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Analytical solution for nonlinear postbuckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers
Accepted Manuscript Analytical solution for nonlinear postbuckling of functionally graded carbon nanotubereinforced composite shells with piezoelectric layers R. Ansari, T. Pourashraf, R. Gholami, A. Shahabodini PII:
S1359-8368(15)00744-1
DOI:
10.1016/j.compositesb.2015.12.012
Reference:
JCOMB 3931
To appear in:
Composites Part B
Received Date: 31 August 2015 Revised Date:
2 December 2015
Accepted Date: 24 December 2015
Please cite this article as: Ansari R, Pourashraf T, Gholami R, Shahabodini A, Analytical solution for nonlinear postbuckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers, Composites Part B (2016), doi: 10.1016/j.compositesb.2015.12.012. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Analytical solution for nonlinear postbuckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers R. Ansari*,a, T. Pourashrafa, R. Gholami*,b, A. Shahabodinia b
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
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a
Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran
Abstract. This paper presents an analytical solution procedure for the nonlinear postbuckling analysis of piezoelectric functionally graded carbon nanotubes reinforced composite (FG-CNTRC) cylindrical shells
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subjected to combined electro-thermal loadings, axial compression and lateral loads. The carbon nanotubes are assumed to be aligned and straight with uniform and functionally graded distributions in
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the thickness direction. The kinematics and constitutive relations are written on the basis of the classic theory and the von Kármán nonlinear strain–displacement relations of large deformation. Applying the Ritz energy approach, analytical solutions are proposed for the nonlinear critical axial load, lateral pressure as well as the load-shortening ratio of the piezoelectric FG-CNTRC shell. Numerical results are presented to study the effects of dimensional parameters, CNT volume fraction, distribution type of the reinforcement and piezoelectric thickness on the nonlinear buckling behavior of the piezoelectric nanocomposite shell. It is revealed that the carrying capacity of the structure increases as the shell is
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integrated by the piezoelectric layers and reinforced by higher CNT volume fraction. Furthermore, FGXand FGO- CNTRC piezoelectric shells are indicated to have higher and lower carrying capacities compared to UD-CNTRC piezoelectric shells, respectively. Keywords: A. Carbon-carbon composites; A. Smart materials; A. Nano-structures; B. Buckling;
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1. Introduction
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C. Analytical modelling.
In recent years, carbon nanotubes as the reinforcement materials in the composite structures have attracted a lot of attention [1, 2]. This is due to the promising thermo-mechanical properties of CNTs such as low density, high strength and stiffness-to-weight ratio. So, replacing the traditional carbon fibers by carbon nanotubes in a matrix can effectively improve composite properties and consequently brings about their application in reinforcing the composites, electronic devices and etc [3-6]. However, the volume fraction of carbon nanotubes in a polymer is limited so that, a more amount may lead to deterioration of their mechanical properties [7]. To overcome this shortcoming, Shen [8] combined the concept of the functionally graded material with CNTs through a nonuniform distribution of CNTs along the thickness *
Corresponding authors. [email protected] (R. Ansari) and [email protected] (R. Gholami).
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of a plate structure. He used a functionally graded distribution of carbon nanotubes within the matrix and improved the load bending moment curves of the plates. Various studies were further carried out on the linear and nonlinear behaviors of the CNT-based functionally graded beam, plate and shell structures. Concerning works are cited below.
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Thermal buckling and postbuckling response of perfect and imperfect functionally graded CNTRC plate [9] and shell [10] were investigated based on a multiscale approach. Shen [11] discussed the postbuckling behavior of CNT-reinforced nanocomposite cylindrical shells under axial compression along with thermal environments based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. They obtained the buckling loads and postbuckling equilibrium paths using a perturbation
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technique. Jafari et al. [12] adopted the first order shear deformation theory to account for the transverse shear deformation in the buckling analysis of the FG-CNTRC plates under uniaxial or biaxial loadings.
