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Nonlinear forced vibration of smart multiscale sandwich composite doubly curved porous shell
Thin-Walled Structures 143 (2019) 106152
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Nonlinear forced vibration of smart multiscale sandwich composite doubly curved porous shell
T
Mahsa Karimiasla, Farzad Ebrahimia,∗, Vinyas Maheshb a b
Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran Department of Mechanical Engineering, Nitte Meenakshi Institute of Technology, Yelahanka, Bangalore, India
ARTICLE INFO
ABSTRACT
Keywords: Nonlinear forced vibration Multiscale composite sandwich curved shell MR SMA Porosity Hygrothermal loading
This is the first research on the nonlinear vibration analysis of composite sandwich doubly curved shell with a flexible core integrated with a piezoelectric layer. By using the higher order shear deformable theory (HSDT) for the face sheets and the third-order polynomial theory for the flexible core, the strains and stresses are obtained. It is assumed a smart model including multiscale composite layers shell with a flexible core and magnetorheological layer (MR) that leading up by the nonlinearity of the in-plane and the vertical displacements of the core. Three-phase composites shells with polymer/Carbon nanotube/fiber (PCF) and polymer/Graphene platelet/fiber (PGF) and Shape Memory Alloy (SMA)/matrix either according to Halpin-Tsai model have been considered. The governing equations of multiscale shell have been derived by implementing Hamilton's principle and solved by multiple scale method. For investigating the correctness and accuracy, this paper is validated by other previous researches. Finally, the effect of different parameters such as temperature rise, various distributions pattern, magnetic fields and curvature ratio are explored in detail.
1. Introduction Vibration and suppression of noise is a major concern in many loadbearing structures such as aerospace, marine vehicles and automotive. However, the vibrations can be suppressed via vibration isolators by shifting the natural frequency. In addition, the sandwich structures with damping layers can also 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, magnetorheological (MR) materials with a flexible core and oil layer applied with the magnetic field played an important role in the smart structures and materials constructions. MR is usually composed of micro/nano-magnetic particles suspended within a fluid, which is generally oil. By applying a magnetic field, the viscosity of the liquid is fairly increased to form a viscoelastic solid. It is noteworthy to mention that, when the liquid is at its active state, it can be precisely controlled by changing the intensity of the magnetic field due to the stress developed. The primary studies on these fluids have been discovered by Jacob [1]. Also, The composite material individual including stiff reinforcement fibers and matrix employ in the aerospace and other industries, are Carbone nanotube–reinforced, Graphene platelet-reinforced (CNTF and GPLF respectively) which are strong and stiff (for their density). A composite ∗
material with most or all of the useful (stiffness, low density, high strength, and toughness) is achieved with few or none of the especially weaknesses of the component materials. Numerous multiscale models are used for micromechanics and atomic simulations to investigate the constitutive properties of different functionalized nanotube materials [2,3]. The mechanical response in a single polymer-CNT using finite element method (FEM) analysis is investigated by Ref. [4]. Although shape memory alloys (SMAs) have been used for varied smart applications and components of adaptive structures, one of the most significant forms of SMAs application is employing fibrous sensors or actuators. The usage of the fibrous shape memory alloy composites will lead to precise sensing, weight savings, some control over directionality in actuation and the resulting SMA effects. In recent years, it has been found that the nonlinear vibration of smart composite structures has a lot of applications in different industries such as automotive, robotics, civil structures, aircraft, and spacecraft, etc. The vibration of sandwich shells made of thin layers and a moderately thick core are investigated by Singh [5]. The natural frequencies were estimated by implementing Rayleigh-Ritz method. The nonlinear vibration of composite shells in hygrothermal environments is investigated by Ref. [6]. In framework of first-order shear deformation theory and Green–Lagrange type nonlinear displacement, the strains have been obtained. They discussed the effects of the thin cylindrical
Corresponding author. E-mail address: [email protected] (F. Ebrahimi).
https://doi.org/10.1016/j.tws.2019.04.044 Received 16 November 2018; Received in revised form 20 March 2019; Accepted 24 April 2019 Available online 15 June 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
Thin-Walled Structures 143 (2019) 106152
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panel and curvature ratios on the nonlinear frequency. Yazdi [7], presented the nonlinear vibration behavior of doubly curved shell based on Donnell's shell theory [8]. investigated the nonlinear vibration behavior of doubly curved composite panel based on higher order shear deformation theory. Finally, they discussed the influences of several parameters such as aspect ratio, curvature ratio, stacking sequence on nonlinear frequency. Nonlinear vibration of cylindrical shell in framework of higher order shear deformation theory are presented by Ref. [9]. They illustrated the significant role of nonlinear term the to predict the nonlinear response of composite shell. Alijani et al. [10] studied primary and subharmonic responses of FGM shallow shell by multiple scales analytical method based on Donnell's type nonlinear strain–displacement relationships. They found that two-to-one internal resonance may be taken as a measure in doubly curved functionally graded material (FGM) shells by kind of the volume fraction exponent. Continued from previous work, large amplitude forced vibrations of rectangular plates via HSDT have been investigated by Ref. [11]. From the experimental and analytical method, they are presented that nonlinear frequency results in important effect in nonlinear to linear response of plates. Meanwhile, the fundamental frequency of functionally graded material doubly curved shallow shell is studied by Chorfi and Houmat [12] with the aid of finite element methods. They investigated the influence of thickness ratio, volume fraction versus nonlinear to linear vibration. The fundamental frequency of FGM doubly curved shell embedded in elastic foundation is presented by Shen and Yang [13] in frame work shear deformation theory and von-Karman nonlinear strain-displacement. The influence of volume fraction index, Pasternak foundation, curvature ratio, and other parameter have been investigated. Free vibration of MR damped beams is investigated by Yeh [14]. The equations solved by FEM. Thai et al. presented the bending and free vibration behaviors of functionally graded plates based on higher-order shear deformation theory. They expressed that their theory can achieve the same accuracy of the existing higher-order shear deformation theories with more number of unknowns. Meziane et al. [_Meziane_et_al_2014] probed the vibration and buckling of exponentially graded material sandwich plate resting on elastic foundations under various boundary conditions. Al Basyouni et al. (2015) presented the bending and dynamic response of functionally graded (FG) microbeam via a novel unified beam formulation and a modified couple stress theory. Some results are presented to show the effects of the power law index, material length scale parameter and shear deformation on the bending and dynamic behaviors of FG microbeams. The bending, buckling and vibration behaviors of functionally graded (FG) nanobeams based on the nonlocal theory have been studied by Zemri et al.. Yahia et al. presented the wave propagation of functionally graded plates with porosities employing different higher-order shear deformation plate theories. They presented the influence of the porosity volume fraction, volume fraction distributions on wave propagation of functionally graded plate Belkorissat et al. investigated the free vibration response of functionally graded (FG) plates based on a new nonlocal hyperbolic refined plate model. They presented different effects of the plate thickness, nonlocal parameter, various material compositions, the plate aspect ratio on the dynamic behavior of the FG nanoplate. Nonlinear free vibration response of shear deformable laminated composite doubly shell panel in the hygrothermal environment have been studied by Mahapatra et al. [15]. Nonlinear free vibration behavior of functionally graded single/doubly curved shell panels based on higher order shear deformation theory and Green-Lagrange type nonlinearity have been studied by Ref. [16]. [17] presented the large amplitude free vibration response of functionally graded (FG) spherical shell in the framework of the higher order shear deformation theory [18]. analyzed the nonlinear free vibration behavior of functionally graded curved shell panels based on higher-order shear deformation theory. The large amplitude flexural response of laminated composite doubly curved shell under hygro-thermo-mechanical loading via higher-order shear
deformation theory has been investigated by Mahapatra et al. [19]. Nonlinear vibration behavior of laminated composite spherical shell panel subjected to the hygrothermal environment based on higherorder shear deformation theory has been presented by Ref. [20]. Nonlinear free vibration response of shear deformable laminated composite single/doubly curved shell panel in the hygrothermal environment have been studied by Ref. [21]. [22] analyzed nonlinear free vibration response of laminated composite curved panel under hygrothermal environment based on higher-order midplane kinematics. Mahapatra et al. [23] presented the large amplitude free vibration response of laminated composite doubly curved shell panel in the hygrothermal environment based on higher-order shear deformation theory. Singh et al. [24] studied the geometrically nonlinear behavior of the smart laminated composite plate under the coupled electromechanical load using higher-order kinematic theory. The geometrically large amplitude thermomechanical behavior of the functionally graded curved structure based on higher-order shear deformation theory has been investigated by Mahapatra et al. [25]. The large amplitude bending behavior of functionally graded (FG) curved panel under combined thermomechanical loading have been investigated by Ref. [16]. They discussed the influence of different parameters such as support conditions, volume fractions, aspect ratios, curvature ratios, thickness ratios and temperature on the nonlinear bending response. The free vibration behaviors of composite sandwich panel structure via Green–Lagrange strain kinematics have been presented by Katariya et al. [26]. Thermal buckling strength of smart sandwich composite structure embedded in SMA based on higher-order finite element model has been presented by Katariya et al. [27]. Free vibration of composite sandwich thick shells under thermomechanical load is investigated by Khare et al. [28]. The displacements were presented through higher-order shear deformation theories. Heydari et al. [29] researched the nonlinear bending of functionally graded/CNT plates subjected to uniform pressure and embedded in an elastomeric medium via first-order shear deformation plate theory based on generalized differential quadrature method. The nonlinear bending of hybrid plates including carbon nano-tube reinforced composite (CNTRC) layers embedded in elastic foundations considering the influence of matrix cracks was studied by Fan and Wang [30]. Rajamohan et al. [31] investigated the large amplitude forced vibration of the MR sandwich beam subjected to under harmonic force excitation. It was found that the natural frequencies increased by increasing the magnetic field until the values of the peak deflections decreased. Garg et al. [32] studied the free vibration of composite sandwich doubly curved shells. In frame work Sander's theory they considered a parabolic distribution of transverse shear strains through the shell thickness. According to the FSDT, postbuckled behavior of thermally loaded plates was studied by Lee and Lee [33]. Wu et al. [34] studied the nonlinear dynamic instability behavior of FG/polymer/GPL nanocomposite by using Timoshenko Beam theory. The vibration behavior of sandwich plates with composite face sheets was presented by Ref. [35]. Recently, based on the nonlocal strain gradient theory [36] investigated the buckling and postbuckling of multilayer GPLRC nanoshell. The electroelastic response of a functionally graded plate resting on WinklerPasternak in the theoretical framework provided by a two-variable sinusoidal shear deformation theory has been presented by Arefi et al. [37]. Recently, large amplitude vibration of graphene-reinforced composite cylindrical shell subjected to thermal environment is investigated by Shen et al. [38]. In frame work Reddy's third order shear deformation theory and von-Karman theory the linear and nonlinear relationship equations. The equations of motion are solved by perturbation method. Two end conditions such as movable and unmovable are assumed. They evaluated the effect of several parameters such as temperature rising, different distribution pattern, end condition situation, stacking sequence. Aguib et al. [39] experimentally and numerically studied vibrational behavior of magnetorheological elastomer sandwich beam with two layer aluminum 7075T6, that first layer subjected to a 2
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
variable magnetic field perpendicular to the layer of the beam, and second layer to a harmonic excitation by magnetic force applied at the free end. Forced stability behavior of MR beam implementing the Galerkin method have been presented by Nayak et al. [40]. Based on numerical results of this research it is shown that the stability of MR beam by applying the magnetic field increasing. Recently large amplitude bending of functionally graded porous micro and nanobeams reinforced with graphene platelets according to the nonlocal strain gradient theory are studied by Sahmani et al. [41]. They investigated kind of parameters such as volume fractions, strain gradient and nonlocal effect and porosity. Arefi [42], investigated bending behavior a doubly curved piezoelectric shell resting on Pasternak's foundation is presented based on first-order shear deformation theory and modified couple stress formulation. They investigated the influence of different parameters such as applied voltage nonlocal coefficient, geometries and two parameters of Pasternak's foundation on the deflection of nanoshell [43]. presented the bending behavior of a sandwich microbeam subjected to the elastic micro-core and two piezoelectric micro face-sheets based on higher-order sinusoidal shear deformation. Also, The electroelastic analyses of the piezoelectric doubly curved shells based on firstorder shear deformation theory resting on Winkler's foundation have been studied [44]. Wei et al. [45] studied dynamic and vibration behaviors of MR sandwich beam with flexible core. The effects of skin–core thickness ratio, applied magnetic field, the speed of axial movement and axial force have been investigated. Free vibration of MR damped sandwich composite multiscale beam under external magnetic field is presented by Ghorbanpour Arani et al. [46]. They found that their modal with loss factor decreases by increasing magnetic field intensity, unlike natural frequency. The analytical solutions for bending response of the three-layered curved nanobeams via sinusoidal shear deformation theory have been studied by Ref. [47]. They discussed the influence of different parameters such as spring and shear parameters, nonlocal parameter, applied voltage geometries on the deflection of nanobeam. The bending analysis a sandwich curved beam based on first-order shear deformation theory resting on Pasternak foundation and subjected to the applied electric field has been investigated by Zenkuor et al.. Also, the nonlocal transient magnetoelectro-elastic formulation of a sandwich curved nanobeam consist of two piezo-magnetic face-sheets and nano-core, subjected to transverse mechanical loads and resting on Pasternak foundation have been presented Arefi et al. [47]. Belabed et al. studied the sound transmission through corrugated core FGM sandwich plates filled with porous material via three-unknown hyperbolic shear deformation theory. They studied the influence of several parameters such as temperature and volume fraction distributions on the wave propagation of functionally graded sandwich plate. Arefi et al. [48] presented the free vibration response of porous sandwich nanoplate resting on Pasternak's foundation, based on nonlocal strain gradient theory. They investigated the influence of different parameters such as nonlocal co-efficient, porosity, parameters of the foundation, volume fraction and length ratios on the free vibration characteristics. The bending response of a doubly curved nano shell subjected to electric field have been studied via nonlocal elasticity theory and first order shear deformation theory by Arefi, [49]. In continues, Arefi [50], investigated the electro-elastic behavior of the piezoelectric doubly curved shells resting on Winkler foundation through first-order shear deformation theory. Arefi et al. [51] investigated the magnetic and electric buckling loads of three-layered elastic nanoplate with exponentially graded core and piezomagnetic face-sheets based on first-order shear deformation theory [52]. presented the nonlinear flexural strength and stress response of nano sandwich graded structural shell panel under the combined thermomechanical loading based on higher-order displacement polynomial theory [52]. researched the nonlinear response of a curved sandwich panel structure adopting the higher-order kinematic theory. They discussed the effect of various design parameters on the nonlinear natural frequency. Katariya et al. [27] studied the nonlinear static behavior of
sandwich flat/curved shell panel subjected to the unvarying transverse mechanical load based on a higher-order kinematic theory including the stretching term effect in the displacement field variable. Sharma et al. [53] presented the vibro-acoustic behavior of laminated composite curved panels via a novel higher-order finite-boundary element model. The elastic response of microbeam with micro-core and two piezoelectric micro-face-sheets sandwich subjected to transverse loads and two-dimensional electric potential based on higher-order sinusoidal shear deformation beam theory have been studied by Arefi et al. [54]. [55] researched the free vibration response of a doubly curved piezoelectric nano shell based on first-order shear deformation theory and nonlocal elasticity theory. The analytical approach for thermal stress and deformation analysis of a curved nanobeam resting on Pasternak's foundation subjected to transverse mechanical and thermal loads based on Sinusoidal shear deformation theory have been studied by Ref. [55]. However, the large amplitude vibration of the smart multi scale doubly curved shell with MR fluid layer and flexible core subjected to magnetic loading has not been reported thus far. The novelty and significant contribution of this article lies in considering a porous multiscale doubly curved sandwich shell with flexible core and MR smart layer. The properties of the flexible core will be varied through the thickness direction of the shell based on the electric and magnetic potentials applied to the magnetorheological face sheets. The aim of this research is to develop a new smart model of the sandwich composite shell. Furthermore, motion equations are derived using Hamilton's principle and solved according to the multiple scale perturbation method. The effect of important parameters on the nonlinear frequency of the sandwich composite shell are also considered. 2. Theory and formulation Fig. 1 illustrates the multiscale doubly curved composite shell with the length of l, the thickness of h and shell curvatures of . The shell is embedded in a distributed hygrothermal load, which is considered in the symmetry plane of the shell cross-section, i.e. in the x-y plane. The first layer consists of PCF and second layer including PGF and third layer consist of SMA/matrix assumed in this research. 2.1. Multiscale model for sandwich composite shell According to Fig. 2, a porous shell in which the porosity is distributed uniformly (pattern B) and patterns of C and D are considered for multiscale composite porosity dispersion. The Young's modulus of CNT/GPL, G12 shear modulus and is mass density, can be presented as [54]:
E˜11 (z ) = E11 [1
p
s (z )]
(1)
E˜22 (z ) = E22 [1
p
s (z )]
(2)
G˜ 12 (z ) =
E˜11 2(1
˜ (z ) = [1
(3)
(z )) m s (z )]
+ Vmcn/ mgpl
mcn / mgpl
(4)
where E˜11, E˜22 are the Young's modulus of CNT/GPL, G˜ 12 shear modulus and ˜ is mass density, respectively.where:
s0
for pattern B
( + ) cos ( )
s = cos
z 2h
4
for pattern C
z h
for pattern D
(5)
According to the Gaussian Random Field scheme [56] the mass density coefficient m can be related to the porosity coefficient p as: m
3
=
1.121[1
(1 s (z )
1 p s (z ) 2.3 ]
(6)
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
Fig. 1. Geometry of doubly curved multiscale and SMA/matrix composite piezoelectric shell with flexible core and MR layer in hygrothermal environment.
Fig. 2. Porosity distributions patterns.
4
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M. Karimiasl, et al.
Corresponding to the closed-cell Gaussian Random Field [56] the Poisson's ratio of the porous shell corresponding to the different patterns of the porosity dispersion can be written as:
˜ (z )
˜ 12 = 0.221 1
+
12
1 + 0.342 1
˜ (z )
2
1.21 1
˜ (z )
s0 =
h 2 ˜ (z )
1
h 2
2.3
dz + 0.121
The effective constituents of the multi-layer PCF and PGF multiscale composite can be presented via Halpin-Tsai model [57] and micromechanics approaches of scheme have been expressed by Shen, [58]. The maximum Young's modulus and the maximum shear modulus and the maximum mass density are concentrated to be orthotropic can be presented as [58]:
Vmcn / mgpl 1 1 = F + E22 Emcn / mgpl E11 V 2f Emcn / mgpl F E22
+
Emcn / mgpl
= Vf
+ Vmcn/ mgpl
f
(13)
Emcn / mgpl
F E22
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
) )
lcn / gpl
d cn / gpl
Vcn / gpl Ecn / gpl
1 2
cn / gpl
+ vm Em
m
vcn / gpl Ecn/ gpl + vm Em m Vm
+ (1 + vcn / mgpl )
Vf E11f + Vmcn / mgpl Emcn/ mgpl
11
=
22
= (1 + Vmcn / mgpl ) Vmcn / mgpl
(1
mcn / mgpl
v12
11
(21)
vmcn / mgpl )
cn / mgpl Vcn / mgpl
(22)
m
Vf E11f + Vmcn / mgpl Emcn / mgpl m
(23a)
v12
11
(23b)
m E11 = Vm E11 + Vs Es
(24)
m Es E22 m Vs E22 + Vm Es
(25)
E22 =
G12 = G23 = 12
ll Vcn / gpl
ll Vcn / gpl
+ (1 + Vmnc / mgpl ) Vmcn/ mgpl
The properties of SMA are expressed by Park et al. [61]. In this present research, it is considered that the SMA fibers are uniformly distributed. The elastic properties of the composite doubly curved shell with SMA fibers can be expressed by:
(12)
1+2 1 + 2 dd Vcn / gpl E = M 5 +3 8 1 1 dd Vcn / gpl
=
f 22
(20)
2.2. Properties of SMA in a matrix of the porous composite
where are the maximum Young's modulus of CNT/GPL, G12 the maximum shear modulus and is the maximum mass density, 12 the maximum 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, the composite tensile modulus can be expressed [59]:
E11F ,
mcn / mgpl
where mcn / mgpl , mcn / mgpl , are the thermal expansion and moisture coefficients of the epoxy resin CNT and GPL matrix and cn / gpl are the thermal expansion coefficients of the CNT and GPL, respectively.
