cylindrical shells. So that, increasing the magnetic intensity, volume fraction and Winkler

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spring constant lead to increase the dimensionless natural frequency and stability of micro shells, while this parameter reduce by increasing the temperature changes. It is noted that sandwich

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structures conveying fluid flow are used as a sensors and actuators in smart devices and aerospace industries. Moreover, carotid arteries play an important role to high blood rate control that they have a

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similar structure with flow conveying cylindrical shells. In fact, the present study can be provided a valuable background for more research and further experimental investigation. Keywords:

Vibration and stability analyses; Modified couple stress theory; Reddy micro cylindrical shells; orthotropic foundation; temperature-dependent material properties.

*

Corresponding author : E-mail: [email protected], Tel: +98 31 55912423; Fax: +98 31 55912424

1. Introduction 2

ACCEPTED MANUSCRIPT Sandwich structures conveying fluid flow due to existence of coupling effect between stress and electric or magnetic displacement fields are used as a sensors and actuators in various applications such as defense and aerospace industries and smart devices. Also these materials due to their special mechanical and thermal properties have attracted much attention in last decades because they have high values of stiffness and strength to weight ratios [1-3]. The

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composites are one of the most examples of magneto electro elastic material, which is made of matrix and reinforcements including carbon nanotubes (CNTs) and boron nitride nanotubes (BNNTs). Carbon nanotubes are considered as ones of the most important reinforcement materials for high performance structural composites in nanotechnology such

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as bar, beam, plate and shell. They can be used at micro or nano scales for molecular reactors, micro-resonator, drug delivery and fluid transport. It is reported in some experimental studies that the structures become stiffer in smaller scales [4-7]. Moreover boron nitride nanotubes

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have the same atomic structure as CNTs while they have suitable resistance to oxidation and more stable electronic property at high temperatures. Recently, a large amount researches

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about the vibration of the micro/ nano structures with and without conveying fluid are

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published. A dynamic extended iso-geometric analysis (XIGA) for transient fracture of cracked magneto electro elastic (MEE) solids under coupled electro-magneto-mechanical

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loadings is developed by Bui et al. [8]. Electro-thermo-mechanical nonlinear vibration and instability of embedded BNNTs conveying viscose fluid is studied by Khodami Maraghi et

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al. [9] and Ghorbanpour Arani et al. [10-11]. The results of their study indicated that the elastic medium, small scale parameter, volume percent of BNNTs, electric potential and temperature change have significantly effect on the dimensionless natural frequency and critical fluid velocity. Ke and Wang [12] investigated the vibration and instability analysis of fluid-conveying double-walled carbon nanotubes (DWCNTs) based on modified couple stress theory (MCST). Their results showed that the imaginary component of the frequency

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ACCEPTED MANUSCRIPT and the critical flow velocity of the fluid-conveying DWCNTs increase with an increase in the length scale parameter. Ansari et al. [13] predicted size-dependent coupled longitudinaltransverse rotational free vibration behavior of post-buckled functionally graded (FG) micro and nano beams based on the most general Mindlin’s strain gradient theory. They are demonstrated that an increase in the material gradient index and aspect ratio gives lower

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critical axial buckling loads and this behavior is more pronounced for FG small-scale beams with clamped-clamped (C-C) end supports. The effects of magnetic field, length of magneto rheological elastomer (MRE) patch, core thickness, percentage of iron particles and carbon blacks are studied by Nayak et al. [14-15] on the dynamic analysis of a three-layered

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symmetric sandwich beam with MRE embedded viscoelastic core and conductive skins subjected to a periodic axial load. Banerjee et al. [16] considered an accurate dynamic stiffness model for a three-layered sandwich beam of unequal thicknesses and they predicted

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the free vibration behavior of system. Ni et al. [17] presented the free vibration problem of pipes conveying fluid with several typical boundary conditions using differential

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transformation method (DTM). Their results are compared with those predicted by the

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differential quadrature method (DQM) and demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of pipes conveying fluid flow.

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Ghavanloo et al. [18] studied the vibration and instability analyses of a CNT resting on a linear viscoelastic Winkler foundation based on the classical Euler–Bernoulli beam model

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(EBT) and using finite element method (FEM). They investigated the effects of the modulus and the damping factor of the linear viscoelastic Winkler foundation and the fluid velocity on the resonance frequencies of CNT. Mohammadimehr et al. [19] examined the bending, buckling and free vibration responses of micro composite plates reinforced by functionally graded single-walled carbon nanotubes (FG-SWCNT) based on modified strain gradient theory (MSGT). The obtained results from this study showed that the material length scale

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ACCEPTED MANUSCRIPT parameters lead to increase the stiffness of system. Also it is demonstrated that the effect of moisture on the micro composite plate is similar to thermal effect. Arefi et al. [20] studied two-dimensional thermo elastic analysis of a FG cylindrical pressure vessel subjected to axially variable thermal and mechanical loads. Their results indicate that the boundary conditions of the cylinder have significant effect on thermo elastic response of the vessel.

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Rouhi et al. [21] developed a size dependent continuum model in order to investigate the free vibrations of nano scale cylindrical shells based on the Gurtin-Murdoch elasticity theory and first order shear deformation theory (FSDT). Their results predicted that the surface stress effect is more pronounced for nano shells under softer end conditions. Moreover, the size-

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dependent behavior of nano shell was intensified and increased as the surface residual tension increases. Mohammadimehr et al. [22] discussed about the nonlocal biaxial buckling load and bending analysis of polymeric piezoelectric nano plate reinforced by CNTs with considering

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the effects of surface stress layer. The results of their study demonstrated that the effect of residual surface stress constant on the surface biaxial critical buckling load ratio is higher

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than the other surface stress parameters on it. Bahadori and Najafizadeh [23] studied the

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dynamic behavior of moderately thick FG cylindrical shell based on FSDT. They are concluded that the natural frequencies of a 2D-FG cylindrical shell on Winkler–Pasternak

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elastic foundation enhance faster when the power law index in z direction increases. Also they are presented that the natural frequencies of 2D-FG cylindrical shells are higher than

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1D-FG cylindrical shells. Tadi Beni et al. [24] predicted size-dependent equations of motion for FG cylindrical shell using shear deformation model and rotation inertia. The obtained results through the MCST are indicative of the considerable effect of the size parameter, particularly in bigger thicknesses and shorter lengths of nanotubes on the natural frequency. Murmu et al. [25] reported an analytical approach to study the effect of a longitudinal magnetic field on the transverse vibration of a magnetically sensitive DWCNT based on

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ACCEPTED MANUSCRIPT nonlocal elasticity theory. The numerical results from this model showed that the longitudinal magnetic field increases the natural frequencies of the DWCNT. Wang et al. [26] established a unified vibration analysis approach for the FG moderately thick doubly-curved shells and panels of revolution with general boundary conditions based on FSDT and using Ritz method. Kolahchi et al. [27] developed the effects of magnetic field, volume fractions and

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distribution type of nanotubes, aspect ratio and temperature changes on the nonlinear buckling of the polymeric temperature-dependent micro plate reinforced by SWCNTs resting on an elastic matrix as orthotropic temperature-dependent elastomeric medium based on Eringen’s nonlocal theory. Their results indicated that the critical buckling load increases

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with increasing magnetic field. Nashihatgozar et al. [28] studied the effects of volume fraction of nanotube, geometrical characteristics as well as two loading types of axial and biaxial on the buckling analysis of piezoelectric cylindrical composite panels reinforced with

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CNTs. Their results predicted that with increasing of the volume fraction of nanotube eventuates, the buckling load enhances. Snap-through buckling of shallow clamped spherical

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shells made of FGM and surface-bonded piezoelectric actuators under the thermo-electro-

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mechanical loadings is considered by Sabzikar Boroujerdy and Eslami [29]. They are demonstrated that the intensity of buckling is dependent on the geometry of the shell, value of

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thermo-electro-mechanical loadings and type of thermal loadings. Sofiyev [30] investigated the vibration and buckling of sandwich cylindrical shells covered by different coatings

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subjected to the hydrostatic pressure. He is discussed about the influences of compositional profiles of coatings, shear stresses and sandwich shell characteristics on the non-dimensional frequencies and critical hydrostatic pressures for FG sandwich cylindrical shells. In this article, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of symmetric thick-walled double-bonded micro composite sandwich cylindrical shells reinforced by CNTs conveying fluid flow under magneto-thermo-

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ACCEPTED MANUSCRIPT mechanical loadings embedded in an orthotropic elastic medium are presented. Moreover, the MCST and classical theory (CT) are applied to obtain the governing equations of motion using Hamilton’s principle and energy method. The generalized differential quadrature method (GDQM) is used to discretize the equations of motions and solve them. Numerical results are considered to investigate the effects of the length-to-thickness ratio, the thickness

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ratio (face sheets to core thickness), Knudsen number, orthotropic elastic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of double-bonded micro composite sandwich cylindrical shells conveying fluid flow.

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It can be say that in the previous studies did not use the third order shear deformation theory (Reddy) for sandwich micro cylindrical shells conveying fluid flow. Also, this system is simulated by orthotropic elastic medium and the material properties are considered as

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2. Geometry and Simulation

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temperature dependent that they are the other novelty of this work.