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Based on the same theory and solution method, postbuckling of CNTRC cylindrical shells subjected to combined axial and radial mechanical loads in thermal environments was analyzed by Shen and Xiang [13]. Arani et al. [14] discussed static stress analysis of CNTRC cylinder under non-axisymmetric thermo-mechanical loads and uniform electro-magnetic fields. Lei et al. [15] employed the element free kp Ritz method to describe the buckling response of the FG-CNTRC plates under various in-plane loading conditions. Van Dung [16] studied torsional buckling and postbuckling of FG shells in thermal environment. Rafiee et al. [17] proposed an exact closed form solution for the thermal buckling and
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postbuckling of piezoelectric FG-CNTRC Timoshenko beam. They found that a functionally graded distribution of the reinforcement material can increase the bucking load capacity. With the use of Ritz method, Liew et al. [18] analyzed the postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels. A three dimensional solution approach was proposed by Wu and Chang [19] to the
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determination of the critical buckling load of a plate integrated by two piezoelectric layers. Based upon the large deformation theory with the Von Karman–Donnell-type of kinematic non-linearity, Sofiyev [20] investigated non-linear buckling behavior of truncated FG conical shells. Shen and Xiang [21] analyzed
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the postbuckling of axially compressed cylindrical panels reinforced by aligned single-walled carbon nanotubes resting on elastic foundation in thermal environment according to a higher-order shear deformation theory and von Kármán assumptions. Based on the similar theory, they [22] also investigated the postbuckling behavior of nanocomposite cylindrical panels resting on elastic foundations and subjected to uniform temperature rise. The nonlinear bending and postbuckling of a plate composed of conventional fiber reinforced composite (FRC) layers and CNTRC layers were investigated by Fan et al. [23]. In addition to the mentioned works above, another research concerning the linear and nonlinear buckling of the functionally graded carbon nanotube-reinforced composite cylindrical shells under different thermal and mechanical loadings namely radial pressure, combined radial and axial loads and
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torsional load can be found in the works of Shen and his coworkers [24-26]. The results of these papers showed that a symmetric distribution of the reinforcement material with respect to the mid-surface of the shell may increase the strength of it against the buckling. He also found that the CNT volume fraction has a profound effect on the buckling load and postbuckling strength of CNTRC shells. Muthu et al. [27]
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studied the characterization of the glass reinforced fiber composites (GRPs) hybridized with functionalized multiwall carbon nanotube. Tensile testing and mechanical characterization of three different epoxy resins reinforced by different concentrations of multiwall carbon nanotubes were investigated by Rousakis et al. [28].
Based on the first order theory of shells, Jam and Kiani [29] studied the buckling of FG-CNTRC conical
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shells under pressure loading. Sofiyev [30] studied the non-linear dynamic buckling response of the FG coated truncated conical shells to predict the effects of compositional profiles of coatings, variation of
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truncated conical shell parameters and loading speed on the dimensionless linear and non-linear critical time parameters. Alizada et al. [31] presented a stress analysis of a substrate coated by nanomaterials with a vacancy under uniform extension load to determine the contact pressure, stresses and damage conditions. Based on the Timoshenko beam theory and von Kármán geometric nonlinearity, Ansari et al. [32] analyzed the forced vibration of nanocomposite beams reinforced by single-walled carbon nanotubes. They used a numerical strategy including the differential quadrature and Galerkin approaches and pseudo-arc length continuum technique to solve the nonlinear governing equations. Ansari et al. [33]
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employed the same strategy to investigate the forced vibration behavior of carbon nanotube-reinforced composite plates. Their results revealed that when the mid-plane of plate is CNT-rich the hardening behavior of the plate is intensified.
Integrating the engineering structures with piezoelectric layers is practically of much importance. These
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structures are smart and have application in different fields such as structural vibration control, thermal stress control and structural health monitoring [34]. Prompted by this, the present paper deals with the postbuckling of functionally graded carbon nanotube-reinforced composite cylindrical shells with surface-
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bonded piezoelectric layers under combined action of axial compression and lateral loads. The geometrical nonlinearity of von-Kármán and Donnell kinematic assumptions are used to derive the governing equation. Using Airy’s stress function and Ritz energy approach, analytical relationships are presented to study the effects of various model parameters on the postbuckling strength of piezoelectric FG-CNTRC shell.
