Large amplitude vibration of viscoelastically damped multiscale composite doubly curved sandwich shell with flexible core and MR layers

Thin-Walled Structures 144 (2019) 106128

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Large amplitude vibration of viscoelastically damped multiscale composite doubly curved sandwich shell with flexible core and MR layers

T

Mahsa Karimiasl, Farzad Ebrahimi∗ Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

ARTICLE INFO

ABSTRACT

Keywords: Nonlinear vibration behavior Multiscale composite doubly curved shell Flexible core MR TSDT HPM applied magnetics voltage

This is the first research on the nonlinear vibration of smart viscoelastic composite doubly curved sandwich shell with flexible core and MR layers with different distribution patterns. By using Reddy's third order shear deformation theory (TSDT), the strains and stresses are obtained. This smart model including multiscale composite layers, flexible core and magnetorheological layer (MR). According to the Halpin-Tsai model three-phase composites layers have been considered. The governing equations of the multiscale doubly curved shell have been derived by implementing the Hamilton's principle. For investigating correctness and accuracy, this paper is validated by other previous studies. The results of this work are validated through the comparison with the results available researches. Finally, the influence of different parameters such as temperature rise, various distributions pattern, magnetic fields, and curvature ratios on nonlinear frequency response are investigated in the details. The results presented that the multiscale doubly curved sandwich shell with flexible core show a complex behavior.

1. Introduction Vibration and suppression of noise is a major concern in many loadbearing structures such as aerospace, marine vehicles and automotive. However, the vibration can be suppressed by vibration isolator by shifting the natural frequency, the sandwich structures with damping layers can be effectively used to overcome this issue for wide frequencies. Amongst these structures, sandwich cylindrical shells or panels are widely used for many applications such as pressurized gas tanks, boilers and aircraft fuselage. Recently, using smart sandwich shells are significant because of its applicable in construction. Magnetorheological layers (MR) are one of the intelligent materials; furthermore, it has been found that nonlinear forced vibration of smart composite structures has a lot of applications in different industrial such as automotive, robotics, civil structures, aircraft and spacecraft, etc. One of the most significant forms of smart structures is their application, employing MR and flexible core in smart structures. The intelligence of the smart structures will make sense, weight savings, some control over directionality in actuation and the resulting MR effects. In recent years, it has been found that nonlinear vibration of smart composite structures has many applications in various industrial applications such as automotive, robotics. The primary studies on MR fluids have been discovered by Jacob

∗

Rabinow (1984). Garg et al. [1] studied the free vibration of composite doubly curved sandwich shells. In frame work Sander's theory they considered a parabolic distribution of transverse shear strains through the shell thickness. According to the first shear deformation theory (FSDT), the thermally postbuckled plates has been studied by Lee and Lee [2].Vibration of sandwich shells made of thin layers and a moderately thick core are investigated by Singh [3]. He obtained natural frequencies by implementing used Rayleigh–Ritz method. The nonlinear vibration of composite shells in hygrothermal environments is investigated by Naidu et al. [4]. The chaotic and nonlinear dynamics orthotropic functionally graded material (FGM) rectangular plate subjected to the thermal environment via third order shear deformation theory has been presented by Zhang et al. (2009). In continue, Zhang et al. (2011) presented the nonlinear dynamics of FGM circular cylindrical shell subjected to thermal environment based on first order shear deformation shell theory (FSDT) and von- Karman geometric nonlinearity. Ebrahimi et al. [5] investigated dynamic behavior of nonhomogenous piezoelectric nanobeams under magnetic field. Ebrahimi et al. [6] presented buckling analysis of nonlocal third-order shear deformable piezoelectric nanobeams embedded in elastic medium. Nonlinear vibration of cylindrical shell in frame work higher order shear deformation theory are presented by Amabili et al. [7]. Continued from previous work, large amplitude forced vibrations of rectangular

Corresponding author. E-mail address: [email protected] (F. Ebrahimi).

https://doi.org/10.1016/j.tws.2019.04.020 Received 5 October 2018; Received in revised form 17 March 2019; Accepted 15 April 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

plates Via higher order shear deformation theory have been investigated by Alijani et al. [8]. From the experimental and analytical method they are presented that nonlinear frequency result in important effect in nonlinear to linear response of plates. Yazdi et al. [9] presented the nonlinear vibration behavior of doubly curved cross-ply shell via Donnell's shell theory. Singh et al. [10] investigated nonlinear vibration behavior of composite doubly curved shell based on higher order shear deformation theory. They studied influences of aspect ratio, curvature ratio, stacking sequence on nonlinear frequency response. Alijani et al. [11] studied primary and subharmonic responses of functianly graded materials (FGM) shallow shell by multiple scales analytical method. They found that two-to-one internal resonance can be taken measure in doubly curved FGM shells by kind of the volume fraction exponent. In order to, fundemental frequency of functionaly graded material doubly curved shallow shell is studied by Chofi (2010). Based on finite element method (FEM) method they established their results. And they investigated the influence of thickness ratio, volume fraction on nolinear frequency response. Rajamohan et al. [12] presented large amplitude forced vibration of the MR sandwich beam subjected to the harmonic force excitation. It is fund that the natural frequencies increased by increasing the magnetic field until the values of the peak deflections decreased. Free vibration of composite sandwich thick shells under thermo-mechanical loading has been investigated by Khare et al. [13]. Free vibration of MR damped beams are investigated by Yeh [14].Chandra (2011) et al. studied vibration behavior of graphene/ polymer composites with multiscale approach. Hao et al. (2010) investigated chaotic, periodic and quasi-periodic motions response of functionally graded materials (FGM) rectangular plate subjected to the thermal loading based on third order shear plate theory. The nonlinear dynamics of functionally graded cylindrical shell subjected to the thermal environment via first order shear deformation theory has been studied by Hao et al. [15]. Rafiee et al. (2014b) investigated large amplitude vibration of multi-scale carbon nanotubes/fiber/polymer composite (CNTFPC) plates in frame work first-order shear deformation plate theory (FSDT) and von Kármán geometrical nonlinearity. Thomas et al. (2015) studied the vibration response of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shell based on Mindlin's theory. The flexural and free vibration response of carbon nanotube cylindrical Panels based on first order shear deformation shell theory (FSDT) have been presented by Zhang et al. [16]. Ebrahimi and Barati [17–19]. applied the nonlocal strain gradient theory in analysis of nanobeams. They mentioned that mechanical characteristics of nanostructures are significantly affected by stiffness-softening and stiffness-hardening mechanisms due to the nonlocal and strain gradient effects, respectively. Ebrahimi et al. (2016d) extended the nonlocal strain gradient theory for analysis of nanoplates to obtain the wave frequencies for a range of two scale parameters. So, it is crucial to incorporate both nonlocal and strain gradient effects in analysis of graphene sheets for the first time. Vibration and buckling of piezoelectric and piezomagnetic nanobeams based on various beam models are verified by Ebrahimi and Barati [20–23]. Heydari et al. [24] researched the nonlinear bending of functionally graded/carbon nanotube plates via first order shear deformation plate theory subjected to the uniform pressure based on generalized differential quadrature method. Nonlinear free and forced vibration behavior of carbon nanotubes (CNTs)/ fiber/polymer multiscale composite beams via Euler-Bernoulli beam theory and von Kármán geometric nonlinearity have been presented by He et al. [25]. Rafiee et al. [27] investigated the nonlinear stress response of piezoelectric CNTs/fiber/polymer composite (CNTFPC) plates subjected to electro-mechanical loading via first-order shear deformation plate theory (FSDT) and von Kármán geometrical nonlinearity. Large amplitude vibration of smart nanocomposite plates subjected to thermo-electro- mechanical loading in frame work first order shear deformation plate theory (FSDT) and von- Karman geometric nonlinearity has been presented by Rafiee et al. [28]. The nonlinear bending of hybrid plates including carbon nanotube reinforcement