A three layered symmetric double-bonded sandwich cylindrical shells conveying

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incompressible, irrotational, inviscid and isentropic fluid flow under magneto-thermo-

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mechanical loadings with length L , radius R and thickness h rested in an orthotropic elastic foundation with Winkler spring constant k w and Pasternak shear modulus k g  and k g in 

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and  local direction is shown in Fig. 1. In this work, the cores of structure are made of foam material (Divinycell-H60) with thickness hc that their properties are written in Table. 1. Also the bottom and top face sheets with thickness hb and ht are made of PmPV (Poly {(mphenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene)vinylene]}) as matrix and

carbon

nanotubes (CNTs) as reinforcements that the mixture rule has been handled for estimate of

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ACCEPTED MANUSCRIPT equal properties. The properties of these materials including PmPV and SWCNTs are considered temperature-dependent and they are shown in Fig. 2 and Fig. 3. Based on mixture method the equal mechanical coefficients of micro-composite materials reinforced by nanotubes are written as follows [19]: E11  1V CNT E11CNT V m E m

E 22

3 G12



V CNT V m  CNT E 22 Em



V CNT V m  G12CNT G m

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2

(1a)

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 V CNT CNT V m m

12 V CNT 12CNT V mm

11 V CNT 11CNT V m  m

(1b)

(1c) (1d) (1e) (1f)

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where 1 , 2 , 3 are the force transformation between SWCNTs and PmPV matrix that

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they are defined in Table. 2. Also, V m and V CNT are the volume fractions of matrix and CNTs, respectively, that there is the following formulation between them [13,24]: (2)

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V m V CNT  1

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It is noted that the carbon nanotubes volume fraction (V CNT ) for uniform distribution of CNTs can be written as follows [19]:

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* V CNT (z ) V CNT

(3)

where:

* V CNT 

w CNT  (

w CNT

CNT )(1 w CNT ) m

(4)

where w CNT ,  m and CNT are CNTs mass fraction, matrix density and carbon nanotubes density, respectively.

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ACCEPTED MANUSCRIPT 3. The governing equations of motion In this article, the energy method is used to obtain the governing equations of motions. According to this method, the actual motion minimizes the difference of the kinetic and total strain energy for a system with prescribed configurations at t  0 and t  t 1 that is [26,33]:  (U V )

(5a)

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 T t1

   dt  0   (T U V )  0 0

(5b)

where T, U and V are the total kinetic energy, strain energy, and work done by external

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forces such as orthotropic elastic medium, magnetic field, and temperature changes, respectively.

After defining the Hamilton’s principle, to obtain the governing equations of motions, it is necessary to determine the displacement field equations. According to the third order shear

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deformation theory (TSDT), these equations for the equivalent single layer (ESL) sandwich

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double-bonded micro cylindrical composites shells are written as follows [34]:

(6)

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 (i ) w 0(i ) (x ,  , t ) 4z 3 ( i ) (i ) (i ) ) u  x ,  , z , t   u 0  x ,  , t   z x  x ,  , t    2 (x  x ,  , t   3h x   4z 3 ( i ) 1 w 0(i ) (x ,  , t )  (i ) (i ) (i ) v x ,  , z , t  v x ,  , t  z  x ,  , t   (  x ,  , t  )          0   3h2 R   w (i ) x ,  , z , t  w (i ) x ,  , t   0     

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where u 0(i ) , v 0(i ) , w 0(i ) , x(i ) and (i ) refer to the displacement components and the angle of

rotation of the cross-section of any point of the middle surface of the micro cylindrical shell in the axial, circumferential and radial directions, respectively. Moreover, i  1,2 represent the lower and upper micro composite sandwich cylindrical shells and  b ,c ,t introduce the bottom face sheet, core and top face sheet of each micro cylindrical shell, respectively.

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ACCEPTED MANUSCRIPT It is noted that, if   0 , the displacement equations reduce to the FSDT. Also, because of the micro cylindrical shells are considered symmetric, thus, it can be say that  /  are equal to zero. 3.1.Kinetic energy

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The virtual kinetic energy for TSDT (Reddy) double-bonded sandwich micro composite cylindrical shells conveying isentropic, incompressible, irrotational and inviscid fluid flow can be expressed as follows [35]:

T   [  (i ) (u ,(ti )  u ,(ti )  v ,(ti ) v ,(ti ) w ,(ti )w ,(ti ) )  f (w ,(ti )w ,(ti )  u f w ,(xi )w ,(xi ) )] R dx d  dz

(7)

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

where,  (i ) ,  f and u f are the density of upper and lower micro cylindrical shells, fluid density and no-slip velocity of fluid flow, respectively.

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Also, it is noted that the mechanical behavior of flow depends on the mass density and viscosity of fluid flow. Knudsen number (Kn) is the one of the dimensionless parameters that

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it is defined as ratio of the mean free path of the fluid molecules to a characteristic length of

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the flow geometry [36]. Based on Knudsen number, four flow regimes are defined as continuum flow ( 0  Kn  0.01 ), slip flow ( 0.01 Kn  0.1 ), transition flow ( 0.1 Kn 10 )

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and free molecular ( Kn 10 ) regimes that for the micro/ nanotubes conveying fluid flow, is

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considered the slip flow regime. Thus the no-slip boundary condition ( u f ) is incorrect and should be used the modified flow velocity ( u m ) where the effect of flow regime ( 0.01  Kn  0.1 ) are considered for it. It this means that [37]: u m VCF u f

where u m

(8) and VCF are the modified fluid velocity and average velocity correction factor,

respectively. This factor can be written as follows [37]:

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VCF  (1   Kn ) (1  4(

2  v

v

)(

Kn )) Kn  1

(9)

In the Eq. (9), v is tangential momentum coefficient and it is assumed be 0.7. Also, the  coefficient is expressed as follows [11]: 2

   0 ( ) arctan(1Kn  ) 

(10)

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0 

4 3 (1  ) b

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In which   0.4 , 1  4 and:

; b  1

(11)

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By combining the Eqs. (8-11) and substituting the modified fluid velocity ( u m ) and Eq. (6) into the Eq. (7), the first variation of kinetic energy of double-bonded micro composite sandwich cylindrical shells conveying fluid flow reinforced by CNTs are defined as follows:

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T 

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where:

(12)

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 ( i )   (i )  (i )  (i ) (i ) (i ) (i ) (i ) (i )  I 0 u 0,tt   I 3 w 0 ,xtt  (I 1   I 3 ) x ,tt   u 0   I 0 v 0,tt  (I 1   I 3 )  ,tt  v 0      (i )  (i ) (i ) 2 (i ) (i ) (i )    I u  ( I  I ) w   I w  2 I u w  I u w   ( I  I )   w   0 f 0,tt 5 0 , xxtt f m 0 , xt f m 0, xx 4 5 x , xtt 0    3 0 ,xtt   R dx d    A   (i )   (i ) (i ) (i ) (i ) (i ) (i ) (i )  ( I   I ) u   ( I  I ) w  ( I  2  I   I )   3 0,tt 4 5 0 , xtt 2 4 5 x ,tt  x   1       (i )   (i ) (i ) (i ) (i ) (i )    (I 1   I 3 )v 0,tt  (I 2  2 I 4   I 5 )  ,tt   

(13a)

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hc ht  hb  (I 0(i ) , I 1(i ) , I 2(i ) )    2hb b( i ) (1, z , z 2 )dz   2hc c( i ) (1, z , z 2 )dz   2ht t( i ) (1, z , z 2 )dz    2 2  2 

hc  h2b (i ) 4z 3 4z 3 4z 3 4z 3  (i ) 2   hb b (1, z , 2 ) 2 dz   hc c (1, z , 2 ) 2 dz  3hb 3hb 3hc 3hc 2  (I 3(i ) , I 4( i ) , I 5( i ) )   2h 3 3  t  4z 4z    2ht t(i ) (1, z , 2 ) 2 dz  3ht 3ht   2 

(13b)

(13c)

I f  2 f dz R

0

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ACCEPTED MANUSCRIPT 3.2. Total strain energy Based on the MCST that it is introduced first by Yang et al. [38] the strain energy of each system is a function of strain and gradient of rotation tensors. According to this theory, the first variation of strain energy can be defined as follows [11,24]:



ij

 m ij ij  d 

where  ij , m ij ,

(14)

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U    ij 

and  ij represent the Cauchy stress, higher order stress, strain and

ij

symmetric rotation gradient tensors, respectively that they are defined in Eqs. (15-20). strain tensor [23-24]

 x  u ,x ,  

w ,  z  w ,z R

u 1 w v 1 1  ( ,  v ,z ) ,  xz  (w ,x  u ,z ) ,  x   (v ,x  , ) 2 R 2 2 R

(15)

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 z

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

By substituting the Eq. (6) into the Eq. (15), the relation between displacement and strain

w0 R

(16b)

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( ) 

(16a)

4z 3 4z )    w 0,xx x ,x 3h2 3h2

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 xx( )  u 0,x  z (1   z

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tensor can be written as follows:

1 2 

V0  4z )     h2 R 

(16c)

1 2 

 4z )(x  w 0,x )  2 h 

(16d)

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 z( )  (1   z

 xz( )  (1   z 1 2 

 x( )  v 0,x  z (1   z

 4z )    , x 3h2 

(16e)

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symmetric rotation gradient tensors [24]

1 2

ij  (e ipq  qj ,x  e jpq  qi ,x ) p

(17)

p

where e ipq is the alternate tensor that it is equal to the Eq. (18). (18a)

e z  x  e xz   e xz   1

(18b)

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e x  z  e zx  e zx  1

Also, by substituting the Eqs. (16,18) into the Eq. (17), the components of symmetric rotation gradient tensor are described as follows:

( ) 

4z ) ,x h2

v 0,x 2R 1 2 

v  4z ) ,x  0,x  2 h R 

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 z(z )  (1   z

(19a)

(19b)

(19c)

 1 1 4z 4z   2 )w 0,x   2 x  2  R h h 

(19d)

1 4 

(19e)

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 z( )  (

1 4 

4z 4z  ) ,xx  2 2   2 3h h 

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 x(z )  v 0,xx  z (1   z

 4z 4z )(w 0,xx )  (1   z 2 )x ,x  2 h h 

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 x( )   (1   z

(19f)

Cauchy and higher stress tensor [39,40]

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

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1 2

 xx( )   (1   z

0 0 0   x   x  Q11 Q12      Q12 Q 22 0 0 0            0 Q 44 0 0   z    z     0     0 0 0 Q 55 0   xz   xz       0 0 0 0 Q    66   x    x   

(20)

m ij  2G (z )l 22 ij

(21)