(11)
= Vf vf + Vmcn / mgpl vmcn / mgpl
12
+ Vmcn / mgpl Emcn / mgpl
Vf E11f + Vmcn / mgpl Emcn/ mgpl
+ (1 + vm )
(10)
mcn / mgpl
f 11
= (1 + Vf ) Vf
2Vf Vmcn / mgpl
Vf Vmcn/ mgpl 1 = F + G12 Gmcn/ mgpl G11
Vf E11f
22
mcn
(9)
F Vf E22 + Vmcn / mgpl Emcn/ mgpl
(19)
=
Vf Vmnc / mgpl 2 Vmcn / mgpl Emcn / mgpl
(18)
2(1 + vmcn/ mgpl )
11
(8)
E11 = Vf E11F + Vmcn/ mgpl Emcn / mgpl
(17)
m
where vm ,vmcn/ mgpl denotes the Poisson's ratio of the matrix, CNT and GPL, respectively. In addition, 11 and 22 refer to the thermal expansion coefficients of longitudinal and transverse directions, respectively [58]. Therefore, 11f is the thermal expansion coefficient of longitudinal fiber whereas, 22f refers to thermal expansion coefficient in transverse directions of the fiber. mcn / mgpl can be expressed as [60]:
1.121
p
+ vm
Vmcn / mgpl = Vm
Furthermore, when the total masses of the shell with different porosity distributions are the same, the value of the s0 can be expressed as: 1 h
cn / gpl
Emnc / mgpl
Gmnc / mgpl =
(7)
1
= Vcn / gpl
mnc / mgpl
11
(14)
22
11
(15)
22
m G12 Gs
m G12 Gs + Vm Gs
(26)
m G23 Vm
(27)
+ Vs Gs
(28)
m = Vm v12 + Vs vs
=
m m Vm E11 11
+ Vs Es E11
= Vm
=
=
m 22
+ Vs
m m Vm E11 11
+ Vs Es
= Vm
m
+ Vs
+ Vs
s
s
(29) (30)
s
E11 m Vm 22
s
s
(31) (32) (33)
where are the Young's modulus of the matrix, G12, G23 are the is mass density, 12 is the Poisson's ratio, reshear modulus and spectively. The corresponding properties of the isotropic matrixes of SMA composite are represented with Es, Gs, s and Vs . Similarly, the volume fractions and mass density of the matrix are denoted by Vm, m m m m , 22 , and 11 , 22 and m , respectively. Finally, 11 are the thermal expansion and moisture coefficients of the matrix and s and s are the thermal expansion and moisture coefficients of the SMA, m E11 ,
(16)
cn / gpl , refers to the Young's modulus, hcn / gpl , dcn/ gpl , lcn /gpl rewhere E11 presents thickness, outer diameter, length, respectively. Meanwhile, Vcn/ gpl are the volume fractions of Carbon Nanotubes and Graphene Platelet, respectively. Further, Vmcn /gpl and Emcn / mgpl are the volume fractions of the matrixes and Young's modulus, respectively. The mass densities of CNT and GPL can be presented as:
5
m E22
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M. Karimiasl, et al.
respectively.
The relationship between displacement-dependent parameters of the middle core by implementing compatibility Eq. (37) can be written as:
2.3. Kinematic relations In the framework of the third-order theory, the displacement fields at an arbitrary point in the composite shell can be expressed as:
ui = u0i + z i
vi = v0i + z i w=
4 3 zi 3h i 2
i x
i y
i x
4 3 zi 3hi 2
i y
(34a)
v 2c =
w0i
+
In these equations, u 0 , v0 , and w0 are the original displacements of the shell in the x, y directions; the rotations of the transverse normal at the mid-plane in the x and y axes are represented by x and y , respectively. u0n , v0n, w0n, xn , yn are functions of the five original variables which describe the shell displacements and n referred to the number of the multiscale composite doubly curved shell layer. In-plane displacements and normal displacement of the magnetorheological core is in the form of third order polynomial, respectively can be presented by:
vc = v0c + z c v1c +
+
vMR zML
wMR zML =
u1c
h2 hc + u2c c 2 4
u3c
w2c =
n xx n yy n xy
v0c
v1c
h2 hc + v2c c 2 4
v3c
3
hc3 h = v0MR + MR 8 2 4 hMR3 + 3hMR 2 8
w0c
w1c
hc2
hc + w 2c = w0MR 2 4
+
w0MR + y
3hc3 u3c + 8hMR
MR x
3hc2 w 0MR y
12v0c + 12v0MR + 6hc v1c
hMR
3hc3 v3c + 8hMR
MR y
3hc2
4u0MR
4w0MR + 2w1c hc2 hc2
n0 xx n0 yy
=
+ zn
n0 xy
(38f)
n0 k xx n0 k yy
+ zn
u0n
n2 k xx n2 k yy
3
n0 k xy
+
x v0n
=
+
y
n2 k xy
u 0n n x
x
n y
+ zn n x
n yz n xz
n0 yz
=
+
+ zn
n0 xz
2
n y
4 z n3 3hn 2
y y
n y
y n x
x
y
n1 k yz n1 k xz
=
n y n x
+
w0n
+
w0n
+
R1 1 2
+
( ) w0n w0n x
y
2w n 0 x2
+
2w n 0 y2
n y
( )
w0n 2 x
+
y
+
x
1 2
w0n 2
R2 u0 x
(39a)
x y
n y n x
x
+
w0n
2w n 0
+2
4 3 z h2
y
+
w0n
+ +
w0n y w0n
(39b)
x
According to the core strain relationship can be expressed as: c xx c yy c zz
u 0n x v0n
=
y
+ zc + zc
u1n x v1n y
u2n
+ z n2 +
x vn z n2 y2
u3n
+ zn3
x v3n
+ zn3
y
(40a)
w1 + 2z c w2
c xy c xz c yz
(37a)
MR y MR y
12u0c + 12u0MR + 6hc u1c
n x
w0MR x
hMR
(38c)
x
MR x MR x
+
y
(36c)
hc3 h = u0MR + MR 8 2 4 hMR 3hMR 2 8
x
w2c hc2
The consistent Donnell nonlinear strains of third-order shear deformation theory from the middle surface displacement–strain relations are [62]:
(36b)
hc 2
+
w 0MR
4w0MR 2hc2
(38e)
where uMR , vMR and wMR indicated the displacement of the magnetorheological layers. By substituting Eqs. (34) And (35) in Eq. (36), compatibility relations can be defined as:
u0c
+
(36a)
hc 2
hMR = wc z c = 2
4u0MR
v3c =
2.3.1. Compatibility conditions of the displacements The sheets are assumed to be ideally attached to the core, such that 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:
hc 2
MR y
6hc2
2 2
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.
h = MR = vc z c = 2
24v0c + 24v0MR + 12hc v1c + 3hc3 v3c + 16hMR
hMR
(38d)
(35c)
wc = w0c + z c w1c + z c2 w2c
uMR zMR
y
u3c =
(35b)
z c3 v3c
h = MR = uc z c = 2
4
2 2
(35a)
z c3 u3c
+
w0MR
w1c =
(35)
i = n , MR
z c2 v2c
MR x
6hc2
(38b)
(34c)
uc = u 0c + z c u1c +
24u0c + 24u0MR + 12hc u1c + 3hc3 u3c + 16hMR
hMR
(34b)
y
w0i
z c2 u2c
x
(38a)
w0i x
+
u2c =
w0MR
4
u 0n y
(37b)
=
+ zc
u1n y
+ z n2
u2n y
+ zn3
u1 + 2z c u2 +
u3n y
3z c 2u3
+ +
v 1 + 2 z c v 2 + 3 z c 2v 3 + (37c)
v0n x w0 x w0 y
+ zc + +
v1n x
w z c x1 w z c y1
+ zn 2 + +
v2n x
+ z n3
v3n x
w z c 2 x2 w z c 2 y2
(40b) 6
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
2.4. Strain-stress relationship
Ey =
According to the magnetorheological layer model considered, the following conditions are assumed:
Ez =
1. No slip condition existed between the core and magnetorheological layers it is assumed. 2. It is considered that transverse displacement w and normal rotational degrees ( x and y ) are the equal on a hypothetical crosssection on the multiscale composite. 3. In the magnetorheological layer, no normal stress exist 4. The Magnetorheological material is modeled as a linear viscoelastic material in the pre-yield condition.
ML xz
w 0n x
=
w0n y
+
uMR z
+
vMR z
n xx n yy n yz n xy n xz T
=
= G2
MR xz ,
MR yz
and
uML z
hn , 2
n x
2hMR
= G2
c yz
,
n
Q¯ 12 n Q¯ 22 0 0 0
vML z
0 0 0 q15 0
can be expressed as:
vMR 1 = v0n z hMR
n y
hn 2
(42)
MR xz
hn ,
w0n vn + 0 y hMR
=
n y
2hMR
hn
Bx By Bz 0 0
MR yz
(43)
s11 0 0 0 0
(44)
= G2
c yz
=
0 s11 0 0 0
0 0 e32 0 0
0 0 s33 0 0
0 0 0 0 0
G = G =
0.9B2 + 0.8124 × 103B + 0.1855 × 106
+ 4.9975 ×
103B
cos( z ) (x , t ) +
cos( z ) (x , t ) +
(
1 z R1
)
x
cos( z ) (48b)
2 h
sin( z )
0 0 0 n Q¯55
0 0 0 0 n 0 Q¯66
(48c)
0 0 0 e15 0
T
11 (T (z )
T0)
11 (H
(z )
H0)
22 (T (z )
T0 )
22 (H (z )
H0 )
0 0 e42 0 0
e13 e23 0 0 0
Ex Ey Ez 0 0
T
+ 0.873 ×
(46a)
106
Bx By Bz 0 0
(46b)
2z V h
cos( z ) Qx =
(
1 z R1
)
x
Qz 0 0
e15 0 0 0 0
(49a)
T
xx
0 0 0 0 0
3 (T (z )
T0)
yy yz xy xz
0 0 0 0 0
0 0 q32 0 0 0 0 0 0
(47b)
R1 1 +
=
0 0 q31 0 0
11
(47a)
2z h
0 e24 0 0 0
T
Qx Qy
q13 q23 0 0 0
d11 0 0 0 0
where = / h . Also, V and are the external electric voltage and the magnetic potential applied. Via Maxwell's equation, the relations between the electric field (Ex , E y, Ez ) and electric potential also the magnetic field (Qx , Qy , Qz ) and magnetic potential can be expressed as [47]:
R1 1 +
x
Ez 0 0
Based on Maxwell's equation, the electric and magnetic potential distributions can be obtained as:
Ex =
)
Ex Ey
(45)
3.3691B2
(x , z , t ) =
0 0 n ¯ Q44 0 0
0 0 q42 0 0
0 0 e31 0 0
where G and G are the loss modulus and storage, respectively, which can be expressed as a polynomial function of magnetic field, B (in Gauss), for expressed magneto-rheological material as:
(x , z , t ) =
=
z R2
xz
By considering that the magneto-rheological material behaves similar to the viscoelastic material in the pre-yield region, Rajamohan et al. proposed a relation that suggests the shear modulus is complex and depends on the intensity of the magnetic field. The complex shear modulus of viscoelastic material can be expressed as:
Gc = G + i G
,z
(
1
R2 1 +
xy
Now, the relationship between transverse stresses and strains in the magneto-rheological layer can be defined as: MR xz
n
Q¯11 n Q¯ 12 = 0 0 0
(41)
n x
w0n un + 0 x hMR
2v Qz = h
sin( z )
n xx n yy
By substituting Relations (42) in (41), components of strain in the magneto-rheological layer can be presented as: MR xz
)
cos( z ) Q y =
x
xz
Relations between
uMR 1 = u 0n z hMR
z R2
The constitutive relation of the composite shell with a piezoelectric layer, if the fiber angle with the geometric x-axis is expressed by θ, the relation stress-strain can be transferred to the geometric coordinates as:
Components of transverse strain in the core can be mentioned as follows: ML yz
(
1
R2 1 +
0 d11 0 0 0
0 0 d33 0 0
0 0 0 0 0
0 q24 0 0 0
q15 0 0 0 0
0
0 0 0 0 0 0 33 0 0 0 0 0 0 0 0
11
0 0 0
0 0 0 0 0
Qx Qy
0 0 0 0 0
Qz 0 0 xx yy yz xy xz
(49b)
d11 0 0 0 0
0 d11 0 0 0
0 0 d33 0 0
0 0 0 0 0
0 0 0 0 0
Ex Ey Ez 0 0
Qx Qy Qz 0 0
(49c)
ij , Dij , Bij , ij, En and Qn expresses the magnitude of stress, electric displacement, magnetic induction, nonlinear strain, electric field and magnetic field. Also, sij and ij are the components of dielectric and magnetic permittivity coefficients; furthermore, eij , qij and dij are the piezoelectric, piezo-magnetic and magneto-electric-elastic coefficients, respectively. The reduced stiffness modulus of the sandwich composite can be expressed by:
cos( z ) (48a) 7
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
E11
Q11 =
1
12 E22
, Q12 =
12 21
1
, Q22 =
12 21
E22 1
12 21
, Q44
a N n=1 0
T= (50)
= G23 , Q55 = G13 , Q66 = G12
+
The principal coordinates of the transformed shell are expressed in Appendix A. Now via Hamilton's principle, it can be written as:
x u0
b
nh
n
41 120R1
+
2 15R2
0
(
w02
+
t
(U + V
T ) dt = 0
(51)
0
where, U is strain energy, V is the work done by external energy and T is kinetic energy. The strain energy is expressed as:
U=
N
1 2
a
b
n=1 0
n n xx xx
+
n n yy yy
MR MR xx xx
+
MR MR yy yy
hML
0
0 hML 1 a b
+
n n xz xz
+
+
0
+
MR MR yz yz
+
c c xx xx
c c yy yy
+
c c yz yz
+
c c xz xz
+
+
2
MR xy
b
o
0 hp 1
[ Dx E x z R1
1+
Dy E y
Dz Ez
Bx Q x
c c xy xy
By Qy
N n=1
a b
Bz Q z ]
(52)
a 0 MR Myy
+
b 0 2 k yy
a 0
c 2 + M yy k yy +
0 xx
0 yz
a 0
MR 1 R yy k yz
+
c 0 c N xx xx + Mxx 0 c 2 c 0 Pyy k yy + N xx xy
b 0
0 xz
+
0 xy
MR K xx
0 k xx
+
hp
0 xz
c Pxx
Dx cos( z )
hp 1
(
h /2
+
2 c k xx + N yy
z R1
x z R1
), q
2
(
= c.2 1 +
Z R2
hn
NT =
8 315
R2
)
)
y
8 315
u c0 + v c0 v c0 + w c0 w c0)
w MR 0 ) dc1 dc2 + IML
MR xz
MR xz
MR yz
+
MR yz
w0 w +q 0 x x
w0 dc1 dc2 x
(55)
(56)
0 yz
1
[Q¯ 11
11
+ Q¯ 12
12](
[Q¯ 11
11
+ Q¯ 12
12](H
T = T1 + T 1
2
T
T1) dz
(57c)
H1) dz
(57d)
1 z + 2 2 h
Cos
(58)
u0c , By setting the coefficients of v1c , u1c , v2c and w0c t o zero and substituting Eqs. (53)–(55) into Eq. (51) may be stated as:
u0n ,
y
)(1 + ) dc dc
(57b)
And T T1, H H1 are variation of temperature and moisture, T can be defined by sinusoidal temperature following as:
2
y
B y cos( z ) z R2
2 dz h
hn 1
)(1 + ) dc dc 1
hn
NH =
0 yy
z R2
(57a)
hn 1
(53)
n Nxx x
where, for convenience, a shell with a rectangular base of dimension a and b in c1 and c2 directions, has been considered. q1 and q2 are the Lame coefficients of the shell which can be expressed as. Z R1
q31
MR 1 Rxx k xz
Dy cos( z )
x
Bx cos( z ) dz × 1 +
x
v02
2V dz h
h /2
NQ =
0 yz
0 yy
c 2 c 2 c + Mxy k xy + Pxy k xy + K yy
(
+ Bz sin( z )
(
e31
n0 yy
MR 2 MR 2 + Mxy k xy + Pxy k xy
c 1 + Rxx k xz × 1 +
+ Dz sin( z )
q1 = c1 1 +
0
1 + Rxx k xz
MR 0 MR 2 MR + Mxx k xx + Pxx k xx + Nyy
b
c 1 c + R yy k yz + K xx
+
0 xz
MR 2 MR + Pyy k yy + Nxx
MR K yy
+ +
MR Nxx
w0 x
N x0
h /2
n 2 n 2 n + Mxy k xy + Pxy k xy + K xy
n 1 + Rxy k yz + Kxx
+
h
NE =
n n0 n n2 n + Mxx k xx + Pxx k xx + N yy 0 xy
v0 30R1
+
R1
where:
0 0
n n2 n n2 n + Nyy k yy + Pyy k yy + Nxy
u0 30R2
u02
MR
h /2 n0 xx
c c (u 0
1 4
N x0 = N E + N Q + N T + N H
z dc1 dc2 dz R2
n Nxx
ch
+
)+ (
It is assumed that the doubly curved composite shell is exposed to the external electric voltage V, magnetic potential and variation of temperature and moisture. q is a the dynamic force and N x0 can be written as:
The first variation can be obtained as:
U=
c
V=
hp
a
b 0
u0 120R1
v0 120R2
+
2 15R1
+
2 y
+
and where are the mass densities and IMR is inertia moment of the magneto rheological layer. The first variation of work can be expressed in the following form:
0 hC 1
+
a 0
41 120R2
2 x
(54)
n n xy xy
MR MR xz xz
+
+
(
w0 252q1 c1
w0 252q2 c1
MR MR h MR (w 0
0
hC
+
× 1+
n n yz yz
+
w0 q2 c2
(
y v0
0 0
0 hn 1
b
MR xy
+
n,
hn
a
a b
+
)+
w0 q1 c1
+
12R1 R2
17 315
u0n u0n + v0n v0n + w0n w0n + h2
= I¯0n +
).