(CNTRC) embedded in elastic foundations has been researched by Fan and Wang [29]. The fundamental frequency of FGM doubly curved shell embedded in elastic foundation has been presented by Shen et al. [30]. They discussed the inflence of volume fraction index, Pasternak foundation, curveture ratio and other parameter on the nonlinear frequency. Shen et al. [31] investigated the nonlinear vibration of composites functionally graded-Graphene reinforcement plates resting on elastic foundation in thermal environments. Forced stability behavior of MR beam has been presented by Nayak et al. (2011). The buckling responses of cylindrical shells and tubes based on Generalized Beam Theory (GBT) have been studied by Silvestre [32]. Sofiyev [33] investigated the buckling and vibration response of sandwich cylindrical shells in frame work first order shear deformation theory (FOSDT). They discussed effects of shear stresses, compositional profiles on frequencies of sandwich shell. The critical buckling behavior of singlewalled carbon nanotubes (CNTs) via various shell theories have been investigated by Wang et al. (2010). Ebrahimi and Barati [34,35] proposed a nonlocal third-order shear deformable beam model for buckling and vibration study of inhomogeneous nanosize beams. They mentioned that using a higher order beam model provide more accurate results at smaller slenderness ratios or higher values of nanobeam thickness. Yang et al. (2017) studied the nonlinear dynamic instability behavior of FG/polymer/GPL nanocomposite by using Timoshenko Beam theory. The vibration behavior of sandwich plates has been presented by Shiau et al. (2006). Recently, based on the nonlocal strain gradient theory Sahmani et al. [36] investigated the buckling and postbuckling analysis of multilayer graphene platelet reinforcement (GPLRC) nanoshell. Large amplitude vibration of graphene-reinforced composite cylindrical shell subjected to thermal environment in frame work Reddy's third order shear deformation theory has been investigated by Shen et al. [31]. They carried out the effect of several parameters such as temperature rising, different distribution pattern, end condition situation, stacking sequence on nonlinear frequency response. Aguib et al. [37] studied, experimentally and numerically vibration behavior of magnetorheological elastomer sandwich beam with two-layer aluminum 7075T6. Wei et al. [38] studied dynamic and vibration behaviors of MR sandwich beam with flexible core. They discussed the effects of skin–core thickness ratio, applied magnetic field, the speed of axial movement and axial on dynamics response. Free vibration of MR multiscale composite sandwich beam under external magnetic field has been presented by Ghorbanpour Arani et al. [39]. They found that their modal with loss factor decreases by increasing magnetic field intensity, unlike natural frequency. Sofiyev [40] presented dynamic stability of viscoelastic cylindrical shells subjected to axial load with different boundary conditions. They analyzed various parameters such as the FGM, boundary conditions, different geometric characteristics of the cylindrical shells on dynamics response. Recently, Belabed et al. [41] presented vibration of functionally graded sandwich plate with new 3 unknown hyperbolic shear deformation theories. Hajmohammad et al. [42] studied dynamic response of reinforcement carbon nanotubes (CNTs) cylindrical shells subjected to hygrothermal loading based on Mori-Tanaka model. Mehar et al. [43] investigated nonlinear vibration of functionally graded carbon nanaotube reinforced sandwich panel subjected to the temperature field. They presented the effects of temperature rise with uniformed distributions. Hasrati and et al. [44] investigated nonlinear forced vibration of cylindrical shell subjected to thermal environment with different distribution patterns. They studied the influence of several parameters such as distribution patterns, volume fractions of CNT, boundary conditions, thermal loading and geometrical parameters on the nonlinear frequency response. The dynamic behaviors of functionally graded CNT plate and shell structures based on higher order shear deformation theory have been studied by Frikha et al. [45]. However; large amplitude vibration of smart multi scale doubly curved sandwich shell with MR oil layer and flexible core subjected to magnetic loading has not been reported thus far. So it is significant reported that influence of several parameters 2

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi cn / gpl E11

ll

=

EM cn / gpl E11

EM cn / gpl E11

dd

=

EM cn / gpl E11

EM

( +(

d cn / gpl 4t cn / gpl lcn / gpl 2hcn / gpl

( +(

) )

dcn / gpl 4hcn / gpl dcn / gpl 2hcn / gpl

(7)

) )