13

ACCEPTED MANUSCRIPT In Eqs. (20,21), Q ij and G are the elastic and shear modulus constants, respectively. It is noted that, the elastic coefficient for the composite materials are calculated based on Eq. (22). Q11 

E 11

1  1221

Q 44  G 23

, Q12 

21E 11 E 22 , Q 22  1  1221 1  1221

, Q 55  G13

(22)

Q 66  G12

,

variations of strain potential energy can be written as follows: U 

CR IP T

Thus, after defining the above equations and substituting them into the Eq. (14), the first

(23)

ED

M

AN US

    Q(1) 1 (i )  m xz ,xx  v 0(i )   N x , x   u 0   N x  , x   R 2      N       Q x(1),x   (Q x(2),x  J x ,xx )     R  (i )  w 0   1  4z 1 4z     (1   z 2 )m x  ,xx  (   2 )m z  ,x   h R h    2    M  Q (1)   (Q (2)  J )  R dx d   x x ,x A   x ,x x   (i )   1   4z 4z x     (1   z 2 )m x  ,x   2 m z    h h    2     1 4 z   M x  ,x  Q(1)   (Q(2)  J x  ,x )  (1   z 2 )(m xx ,x  m zz ,x )   2 h   ( i )      1      z (1   z 4z2 )m xz ,xx   4z2 m xz   3h h   2  

PT

where: hb

(i ) (N x(i ) , M x(i ) , J x(i ) )   2hb  xb (1, z ,

hb 2 h  b 2

N  

(N

(i ) x

,M

(i ) x

hc 2 h  c 2

ht 2 h  t 2

,J

(i ) x

hb 2 h  b 2

) 

(i ) c

h

(i ) x b

hb



2

R1

(i ) (Q(1)(i ) ,Q(2) ( i ) )   K s  z  b (1, z R

h

c t 4z 3 4z 3 4z 3 (1, z , 2 ) dz   2hc  x(i)c (1, z , 2 ) dz   2ht  x(i)t (1, z , 2 ) dz   3hb 3hc 3ht 2 2

(i ) (Q x(1)(i ) ,Q x(2)(i ) )   2hb K s xzb (1, z

(24a) (24b)

 dz    dz    (ti )dz (i ) b

AC

(i )

2

CE



h

t hc 4z 3 4z 3 4z 3 (i ) (i ) 2 2 ) dz   (1, z , ) dz   (1, z , ) dz  h2c xc  h2t xt 3hb2 3hc2 3ht2

h

h

c t 4z 4z 4z (i ) (i ) 2 2 ) dz  K  (1, z ) dz  ) dz hc ht K s xzt (1, z s xzc 2 2     hb hc ht2 2 2

R2 R3 4z 4z 4z ) dz   K s z(i)c (1, z 2 ) dz   K s z(i)t (1, z 2 ) dz 2 R R 1 2 hb hc ht

14

(24c)

(24d)

(24e)

ACCEPTED MANUSCRIPT

where K s is the shear correction factor and in this study, it is considered equal to 1 for third order displacement shear deformation theory. 3.3.Virtual work done by external work In this article, it is assumed that the magneto-thermo-mechanical loadings are applied on

CR IP T

the TSDT (Reddy) double-bonded micro composite sandwich cylindrical shells conveying fluid flow and reinforced by CNTs. Also, the micro structure is rested in an orthotropic elastic medium with Winkler spring constant K w and Pasternak shear modulus K g  and K g in local  and  directions with angle  . Thus, it can be say that: V

Orthotropic

where V TH , V

FL

Orthotropic

andV

FL

(25)

AN US

V V TH V

are the external works done by temperature changes,

orthotropic foundation and magnetic field that they are defined as follows [11,22,40]:

Orthotropic

V

FL





1 F Orthotropicw dA  A 2

ED

V

M

1 w 2 NT( ) dA  A 2 x

(26a)

(26b)

(26c)

1 ( f xl u  f  lv  f zlw ) dA 2 A

PT

V TH 

CE

where:

1) based on the thermo-elasticity theory [11,41]: hb 2 h  b 2



AC N

T

Q  (i ) 11b

(i ) 11b

ht 2 h  t 2

T dz  

Q11(it)11( it) T dz

(27)

In the Eq. (27), Q11(i ) , 11(i ) and T are the mechanical constant, thermal expansion

coefficient and temperature changes, respectively. It is noted that, in this work, T are considered equal to T T 0 that T 0  300 K . 2) F Orthotropic is the orthotropic elastic medium force that it is defined as follows [42]: 15

ACCEPTED MANUSCRIPT F Orthotropic  F1Orthotropic  F2Orthotropic

(28a)

F1Orthotropic  K w w 1  K w (w 1 w 2 )  K g  (w 2,xx w 1,xx ) cos2   K g (w 2,xx w 1,xx )sin 2 

(28b)

F2Orthotropic  K w w 2  K w (w 2 w 1 )  K g  (w 1,xx w 2,xx ) cos2   K g (w 1,xx w 2,xx )sin 2 

(28c)

In the Eq. (28), KW , K G  , K G and  correspond Winkler foundation parameter, shear

CR IP T

foundation parameters in  and  directions and the local  direction of orthotropic foundation with respect to the global x-axis of the micro cylindrical shells, respectively.

3) The external work done by magnetic field based on electro-dynamic Maxwell relation and Lorentz function can be expressed as follows [22]:

AN US

H  (0,0, H z ) h   (U  H )

f l  (f xl , f  l ,0)  ( j  H )

M

j   h

(29a) (29b) (29c) (29d)

ED

where H , h , U , j , f l and  are the magnetic intensity, perturbation of magnetic field,

respectively.

PT

displacement, electric current density vectors, Lorentz forces and magnetic permeability,

CE

Finally, by combining the Eqs. (26-29) and substituting them into the Eq. (25) and applying the variation principle, the first variation of work done by external works are

AC

described as follows: V 

   (i ) (i )  (i ) (i ) (i ) (i ) (i ) (i ) (i ) (i ) (i ) (i ) Y 0 u 0,xx   Y 3 w 0, xxx  2  Y 6 w 0, x  (Y 1   Y 3 )  x , xx  2  Y 6  x   u 0    (i ) (i ) (i ) (i ) (i ) (i ) (i ) (i ) (i ) T (i ) (i ) (i ) (i )   Y u  ( Y  Y )   2 Y   Y w ( N  2 Y ) w  w   R d x d   3 0, xxx 4 5 x , xxx 8 x , x 5 0, xxxx 8 0, xx 0   A  (i ) (i ) (i ) (i ) (i ) (i ) (i ) (i ) (i )   (Y 1   Y 3 ) u 0,xx   (Y 5 Y 4 )w 0, xxx  2  (Y 8 Y 7 )w 0, x  (i )   x     (i ) (i ) (i ) (i ) (i ) (i ) (i )    (Y 2   (Y 5  2Y 4 )) x ,xx  2  (Y 8 Y 7 )  x  

16

(30)

ACCEPTED MANUSCRIPT

 1 (2) 1 (2) 1 (2)   (1) 2 (1) 2  K w (2w 0  2 w 0 )  K g  cos ()( 2 w 0,xx w 0, xx )  K g sin ()( 2 w 0, xx     w 0(1)   w (1) )  1 (K w (2)  K cos 2 ()w (2)  K sin 2 ()w (2) )    w 0 g 0, xx g 0, xx  0,xx  2     R dx d  1 (1) 1 (1) 1 (1)  (2) 2 ( 2) 2 A    K (2 w  w )  K cos (  )( w  w )  K sin (  )( w 0 0 g 0, xx 0, xx g 0, xx    w  2 2 2  w 0(2)       w 0,(2)xx )  1 (K w w 0(1)  K g  cos 2 ( )w 0,(1)xx  K g sin 2 ()w 0,(1)xx )   2    

as follows: hb

hc

(Y 0(i ) ,Y 1(i ) ,Y 2(i ) )   2hb  H z2 (1, z , z 2 )dz   2hc  H z2 (1, z , z 2 )dz

(Y

(i ) 3

,Y

(i ) 4

,Y

(i ) 5



2

hb 2 h  b 2

)

h

t 4z 3 4z 3 4z 3 4z 3  H (1, z , 2 ) 2 dz   2ht  H z2 (1, z , 2 ) 2 dz  3hb 3hb 3ht 3ht 2

2 z

hb

(Y 6(i ) ,Y 7(i ) ,Y 8(i ) )   2hb  H z2 (1, z , 2

h

t 4 z 3 4z 4 z 3 4z 2 2 ) dz   H (1, z , ) dz  h2t z 3hb2 hb2 3ht2 ht2

(31a)

(31b)

(31c)

M



2

AN US



CR IP T

whereY j(i ) (i  1, 2) and j  0,1,...,8 are the magnetic constants and they are considered

ED

3.4.The governing equations of motion

PT

According to the Hamilton’s principle, the governing equations of TSDT (Reddy) flowconveying double-bonded micro sandwich composite cylindrical shells with foam core and

CE

composites layer reinforced by CNTs are dependent to strain potential, kinetic energy of fluid and micro cylindrical shells, magnetic field, orthotropic elastic medium and

AC

temperature changes that they are defined in previous sections. Thus, by substituting the Eqs. (12, 23, 30) into the Eq. (5-b), the governing equations of motion are obtained as follows:

 u 0( i ) : Y 1 ( i )  B 11( i )  (i )  A12( i ) (i ) (i ) (i ) 0( i ) (i ) (i ) (i )  (i ) Y  A u   F  Y w   2  Y w    0  11  0,xxx  R (i )  0,x  11  0, xx 3 4 0( i ) ( i )  x , xx   ( F  Y )   11 3   (i )

(i )

(i )

  2  Y 4( i )  x( i )  I 0( i ) u 0,tt   I 3( i ) w 0, xtt   I 1( i )   I 3( i )   x ,tt  0

17

(32a)

