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Nonlinear vibration of imperfect FG-CNTRC cylindrical panels under external pressure in the thermal environment
Composite Structures 227 (2019) 111310
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Nonlinear vibration of imperfect FG-CNTRC cylindrical panels under external pressure in the thermal environment
T
K. Foroutana, H. Ahmadia, E. Carrerab,
⁎
a b
Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran Mul2 Group, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy
ARTICLE INFO
ABSTRACT
Keywords: Composite cylindrical panels FG carbon nanotube Nonlinear vibration analysis Initial imperfection Thermal environment
An analytical approach for the nonlinear vibration analysis of imperfect functionally graded carbon nanotubereinforced composite (FG-CNTRC) cylindrical panels is presented. The FG-CNTRC cylindrical panel is subjected to external pressure in the thermal environment. The nonlinear temperature distribution in the thickness direction is assumed. The classical thin shell theory with the von-Kármán strain-displacement kinematic nonlinearity is employed in the constitutive laws of the shell. The governing equation is solved by utilizing Galerkin’s method in conjunction with the stress function concept. Finally, to find the nonlinear dynamic responses, the fourth order Runge-Kutta method is used. The effect of the four various types of the CNTs distributions, such as UD, FG-Λ, FG-X, and FG-O is considered on the cylindrical panel. The influence of material parameters, initial imperfection, and temperature on the nonlinear vibration response of functionally graded CNTRC cylindrical panel is presented.
1. Introduction Recently, the research on the analysis of advanced materials has been of interest of scientists. The carbon nanotubes (CNTs) as one of the advanced materials with extraordinary stiffness and tensile strength are utilized as reinforcing components for composites. Also, these structures have been extensively used in a wide range of engineering industrials consist of submarines, aircraft, and aerospace engineering [1–4]. Therefore, the research on the vibration behavior of these structures has been one of the interests of scientists from many years ago, and the great number of researches has been done on the nonlinear vibration behavior of CNTRC shell structures under mechanical loading. Some studies are concentrated in the field of the vibration analysis of cylindrical shell and plate. Thai and Kim [5] reported a review of theories for the modeling and analysis of functionally graded plates and shells. Akgöz and Civalek [6] analyzed the nonlinear vibration analysis of laminated plates resting on nonlinear two-parameter elastic foundations. The vibration behavior of imperfect eccentrically stiffened FG cylindrical shells with an elastic medium under mechanical and damping loads was studied by Duc and Thang [7]. Dung and Nam [8] addressed the vibration behavior of stiffened functionally graded cylindrical shells surrounded by an elastic medium under external pressure. Bich et al. [9] reported the dynamic and static behaviors of ⁎
stiffened FG cylindrical shells. Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation with a combination of harmonic differential quadrature-finite difference methods was investigated by Civalek [10]. Foroutan et al. [11] addressed the vibration behavior of functionally graded cylindrical shells with spiral stiffeners surrounded by nonlinear elastic foundation and damping under axial loading. Some studies have analyzed the vibration of composite cylindrical shells. Sofiyev et al. [12] analyzed the vibration analysis of the composite cylindrical shells surrounded by the elastic mediums and using the SDT. Vibration analysis of the composite cylindrical shells subjected to the periodic radial and axial loading was addressed by Dey and Ramachandra [13]. Choe et al. [14] reported the vibration behavior of the composite shell by considering axis-symmetric geometry with doubly-curved using the unified Jacobi-Ritz method. The dynamic response of composite cylindrical shell was investigated by Zhang et al. [15]. In the above-mentioned studies, the influences of the carbon nanotube on the dynamic behavior of cylindrical shells have not been considered. Some researchers have been done on the dynamic behavior of cylindrical shells reinforced by carbon nanotube. The vibration behavior of functionally graded CNTRC shell was addressed by Zghal et al. [16]. Shen et al. [17] presented the vibration response of FG composite cylindrical panels reinforced by graphene
Corresponding author. E-mail address: [email protected] (E. Carrera).
https://doi.org/10.1016/j.compstruct.2019.111310 Received 15 June 2019; Received in revised form 4 August 2019; Accepted 5 August 2019 Available online 08 August 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
with elastic foundation under thermal loading. The vibration analysis of functionally graded CNTRC cylindrical shells in the thermal environment was addressed by Song et al. [18]. Kiani et al. [19] reported the vibration of functionally graded CNTRC skew cylindrical shells utilizing the Chebyshev-Ritz formulation. Civalek [20] studied the free vibration of carbon nanotubes reinforced and functionally graded shells and plates based on FSDT via discrete singular convolution method. The vibration response of CNTRC cylindrical shells in thermal environments was presented by Shen and Xiang [21]. Shen et al. [22] investigated the vibration response of composite cylindrical shells reinforced by FG graphene under the thermal environments. Duc et al. [23] analyzed the nonlinear dynamic and free vibration response of imperfect functionally graded CNTRC shallow shells under the thermal environment. The stability and vibration analysis of the FG-CNTRC cylindrical panels utilizing the semi-analytical approach reported by Chakraborty et al. [24]. Qin et al. [25] presented the vibration behavior of rotating functionally graded CNTRC cylindrical shells with arbitrary boundary conditions. Review of the literature shows that there is no study on the nonlinear vibration analysis of imperfect FG-CNTRC cylindrical panels subjected to the external pressure in the thermal environment. Therefore, the novelties of the present paper may be summarized as follows: (1) investigation of the nonlinear vibration analysis of FGCNTRC cylindrical panels via the semi-analytical method, (2) incorporation of the effect of initial small geometric imperfection on the cylindrical panels, (3) the system is considered under the external pressure and thermal environment, (4) The nonlinear temperature distribution in the thickness direction is assumed. The four various types of the CNTs distributions, such as UD, FG-Λ, FG-X, and FG-O are considered. Effect of the von Kármán strain-displacement kinematic nonlinearity based on the classical thin shell theory is included in the constitutive laws of the shell. The Galerkin method is applied to the nonlinear vibration problem. In continue the fourth-order Runge-Kutta method is utilized to find the vibration responses. To valid the formulations, comparisons are made with the previous researches. Results are presented to evaluate the influence of initial imperfection, material parameters, and temperature on the nonlinear vibration response of FGCNTRC cylindrical panel.
Fig. 2. Configuration of the FG-CNTRC cylindrical panel.
properties are determined for CNT reinforced FGM materials as illustrated in Fig. 2 which for four patterns of CNT distribution are considered such as UD CNTRC, FG-Λ CNTRC, FG-X CNTRC, and FG-O CNTRC. It should be noted that the CNT volume fractions VCNT for each types of FG-CNTRC are different which is shown in Eq. (3). We assume that a CNTRC layer is made of a mixture of single-wall carbon nanotubes (SWCNTs) and the matrix is assumed isotropic [2,25,26]. The SWCNT reinforcement is either functionally graded (FG) or uniformly distributed (UD) across the shell thickness, as shown in Fig. 2. At nanoscale, the carbon nanotube structure strongly effects on the overall the composite properties. Various micromechanical models have been developed to predict the effective material CNTRCs properties, e.g. the model of Mori-Tanaka [27,28] and the model of Voigt as the rule of the mixture [29,30]. In [31,32] were used the approach of Eshelby–Mori–Tanaka to compute the elastic stiffness properties of nanocomposite materials reinforced by graded oriented, straight CNTs. The Mori-Tanaka approach is suitable to nanoparticles and the mixture rule is simple and convenient to apply the predicting for the overall properties of material and the structures responses. The accuracy of the mixture rule was considered and a remarkable synergism between the MoriTanaka approach and the mixture rule for functionally graded metal–ceramic beams was investigated in [31]. With regard to the mixture rule, the effective shear modulus, Young's modulus, and mass density of CNTRCs are as follows [22,33,34]