(8)

dcn / gpl , lcn/ gpl and Vcn/ gpl refer to the Young's where modulus, thickness, outer diameter, length and volumes fraction of Carbon Nanotubes and Graphene Platelet, andVmcn / gpl andEmcn / mgpl are the volumes fraction of the matrixes and Young's modulus, respectively. For investigating different distribution patterns of doubly curved shell, the weight fraction of CNT and GPL changes layerwise in accordance with the corresponding distribution pattern such as U, X, A, and O are studied. CNT and GPL volume fraction of n-th layer corresponding to each distribution pattern can be presented as (Feng, 2017): cn / gpl E11 ,

Fig. 1. Geometry of doubly curved multiscale composite doubly curved shell with flexible core and MR layer.

such as MR and core thickness, various distributions pattern, magnetic fields and curvature ratio are considered in this article. The novelty and contribution of this article is considering a sandwich multiscale panel with different distributions pattern and smart layers such as flexible core and MR. Furthermore, motion equations are deriving using Hamilton's principle and are solving according to homotopy perturbation method. The results of this study can be used to design more efficient smart structures in the future such as aerospace technologies.

hcn / gpl ,

n U : Vcn / gpl = Vcn / gpl

(9a)

2n

n X : Vcn / gpl = 2Vcn / gpl

Fig. 1 illustrated multiscale composite doubly curved shell with length of l, thickness h and shell curvatures of R1, R2 in the x-y plane. It is assumed a smart model including multiscale composite layers shell with a flexible core and magnetorheological layers (MR).

1 (9b)

2n

n O: Vcn / gpl = 2Vcn / gpl 1

2. Theory and formulation

nt nt

nt nt

1 (9c)

2n 1 nt

n A: Vcn / gpl = Vcn / gpl

(9d)

where the total number of layers can be expressed by nt and the total volumes fraction of CNT GPL can be presented by Ref. [49]:

2.1. Multiscale model for sandwich composite

Vcn/ gpl =

wcn +

wcn / gpl

(

cn / gpl

) (

cn / gpl

)w

cn

(10)

The effective constituent of the multi-layer of the polymer/Carbon nanotube/fiber (PCF) and polymer/Graphene platelet/fiber (PGF) multiscale composite doubly curved shell has been presented via Halpin-Tsai model [46] and micromechanics approaches of scheme have been expressed by Shen et al. [47]. The properties of the (PCF) and (PGF) doubly curved shells can be presented as [47]:

where cn /gpl are the mass densities of the CNT and GPL and m is epoxy resin matrix, wcn/ gpl is the mass fraction of the CNT and GPL, respectively. The mass densities of CNT and GPL can be presented as:

E11 = Vf E11F + Vmcn/ mgpl

Gmnc / mgpl =

Vmcn / mgpl 1 1 = F + E22 Emcn / mgpl E11

Vf Vmnc / mgpl

mnc / mgpl

(1) V2 Emcn / mgpl V2 E f mcn / mgpl mcn / mgpl + 2Vf Vmcn / mgpl F E22 Emcn / mgpl F Vf E22 + Vmcn / mgpl Emcn / mgpl

= Vf 12

f

+ Vmcn/ mgpl

(5)

Emcn / mgpl

F E22

( )

1+2 1 + 2 dd Vcn / gpl E = M 5 +3 8 1 1 dd Vcn / gpl

lcn / gpl

d cn / gpl

vi = v0i + z i w = w0i

Emnc / mgpl (12) (13)

i x

i y

4 3 zi 3hi2

4 3 zi 3hi2

i x

i y

+

+

w0i x

(14a)

w0i y

(14b) (14c)

i = n , MR In these equations, u 0 , v0 , and w0 are the original displacements of the doubly curved shell in the x, y directions; the rotations of transverse normal at the mid-plane in the x and y axes represented by x and y . In-plane displacements and normal displacement of the magnetorheological core in the form of third order polynomial theory can be presented by:

ll Vcn / gpl

ll Vcn / gpl

(11)

m

2(1 + vmcn/ mgpl )

ui = u0i + z i

where are the Young's modulus of CNT/GPL, G12 shear modulus and is mass density, 12 Poisson's ratio of fibers, respectively, the corresponding properties of the isotropic matrixes of CNT/GPL composite presented with Emcn / mgpl , Gmcn / mgpl, mcn / mgpl and Vmcn / mgpl and Volume fractions of the fiber presented by Vf . Via Halpin–Tsai model, composites tensile modulus has been expressed [48]:

E11F ,

+ vm

2.1.1. Kinematic relations In frame work, third-order shear deformation theory, the displacement fields at an arbitrary point in the composite shell can be expressed as:

(4)

= Vf vf + Vmcn / mgpl vmcn / mgpl

cn / gpl

where vm , vmcn / mgpl Poisson's ratio of the matrix, CNT, GPL.

(3)

mcn / mgpl

= Vcn / gpl

m

Vmcn / mgpl = Vm

(2)

Vf Vmcn/ mgpl 1 = F + G12 Gmcn/ mgpl G11

m

(6) 3

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

uc = u 0c + z c u1c + z c2 u2c + z c3 u3c

(15a)

vc = v0c + z c v1c + z c2 v2c + z c3 v3c

(15b)

where u 0c , u1c , u2c , u3c , v0c , v1c, v2c, v3c, w0c , w1c and w2c are the eleven original variables which describe the core displacements and coefficient of z c is unknown must be determined.

wMR zML

hMR = uc z c = 2

hc 2

(16a)

h = MR = vc z c = 2

hc 2

(16b)

h = MR = wc z c = 2

hc 2

u 0c

u1c

hc + u2c 2 4

u3c

hc3 8

= u0MR +

hMR 2

MR x

4 2 8 3hMR

+

MR x

v0c

h3 h v3c c = v0MR + MR 8 2

MR y

+

3 hMR

4 2 8 3hMR

MR y

+

+ + +

y

w 0n

+

R1 1 2

w0n w 0n

n y

+

u2c =

w0MR x

hMR

u 0n

c xx c yy c zz

x v0n

=

y

y

(17c)