ACCEPTED MANUSCRIPT v 0( i ) :  M (i )  M 0( i ) ( i ) 1 ) v 0, xxxx   02  A 66( i ) v 0,( ixx)   M 1( i )   M 3( i )  ( i, xxxx  2R  8 8  (i ) 

(32b)

(i ) (i ) 1 0( i ) (i ) (i ) (i ) (i ) (i )  F  A   I v  I   I  0 0, tt  44 1 44   0 3   ,tt R (i )

CR IP T



 M 0( i ) (i ) 0( i )   4R ( i )  B 66   ( F66    (i )   M 7( i ) M 6( i )   ,xx   )   (i ) 4  4R 

w 0(1) :

M (1) 3M 9(1) A12(1) (1)  M 0(1)  u 0,x    ( 7   F112(1) Y 5(1) ) w 0,(1)xxxx R 4 8  8  (1)  (1) M 0 T 2 2 2   A 55  2 R 2  N (1)  K g  cos ( )  K g sin ( )  I f u m   (1)  w 0,(1)xx   A 222  2 K w w 0(1)  (1) (1) 0(1)   M M 2F12  R   2 F550(1)  F552(1)  8  6  2Y 6(1) )    (   R 2 R  (1) 3 M 9(1)  M0     (  F111(1)  (1) 2 2 ( 2) ( 2)    K g  cos ( )  K g sin ( ) w 0, xx  K w w 0   8 8   x ,xxx   F 2(1) Y (1) Y (1) )  11 7 5  

AN US

 Y 3(1)  F110(1) u 0,(1)xxx 

(1)

(1)

(1)

M

 M (1) M (1) B (1) F 0(1)   A 55(1)  12   ( 12  F552(1)  2 F550(1)  8  6  2Y 6(1) R R 2 2R 

(32c)

 )  x(1),x 

(1)

(1)

ED

  I 3(1) u 0, xtt   I 5(1) w 0,xxtt  2 I f u m w 0,xt   I 0(1)  I f w 0,tt   (I 4(1)  I 5(1) )  x , xtt  0

PT

x( i ) :

3M 9( i )  M 0( i )   (  F112( i )  ( i )  w 8 Y 1  B   (F Y ) u   8  0, xxx   F 1( i ) Y ( i ) Y ( i ) )  7 5  11  ( 2) (i ) (i ) 0( i ) M M F  B     12  A 55( i )   ( 12  2 F550( i )  F552( i )  8  6  2Y 8( 2)  2Y 6( 2) ) w 0,( ix) R 2 2R  R  (i ) (i ) (i ) M 3M 9 M7     D1(1i )  0 Y 2( i )   (    2F111( i )  F112( i )  2Y 7( i ) Y 5( i ) )   x( i, xx) 8 8 4   (i ) 11

0( i ) 11

(i ) 3

(i ) 0, xx

AC

CE

(i )

M (i ) F 0( 2)     A 55( i )   ( 12  F552( i )  2 F550( i )  8  2Y 8( i )  2Y 6( i ) )  x( i ) R 2   (i )

(i )

(i )

  I 1( i )   I 3( i )  u 0,tt    I 5( i )  I 4( i )  w 0, xtt  (I 2( i )   (2 I 4( i )  I 5( i ) ))  x ,tt  0

18

(32d)

ACCEPTED MANUSCRIPT w 0( 2) : A12( 2) ( 2) u 0, x   K g  cos 2 ( )  K g sin 2 ( ) w 0,(1)xx  K w w 0(1) R  ( 2) M 0( 2)  T 2   A 55  2 R 2  N ( 2)  K g  cos ( )  ( 2) ( 2) ( 2)   M 3M 9  M0   ( 2)  ( 7  2F120( 2) ( 2) 2 2    8 w 0, xxxx    K g sin ( )  I f u m   ( 4 8 w 0, xx   R   F 2( 2) Y ( 2) )    5  11   M 8( 2) M 6( 2) 0( 2) 2( 2) ( 2)   2 F55  F55  2  R  2Y 6 )    ( 2) ( 2) 3M 9  M   F111( 2)  ( 2)  A 22( 2)  ( 2)   0   (   2  2 K w w 0   8 8   x , xxx  R    F 2( 2) Y ( 2) Y ( 2) )  11 7 5  

 Y 3( 2)  F110( 2) u 0,( 2)xxx 

( 2)

( 2)

( 2)

( 2)

CR IP T

 M ( 2) M ( 2) B ( 2) F 0( 2)   A 55( 2)  12   ( 12  F552( 2)  2 F550( 2)  8  6  2Y 6( 2) R R 2 2R 

(32e)

 )   x( 2), x 

( 2)

AN US

  I 3( 2) u 0, xtt   I 5( 2) w 0,xxtt  2 I f u m w 0,xt   I 0( 2)  I f w 0,tt   (I 4( 2)  I 5( 2) )  x , xtt  0

( i ) :

ED

M

(i ) (i ) M 6( i )  ( i ) 1  M 2   (M 5  ( i ) 1  (i ) (i ) (i ) (i ) 0( i ) M   M v   B   ( F  )  v   1   , xxxx 3  0, xxxx 66  66  0, xx 8  (i ) 8 4  2 M )    4  (i ) M  (i )  A 44   ( 8  ( i ) 3M 7( i )  M 0( i ) (i ) 1( i ) 2( i ) (i )  (i )    D 66   ( 2F66  F66   2M 9 )   , xx  2    2( i ) 2 2 0( i )     F44  2F44 )  (i )

(32f)

(i )

  I 1( i )   I 3( i ) v 0,tt  (I 2( i )   (I 5( i )  2I 4( i ) ) )   ,tt  0

PT

where A ij , B ij , D ij , FijK and M j are the mechanical constant in the z-direction and they

CE

are written as follows: hb

hc

(A ij( i ) , B ij( i ) , D ij( i ) )   2hb Q ijb( i ) (1, z , z 2 ) dz   2hc Q ijc( i ) (1, z , z 2 )dz

AC



hb



2

ht 2 h  t 2

  Q (1, z , z )dz (i ) ijt

hc

2

2

(33a)

(i , j  1, 2, 6) ht

Aij( i )   2hb K sQ ijb( i ) dz   2hc K s Q ijc( i )dz   2ht K s Q ijt( i )dz 

2



2



2

19

(i , j  4,5)

(33b)

ACCEPTED MANUSCRIPT hb

(Fij0( i ) , Fij1( i ) , Fij2( i ) )   2hb Q ijb( i ) (1, z , 2

ht 2 h  t 2



0( i ) ij

(F

1( i ) ij

,F

2( i ) ij

,F

4z 3 4z 3 Q ijt( i ) (1, z , 2 ) 2 dz 3ht 3ht hb 2 h  b 2

hc 4z 2 4z 2 4z 2 4z 2 K s Q (1, z , 2 ) 2 dz   2hc K s Q ijc( i ) (1, z , 2 ) 2 dz  hb hb hc hc 2

ht 2 h  t 2

4z 2 4z 2 K s Q ijt( i ) (1, z , 2 ) 2 dz ht ht

) 

(33c)

(i , j  1, 2, 6)

(i ) ijb

hb

(33d)

(i , j  4,5)

hc

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

hc 4z 3 4z 3 4z 3 4z 3 (i ) 2 ) dz  Q (1, z , ) dz h ijc c  2 3hb2 3hb2 3hc2 3hc2

(M 0( i ) , M 1( i ) , M 2( i ) )   2hb 2G b l 22 (1, z , z 2 ) dz   2hc 2G c l 22 (1, z , z 2 )dz 



ht 2 h  t 2

2G t l 22 (1, z ,

4z 3 4z 3 ) dz 3ht2 3ht2

2

hc 2 h  c 2

,M

,M

hb 2 h  b 2

)

2G b l 22 (1, z ,

4z 4z 3 4z , ) dz hb2 3hb2 hb2

ht 4z 4z 3 4z 4z 4z 3 4z 2G c l (1, z , 2 , 2 ) 2 dz   2ht 2G t l 22 (1, z , 2 , 2 ) 2 dz  hc 3hc hc ht 3ht ht 2 2 2

(33e)

(33f)

(33g)

ED



,M

(i ) 9

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hc 4z 3 4z 3 4z 3 4z 3 2 2 ) dz  2 G l (1, z , ) dz h c 2  2c 3hb2 3hb2 3hc2 3hc2

M

(M

(i ) 8

2G t l 22 (1, z , z 2 )dz

2G b l 22 (1, z ,



(i ) 7

2

hb 2 h  b

(M 3( i ) , M 4( i ) , M 5( i ) )  

(i ) 6



2

ht 2 h  t 2

PT

4. Numerical Solution method

In the present study, GDQM is used to solve the governing equations of TSDT double-

CE

bonded sandwich micro composite cylindrical shells conveying fluid flow based on MCST.