2. Description of the FG-CNTRC cylindrical panel
E11 =
A schematic view of a CNTRC cylindrical panel with its coordinate system (x, y, z) is shown in Fig. 1, where xy is the mid-plane of the panel and z is the thickness coordinate. The point “O” in Fig. 1 shows the coordinate system origin. The cylindrical panel has the length in the ydirection b, length in the x-direction a, thickness h, and radius of curvatures R. The distribution of CNTs in the matrix may be uniform or functionally graded in the thickness direction of the shell layer. So material
2
E22 3
G12
= =
CNT 1 VCNT E11 VCNT Vm CNT + E m E22 VCNT CNT G12
= VCNT
+
CNT
+ Vm E m
Vm Gm
+ Vm
(1)
m
Vm and VCNT are the volume fractions of the matrix and the CNT which satisfy the relationship of Vm + VCNT = 1. The coefficients of thermal expansion in the transverse and longitudinal directions can be written as CNT + Vm m 11 CNT CNT (1 + 12 ) VCNT 22 CNT * 12 + Vm m VCNT
11
= VCNT
22
=
12
=
+ (1 + v m) Vm
m
12 11
(2)
The transferring load between the polymeric and nanotube phases is less than perfect (for example, intermolecular coupled stress effects, the surface effects, strain gradients effects, etc.). Thus, it is defined i (i = 1, 2, 3) (the parameter of CNT efficiency) into Eq. (1) to consider the influence of small scale and other influences on the CNTRCs material properties. In Fig. 2, the four various types of CNTs distributions are depicted. We assume that the CNTRC is made from a mixture of single-wall carbon nanotubes (SWCNT), graded distribution in the thickness direction of the shell layer, and matrix which is assumed to be isotropic. The relation of CNTs volume fraction in each type is explained as:
Fig. 1. Configuration of FG-CNTRC cylindrical panel. 2
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Table 1 Comparison of the dimensionless natural frequencies (a b = 1, h a = 0.1, Em = 7 × 1010N m2, Ec = 38 × 1010N m2, Present
k
a b
FGM plate 0
0
FGM cylindrical plate 0.5 0
* VCNT
UD: VCNT (z ) =
(
Matsunaga [48]
Alijani et al. [49]
Error (%)
Error (%)
Error (%)
0.057
3.4
0.058
1.7
0.059
0.0
0.064
0.063
1.5
0.062
3.1
0.064
0.0
( ) 2z h
(
xy
) 2 | z| h
)
1 E = 22 E11
( ) ( )w
wCNT +
CNT
m
m
(4)
CNT
T (z ) =
z x,
y
0 y
=
z y,
xy
=
0 xy
xy
= v,y
0 xy
= u, y + v , x + w , x w , y = w , xx , y = w, yy , xy = w, xy
w R
(5)
0 y, xx
21 E11
; Q12 =
1
12 21
; Q66 = G12;
(
To )ln(R + z ) + ln R
(
h 2
ln R
)
(
h 2
)T
ln R +
(
h 2
ln R +
o h 2
0 xy, xy
= w ,2xy
21
)
)T
i
(10)
0 x
+ (E12, E22 , 0)
0 y
+ (0, 0, E33 )
*) (0, 0, 2E33
y
xy
0 xy
* , E12 * , 0) (E11
+ (1, 0, 0)
1
+ (0, 1, 0)
0 x
* , E*22 , 0) + (E12
**, E22 **, 0) (E12
y
0 y
+ (0, 0, E*33)
(0, 0, 2E** 33 )
xy
0 xy
2
**, E12 **, 0) (E11
+ (1, 0, 0)
3
+ (0, 1, 0)
4
(12)
1
+ 2 w ,2y
where Eij are the coupling, bending and extensional stiffness components and i (i = 1, 2, , 4) are the thermal components of CNTRC cylindrical panel shell that are as follows
(6)
(Eij , Eij*, Eij**) =
strain, and y0 , x0 are normal strains. Also, x , y, xy are the shell curvatures changes and twist, respectively. Considering Eq. (6), the compatibility equation is derived as follows
+
12 21
(Mx , My , Mxy )
In Eq. (6), w = w (x , y ), v = v (x , y ) and u = u (x , y ) are the compo0 nents of displacement inz , y and x directions, respectively. xy is a shear
0 x , yy
1
(9)
* , E22 * , 0) (E12
x
0 y
(8)
(11)
1
= u, x + 2 w ,2x
E22
; Q22 =
* , E12 * , 0) = (E11
0 x
T
22
12
(Ti
x
Where
x
12 21
= (E11, E12, 0)
2z
0
xy
(Nx , Ny , Nxy )
With regard to the von Kármán kinematic nonlinearity [40], the strain-displacement relations are as follows 0 x
0
0 0
where To and Ti are outside and inside temperature of the cylindrical panel shell and how to calculate T (z ) is also presented in the Appendix. To derive the resultant forces (Nx , Ny , Nxy ) for FG-CNTRC cylindrical panel, the stress-strain equations (Eq. (8)) are integrated through the thickness. Also, to derive the resultant moments (Mx , My , Mxy ), first multiply the stress-strain equations (Eq. (8)) in z and then integrate through the thickness as
3. Governing equation of motion
=
11
y
21 and 12 are the Poisson’s ratios. y, x and xy are the normal stress and shear stress of cylindrical panel, respectively. The thermal distribution in Eq. (8) is obtained as
Research groups have demonstrate that the material properties of CNT-reinforced materials are very dependent on the structure of CNTs [28,29]; effective material properties of the CNTRC are obtained through a several of micromechanics techniques, such as Eshelby-MoriTanaka approach [35,36] and the mixture rule [37,38]. The approach of Eshelby-Mori-Tanaka, according to the equivalent elastic inclusion idea of Eshelby and the concept of average stress in the matrix due to Mori and Tanaka, is also known as the equivalent inclusion-average stress method. With regard to the Benveniste’s revision [39], effective elastic module has been achieved. Considering Fig. 2, in the case of FG-X case, the inner and the outer surfaces are CNT-rich while for the FG-O, the middle surface of the cylindrical panel is CNT-rich. For the FG-Λ case, the outer surface is free of the CNTs, and the inner surface of the cylindrical panel is CNT-rich. Finally, the CNTs that is uniformly distributed along the thickness direction of the CNTRC cylindrical panel which is named the UD.
x
E11
Q11 =
wCNT
CNT
x
where
(3)
where
* = VCNT
Q11 Q12 0 = Q12 Q22 0 0 0 Q66
y
|z| h
* 1 2VCNT
FG O:
= 3800kg m3 ).
m
0.059
* 1 : VCNT
FG
= 2796kg m3,
Chorfi and Houmat [47]
x
* FG X: 4VCNT
m
1 w , xx R
w , xx w , yy
(7)
The relations of stress-strain for FG-CNTRC cylindrical panel are defined as
( 1,
2)
=
( 3,
4)
=
as 3
h 2 h 2 h 2 h 2
h 2 h 2
Qij (1, z , z 2) dz
;
i, j = 1, 2, 3
(Q11
11
+ Q12
22,
Q12
11
+ Q22
22 )
Tdz
(Q11
11
+ Q12
22,
Q12
11
+ Q22
22 )
Tzdz = 0
(13)
With regard to Eq. (11), the components of strain can be rewritten
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Table 2 Comparison of the natural frequencies (Hz) for isotropic cylindrical panels (a = 0.2794, b = 0.2286, E = 7.2 × 1010N m2,
= 2800kg m3 ).
= 1 3,
R (m)
h (mm)
(m , n)
Present
Shen and Xiang [21]
Soedel [50]
Nath [51]
Chern and Chao [52]
Bardell and Mead [53]
2.4384 2.4384 1.8288 1.8288 1.2192 1.2192
0.7112 1.2190 0.7112 1.2190 0.7112 1.2190
(1,1) (1,1) (1,2) (1,1) (1,2) (1,2)
144.9 162.2 168.0 199.9 182.0 280.6
143.8 163.2 167.4 200.8 181.7 281.9
144.3 163.7 167.8 201.4 182.1 282.3
144.5 163.9 167.8 201.7 182.2 282.3
142.1 161.4 166.7 198.5 180.8 278.1
143.8 163.1 167.2 200.8 181.6 281.8
Table 3 The CNT/matrix efficiency parameters. * VCNT 0.12 0.17 0.28
1
0.137 0.142 0.141
( x0,
0 y,
33 ) Nxy 2
1.022 1.626 1.585
3
*) 2 33
0.715 1.138 1.109
xy
0 xy )
=(
*, + ( 11 +(
12 2
22 ,
12 ,
*21, 0)
x
22 1,
0) Nx + ( *, + ( 12
12 ,
*22 , 0)
0, 0) + (0,
11, y
0) Ny + (0, 0,
+ (0, 0,
12 1
11 2 ,
0)
(14)
Then, Eq. (14) is substituted into Eq. (12) in the following form
Fig. 4. The nonlinear vibration responses of the cylindrical panel with and without CNT. 4
Composite Structures 227 (2019) 111310
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density 1
1
are
h 2
=
h 2
(z ) dz
(17)
With regard to the first two Eq. (16), stress function ( ) is defined as
Nx =
yy ,
Ny =
xx ,
Nxy =
(18)
xy
By substituting Eq. (14) in Eq. (7) and Eq. (15) in the third part of Eq. (16) and then by utilizing Eq. (6) and (18), the following equations of the system can be derived as 11 , xxxx
+(
* , xxxx 21
w, xx + 2
+(
+
1 w , tt
33
1 (w , xx R
, xxxx
Fig. 6. The nonlinear dynamic response of FG-Λ CNTRC cylindrical panel.