+
Further, R1 and R2 are the principal radii of curvature in q1 and q1 directions, respectively. The Kinetic energy can be presented as:
+
n Nxy
+
w0MR
1 hMR t 2 hc4
2
2u
8
4 hc3
I3c
0c
t2
8 hc5
2u n 0 t2
+2
2u n 0 t2
2I5c
+ 4I5c
1 2 hMR
2u n 0 t2
x
M2cxy
+
( ) + (I 2
2 n x t2
4
hn I5c
2 n x t2
I4c
c 2
hc2
2u
0c
t2
+
2u
+ s1 I¯3n 2u
0c
t2
+ 2I6c 8I6c hc6
0c
+
t2
+
w0n ,
n x,
n y,
2MQc 1 xz
y
2 n x t2
hn 2 2hMR
2u 1c t2
+ hn
2u n 0 t2
M2cxx
2 hc2
J¯1n +
+ +
x
+ +
2u n 0 t2
1 2 hMR 2
2I4c
1 Q MR hMR xz
y
v0n ,
( ) ) w0n x
2u I3c 21c t
8 c I hc4 5
2u
2
t2
2u n 1 t2
0c
t2
2
2u n 0 t2
hn
2 n x t2
(59a)
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al. n Nxy
n N yy
+
x
( )+ v0n
1 2 hMR
= I¯0n +
w0MR
2I4c
+
8
+ +
4
n K¯ xx x
n K¯ yy
+
1 hc
+
y
+
n Nxy
(
=
+ I3
t2
+
hMR t 2
+
R1
u0n
x
R1
2P n xy
hc3
n x
4 t2
y
hn 2 2hMR
2w c 0 t2
2w n 0 t2
n y
y
hc2 2w c 0 t2
hc4
= I1c
MQc xz 2 y
2
+ s1 I3
+
1 c I hc 1
2
2w c 0 t2
t2
c N yy
( ) u0n
y
x
n ¯ xy M
+
hn Q MR 2hMR yz
n K¯ xx +
y
2hn 2u n 0 t2
hn 2 2hMR
+
+4 2
n ¯ xy M
x
( )+
4
+
c
I hc2 3
2w n 0 t2
hc3
(
I3c
n ¯ yy M
+
2u n 0 t2
2
hc4
2u
0c t2
2hn hc3
=
2v n 0 t2
J1n
hn 2 2hMR
+
2
+
( )+ 2v
2
hc4
hn I5c
hc3
(I
c 3
2v
0c t2
+ I4c
x
hn
2u
+4
)+4
0c
hn
M3cxy
2 n y t2
hn 2v
1c t2
2 n y t2
hn
hc2
)+4
c 2
2v n 0 t2 2v n 0 t2
hn I6c hc6
2v
0c
t2
M2cyy
2u c 1 t2
(59d)
= I0c
2w c 0 t2
+ hn
2 n x t2
4
2
2u n 0 t2
hn
2 n x t2
2hc
c N xy
4
M2cyy
x
hc2
y
2v c 1 t2
2
+
c
I hc3 3 c
4
2
2v n 0 t2
hn
2 n y t2
2h c
4
c
2
4
y
hc2 2w n 0 t2
hc4
h h c hMR hn
2v n 0 t2
2v n 0 t2
2I2c hc2
2w n 0 t2
cos( z )
2v
4 2hc
(59h)
M3cxy x
2v
M3cyy
+
y
2
4
Ic hc2 4
2v n 0 t2
0c
t2 2v
0c
t2
2v
0c
t2
(59i)
MQc yz 2 y
MQc xz 2 x
2w n 0 t2
2
1c
t2
2h c
2 n y t2
2Mzc + +
0c
2 n y hn 2 t
2 n y t2
+ hn hn
+
2 n y t2
hn
2v n 0 t2
c Q yz
I1c
2
hc2
2v n 0 t2
2 2
I3c
2u n 1 t2
t2
3MQc 2 yz +
hc2
+
2v
2v n 0 t2
4
2h c
2 n y t2
2v c 1 t2
hc
2 n y t2
hn
(59g)
2 n y t2
+ hn
+ hn
I hc3 4
+
2v n 0 t2
2
c Qyz
+ I2c
2v n 0 t2
2
2u n 1 t2
2MQc 1 yz
x
2v n 0 t2
y
+
I2c
hc2
2u n 1 t2
2u n 0 t2
2
M1yy
2
+
2h c
M2cxy
+
2u n 0 t2
4
2w n 0c t2
4I3c hc3
2w n 0 t2
2w n 0c t2
(59j)
Dy Dx + cos( z ) + x y
sin( z ) Dz dz = 0 (59k)
h h c hMR hn + hp
1c
h h c hMR hn
cos( z )
By Bx + cos( z ) + x y
sin( z ) Bz dz = 0 (59l)
1c
hn I6c
2v
where:
1c t2
hn
2 n x t2
hn
y
y
t2
hc5
+
M3cxy
+
2 n hn 2x t
2u n 0 t2
8I4c
t2
2v
2u n 0 t2
2
(59f)
x
2
x
( ) +I ) ( ) ( )
2
hc2
2u n 1 t2
M3cxx
4 c I hc4 5
M2cxy
w0n
2v
c 3
+
2u n 0 t2
2u n 1 t2
2u n 0 t2
2 c I hc3 4
8 c I hc5 6 c Qxz
I3c
2u n 0 t2
4
2hc
3MQc 2 xz +
hc2
4 c I hc4 5
x
+
y
4
c Qxz
hn
+
+
2u 1c t2
hc4
2v n 0 t2
2h c
h h c hMR hn + hp
hn I5c
+8
2 n x t2
2u 1c t2
hn 2hMR t 2
4
2
2v c 0 t2
x
2
y
+
w0n
y w0n
+2
x
M3cyy
( ) + (I
+4
M1xy
y
2 n hn t 2x
2u n 0 t2
x
hn
2u 1c t2
c 3
0c
t2
2MQc 1 yz +
hc2
M2cxy
( ) +I ) 4 ( ) +8 ( )
2
hc6
2u n 0 t2
8 c I hc5 5
= I1c
hc5
hn I6c
2
I hc4 4
2 hn 2hMR t 2
2u
4
4
+
x
hn I6c
0c t2
s1 J4n 2 t
hn
M2cxx
hn I5c hc4
2u
+2
+
2 n y t2
c 2
t2
2
hn2 2 4hMR
2v n 0 t2
w0n
hc2
3MQc 2 yz +
2v n 0 t2
2
hc4
hn
1c
t2
hn I4c
+4
k2n
y
( ) + (I
2u 1c t2
2 n y t2
x 2
2 n x t2
hn Q MR 2hMR yz
M3cyy
t2
hn I4c
+
n K¯ yy +
y
2 n hn t 2x
2u n 0 t2
2
2 n x t2
M3cxy
2 n x t2
2v n 0 t2
2w n 0 t2
+2
2MQc 1 xz +
hc2
s1 J4n
hn2 2 4hMR
2u 1c t2
hn I5c
hn
2 n x t2
+ k 2n
hn I4c hc4
hn
3MQc 2 yz +
hc3
= J1n
+
+ hn
+ I1c
(59c) n ¯ xx M x
2u n 0 t2
+
2v c 0 t2
= I0c
2w n 0 t2
2 n x t2
2
8 c I hc5 6
R2
2u n 0 t2
2
2 n x t2
+ hn
4 c I hc4 4
2u n 0 t2
n N yy
2u n 0 t2
2
hc2
2MQc 1 xz
y
2 c I hc3 3
+ I2c
4 c I hc2 4
I2c
+
M2cxy
+
x
+
y
2u c 0 t2
M2cxx
2u c 1 t2
M1xy
+
)
2u n 0 y2
I2c
4I4c
+
+
n Nxx R1
y
+
y
R2
M1xx x
(59b)
)
R2 v0n
2 n x
y
t2
2Mzc +
y
+
+
x
hn
v0n
y
2
MQc xz 2 x
w0n
w0n
2w n 0 x2
2 s12 I6n 2 t 2
hc2
2P n yy y2
+
x y
2
+
hc2
8 c I hc5 5
2v n 0 t2
2
4
y
2u n 0 t2
4 c I hc2 3
1c
t2
c N xy
+ I1c
0c t2
hc6
n xy
2v
2u c 0 t2
= I0c
y
+
2v
8I6c
n yy
+2
v0n
2v n 1 t2
)+N ( )+N (
x
w0n
8 c I hc4 5
MQc xz 1 y
w 0n
t2
+ I3c
+ 2I6c
+
+
( )+J ( )+
4I3c
0c
u0n
y
2
1
y t2
( ) )
2
+ s1 I¯3n
c 2v 0c 2 t2
+
2
t2
w 0n
v0n
2
2v
MQc xz 1 x
2w n 0 t2
+ IMR)
2v n 0 t2
4 hn I5c
2P n yy x2
+ s1 (I0n
y
I4c
(
2
hc2
c N xx x
2MQc 1 yz
y
2 n y t2
1c
t2
2v n 0 t2
n Nxx
M2cxy
+
hn 2 2hMR
t2
Rzc + x
x
2v
2v n 0 t2
+ 4I5c
M2cyy
( ) + (I 2
+ hn
2I5c
hc5
2v n 0 t2
I3c
hc3
2v n 0 t2
2
hc4
1 2 hMR
+
y
2 hc2
J¯1n +
t2
2 1 hMR t 2
+
1 Q MR hMR yz
y
2 n y t2
¯ ijn = Mijn M K¯ ijn
(59e) 9
=
K ijn
s1 Pijn s2 Rijn
(ij = x , y ), (ij = x , y )
s1 =
4 , s2 = 3s1 3h2
(60a) (60b)
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al. n n n n n n , Nyy , Nxy , Myy , Mxy Here, Nxx and Mxx express the total in-plane force n n n n n , Pyy , Pxy , R yy and moment resultants and Pxx and R xx are the third order stresses resultants which can be written as:
hn /2
n n n Nxx , Mxx , Pxx =
n x (1,
zn , z n3) dz
n y (1,
zn , z n3) dz
hn /2 hn /2
n n n Nyy , Myy , Pyy =
hn /2
n xy (1,
zn , zn3) dz
hn /2
hn /2
n n Pxx , R xx =
n xz (1,
z n, zn3) dz
n yz (1,
zn , z n3) dz
hn /2 hn /2
n n Pyy , R yy =
hn /2
h c /2
c x (1,
z c ) dz
hc /2 h c /2
c c N yy , M yy =
c y (1,
z c ) dz
c xy (1,
z c ) dz
hc /2 h c /2
c c N xy , Mxy =
hc /2
c c Qxz , Mnxz =
h c /2
c xz (1,
z c ) dz
h c /2
c c Qyz , M yz =
hn /2
n yz (1,
z c ) dz
hn /2
Rzc , Rzc =
c x (1,
z c ) dz
(61a)
hn /2
(61b)
(Aij , Bij , Dij , Eij, Fij ) =
MR Qyz =
MR xz (1,
zMR ) dz
(61c)
( ij,
MR yz (1,
zMR) dz
=
MR Q¯ij (1, zMR ) dz (i, j = 1,2,6)
A¯ ij = Aij
3s1 Dij , D¯ ij = Aij
3s1 Fij , Aˆ ij = A¯ ij
hn /2
Jin = Iin
s1 Iin+ 2,
K¯ 2n = I2n
2s1 I4n + s12 I6n
(i = 1,4)
s s I¯0n = I0n + 2 1 I3n + 1 R1 R1
Bˆij = Bij
s1 Eij , Fˆij = Fij
s1 Hij , Dˆ ij = Dij
(62a)
The boundary conditions of the multiscale composite have been considered to be simply-supported (S-S) whose constraints can be mentioned as follows:
(62b)
u 0 (x , 0, t ) = u 0 (x , b, t ) = 0 ,
v0 (0, y, t ) = v0 (x , b, t ) = 0
(62c)
y (0,
y, t ) =
y (a ,
y , t ) = 0,
x (x ,
0, t ) =
¯ x (x , 0, t ) = M ¯ x (x , b , t ) = 0, M ¯ y (0, y, t ) = M ¯ y (a , y , t ) = 0 M
(66f)
{umn, uocnm, u1cnm}cos
{v0, v0c , v1c } =
(62g)
{vmn, v0cnm, v1c}sin
{w0, w0cnm} =
(62h)
x
=
{wmn, w0cnm}sin
=
(63a)
n x sin(my ) b
(67b) (67c) (67d)
y mn sin
n x cos(my ) b
(67e)
mn (t )cos
n x sin(my ) b
(67f)
mn (t )cos
n x sin(my ) b
(67g)
n= 1 m = 1
(x , y , t ) =
Umn (t ), Vmn (t ), Wmn (t ), x mn (t ), y mn (t ), u0c (t ) ,v1c (t ), u1c (t ) , where v2c (t ) and w0c (t ) refer to the unknown functions of the time; n and m are the number mode of frequency in the x and y directions, respectively. n x Here, b = l are assumed. By substituting Eq. (67) into Eq. (59) and driving the Navier procedure, the following expressions can be expressed:
(63d)
(63f)
n x cos(my ) b
(67a)
n x sin(my ) b
(x , y , t ) =
(63b)
n x sin(my ) b
x mn cos n=1 m=1
n=1 m=1
(63e)
(66c) (66e)
n=1 m =1
s I¯3n = I3n + 1 I5n R1
b, t ) = 0
The displacements are assumed as the series of double trigonometric functions that satisfy the boundary conditions:
(62e)
s J¯1n = J1n + 1 I4n R1
x (x ,
Ny (0, y, t ) = Ny (a , y , t ) = 0
Nx (x , 0, t ) = Nx (x , b, t ) = 0 , (62d)
(66a) (66b)
w0 (x , 0, t ) = w0 (x , b, t ) = 0
n=1 m=1
I6n
(65b)
s1 Fij
3. Solution procedure
(63c) 2
3s1 D¯ ij (i, j = 4,5) (65a)
(61e)
y
(i = 0, …, 6)
(64b)
where the cross-sectional rigidities can be expressed as follows:
(61d)
hn /2 n z idz ,
n=1
ij)
hMR /2
The mass inertias of the composite shell can be expressed in the following form:
Iin =
(i , j = 1,2,6) (64a)
hMR /2
n=1 m=1
hMR /2
N
n Q¯ij (1, z, z 2 , z 3, z 4 , z 6) dz,
n=1 m=1
hMR /2 hMR /2
(63g)
Equations of motion of multiscale composite shell can be expressed in terms of u0n , v0n , w0n , xn , yn , u 0c , v1c , u1c , v2c and w0c displacements that are obtained by substituting Eq. (34) into (59) yields the relations expressed in Appendix 2
(62f)
hn /2
QxzMR =
(i = 0, .., 5)
{u, uoc , u1c } =
hn /2
hMR /2
c z idz , h c /2
Following integrals were regarded as stress results within the core and magneto-rheological layers which can be written as: c c N xx , Mnxx =
h c /2
hn /2
hn /2
n n n Nxy , Mxy , Pxy =
Iic =
10
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al. n n n Lij Umn (t ) + Lij Vmn (t ) + Lij Wmn (t ) + Lij
n x mn (t )
n y mn (t )
+ Lij
+ Lij v0c (t ) + Lij u1c (t ) + Lij v1c (t ) + Lij w0c (t ) + Lij
mn
+ Lij u0c (t )
+ Lij
mn
0:
D02 w0 +
2w
1:
D02 w1 +
2w
n n ¨ mn = Mij U¨mn (t ) + Mij V¨mn (t ) + Mij W (t ) + Mij ¨x mn (t ) + Mij ¨y n (t )
+ L36
+
n L32 Vmn (t )
(68a)
n y mn
+
n L33 Wmn (t )
+
3 L34 Wmn (t )
+
mn
+ L313 +
mn
, ( i= 3, j= 1: 10)
(68b)
2i
where the operators Lij are defined as in Appendix 3. By substituting Eq. (68a) into Eq. (68b) to solve the unknown functions Umn (t ), Vmn (t ), x mn (t ), y mn (t ) , u0c (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:
d 2Wmn (t ) + µw + P1 Wmn (t ) + P2 Wmn 2 (t ) + P3 Wmn3 (t ) = q cos t dt 2
M33 + M34 + M35 a34
=
(69)
(72)
Next, the system response during the resonance excitation, 1, is considered. The proximity of the excitation frequency to the system natural frequency is expressed as (73)
=1+
2w
2 (T0,
T1, T2, …)
a
3 P3 3 a 8 0
=
)
1 F¯ cos( T1 2 0
(82)
)
(83)
(84)
(75a)
T1 = t
(75b)
1 q¯ sin 2 0
a =
µa +
a
3 P3 3 a 8 0
=
µa +
(85)
1 q¯ cos 2 0
(86)
1 q¯ sin 2 0
3 P3 3 a = 8 0
a
(87)
1 q¯ cos 2 0
(88)
The fixed points of Eqs. 87 and 88 correspond to solutions with constant amplitude and phase. These solutions satisfy
(74)
T0 = t
µa =
1 q¯ sin 2 0
(89)
1 q¯ cos 2 0
3 P3 2 a = 8 0
(90)
Squaring and adding these equations, one may obtain the frequency response equation:
After substituting Eq. (74) into Eq. (69) and noting Eqs. (76) and (75), the following hierarchy of equations can be obtained for O(1) and O(e), respectively.
+ 2D0 D1)
(80)
(81)
1 F¯ sin( T1 2 0
µa +
a=
where the fast timescale T0 and slow timescale T1 are given by
2 (D 2 1
(79)
The point at which a = 0 and = 0 corresponds to a singular point of the system and shows the steady-state motion of the system. So, in the steady-state condition it can be written:
where is called the detuning parameter, which is a measure of how close the excitation frequency is to the natural frequency. Noting that the steady-state solution for the forced linear oscillator is acos ( t + ) , where the amplitude a and phase f are slowly varying quantities. This analytical approximation is an example of a generalized asymptotic series, as the coefficients are also functions of the asymptotic ordering parameter, which is e in this case. Such analytical approximations can be constructed by using the method of multiple scales or the method of averaging. This construction is illustrated here by using the method of multiple scales. Let:
d = D02 + 2 D0 D1 + dt
+ T1)
and substituting Eqs. 82 and 83 in Eq. (84) leads to:
t=0
d = D0 + D1 dt
1 q exp( i T1) = 0 2
= T1
=0
w = w0 (T0, T1, T2, …) + w1 (T0, T1, T2, …) +
0 T0
Term T1 can be eliminated by transforming Eqs. 82 and 83 to an autonomous system considering:
(71)
¯ dWmn (t ) W , h dt
kcos (
1 a exp(i ) 2
a =
where the initial conditions are illustrated by:
Wmn (0) =
P3 w03
where a and are real. Separating real and imaginary parts of the derived equation, it yields
(70)
p1
¯ + µA) + 3P3 A2 A
0 (A
A=
And the linear frequency of the composite shell is expressed by: l
µD0 w0
Let A be in the polar form:
where:
P3 =
2D0 D1 w0
where A is an unknown complex function and A¯ is the complex conjugate of A. The governing equations for A are obtained by requiring w1 to be periodic in T0 and extracting secular terms which are coefficients of e ±i 0 T0 , the solvability equation will be determined as:
n n = M33 Wmn (t ) + M34 ¨x mn (t )
M35 ¨ y n mn
=
w0 (T0, T1, T2, …) = exp(iT0 ) + A¯ exp( iT0 )
n L35 x mn (t )
+ L37 u 0c (t ) + L38 v0c (t ) + L39 u1c (t ) + L310 v1c (t )
+ L311 w0c (t ) + L312
1
(77)
With this approach, it turns out to be convenient to write the solution of Eq. (65) as:
+ Mij u¨ 0c (t ) + Mij u¨1c (t ) + Mij v¨0c (t ) + Mij v¨1c (t ) + Mij w¨ 0c (t )
n L31 Umn (t )
=0
(78)
mn
(i = 1: 12 Except 3, j = 1: 12)
0
3 P3 2 a 8 0
2
+ µ2 a2 =
q¯2 4 02
(91)
(76a)
Substituting Eq. (90) into Eq. (88) and substituting the resultant equation in Eq. (74), one may obtain the first approximation:
(76b)
w= a cos(
0
t+
t
) + O( )
(92)
With this, the amplitude response (magnification factor) can be obtained as:
Substituting Eq. (76) into Eq. (69) and equating the coefficients of to zero yield the following differential equations: 11
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
M=
a = q¯
1 2
0
(
)
3 P3 2 2 a 8 0
+ µ2
Table 2 Material properties of SMA [61].
(93)
Composite matrix Properties
Similar to the case of the linear oscillator, the maximum value of the magnification factor can be found from
dM d 2M = 0, 2 d d
1 a (3P3 a2 32
8
8) 3P3
which can be solved for
da = d 27P32 a4
da d
da d
8
8) = 0ap =
8(
dM ) d
1)2)
1)
(95)
96(
(97)
1)
=
e31 = −2.2, e33 = 9.3, e15 = 5.8 d11 = 5.64, d33 = 6.35 q15 = 275, q31 = 290.1, q33 = 349.9 s11 = 5.367, s33 = 2737.5
Face sheets
E11 = E22 = E33(GPa) = 0.10363 G12 = G23 = G11=(GPa) = 0.05
E11 = 24.51 (GPa) E22 = E33(GPa) = 7.77 G12 = G13(GPa) = 3.34
( ) = 130 kg m3
12
1 8 + 6P3 a2 8
6
6
Core
whose roots provide 1,2
0.07 × 10
9P32 a4
64µ2
6
Table 4 Material properties of sandwich panel [64].