4

y

hMR

24v0c + 24v0MR + 12hc v1c + 3hc3 v3c + 16hMR

MR x

MR y

w1c =

2 2 u3c =

+

w 0MR x

w2c hc2

+

hMR

12u0c + 12u0MR + 6hc u1c

u0n y

+ zc + zc

+ zc

2 2 v3c =

y

hMR

12v0c +

+ 6hc v1c

3hc3 v3c

+

w2c =

+

2w1c hc2

w 0n

=

x

x v1n y

u2n

+ z n2 +

u1n y

w0n y

+

uMR z

+

vMR z

uMR 1 = u 0n z hMR

+

4 3 z h2

y w 0n x

n y

+

n x

+

w0n y w0n

(19b)

x

x n 2 v2 zn y

u3n

+ zn3 +

x n 3 v3 zn y

(20a)

u2n

+ zn2

+ zn3

y

u3n y

+

v0n

+ zc

x w0 x w0 y

v1n

+ zn2

x

w1 x w z c y1

v2n x

+ zn3

v3n x

w2 x w z c2 y2

+ zc

+ z c2

+

+

n x

(21) uML and vML z z

hn , 2

can be expressed as:

vMR 1 = v0n z hMR

n y

hn 2

(22)

By substituting Relations (22) in (21), components of strain in magneto-rheological layer can be presented as:

(18e)

4w0MR hc2

u1n

Relations between

8hMR yMR

3hc2

4u0MR

n x

w 0n

v1 + 2z c v2 + 3z c2 v3 +

ML xz

(18d)

12v0MR

=

+

u1 + 2z c u2 + 3z c2 u3 +

=

ML yz

MR x

3hc2 w 0MR

n1 k xz

n y

Components of transverse strain in the core were as follows:

(18c)

3hc3 u3c + 8hMR

n1 k yz

(19a)

x y

1. It is assumed that no slip condition exists between the core and the magnetorheological layers. 2. It is considered that transverse displacement w and normal rotational degrees ( x and y ) were the equal on a hypothetical crosssection on the multiscale composite shell. 3. In magnetorheological layer no normal stress are existed 4. Magnetorheological material is modeled as a linear viscoelastic material in the pre-yield condition (Mohammadi et al. [50].

(18b)

4w0MR 2hc2

2w n 0

+2

x

In order to model the magneto-rheological layer, the following cases were assumed:

6hc2

4u0MR

2w n 0 y2

2.2. Strain-stress relationship

(18a)

v 2c =

+

(20b)

6hc2 w0MR

2w n 0 x2

x

w1 + 2z c w2

c xy c xz c yz

w0MR

24u0c + 24u0MR + 12hc u1c + 3hc3 u3c + 16hMR

+

n y

+

The core strain relationships can be expressed as:

The relationship between displacement-dependent parameters of the middle core by implementing compatibility Eq. (17) can be written as:

4

n x

y

n y

+

zn2

(17b)

w0c

y

u0 x

y n x

n y

+ zn

R2

+

y

x

w0n

+

y

x

h2 h w1c c + w2c c = w0MR 2 4

n x

x

y

n0 xz

n2 k xy

w0n 2

4 z3 3hn2 n

=

n2 k yy

( )

1 2

w0n 2

n0 yz

+

zn3

n0 k xy

n x

n yz n xz

n2 k xx

n0 k yy

+ zn

( ) x

can be given as:

xy n0 k xx

x

(17a) h2 h v1c c + v2c c 2 4

y

and

yy

n0 xy

u0n

w0MR

+

v0n

=

(16c)

3 hMR

=

x

where uMR , vMR and indicated the displacement of the magnetorheological layers. By substituting equations (14) and (15) in equation (16), compatibility relations can be defined as: hc2

n xx n yy n xy

n0 xx n0 yy

u0n

2.1.2. Compatibility conditions of the displacements According to state of ideally, the sheets have been attached to the core. So, the displacement components of the upper sheet of the core as well as the core were equal at their intersections. So, displacement compatibility conditions at the intersection of the upper sheet with core can be presented as:

vMR zML

xx ,

(15c)

wc = w0c + z c w1c + z c2 w2c

uMR zMR =

components

(18f)

MR xz

Based on von-Karman type geometric nonlinearity, the strain 4

=

w0n un + 0 x hMR

n x

2hMR

hn ,

MR xz

=

w0n vn + 0 y hMR

n y

2hMR

hn

(23)

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

Now, the relationship between transverse stresses and strains in magneto-rheological layer can be defined as: MR xz

= G2

MR xz

MR yz

,

MR yz

= G2

0 xz

c yz

,

c yz

= G2

a b

(26a)

G =

0.9B2 + 0.8124 × 103B + 0.1855 × 106

(26b)

=

n Q11 n Q12 0 0 0

n Q12 n Q22 0 0 0

0 0 0 n Q55 0

0 0 n Q44 0 0

n xx n yy n yz n xy n xz

0 0 0 0 n Q66

a b

=

n Q¯ 11 n Q¯ 12

n Q¯ 12 n Q¯ 22

0 0 0 0 n 0 ¯ 44 Q 0 ¯n 0 0 Q55 0 0 0

0 0 0

0 0 0 0 n Q¯66

T

E11 1

12 21

, Q12 =

12 E22

1

12 21

(

1 2

(28)

n=1 0 a

b

0 hn 1 hML

+ 0 MR xy

0 hML 1 a b

MR MR xx xx hC

+ 0

z × 1+ R1

n n yy yy

+

MR MR yy yy

c c xx xx

+

z R2

n n yz yz

+

+

c c yy yy

1

+

+

Z R1

), q

2

(

= c.2 1 +

hn /2

Z R2

12 21

zn , z n3) dz

n y (1,

zn , z n3) dz

MR MR yz yz

c c yz yz

hn /2

n xy (1,

zn , zn3) dz

n xz (z n ,

zn3) dz

n yz (z n ,

zn3) dz

hn /2

n n K yy , R yy =

hn /2 hn /2

c c N xx , Mnxx =

+ +

c c xz xz

c c N yy , M yy =

n n xy xy

MR MR xz xz

+

+

h c /2

c x (1,

z c ) dz

h c /2

c y (1,

z c ) dz

c xy (1,

z c ) dz

hc /2

c c N xy , Mxy =

MR xy

h c /2 hc /2

c c xy xy

c c K xz , Mnxz =

0 hC 1

z 1+ dc1 dc2 dz R2

c 1 c + R yy k yz + K xx

0 xz

c 1 + R xx k xz

(33a)