2H33 ) 0)
xy
+ (H13, 0, 0)
w, xy
*21, 0) Nx + ( 12 * , *22 , 0) Ny + (0,
1
x
(H21, H22, 0)
+ (0, H23, 0)
2
+ (1, 0, 0)
3
(15)
where Hij , ij , Hij*, ij* are presented in Appendix. The nonlinear governing equations of motions of the assembly may be found by applying Donnell’s shell theory, in the following form [9,41,42]
(
=
1 w , tt
*) 2 33
2
12 ) , xxyy
+
2 *33)(w , xxyy w ,*xx )
22 , yyyy
(19)
* , yyyy 12
, xxyy
, xx , yy
R
(20)
[w,2xy
w, xx w, yy ] + [w,*xy2
w,*xxxx ) + w ,*yyyy ) w ,*xxw ,*yy] = 0 (21)
w,*xxxx ) + (H12 + H21 + 4H33)
w ,*xxyy ) + H22 (w , yyyy * + *22 ( 11 , xx w , yy
* (w, xxxx + 12
* (w, yyyy w,*xxyy ) + 21
2 *33)
w,*yyyy )
, xxyy
* , yyyy 12
*21 , xx
R
, yy w , xx
+2
, xy
(22)
q (t ) = 0
w = 0, Mx = 0, Nx, y = 0, at x = 0; a w = 0, My = 0, Nx , y = 0 at y = 0; b
(23)
Considering the boundary condition, the total deflection and initial imperfection of cylindrical panel is expressed as follows [42–46]
Nx , x + Nxy, y = 0 Nxy, x + Ny, y = 0 Mx, xx + 2Mxy, xy + My, yy + Nx w , xx + 2Nxy w , xy + Ny w , yy +
w , xx w , yy] = 0
A simply supported imperfect FG-CNTRC cylindrical panel under external pressure in the thermal environment is considered. The applied boundary conditions are as follows
+ (0, 1,
4
1 R
2 *33)
4. Nonlinear vibration analysis of the FG-CNTRC cylindrical panel
(0, 0,
y
* w, xxxx + ( 11 * + 22 * + 12
q (t ) = 0
, xx w , yy
+ 2 1 c w , t + H11 (w, xxxx
(w, xxyy
(H11, H12 , 0)
* + 22 * ( 11
, xy w , xy
* + *22 ( 11
* ) Nxy 0, 33
22 , yyyy
For the initial small geometric imperfection of CNTRC cylindrical panel: The initial small geometric imperfection of the cylindrical panel assumed here can be considered as a small deviation of the middle surface of the panel relative to the perfect shape. With regard to this fact that an initial deflection is very small compared with the dimensions of the panel, whereas may be compared with the thickness of the panel. Let w* = w * (x , y ) show a known small imperfection, resulting from the Eqs. (19) and (20) for a perfect CNTRC cylindrical the panel. Considering Volmir’s approach [42] for an imperfection panel, the system of motion equations can be formulated for an imperfect CNTRC cylindrical panel as 11 , xxxx
*, (Mx , My , Mxy ) = ( 11
+
1 + w, xx + [w ,2xy R
+ 2 1 c w , t + H11w, xxxx + (H12 + H21 + 4H33) w , xxyy + H22
w, yyyy Fig. 5. The nonlinear dynamic responses of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
12 ) , xxyy
*21w , yyyy
w, xxyy + 1 w , tt
2
33
w (x , y , t ) = W (t ) sin m x sin w * (x , y ) = W0 sin m x sin n y
) + q (t )
m a
n b
ny
(24)
where m = and n = , subscript m is the number of half wave in the axial direction and n is the number of full wave in the circumferential direction. Also, Wmn (t ) represents the unknown deflection amplitude, and W0 is the known initial amplitude.
+ 2 1 c w,t (16)
where q (t ) is the external pressure, c is coefficient of damping and mass
5
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Fig. 7. The phase plane of nonlinear forced vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
The i (i = 1, 2, 3) coefficients may be obtained by equating the resulting identical trigonometric expressions, to reach at 1
=
3
=
2 n
2 (W 32 11 m
W02 ),
2
2
2 m
(W 32 22 n2
=
4 +( 2 2+ 2 33 *21 m * + 22 * * ) m * n4 11 n 12 4 2 2 4 11 m + ( 33 2 12 ) m n + 22 n
2 m R
W02 )
2
W
(26)
After substitution of Eqs. (24) and (25) into Eq. (22), the following results may be obtained as follows by applying the Galerkin method 4 m
(H11
¨ + 2cW + W
2 2 m n
+ (H12 + H21 + 4H33 )
+ H22
4 n)
1
+
* ( 21
4 m
* + *22 + ( 11 1(
* m4 21
4 11 m
Fig. 8. The phase plane of nonlinear forced vibration of FG-Λ CNTRC cylindrical panel.
(W 2
Eq. (24) is substituted into Eq. (21), to seek the following biharmonic solution for
=
1cos2 m x
+
2 cos2 n y
y2 3sin m x sin n y + Nx 0 2
* 12
+(
11
2
33
6 41
W02)
mn
2 2
* n4 12 +
2 m
11R
2 m
q (t ) + 1
(W 2 2 m 1
W02 ) +
Nx 0 (W
R) 2
4 22 n )
2 2 2 * n4 m n + 12 m 2 2 4 12 ) m n + 22 n )
2 n
22
2 2 m n + 2 2 12 ) m n
*) 2 33
* 21
+
2
33
* + *22 + ( 11 4 1 ( 11 m
2 3
+(
*) 2 33
R
1 16 1
W0 ) = 0
(W
W0 )
W (W 4 m 22
+
W0 ) 4 n
81 3mn
2 2
1
2 2 1 2 m n 2 1 mn
W
11
(27)
where
(25)
1
= ( 1)m
1
2
= ( 1) n
1
(28)
The condition expressing the immovability on the boundary edges 6
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Fig. 9. Effect of temperature on FG-CNTRC cylindrical panel for different types of the CNTs distributions.
of the shell, i.e. u = 0 at x = 0, L is justified in an average sense as [7] b
a
0
0
u dxdy = x
b
a
0
0
1 2
0 x
2
w x
w w* dxdy = 0 x x
2 n
8
12 2
W (W + 2µh) +
22
4 m
+ (H12 + H21 + 4H33)
2 2 m n
(30)
+
4 m
* + *22 + ( 11 1(
* m4 21
(W 2
* 12
+(
+
11
W02)
4 11 m
+(
2
33
* + *22 + ( 11 1(
2 3
4 11 m
*) 2 33 *) 2 33 2
33
+ H22
2 11R m
6
22
41 mn
2 2
q (t ) + 1
* n4 12 +
2 m
2 m 1
W02 )
H (T )(W
(
h 2
ln R
)
(
ln R +
h 2
)T
o
(
ln R +
h 2
)T
i
)
dz
1 + 16 1
4 m
+ (H12 + H21 + 4H33)
2 2 m n
+ H22
* n4 12
2 m
4 n
1
R) 2
4 22 n )
2 2 2 * n4 m n + 12 m 2 2 4 + ) 22 n m n
(W 2
22
H11
¨ + 2cW + W
4 n
R
(W
W0 ) +
W (W
12 )
2 n
* 21
2 2 m n + 2 2 12 ) m n
h 2
h 2
To )ln(R + z ) + ln R
Consider the FG-CNTRC cylindrical panel under external pressure with the q (t ) = sin t , Eq. (31) become
1
* ( 21
(
(Ti
Eq. (31) is utilized to analysis of the nonlinear vibration behavior of imperfect FG-CNTRC cylindrical panel in the thermal environment.
Substituting Eq. (30) in to Eq. (27) leads to H11
22 1 )
12 2
4.1. Forced vibration analysis
22 1 22
¨ + 2cW + W
(
(32)
(29)
Substituting Eqs. (14), (18), (24) and (25) into Eq. (29), this integral can be obtained as
Nx 0 =
h 2
H (T ) =
4 m 22
W0) = 0
+
W0) + 4 n
81 3mn
2 2
+
* ( 21
4 m
1(
1
* m4 21
2 2 1 2 m n 2 mn 1
* + *22 + ( 11 4 11 m
W
11
(W 2
(31)
where tion.
where 7
* 12
+
11
W02)
+(
2
33
* + *22 + ( 11 4 1 ( 11 m
2 3
+(
*) 2 33
2
33 2 n
6 41
mn
2 2
+
11R
2 m
2 2 2 * n4 m n + 12 m 2 2 4 12 ) m n + 22 n )
sin t + 1
W02 ) +
(W 2 2 m
R) 2
4 22 n )
*) 2 33
* 21 22
2 2 m n + 2 2 12 ) m n
1 16 1
H (T )(W
1
is the excitation amplitude, and
R
(W
W0 ) +
W (W 4 m
+
22
W0) = 0
W0) + 4 n
81 3mn
2 2
1
2 2 1 2 m n 2 1 mn
W
11
(33)
is the frequency of excita-
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Fig. 10. The effect of temperature on the phase plane of nonlinear forced vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
Utilizing this equation, the nonlinear forced vibration analysis of the FG-CNTRC cylindrical panel is taken into account. The second order nonlinear governing differential equation (Eq. (33)) is solved by the fourth order Runge-Kutta method.
where
5.1. Verification of the results For validation, present results of the dimensionless natural frequencies are compared in Table 1 with those of the nonlinear vibration analysis accomplished already by, Chorfi and Houmat [47], Matsunaga [48], and Alijani et al. [49]. It may readily be noted that there is a good agreement between the results. The dimensionless frequency is defined in the following form
To validate the present formulation, the free and linear vibration behavior of FG-CNTRC cylindrical panel without initial imperfection and damping, Eq. (33) is obtained as
H11
4 m
+ (H12 + H21 + 4H33 )
2 2 m n
+ H22
4 n
1
+
( *21
4 m
* + *22 + ( 11 1(
4 11 m
+(
*) 2 33 2
33
2 2 m n + 2 2 12 ) m n
4 n
* 12
2 m
R) 2
4 22 n )
+
2 m
+
mn
H (T ) W
The fundamental natural frequency of FG-CNTRC cylindrical panel in the thermal environment can be determined by H11
=
4 m
+ (H12 + H21 + 4H33)
2 2 m n
+ H22
4 n
( *21
4 m
* + 22 * + ( 11 1(
4 11 m
+(
*) 2 33 33
2
2 2 m n + 2 2 ) 12 m n
* n4 12 +
4 22 n )
mn h
c
Ec
(36)
5.2. Nonlinear vibration analysis
1
+
=
Also, Table 2 shows the isotropic cylindrical panels frequencies that obtained in this study, to compare with the results reported in Shen and Xiang [21], Soedel [50], Nath [51], Chern and Chao [52], and Bardell and Mead [53]. Also, these comparisons confirm a good agreement of the results obtained in this paper.
1
(34)
mn
is the dominant natural frequency of the cylindrical shell.
5. Numerical results
4.2. Free vibration analysis
¨ + W
mn
2 m
R) 2
+
2 m
1 2
Here, the nonlinear vibration responses of imperfect FG-CNTRC cylindrical panels are illustrated. The effect of the initial imperfection, the material parameters, and the temperature are presented. Poly
H (T )
1
(35) 8
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K. Foroutan, et al.