AC

At the first, according to the separation variable method the solution of motion equations can be considered as follows [11]: u 0( i ) (x ,  , t )  u 0( i ) (x ,  ) e  t  (i ) (i ) t v 0 (x ,  , t )  v 0 (x ,  ) e  (i ) (i ) t w 0 (x ,  , t )  w 0 (x ,  ) e  (i ) (i ) t x (x ,  , t )   x (x ,  ) e  ( i ) (x ,  , t )   ( i ) (x ,  ) e  t   

(34)

Moreover, the Chebyshev-Gauss-Lobatto grid points are employed to perform this method. Thus, the generate grid points in x-direction can be written as follows [33]: 20

ACCEPTED MANUSCRIPT

  a 

b a  1 (1  cos( ) ) 2 N 1

  1, 2,..., N

;

(35)

where N, a and b are the grid point and two end of the each micro cylindrical shells along the x-direction that a and b are equal to zero and L, respectively. Also, based on the GDQ method, the displacement functions are approximated as the

k u (i ) ,v 0(i ) ,w 0(i ) , x(i ) , (i ) |  k  0 x x j x N

C m 1

(k ) jm

(x ) u (x m , t ),v (i ) 0m

(i ) 0m

(x m , t ),w

(i ) 0m

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following form [11]:

(x m , t ),  (x m , t ),  ( x m , t ) (i ) xm

(36)

(i ) m

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(k ) where C jm denotes the weighting coefficients corresponding to the first and higher order

derivatives can be described as follows [33]:

,  0

  1  , i , j  1, 2,..., N i  j 

CE

PT

ED

M

I x  N   (x i  x j )  j 1; j  i  N  (x i  x j )  (x j  x i )  i 1;i  j  ( ) C x    ij  C (  1)   C ij(1)C ii(  1)  ij    x i  x j     N   C ij(  )  j 1; j i 

  2,3,..., N  1  , i , j  1, 2,..., N i  j 

(37)

  1, 2,..., N  1  , i , j  1, 2,..., N i  j 

AC

where I x is N  N identity matrix. Also, the GDQ form of mechanical clamped– clamped

boundary conditions can be assumed as follows: (1) (1) (1) u 01 v 01 w 01  x(1)1  (1)1  0 ;  (1) (1) (1) (1) (1) u 0 N v 0 N w 0 N  xN   N  0 ;  (2) (2) (2) (2) (2) u 01 v 01 w 01  x 1   1  0 ; u (2) v (2) w (2)   (2)   (2)  0 ; 0N 0N xN N  0N

x a x b

(38)

x a x b

21

ACCEPTED MANUSCRIPT By applying the first and higher order derivatives and boundary conditions the discretized form of governing equations of motions using the generalized differential quadrature technique are obtained as follows:   M d  X d ,tt  C d  X d ,t   K d  X d  0  T (i ) (i ) (i ) (i ) X d  u 01 , ...,u 0(iN) ,v 01 , ...,v 0(iN) ,w 01 , ...,w 0(iN) ,..., x(i1) ,..., xN ,..., (i1) ,..., (iN)  , i  1, 2 

(39)

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where, M d , C d , K d and X d denote the domain mass, damping and stiffness matrices and displacement vector, respectively.

By solving the Eq.(39) and reducing it to the standard form of eigenvalue problems

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(Eq.(40)), the complex frequencies (    (m  ni ) ) are obtained that these frequencies contain imaginary ( n ) and real ( m ) parts which are corresponding to the natural frequencies and damping of TSDT flow-conveying double-bonded sandwich micro composite cylindrical

 0

   M

1

I x  K    M

  C  

(40)

PT

5. Validation

1

ED



Q   

M

shells reinforced by CNTs, respectively.

In this section, the governing equations of motions and GDQ MATLAB program are

CE

compared with the other previous researches. In order to, the Navier’s solution types are used for

AC

the simply-supported boundary condition and the obtained results from this method are compared with Tadi Beni et al., [24], Alibeigloo and Shaban, [43], Soldatos and Hadjigeorgiou, [44] studies. Based on the Navier’s type, the following expansion of the displacement field with considering the  /  (the governing equations with considering  /  for   0 are written in Appendix A) is defined as follows:

22

ACCEPTED MANUSCRIPT     m  i t u 0   U mn cos  L x  cos  n  e   n 1 m 1     m    x  sin  n  e i t v 0   V mn sin  L   n  1 m 1       m  x  cos  n  e i t w 0   W mn sin  L   n  1 m 1      m   x    xmn cos  x  cos  n  e i t L    n 1 m 1    m       mn sin  x  sin  n  e i t   L  n 1 m 1

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(41)

where U mn , V mn , W mn , xmn and  mn are the undetermined Fourier coefficients, respectively. Also,

 is the dimensionless vibration frequency and i  1 .

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Table 3 shows the output results from this method for an isotropic single-layered micro cylindrical shell based on CT and MCST. The used material properties in this Table are considered as follows: R  2 nm , L  R , E  1060GPa,   0.3,   2.3 g / cm 3.

M

Also, in Fig. 4. the obtained results from GDQ method for clamped-clamped boundary conditions are compared by Ni et al. [17] study using the differential transformation method

ED

(DTM) for fluid flow analysis.

PT

The results of these Table and Figure demonstrated that there is a good agreement between the present results and the other studies. Thus, it is obvious that the present method is the

CE

acceptable for fluid flow analysis.

AC

6. Results and discussion In this work, based on the MCST, the effects of length scale parameters, magnetic field,

length to radius ratio, thickness ratio (thickness of face sheets to thickness of core), Knudsen number, volume fraction of CNTs, orthotropic elastic medium and temperature changes on the stability and free vibration behaviors of the flow-conveying TSDT double-bonded micro

23

ACCEPTED MANUSCRIPT composite sandwich cylindrical shells reinforced by CNTs embedded in an orthotropic foundation are presented. The used parameters are considered as the following form: h  2l ,

R  h , 2h ,

k g  2000 N / m ,

L  3h ,

Kn  0.02 ,

   /10 rad ,

k w  20(GN / m 3 ) ,

T  10 , H z  20 A /  m ,

k g   1500 N / m ,

  4 107 H / m ,

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u f  50 m / s and l 2  l . It is noted that the material length scale parameter of each micro cylindrical shells are equal to l  17.6  m . Also, the dimensionless frequencies and fluid flow velocity are

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considered as follows:

   h D ; U m u m D

(42)

In the Eq. (42), D is defined as following form:

M

D  I 10 / A110

hb

ED

where: hc

ht

(44a)

I 10   2hb b dz   2hc c dz   2ht t dz 

2

A110  

hc 2 h  c 2

E mb dz  

CE

hb 2 h  b 2



2

2

PT



(43)

ht 2 h  t 2

E c dz  

(44b)

E mt dz

AC

At the first, it is necessary to be considered the convergence of the GDQ domain solution. Fig. 5. shows the effect of GDQ domain grid number (N) on the first dimensionless natural frequencies of the TSDT double-bonded micro sandwich cylindrical shells conveying fluid flow with composite face sheets reinforced by CNTs based on both of the classic and modified couple stress theories. It is depicted that the dimensionless natural frequencies are converged when N increases, so that with increasing N and for N 10 , the dimensionless natural frequencies are the same and it is means that the obtained results are independent to 24

ACCEPTED MANUSCRIPT number of grids for N 10 . It is noted that N 15 are used to calculate the output results in the present work. Fig. 6. depicts the various modes changes of the flow-conveying TSDT double-bonded micro composite sandwich cylindrical shells versus modified fluid flow velocity. It is

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observed that the dimensionless natural frequencies decrease with increasing of the flow velocity. Moreover, it is demonstrated that there is not damping behavior of micro structures for the small fluid flow velocity. But with increasing the flow velocity the dimensionless natural frequency becomes equal to zero that in this zone, the real part of dimensionless

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frequencies are divergence ( u m  372.93 m / s ) and system is unstable (Fig. 6a). Therefore, the micro cylindrical shells become susceptible to buckling and then stability of system decreases. Also, it is illustrated that flutter phenomenon is occurred for the zones that (1,2), (2,3) and (3,4) modes are equal. It means that the flutter phenomenon causes instability of for

the

505.26  u m (m / s )  649.62 ,

M

system

667.67  u m (m / s )  866.17

and

ED

881.20  u m (m / s ) 1013.53 , respectively. Thus, it can be say that increasing the fluid flow velocity leads to decrease the micro structure stiffness and when this velocity larger than

PT

allowable amount (critical flow velocity), the system will be unstable and buckling or flutter

CE

phenomenon appears.

The effects of material length scale parameters on the imaginary and real parts of

AC

dimensionless frequencies and stability of Reddy double-bonded sandwich micro composite cylindrical shells based on CT and MCST are illustrated in Fig. 7. It is depicted that the effect of material length scale parameters is noticeable for lower values of thickness to material length scale ratio. So that, the size dependent effects including CT and MCST converged by increasing the thickness to material length scale ratio ( h  10 l ). Also, considering the material length scale parameters lead to increase stiffness of micro structure,

25

ACCEPTED MANUSCRIPT therefore increase the dimensionless natural frequencies and stability of system for MCST. Moreover, the critical flow velocity for the CT and MCST are obtained equal to 112.15 m/s and 283.88 m/s, respectively. Thus, it can be say that the material length scale parameter causes to delay the buckling and flutter phenomenon. Fig. 8. indicates that the micro structure stiffness has inverse relation with length of

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double-bonded micro cylindrical shells. The dimensionless natural frequency and critical flow velocity decrease with increasing of the length to thickness ratio. Moreover, when the dimensionless natural frequencies are equal to zero, the micro composite sandwich cylindrical shells become unstable due to the divergence via a pitchfork bifurcation in Fig.

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8b. Thus with enhancing the length-to-thickness (radius) ratio the buckling and flutter phenomenon is occurred for the lower values of fluid flow velocity. Also, it is depicted that system stability reduces from u m  564.22 m / s for L  3h to u m 158.26 m / s for L  5h .

M

Effects of magnetic field and temperature changes on the dimensionless natural frequencies and stability of TSDT flow-conveying double-bonded micro composite sandwich

ED

cylindrical shells investigated in Fig. 9. and Table. 4. It is showed that the dimensionless

PT

natural frequency and critical flow velocity enhance by increasing the magnetic field and vise versa decrease with enhancing temperature changes. Moreover, this figure demonstrates that

CE

the magnetic field has a special role on the micro structures behaviors because when

AC

increases the magnetic intensity ( H z 150 A /  m ), the curves convergence to each other for the various temperatures and it means that temperature changes has no effect on the dimensionless natural frequency and stability by increasing the magnetic field. Thus, it can be say that micro cylindrical shells stiffness enhances in the presence of magnetic field and it is causes that the influences of temperature changes reduce on the micro structure vibration and buckling (stability) response.