(33b)

(33c)

(33d)

(33e)

Following integrals were regarded as stress results within core and magneto-rheological layers can be written as:

hc /2

+

c 2 c 2 c k yy k yy + M yy + Pyy + N xx

)

n x (1,

hn /2

hn /2

n n K xx , Rxx =

,

n n xz xz

0 yy

2

hn /2

(29)

+

MR 1 MR k yz + K xx + R yy

)(1 + ) dc dc

n n n Nxy , Mxy , Pxy =

(30)

n n xx xx

z R1

0 yz

hn /2

1

0 yz

. R1and R2 are the principal radii of curvature in q1 and q1 directions, respectively. n n n n n n , Nyy , Nxy , Myy , Mxy and Mxx expressed the total in-plane Here, Nxx n n n , Pyy , Pxy forced and moment resultants and Pxx are the third orders n n n n stresses resultants, furthermore K xx , K yy and R xx , R yy are first and third shear stress resultants, these Eqs. can be written as:

where, U is strain energy and T is kinetic energy. The strain energy is expressed as:

U=

c 0 c 2 c k xx k xx + Mxx + Pxx + N yy

n n n Nyy , Myy , Pyy =

T ) dt = 0

hn

2 2 MR MR MR k xy k xy + Mxy + Pxy + K yy

hn /2

0

b

0 xy

n n n Nxx , Mxx , Pxx =

E22

2 MR MR k yy + Myy + Pyy

where, for convenience a doubly curved shell by rectangular base in dimension a and b in c1 and c2 directions, has been considered. q1, q2 are the Lame coefficients of the shell can be expressed as

(27)

, Q22 =

0 yy

(32)

t

a

0 2 MR MR MR k xx k xx + Mxx + Pxx + Nyy

2 2 c c c + Mxy k xy + Pxy k xy + K yy

× 1+

The transformed geometric coordinates have been expressed in Appendix A. Now via Hamilton's principle can be written:

N

0 xx

(

Q44 = G23, Q55 = G13, Q66 = G12

(U

c N xx

0 xy

The reduce stiffness modulus of composite sandwich shell can be expressed by:

Q11 =

n 1 k yz + Kxx + Rxy

MR 1 + R xx k xz

q1 = c1 1 +

n xx n yy n yz n xy n xz T

0 yz

n n2 + N yy k yy

0 0

If the fiber angle with the geometric x axis is expressed by θ, the relation (27) can be transferred to the geometric coordinates as: n xx n yy n yz n xy n xz T

0 xx

MR Nxx

0 xz

The constitutive relation of the multiscale composite doubly curved shell can be expressed as: n xx n yy n yz n xy n xz

2 2 n n n k xy k xy + Mxy + Pxy + K xy

n0 yy

1 + Rxx k xz

2 MR k yy + Nxx

where G and G are loss modulus and storage, respectively, and described as a polynomial function of magnetic field, B (in Gauss), for expressed magneto-rheological material as:

3.3691B2 + 4.9975 × 103B + 0.873 × 106

0 xy

n n0 n n2 n + Mxx k xx + Pxx k xx + Nyy

0 0

(25)

G =

n0 xx

n Nxx

0 0

n n2 n k yy + Pyy + Nxy

(24)

By considering magneto-rheological material like as viscoelastic material in the pre-yield region, based on Rajamohan et al. (2010) relation proposed the shear modulus is complex and depended on the intensity of the magnetic field. Complex shear modulus of viscoelastic material can be expressed as:

Gc = G + i G

a b

N n=1

U=

h c /2

c xz (1,

z c ) dz

h c /2

(31)

c c K yz , M yz =

The first variation of strain energy can be obtained as:

hn /2 hn /2

5

n yz (1,

z c ) dz

(34a)

(34b)

(34c)

(37d)

(34e)

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi hn /2

Rzc , Rzc =

c x (1,

M1xy

z c ) dz

hMR /2

MR K xz

MR xz (1,

=

c Qxz

zMR ) dz

MR K yz =

MR yz (1,

x

(34g)

hMR /2

hMR /2

x

(34f)

hn /2

zMR ) dz

The Kinetic energy can be presented as: N

a

T=

n n (u 0

u0n

v0n

+

v0n

w0n

+

w0n)

n=1 0 0 a b ch

+ 0

c c (u 0

+ IMR

MR xz

a

b

0

0

u c0 + v c0 v c0 + w c0 w c0) +

MR MR h MR (w 0

0 MR xz

MR yz

+

1 MR Q xz hMR

w MR 0 )

MR yz

(35)

M2cxy M2cxx + x y

2 hc2

n N yy

+

x

1 MR Qyz hMR

y n K¯ yy

n K¯ xx + x

1 + y hc

Rzc +

x

+

y

2MQc 1 xz

+ s1

n Nxy

w0n y

v0n R2

n ¯ xy M

x

+

n ¯ yy M

n K¯ yy +

y

2hn hc3

3MQc 2 yz

c c N xy N xx + x y

M1xy M1xx + x y c N yy

y

+

c N xy

x

4 hc2

+

M3cxy

M3cyy

x

y

M3cxy

M3cyy

x

y

M2cxy M2cxx + x y

4 hc2

M2cyy y

= uoc = u1c = w0c =

M2cxy x

=

y

=

x = 0, a (37a)

= 0 , y = 0, b

= uoc = u1c = v0c = v1c = w0c =

=

=0, (37b)

F: u 0 = v0 = w 0 =

y

= v0c = v1c = w0c =

x = 0, a u 0 = v0 = w0 =

x

=

=0

= uoc = u1c = w0c =

, =

= 0 , y = 0, b (37c)

The motion equations can be written in operations forms as:

(36b)

n n n Lij Umn (t ) + Lij Vmn (t ) + Lij Wmn (t ) + Lij

n x mn (t )

+ Lij

n y mn (t )

+ Lij u0c (t )

n n = M33 Wmn (t ) + M34 ¨x mn (t ) + M35 ¨y n , ( i= 3, j= 1: 10) mn

= IW

M2cyy y

+

y

d 2Wmn (t ) 2 3 + P1 Wmn (t ) + P2 Wmn (t ) + P3 Wmn (t ) = 0 dt 2

M2cxy

+

2MQc 1 yz = Iv0c

y

M33 + M34 + M35 a34

(40)