Fig. 11. Effect of initial imperfection on the nonlinear dynamic responses of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
(methyl methacrylate), referred to as PMMA, is selected for the matrix and the (10,10) SWCNTs are selected as reinforcements. In this work, m = 1, n = 5, and the CNT/matrix efficiency parameters are presented * is in Table 3 [14,35] which in this present study, the parameter VCNT * = 0.12 . Also, the material properties of CNTs and the taken as VCNT composite matrix are in the following form Composite matrix: m
= 1150 kg m3,
E m = (3.52
m
= 0.34,
m
The nonlinear vibration responses of the cylindrical panel with and without CNT are investigated in Fig. 4. The excitation frequencies equal to PX = 106sin(300t ) are much smaller than fundamental frequencies of natural vibration. According to Fig. 4a, it can be shown that the CNT strongly decreased the nonlinear amplitude vibration of the cylindrical shell when excitation frequency is far from natural frequency. This behavior is logical because applying the CNT on the structure leads to increasing the Young modulus, therefore, due to this fact stiffens and rigidity of the system is increased, and consequently, the amplitude vibration is decreased. Also, in Fig. 4b, the mode shape of the cylindrical panel in three-dimensional (3D) is shown. The nonlinear dynamic response of FG-CNTRC cylindrical panel for various types of the CNTs distributions is investigated in Fig. 5. It is observed that the nonlinear amplitude vibration of FG-CNTRC cylindrical panel for the UD CNTRC and FG CNTRC are the most and less than the other cases, respectively. According to this figure, the time response curve of the cylindrical panel with the FG-Λ CNTRC is smoother than the other cases. Due to this fact that stiffens and rigidity of the system is increased, and consequently the amplitude vibration is decreased. Fig. 6 illustrates the nonlinear vibration behavior of FG-Λ CNTRC cylindrical panel when the excitation frequency is near to the fundamental natural frequency of system. The excitation frequency equal to PX = 106sin(1500t ) is close to fundamental natural frequency of system 1577.6 rad/s of FG-Λ CNTRC cylindrical panel. By comparing Figs. 5 and 6, it can be seen when the excitation frequencies are near to the
= 45(1 + 0.0005 T ) × 10 61 K,
0.0034 T ) GPa
In such a way, m = 45 × 10 CNTs at T = 300K :
6
and Em = 2.5GPa at T = 300K .
CNT CNT CNT G12 = 1.9445 TPa, E22 = 7.0800 TPa, E11 = 5.6466 TPa,
CNT 12
= 0.175 CNT 22
= 5.1682 × 10 61 K,
CNT 11
= 3.4584 × 10 61 K
CNTs at T = 400 K : CNT G12 CNT 22
CNT CNT = 1.9703TPa, E22 = 6.9814TPa, E11 = 5.5679TPa,
= 5.0905 × 10 61 K,
CNT 11
CNT 12
= 0.175
= 4.1496 × 10 61 K
The geometrical characteristics of FG-CNTRC cylindrical panel are in the following form
h = 2.54 × 10 3m, a = b = 0.4m, R = 0.254m 9
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K. Foroutan, et al.
Fig. 12. The effect of damping coefficient on the free nonlinear vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
fundamental natural frequency of the system, the nonlinear amplitude vibration increased, so that the range of the nonlinear amplitude vibration, from about 0.005 to about 0.015 is changed. Also, it can be shown that when the excitation frequencies are near to the fundamental natural frequency of system, the interesting phenomenon like the harmonic beat phenomenon of vibration is observed. As can be seen, the amplitude of beats increases rapidly when the excitation frequency approaches to the natural frequencies. The phase plane of nonlinear forced vibration of the system for various types of the CNTs distributions is shown in Fig. 7. As can be seen, for the cylindrical panel with FG CNTRC, the nonlinear amplitude vibration and velocity are less than the other cases. Therefore the effect of CNT on the cylindrical panel is very prominently in vibration reduction in the system. The phase plane of nonlinear forced vibration of FG-Λ CNTRC cylindrical panel is plotted in Fig. 8. Considering this figure, increasing the excitation force (Q = 2 × 106N m2 ) leads to increasing the disorderly of the deflection-velocity curve. Fig. 9 illustrates the influence of temperature on the nonlinear dynamic responses of the system for various types of the CNTs distributions. According to this figure, by increasing the temperature, the nonlinear amplitude vibration is increased. Also, it is observed that the nonlinear amplitude vibration of FG-CNTRC cylindrical panel with UD CNTRC is less than other cases. By comparing Figs. 5 and 9, it can be seen when the system is not considered into the thermal environment, the nonlinear amplitude vibration of FG-CNTRC cylindrical panel with
FG CNTRC is less than other cases. But, when the system is considered into the thermal environment, the nonlinear amplitude vibration of FG-CNTRC cylindrical panel with UD CNTRC is less than other cases. Due to this fact, when the system is considered into the thermal environment, stiffens and rigidity of the system is decreased, and consequently, the amplitude vibration is increased. The influence of temperature on the phase plane to nonlinear forced vibration of the system for various types of the CNTs distributions is illustrated in Fig. 10. It is observed that applying the temperature in system leads to increasing the disorderly on the deflection-velocity curve. Also, by increasing the temperature, the nonlinear amplitude vibration and velocity of FG-CNTRC cylindrical panel are increased. As can be seen, the nonlinear amplitude vibration and velocity of FGCNTRC cylindrical panel with UD CNTRC is less than other cases. The influence of initial imperfection on the nonlinear dynamic responses of FG-CNTRC cylindrical panel for various types of the CNTs distributions is shown in Fig. 11. According to this figure, by increasing the initial imperfection, the nonlinear amplitude vibration is decreased. Also, it is observed that the nonlinear amplitude vibration of FG-CNTRC cylindrical panel for the UD CNTRC and FG-Λ CNTRC are the most and less than other cases, respectively. As can be seen, although the initial imperfection is considered in the system, the time response curve of the cylindrical panel with the FG-Λ CNTRC is smoother than the other cases. According to Fig. 11b, in the absence of initial imperfection, the maximum nonlinear amplitude is about the 0.0025 and for W0 = 0.002 , the maximum nonlinear amplitude is about the 0.0027 and for 10
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K. Foroutan, et al.
Fig. 13. Effect of damping coefficient on the phase plane of nonlinear forced vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
W0 = 0.004 , the maximum nonlinear amplitude is about the 0.0035. The influence of damping coefficient on the free nonlinear vibration of the system for various types of the CNTs distributions is shown in Fig. 12. According to this figure, by considering the damping in the system, the nonlinear amplitude vibration is strongly decreased. Also, it is observed that the nonlinear amplitude vibration of the damped FGCNTRC cylindrical panel with various types of the CNTs distributions is decreased similarly. The influence of damping coefficient on the phase plane of forced nonlinear vibration of the system for various types of the CNTs distributions is shown in Fig. 13. It is observed that, by considering the damping in system, the curve of the phase plane is scaled down. Also, this figure shows the velocity of the damped FG-CNTRC cylindrical panel with FG-Λ CNTRC is more than other various types of CNTs distributions.
kinematic nonlinearity was adopted to enable large deformations analyses. Then, the coupled nonlinear problem was solved utilizing the concept of the stress function, and Galerkin’s orthogonality. Also, the fourth order Runge-Kutta method was utilized to obtain the nonlinear vibration response of the FG-CNTRC cylindrical panel. The effects of material parameters, initial imperfection, and temperature on the nonlinear vibration response of FG-CNTRC cylindrical panel are presented. The main conclusions can be summarized as follows:
• The nonlinear amplitude vibration of FG-CNTRC cylindrical panel • •
6. Conclusions
•
The analytical method was utilized to analyze the nonlinear vibration analysis of imperfect FG-CNTRC cylindrical panel under external pressure in the thermal environment. The nonlinear temperature distribution in the thickness direction is assumed. The von-Kármán-type of
•
11
CNTRC are the most and less than for the UD CNTRC and FG the other cases, respectively. By increasing the temperature, the nonlinear amplitude vibration and velocity of FG-CNTRC cylindrical panel are increased. The nonlinear amplitude vibration and velocity of FG-CNTRC cylindrical panel with UD CNTRC are less than other cases in the thermal environment. By increasing the initial imperfection, the nonlinear amplitude vibration of FG-CNTRC cylindrical panel for the UD CNTRC and FG-Λ CNTRC are the most and less than other cases, respectively. The velocity of the damped FG-CNTRC cylindrical panel with FG-Λ CNTRC is more than other types of the CNTs distributions.