Consider a carbon nanotube-reinforced composite cylindrical shell of length , mean radius and 2. Mathematical formulation
thickness ℎ whose inner and outer surfaces are perfectly bonded with piezoelectric actuator layers of
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ACCEPTED MANUSCRIPT thickness ℎ . In this way, the total thickness of the shell is equal to = ℎ + 2ℎ . A coordinate system
, , with the origin located at the left end on the middle plane is considered in which the axes and
are selected in the longitudinal and circumferential directions of the shell and is in the direction of the
CNTRC shell along , and axes are denoted by , and , respectively and the associated
inward normal to the middle surface. Displacement components of an arbitrary point of piezoelectric FG-
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components on the middle plane (physical neutral surface) are represented by , and . 2.1 Material properties of CNTRC
The present nanocomposite is assumed to be made of a mixture of single wall carbon nanotube (SWCNT)
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as the reinforcement material and an isotropic matrix. The distribution profile of carbon nanotubes through the shell thickness is taken to be either uniformly distributed (referred to UD) or functionally
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graded (referred to FG). In the case of functionally graded-CNTRC, the reinforcement has a linear distribution in the thickness direction of two forms FGX and FGO. For type FGX a mid-plane symmetric graded distribution of carbon nanotube is achieved and both outer and inner surfaces are CNT-rich and for type FGO the middle plane is CNT-rich, only. Accordingly, the CNT volume fraction for the considered nanocomposite shell takes the following forms ∗ = UD
= 2 1 −
∗ =
% % + $ ( − $ ( %&' %&'
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where
2|| ∗ FGX ℎ
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= 2
2|| ∗ FGO ℎ
(1a) (1b) (1c)
(2)
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and indicates the mass fraction of the carbon nanotubes, % and %&' are the mass densities of the
carbon nanotube and matrix, respectively. The structure of CNT considerably affects the properties of the resulting composite. The effective material properties of the two-phase nanocomposites, combination of
CNT and a matrix, are estimated through the well-known Mori-Tanaka scheme [35] or the rule of mixtures [36]. Predicting the overall material properties and consequently accounting for the impact of carbon nanotubes is simple and convenient by applying the rule of mixtures. To take the small scale efficiency parameters *+ , = 1,2,3 calculated by matching the effective properties of CNTRC effects into account, this rule needs to be extended. This is accomplished by introducing the CNT
determined from the MD simulations with those from the rule of mixture. Based on the extended version of this rule, the effective Young’s modulus and shear modulus of CNTRCs can be expressed as [11]
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(3a)
*1 '& = + ' 2/0& 2/0 2&
(3c)
*0 '& = + ' .00& .00 .&
(3b)
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in which .// , .00 and 2/0 represent Young’s modulus and shear modulus of SWCNTs , respectively
and .&' and 2&' stand for the relevant properties of the isotropic matrix of the host. Moreover, in Eq. (3),
the CNT and matrix volume fractions are related by + '& = 1. The thermal expansion coefficients in 3//& = 3// + '& 3&'
& 300& = 41 + 5/0 6 300 + 1 + 5&' '& 3&' − 5/0 3//&
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the longitudinal and transverse directions are given by
(4a) (4b)
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where 3// , 300 and 3&' are thermal expansion coefficients and 5/0 , 5&' denote Poisson’s ratios,
respectively, of the carbon nanotube and matrix.
2.2 Fundamental equations
In the current paper, the classic theory including some assumptions such as the thin shell assumption, the Kirchhoff–Love hypotheses, the shallow shell approximations and etc [37] is adopted to describe the dimensional displacement components , and take the form [38]
deformation of piezoelectric FG-CNTRC cylindrical shells. According to this theory, the three-
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8 , , 8 8 , , , = , − − 7 , 8 , , = , − − 7
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, , = , .