(98)
1) 2)
S
Piezoelectric/(C・m−2) Dielectric/(10−9C・V−1・m−1) Piezomagnetic/(N・A−1・m−1) Magnetoelectric/(10−12Ns・V−1・C−1)
1)
1) P3 a2 + 64(µ2 + (
1) = 10.26 × 10 = 0.33
11 (C
Properties BiTiO3-CoFe2O4
(96)
To find the values of the critical points 1 and 2 , these points correspond to vertical tangencies of the response curve; that is, where d = 0 : This condition can be found by equating the denominator of Eq. dM (97) to zero, which translates to:
27P32 a4
=
ES From Fig. 1, (Park et al. , 2004) From Fig. 1(b) [61],
Table 3 Material properties of BiTiO3-CoFe2O4 [63].
when:
3P3
1)
1 22 (C ) = 30.1 × 10
as:
8a (3P3 a2 8 8) 1) P3 a2 + 64(µ2 + (
96(
11 (C
da =0 d
r
G23(GPa) = 3.25 12 = 0.22
yields:
4 + µ2 + (a2 )
This derivative vanishes (and so does
(3P3 a2
E11(GPa) = 155 E22(GPa) = 8.07 G12(GPa) = 4.55
(94)
Differentiating Eq. (93) with respect to
SMA
G23(GPa) = 1.34 12 = 13 = 0.078 23 = 0.49
= 0.33
(99)
( ) = 1800 kg m3
4. Results and discussion Table 5 Convergence study of linear/nonlinear frequency parameter of sandwich composite shell under hygrothermal environment.
Numerical results of the nonlinear vibration behavior of doubly curved shell with MR layers and flexible core are presented in this section. The properties of the smart shell are established in Tables 1–4. Furthermore, Elliptic paraboloid shell (R1 R2) is considered and the material properties of which are assumed to be tcnt = 0.0348 nm and tgpl = 0.0348 nm and G13 = G23 = 0.5G12. Table 5 illustrates the linear and nonlinear frequency behavior of sandwich composite shell in two sets of hygrothermal conditions under the influence of curvature ratios (R/a) and volume fractions of SMA. It can be noticed that the temperature rise has a minute effect on the nonlinear forced vibration. Also, it can be witnessed that by increasing the SMA volume fraction, the fundamental natural frequency of the shell will increase.
%SMA
R/a
T= 300, l
0
5 10 50 5 10 50
10
T= 400,
H= 0 nl
13.0634 12.8629 12.7288 13.1882 12.9321 12.8361
H= 0.5
l
13.1491 13.0762 12.9522 13.3192 13.1632 13.0981
nl
13.0805 12.8793 12.7618 13.2109 12.9842 12.8824
13.1523 13.0790 12.9903 13.3723 13.2279 13.1203
Table 1 The properties of multiscale composite shell [13,36]. Carbon (fiber)
Epoxy (matrix)
E11f (GPa ) = 233.05
v m = 0.3 m
E11f (GPa ) = 23.1
= 0.6
f
( ) kg m3
m
11 (K
1)
=
22 (K
1)
= 10.8 × 10
E cn (GPa)= 640(1
0.54 × 10
0.0005 T ) 9
t cn (m) = 0.34 × 10
0.0034T + 0.142H )
(K 1) = 45(1 + 0.001T ) × 10
= 2.68 × 10
= 0.2
Graphene platelet
dcn (m) = 1.4 × 10
(kgm) = 1200
Em (Gpa) = (3.51
G12f (GPa) = 8.96 f
Carbon nanotube
lcn (m) = 25 × 10
6/K
3wt% 1
lcn
6
6
12
9
6
(m) = 0.25 × 10
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 143 (2019) 106152
M. Karimiasl, et al.
Table 6 Comparison of dimensionless frequencies of (0°/90°/0°/core/0°/90°/0°). Dimensionless frequency ( ) Mode(m, n)
Rahmani et al. [64]
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
Table 7 Comparison of nonlinear frequency ratio of [0°/90°/90°/0°] laminated plate (R1 = R2 = ). Applied voltage
nl l
50
[65] Present [65] Present [65] Present
0 −50
0.2
0.4
0.6
0.8
1
1.019 1.01919 1.01921 1.01993 1.020 1.02018
1.075 1.07621 1.075 1.07590 1.075 1.07601
1.161 1.16284 1.163 1.16401 1.164 1.16591
1.272 1.27491 1.274 1.27521 1.276 1.27783
1.402 1.40261 1.405 1.40626 1.408 1.40889
Fig. 3. Influence of different distributions patterns on large deflection versus a b nonlinear of doubly curved shell with R = 0.1m, R = 0.05m q = 0.015 and (m, n = 1,2).
y,
y¯ =
y , x¯ = b
x , a (100)
Fig. 3, illustrates the effect of different porosity distributions pattern such as A, B, C, D versus nonlinear to the linear frequency with a b a a = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1, B = 500 and the cross R1 2 ply [0/MR/core/MR/0] stacking sequence. Composite shell mode (m, n) = (1,1) is considered. It can be observed from the evaluation that the frequency of the C distribution is highest and D is the lowest value. Influence of curvature ratio ( b ) on the fundamental frequency of R2 cross ply [0/MR/core/MR/0] multiscale doubly curved shell with, a = 0.1, h = 2 mm is presented in Fig. 4. The frequency mode is taken R1 to be (m, n = 1, 1). It is found that increasing the curvature value reduces the nonlinear frequency. It is important to express that the effect of curvature ratios has a significant role in the numerical results. It can be found that when the circumferential wave number decreases, the hardening effect declines. Furthermore when the circumferential wave number stays constant, the hardening effect increases. This is because of the fact that an increase in nonlinear curvature ratio ( b ) leads to a R2 decrease in the stiffness of structure which causes natural frequency and stability to decrease. The influence of magnetic field on the frequency-amplitude curve,
Table 8 Comparison of the curvature ratios (R/a) and volume fractions of SMA with different amplitude ratios versus nonlinear frequency of composite spherical a panel with h and R1 = R2 . R/a
2
u0 v0 w , v¯0 = , w¯ 0 = 0 , ¯x = x , ¯y = a b R z ¯ R1 ¯ R2 R2 T z¯ = , R1 = , R2 = , = l h R R h ET
u¯ 0 =
As a part of the validation of the present method, the correctness of linear frequency of doubly curved shell with h/a = 0.10 and hc/ h = 0.88 based on first shear deformable theory are compared with Rahamni et al. (2010) and presented in Table .6. Due to different selections of displacement field of the core, there is little difference between the results of the present work with Rahamni et al. (2010) researches. The nonlinear frequency ratio of the laminated plate is presented in Table 7 based on TSDT, in which a comparison is made with the results reported in Ref. [65]. It is brightly shown that the numerical results are very similar. Finally, the influence of curvature ratios (R/a) and volume fractions of SMA with different amplitude ratios on the nonlinear frequency of a composite spherical panel with h and R1 = R2 are established in Table 8. Also, the results are compared with [66] based on the HSDT. These comparisons show that the results from the present method are in good agreement with existing results. The dimensionless parameters are adopted as:
%SMA
1
nl
A 1.2
0
5
10
5 10 50 100 5 10 50 100 5 10 50 100
3.4
5.8
Parhi (2016)
Present
Parhi (2016)
Present
Parhi (2016)
Present
39.83 30.28 20.52 18.29 41.59 30.44 20.52 17.09 41.58 34.10 25.99 19.96
38.77 29.14 20.01 17.55 40.70 29.84 20 16.49 40.66 29.78 25.83 19.24
35.57 27.46 19.96 17.01 35.81 26.82 19.96 17.81 34.54 32.94 25.94 21.71
35.11 26.44 19.80 16.43 35.03 26.13 19.83 17.09 33.88 32.17 25.80 20.81
28.69 21.84 17.65 14.86 27.25 23.20 17.65 14.94 32.66 22.32 21.53 13.82
28.24 21.10 17.57 14.16 26.87 22.53 17.49 14.32 32.21 21.75 21.41 13.14
Fig. 4. Influence of different curvature ratio and magnetic field on large dea b flection versus nonlinear of doubly curved shell with R = 0.1m, R = 0.05m , h = 2 mm, q = 0.015 and (m, n = 1,2).
13
1
2
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
Fig. 7. Influence of temperature rise and magnetic potential on large deflection a b versus nonlinear of doubly curved shell with R = 0.1m, R = 0.05m , h = 2 mm,
Fig. 5. Influence of different magnetic field on large deflection versus nonlinear a b of doubly curved shell with R = 0.1m, R = 0.05m , h = 2 mm, q = 0.015 and 1
(m, n = 1,2). a R1
q = 0.015 and (m, n = 1,2).
2
b
1
2
= 0.1, R = 0.05, h = 2 mm, h = 10, b = 1 , q = 0.015 and (m, 2 n = 1,1) is shown in Fig. 5. Stacking sequence considered is cross ply [0/MR/core/MR/0]. It is brightly shown that the nonlinear frequency parameters increase by increasing the magnetic field. Furthermore, the nonlinear frequency is not only the function of stiffness and mass matrix but also depend up on the amplitude ratio. Fig. 6 depicts the effect of 2hMR with U distribution pattern versus a
frequency-amplitude a h
= 10,
a b
curve
h
with
a
a R1
= 0.1,
b R2
= 0.05,
h = 2 mm,
= 1, q = 0.015 and stacking sequence [0/MR/core/MR/0].
The results confirm that 2hMR have a minimal effect on the fundamental h frequency ratios of the doubly curved shells. Further, it can be observed that by increasing the MR layer thickness, the nonlinear frequency decrease due to a reduction in the stiffness and dynamics stability of the sandwich structure. The influence of temperature rise and different magnetic field on the a b frequency-amplitude curve with R = 0.1, R = 0.05, h = 2 mm, 1
Fig. 8. Influence of curvature ratio and applied voltage on large deflection a b versus nonlinear of doubly curved shell with R = 0.1, R = 0.05, h = 2 mm,
2
= 10, b = 1, ΔH = 1 and (m, n = 1,1) is shown in Fig. 7. Stacking sequence is considered cross ply [0/MR/core/MR/0]. Two set of T = 300 and 400 K and B = 0, B = 50 are considered for evaluation. It is clearly shown that nonlinear frequency increases with rising temperature and decrease by increasing of magnetic field. In addition, according to the observation made, it can be revealed that the a h
a
T = 300, ΔH = 1 q = 0.015 and (m, n = 1,2).
curved shell with n = 1,2).
2hMR h a = R1
nonlinear amplitude have been presented in Fig. 9. With R = 0.1, 1 h = 2 mm, T = 300, ΔH = 1, V = 50, q = 0.015 and (m, n = 1,2). PGF PCF Stacking sequence is considered cross ply [0 /90 ]S. It is brightly observed that the curvature ratio has a reducing effect on nonlinear frequency. Furthermore, negative magnetic potential has a lower influence on nonlinear amplitude curve than positive magnetic potential magnitude. It can be found that when the circumferential wave number
ratio on large deflection versus nonlinear of doubly
0.1m,
b R2
2
temperature variation only has a small influence on the nonlinear frequency ratios of the multiscale doubly curve shell. Influence of curvature ratio ( b ) and applied voltage with positive R2 and negative magnitude under the fundamental frequency of cross ply a PGF PCF [0 /90 ]S multiscale doubly curved shell with R = 0.1, h = 2 mm, 1 T = 300, ΔH = 1, q = 0.015 and = 0.2 is presented in Fig. 8. The frequency mode is taken to be (m, n = 1,1). It is found that increasing of curvature value leads to a decrease in the nonlinear frequency. Meanwhile, the positive magnitude of applied voltage has the significant effect on nonlinear frequency whereas the negative applied voltage has the lowest magnitude. This is because of the fact that an increase in nonlinear curvature ratio ( b ) leads to a decrease in the R2 stiffness of structure which causes natural frequency and stability to decrease. For investigated effects of different magnetic potential and curvature ratio ( b ), the numerical results of nonlinear frequency versus R2
Fig. 6. Influence of
1
= 0.05m , h = 2 mm, q = 0.015 and (m,
14
a
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
Fig. 9. Influence of curvature ratio and magnetic potential versus on large a b deflection versus nonlinear of doubly curved shell with R = 0.1, R = 0.05, h = 2 mm, T = 300, ΔH = , q = 0.015 and (m, n = 1,2).
1
Fig. 11. Influence of aspect ratio and magnetic potential on large deflection a b versus nonlinear frequency of doubly curved shell with R = 0.1, R = 0.05,
2
1
2
Fig. 12. Influence of the SMA volume fraction on large deflection versus nona linear frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300,
Fig. 10. Influence of aspect ratio and applied voltage on large deflection versus a b nonlinear of doubly curved shell with R = 0.1, R = 0.05, h = 2 mm, T = 300, ΔH = 1,q = 0.015 and (m, n = 1,2).
1
h = 2 mm, T = 300, ΔH = 1 q = 0.015 and (m, n = 1,2).
1
ΔH = 1, q = 0.015 and (m, n = 1,1).
2
potential. The reason is that, by increasing the aspect ratio, the dynamic stability of the structure increase. Fig. 12 depicts the effects of different SMA volume fractions on the nonlinear forced vibrations of the doubly curved shell subjected to the hya b a a grothermal loading with R = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1,
decreases, the hardening effect declines. Furthermore when the circumferential wave number stays constant, the hardening effect increases. Fig. 10, investigated the effect of aspect ratio and kind of applied voltage versus frequency-amplitude curve of multiscale doubly curved a b cross-ply [0 PCF /90 PGF ]S shell with R = 0.1, R = 0.05, h = 2 mm, 1 2 T = 300, ΔH = 1, q = 0.015 and = 0.2 . It is clear that aspect ratio is the important factor to study of thick curved panel structures. The nonlinear to linear frequency increase by increasing the aspect ratio and positive magnitude applied voltage. Furthermore, by increasing the value of aspect ratio leads to an increase in the fundamental frequency. In addition, the decrease in the circumferential wave number causes a decrease in the hardening effect. The reason is that, by increasing the applied voltage, the dynamic stability of the structure increase. Thereafter, the influence of aspect ratio and different magnetic a b potential versus nonlinear amplitude curve with R = 0.1, R = 0.05, 1 2 h = 2 mm, T = 300, ΔH = 1,V = 50, (m, n = 1,1), q = 0.015 and the PGF PCF cross ply [0 /90 ]S multiscale doubly curved shell are study in Fig. 11. Via numerical results, it is observed that by increasing the aspect ratio and the magnetic potential, the nonlinear frequency increases and the most influence refer to positive magnitude of magnetic
1
2
T = 300 ΔH = 1,q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S, where the dimensionless vibration amplitude ratio is plotted versus the excitation frequency ratio. It is brightly shown that by increasing the SMA volume fraction the non-dimensional amplitude and hardening effects decline. Furthermore, by increasing the SMA volume fraction the fundamental natural frequency of the shell will increase. It indicates that the higher SMA volume fraction results in the larger stiffness of the doubly curved shell. In addition to, the corresponding circumferential and longitudinal wave numbers are reported in the figure. The effects of temperature rise and the SMA volume fraction influence versus the frequency response curves of the doubly curved shell a b = 0.1, R = 0.05, have been shown in Fig. 13 with R a
a
1
2
h = 2 mm, h = 10, b = 1, T = 300 ΔH = 1, q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S. The results show in which by increasing of thermal loading yields to the reduction of the fundamental natural frequency of the shell. In addition to, it is shown that the temperature rise have very small effects on nonlinear forced vibration. Also, by increasing of SMA volume fraction the fundamental natural frequency of the shell will increase.
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Fig. 15. Influence of curvature ratio and SMA on large deflection versus nona linear frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300,
Fig. 13. Influence of temperature rise and SMA on large deflection versus a nonlinear frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300,
ΔH = 1, q = 0.015 and (m, n = 1,1).
1
ΔH = 1, q = 0.015 and (m, n = 1,1).
1
Fig. 16. SMA modulus of elasticity versus temperature.
Fig. 14. Influence of aspect ratio and SMA on large deflection versus nonlinear a frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300, ΔH = 1, q = 0.015 and (m, n = 1,1).
1
Fig. 14, illustrate the influence of curvature ratios on the nonlinear a b frequency of doubly curved shells with R = 0.1, R = 0.05, a
1
a
2
h = 2 mm, h = 10, b = 1, T = 300 ΔH = 1,q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S. It can be seen that, increasing the curvature ratio accompanies with the decrease of the fundamental natural frequencies and dimensionless maximum deflection and also yields the change in the circumferential wave number. It can be found that when the circumferential wave number decreases, the hardening effect declines. Furthermore when the circumferential wave number stays constant, the hardening effect increases. Fig. 15 shows the influence of aspect ratio on the nonlinear frea b a a quency with R = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1, T = 300 1
Fig. 17. SMA recovery stress versus temperature.
Fig. 18, investigates the effect of aspect ratio on the linear and nonlinear frequency of multiscale doubly curved cross-ply a b [0 PCF /90 PGF ]S shell with R = 0.1, R = 0.05, h = 2 mm, T = 300, 1 2 ΔH = 1, q = 0.015. It is clear that aspect ratio is the important factor to study of thick curved panel structures. The nonlinear and linear frequency increases by increasing the aspect ratio. Influence of curvature ratio ( b ) on the linear and nonlinear freR2 quency of cross ply [0/MR/core/MR/0] multiscale doubly curved shell a with, R = 0.1, h = 2 mm, T = 300, Vs = 0%, B = 0 and ΔH = 1is 1 presented in Fig. 19. The frequency mode is taken to be (m, n = 1, 1). It can be observed that when the composite shell becomes deep, its
2
ΔH = 1, q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S. The results indicate that with increase the aspect ratio, the circumferential wave number and corresponding vibrational mode shape stay constant, by increasing hardening effects. In addition to, the decrease in the circumferential wave number causes to decrease in the hardening effect. Figs. 16 and 17 depict the variation of the SMA Young's modulus and the recovery stress versus temperature rise with high nonlinearity. The influence of the initial stresses and initial deflections are considered in the incremental method.
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Fig. 18. Influence of aspect ratio on large deflection versus nonlinear frequency a of doubly curved shell with R = 0.1, h = 2 mm, T = 300, ΔH = 1, q = 0.015 and (m, n = 1,1).
1
Fig. 19. Influence of curvature ratio on large deflection versus nonlinear frea quency of doubly curved shell with R = 0.1, h = 2 mm, T = 300, ΔH = 1, q = 0.015 and (m, n = 1,1).