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Appendix How to calculate T (z ) : 2T
=0
r
d 2T dT + =0 dr 2 dr
d dT r dr dr
=0
r
dT = c1 dr
T = c1 ln(r ) + c2
where
at r =
h 2
2 E12 ,
= E11E22 E11
h 2
T = To and at r =
,
E22
=
,
12
=
(A.1)
E12
E* 1 * = 33 , 33 E33 E33
11
=
* 11
=
* 22 E11
*, 12 E12
* 12
=
* 22 E12
* 12 E22
*21 =
* 11E12
*, 12 E11
*22 =
11E* 22
* 12 E12
33
=
22
T = Ti
** H11 = E11
* E11 * 11
* E12 *, 21
** H12 = E12
* E11 * 12
* E12 * 22
** H21 = E12
* E12 * 11
*21E22 * , H22 = E** 22
* E12 * 12
* E22 * 22
*( H13 = E11
12 2
22 1)
*( + E12
12 1
11 2 )
*( H23 = E12
12 2
22 1)
+ E*22 (
12 1
11 2)
H33 = E** 33
(A.2)
* E33 * 33
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Nonlinear vibration of imperfect FG-CNTRC cylindrical panels under external pressure in the thermal environment
T
K. Foroutana, H. Ahmadia, E. Carrerab,
⁎
a b
Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran Mul2 Group, Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy
ARTICLE INFO
ABSTRACT
Keywords: Composite cylindrical panels FG carbon nanotube Nonlinear vibration analysis Initial imperfection Thermal environment
An analytical approach for the nonlinear vibration analysis of imperfect functionally graded carbon nanotubereinforced composite (FG-CNTRC) cylindrical panels is presented. The FG-CNTRC cylindrical panel is subjected to external pressure in the thermal environment. The nonlinear temperature distribution in the thickness direction is assumed. The classical thin shell theory with the von-Kármán strain-displacement kinematic nonlinearity is employed in the constitutive laws of the shell. The governing equation is solved by utilizing Galerkin’s method in conjunction with the stress function concept. Finally, to find the nonlinear dynamic responses, the fourth order Runge-Kutta method is used. The effect of the four various types of the CNTs distributions, such as UD, FG-Λ, FG-X, and FG-O is considered on the cylindrical panel. The influence of material parameters, initial imperfection, and temperature on the nonlinear vibration response of functionally graded CNTRC cylindrical panel is presented.
1. Introduction Recently, the research on the analysis of advanced materials has been of interest of scientists. The carbon nanotubes (CNTs) as one of the advanced materials with extraordinary stiffness and tensile strength are utilized as reinforcing components for composites. Also, these structures have been extensively used in a wide range of engineering industrials consist of submarines, aircraft, and aerospace engineering [1–4]. Therefore, the research on the vibration behavior of these structures has been one of the interests of scientists from many years ago, and the great number of researches has been done on the nonlinear vibration behavior of CNTRC shell structures under mechanical loading. Some studies are concentrated in the field of the vibration analysis of cylindrical shell and plate. Thai and Kim [5] reported a review of theories for the modeling and analysis of functionally graded plates and shells. Akgöz and Civalek [6] analyzed the nonlinear vibration analysis of laminated plates resting on nonlinear two-parameter elastic foundations. The vibration behavior of imperfect eccentrically stiffened FG cylindrical shells with an elastic medium under mechanical and damping loads was studied by Duc and Thang [7]. Dung and Nam [8] addressed the vibration behavior of stiffened functionally graded cylindrical shells surrounded by an elastic medium under external pressure. Bich et al. [9] reported the dynamic and static behaviors of ⁎
stiffened FG cylindrical shells. Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation with a combination of harmonic differential quadrature-finite difference methods was investigated by Civalek [10]. Foroutan et al. [11] addressed the vibration behavior of functionally graded cylindrical shells with spiral stiffeners surrounded by nonlinear elastic foundation and damping under axial loading. Some studies have analyzed the vibration of composite cylindrical shells. Sofiyev et al. [12] analyzed the vibration analysis of the composite cylindrical shells surrounded by the elastic mediums and using the SDT. Vibration analysis of the composite cylindrical shells subjected to the periodic radial and axial loading was addressed by Dey and Ramachandra [13]. Choe et al. [14] reported the vibration behavior of the composite shell by considering axis-symmetric geometry with doubly-curved using the unified Jacobi-Ritz method. The dynamic response of composite cylindrical shell was investigated by Zhang et al. [15]. In the above-mentioned studies, the influences of the carbon nanotube on the dynamic behavior of cylindrical shells have not been considered. Some researchers have been done on the dynamic behavior of cylindrical shells reinforced by carbon nanotube. The vibration behavior of functionally graded CNTRC shell was addressed by Zghal et al. [16]. Shen et al. [17] presented the vibration response of FG composite cylindrical panels reinforced by graphene
Corresponding author. E-mail address: [email protected] (E. Carrera).
https://doi.org/10.1016/j.compstruct.2019.111310 Received 15 June 2019; Received in revised form 4 August 2019; Accepted 5 August 2019 Available online 08 August 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
with elastic foundation under thermal loading. The vibration analysis of functionally graded CNTRC cylindrical shells in the thermal environment was addressed by Song et al. [18]. Kiani et al. [19] reported the vibration of functionally graded CNTRC skew cylindrical shells utilizing the Chebyshev-Ritz formulation. Civalek [20] studied the free vibration of carbon nanotubes reinforced and functionally graded shells and plates based on FSDT via discrete singular convolution method. The vibration response of CNTRC cylindrical shells in thermal environments was presented by Shen and Xiang [21]. Shen et al. [22] investigated the vibration response of composite cylindrical shells reinforced by FG graphene under the thermal environments. Duc et al. [23] analyzed the nonlinear dynamic and free vibration response of imperfect functionally graded CNTRC shallow shells under the thermal environment. The stability and vibration analysis of the FG-CNTRC cylindrical panels utilizing the semi-analytical approach reported by Chakraborty et al. [24]. Qin et al. [25] presented the vibration behavior of rotating functionally graded CNTRC cylindrical shells with arbitrary boundary conditions. Review of the literature shows that there is no study on the nonlinear vibration analysis of imperfect FG-CNTRC cylindrical panels subjected to the external pressure in the thermal environment. Therefore, the novelties of the present paper may be summarized as follows: (1) investigation of the nonlinear vibration analysis of FGCNTRC cylindrical panels via the semi-analytical method, (2) incorporation of the effect of initial small geometric imperfection on the cylindrical panels, (3) the system is considered under the external pressure and thermal environment, (4) The nonlinear temperature distribution in the thickness direction is assumed. The four various types of the CNTs distributions, such as UD, FG-Λ, FG-X, and FG-O are considered. Effect of the von Kármán strain-displacement kinematic nonlinearity based on the classical thin shell theory is included in the constitutive laws of the shell. The Galerkin method is applied to the nonlinear vibration problem. In continue the fourth-order Runge-Kutta method is utilized to find the vibration responses. To valid the formulations, comparisons are made with the previous researches. Results are presented to evaluate the influence of initial imperfection, material parameters, and temperature on the nonlinear vibration response of FGCNTRC cylindrical panel.
Fig. 2. Configuration of the FG-CNTRC cylindrical panel.
properties are determined for CNT reinforced FGM materials as illustrated in Fig. 2 which for four patterns of CNT distribution are considered such as UD CNTRC, FG-Λ CNTRC, FG-X CNTRC, and FG-O CNTRC. It should be noted that the CNT volume fractions VCNT for each types of FG-CNTRC are different which is shown in Eq. (3). We assume that a CNTRC layer is made of a mixture of single-wall carbon nanotubes (SWCNTs) and the matrix is assumed isotropic [2,25,26]. The SWCNT reinforcement is either functionally graded (FG) or uniformly distributed (UD) across the shell thickness, as shown in Fig. 2. At nanoscale, the carbon nanotube structure strongly effects on the overall the composite properties. Various micromechanical models have been developed to predict the effective material CNTRCs properties, e.g. the model of Mori-Tanaka [27,28] and the model of Voigt as the rule of the mixture [29,30]. In [31,32] were used the approach of Eshelby–Mori–Tanaka to compute the elastic stiffness properties of nanocomposite materials reinforced by graded oriented, straight CNTs. The Mori-Tanaka approach is suitable to nanoparticles and the mixture rule is simple and convenient to apply the predicting for the overall properties of material and the structures responses. The accuracy of the mixture rule was considered and a remarkable synergism between the MoriTanaka approach and the mixture rule for functionally graded metal–ceramic beams was investigated in [31]. With regard to the mixture rule, the effective shear modulus, Young's modulus, and mass density of CNTRCs are as follows [22,33,34]
2. Description of the FG-CNTRC cylindrical panel
E11 =
A schematic view of a CNTRC cylindrical panel with its coordinate system (x, y, z) is shown in Fig. 1, where xy is the mid-plane of the panel and z is the thickness coordinate. The point “O” in Fig. 1 shows the coordinate system origin. The cylindrical panel has the length in the ydirection b, length in the x-direction a, thickness h, and radius of curvatures R. The distribution of CNTs in the matrix may be uniform or functionally graded in the thickness direction of the shell layer. So material