(5a) (5b) (5c)
in which 7 denotes the coordinate associated with the physical neutral surface. From the nonlinearity of von-Kármán, the strain–displacement relationships are stated as follows
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;< ;<7 B< 7 ; B ; : = ? = @ = A + − 7 : = ? ><= B<= >7 in which
(6)
<
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ACCEPTED MANUSCRIPT 8 0 8 1 8 0 F − 0 I F + I 8 8 2 8 D D ;<7 B< D 8 0 D D D8 1 8 D D 0 7 B − +
; − , : = ? = @ =A = 0 E8 2 8 H B<= E 8 H ><7 0 D 8 8 8 8 D D D D D−2 8 D D + + C 8 8 8 8 G C 8 8 G
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(7)
In the presence of thermal and electrical loadings, the constitutive relations for matrix and piezoelectric J< L// : J= ? = KL/0 J<= 0
;< 0 3// 0 0 V1/ .< 0 O P: ;= ? − Q300 R ∆TU − K0 0 V10O @.= A ><= 0 LNN .W 0 0 0
layers are given by
L/0 L00 0
.//& .00& 5/0& ./0& , L00& = , L/0& = ,L = 2/0& 1 − 5/0& 50/& 1 − 5/0& 50/& 1 − 5/0& 50/& NN& .// , 1 − 5 0
for the CNTRC host and L// = L00 = 3// = 300
L/0 =
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L//& =
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where L+X are the reduced elastic modulus given by
5 .// .// , LNN = 2/0 = 0 1 − 5 2 1 + 5
(8)
(9)
(10)
for piezoelectric layer. ∆T = T − T7 is the temperature difference measured from the reference
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temperature T7 . Moreover, V1/ and V10 are piezoelectric constants and .< , .= and .W stand for the
displacement electric field. The electric potential is assumed to linearly distribute in the thickness direction. Since the piezoelectric layer is very thin the self-induced electric potential is much smaller than between applied voltage and electric field intensity [39] within a piezoelectric actuator can be stated as ℎ
.W =
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follows [40]
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the applied voltage and the voltage is applied to the actuator along the thickness only, the relationship
(11)
The force, moment and transverse shear force resultants per unit length are related to stress components as Y< =
&/0
Z J<& [ ]&/0
Y= = Z
&/0
]&/0
J=& [
+Z
]&/0
& ] ^&_ 0
+Z
]&/0
& ] ^&_ 0
J< [
J= [
+Z
&
^&_ 0
&/0
+Z
&
^&_ 0
&/0
J< [
(12a)
J= [
(12b)
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&/0
Z J<& ]&/0
`= = Z
&/0
]&/0
`<= = Z
J=&
&/0
+Z
]&/0
& ] ^&_ 0
− 7 [ + Z
J<= [ ]&/0
& ] ^&_ 0 ]&/0
− 7 [ + Z
& ] ^&_ 0
& J<=
− 7 [ + Z
]&/0
+Z
&/0
J<
J=
& ] ^&_ 0
]&/0
&
^&_ 0
J<= [
(12c)
− 7[ + Z
&
^&_ 0
&/0
− 7 [ + Z
J<=
&
^&_ 0
&/0
− 7 [ + Z
d// cd b /0 0 <= =b ` e E < H b // D D `= D D be/0 C`<= G a 0
d/0 d00 0 e/0 e00 0
0 0 dNN 0 0 eNN
e// e/0 0 f// f/0 0
e/0 e00 0 f/0 f00 0
J= − 7 [
J<= − 7 [
Y< Y 7 0 − 0 <= − 0 h E B H E`
(12d)
(12e)
(12f)
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Y< FY I = D DY D D
&
^&_ 0
&/0
Substituting Eqs. (6) and (8) into Eqs. (12) gives
J< − 7 [
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`< =
& Z J<= [ ]&/0
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Y<= =
&/0
(13)
where the subscripts ‘T’ and ‘k’ signal the thermal and electric loads, respectively and the coefficients of stretching stiffness d+X , stretching-bending coupling stiffness e+X and bending stiffness f+X are given by l 0
l ] 0
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4d+X , e+X , f+X 6 = Z L+X 1, − 7 , − 7 0 [ ; ,, n = 1,2,6
(14)
It is remarkable that for symmetric distribution of the carbon nanotubes through thickness direction of the nano-composite shells, the stretching–bending coupling effect will not be appeared. From the physical = 7 [40] &/0
p]&/0 .// [ &/0
(15)
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7 =
p]&/0 .// [
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neutral surface concept and constant Poisson’s ratio, the physical neutral surface of a shell is obtained by
Thermal force and piezoelectric resultants are calculated as
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L// + Z qL/0 0 &/0