And the linear frequency of the composite shell is expressed by:

(36f)

= Iu1c

(39)

where:

x

l

M3cxy

(38b)

(36d)

(36e)

2MQc 1 xz = Iu 0c

x

n y mn

where the operators Lij are defined as in Appendix 3. By substituting Eqs, (37a) into Eq. (37b) to solve the unknown functions Umn (t ), Vmn (t ), xmn (t ), ymn (t ) , u 0c (t ) ,u1c (t ), v0c (t ), v1c (t ) and w0c (t ) in terms of Wmn (t ) the nonlinear differential equation of composite doubly curved shell can be driven as:

P3 =

M3cxx

+ a36

(36c)

M2cxy M2cxx + x y

x

2MQc 1 yz + =I

n x mn (t )

+ a37 u 0c (t ) + a38 v0c (t ) + a39 u1c (t ) + a310 v1c (t ) + a311 w0c (t )

R2

=I

(38a)

n n n 3 Lij Umn (t ) + Lij Vmn (t ) + a33 Wmn (t ) + a34 Wmn (t ) + a35

2MQc 1 xz +

3MQc 2 xz +

+

x

=0,

= 1:10 Except 3 , j= 1:10)

n Nyy

n Nxx R1

hn h MR Qyz + n2 2hMR hc

c Qxz

4 hc2

2P n yy y2

hn MR h Qyz + n2 2hMR hc

3MQc 2 yz +

x

y

u0n n + Nyy R1

2hn hc3

u 0 = w0 =

=

mn

w0n x

n K¯ xx +

(36j)

+ Mij u¨ 0c (t ) + Mij u¨1c (t ) + Mij v¨0c (t ) + Mij v¨1c (t ) + Mij w¨ 0c (t ) ( i

v0n R2

n n ¯ xy ¯ xx M M + x y

= Iw0c

n n ¨ mn = Mij U¨mn (t ) + Mij V¨mn (t ) + Mij W (t ) + Mij ¨x mn (t ) + Mij ¨y n (t )

w0n y

x y

y

= v0c = v1c = w0c =

MQc 2 xz

+

x

(36i)

+ Lij v0c (t ) + Lij u1c (t ) + Lij v1c (t ) + Lij w0c (t )

n Nxx

+2

MQc 2 yz

y

y

u0n n + Nxy R1

2P n xy

MQc 2 xz

= Iv1c

y

x = 0, a , y = 0, b

= IU

2MQc 1 yz = IV

y

w0n x

2P n yy x2

2Mzc +

S: v0 = w 0 =

MQc 1 xz

+

x

M2cxy

+

x

MQc 1 xz

MQc 2 xz 2 + 2 2Mzc + x hc +

M2cyy

2 hc2

y

C: u 0 = v0 = w 0 =

(36a) n Nxy

4 hc2

x

M3cyy

+

Furthermore for obtain the boundary conditions, the displacement of the composite doubly curved shell can be driven as:

where n , c and MR are the mass densities and IMR is inertia moment of magnetorheological layer. By setting the coefficients of u0n , v0n w0n , xn , yn , u0c , v1c , u1c , v2c and w0c t o zero and substituting equations (33) and (35) into equation (32) may be stated as: n n Nxy Nxx + x y

c Qyz

M3cxy

3MQc 2 yz +

3. Solution procedure

b nh

+

y

4 hc2

c Qyz

Motions equations of multiscale composite shell can be expressed in terms of u0n , v0n ,w0n , xn , yn , u 0c ,v1c , u1c , v2c and w0c and displacements are obtained by Substituting Eq. (14) into (36) that have been expressed in Appendix 2.

(34h)

hMR /2

M1yy

+

=

(41)

p1

where initial conditions can be expressed by:

Wmn (0) =

(36g)

¯ dWmn (t ) W , h dt

=0 t=0

(42)

In addition to consider understandable and clear solution approach for the unknown function Wmn (t ) is replaced with g (t) and so Eqs. 41 and 42 can be expressed by:

(36h) 6

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

d 2g (t ) + P1 {g (t ) + g 3 (t )} = 0 dt 2

And by considering a =

(43)

In which:

nl

P = 1 P2

d 2g (t )

2g (t )

+

dt 2

2M ij

d 2g (t )

2 ) g (t )

+ {(P1

2g (t )

+

dt 2

+ P1 g 3 (t )} = 0

=0

2

2

(46)

g2 (t ) + …

dt 2 d 2g1 (t ) dt 2

2g

+

¯ dg0 (t ) W , 0 (t ) = 0 , g0 (0) = h dt

2 g (t ) 1

+

+

+

P1 g30 (t )

¯ dg1 (t ) W , h dt

=0

2) g

(P1 =

=0 t=0

0 (t )

(47)

(48)

+

1+

l

(51)

3 2 a =0 4

+

(59)

=

a2 h2 a2 CR h2

l

1+

KnlR 2 h, M l2

u0 , v¯0 = a z R z¯ = , R¯1 = 1 , h R

(60)

CI =

KnlI M

l

2 l

andCI

h2

(61)

KnlI

v0 w , w¯ 0 = 0 , ¯x = b R R2 R2 ¯ R2 = , l = l R h

x,

¯y =

y,

y , x¯ = b

y¯ =

x , a

T

(62)

ET

4. Results and discussion

where, the nonlinear frequency of the multiscale composite doubly curved shell can be presented as:

=

(58)

KnlR a2

1 + CI

u¯ 0 =

(50)

nl

+ KnlI a2

In which , nl , l , and introduced of imaginary frequency, non-linear and linear loss factor, real and imaginary constant of stiffness. The dimensionless parameters are adopted as:

By eliminating terms of g0 (t) can be written as: 2

Re ( ) = Mij

2

2

KnlR

3 2 1 a P1 acos( t ) + P1 a3 cos(3 t ) = 0 4 4

3 + a2 P1 = 0 4

(57)

l

CR =

By considering a = where a is dimensionless deflection and is the nonlinear frequency of the composite doubly curved shell. By using Eq. (45), Eq. (46) can be expressed: 2

be determined as:

Non-dimensional constants have been presented instead of the real and imaginary parts of the nonlinear stiffness for simplicity.