26

ACCEPTED MANUSCRIPT Tables. 5, 6. and Figs. 10,11. show the effects of orthotropic elastic foundation and CNTs volume fraction on the dimensionless natural frequency and critical flow velocity of fluidconveying TSDT double-bonded micro composite sandwich cylindrical shells. Fig. 10. and Table. 5. depict that there is a direct relation between the Winkler spring

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constant, dimensionless natural frequency and critical flow velocity. These parameters increase by increasing the Winkler constant, because this constant leads to enhance the stiffness of micro structures. Also, It is observed that the carbon nanotubes volume fraction have a great effect on the increasing frequency and stability. On the other hands, when the

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CNTs volume fraction are equal V CNTs  0.14 , the dimensionless natural frequency is obtained   0.2878 for K w 100(GN / m 3 ) , but when this parameter increases to

V CNTs  0.17 ,  is calculated 0.3547 .

M

Also, according to the Fig. 11. and Table 6., the Pasternak shear modulus has a similar behavior with Winkler spring constant and CNTs volume fraction. It is indicated that with

ED

enhancing the Pasternak shear modulus in both of 

and  directions increases the

PT

imaginary part of dimensionless frequency and critical flow velocity. Thus, it can be concluded that by using the reinforcements and orthotropic elastic medium, this system will

CE

be unstable for the larger values of fluid velocity and lead to delay buckling and flutter

AC

phenomenon because the stability zone of micro structures increased by increasing the orthotropic elastic constants and CNTs volume fraction. Fig. 12. illustrates the effect of Knudsen number on fluid flow velocity. It is seen that the

dimensionless natural frequencies of flow-conveying TSDT double-bonded micro sandwich cylindrical shells reduces by enhancing this parameter. Hence, for smaller Knudsen number, the frequencies changes are less than when Kn is large. Thus, it can be concluded that the micro structures stability increases with a decrease in the Knudsen number. Moreover, it is 27

ACCEPTED MANUSCRIPT demonstrate that the effect of Knudsen number is important for the large values of flow velocity. So that curves slopes increase by increasing the flow velocity and Knudsen number. In the other word, when the Knudsen number increases the micro structures will be unstable for the lower values of flow velocity and the real part of dimensionless frequencies are

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divergence for smaller velocity. Fig. 13. predicts the free vibration behavior of TSDT (Reddy) double-bonded sandwich micro composite cylindrical shells based on the presence of fluid flow in the each micro shells versus radius to thickness ratio. It is observed that the fluid flow has very important

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effect on the dimensionless natural frequencies. So that when the fluid exist in the system the imaginary of dimensionless natural frequencies reduce form 1.00 to 0.38 because using the fluid flow leads to enhance the force slip between the fluid-solid boundary conditions and decreases the micro structure stiffness. Thus, this stiffness decreases with increasing the

M

cross-section of flow and it causes that reduces the dimensionless natural frequency. Also, it

ED

is depicted that if both of micro cylindrical shells conveyed static fluid ( u f  0 ) or one of them conveying flow with u f  0 converged the dimensionless natural frequency for R  5h .

PT

In the other hand, the effect of static fluid flow in the upper and lower micro cylindrical shells

CE

in comparison with influences of fluid flow in the one of them is the same for the moderately thick-walled micro cylindrical shell while these changes are different for thick-walled micro

AC

shells.

Effects of thickness ratio (the face sheets thickness to core thickness) of TSDT sandwich

double-bonded micro composite cylindrical shells are presented in Figure 14. As can be observed from this figure, the dimensionless natural frequency and critical flow velocity increase when the thickness ratio increases and this increasing leads to enhance the stability

of micro structure. Thus, it can be say that micro cylindrical shells stiffness increases with 28

ACCEPTED MANUSCRIPT increasing the thickness ratio because the face sheets are stiffer than foam core and magnetic field are applied on the bottom and top face sheets. Also, it is depicted that bifurcation instability of system are divergence for the larger values of flow velocity and micro structures will be unstable for the lower ratio of thickness. So that, the critical flow velocity is obtained

u m 184.49 m / s , u m  231.61m / s and u m  304.03 m / s for the hf  hc / 9 , hf  hc / 4 and

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hf  3hc / 7 , respectively. 7. Conclusion

In the present study, the stability and free vibration analyses of TSDT (Reddy) symmetric

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flow-conveying double-bonded micro composite sandwich cylindrical shells reinforced by CNTs rested in an orthotropic elastic foundation based on MCST are investigated. It is noted that in the previous studies do not use the third order shear deformation theory (Reddy) for

M

sandwich micro cylindrical shells conveying fluid flow. Also, the system is simulated by orthotropic elastic medium and the material properties are considered as temperature

ED

dependent that they are the other novelty of this work. Moreover, it is assumed that the magnetic field is applied on the thickness of face sheets and temperature-dependent material

PT

properties. The governing equations of motion are obtained using Hamilton’s principle and

CE

GDQ method to discretize these equations and solve them. Numerical results are considered to predict the effects of various parameter such as length-to-thickness ratio, Knudsen number,

AC

orthotropic elastic medium, temperature changes, thickness ratio and CNTs volume fraction on the natural frequencies and critical flow velocity of flow-conveying double-bonded micro composite sandwich cylindrical shells. The results of this work are demonstrated as follows: 1) The dimensionless natural frequencies reduced in the presence of fluid flow. It is due to decrease the stiffness of micro structures. So that when the static fluid is existed in

29

ACCEPTED MANUSCRIPT the system, the imaginary part of dimensionless natural frequency reduced from 1.0 to 0.3. 2) Stability and frequencies of TSDT double-bonded micro sandwich cylindrical shells decreased by increasing the flow velocity. Also, it is indicated that when the dimensionless natural frequencies are equal to zero, the system was unstable and

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occur the flutter phenomenon and appeared the bifurcation curves on the real part of frequencies.

3) The effect of magnetic field on the dimensionless natural frequency and critical flow velocity was more than the other parameters. In the other hand, when the micro

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structure reinforced by nanotubes and applied the magnetic field leaded to enhance the stiffness. Thus, it can be say that using the CNTs had a great role to delay the buckling and flutter phenomenon.

M

4) Because the face sheets are very stiffer than foam cores, it was reasonable that with

structures.

ED

increasing the thickness ratio increase the natural frequencies and stability of micro

PT

5) The material length scale parameter leaded to enhance the dimensionless natural frequency, stability and critical flow velocity of TSDT double-bonded micro

CE

composite sandwich cylindrical shells. Moreover, it was demonstrated that the effect of this parameter was important for the lower values of thickness to length scale ratio.

AC

Thus, when the h / l was more than 10, both of CT and MCST are convergence.

6) The influences of orthotropic elastic constants and temperature changes were less in comparison with presence of fluid flow, magnetic field and CNTs volume fraction. However, the dimensionless natural frequencies and buckling behavior of micro structures enhanced by increasing the Winkler and Pasternak constants while reduces with increasing temperature changes.

30

ACCEPTED MANUSCRIPT Acknowledgments The authors would like to thank the reviewers for their valuable comments. They are also grateful to the Iranian Nanotechnology Development Committee for their financial support and the University of Kashan for supporting this work by Grant No. 574602/12.

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ACCEPTED MANUSCRIPT [10] A. Ghorbanpour Arani, E. Haghparast, M. Heidari Rarani and Z. Khoddami Maraghi, Strain gradient shell model for nonlinear vibration analysis of visco-elastically coupled Boron Nitride nano-tube reinforced composite micro-tubes conveying viscous fluid, Computational Materials Science., 96 (2015) 448-458. [11] A. Ghorbanpour Arani and S. Amir, Electro-thermal vibration of visco-elastically coupled BNNT systems conveying fluid embedded on elastic foundation via strain gradient theory, Physica B, 419 (2013) 1-6. [12] L.L. Ke and Y.Sh. Wang, Flow-induced vibration and instability of embedded double-walled carbon

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nanotubes based on a modified couple stress theory, Physica E, 43 (2011) 1031-1039. [13] R. Ansari, R. Gholami, M. Faghih Shojaei, V., Mohammadi and M.A. Darabi, Coupled longitudinaltransverse-rotational free vibration of post-buckled functionally graded first-order shear deformable microand nano-beams based on the Mindlin's strain gradient theory, Applied Mathematical Modelling, 40 (2016)

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magneto rheological elastomer core, European Journal of Mechanics A/Solid, 47 (2014) 143-155. [16] J.R. Banerjee, C.W. Cheung, R. Morishima, M. Perera and J. Njuguna, Free vibration of a three-layered

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sandwich beam using the dynamic stiffness method and experiment, international Journal of Solids Structures, 44 (2007) 7543-7563.

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[17] Q. Ni, Z.L. Zhang and L. Wang, Application of the differential transformation method to vibration analysis of pipes conveying fluid, Applied Mathematics and Computation, 217 (2011) 7028-7038.

CE

[18] E. Ghavanloo, F. Daneshmand and M. Rafiei, Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation, Physica E, 42 (2010) 2218-2224.

AC

[19] M. Mohammadimehr, M. Salemi and B. Rousta Navi, Bending, buckling and free vibration analysis of MSGT microcomposite Reddy plate reinforced by FG-SWCNTs with temperature-dependent material properties under hydro-thermo-mechanical loadings using DQM, Composite .Structures, 138 (2016) 361380. [20] M. Arefi, R. Koohi Faegh and A. Loghman, The effect of axially variable thermal and mechanical loads on the 2D thermoelastic response of FG cylindrical shell , Journal of Thermal Stresses, 39 (2016) 1539-1559.

32

ACCEPTED MANUSCRIPT [21] H. Rouhi, R. Ansari and M. Darvizeh, Size-dependent free vibration analysis of nano shells based on the surface stress elasticity, Applied Mathematical Modelling, 40 (2016) 3128-3140. [22] M. Mohammadimehr, B. Rousta Navi and A. Ghorbanpour Arani, Surface stress effect on the nonlocal biaxial buckling and bending analysis of polymeric piezoelectric nano plate reinforced by CNT using Eshelby-Mori-Tanaka approach, Journal of Solid Mechanics., 7 (2015) 173-190. [23] R. Bahadori and M.M. Najafizadeh, Free vibration analysis of two-dimensional functionally graded

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axisymmetric cylindrical shell on Winkler–Pasternak elastic foundation by First-order Shear Deformation Theory and using Navier-differential quadrature solution methods, Applied Mathematical Modelling, 139 (2015) 4877-4894.