2
E22 3
G12
= =
CNT 1 VCNT E11 VCNT Vm CNT + E m E22 VCNT CNT G12
= VCNT
+
CNT
+ Vm E m
Vm Gm
+ Vm
(1)
m
Vm and VCNT are the volume fractions of the matrix and the CNT which satisfy the relationship of Vm + VCNT = 1. The coefficients of thermal expansion in the transverse and longitudinal directions can be written as CNT + Vm m 11 CNT CNT (1 + 12 ) VCNT 22 CNT * 12 + Vm m VCNT
11
= VCNT
22
=
12
=
+ (1 + v m) Vm
m
12 11
(2)
The transferring load between the polymeric and nanotube phases is less than perfect (for example, intermolecular coupled stress effects, the surface effects, strain gradients effects, etc.). Thus, it is defined i (i = 1, 2, 3) (the parameter of CNT efficiency) into Eq. (1) to consider the influence of small scale and other influences on the CNTRCs material properties. In Fig. 2, the four various types of CNTs distributions are depicted. We assume that the CNTRC is made from a mixture of single-wall carbon nanotubes (SWCNT), graded distribution in the thickness direction of the shell layer, and matrix which is assumed to be isotropic. The relation of CNTs volume fraction in each type is explained as:
Fig. 1. Configuration of FG-CNTRC cylindrical panel. 2
Composite Structures 227 (2019) 111310
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Table 1 Comparison of the dimensionless natural frequencies (a b = 1, h a = 0.1, Em = 7 × 1010N m2, Ec = 38 × 1010N m2, Present
k
a b
FGM plate 0
0
FGM cylindrical plate 0.5 0
* VCNT
UD: VCNT (z ) =
(
Matsunaga [48]
Alijani et al. [49]
Error (%)
Error (%)
Error (%)
0.057
3.4
0.058
1.7
0.059
0.0
0.064
0.063
1.5
0.062
3.1
0.064
0.0
( ) 2z h
(
xy
) 2 | z| h
)
1 E = 22 E11
( ) ( )w
wCNT +
CNT
m
m
(4)
CNT
T (z ) =
z x,
y
0 y
=
z y,
xy
=
0 xy
xy
= v,y
0 xy
= u, y + v , x + w , x w , y = w , xx , y = w, yy , xy = w, xy
w R
(5)
0 y, xx
21 E11
; Q12 =
1
12 21
; Q66 = G12;
(
To )ln(R + z ) + ln R
(
h 2
ln R
)
(
h 2
)T
ln R +
(
h 2
ln R +
o h 2
0 xy, xy
= w ,2xy
21
)
)T
i
(10)
0 x
+ (E12, E22 , 0)
0 y
+ (0, 0, E33 )
*) (0, 0, 2E33
y
xy
0 xy
* , E12 * , 0) (E11
+ (1, 0, 0)
1
+ (0, 1, 0)
0 x
* , E*22 , 0) + (E12
**, E22 **, 0) (E12
y
0 y
+ (0, 0, E*33)
(0, 0, 2E** 33 )
xy
0 xy
2
**, E12 **, 0) (E11
+ (1, 0, 0)
3
+ (0, 1, 0)
4
(12)
1
+ 2 w ,2y
where Eij are the coupling, bending and extensional stiffness components and i (i = 1, 2, , 4) are the thermal components of CNTRC cylindrical panel shell that are as follows
(6)
(Eij , Eij*, Eij**) =
strain, and y0 , x0 are normal strains. Also, x , y, xy are the shell curvatures changes and twist, respectively. Considering Eq. (6), the compatibility equation is derived as follows
+
12 21
(Mx , My , Mxy )
In Eq. (6), w = w (x , y ), v = v (x , y ) and u = u (x , y ) are the compo0 nents of displacement inz , y and x directions, respectively. xy is a shear
0 x , yy
1
(9)
* , E22 * , 0) (E12
x
0 y
(8)
(11)
1
= u, x + 2 w ,2x
E22
; Q22 =
* , E12 * , 0) = (E11
0 x
T
22
12
(Ti
x
Where
x
12 21
= (E11, E12, 0)
2z
0
xy
(Nx , Ny , Nxy )
With regard to the von Kármán kinematic nonlinearity [40], the strain-displacement relations are as follows 0 x
0
0 0
where To and Ti are outside and inside temperature of the cylindrical panel shell and how to calculate T (z ) is also presented in the Appendix. To derive the resultant forces (Nx , Ny , Nxy ) for FG-CNTRC cylindrical panel, the stress-strain equations (Eq. (8)) are integrated through the thickness. Also, to derive the resultant moments (Mx , My , Mxy ), first multiply the stress-strain equations (Eq. (8)) in z and then integrate through the thickness as
3. Governing equation of motion
=
11
y
21 and 12 are the Poisson’s ratios. y, x and xy are the normal stress and shear stress of cylindrical panel, respectively. The thermal distribution in Eq. (8) is obtained as
Research groups have demonstrate that the material properties of CNT-reinforced materials are very dependent on the structure of CNTs [28,29]; effective material properties of the CNTRC are obtained through a several of micromechanics techniques, such as Eshelby-MoriTanaka approach [35,36] and the mixture rule [37,38]. The approach of Eshelby-Mori-Tanaka, according to the equivalent elastic inclusion idea of Eshelby and the concept of average stress in the matrix due to Mori and Tanaka, is also known as the equivalent inclusion-average stress method. With regard to the Benveniste’s revision [39], effective elastic module has been achieved. Considering Fig. 2, in the case of FG-X case, the inner and the outer surfaces are CNT-rich while for the FG-O, the middle surface of the cylindrical panel is CNT-rich. For the FG-Λ case, the outer surface is free of the CNTs, and the inner surface of the cylindrical panel is CNT-rich. Finally, the CNTs that is uniformly distributed along the thickness direction of the CNTRC cylindrical panel which is named the UD.
x
E11
Q11 =
wCNT
CNT
x
where
(3)
where
* = VCNT
Q11 Q12 0 = Q12 Q22 0 0 0 Q66
y
|z| h
* 1 2VCNT
FG O:
= 3800kg m3 ).
m
0.059
* 1 : VCNT
FG
= 2796kg m3,
Chorfi and Houmat [47]
x
* FG X: 4VCNT
m
1 w , xx R
w , xx w , yy
(7)
The relations of stress-strain for FG-CNTRC cylindrical panel are defined as
( 1,
2)
=
( 3,
4)
=
as 3
h 2 h 2 h 2 h 2
h 2 h 2
Qij (1, z , z 2) dz
;
i, j = 1, 2, 3
(Q11
11
+ Q12
22,
Q12
11
+ Q22
22 )
Tdz
(Q11
11
+ Q12
22,
Q12
11
+ Q22
22 )
Tzdz = 0
(13)
With regard to Eq. (11), the components of strain can be rewritten
Composite Structures 227 (2019) 111310
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Table 2 Comparison of the natural frequencies (Hz) for isotropic cylindrical panels (a = 0.2794, b = 0.2286, E = 7.2 × 1010N m2,
= 2800kg m3 ).
= 1 3,
R (m)
h (mm)
(m , n)
Present
Shen and Xiang [21]
Soedel [50]
Nath [51]
Chern and Chao [52]
Bardell and Mead [53]
2.4384 2.4384 1.8288 1.8288 1.2192 1.2192
0.7112 1.2190 0.7112 1.2190 0.7112 1.2190
(1,1) (1,1) (1,2) (1,1) (1,2) (1,2)
144.9 162.2 168.0 199.9 182.0 280.6
143.8 163.2 167.4 200.8 181.7 281.9
144.3 163.7 167.8 201.4 182.1 282.3
144.5 163.9 167.8 201.7 182.2 282.3
142.1 161.4 166.7 198.5 180.8 278.1
143.8 163.1 167.2 200.8 181.6 281.8
Table 3 The CNT/matrix efficiency parameters. * VCNT 0.12 0.17 0.28
1
0.137 0.142 0.141
( x0,
0 y,
33 ) Nxy 2
1.022 1.626 1.585
3
*) 2 33
0.715 1.138 1.109
xy
0 xy )
=(
*, + ( 11 +(
12 2
22 ,
12 ,
*21, 0)
x
22 1,
0) Nx + ( *, + ( 12
12 ,
*22 , 0)
0, 0) + (0,
11, y
0) Ny + (0, 0,
+ (0, 0,
12 1
11 2 ,
0)
(14)
Then, Eq. (14) is substituted into Eq. (12) in the following form
Fig. 4. The nonlinear vibration responses of the cylindrical panel with and without CNT. 4
Composite Structures 227 (2019) 111310
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density 1
1
are
h 2
=
h 2
(z ) dz
(17)
With regard to the first two Eq. (16), stress function ( ) is defined as
Nx =
yy ,
Ny =
xx ,
Nxy =
(18)
xy
By substituting Eq. (14) in Eq. (7) and Eq. (15) in the third part of Eq. (16) and then by utilizing Eq. (6) and (18), the following equations of the system can be derived as 11 , xxxx
+(
* , xxxx 21
w, xx + 2
+(
+
1 w , tt
33
1 (w , xx R
, xxxx
Fig. 6. The nonlinear dynamic response of FG-Λ CNTRC cylindrical panel.
2H33 ) 0)
xy
+ (H13, 0, 0)
w, xy
*21, 0) Nx + ( 12 * , *22 , 0) Ny + (0,
1
x
(H21, H22, 0)
+ (0, H23, 0)
2
+ (1, 0, 0)
3
(15)
where Hij , ij , Hij*, ij* are presented in Appendix. The nonlinear governing equations of motions of the assembly may be found by applying Donnell’s shell theory, in the following form [9,41,42]
(
=
1 w , tt
*) 2 33
2
12 ) , xxyy
+
2 *33)(w , xxyy w ,*xx )
22 , yyyy
(19)
* , yyyy 12
, xxyy
, xx , yy
R
(20)
[w,2xy
w, xx w, yy ] + [w,*xy2
w,*xxxx ) + w ,*yyyy ) w ,*xxw ,*yy] = 0 (21)
w,*xxxx ) + (H12 + H21 + 4H33)
w ,*xxyy ) + H22 (w , yyyy * + *22 ( 11 , xx w , yy
* (w, xxxx + 12
* (w, yyyy w,*xxyy ) + 21
2 *33)
w,*yyyy )
, xxyy
* , yyyy 12
*21 , xx
R
, yy w , xx
+2
, xy
(22)
q (t ) = 0
w = 0, Mx = 0, Nx, y = 0, at x = 0; a w = 0, My = 0, Nx , y = 0 at y = 0; b
(23)
Considering the boundary condition, the total deflection and initial imperfection of cylindrical panel is expressed as follows [42–46]
Nx , x + Nxy, y = 0 Nxy, x + Ny, y = 0 Mx, xx + 2Mxy, xy + My, yy + Nx w , xx + 2Nxy w , xy + Ny w , yy +
w , xx w , yy] = 0
A simply supported imperfect FG-CNTRC cylindrical panel under external pressure in the thermal environment is considered. The applied boundary conditions are as follows
+ (0, 1,
4
1 R
2 *33)
4. Nonlinear vibration analysis of the FG-CNTRC cylindrical panel
(0, 0,
y
* w, xxxx + ( 11 * + 22 * + 12
q (t ) = 0
, xx w , yy
+ 2 1 c w , t + H11 (w, xxxx
(w, xxyy
(H11, H12 , 0)
* + 22 * ( 11
, xy w , xy
* + *22 ( 11
* ) Nxy 0, 33
22 , yyyy
For the initial small geometric imperfection of CNTRC cylindrical panel: The initial small geometric imperfection of the cylindrical panel assumed here can be considered as a small deviation of the middle surface of the panel relative to the perfect shape. With regard to this fact that an initial deflection is very small compared with the dimensions of the panel, whereas may be compared with the thickness of the panel. Let w* = w * (x , y ) show a known small imperfection, resulting from the Eqs. (19) and (20) for a perfect CNTRC cylindrical the panel. Considering Volmir’s approach [42] for an imperfection panel, the system of motion equations can be formulated for an imperfect CNTRC cylindrical panel as 11 , xxxx
*, (Mx , My , Mxy ) = ( 11
+
1 + w, xx + [w ,2xy R
+ 2 1 c w , t + H11w, xxxx + (H12 + H21 + 4H33) w , xxyy + H22
w, yyyy Fig. 5. The nonlinear dynamic responses of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
12 ) , xxyy
*21w , yyyy
w, xxyy + 1 w , tt
2
33
w (x , y , t ) = W (t ) sin m x sin w * (x , y ) = W0 sin m x sin n y
) + q (t )
m a
n b
ny
(24)
where m = and n = , subscript m is the number of half wave in the axial direction and n is the number of full wave in the circumferential direction. Also, Wmn (t ) represents the unknown deflection amplitude, and W0 is the known initial amplitude.