(49)

+ P1 g1 (t ) + P1

nl can

CR

¯ W h

dt 2

(56)

nl )

Im ( ) Re ( )

Im ( ) = Mij

nl

¯ W cos( t ) h

d 2g1 (t )

=

= 0 , g1 (0)

t=0

+i

By substituting Eq. 58 and 59 into Eq. (57) yields to Eq. (60):

By solving Eq. 47 and 48 has been yielded:

g0 (t) =

2 nl (1

In which Im ( ) and Re ( ) can be expressed as:

Substituting Eq. (40) into Eq. (43) can be given:

d 2g0 (t )

=

The modal loss factor

(45)

g1 (t ) +

(55)

Where q is the complex eigenvector and the complex eigenvalue can be written as:

nl

g (t ) = g0 (t ) +

(54)

q + Kl q + Knl q3 = 0

Kl = KlR + iKlI , Knl = KnlR + iKnlI

(44)

Where:

P1

(53)

The physical meaning of the real and imaginary parts of the modal stiffness constants:

In which [0,1] is an embedding parameter. When ξ = 0, Eq. (43) change to the linear differential equation is written by:

1:

3 2 h (a ) 2 = 0 4

1+

l

Eq. (52) can be presented as:

The equation of motion for sandwich panel can be expressed as:

To obtain the solution of Eq. (43) the homotopy perturbation approach is used [51,52]. Can be presented by:

0:

=

¯ W h2

Numerical results of the nonlinear vibration of doubly curved shell with MR layers and flexible core are presented in this section. The properties of multiscale composite doubly curved shell, MR layers and flexible core are established in Table 1 and 2, further more we assumed

(52)

Table 1 The properties of multiscale composite shell (Shen and et al., 2015; Sahmani and et al., 2017). Carbon (fiber)

Epoxy (matrix)

Carbon nanotube

E11f (GPa ) = 233.05

v m = 0.3

E cn (Gpa)= 640(1

E11f (GPa ) G12f (GPa) f = 0.6 f

m

= 23.1

Em (Gpa) = (3.51

= 8.96

m

( ) = 0.2

11 (K

1)

22 (K

1)

=

0.54 × 10

= 10.8 × 10

t cn

0.0034T + 0.142H )

(K 1) = 45(1 + 0.001T )× 10

= 2.68 × 10 3wt%

kg m3

dcn

(kgm) = 1200

1

0.0005 T ) 9

(m) = 1.4 × 10 (m) = 0.34 × 10

lcn (m) = 25 × 10

6/K

Graphene platelet

6

lcn (m) = 0.25 × 10

6

6

7

9

9

E gpl (Gpa)= (3.52

0.0034T )

d gpl (m) = 14.76 × 10

9

t gpl

(m) = 14.77 × 10

9

h gpl (m) = 0.188 × 10 12 = 0.177

9

12

= 0.33

gpl

cn

(kg /m3) = 1350

11 (K

1)

=

0.9 × 10

22 (K

1)

=

0.95 × 10

11 (K

1)

= 4.5361 × 10

6

22 (K

1)

= 4.6677 × 10

6

(kg /m3) = 4118 6 6

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

Table 2 Material properties of sandwich panel [53].

Table 5 Dimensionless frequencies of a b = 0.1m, R = 0.05m , h = 2 mm. R

Core

Face sheets

E11 = E22 = E33(Gpa) = 155 E22(Gpa) = 8.07 G12 = G23 = G11=(Gpa) = 0.05

E11 = 24.51 (Gpa) E22 = E33(Gpa) = 7.77 G12 = G13(Gpa) = 3.34 G23(Gpa) = 1.34

( ) = 130 kg m3

12

= 0.036

12

23

1

= 13 = 0.078 = 0.49 kg m3

Dimensionless frequency ( ) Mode(m,n)

Rahmani et al. [54]

Present

(1,1) (1,2) (2,1) (2,2)

14.27 26.31 27.04 34.95

14.331 26.360 27.108 35.001

(0/MR/core/MR/0)

(1,1) (1,2) (2,1) (2,2)

23.59791 28.529738 30.416181 35.369247

25.54721 20.19237 32.80093 25.16923

a

a/h

Singh. (2014)

Present

10 50 100

1.1371 1.0118 1.0046

1.1102 1.0199 1.0007

10 50 100

1.4168 1.0340 1.0118

1.3982 1.0185 1.0009

10 50 100

1.8669 1.1058 1.0342

1.7799 1.0997 1.0172

h

b

a

a

h

frequency and increasing each value of hc yields loss factor decrease. It h is brightly show that the nonlinearity due to the relatively small amplitude can desirably decrease the structural loss factor. The structural loss factor for SS boundary condition is greater than that for CC boundary conditions and CS more than CC and CCS more than CF. the transverse shear strains in the core layer are greater in SS boundary condition indicating more energy dissipating in the structure. Fig. 3, presented the effect of the different distributions pattern such a b as X, A, U, O versus nonlinear to frequency with R = 0.1, R = 0.05, 1

2

h = 2 mm h = 10, b = 1, B = 500 and the cross ply [0/MR/core/MR/ 0]. Composite doubly curved shell mode is considered (m, n)=(1,1). Unlike linear frequency, it is observed that frequency of the O distribution is highest and X is the lowest value. For X, A, U, O distributions pattern loss factor decrease. For the clamped (CC and CFCF) and simply-supported boundary conditions, at large displacements the nonlinear frequency increases and for loss factor decreases SS > CS > CC > CF. Influence of curvature ratio ( b ) under fundamental frequency of R2 cross ply [0/MR/core/MR/0] multiscale doubly curved shell with, a = 0.1, h = 2 mm has been presented in Fig. 4. The frequency mode is R1 taken to be (m, n = 1,1). It is found that increasing of curvature value leads to decrease the nonlinear frequency. It is important to express that the effect of curvature ratios has been significant role in the numerical results. Furthermore, by increasing the value of the curvature ratio, the loss factor decrease for the whole of the boundary conditions and simply supported is more than CF and CC is more than CF. The influence of magnetic field on the frequency curve with difa b = 0.1, R = 0.05, ferent boundary conditions, h = 2 mm, R a

a

1

2

= 10, b = 1 and (m, n = 1,1) have been shown in Fig. 5. Stacking sequence is considered cross ply [0/MR/core/MR/0]. It is brightly shown that the nonlinear frequency parameters increase by decreasing magnetic field. For both of magnetic fields loss factor decrease. For the clamped (CC and CFCF) and simply-supported boundary conditions (SS and CS), at large displacements the nonlinear frequency increases and for loss factor decreases SS > CS > CC > CF. Fig. 6 illustrated the effect of 2hMR with U distribution pattern versus a h

Table 4 Comparison of dimensionless frequencies for 4-layer cylindrical shell (a/ b = 1,R/a = 5).