1
bending energy is low as compared to stretch in energy and it subsequently affect the final responses of the structure. Furthermore, the nonlinear frequency is not only the function of stiffness and mass matrix but also depend up on the amplitude ratio. It is found that increasing the curvature value lead to a decrease in the linear and nonlinear frequency. It is important to express that the effect of curvature ratios has significant role in the numerical results. 5. Conclusions Numerical investigation of nonlinear forced vibration of intelligent multiscale SMA/matrix composite doubly curved shell in the hygrothermal environment is studied in this research. Micromechanical model is used to estimate the temperature dependent material properties of different materials. The major difference between the present model and the previous one is that the present model includes the smart composite material with a flexible core. Three types of porosity are considered. The results reveal that the C pattern in CNT/GPL as the reinforcement through the thickness results in a stiffer shell when compared to other types of porosity, also the result presented volume fraction of SMA, Positive applied voltage and magnetic potential have a significant effect on the fundamental frequency of composite shell. Based on observation it is found that increasing of SMA volume fraction yields decreases of nonlinear frequency. 17
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Full length article
Nonlinear forced vibration of smart multiscale sandwich composite doubly curved porous shell
T
Mahsa Karimiasla, Farzad Ebrahimia,∗, Vinyas Maheshb a b
Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran Department of Mechanical Engineering, Nitte Meenakshi Institute of Technology, Yelahanka, Bangalore, India
ARTICLE INFO
ABSTRACT
Keywords: Nonlinear forced vibration Multiscale composite sandwich curved shell MR SMA Porosity Hygrothermal loading
This is the first research on the nonlinear vibration analysis of composite sandwich doubly curved shell with a flexible core integrated with a piezoelectric layer. By using the higher order shear deformable theory (HSDT) for the face sheets and the third-order polynomial theory for the flexible core, the strains and stresses are obtained. It is assumed a smart model including multiscale composite layers shell with a flexible core and magnetorheological layer (MR) that leading up by the nonlinearity of the in-plane and the vertical displacements of the core. Three-phase composites shells with polymer/Carbon nanotube/fiber (PCF) and polymer/Graphene platelet/fiber (PGF) and Shape Memory Alloy (SMA)/matrix either according to Halpin-Tsai model have been considered. The governing equations of multiscale shell have been derived by implementing Hamilton's principle and solved by multiple scale method. For investigating the correctness and accuracy, this paper is validated by other previous researches. Finally, the effect of different parameters such as temperature rise, various distributions pattern, magnetic fields and curvature ratio are explored in detail.
1. Introduction Vibration and suppression of noise is a major concern in many loadbearing structures such as aerospace, marine vehicles and automotive. However, the vibrations can be suppressed via vibration isolators by shifting the natural frequency. In addition, the sandwich structures with damping layers can also 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, magnetorheological (MR) materials with a flexible core and oil layer applied with the magnetic field played an important role in the smart structures and materials constructions. MR is usually composed of micro/nano-magnetic particles suspended within a fluid, which is generally oil. By applying a magnetic field, the viscosity of the liquid is fairly increased to form a viscoelastic solid. It is noteworthy to mention that, when the liquid is at its active state, it can be precisely controlled by changing the intensity of the magnetic field due to the stress developed. The primary studies on these fluids have been discovered by Jacob [1]. Also, The composite material individual including stiff reinforcement fibers and matrix employ in the aerospace and other industries, are Carbone nanotube–reinforced, Graphene platelet-reinforced (CNTF and GPLF respectively) which are strong and stiff (for their density). A composite ∗
material with most or all of the useful (stiffness, low density, high strength, and toughness) is achieved with few or none of the especially weaknesses of the component materials. Numerous multiscale models are used for micromechanics and atomic simulations to investigate the constitutive properties of different functionalized nanotube materials [2,3]. The mechanical response in a single polymer-CNT using finite element method (FEM) analysis is investigated by Ref. [4]. Although shape memory alloys (SMAs) have been used for varied smart applications and components of adaptive structures, one of the most significant forms of SMAs application is employing fibrous sensors or actuators. The usage of the fibrous shape memory alloy composites will lead to precise sensing, weight savings, some control over directionality in actuation and the resulting SMA effects. In recent years, it has been found that the nonlinear vibration of smart composite structures has a lot of applications in different industries such as automotive, robotics, civil structures, aircraft, and spacecraft, etc. The vibration of sandwich shells made of thin layers and a moderately thick core are investigated by Singh [5]. The natural frequencies were estimated by implementing Rayleigh-Ritz method. The nonlinear vibration of composite shells in hygrothermal environments is investigated by Ref. [6]. In framework of first-order shear deformation theory and Green–Lagrange type nonlinear displacement, the strains have been obtained. They discussed the effects of the thin cylindrical
Corresponding author. E-mail address: [email protected] (F. Ebrahimi).
https://doi.org/10.1016/j.tws.2019.04.044 Received 16 November 2018; Received in revised form 20 March 2019; Accepted 24 April 2019 Available online 15 June 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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M. Karimiasl, et al.
panel and curvature ratios on the nonlinear frequency. Yazdi [7], presented the nonlinear vibration behavior of doubly curved shell based on Donnell's shell theory [8]. investigated the nonlinear vibration behavior of doubly curved composite panel based on higher order shear deformation theory. Finally, they discussed the influences of several parameters such as aspect ratio, curvature ratio, stacking sequence on nonlinear frequency. Nonlinear vibration of cylindrical shell in framework of higher order shear deformation theory are presented by Ref. [9]. They illustrated the significant role of nonlinear term the to predict the nonlinear response of composite shell. Alijani et al. [10] studied primary and subharmonic responses of FGM shallow shell by multiple scales analytical method based on Donnell's type nonlinear strain–displacement relationships. They found that two-to-one internal resonance may be taken as a measure in doubly curved functionally graded material (FGM) shells by kind of the volume fraction exponent. Continued from previous work, large amplitude forced vibrations of rectangular plates via HSDT have been investigated by Ref. [11]. From the experimental and analytical method, they are presented that nonlinear frequency results in important effect in nonlinear to linear response of plates. Meanwhile, the fundamental frequency of functionally graded material doubly curved shallow shell is studied by Chorfi and Houmat [12] with the aid of finite element methods. They investigated the influence of thickness ratio, volume fraction versus nonlinear to linear vibration. The fundamental frequency of FGM doubly curved shell embedded in elastic foundation is presented by Shen and Yang [13] in frame work shear deformation theory and von-Karman nonlinear strain-displacement. The influence of volume fraction index, Pasternak foundation, curvature ratio, and other parameter have been investigated. Free vibration of MR damped beams is investigated by Yeh [14]. The equations solved by FEM. Thai et al. presented the bending and free vibration behaviors of functionally graded plates based on higher-order shear deformation theory. They expressed that their theory can achieve the same accuracy of the existing higher-order shear deformation theories with more number of unknowns. Meziane et al. [_Meziane_et_al_2014] probed the vibration and buckling of exponentially graded material sandwich plate resting on elastic foundations under various boundary conditions. Al Basyouni et al. (2015) presented the bending and dynamic response of functionally graded (FG) microbeam via a novel unified beam formulation and a modified couple stress theory. Some results are presented to show the effects of the power law index, material length scale parameter and shear deformation on the bending and dynamic behaviors of FG microbeams. The bending, buckling and vibration behaviors of functionally graded (FG) nanobeams based on the nonlocal theory have been studied by Zemri et al.. Yahia et al. presented the wave propagation of functionally graded plates with porosities employing different higher-order shear deformation plate theories. They presented the influence of the porosity volume fraction, volume fraction distributions on wave propagation of functionally graded plate Belkorissat et al. investigated the free vibration response of functionally graded (FG) plates based on a new nonlocal hyperbolic refined plate model. They presented different effects of the plate thickness, nonlocal parameter, various material compositions, the plate aspect ratio on the dynamic behavior of the FG nanoplate. Nonlinear free vibration response of shear deformable laminated composite doubly shell panel in the hygrothermal environment have been studied by Mahapatra et al. [15]. Nonlinear free vibration behavior of functionally graded single/doubly curved shell panels based on higher order shear deformation theory and Green-Lagrange type nonlinearity have been studied by Ref. [16]. [17] presented the large amplitude free vibration response of functionally graded (FG) spherical shell in the framework of the higher order shear deformation theory [18]. analyzed the nonlinear free vibration behavior of functionally graded curved shell panels based on higher-order shear deformation theory. The large amplitude flexural response of laminated composite doubly curved shell under hygro-thermo-mechanical loading via higher-order shear
deformation theory has been investigated by Mahapatra et al. [19]. Nonlinear vibration behavior of laminated composite spherical shell panel subjected to the hygrothermal environment based on higherorder shear deformation theory has been presented by Ref. [20]. Nonlinear free vibration response of shear deformable laminated composite single/doubly curved shell panel in the hygrothermal environment have been studied by Ref. [21]. [22] analyzed nonlinear free vibration response of laminated composite curved panel under hygrothermal environment based on higher-order midplane kinematics. Mahapatra et al. [23] presented the large amplitude free vibration response of laminated composite doubly curved shell panel in the hygrothermal environment based on higher-order shear deformation theory. Singh et al. [24] studied the geometrically nonlinear behavior of the smart laminated composite plate under the coupled electromechanical load using higher-order kinematic theory. The geometrically large amplitude thermomechanical behavior of the functionally graded curved structure based on higher-order shear deformation theory has been investigated by Mahapatra et al. [25]. The large amplitude bending behavior of functionally graded (FG) curved panel under combined thermomechanical loading have been investigated by Ref. [16]. They discussed the influence of different parameters such as support conditions, volume fractions, aspect ratios, curvature ratios, thickness ratios and temperature on the nonlinear bending response. The free vibration behaviors of composite sandwich panel structure via Green–Lagrange strain kinematics have been presented by Katariya et al. [26]. Thermal buckling strength of smart sandwich composite structure embedded in SMA based on higher-order finite element model has been presented by Katariya et al. [27]. Free vibration of composite sandwich thick shells under thermomechanical load is investigated by Khare et al. [28]. The displacements were presented through higher-order shear deformation theories. Heydari et al. [29] researched the nonlinear bending of functionally graded/CNT plates subjected to uniform pressure and embedded in an elastomeric medium via first-order shear deformation plate theory based on generalized differential quadrature method. The nonlinear bending of hybrid plates including carbon nano-tube reinforced composite (CNTRC) layers embedded in elastic foundations considering the influence of matrix cracks was studied by Fan and Wang [30]. Rajamohan et al. [31] investigated the large amplitude forced vibration of the MR sandwich beam subjected to under harmonic force excitation. It was found that the natural frequencies increased by increasing the magnetic field until the values of the peak deflections decreased. Garg et al. [32] studied the free vibration of composite sandwich doubly curved shells. In frame work Sander's theory they considered a parabolic distribution of transverse shear strains through the shell thickness. According to the FSDT, postbuckled behavior of thermally loaded plates was studied by Lee and Lee [33]. Wu et al. [34] studied the nonlinear dynamic instability behavior of FG/polymer/GPL nanocomposite by using Timoshenko Beam theory. The vibration behavior of sandwich plates with composite face sheets was presented by Ref. [35]. Recently, based on the nonlocal strain gradient theory [36] investigated the buckling and postbuckling of multilayer GPLRC nanoshell. The electroelastic response of a functionally graded plate resting on WinklerPasternak in the theoretical framework provided by a two-variable sinusoidal shear deformation theory has been presented by Arefi et al. [37]. Recently, large amplitude vibration of graphene-reinforced composite cylindrical shell subjected to thermal environment is investigated by Shen et al. [38]. In frame work Reddy's third order shear deformation theory and von-Karman theory the linear and nonlinear relationship equations. The equations of motion are solved by perturbation method. Two end conditions such as movable and unmovable are assumed. They evaluated the effect of several parameters such as temperature rising, different distribution pattern, end condition situation, stacking sequence. Aguib et al. [39] experimentally and numerically studied vibrational behavior of magnetorheological elastomer sandwich beam with two layer aluminum 7075T6, that first layer subjected to a 2
Thin-Walled Structures 143 (2019) 106152
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variable magnetic field perpendicular to the layer of the beam, and second layer to a harmonic excitation by magnetic force applied at the free end. Forced stability behavior of MR beam implementing the Galerkin method have been presented by Nayak et al. [40]. Based on numerical results of this research it is shown that the stability of MR beam by applying the magnetic field increasing. Recently large amplitude bending of functionally graded porous micro and nanobeams reinforced with graphene platelets according to the nonlocal strain gradient theory are studied by Sahmani et al. [41]. They investigated kind of parameters such as volume fractions, strain gradient and nonlocal effect and porosity. Arefi [42], investigated bending behavior a doubly curved piezoelectric shell resting on Pasternak's foundation is presented based on first-order shear deformation theory and modified couple stress formulation. They investigated the influence of different parameters such as applied voltage nonlocal coefficient, geometries and two parameters of Pasternak's foundation on the deflection of nanoshell [43]. presented the bending behavior of a sandwich microbeam subjected to the elastic micro-core and two piezoelectric micro face-sheets based on higher-order sinusoidal shear deformation. Also, The electroelastic analyses of the piezoelectric doubly curved shells based on firstorder shear deformation theory resting on Winkler's foundation have been studied [44]. Wei et al. [45] studied dynamic and vibration behaviors of MR sandwich beam with flexible core. The effects of skin–core thickness ratio, applied magnetic field, the speed of axial movement and axial force have been investigated. Free vibration of MR damped sandwich composite multiscale beam under external magnetic field is presented by Ghorbanpour Arani et al. [46]. They found that their modal with loss factor decreases by increasing magnetic field intensity, unlike natural frequency. The analytical solutions for bending response of the three-layered curved nanobeams via sinusoidal shear deformation theory have been studied by Ref. [47]. They discussed the influence of different parameters such as spring and shear parameters, nonlocal parameter, applied voltage geometries on the deflection of nanobeam. The bending analysis a sandwich curved beam based on first-order shear deformation theory resting on Pasternak foundation and subjected to the applied electric field has been investigated by Zenkuor et al.. Also, the nonlocal transient magnetoelectro-elastic formulation of a sandwich curved nanobeam consist of two piezo-magnetic face-sheets and nano-core, subjected to transverse mechanical loads and resting on Pasternak foundation have been presented Arefi et al. [47]. Belabed et al. studied the sound transmission through corrugated core FGM sandwich plates filled with porous material via three-unknown hyperbolic shear deformation theory. They studied the influence of several parameters such as temperature and volume fraction distributions on the wave propagation of functionally graded sandwich plate. Arefi et al. [48] presented the free vibration response of porous sandwich nanoplate resting on Pasternak's foundation, based on nonlocal strain gradient theory. They investigated the influence of different parameters such as nonlocal co-efficient, porosity, parameters of the foundation, volume fraction and length ratios on the free vibration characteristics. The bending response of a doubly curved nano shell subjected to electric field have been studied via nonlocal elasticity theory and first order shear deformation theory by Arefi, [49]. In continues, Arefi [50], investigated the electro-elastic behavior of the piezoelectric doubly curved shells resting on Winkler foundation through first-order shear deformation theory. Arefi et al. [51] investigated the magnetic and electric buckling loads of three-layered elastic nanoplate with exponentially graded core and piezomagnetic face-sheets based on first-order shear deformation theory [52]. presented the nonlinear flexural strength and stress response of nano sandwich graded structural shell panel under the combined thermomechanical loading based on higher-order displacement polynomial theory [52]. researched the nonlinear response of a curved sandwich panel structure adopting the higher-order kinematic theory. They discussed the effect of various design parameters on the nonlinear natural frequency. Katariya et al. [27] studied the nonlinear static behavior of
sandwich flat/curved shell panel subjected to the unvarying transverse mechanical load based on a higher-order kinematic theory including the stretching term effect in the displacement field variable. Sharma et al. [53] presented the vibro-acoustic behavior of laminated composite curved panels via a novel higher-order finite-boundary element model. The elastic response of microbeam with micro-core and two piezoelectric micro-face-sheets sandwich subjected to transverse loads and two-dimensional electric potential based on higher-order sinusoidal shear deformation beam theory have been studied by Arefi et al. [54]. [55] researched the free vibration response of a doubly curved piezoelectric nano shell based on first-order shear deformation theory and nonlocal elasticity theory. The analytical approach for thermal stress and deformation analysis of a curved nanobeam resting on Pasternak's foundation subjected to transverse mechanical and thermal loads based on Sinusoidal shear deformation theory have been studied by Ref. [55]. However, the large amplitude vibration of the smart multi scale doubly curved shell with MR fluid layer and flexible core subjected to magnetic loading has not been reported thus far. The novelty and significant contribution of this article lies in considering a porous multiscale doubly curved sandwich shell with flexible core and MR smart layer. The properties of the flexible core will be varied through the thickness direction of the shell based on the electric and magnetic potentials applied to the magnetorheological face sheets. The aim of this research is to develop a new smart model of the sandwich composite shell. Furthermore, motion equations are derived using Hamilton's principle and solved according to the multiple scale perturbation method. The effect of important parameters on the nonlinear frequency of the sandwich composite shell are also considered. 2. Theory and formulation Fig. 1 illustrates the multiscale doubly curved composite shell with the length of l, the thickness of h and shell curvatures of . The shell is embedded in a distributed hygrothermal load, which is considered in the symmetry plane of the shell cross-section, i.e. in the x-y plane. The first layer consists of PCF and second layer including PGF and third layer consist of SMA/matrix assumed in this research. 2.1. Multiscale model for sandwich composite shell According to Fig. 2, a porous shell in which the porosity is distributed uniformly (pattern B) and patterns of C and D are considered for multiscale composite porosity dispersion. The Young's modulus of CNT/GPL, G12 shear modulus and is mass density, can be presented as [54]:
E˜11 (z ) = E11 [1
p
s (z )]
(1)
E˜22 (z ) = E22 [1
p
s (z )]
(2)
G˜ 12 (z ) =
E˜11 2(1
˜ (z ) = [1
(3)
(z )) m s (z )]
+ Vmcn/ mgpl
mcn / mgpl
(4)
where E˜11, E˜22 are the Young's modulus of CNT/GPL, G˜ 12 shear modulus and ˜ is mass density, respectively.where:
s0
for pattern B
( + ) cos ( )
s = cos
z 2h
4
for pattern C
z h
for pattern D
(5)
According to the Gaussian Random Field scheme [56] the mass density coefficient m can be related to the porosity coefficient p as: m
3
=
1.121[1
(1 s (z )
1 p s (z ) 2.3 ]
(6)
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Fig. 1. Geometry of doubly curved multiscale and SMA/matrix composite piezoelectric shell with flexible core and MR layer in hygrothermal environment.
Fig. 2. Porosity distributions patterns.
4
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Corresponding to the closed-cell Gaussian Random Field [56] the Poisson's ratio of the porous shell corresponding to the different patterns of the porosity dispersion can be written as:
˜ (z )
˜ 12 = 0.221 1
+
12
1 + 0.342 1
˜ (z )
2
1.21 1
˜ (z )
s0 =
h 2 ˜ (z )
1
h 2
2.3
dz + 0.121
The effective constituents of the multi-layer PCF and PGF multiscale composite can be presented via Halpin-Tsai model [57] and micromechanics approaches of scheme have been expressed by Shen, [58]. The maximum Young's modulus and the maximum shear modulus and the maximum mass density are concentrated to be orthotropic can be presented as [58]:
Vmcn / mgpl 1 1 = F + E22 Emcn / mgpl E11 V 2f Emcn / mgpl F E22
+
Emcn / mgpl
= Vf
+ Vmcn/ mgpl
f
(13)
Emcn / mgpl
F E22
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
) )
lcn / gpl
d cn / gpl
Vcn / gpl Ecn / gpl
1 2
cn / gpl
+ vm Em
m
vcn / gpl Ecn/ gpl + vm Em m Vm
+ (1 + vcn / mgpl )
Vf E11f + Vmcn / mgpl Emcn/ mgpl
11
=
22
= (1 + Vmcn / mgpl ) Vmcn / mgpl
(1
mcn / mgpl
v12
11
(21)
vmcn / mgpl )
cn / mgpl Vcn / mgpl
(22)
m
Vf E11f + Vmcn / mgpl Emcn / mgpl m
(23a)
v12
11
(23b)
m E11 = Vm E11 + Vs Es
(24)
m Es E22 m Vs E22 + Vm Es
(25)
E22 =
G12 = G23 = 12
ll Vcn / gpl
ll Vcn / gpl
+ (1 + Vmnc / mgpl ) Vmcn/ mgpl
The properties of SMA are expressed by Park et al. [61]. In this present research, it is considered that the SMA fibers are uniformly distributed. The elastic properties of the composite doubly curved shell with SMA fibers can be expressed by:
(12)
1+2 1 + 2 dd Vcn / gpl E = M 5 +3 8 1 1 dd Vcn / gpl
=
f 22
(20)
2.2. Properties of SMA in a matrix of the porous composite
where are the maximum Young's modulus of CNT/GPL, G12 the maximum shear modulus and is the maximum mass density, 12 the maximum 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, the composite tensile modulus can be expressed [59]:
E11F ,
mcn / mgpl
where mcn / mgpl , mcn / mgpl , are the thermal expansion and moisture coefficients of the epoxy resin CNT and GPL matrix and cn / gpl are the thermal expansion coefficients of the CNT and GPL, respectively.