+ 2 1 c w,t (16)
where q (t ) is the external pressure, c is coefficient of damping and mass
5
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Fig. 7. The phase plane of nonlinear forced vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
The i (i = 1, 2, 3) coefficients may be obtained by equating the resulting identical trigonometric expressions, to reach at 1
=
3
=
2 n
2 (W 32 11 m
W02 ),
2
2
2 m
(W 32 22 n2
=
4 +( 2 2+ 2 33 *21 m * + 22 * * ) m * n4 11 n 12 4 2 2 4 11 m + ( 33 2 12 ) m n + 22 n
2 m R
W02 )
2
W
(26)
After substitution of Eqs. (24) and (25) into Eq. (22), the following results may be obtained as follows by applying the Galerkin method 4 m
(H11
¨ + 2cW + W
2 2 m n
+ (H12 + H21 + 4H33 )
+ H22
4 n)
1
+
* ( 21
4 m
* + *22 + ( 11 1(
* m4 21
4 11 m
Fig. 8. The phase plane of nonlinear forced vibration of FG-Λ CNTRC cylindrical panel.
(W 2
Eq. (24) is substituted into Eq. (21), to seek the following biharmonic solution for
=
1cos2 m x
+
2 cos2 n y
y2 3sin m x sin n y + Nx 0 2
* 12
+(
11
2
33
6 41
W02)
mn
2 2
* n4 12 +
2 m
11R
2 m
q (t ) + 1
(W 2 2 m 1
W02 ) +
Nx 0 (W
R) 2
4 22 n )
2 2 2 * n4 m n + 12 m 2 2 4 12 ) m n + 22 n )
2 n
22
2 2 m n + 2 2 12 ) m n
*) 2 33
* 21
+
2
33
* + *22 + ( 11 4 1 ( 11 m
2 3
+(
*) 2 33
R
1 16 1
W0 ) = 0
(W
W0 )
W (W 4 m 22
+
W0 ) 4 n
81 3mn
2 2
1
2 2 1 2 m n 2 1 mn
W
11
(27)
where
(25)
1
= ( 1)m
1
2
= ( 1) n
1
(28)
The condition expressing the immovability on the boundary edges 6
Composite Structures 227 (2019) 111310
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Fig. 9. Effect of temperature on FG-CNTRC cylindrical panel for different types of the CNTs distributions.
of the shell, i.e. u = 0 at x = 0, L is justified in an average sense as [7] b
a
0
0
u dxdy = x
b
a
0
0
1 2
0 x
2
w x
w w* dxdy = 0 x x
2 n
8
12 2
W (W + 2µh) +
22
4 m
+ (H12 + H21 + 4H33)
2 2 m n
(30)
+
4 m
* + *22 + ( 11 1(
* m4 21
(W 2
* 12
+(
+
11
W02)
4 11 m
+(
2
33
* + *22 + ( 11 1(
2 3
4 11 m
*) 2 33 *) 2 33 2
33
+ H22
2 11R m
6
22
41 mn
2 2
q (t ) + 1
* n4 12 +
2 m
2 m 1
W02 )
H (T )(W
(
h 2
ln R
)
(
ln R +
h 2
)T
o
(
ln R +
h 2
)T
i
)
dz
1 + 16 1
4 m
+ (H12 + H21 + 4H33)
2 2 m n
+ H22
* n4 12
2 m
4 n
1
R) 2
4 22 n )
2 2 2 * n4 m n + 12 m 2 2 4 + ) 22 n m n
(W 2
22
H11
¨ + 2cW + W
4 n
R
(W
W0 ) +
W (W
12 )
2 n
* 21
2 2 m n + 2 2 12 ) m n
h 2
h 2
To )ln(R + z ) + ln R
Consider the FG-CNTRC cylindrical panel under external pressure with the q (t ) = sin t , Eq. (31) become
1
* ( 21
(
(Ti
Eq. (31) is utilized to analysis of the nonlinear vibration behavior of imperfect FG-CNTRC cylindrical panel in the thermal environment.
Substituting Eq. (30) in to Eq. (27) leads to H11
22 1 )
12 2
4.1. Forced vibration analysis
22 1 22
¨ + 2cW + W
(
(32)
(29)
Substituting Eqs. (14), (18), (24) and (25) into Eq. (29), this integral can be obtained as
Nx 0 =
h 2
H (T ) =
4 m 22
W0) = 0
+
W0) + 4 n
81 3mn
2 2
+
* ( 21
4 m
1(
1
* m4 21
2 2 1 2 m n 2 mn 1
* + *22 + ( 11 4 11 m
W
11
(W 2
(31)
where tion.
where 7
* 12
+
11
W02)
+(
2
33
* + *22 + ( 11 4 1 ( 11 m
2 3
+(
*) 2 33
2
33 2 n
6 41
mn
2 2
+
11R
2 m
2 2 2 * n4 m n + 12 m 2 2 4 12 ) m n + 22 n )
sin t + 1
W02 ) +
(W 2 2 m
R) 2
4 22 n )
*) 2 33
* 21 22
2 2 m n + 2 2 12 ) m n
1 16 1
H (T )(W
1
is the excitation amplitude, and
R
(W
W0 ) +
W (W 4 m
+
22
W0) = 0
W0) + 4 n
81 3mn
2 2
1
2 2 1 2 m n 2 1 mn
W
11
(33)
is the frequency of excita-
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Fig. 10. The effect of temperature on the phase plane of nonlinear forced vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
Utilizing this equation, the nonlinear forced vibration analysis of the FG-CNTRC cylindrical panel is taken into account. The second order nonlinear governing differential equation (Eq. (33)) is solved by the fourth order Runge-Kutta method.
where
5.1. Verification of the results For validation, present results of the dimensionless natural frequencies are compared in Table 1 with those of the nonlinear vibration analysis accomplished already by, Chorfi and Houmat [47], Matsunaga [48], and Alijani et al. [49]. It may readily be noted that there is a good agreement between the results. The dimensionless frequency is defined in the following form
To validate the present formulation, the free and linear vibration behavior of FG-CNTRC cylindrical panel without initial imperfection and damping, Eq. (33) is obtained as
H11
4 m
+ (H12 + H21 + 4H33 )
2 2 m n
+ H22
4 n
1
+
( *21
4 m
* + *22 + ( 11 1(
4 11 m
+(
*) 2 33 2
33
2 2 m n + 2 2 12 ) m n
4 n
* 12
2 m
R) 2
4 22 n )
+
2 m
+
mn
H (T ) W
The fundamental natural frequency of FG-CNTRC cylindrical panel in the thermal environment can be determined by H11
=
4 m
+ (H12 + H21 + 4H33)
2 2 m n
+ H22
4 n
( *21
4 m
* + 22 * + ( 11 1(
4 11 m
+(
*) 2 33 33
2
2 2 m n + 2 2 ) 12 m n
* n4 12 +
4 22 n )
mn h
c
Ec
(36)
5.2. Nonlinear vibration analysis
1
+
=
Also, Table 2 shows the isotropic cylindrical panels frequencies that obtained in this study, to compare with the results reported in Shen and Xiang [21], Soedel [50], Nath [51], Chern and Chao [52], and Bardell and Mead [53]. Also, these comparisons confirm a good agreement of the results obtained in this paper.
1
(34)
mn
is the dominant natural frequency of the cylindrical shell.