L/0& L00& 0
l/0
L// + Z qL/0 0 ]l/0 ]&/0
Y<=
L/0 L00 0
0 3// 0 r :300 ? 41, − 7 6∆T[ 0 LNN
0 3// 0 r :300 ? 41, − 7 6∆T[ 0 LNN
(16)
l/0 ]&/0 `< V1/ V1/ u `= t = Z QV10 R 41, − 7 6 [ + Z QV10 R 41, − 7 6 [ ℎ ℎ 0 0 &/0 ]l/0 `<=
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Y< s Y=
L/0 L00 0
0 3//& 0 O Q300& R 41, − 7 6∆T[ 0 LNN&
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Y
(17)
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Let the same constant excitation voltage = v7 be applied to both the top and down piezoelectric
actuators. In this manner, by means of Eq. (17), one can obtain the force per unit length of the shell as Y< = Y= = Yv7 = 2v7 V1/
(18)
Using Eq. (7), the compatibility equation is attained as
7 8 0 ;<7 8 0 ;=7 8 0 ><= 80 0 80 80 1 8 0 + − =
− 0 − 8 0 8 0 8 8 8 8 8 8 0 8 0
Airy’s stress function w is defined such that 8 0w , 8 0
Y= =
80 w , 8 0
Y<= = −
From Eqs. (13) and (20), one has
80 w 80w − d + d00 Y
;=7 = x7 P−d/0
80 w 80w + d + d// 4Y=j + Y=y 6 − d/0 Y
80w 8 8
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7 ><= = x/
(20)
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;<7 = x7 Pd00
80 w 8 8
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Y< =
in which x7 =
/ , z{{ z|| ]z|{|
relation becomes
x/ = −
(19)
(21a)
(21b)
(21c) / . z}}
By inserting relationships (21) into Eq. (19), the compatibility
8 ~ w −2d/0 x7 − x/ 8 ~ w d00 8 ~ w 1 80 8 0 8 0 1 80 + + = [ − − ] 8 ~ x7 d// 8 0 8 0 d// 8 ~ x7 d// 8 8 8 0 8 0 8 0 0
The strain energy of the FG-CNTRC shell is evaluated as
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(22)
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П+
1 = 4J< ;< + J= ;= + <= ><= 6[ [ [ 2
(23)
where is the volume of body. With regard to Eqs. (6), (8) and (21), the strain energy is rewritten as 8 0w 80 w 80w 80w 80w = Z Z[/ 0 + 0 0 + 1 0 0 + ~ − / Y
0
− 0 4Y=j + Y=y 6 − 1 Y
80 80 80 + 0 + ] [ [ 8 8 0 8 8
0
0
f00 f// = , N = , = f/0 , = fNN 2 2
(24)
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0
80 80 + N 8 0 8 0
d00 d// d/0 1 0 , 0 = 0 , 1 = − 0 , ~ = d , 2 d// d00 − d/0 2 d// d00 − d/0 d// d00 − d/0 NN
where / =
0
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7
0
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П+
0
(25)
Assume that the shell is synchronously subjected to average axial stress J7< on two ends and uniform radial pressure . Using Eqs. (7) and (21), the work done by the external forces is calculated as 0
0
П< = Z Z [ [ − J7< ℎ Z Z P 7
0
7
7
0
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7
8 − x7 d00 Y
= Z Z [ [ − J7< ℎ Z Z[x7 d00 7
7
7
П j = П+ − П<
0
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Then, the total potential energy of the system is
7
8 w 8 w 1 8 − d/0 0 − 0 ] [ [ 0 8 8 2 8 0
(26)
(27)
Regarding relationships (7) and (21), the following circumferential closed condition should be satisfied for a cylindrical shell 0 7
7
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8 Z Z $ ( [ [ 8 0
= Z Z[x7 P−d/0 7
7
8 w 8 w 1 8 + d// 0 + d// 4Y=j + Y=y 6 − d/0 Y
0
In like manner, the average end-shortening ratio of the shell < may be stated as
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(28)
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1 8 < = − Z Z $ ( [ [ 2 8 7
7
0
80 w 80w 1 =− Z Z[x7 Pd00 0 − d/0 0 + d00 Y
7
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1 8 − 0 ] [ [ 2 8
(29)
3. Analytical solution procedure
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Herein, According to [41], the deflection of the piezoelectric FG-CNTRC shell can be expanded into a series of multi-items
, = 7 + / sin 3 sin + 0 sin 3 0 in which 3 =
'
(30)
and = with , as the axial half-wave numbers in the direction and the wave
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numbers in the direction. Also, 7 , / and 0 are unknown amplitudes which are associated with the pre-
buckling, linear buckling and nonlinear buckling states, respectively. On substitution of Eq. (30) into Eq.