[24] Y. Tadi Beni, F. Mehralian and H. Razavi, Free vibration analysis of size-dependent shear deformable

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functionally graded cylindrical shell on the basis of modified couple stress theory, Composite Structures,

[25] T. Murmu, M.A. McCarthy and S. Adhikari, vibration response of double-walled carbon nanotubes subjected to an externally applied longitudinal magnetic field: A nonlocal elasticity approach, Journal of Sound and Vibration, 331 (2012) 5069-5086.

M

[26] Q. Wang, D. Shi, Q. Liang and F. Pang, Free vibration of four-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundary conditions, Applied

ED

Mathematical Modelling, 42 (2017) 705-734. [27] R. Kolahchi, M. Rabani Bidgoli, Gh. Beygipoor and M.H. Fakhar, A nonlocal nonlinear analysis for

PT

buckling in embedded FG-SWCNT-reinforced microplates subjected to magnetic field, Journal of Mechanical Science and Technology, 29 (2015) 3669-3677.

CE

[28] M. Nasihatgozar, V. Daghigh, M. Eskandari, K. Nikbin and A. Simoneau, Buckling analysis of piezoelectric cylindrical composite panels reinforced with carbon nanotubes, International Journal of

AC

Mechanical Sciences, 107 (2016) 69-79.

[29] M. Sabzikar Boroujerdy and M.R. Eslami, Axisymmetric snap-through behavior of piezo-FGM Shallow clamped spherical shells under thermo-electro-machanical loading, International Journal of Pressure Vessels and Piping, 120-121 (2014) 19-26. [30] A.H. Sofiyev, The vibration and buckling of sandwich cylindrical shells covered by different coatings subjected to the hydrostatic pressure, Composite Structures, 117 (2014) 124-134.

33

ACCEPTED MANUSCRIPT [31] A.L. Araujo, V.S. Carvalho, C.M. Mota Soares, J. Belinha and A.J.M. Ferreira, Vibration analysis of laminated soft core sandwich plates with piezoelectric sensors and actuators, Composite Structures, 151 (2016) 91-98. [32] P. Zhu, Z.X. Lei and K.M. Liew, Static and free vibration analysis of Carbon nanotubes-reinforced composite plates using finite element method with first order shear deformation plate theory, Composite Structures, 94 (2012) 1450-1460.

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[33] R. Ansari, M. Faghih Shojaei, V. Mohammadi, R. Gholami and F. Sadeghi, Nonlinear forced vibration analysis of functionally graded carbon nanotube-reinforced composite Timoshenko beams, Composite Structures, 113 (2014) 316-327.

[34] Zh. Lang and L. Xuewu, Buckling and vibration analysis of functionally graded magneto-electro-thermo-

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elastic circular cylindrical shells, Applied Mathematical Modelling, 37 (2013) 2279-2292.

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M

Mechanical Engineering, 272 (2014) 100-120.

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of conveying fluid functionally graded nano shells based on Mindlin’s strain gradient theory, Thin Walled Structures, 105 (2016) 172-184.

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[38] F. Yang, A.C.M. Chong, D.C.C. Lam and P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solid Structures, 39 (2002) 2731-2743.

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[39] T.I. Thinh and M.C. Nguyen, Dynamic Stiffness Method for free vibration of composite cylindrical shells containing fluid, Applied Mathematical Modelling, 40 (2016) 9286-9301.

AC

[40] Y. Xu, Y. Qian and G. Song, Stochastic finite element method for free vibration characteristics of random FGM, Applied Mathematical Modelling, 40 (2016) 10238-10253.

[41] Y.Z. Wang, F.M. Li and K. Kishimoto, Thermal effects on vibration properties of double-layered nanoplates at small scales, Composites: Part B, 42 (2011) 1311-1317. [42] A. Kutlu and M.H. Omurtag, Large deflection bending analysis of elliptic plates on orthotropic elastic foundation with mixed finite element method, International Journal of Mechanical Sciences, 65 (2012) 6474.

34

ACCEPTED MANUSCRIPT [43] A. Alibeigloo and M. Shaban, Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity, Acta Mechanica, 224 (2013) 1415-1427. [44] K.P. Soldatos and V.P. Hadjigeorgiou , Three-dimensional solution of the free vibration problem of

Appendix A

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homogeneous isotropic cylindrical shells and panels, Journal of Sound and vibration, 137 (1990) 369-384.

The governing equations of motion for single-layered micro cylindrical shells based on MCST and using   0 can be written as follows:

AN US

u 0 : 

M0 M A M M u  02 u 0, xx   A11 u 0, xx  662 u 0,  0 v 0, xxx   03 v 0, x  4 0, 8R 8R R 8R 8R



M A M M 1  A12  A 66 v 0,x   03 w 0,x   12 w 0,x  12 x ,xx   14 x , R 4R R 8R 8R B 66 M M 1 x ,  1  ,xxx   13  ,x    B 12  B 66   ,x  2 R 8R 8R R

M

 B 11 x ,xx 

(A-1)

ED

 I 0 u 0  I1  x  0

PT

v 0 :

M0 M M M 1 u  0 u 0,xxx    A12  A 66 u 0,x   0 v 0, xxxx  02 v 0, xx  3 0, x  8R 8R R 8 8R M0 M A 1 M  A 66 )v 0,xx  2 ( 02  A 22 )v 0,  442 v 0  04 w 0, 2 2R R 8R R 8R

CE

(

AC

7M 0 M M 1  w 0,xx   2  A 22  A 44 w 0,  8 x ,xxx   13 x , x  2 8R R 8R 8R  M0  M  M1 M   ,xxxx  12  ,xx    B 66  0   ,xx  B 12  B 66  2  x ,x   8R  8 8R 4R    M  A 1   2  B 22  0   ,  44   I 0 v 0  I 1    0 R  8R  R 

1 R

35

(A-2)

ACCEPTED MANUSCRIPT w 0 : 

M0 M 7M 0 A 1 u  12 u 0, x  04 v 0,  v 0, xx   2  A 22  A 44 v 0, 3 0, x  2 4R R 8R 8R R

M  M0 M 5M 0 A w 0,xxxx  04 w 0,  w 0, xx    02  A 55 w 0, xx  442 w 0, 2 8 8R 4R R  2R  M A B 12  1  7M 0 M 1    222 w 0  0 x ,xxx    x ,x   x ,x    A 55  R 8 2R  2 R  R   M B 22  1  3M 0 M 1  1  03  ,      ,  I 0 w 0  0   ,xx    A 44  8R 2 2 R  R R  

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x :

B M1 M M M u  12 u 0, xx   B 11 u 0, xx  662 u 0,  1 v 0, xxx   13 v 0, x  4 0, 8R 8R R 8R 8R

AN US



(A-3)

M0  M0 1 1 7 M 0 M 11 w 0, xxx  (  )w 0, x   B 12  B 66  v 0,x   R 8R  8 2R 2 R M B M M  ( 12  A 55 )w 0,x  24 x ,  22 x ,xx   (D11  0 ) x ,xx R 8R 8R 8 

3M 0 M M 1 (D 66  ) x ,  A 55x  2  ,xxx   23  ,x  2 R 2 8R 8R



1 R

M



(A-4)

PT

ED

3M 0    D12  D 66    ,x   I 1 u 0  I 2  x  0 8  

 :

CE

M1 M M M 1 u  1 u 0,xxx    B 12  B 66  u 0,x   1 v 0, xxxx  12 v 0, xx  3 0, x  8R 8R R 8 8R M0 M A 1 M )v 0,xx  2 ( 12  B 22 )v 0,  44 v 0  03 w 0, 2 4R R 8R R 8R

AC

 (B 66  

1 3M 0 M 1 1 (  )w 0,xx   4R 2 R R

M M  B 22   A 44 w 0,  2 x ,xxx   23 x ,x   8R 8R  R  M  M M   2  ,xxxx  2  ,xx    D 66  0   ,xx 8 8R 2  

3M 0    D12  D 66   x ,x  8   M  1   2  D 22  0   ,  A 44   I 1 v 0  I 2    0 R  8  

1 R

36

(A-5)

AC

CE

PT

ED

M

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ACCEPTED MANUSCRIPT

37

ACCEPTED MANUSCRIPT

Winkler Spring Constant (Kw) ht1

PmPV Carbon nanotubes(CNTs)

hc1

Foam core Magnetic Field

hb1

PmPV

Fluid Flow (in)

R Fluid Flow (out)

R

Foam core PmPV

Pasternak Shear Modulus (KG)

Orthotropic Elastic Medium PmPV Carbon nanotubes(CNTs)

Foam core Magnetic Field

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PmPV

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PmPV

Fluid Flow (in)

X

R R

PmPV

Foam core

hb2 hc2 ht2

M

PmPV

Fluid Flow (out)

ED

Winkler Spring Constant (Kw)

5.7

AC

5.65

CE

PT

Figure 1: Schematic of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow in presence of magnetic field rested in an orthotropic medium

-7

7.1 (9.3125*10-7*T2)-(1.471*10-3*T)+7.4375

7.05

2

-3

(TPa) 22

5.55

E

E

11

(TPa)

(7.425*10 *T )-(1.173*10 *T)+5.9317

5.6

5.5 5.45 300

7 6.95 6.9

350

400

450

500

550

600

650

6.85 300

700

Temperature (K)

350

400

450

500

550

Temperature (K)

38

600

650

700

ACCEPTED MANUSCRIPT 5

1.98 (-2.4625*10-7*T2)+(2.96*10-4*T)+1.8779

4.5

11 (10-6/K)

G

12

1.96

1.95

4

3.5

1.94 300

350

400

450

500

550

600

650

3 300

700

Temperature (K)

350

0.5 0 300

650

700

400

500

600

Em/(2*(1+ m))

m

0.4

0.2

0 300

700

400

CE

700

600

700

54

m (10-6/K)