(11)
= Vf vf + Vmcn / mgpl vmcn / mgpl
12
+ Vmcn / mgpl Emcn / mgpl
Vf E11f + Vmcn / mgpl Emcn/ mgpl
+ (1 + vm )
(10)
mcn / mgpl
f 11
= (1 + Vf ) Vf
2Vf Vmcn / mgpl
Vf Vmcn/ mgpl 1 = F + G12 Gmcn/ mgpl G11
Vf E11f
22
mcn
(9)
F Vf E22 + Vmcn / mgpl Emcn/ mgpl
(19)
=
Vf Vmnc / mgpl 2 Vmcn / mgpl Emcn / mgpl
(18)
2(1 + vmcn/ mgpl )
11
(8)
E11 = Vf E11F + Vmcn/ mgpl Emcn / mgpl
(17)
m
where vm ,vmcn/ mgpl denotes the Poisson's ratio of the matrix, CNT and GPL, respectively. In addition, 11 and 22 refer to the thermal expansion coefficients of longitudinal and transverse directions, respectively [58]. Therefore, 11f is the thermal expansion coefficient of longitudinal fiber whereas, 22f refers to thermal expansion coefficient in transverse directions of the fiber. mcn / mgpl can be expressed as [60]:
1.121
p
+ vm
Vmcn / mgpl = Vm
Furthermore, when the total masses of the shell with different porosity distributions are the same, the value of the s0 can be expressed as: 1 h
cn / gpl
Emnc / mgpl
Gmnc / mgpl =
(7)
1
= Vcn / gpl
mnc / mgpl
11
(14)
22
11
(15)
22
m G12 Gs
m G12 Gs + Vm Gs
(26)
m G23 Vm
(27)
+ Vs Gs
(28)
m = Vm v12 + Vs vs
=
m m Vm E11 11
+ Vs Es E11
= Vm
=
=
m 22
+ Vs
m m Vm E11 11
+ Vs Es
= Vm
m
+ Vs
+ Vs
s
s
(29) (30)
s
E11 m Vm 22
s
s
(31) (32) (33)
where are the Young's modulus of the matrix, G12, G23 are the is mass density, 12 is the Poisson's ratio, reshear modulus and spectively. The corresponding properties of the isotropic matrixes of SMA composite are represented with Es, Gs, s and Vs . Similarly, the volume fractions and mass density of the matrix are denoted by Vm, m m m m , 22 , and 11 , 22 and m , respectively. Finally, 11 are the thermal expansion and moisture coefficients of the matrix and s and s are the thermal expansion and moisture coefficients of the SMA, m E11 ,
(16)
cn / gpl , refers to the Young's modulus, hcn / gpl , dcn/ gpl , lcn /gpl rewhere E11 presents thickness, outer diameter, length, respectively. Meanwhile, Vcn/ gpl are the volume fractions of Carbon Nanotubes and Graphene Platelet, respectively. Further, Vmcn /gpl and Emcn / mgpl are the volume fractions of the matrixes and Young's modulus, respectively. The mass densities of CNT and GPL can be presented as:
5
m E22
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respectively.
The relationship between displacement-dependent parameters of the middle core by implementing compatibility Eq. (37) can be written as:
2.3. Kinematic relations In the framework of the third-order theory, the displacement fields at an arbitrary point in the composite shell can be expressed as:
ui = u0i + z i
vi = v0i + z i w=
4 3 zi 3h i 2
i x
i y
i x
4 3 zi 3hi 2
i y
(34a)
v 2c =
w0i
+
In these equations, u 0 , v0 , and w0 are the original displacements of the shell in the x, y directions; the rotations of the transverse normal at the mid-plane in the x and y axes are represented by x and y , respectively. u0n , v0n, w0n, xn , yn are functions of the five original variables which describe the shell displacements and n referred to the number of the multiscale composite doubly curved shell layer. In-plane displacements and normal displacement of the magnetorheological core is in the form of third order polynomial, respectively can be presented by:
vc = v0c + z c v1c +
+
vMR zML
wMR zML =
u1c
h2 hc + u2c c 2 4
u3c
w2c =
n xx n yy n xy
v0c
v1c
h2 hc + v2c c 2 4
v3c
3
hc3 h = v0MR + MR 8 2 4 hMR3 + 3hMR 2 8
w0c
w1c
hc2
hc + w 2c = w0MR 2 4
+
w0MR + y
3hc3 u3c + 8hMR
MR x
3hc2 w 0MR y
12v0c + 12v0MR + 6hc v1c
hMR
3hc3 v3c + 8hMR
MR y
3hc2
4u0MR
4w0MR + 2w1c hc2 hc2
n0 xx n0 yy
=
+ zn
n0 xy
(38f)
n0 k xx n0 k yy
+ zn
u0n
n2 k xx n2 k yy
3
n0 k xy
+
x v0n
=
+
y
n2 k xy
u 0n n x
x
n y
+ zn n x
n yz n xz
n0 yz
=
+
+ zn
n0 xz
2
n y
4 z n3 3hn 2
y y
n y
y n x
x
y
n1 k yz n1 k xz
=
n y n x
+
w0n
+
w0n
+
R1 1 2
+
( ) w0n w0n x
y
2w n 0 x2
+
2w n 0 y2
n y
( )
w0n 2 x
+
y
+
x
1 2
w0n 2
R2 u0 x
(39a)
x y
n y n x
x
+
w0n
2w n 0
+2
4 3 z h2
y
+
w0n
+ +
w0n y w0n
(39b)
x
According to the core strain relationship can be expressed as: c xx c yy c zz
u 0n x v0n
=
y
+ zc + zc
u1n x v1n y
u2n
+ z n2 +
x vn z n2 y2
u3n
+ zn3
x v3n
+ zn3
y
(40a)
w1 + 2z c w2
c xy c xz c yz
(37a)
MR y MR y
12u0c + 12u0MR + 6hc u1c
n x
w0MR x
hMR
(38c)
x
MR x MR x
+
y
(36c)
hc3 h = u0MR + MR 8 2 4 hMR 3hMR 2 8
x
w2c hc2
The consistent Donnell nonlinear strains of third-order shear deformation theory from the middle surface displacement–strain relations are [62]:
(36b)
hc 2
+
w 0MR
4w0MR 2hc2
(38e)
where uMR , vMR and wMR indicated the displacement of the magnetorheological layers. By substituting Eqs. (34) And (35) in Eq. (36), compatibility relations can be defined as:
u0c
+
(36a)
hc 2
hMR = wc z c = 2
4u0MR
v3c =
2.3.1. Compatibility conditions of the displacements The sheets are assumed to be ideally attached to the core, such that 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:
hc 2
MR y
6hc2
2 2
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.
h = MR = vc z c = 2
24v0c + 24v0MR + 12hc v1c + 3hc3 v3c + 16hMR
hMR
(38d)
(35c)
wc = w0c + z c w1c + z c2 w2c
uMR zMR
y
u3c =
(35b)
z c3 v3c
h = MR = uc z c = 2
4
2 2
(35a)
z c3 u3c
+
w0MR
w1c =
(35)
i = n , MR
z c2 v2c
MR x
6hc2
(38b)
(34c)
uc = u 0c + z c u1c +
24u0c + 24u0MR + 12hc u1c + 3hc3 u3c + 16hMR
hMR
(34b)
y
w0i
z c2 u2c
x
(38a)
w0i x
+
u2c =
w0MR
4
u 0n y
(37b)
=
+ zc
u1n y
+ z n2
u2n y
+ zn3
u1 + 2z c u2 +
u3n y
3z c 2u3
+ +
v 1 + 2 z c v 2 + 3 z c 2v 3 + (37c)
v0n x w0 x w0 y
+ zc + +
v1n x
w z c x1 w z c y1
+ zn 2 + +
v2n x
+ z n3
v3n x
w z c 2 x2 w z c 2 y2
(40b) 6
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2.4. Strain-stress relationship
Ey =
According to the magnetorheological layer model considered, the following conditions are assumed:
Ez =
1. No slip condition existed between the core and magnetorheological layers it is assumed. 2. It is considered that transverse displacement w and normal rotational degrees ( x and y ) are the equal on a hypothetical crosssection on the multiscale composite. 3. In the magnetorheological layer, no normal stress exist 4. The Magnetorheological material is modeled as a linear viscoelastic material in the pre-yield condition.
ML xz
w 0n x
=
w0n y
+
uMR z
+
vMR z
n xx n yy n yz n xy n xz T
=
= G2
MR xz ,
MR yz
and
uML z
hn , 2
n x
2hMR
= G2
c yz
,
n
Q¯ 12 n Q¯ 22 0 0 0
vML z
0 0 0 q15 0
can be expressed as:
vMR 1 = v0n z hMR
n y
hn 2
(42)
MR xz
hn ,
w0n vn + 0 y hMR
=
n y
2hMR
hn
Bx By Bz 0 0
MR yz
(43)
s11 0 0 0 0
(44)
= G2
c yz
=
0 s11 0 0 0
0 0 e32 0 0
0 0 s33 0 0
0 0 0 0 0
G = G =
0.9B2 + 0.8124 × 103B + 0.1855 × 106
+ 4.9975 ×
103B
cos( z ) (x , t ) +
cos( z ) (x , t ) +
(
1 z R1
)
x
cos( z ) (48b)
2 h
sin( z )
0 0 0 n Q¯55
0 0 0 0 n 0 Q¯66
(48c)
0 0 0 e15 0
T
11 (T (z )
T0)
11 (H
(z )
H0)
22 (T (z )
T0 )
22 (H (z )
H0 )
0 0 e42 0 0
e13 e23 0 0 0
Ex Ey Ez 0 0
T
+ 0.873 ×
(46a)
106
Bx By Bz 0 0
(46b)
2z V h
cos( z ) Qx =
(
1 z R1
)
x
Qz 0 0
e15 0 0 0 0
(49a)
T
xx
0 0 0 0 0
3 (T (z )
T0)
yy yz xy xz
0 0 0 0 0
0 0 q32 0 0 0 0 0 0
(47b)
R1 1 +
=
0 0 q31 0 0
11
(47a)
2z h
0 e24 0 0 0
T
Qx Qy
q13 q23 0 0 0
d11 0 0 0 0
where = / h . Also, V and are the external electric voltage and the magnetic potential applied. Via Maxwell's equation, the relations between the electric field (Ex , E y, Ez ) and electric potential also the magnetic field (Qx , Qy , Qz ) and magnetic potential can be expressed as [47]:
R1 1 +
x
Ez 0 0
Based on Maxwell's equation, the electric and magnetic potential distributions can be obtained as:
Ex =
)
Ex Ey
(45)
3.3691B2
(x , z , t ) =
0 0 n ¯ Q44 0 0
0 0 q42 0 0
0 0 e31 0 0
where G and G are the loss modulus and storage, respectively, which can be expressed as a polynomial function of magnetic field, B (in Gauss), for expressed magneto-rheological material as:
(x , z , t ) =
=
z R2
xz
By considering that the magneto-rheological material behaves similar to the viscoelastic material in the pre-yield region, Rajamohan et al. proposed a relation that suggests the shear modulus is complex and depends on the intensity of the magnetic field. The complex shear modulus of viscoelastic material can be expressed as:
Gc = G + i G
,z
(
1
R2 1 +
xy
Now, the relationship between transverse stresses and strains in the magneto-rheological layer can be defined as: MR xz
n
Q¯11 n Q¯ 12 = 0 0 0
(41)
n x
w0n un + 0 x hMR
2v Qz = h
sin( z )
n xx n yy
By substituting Relations (42) in (41), components of strain in the magneto-rheological layer can be presented as: MR xz
)
cos( z ) Q y =
x
xz
Relations between
uMR 1 = u 0n z hMR
z R2
The constitutive relation of the composite shell with a piezoelectric layer, if the fiber angle with the geometric x-axis is expressed by θ, the relation stress-strain can be transferred to the geometric coordinates as:
Components of transverse strain in the core can be mentioned as follows: ML yz
(
1
R2 1 +
0 d11 0 0 0
0 0 d33 0 0
0 0 0 0 0
0 q24 0 0 0
q15 0 0 0 0
0
0 0 0 0 0 0 33 0 0 0 0 0 0 0 0
11
0 0 0
0 0 0 0 0
Qx Qy
0 0 0 0 0
Qz 0 0 xx yy yz xy xz
(49b)
d11 0 0 0 0
0 d11 0 0 0
0 0 d33 0 0
0 0 0 0 0
0 0 0 0 0
Ex Ey Ez 0 0
Qx Qy Qz 0 0
(49c)
ij , Dij , Bij , ij, En and Qn expresses the magnitude of stress, electric displacement, magnetic induction, nonlinear strain, electric field and magnetic field. Also, sij and ij are the components of dielectric and magnetic permittivity coefficients; furthermore, eij , qij and dij are the piezoelectric, piezo-magnetic and magneto-electric-elastic coefficients, respectively. The reduced stiffness modulus of the sandwich composite can be expressed by:
cos( z ) (48a) 7
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
E11
Q11 =
1
12 E22
, Q12 =
12 21
1
, Q22 =
12 21
E22 1
12 21
, Q44
a N n=1 0
T= (50)
= G23 , Q55 = G13 , Q66 = G12
+
The principal coordinates of the transformed shell are expressed in Appendix A. Now via Hamilton's principle, it can be written as:
x u0
b
nh
n
41 120R1
+
2 15R2
0
(
w02
+
t
(U + V
T ) dt = 0
(51)
0
where, U is strain energy, V is the work done by external energy and T is kinetic energy. The strain energy is expressed as:
U=
N
1 2
a
b
n=1 0
n n xx xx
+
n n yy yy
MR MR xx xx
+
MR MR yy yy
hML
0
0 hML 1 a b
+
n n xz xz
+
+
0
+
MR MR yz yz
+
c c xx xx
c c yy yy
+
c c yz yz
+
c c xz xz
+
+
2
MR xy
b
o
0 hp 1
[ Dx E x z R1
1+
Dy E y
Dz Ez
Bx Q x
c c xy xy
By Qy
N n=1
a b
Bz Q z ]
(52)
a 0 MR Myy
+
b 0 2 k yy
a 0
c 2 + M yy k yy +
0 xx
0 yz
a 0
MR 1 R yy k yz
+
c 0 c N xx xx + Mxx 0 c 2 c 0 Pyy k yy + N xx xy
b 0
0 xz
+
0 xy
MR K xx
0 k xx
+
hp
0 xz
c Pxx
Dx cos( z )
hp 1
(
h /2
+
2 c k xx + N yy
z R1
x z R1
), q
2
(
= c.2 1 +
Z R2
hn
NT =
8 315
R2
)
)
y
8 315
u c0 + v c0 v c0 + w c0 w c0)
w MR 0 ) dc1 dc2 + IML
MR xz
MR xz
MR yz
+
MR yz
w0 w +q 0 x x
w0 dc1 dc2 x
(55)
(56)
0 yz
1
[Q¯ 11
11
+ Q¯ 12
12](
[Q¯ 11
11
+ Q¯ 12
12](H
T = T1 + T 1
2
T
T1) dz
(57c)
H1) dz
(57d)
1 z + 2 2 h
Cos
(58)
u0c , By setting the coefficients of v1c , u1c , v2c and w0c t o zero and substituting Eqs. (53)–(55) into Eq. (51) may be stated as:
u0n ,
y
)(1 + ) dc dc
(57b)
And T T1, H H1 are variation of temperature and moisture, T can be defined by sinusoidal temperature following as:
2
y
B y cos( z ) z R2
2 dz h
hn 1
)(1 + ) dc dc 1
hn
NH =
0 yy
z R2
(57a)
hn 1
(53)
n Nxx x
where, for convenience, a shell with a rectangular base of dimension a and b in c1 and c2 directions, has been considered. q1 and q2 are the Lame coefficients of the shell which can be expressed as. Z R1
q31
MR 1 Rxx k xz
Dy cos( z )
x
Bx cos( z ) dz × 1 +
x
v02
2V dz h
h /2
NQ =
0 yz
0 yy
c 2 c 2 c + Mxy k xy + Pxy k xy + K yy
(
+ Bz sin( z )
(
e31
n0 yy
MR 2 MR 2 + Mxy k xy + Pxy k xy
c 1 + Rxx k xz × 1 +
+ Dz sin( z )
q1 = c1 1 +
0
1 + Rxx k xz
MR 0 MR 2 MR + Mxx k xx + Pxx k xx + Nyy
b
c 1 c + R yy k yz + K xx
+
0 xz
MR 2 MR + Pyy k yy + Nxx
MR K yy
+ +
MR Nxx
w0 x
N x0
h /2
n 2 n 2 n + Mxy k xy + Pxy k xy + K xy
n 1 + Rxy k yz + Kxx
+
h
NE =
n n0 n n2 n + Mxx k xx + Pxx k xx + N yy 0 xy
v0 30R1
+
R1
where:
0 0
n n2 n n2 n + Nyy k yy + Pyy k yy + Nxy
u0 30R2
u02
MR
h /2 n0 xx
c c (u 0
1 4
N x0 = N E + N Q + N T + N H
z dc1 dc2 dz R2
n Nxx
ch
+
)+ (
It is assumed that the doubly curved composite shell is exposed to the external electric voltage V, magnetic potential and variation of temperature and moisture. q is a the dynamic force and N x0 can be written as:
The first variation can be obtained as:
U=
c
V=
hp
a
b 0
u0 120R1
v0 120R2
+
2 15R1
+
2 y
+
and where are the mass densities and IMR is inertia moment of the magneto rheological layer. The first variation of work can be expressed in the following form:
0 hC 1
+
a 0
41 120R2
2 x
(54)
n n xy xy
MR MR xz xz
+
+
(
w0 252q1 c1
w0 252q2 c1
MR MR h MR (w 0
0
hC
+
× 1+
n n yz yz
+
w0 q2 c2
(
y v0
0 0
0 hn 1
b
MR xy
+
n,
hn
a
a b
+
)+
w0 q1 c1
+
12R1 R2
17 315
u0n u0n + v0n v0n + w0n w0n + h2
= I¯0n +
).