5. Numerical results
4.2. Free vibration analysis
¨ + W
mn
2 m
R) 2
+
2 m
1 2
Here, the nonlinear vibration responses of imperfect FG-CNTRC cylindrical panels are illustrated. The effect of the initial imperfection, the material parameters, and the temperature are presented. Poly
H (T )
1
(35) 8
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Fig. 11. Effect of initial imperfection on the nonlinear dynamic responses of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
(methyl methacrylate), referred to as PMMA, is selected for the matrix and the (10,10) SWCNTs are selected as reinforcements. In this work, m = 1, n = 5, and the CNT/matrix efficiency parameters are presented * is in Table 3 [14,35] which in this present study, the parameter VCNT * = 0.12 . Also, the material properties of CNTs and the taken as VCNT composite matrix are in the following form Composite matrix: m
= 1150 kg m3,
E m = (3.52
m
= 0.34,
m
The nonlinear vibration responses of the cylindrical panel with and without CNT are investigated in Fig. 4. The excitation frequencies equal to PX = 106sin(300t ) are much smaller than fundamental frequencies of natural vibration. According to Fig. 4a, it can be shown that the CNT strongly decreased the nonlinear amplitude vibration of the cylindrical shell when excitation frequency is far from natural frequency. This behavior is logical because applying the CNT on the structure leads to increasing the Young modulus, therefore, due to this fact stiffens and rigidity of the system is increased, and consequently, the amplitude vibration is decreased. Also, in Fig. 4b, the mode shape of the cylindrical panel in three-dimensional (3D) is shown. The nonlinear dynamic response of FG-CNTRC cylindrical panel for various types of the CNTs distributions is investigated in Fig. 5. It is observed that the nonlinear amplitude vibration of FG-CNTRC cylindrical panel for the UD CNTRC and FG CNTRC are the most and less than the other cases, respectively. According to this figure, the time response curve of the cylindrical panel with the FG-Λ CNTRC is smoother than the other cases. Due to this fact that stiffens and rigidity of the system is increased, and consequently the amplitude vibration is decreased. Fig. 6 illustrates the nonlinear vibration behavior of FG-Λ CNTRC cylindrical panel when the excitation frequency is near to the fundamental natural frequency of system. The excitation frequency equal to PX = 106sin(1500t ) is close to fundamental natural frequency of system 1577.6 rad/s of FG-Λ CNTRC cylindrical panel. By comparing Figs. 5 and 6, it can be seen when the excitation frequencies are near to the
= 45(1 + 0.0005 T ) × 10 61 K,
0.0034 T ) GPa
In such a way, m = 45 × 10 CNTs at T = 300K :
6
and Em = 2.5GPa at T = 300K .
CNT CNT CNT G12 = 1.9445 TPa, E22 = 7.0800 TPa, E11 = 5.6466 TPa,
CNT 12
= 0.175 CNT 22
= 5.1682 × 10 61 K,
CNT 11
= 3.4584 × 10 61 K
CNTs at T = 400 K : CNT G12 CNT 22
CNT CNT = 1.9703TPa, E22 = 6.9814TPa, E11 = 5.5679TPa,
= 5.0905 × 10 61 K,
CNT 11
CNT 12
= 0.175
= 4.1496 × 10 61 K
The geometrical characteristics of FG-CNTRC cylindrical panel are in the following form
h = 2.54 × 10 3m, a = b = 0.4m, R = 0.254m 9
Composite Structures 227 (2019) 111310
K. Foroutan, et al.
Fig. 12. The effect of damping coefficient on the free nonlinear vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
fundamental natural frequency of the system, the nonlinear amplitude vibration increased, so that the range of the nonlinear amplitude vibration, from about 0.005 to about 0.015 is changed. Also, it can be shown that when the excitation frequencies are near to the fundamental natural frequency of system, the interesting phenomenon like the harmonic beat phenomenon of vibration is observed. As can be seen, the amplitude of beats increases rapidly when the excitation frequency approaches to the natural frequencies. The phase plane of nonlinear forced vibration of the system for various types of the CNTs distributions is shown in Fig. 7. As can be seen, for the cylindrical panel with FG CNTRC, the nonlinear amplitude vibration and velocity are less than the other cases. Therefore the effect of CNT on the cylindrical panel is very prominently in vibration reduction in the system. The phase plane of nonlinear forced vibration of FG-Λ CNTRC cylindrical panel is plotted in Fig. 8. Considering this figure, increasing the excitation force (Q = 2 × 106N m2 ) leads to increasing the disorderly of the deflection-velocity curve. Fig. 9 illustrates the influence of temperature on the nonlinear dynamic responses of the system for various types of the CNTs distributions. According to this figure, by increasing the temperature, the nonlinear amplitude vibration is increased. Also, it is observed that the nonlinear amplitude vibration of FG-CNTRC cylindrical panel with UD CNTRC is less than other cases. By comparing Figs. 5 and 9, it can be seen when the system is not considered into the thermal environment, the nonlinear amplitude vibration of FG-CNTRC cylindrical panel with
FG CNTRC is less than other cases. But, when the system is considered into the thermal environment, the nonlinear amplitude vibration of FG-CNTRC cylindrical panel with UD CNTRC is less than other cases. Due to this fact, when the system is considered into the thermal environment, stiffens and rigidity of the system is decreased, and consequently, the amplitude vibration is increased. The influence of temperature on the phase plane to nonlinear forced vibration of the system for various types of the CNTs distributions is illustrated in Fig. 10. It is observed that applying the temperature in system leads to increasing the disorderly on the deflection-velocity curve. Also, by increasing the temperature, the nonlinear amplitude vibration and velocity of FG-CNTRC cylindrical panel are increased. As can be seen, the nonlinear amplitude vibration and velocity of FGCNTRC cylindrical panel with UD CNTRC is less than other cases. The influence of initial imperfection on the nonlinear dynamic responses of FG-CNTRC cylindrical panel for various types of the CNTs distributions is shown in Fig. 11. According to this figure, by increasing the initial imperfection, the nonlinear amplitude vibration is decreased. Also, it is observed that the nonlinear amplitude vibration of FG-CNTRC cylindrical panel for the UD CNTRC and FG-Λ CNTRC are the most and less than other cases, respectively. As can be seen, although the initial imperfection is considered in the system, the time response curve of the cylindrical panel with the FG-Λ CNTRC is smoother than the other cases. According to Fig. 11b, in the absence of initial imperfection, the maximum nonlinear amplitude is about the 0.0025 and for W0 = 0.002 , the maximum nonlinear amplitude is about the 0.0027 and for 10
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K. Foroutan, et al.
Fig. 13. Effect of damping coefficient on the phase plane of nonlinear forced vibration of FG-CNTRC cylindrical panel for different types of the CNTs distributions.
W0 = 0.004 , the maximum nonlinear amplitude is about the 0.0035. The influence of damping coefficient on the free nonlinear vibration of the system for various types of the CNTs distributions is shown in Fig. 12. According to this figure, by considering the damping in the system, the nonlinear amplitude vibration is strongly decreased. Also, it is observed that the nonlinear amplitude vibration of the damped FGCNTRC cylindrical panel with various types of the CNTs distributions is decreased similarly. The influence of damping coefficient on the phase plane of forced nonlinear vibration of the system for various types of the CNTs distributions is shown in Fig. 13. It is observed that, by considering the damping in system, the curve of the phase plane is scaled down. Also, this figure shows the velocity of the damped FG-CNTRC cylindrical panel with FG-Λ CNTRC is more than other various types of CNTs distributions.
kinematic nonlinearity was adopted to enable large deformations analyses. Then, the coupled nonlinear problem was solved utilizing the concept of the stress function, and Galerkin’s orthogonality. Also, the fourth order Runge-Kutta method was utilized to obtain the nonlinear vibration response of the FG-CNTRC cylindrical panel. The effects of material parameters, initial imperfection, and temperature on the nonlinear vibration response of FG-CNTRC cylindrical panel are presented. The main conclusions can be summarized as follows:
• The nonlinear amplitude vibration of FG-CNTRC cylindrical panel • •
6. Conclusions
•
The analytical method was utilized to analyze the nonlinear vibration analysis of imperfect FG-CNTRC cylindrical panel under external pressure in the thermal environment. The nonlinear temperature distribution in the thickness direction is assumed. The von-Kármán-type of
•
11
CNTRC are the most and less than for the UD CNTRC and FG the other cases, respectively. By increasing the temperature, the nonlinear amplitude vibration and velocity of FG-CNTRC cylindrical panel are increased. The nonlinear amplitude vibration and velocity of FG-CNTRC cylindrical panel with UD CNTRC are less than other cases in the thermal environment. By increasing the initial imperfection, the nonlinear amplitude vibration of FG-CNTRC cylindrical panel for the UD CNTRC and FG-Λ CNTRC are the most and less than other cases, respectively. The velocity of the damped FG-CNTRC cylindrical panel with FG-Λ CNTRC is more than other types of the CNTs distributions.
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Appendix How to calculate T (z ) : 2T
=0
r
d 2T dT + =0 dr 2 dr
d dT r dr dr
=0
r
dT = c1 dr
T = c1 ln(r ) + c2
where
at r =
h 2
2 E12 ,
= E11E22 E11
h 2
T = To and at r =
,
E22
=
,
12
=
(A.1)
E12
E* 1 * = 33 , 33 E33 E33
11
=
* 11
=
* 22 E11
*, 12 E12
* 12
=
* 22 E12
* 12 E22
*21 =
* 11E12
*, 12 E11
*22 =
11E* 22
* 12 E12
33
=
22
T = Ti
** H11 = E11
* E11 * 11
* E12 *, 21
** H12 = E12
* E11 * 12
* E12 * 22
** H21 = E12
* E12 * 11
*21E22 * , H22 = E** 22
* E12 * 12
* E22 * 22
*( H13 = E11
12 2
22 1)
*( + E12
12 1
11 2 )
*( H23 = E12
12 2
22 1)
+ E*22 (
12 1
11 2)
H33 = E** 33
(A.2)
* E33 * 33
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