0.8

(0/MR/core/MR/0)

factor with R = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1 and stacking 1 2 sequence [0/MR/core/MR/0] for various different boundary conditions. It can be found that increasing hc leads to increasing nonlinear

Elliptic paraboloid shell (R1 R2) . Carbon nanotube and graphene platelet with effective thickness tcnt = 0.0348 nm and tgpl = 0.0348 nm are selected as reinforcements and G13 = G23 = 0.5G12 have been considered. The validity of the present study is proved by the means of comparing the dimensionless frequencies of this model by several previous researches. The correctness of linear frequency of the composite doubly curved shell based on first shear deformable theory compared have been presented by Rahamni et al. (2010) in Table .3. The square sandwich panel (R1 = R2 = ) with h/a = 0.10 and hc/ h = 0.88 are considered for this validation. Due to the various selections of the displacement field of the core, there is little difference between the results of the dimensionless frequency in the present method and Rahamni et al. (2010) researches. Table .4 presented the correctness of the nonlinear frequency of the composite doubly curved shell based on first shear deformable theory in which have been compared with Singh [10]. As well as, it is brightly that the results of this comparison are similar. The geometric E and material properties E1 = 40, G12 = G12 = G13 = 0.6E2, G23 = 0.5E2, 2 = = = 0.25 have been considered for compare results with 12 13 23 (2014). The validation effect of MR layer on natural frequency of doubly curved shell, with thickness of hMR mm, density MR and magnetic field a b a a intensity B with R = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1, T = 300 1 2 ΔH = 1 has been reported in Table 5.

0.4

Mode(m,n)

Table .6 illustrated the Natural frequencies and Loss factor of doubly curved sandwich shell with different boundary conditions and h = 2 mm where have been compared with Mohammadi et al. [50]. As it can be realized, good agreement exists between the presented results. Fig. 2, illustrated the effect of hc versus frequency curves and loss

Table 3 Comparison of dimensionless frequencies of (0/90/0/core/0/90/0).

0.2

with

Dimensionless frequency ( )

( ) = 1800

A∗

(0/90/0/core/0/90/0)

2

a

a

h b

a

a

frequency curves with R = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1 1 2 and stacking sequence [0/MR/core/MR/0]. The results confirm that 2hMR only have a small effect on the fundamental frequency ratios of the h

doubly curved shell and increasing each value of 2hMR yields loss factor h decrease. It can be found the nonlinearity due to the relatively small amplitude can desirably increase the structural loss factor. The structural loss factor for SS boundary condition is greater than that for CC boundary conditions and CS more than CC and CCS more than CF. the 8

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

Table 6 Natural frequencies and Loss factor of sandwich cylindrical shell with different boundary conditions h = 2 mm. l

C-C 0.000146 Mohammadi et al. [50]

Fig. 2. Influence of different

C-F 0.000136

hc versus h

0.001312 Mohammadi et al. [50]

C-C 0.00128

C-F

844.8 Mohammadi et al. [50]

a

nonlinear frequency and loss factor of doubly curved shell with R = 0.1m, 1

840.3

b R2

1911.3 Mohammadi et al. [50]

= 0.05m , h = 2 mm and (m, n = 1,2).

a

Fig. 3. Influence of different distributions patterns versus nonlinear frequency and loss factor of doubly curved shell with R = 0.1m, 1

9

1910.98

b R2

= 0.05m and (m, n = 1,2).

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

a

Fig. 4. Influence of different curvature ratio and magnetic field versus nonlinear frequency and loss factor of doubly curved shell with R = 0.1m, 1

h = 2 mm and (m, n = 1,2).

transverse shear strains in the core layer are greater in SS boundary condition, indicating more energy dissipating in the structure.

b R2

= 0.05m ,

Tsai model have been studied in this research. The displacements of this model have been obtained by third order shear deformation theory and von- Karman type geometric nonlinearity. Via Hamilton's principle the governing equation are have been derived and by using the HPM method have been solved numerically. According to the significant numerical result of present research can be expressed as:

5. Conclusions Numerical investigations of large amplitude of multiscale composite doubly curved shell with flexible core and smart MR layers via Halpin-

Fig. 5. Influence of different magnetic field versus nonlinear frequency and loss factor of doubly curved shell with R = 0.1m, a

1

n = 1,2).

10

b R2

= 0.05m , h = 2 mm and (m,

Thin-Walled Structures 144 (2019) 106128

M. Karimiasl and F. Ebrahimi

Fig. 6. Influence of

2hMR ratio h

versus nonlinear frequency and loss factor of doubly curved shell with R = 0.1m, a

1

✓ The nonlinear frequency of composite doubly curved shell decrease by increasing curvature ratio and increase by increasing magnetic fields. ✓ By increasing the value of the curvature ratio, the loss factor decrease for the whole of the boundary conditions ✓ The highest value of the nonlinear frequency is for O distribution pattern and the lowest value is for the X distribution pattern of sandwich shell. ✓ The magnitude of loss factor for SS boundary condition is greater than CC boundary conditions. ✓ By increasing hc/h frequency decreased and by decreasing thickness of MR layer, frequency increased. ✓ Increasing the value of magnetic potentials cause to increase nonlinear frequency. ✓ The transverse shear strains in the core layer are greater in SS boundary condition, indicating more energy dissipating in the structure.

b R2

= 0.05m , h = 2 mm and (m, n = 1,2).

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