+
Further, R1 and R2 are the principal radii of curvature in q1 and q1 directions, respectively. The Kinetic energy can be presented as:
+
n Nxy
+
w0MR
1 hMR t 2 hc4
2
2u
8
4 hc3
I3c
0c
t2
8 hc5
2u n 0 t2
+2
2u n 0 t2
2I5c
+ 4I5c
1 2 hMR
2u n 0 t2
x
M2cxy
+
( ) + (I 2
2 n x t2
4
hn I5c
2 n x t2
I4c
c 2
hc2
2u
0c
t2
+
2u
+ s1 I¯3n 2u
0c
t2
+ 2I6c 8I6c hc6
0c
+
t2
+
w0n ,
n x,
n y,
2MQc 1 xz
y
2 n x t2
hn 2 2hMR
2u 1c t2
+ hn
2u n 0 t2
M2cxx
2 hc2
J¯1n +
+ +
x
+ +
2u n 0 t2
1 2 hMR 2
2I4c
1 Q MR hMR xz
y
v0n ,
( ) ) w0n x
2u I3c 21c t
8 c I hc4 5
2u
2
t2
2u n 1 t2
0c
t2
2
2u n 0 t2
hn
2 n x t2
(59a)
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al. n Nxy
n N yy
+
x
( )+ v0n
1 2 hMR
= I¯0n +
w0MR
2I4c
+
8
+ +
4
n K¯ xx x
n K¯ yy
+
1 hc
+
y
+
n Nxy
(
=
+ I3
t2
+
hMR t 2
+
R1
u0n
x
R1
2P n xy
hc3
n x
4 t2
y
hn 2 2hMR
2w c 0 t2
2w n 0 t2
n y
y
hc2 2w c 0 t2
hc4
= I1c
MQc xz 2 y
2
+ s1 I3
+
1 c I hc 1
2
2w c 0 t2
t2
c N yy
( ) u0n
y
x
n ¯ xy M
+
hn Q MR 2hMR yz
n K¯ xx +
y
2hn 2u n 0 t2
hn 2 2hMR
+
+4 2
n ¯ xy M
x
( )+
4
+
c
I hc2 3
2w n 0 t2
hc3
(
I3c
n ¯ yy M
+
2u n 0 t2
2
hc4
2u
0c t2
2hn hc3
=
2v n 0 t2
J1n
hn 2 2hMR
+
2
+
( )+ 2v
2
hc4
hn I5c
hc3
(I
c 3
2v
0c t2
+ I4c
x
hn
2u
+4
)+4
0c
hn
M3cxy
2 n y t2
hn 2v
1c t2
2 n y t2
hn
hc2
)+4
c 2
2v n 0 t2 2v n 0 t2
hn I6c hc6
2v
0c
t2
M2cyy
2u c 1 t2
(59d)
= I0c
2w c 0 t2
+ hn
2 n x t2
4
2
2u n 0 t2
hn
2 n x t2
2hc
c N xy
4
M2cyy
x
hc2
y
2v c 1 t2
2
+
c
I hc3 3 c
4
2
2v n 0 t2
hn
2 n y t2
2h c
4
c
2
4
y
hc2 2w n 0 t2
hc4
h h c hMR hn
2v n 0 t2
2v n 0 t2
2I2c hc2
2w n 0 t2
cos( z )
2v
4 2hc
(59h)
M3cxy x
2v
M3cyy
+
y
2
4
Ic hc2 4
2v n 0 t2
0c
t2 2v
0c
t2
2v
0c
t2
(59i)
MQc yz 2 y
MQc xz 2 x
2w n 0 t2
2
1c
t2
2h c
2 n y t2
2Mzc + +
0c
2 n y hn 2 t
2 n y t2
+ hn hn
+
2 n y t2
hn
2v n 0 t2
c Q yz
I1c
2
hc2
2v n 0 t2
2 2
I3c
2u n 1 t2
t2
3MQc 2 yz +
hc2
+
2v
2v n 0 t2
4
2h c
2 n y t2
2v c 1 t2
hc
2 n y t2
hn
(59g)
2 n y t2
+ hn
+ hn
I hc3 4
+
2v n 0 t2
2
c Qyz
+ I2c
2v n 0 t2
2
2u n 1 t2
2MQc 1 yz
x
2v n 0 t2
y
+
I2c
hc2
2u n 1 t2
2u n 0 t2
2
M1yy
2
+
2h c
M2cxy
+
2u n 0 t2
4
2w n 0c t2
4I3c hc3
2w n 0 t2
2w n 0c t2
(59j)
Dy Dx + cos( z ) + x y
sin( z ) Dz dz = 0 (59k)
h h c hMR hn + hp
1c
h h c hMR hn
cos( z )
By Bx + cos( z ) + x y
sin( z ) Bz dz = 0 (59l)
1c
hn I6c
2v
where:
1c t2
hn
2 n x t2
hn
y
y
t2
hc5
+
M3cxy
+
2 n hn 2x t
2u n 0 t2
8I4c
t2
2v
2u n 0 t2
2
(59f)
x
2
x
( ) +I ) ( ) ( )
2
hc2
2u n 1 t2
M3cxx
4 c I hc4 5
M2cxy
w0n
2v
c 3
+
2u n 0 t2
2u n 1 t2
2u n 0 t2
2 c I hc3 4
8 c I hc5 6 c Qxz
I3c
2u n 0 t2
4
2hc
3MQc 2 xz +
hc2
4 c I hc4 5
x
+
y
4
c Qxz
hn
+
+
2u 1c t2
hc4
2v n 0 t2
2h c
h h c hMR hn + hp
hn I5c
+8
2 n x t2
2u 1c t2
hn 2hMR t 2
4
2
2v c 0 t2
x
2
y
+
w0n
y w0n
+2
x
M3cyy
( ) + (I
+4
M1xy
y
2 n hn t 2x
2u n 0 t2
x
hn
2u 1c t2
c 3
0c
t2
2MQc 1 yz +
hc2
M2cxy
( ) +I ) 4 ( ) +8 ( )
2
hc6
2u n 0 t2
8 c I hc5 5
= I1c
hc5
hn I6c
2
I hc4 4
2 hn 2hMR t 2
2u
4
4
+
x
hn I6c
0c t2
s1 J4n 2 t
hn
M2cxx
hn I5c hc4
2u
+2
+
2 n y t2
c 2
t2
2
hn2 2 4hMR
2v n 0 t2
w0n
hc2
3MQc 2 yz +
2v n 0 t2
2
hc4
hn
1c
t2
hn I4c
+4
k2n
y
( ) + (I
2u 1c t2
2 n y t2
x 2
2 n x t2
hn Q MR 2hMR yz
M3cyy
t2
hn I4c
+
n K¯ yy +
y
2 n hn t 2x
2u n 0 t2
2
2 n x t2
M3cxy
2 n x t2
2v n 0 t2
2w n 0 t2
+2
2MQc 1 xz +
hc2
s1 J4n
hn2 2 4hMR
2u 1c t2
hn I5c
hn
2 n x t2
+ k 2n
hn I4c hc4
hn
3MQc 2 yz +
hc3
= J1n
+
+ hn
+ I1c
(59c) n ¯ xx M x
2u n 0 t2
+
2v c 0 t2
= I0c
2w n 0 t2
2 n x t2
2
8 c I hc5 6
R2
2u n 0 t2
2
2 n x t2
+ hn
4 c I hc4 4
2u n 0 t2
n N yy
2u n 0 t2
2
hc2
2MQc 1 xz
y
2 c I hc3 3
+ I2c
4 c I hc2 4
I2c
+
M2cxy
+
x
+
y
2u c 0 t2
M2cxx
2u c 1 t2
M1xy
+
)
2u n 0 y2
I2c
4I4c
+
+
n Nxx R1
y
+
y
R2
M1xx x
(59b)
)
R2 v0n
2 n x
y
t2
2Mzc +
y
+
+
x
hn
v0n
y
2
MQc xz 2 x
w0n
w0n
2w n 0 x2
2 s12 I6n 2 t 2
hc2
2P n yy y2
+
x y
2
+
hc2
8 c I hc5 5
2v n 0 t2
2
4
y
2u n 0 t2
4 c I hc2 3
1c
t2
c N xy
+ I1c
0c t2
hc6
n xy
2v
2u c 0 t2
= I0c
y
+
2v
8I6c
n yy
+2
v0n
2v n 1 t2
)+N ( )+N (
x
w0n
8 c I hc4 5
MQc xz 1 y
w 0n
t2
+ I3c
+ 2I6c
+
+
( )+J ( )+
4I3c
0c
u0n
y
2
1
y t2
( ) )
2
+ s1 I¯3n
c 2v 0c 2 t2
+
2
t2
w 0n
v0n
2
2v
MQc xz 1 x
2w n 0 t2
+ IMR)
2v n 0 t2
4 hn I5c
2P n yy x2
+ s1 (I0n
y
I4c
(
2
hc2
c N xx x
2MQc 1 yz
y
2 n y t2
1c
t2
2v n 0 t2
n Nxx
M2cxy
+
hn 2 2hMR
t2
Rzc + x
x
2v
2v n 0 t2
+ 4I5c
M2cyy
( ) + (I 2
+ hn
2I5c
hc5
2v n 0 t2
I3c
hc3
2v n 0 t2
2
hc4
1 2 hMR
+
y
2 hc2
J¯1n +
t2
2 1 hMR t 2
+
1 Q MR hMR yz
y
2 n y t2
¯ ijn = Mijn M K¯ ijn
(59e) 9
=
K ijn
s1 Pijn s2 Rijn
(ij = x , y ), (ij = x , y )
s1 =
4 , s2 = 3s1 3h2
(60a) (60b)
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al. n n n n n n , Nyy , Nxy , Myy , Mxy Here, Nxx and Mxx express the total in-plane force n n n n n , Pyy , Pxy , R yy and moment resultants and Pxx and R xx are the third order stresses resultants which can be written as:
hn /2
n n n Nxx , Mxx , Pxx =
n x (1,
zn , z n3) dz
n y (1,
zn , z n3) dz
hn /2 hn /2
n n n Nyy , Myy , Pyy =
hn /2
n xy (1,
zn , zn3) dz
hn /2
hn /2
n n Pxx , R xx =
n xz (1,
z n, zn3) dz
n yz (1,
zn , z n3) dz
hn /2 hn /2
n n Pyy , R yy =
hn /2
h c /2
c x (1,
z c ) dz
hc /2 h c /2
c c N yy , M yy =
c y (1,
z c ) dz
c xy (1,
z c ) dz
hc /2 h c /2
c c N xy , Mxy =
hc /2
c c Qxz , Mnxz =
h c /2
c xz (1,
z c ) dz
h c /2
c c Qyz , M yz =
hn /2
n yz (1,
z c ) dz
hn /2
Rzc , Rzc =
c x (1,
z c ) dz
(61a)
hn /2
(61b)
(Aij , Bij , Dij , Eij, Fij ) =
MR Qyz =
MR xz (1,
zMR ) dz
(61c)
( ij,
MR yz (1,
zMR) dz
=
MR Q¯ij (1, zMR ) dz (i, j = 1,2,6)
A¯ ij = Aij
3s1 Dij , D¯ ij = Aij
3s1 Fij , Aˆ ij = A¯ ij
hn /2
Jin = Iin
s1 Iin+ 2,
K¯ 2n = I2n
2s1 I4n + s12 I6n
(i = 1,4)
s s I¯0n = I0n + 2 1 I3n + 1 R1 R1
Bˆij = Bij
s1 Eij , Fˆij = Fij
s1 Hij , Dˆ ij = Dij
(62a)
The boundary conditions of the multiscale composite have been considered to be simply-supported (S-S) whose constraints can be mentioned as follows:
(62b)
u 0 (x , 0, t ) = u 0 (x , b, t ) = 0 ,
v0 (0, y, t ) = v0 (x , b, t ) = 0
(62c)
y (0,
y, t ) =
y (a ,
y , t ) = 0,
x (x ,
0, t ) =
¯ x (x , 0, t ) = M ¯ x (x , b , t ) = 0, M ¯ y (0, y, t ) = M ¯ y (a , y , t ) = 0 M
(66f)
{umn, uocnm, u1cnm}cos
{v0, v0c , v1c } =
(62g)
{vmn, v0cnm, v1c}sin
{w0, w0cnm} =
(62h)
x
=
{wmn, w0cnm}sin
=
(63a)
n x sin(my ) b
(67b) (67c) (67d)
y mn sin
n x cos(my ) b
(67e)
mn (t )cos
n x sin(my ) b
(67f)
mn (t )cos
n x sin(my ) b
(67g)
n= 1 m = 1
(x , y , t ) =
Umn (t ), Vmn (t ), Wmn (t ), x mn (t ), y mn (t ), u0c (t ) ,v1c (t ), u1c (t ) , where v2c (t ) and w0c (t ) refer to the unknown functions of the time; n and m are the number mode of frequency in the x and y directions, respectively. n x Here, b = l are assumed. By substituting Eq. (67) into Eq. (59) and driving the Navier procedure, the following expressions can be expressed:
(63d)
(63f)
n x cos(my ) b
(67a)
n x sin(my ) b
(x , y , t ) =
(63b)
n x sin(my ) b
x mn cos n=1 m=1
n=1 m=1
(63e)
(66c) (66e)
n=1 m =1
s I¯3n = I3n + 1 I5n R1
b, t ) = 0
The displacements are assumed as the series of double trigonometric functions that satisfy the boundary conditions:
(62e)
s J¯1n = J1n + 1 I4n R1
x (x ,
Ny (0, y, t ) = Ny (a , y , t ) = 0
Nx (x , 0, t ) = Nx (x , b, t ) = 0 , (62d)
(66a) (66b)
w0 (x , 0, t ) = w0 (x , b, t ) = 0
n=1 m=1
I6n
(65b)
s1 Fij
3. Solution procedure
(63c) 2
3s1 D¯ ij (i, j = 4,5) (65a)
(61e)
y
(i = 0, …, 6)
(64b)
where the cross-sectional rigidities can be expressed as follows:
(61d)
hn /2 n z idz ,
n=1
ij)
hMR /2
The mass inertias of the composite shell can be expressed in the following form:
Iin =
(i , j = 1,2,6) (64a)
hMR /2
n=1 m=1
hMR /2
N
n Q¯ij (1, z, z 2 , z 3, z 4 , z 6) dz,
n=1 m=1
hMR /2 hMR /2
(63g)
Equations of motion of multiscale composite shell can be expressed in terms of u0n , v0n , w0n , xn , yn , u 0c , v1c , u1c , v2c and w0c displacements that are obtained by substituting Eq. (34) into (59) yields the relations expressed in Appendix 2
(62f)
hn /2
QxzMR =
(i = 0, .., 5)
{u, uoc , u1c } =
hn /2
hMR /2
c z idz , h c /2
Following integrals were regarded as stress results within the core and magneto-rheological layers which can be written as: c c N xx , Mnxx =
h c /2
hn /2
hn /2
n n n Nxy , Mxy , Pxy =
Iic =
10
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al. n n n Lij Umn (t ) + Lij Vmn (t ) + Lij Wmn (t ) + Lij
n x mn (t )
n y mn (t )
+ Lij
+ Lij v0c (t ) + Lij u1c (t ) + Lij v1c (t ) + Lij w0c (t ) + Lij
mn
+ Lij u0c (t )
+ Lij
mn
0:
D02 w0 +
2w
1:
D02 w1 +
2w
n n ¨ mn = Mij U¨mn (t ) + Mij V¨mn (t ) + Mij W (t ) + Mij ¨x mn (t ) + Mij ¨y n (t )
+ L36
+
n L32 Vmn (t )
(68a)
n y mn
+
n L33 Wmn (t )
+
3 L34 Wmn (t )
+
mn
+ L313 +
mn
, ( i= 3, j= 1: 10)
(68b)
2i
where the operators Lij are defined as in Appendix 3. By substituting Eq. (68a) into Eq. (68b) to solve the unknown functions Umn (t ), Vmn (t ), x mn (t ), y mn (t ) , u0c (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:
d 2Wmn (t ) + µw + P1 Wmn (t ) + P2 Wmn 2 (t ) + P3 Wmn3 (t ) = q cos t dt 2
M33 + M34 + M35 a34
=
(69)
(72)
Next, the system response during the resonance excitation, 1, is considered. The proximity of the excitation frequency to the system natural frequency is expressed as (73)
=1+
2w
2 (T0,
T1, T2, …)
a
3 P3 3 a 8 0
=
)
1 F¯ cos( T1 2 0
(82)
)
(83)
(84)
(75a)
T1 = t
(75b)
1 q¯ sin 2 0
a =
µa +
a
3 P3 3 a 8 0
=
µa +
(85)
1 q¯ cos 2 0
(86)
1 q¯ sin 2 0
3 P3 3 a = 8 0
a
(87)
1 q¯ cos 2 0
(88)
The fixed points of Eqs. 87 and 88 correspond to solutions with constant amplitude and phase. These solutions satisfy
(74)
T0 = t
µa =
1 q¯ sin 2 0
(89)
1 q¯ cos 2 0
3 P3 2 a = 8 0
(90)
Squaring and adding these equations, one may obtain the frequency response equation:
After substituting Eq. (74) into Eq. (69) and noting Eqs. (76) and (75), the following hierarchy of equations can be obtained for O(1) and O(e), respectively.
+ 2D0 D1)
(80)
(81)
1 F¯ sin( T1 2 0
µa +
a=
where the fast timescale T0 and slow timescale T1 are given by
2 (D 2 1
(79)
The point at which a = 0 and = 0 corresponds to a singular point of the system and shows the steady-state motion of the system. So, in the steady-state condition it can be written:
where is called the detuning parameter, which is a measure of how close the excitation frequency is to the natural frequency. Noting that the steady-state solution for the forced linear oscillator is acos ( t + ) , where the amplitude a and phase f are slowly varying quantities. This analytical approximation is an example of a generalized asymptotic series, as the coefficients are also functions of the asymptotic ordering parameter, which is e in this case. Such analytical approximations can be constructed by using the method of multiple scales or the method of averaging. This construction is illustrated here by using the method of multiple scales. Let:
d = D02 + 2 D0 D1 + dt
+ T1)
and substituting Eqs. 82 and 83 in Eq. (84) leads to:
t=0
d = D0 + D1 dt
1 q exp( i T1) = 0 2
= T1
=0
w = w0 (T0, T1, T2, …) + w1 (T0, T1, T2, …) +
0 T0
Term T1 can be eliminated by transforming Eqs. 82 and 83 to an autonomous system considering:
(71)
¯ dWmn (t ) W , h dt
kcos (
1 a exp(i ) 2
a =
where the initial conditions are illustrated by:
Wmn (0) =
P3 w03
where a and are real. Separating real and imaginary parts of the derived equation, it yields
(70)
p1
¯ + µA) + 3P3 A2 A
0 (A
A=
And the linear frequency of the composite shell is expressed by: l
µD0 w0
Let A be in the polar form:
where:
P3 =
2D0 D1 w0
where A is an unknown complex function and A¯ is the complex conjugate of A. The governing equations for A are obtained by requiring w1 to be periodic in T0 and extracting secular terms which are coefficients of e ±i 0 T0 , the solvability equation will be determined as:
n n = M33 Wmn (t ) + M34 ¨x mn (t )
M35 ¨ y n mn
=
w0 (T0, T1, T2, …) = exp(iT0 ) + A¯ exp( iT0 )
n L35 x mn (t )
+ L37 u 0c (t ) + L38 v0c (t ) + L39 u1c (t ) + L310 v1c (t )
+ L311 w0c (t ) + L312
1
(77)
With this approach, it turns out to be convenient to write the solution of Eq. (65) as:
+ Mij u¨ 0c (t ) + Mij u¨1c (t ) + Mij v¨0c (t ) + Mij v¨1c (t ) + Mij w¨ 0c (t )
n L31 Umn (t )
=0
(78)
mn
(i = 1: 12 Except 3, j = 1: 12)
0
3 P3 2 a 8 0
2
+ µ2 a2 =
q¯2 4 02
(91)
(76a)
Substituting Eq. (90) into Eq. (88) and substituting the resultant equation in Eq. (74), one may obtain the first approximation:
(76b)
w= a cos(
0
t+
t
) + O( )
(92)
With this, the amplitude response (magnification factor) can be obtained as:
Substituting Eq. (76) into Eq. (69) and equating the coefficients of to zero yield the following differential equations: 11
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
M=
a = q¯
1 2
0
(
)
3 P3 2 2 a 8 0
+ µ2
Table 2 Material properties of SMA [61].
(93)
Composite matrix Properties
Similar to the case of the linear oscillator, the maximum value of the magnification factor can be found from
dM d 2M = 0, 2 d d
1 a (3P3 a2 32
8
8) 3P3
which can be solved for
da = d 27P32 a4
da d
da d
8
8) = 0ap =
8(
dM ) d
1)2)
1)
(95)
96(
(97)
1)
=
e31 = −2.2, e33 = 9.3, e15 = 5.8 d11 = 5.64, d33 = 6.35 q15 = 275, q31 = 290.1, q33 = 349.9 s11 = 5.367, s33 = 2737.5
Face sheets
E11 = E22 = E33(GPa) = 0.10363 G12 = G23 = G11=(GPa) = 0.05
E11 = 24.51 (GPa) E22 = E33(GPa) = 7.77 G12 = G13(GPa) = 3.34
( ) = 130 kg m3
12
1 8 + 6P3 a2 8
6
6
Core
whose roots provide 1,2
0.07 × 10
9P32 a4
64µ2
6
Table 4 Material properties of sandwich panel [64].