AC

600

56

0.4

 m=0.34

0.35

500

Temperature (K)

Temperature (K)

m

600

0.6

G (GPa)

ED

1

PT

m

M

3.51-(0.0047*T)

1.5

0.3

45*(1+(0.0005* T))

52 50 48

0.25 0.2 300

550

0.8

2

0.45

500

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2.5

0.5

450

Temperature (K)

Figure 2: The temperature-dependent properties of SWCNTs

E (GPa)

400

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(TPa)

1.97

(-1.1826*10-5*T2)+(1.4849*10-2*T)+(6.7913*10 -2)

46

400

500

600

700

300

Temperature (K)

Figure 3: The temperature-dependent properties of PmPV matrix

39

400

500

Temperature (K)

ACCEPTED MANUSCRIPT

mode1---present work mode1---Ref [17] mode2---present work mode2---Ref [17] mode3---present work mode3---Ref [17]

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175 150 125

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100 75 50 25 0 0

1

2

3 4 5 6 7 Flow Velocity ( m/s )

M

dim.Natural Frequency Im (  )

200

8

9

10

AC

CE

PT

ED

Figure 4: Dimensionless natural frequencies using GDQ (present work) and DT (Ni et al. [17]) methods for clamped- clamped boundary condition

40

ACCEPTED MANUSCRIPT

CT MCST

0.6 0.5 0.4

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0.3 0.2 0.1 0 5

7

9

11 13 15 17 The Number of Grid ( N )

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dim. Natural Frequency Im()

0.7

19

21

Figure 5: Convergence of the GDQ domain solution for the flow-conveying TSDT double-bonded micro composite sandwich cylindrical shells based on CT and MCST

4.5

3

M

PT

2.5

ED

3.5

2 1.5

CE

dim. Natural Frequency Im ()

4

mode 1 mode 2 mode 3 mode 4

AC

1

0.5 0 0

200

400 600 800 Flow Velocity (m/s)

a)

41

1000

1200

ACCEPTED MANUSCRIPT

0.8 mode 1 mode 2

0.4 0.2 0

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dim. Frequency Re (  )

0.6

-0.2 -0.4

-0.8 0

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-0.6 200

400 600 800 Flow Velocity (m/s)

1000

1200

0.5

CT MCST

PT

dim. Natural Frequency Im ()

0.6

ED

M

b) Figure 6: Various mode changes ( a. imaginary and b. real parts) of the sandwich TSDT doublebonded micro cylindrical shells versus dimensionless fluid flow velocity

0.4

AC

CE

0.3 0.2 0.1

0

2

4

6

8 h/l

a)

42

10

12

14

ACCEPTED MANUSCRIPT

CT MCST

0.3

0.2

0.1

0 0

100

200 300 Flow Velocity ( m/s )

b)

0.5 0

-1

C)

500

M

-0.5

-1.5 0

400

AN US

1

CT MCST

ED

dim. Frequency Re ()

1.5

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dim. Natural Frequency Im ()

0.4

100

200 300 Flow Velocity ( m/s )

400

500

AC

CE

PT

Figure 7: Effects of material length scale parameter on the a) dimensionless natural frequency b) critical velocity and c) stability of TSDT double-bonded micro composite sandwich cylindrical shells

43

ACCEPTED MANUSCRIPT

L=3h L=4h L=5h

1 0.8

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0.6 0.4 0.2 0 0

200

a)

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dim. Natural Frequency Im ()

1.2

400 600 Flow Velocity ( m/s )

800

1000

ED

PT

0.1

-0.1

CE

dim. Frequency Re ()

0.3

L=3h L=4h L=5h

1000

M

0.5

800

AC

-0.3

b)

-0.5 0

200

400 600 Flow Velocity ( m/s )

Figure 8: Effects of length to thickness ratio on the dimensionless natural frequency and critical flow velocity

44

ACCEPTED MANUSCRIPT

0.38 T=325 K

0.37

T=340 K T=360 K

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0.36 0.35 0.34 0.33 0

30

60

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dim. Natural Frequency Im ()

T=310 K

90 120 Hz (A/m)

150

180

M

Figure 9: Effects of magnetic field and temperature changes on the dimensionless natural frequencies of TSDT micro composite cylindrical shells

ED

0.36

VCNT=0.14

PT

0.35

VCNT=0.11

0.34

VCNT=0.17

0.311

0.33 0.308

AC

CE

dim. Natural Frequency Im ()

0.37

0.32 0.31

0.305 0.302 250

300

350

400

0.3 0.29 0

100

200 300 Kw (GN/m3)

400

500

Figure 10: Effects of Winkler spring constant and CNTs volume fraction on the imaginary part of dimensionless frequencies 45

ACCEPTED MANUSCRIPT

0.32

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0.3

0.28

Kg=0

Kg=1 (KN/m)

0.26

Kg=2 (KN/m)

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dim. Natural Frequency Im ()

0.34

Kg=10 (KN/m)

0.24 0

0.3

0.6

0.9 1.2 Kg (KN/m)

1.5

1.8

M

Figure 11: Effects of Pasternak shear modulus in  and  directions on the free vibration behavior

AC

CE

PT

ED

of double-bonded sandwich micro cylindrical shells

46

ACCEPTED MANUSCRIPT

0.3 0.24

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0.18 uf = 0

0.12

uf = 60 m/s uf = 100 m/s

0.06 0

uf = 120 m/s

0.02

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dim. Natural Frequency Im ()

0.36

0.04 0.06 0.08 Slip Flow Regime (Kn)

0.1

AC

CE

PT

ED

M

Figure 12: Effects of slip flow regime with defining the Knudsen number on the dimensionless natural frequencies of TSDT micro composite cylindrical shells

47

ACCEPTED MANUSCRIPT

0.9 Double-bonded micro shells without considering flow single micro shell conveying fluid flow Double-bonded micro shells conveying static flow Double-bonded micro shells conveying fluid flow

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0.6

0.3

0 2

2.5

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dim. Natural Frequency Im ()

1.2

3

3.5

4 R/h

4.5

5

5.5

6

hf / hc= 1/9

PT

dim. Natural Frequency Im ()

0.4

ED

M

Figure 13: Comparison of dimensionless natural frequencies of TSDT double-bonded micro composite sandwich cylindrical shells in the presence or absence of fluid flow

hf / hc= 2/8

0.3

AC

CE

hf / hc= 3/7

0.2

0.1 hf=ht+hb

0 0 a)

h=2l=constant

100

200 300 Flow velocity (m/s)

48

400

500

ACCEPTED MANUSCRIPT

0.3 hf / hc= 2/8

0.2

hf / hc= 3/7

0.1 0 -0.1 -0.2

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dim. Frequency Re ()

hf / hc= 1/9

hf=ht+hb h=2l=contsant

-0.3 0

100

200 300 Flow Velocity ( m/s )

400

500

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b) Figure 14: Effect of thickness ratio on the a) imaginary and b) real parts of TSDT double-bonded micro sandwich cylindrical shells reinforced by CNTs

(MPa)

(MPa)

56.0

22.05





Q11 ,Q 22

Q12

Q 44 ,Q55 ,Q 66

(Kg / m 3 )

(GPa)

(GPa)

(GPa)

60.0

60.40

16.31

22.05

ED

G

0.27

PT

E

M

Table 1: The material properties foam cores (Divinycell-H60) [31]

1

2

3

AC

CE

Table 2: i coefficient between SWCNTs and PmPV matrix [32]

0.11

0.149

0.934

0.934

0.14

0.150

0.941

0.941

0.17

0.149

1.381

1.381

* V CNT

Table 3: Comparison of dimensionless natural frequency without fluid flow for   0 based on CT 49

ACCEPTED MANUSCRIPT and MCST Dimensionless Natural Frequency    R

0.1 0.2 0.3

CT (l 2  0)

MCST (l 2  h )

n

Ref. [43]

Ref. [44]

Ref. [24]

1 2 1 2 1 2

0.913 0.762 0.993 0.936 1.112 1.116

0.932 0.774 1.043 0.966 1.173 1.161

0.933 0.776 1.048 0.971 1.181 1.162

Present work 0.934 0.777 1.049 0.971 1.181 1.163

Ref. [24] 1.126 1.069 1.537 1.590 1.878 1.974

Present work 1.116 1.054 1.525 1.574 1.875 1.964

CR IP T

h /R

/E

Table 4: Effects of magnetic field and temperature changes on the critical flow velocity (m/s) of TSDT double-bonded micro cylindrical shells

T (K ) 30.0

100.0

150.0

170.0

175.0

310.0

278.740

279.046

282.503

291.147

299.079

302.028

325.0

272.289

272.489

274.892

284.602

293.712

297.116

340.0

262.585

262.979

267.115

277.950

289.572

292.429

360.0

250.964

251.349

269.069

282.166

287.078

ED

M

0.0

AN US

H z (A /  m )

256.165

Table 5: Effects of CNTs volume fraction and Winkler spring constant on the stability of micro cylindrical shells (m/s)

V CNTs  0.14

V CNTs  0.17

237.148

237.351

281.283

200.0

243.555

243.758

286.774

300.0

246.707

247.012

289.418

252.809

253.216

294.706

AC

CE

0.0

PT

V CNTs  0.11

K w (GN / m 3 )

500.0

Table 6: Effects of Pasternak shear modulus in  and  directions on the critical flow velocity of 50

ACCEPTED MANUSCRIPT TSDT double-bonded micro composite sandwich cylindrical shells

K g (KN / m )

K g   2(KN / m )

K g  0

K g  10(KN / m )

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0.0

203.996

203.996

215.182

208.267

255.453

225.148

1.0

209.691

206.233

220.674

210.105

260.131

226.979

2.0

215.182

208.267

225.962

210.504

264.605

229.114

10.0

255.453

225.148

264.605

229.013

298.367

244.267

AC

CE

PT

ED

M

AN US

CR IP T

  /6

51