(98)
1) 2)
S
Piezoelectric/(C・m−2) Dielectric/(10−9C・V−1・m−1) Piezomagnetic/(N・A−1・m−1) Magnetoelectric/(10−12Ns・V−1・C−1)
1)
1) P3 a2 + 64(µ2 + (
1) = 10.26 × 10 = 0.33
11 (C
Properties BiTiO3-CoFe2O4
(96)
To find the values of the critical points 1 and 2 , these points correspond to vertical tangencies of the response curve; that is, where d = 0 : This condition can be found by equating the denominator of Eq. dM (97) to zero, which translates to:
27P32 a4
=
ES From Fig. 1, (Park et al. , 2004) From Fig. 1(b) [61],
Table 3 Material properties of BiTiO3-CoFe2O4 [63].
when:
3P3
1)
1 22 (C ) = 30.1 × 10
as:
8a (3P3 a2 8 8) 1) P3 a2 + 64(µ2 + (
96(
11 (C
da =0 d
r
G23(GPa) = 3.25 12 = 0.22
yields:
4 + µ2 + (a2 )
This derivative vanishes (and so does
(3P3 a2
E11(GPa) = 155 E22(GPa) = 8.07 G12(GPa) = 4.55
(94)
Differentiating Eq. (93) with respect to
SMA
G23(GPa) = 1.34 12 = 13 = 0.078 23 = 0.49
= 0.33
(99)
( ) = 1800 kg m3
4. Results and discussion Table 5 Convergence study of linear/nonlinear frequency parameter of sandwich composite shell under hygrothermal environment.
Numerical results of the nonlinear vibration behavior of doubly curved shell with MR layers and flexible core are presented in this section. The properties of the smart shell are established in Tables 1–4. Furthermore, Elliptic paraboloid shell (R1 R2) is considered and the material properties of which are assumed to be tcnt = 0.0348 nm and tgpl = 0.0348 nm and G13 = G23 = 0.5G12. Table 5 illustrates the linear and nonlinear frequency behavior of sandwich composite shell in two sets of hygrothermal conditions under the influence of curvature ratios (R/a) and volume fractions of SMA. It can be noticed that the temperature rise has a minute effect on the nonlinear forced vibration. Also, it can be witnessed that by increasing the SMA volume fraction, the fundamental natural frequency of the shell will increase.
%SMA
R/a
T= 300, l
0
5 10 50 5 10 50
10
T= 400,
H= 0 nl
13.0634 12.8629 12.7288 13.1882 12.9321 12.8361
H= 0.5
l
13.1491 13.0762 12.9522 13.3192 13.1632 13.0981
nl
13.0805 12.8793 12.7618 13.2109 12.9842 12.8824
13.1523 13.0790 12.9903 13.3723 13.2279 13.1203
Table 1 The properties of multiscale composite shell [13,36]. Carbon (fiber)
Epoxy (matrix)
E11f (GPa ) = 233.05
v m = 0.3 m
E11f (GPa ) = 23.1
= 0.6
f
( ) kg m3
m
11 (K
1)
=
22 (K
1)
= 10.8 × 10
E cn (GPa)= 640(1
0.54 × 10
0.0005 T ) 9
t cn (m) = 0.34 × 10
0.0034T + 0.142H )
(K 1) = 45(1 + 0.001T ) × 10
= 2.68 × 10
= 0.2
Graphene platelet
dcn (m) = 1.4 × 10
(kgm) = 1200
Em (Gpa) = (3.51
G12f (GPa) = 8.96 f
Carbon nanotube
lcn (m) = 25 × 10
6/K
3wt% 1
lcn
6
6
12
9
6
(m) = 0.25 × 10
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 143 (2019) 106152
M. Karimiasl, et al.
Table 6 Comparison of dimensionless frequencies of (0°/90°/0°/core/0°/90°/0°). Dimensionless frequency ( ) Mode(m, n)
Rahmani et al. [64]
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
Table 7 Comparison of nonlinear frequency ratio of [0°/90°/90°/0°] laminated plate (R1 = R2 = ). Applied voltage
nl l
50
[65] Present [65] Present [65] Present
0 −50
0.2
0.4
0.6
0.8
1
1.019 1.01919 1.01921 1.01993 1.020 1.02018
1.075 1.07621 1.075 1.07590 1.075 1.07601
1.161 1.16284 1.163 1.16401 1.164 1.16591
1.272 1.27491 1.274 1.27521 1.276 1.27783
1.402 1.40261 1.405 1.40626 1.408 1.40889
Fig. 3. Influence of different distributions patterns on large deflection versus a b nonlinear of doubly curved shell with R = 0.1m, R = 0.05m q = 0.015 and (m, n = 1,2).
y,
y¯ =
y , x¯ = b
x , a (100)
Fig. 3, illustrates the effect of different porosity distributions pattern such as A, B, C, D versus nonlinear to the linear frequency with a b a a = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1, B = 500 and the cross R1 2 ply [0/MR/core/MR/0] stacking sequence. Composite shell mode (m, n) = (1,1) is considered. It can be observed from the evaluation that the frequency of the C distribution is highest and D is the lowest value. Influence of curvature ratio ( b ) on the fundamental frequency of R2 cross ply [0/MR/core/MR/0] multiscale doubly curved shell with, a = 0.1, h = 2 mm is presented in Fig. 4. The frequency mode is taken R1 to be (m, n = 1, 1). It is found that increasing the curvature value reduces the nonlinear frequency. It is important to express that the effect of curvature ratios has a significant role in the numerical results. It can be found that when the circumferential wave number decreases, the hardening effect declines. Furthermore when the circumferential wave number stays constant, the hardening effect increases. This is because of the fact that an increase in nonlinear curvature ratio ( b ) leads to a R2 decrease in the stiffness of structure which causes natural frequency and stability to decrease. The influence of magnetic field on the frequency-amplitude curve,
Table 8 Comparison of the curvature ratios (R/a) and volume fractions of SMA with different amplitude ratios versus nonlinear frequency of composite spherical a panel with h and R1 = R2 . R/a
2
u0 v0 w , v¯0 = , w¯ 0 = 0 , ¯x = x , ¯y = a b R z ¯ R1 ¯ R2 R2 T z¯ = , R1 = , R2 = , = l h R R h ET
u¯ 0 =
As a part of the validation of the present method, the correctness of linear frequency of doubly curved shell with h/a = 0.10 and hc/ h = 0.88 based on first shear deformable theory are compared with Rahamni et al. (2010) and presented in Table .6. Due to different selections of displacement field of the core, there is little difference between the results of the present work with Rahamni et al. (2010) researches. The nonlinear frequency ratio of the laminated plate is presented in Table 7 based on TSDT, in which a comparison is made with the results reported in Ref. [65]. It is brightly shown that the numerical results are very similar. Finally, the influence of curvature ratios (R/a) and volume fractions of SMA with different amplitude ratios on the nonlinear frequency of a composite spherical panel with h and R1 = R2 are established in Table 8. Also, the results are compared with [66] based on the HSDT. These comparisons show that the results from the present method are in good agreement with existing results. The dimensionless parameters are adopted as:
%SMA
1
nl
A 1.2
0
5
10
5 10 50 100 5 10 50 100 5 10 50 100
3.4
5.8
Parhi (2016)
Present
Parhi (2016)
Present
Parhi (2016)
Present
39.83 30.28 20.52 18.29 41.59 30.44 20.52 17.09 41.58 34.10 25.99 19.96
38.77 29.14 20.01 17.55 40.70 29.84 20 16.49 40.66 29.78 25.83 19.24
35.57 27.46 19.96 17.01 35.81 26.82 19.96 17.81 34.54 32.94 25.94 21.71
35.11 26.44 19.80 16.43 35.03 26.13 19.83 17.09 33.88 32.17 25.80 20.81
28.69 21.84 17.65 14.86 27.25 23.20 17.65 14.94 32.66 22.32 21.53 13.82
28.24 21.10 17.57 14.16 26.87 22.53 17.49 14.32 32.21 21.75 21.41 13.14
Fig. 4. Influence of different curvature ratio and magnetic field on large dea b flection versus nonlinear of doubly curved shell with R = 0.1m, R = 0.05m , h = 2 mm, q = 0.015 and (m, n = 1,2).
13
1
2
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
Fig. 7. Influence of temperature rise and magnetic potential on large deflection a b versus nonlinear of doubly curved shell with R = 0.1m, R = 0.05m , h = 2 mm,
Fig. 5. Influence of different magnetic field on large deflection versus nonlinear a b of doubly curved shell with R = 0.1m, R = 0.05m , h = 2 mm, q = 0.015 and 1
(m, n = 1,2). a R1
q = 0.015 and (m, n = 1,2).
2
b
1
2
= 0.1, R = 0.05, h = 2 mm, h = 10, b = 1 , q = 0.015 and (m, 2 n = 1,1) is shown in Fig. 5. Stacking sequence considered is cross ply [0/MR/core/MR/0]. It is brightly shown that the nonlinear frequency parameters increase by increasing the magnetic field. Furthermore, the nonlinear frequency is not only the function of stiffness and mass matrix but also depend up on the amplitude ratio. Fig. 6 depicts the effect of 2hMR with U distribution pattern versus a
frequency-amplitude a h
= 10,
a b
curve
h
with
a
a R1
= 0.1,
b R2
= 0.05,
h = 2 mm,
= 1, q = 0.015 and stacking sequence [0/MR/core/MR/0].
The results confirm that 2hMR have a minimal effect on the fundamental h frequency ratios of the doubly curved shells. Further, it can be observed that by increasing the MR layer thickness, the nonlinear frequency decrease due to a reduction in the stiffness and dynamics stability of the sandwich structure. The influence of temperature rise and different magnetic field on the a b frequency-amplitude curve with R = 0.1, R = 0.05, h = 2 mm, 1
Fig. 8. Influence of curvature ratio and applied voltage on large deflection a b versus nonlinear of doubly curved shell with R = 0.1, R = 0.05, h = 2 mm,
2
= 10, b = 1, ΔH = 1 and (m, n = 1,1) is shown in Fig. 7. Stacking sequence is considered cross ply [0/MR/core/MR/0]. Two set of T = 300 and 400 K and B = 0, B = 50 are considered for evaluation. It is clearly shown that nonlinear frequency increases with rising temperature and decrease by increasing of magnetic field. In addition, according to the observation made, it can be revealed that the a h
a
T = 300, ΔH = 1 q = 0.015 and (m, n = 1,2).
curved shell with n = 1,2).
2hMR h a = R1
nonlinear amplitude have been presented in Fig. 9. With R = 0.1, 1 h = 2 mm, T = 300, ΔH = 1, V = 50, q = 0.015 and (m, n = 1,2). PGF PCF Stacking sequence is considered cross ply [0 /90 ]S. It is brightly observed that the curvature ratio has a reducing effect on nonlinear frequency. Furthermore, negative magnetic potential has a lower influence on nonlinear amplitude curve than positive magnetic potential magnitude. It can be found that when the circumferential wave number
ratio on large deflection versus nonlinear of doubly
0.1m,
b R2
2
temperature variation only has a small influence on the nonlinear frequency ratios of the multiscale doubly curve shell. Influence of curvature ratio ( b ) and applied voltage with positive R2 and negative magnitude under the fundamental frequency of cross ply a PGF PCF [0 /90 ]S multiscale doubly curved shell with R = 0.1, h = 2 mm, 1 T = 300, ΔH = 1, q = 0.015 and = 0.2 is presented in Fig. 8. The frequency mode is taken to be (m, n = 1,1). It is found that increasing of curvature value leads to a decrease in the nonlinear frequency. Meanwhile, the positive magnitude of applied voltage has the significant effect on nonlinear frequency whereas the negative applied voltage has the lowest magnitude. This is because of the fact that an increase in nonlinear curvature ratio ( b ) leads to a decrease in the R2 stiffness of structure which causes natural frequency and stability to decrease. For investigated effects of different magnetic potential and curvature ratio ( b ), the numerical results of nonlinear frequency versus R2
Fig. 6. Influence of
1
= 0.05m , h = 2 mm, q = 0.015 and (m,
14
a
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
Fig. 9. Influence of curvature ratio and magnetic potential versus on large a b deflection versus nonlinear of doubly curved shell with R = 0.1, R = 0.05, h = 2 mm, T = 300, ΔH = , q = 0.015 and (m, n = 1,2).
1
Fig. 11. Influence of aspect ratio and magnetic potential on large deflection a b versus nonlinear frequency of doubly curved shell with R = 0.1, R = 0.05,
2
1
2
Fig. 12. Influence of the SMA volume fraction on large deflection versus nona linear frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300,
Fig. 10. Influence of aspect ratio and applied voltage on large deflection versus a b nonlinear of doubly curved shell with R = 0.1, R = 0.05, h = 2 mm, T = 300, ΔH = 1,q = 0.015 and (m, n = 1,2).
1
h = 2 mm, T = 300, ΔH = 1 q = 0.015 and (m, n = 1,2).
1
ΔH = 1, q = 0.015 and (m, n = 1,1).
2
potential. The reason is that, by increasing the aspect ratio, the dynamic stability of the structure increase. Fig. 12 depicts the effects of different SMA volume fractions on the nonlinear forced vibrations of the doubly curved shell subjected to the hya b a a grothermal loading with R = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1,
decreases, the hardening effect declines. Furthermore when the circumferential wave number stays constant, the hardening effect increases. Fig. 10, investigated the effect of aspect ratio and kind of applied voltage versus frequency-amplitude curve of multiscale doubly curved a b cross-ply [0 PCF /90 PGF ]S shell with R = 0.1, R = 0.05, h = 2 mm, 1 2 T = 300, ΔH = 1, q = 0.015 and = 0.2 . It is clear that aspect ratio is the important factor to study of thick curved panel structures. The nonlinear to linear frequency increase by increasing the aspect ratio and positive magnitude applied voltage. Furthermore, by increasing the value of aspect ratio leads to an increase in the fundamental frequency. In addition, the decrease in the circumferential wave number causes a decrease in the hardening effect. The reason is that, by increasing the applied voltage, the dynamic stability of the structure increase. Thereafter, the influence of aspect ratio and different magnetic a b potential versus nonlinear amplitude curve with R = 0.1, R = 0.05, 1 2 h = 2 mm, T = 300, ΔH = 1,V = 50, (m, n = 1,1), q = 0.015 and the PGF PCF cross ply [0 /90 ]S multiscale doubly curved shell are study in Fig. 11. Via numerical results, it is observed that by increasing the aspect ratio and the magnetic potential, the nonlinear frequency increases and the most influence refer to positive magnitude of magnetic
1
2
T = 300 ΔH = 1,q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S, where the dimensionless vibration amplitude ratio is plotted versus the excitation frequency ratio. It is brightly shown that by increasing the SMA volume fraction the non-dimensional amplitude and hardening effects decline. Furthermore, by increasing the SMA volume fraction the fundamental natural frequency of the shell will increase. It indicates that the higher SMA volume fraction results in the larger stiffness of the doubly curved shell. In addition to, the corresponding circumferential and longitudinal wave numbers are reported in the figure. The effects of temperature rise and the SMA volume fraction influence versus the frequency response curves of the doubly curved shell a b = 0.1, R = 0.05, have been shown in Fig. 13 with R a
a
1
2
h = 2 mm, h = 10, b = 1, T = 300 ΔH = 1, q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S. The results show in which by increasing of thermal loading yields to the reduction of the fundamental natural frequency of the shell. In addition to, it is shown that the temperature rise have very small effects on nonlinear forced vibration. Also, by increasing of SMA volume fraction the fundamental natural frequency of the shell will increase.
15
Thin-Walled Structures 143 (2019) 106152
M. Karimiasl, et al.
Fig. 15. Influence of curvature ratio and SMA on large deflection versus nona linear frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300,
Fig. 13. Influence of temperature rise and SMA on large deflection versus a nonlinear frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300,
ΔH = 1, q = 0.015 and (m, n = 1,1).
1
ΔH = 1, q = 0.015 and (m, n = 1,1).
1
Fig. 16. SMA modulus of elasticity versus temperature.
Fig. 14. Influence of aspect ratio and SMA on large deflection versus nonlinear a frequency of doubly curved shell with R = 0.1, h = 2 mm, T = 300, ΔH = 1, q = 0.015 and (m, n = 1,1).
1
Fig. 14, illustrate the influence of curvature ratios on the nonlinear a b frequency of doubly curved shells with R = 0.1, R = 0.05, a
1
a
2
h = 2 mm, h = 10, b = 1, T = 300 ΔH = 1,q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S. It can be seen that, increasing the curvature ratio accompanies with the decrease of the fundamental natural frequencies and dimensionless maximum deflection and also yields the change in the circumferential wave number. It can be found that when the circumferential wave number decreases, the hardening effect declines. Furthermore when the circumferential wave number stays constant, the hardening effect increases. Fig. 15 shows the influence of aspect ratio on the nonlinear frea b a a quency with R = 0.1, R = 0.05, h = 2 mm, h = 10, b = 1, T = 300 1
Fig. 17. SMA recovery stress versus temperature.
Fig. 18, investigates the effect of aspect ratio on the linear and nonlinear frequency of multiscale doubly curved cross-ply a b [0 PCF /90 PGF ]S shell with R = 0.1, R = 0.05, h = 2 mm, T = 300, 1 2 ΔH = 1, q = 0.015. It is clear that aspect ratio is the important factor to study of thick curved panel structures. The nonlinear and linear frequency increases by increasing the aspect ratio. Influence of curvature ratio ( b ) on the linear and nonlinear freR2 quency of cross ply [0/MR/core/MR/0] multiscale doubly curved shell a with, R = 0.1, h = 2 mm, T = 300, Vs = 0%, B = 0 and ΔH = 1is 1 presented in Fig. 19. The frequency mode is taken to be (m, n = 1, 1). It can be observed that when the composite shell becomes deep, its
2
ΔH = 1, q = 0.015 and stacking sequence [0 PCF /90 PGF /0 SMA ]S. The results indicate that with increase the aspect ratio, the circumferential wave number and corresponding vibrational mode shape stay constant, by increasing hardening effects. In addition to, the decrease in the circumferential wave number causes to decrease in the hardening effect. Figs. 16 and 17 depict the variation of the SMA Young's modulus and the recovery stress versus temperature rise with high nonlinearity. The influence of the initial stresses and initial deflections are considered in the incremental method.
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Fig. 18. Influence of aspect ratio on large deflection versus nonlinear frequency a of doubly curved shell with R = 0.1, h = 2 mm, T = 300, ΔH = 1, q = 0.015 and (m, n = 1,1).
1
Fig. 19. Influence of curvature ratio on large deflection versus nonlinear frea quency of doubly curved shell with R = 0.1, h = 2 mm, T = 300, ΔH = 1, q = 0.015 and (m, n = 1,1).
1
bending energy is low as compared to stretch in energy and it subsequently affect the final responses of the structure. Furthermore, the nonlinear frequency is not only the function of stiffness and mass matrix but also depend up on the amplitude ratio. It is found that increasing the curvature value lead to a decrease in the linear and nonlinear frequency. It is important to express that the effect of curvature ratios has significant role in the numerical results. 5. Conclusions Numerical investigation of nonlinear forced vibration of intelligent multiscale SMA/matrix composite doubly curved shell in the hygrothermal environment is studied in this research. Micromechanical model is used to estimate the temperature dependent material properties of different materials. The major difference between the present model and the previous one is that the present model includes the smart composite material with a flexible core. Three types of porosity are considered. The results reveal that the C pattern in CNT/GPL as the reinforcement through the thickness results in a stiffer shell when compared to other types of porosity, also the result presented volume fraction of SMA, Positive applied voltage and magnetic potential have a significant effect on the fundamental frequency of composite shell. Based on observation it is found that increasing of SMA volume fraction yields decreases of nonlinear frequency. 17
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