(22), we get
8 ~w 8 ~w 8~w 0 0 + + 8 ~ 8 0 8 0 8 ~
= [7/ cos 23 + [70 cos 2 + [71 sin 3 sin 0 =
d00 , d//
[7/ =
1 −4 $ 0 3 0 + /0 3 0 0 (, 2x7 d//
1 1 1
/0 3 0 0 , [71 =
/ 3 0 − / 0 3 0 0 , [7~ =
3 0 0 . = 2x7 d// x7 d// x7 d// / 0
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[70
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−2d/0x7 − x/ , x7 d//
where 0 =
+ [7~ sin 33 sin
(31)
The general solution of the stress function w is taken to be of the form
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w = [/ cos 23 + [0 cos 2 + [1 sin 3 sin + [~ sin 33 sin − J7< ℎ 0 − J7= ℎ 2
0 2
(32)
(33)
in which J7= denotes the average circumferential stress, positive when the shell is circumferentially
compressed, and
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[7/ = / 0 + 0 /0 , 163 ~
[70 = 1 /0 , 16 0 ~
[0 =
/ = −
~ = − N =
1
8x7 d// 3 0
,
0 =
0 , 32x7 d// 3 0
3 0 0 , x7 d// 3 ~ + 0 3 0 0 + 0 ~
3 0 0 x7 d// 813 ~ + 9 0 3 0 0 + 0 ~
1 =
=
30 , 32x7 d// 0 0
30 , x7 d// 3 ~ + 0 3 0 0 + 0 ~
(34)
(35)
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in which
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[71 [7~ [1 = ~ = ~ / 0 + / , [~ = = N / 0 . 0 0 0 0 ~ ~
3 + 3 +
813 + 9 0 3 0 0 + 0 ~
With the aid of Eqs. (30) and (33), the relationships of strain energy in Eq. (24) and external work in Eq. 1 П+ = ¡/ −4 Y
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(26) become
+ 0 ¢−44Y=j + Y=y 6 + 32[/0 3 ~ + [10 3 ~ + 81[~0 3 ~ £ 0
+ 1 4−4 Y
+ /0 ~ + N 3 ~ /0 + 800 + /0 3 0 0 + /0 3 0 0
(36)
0 0 + 4ℎ0 4/ J7< + 0 J7= + 1 J7< J7= 6¤
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1 П< = ¥ 27 + 0 + J7< ℎ¦ /0 + 200 3 0 + 8x7 ℎ4d00 J7< − d/0 J7= 6§¨ 4
(37)
Eqs. (36) and (37) are inserted into Eq. (27) to achieve the total potential energy. From Eqs. (28), (30) and (33), the average circumferential stress is obtained by 1
27 + 0 /0 0 + d// 4Y=j + Y=y 6 − d/0 Y
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J7= =
8П jy© 8П jy© 8П jy© = = =0 87 8/ 80
(38)
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By applying the Ritz energy approach one has
From the preceding equation and considering the relationship (38) for J7= , we get 8П jy© = 0 0 ¡40 27 + 0 87 2d// x7
0 0 + ¦4x7d// 1 + x7 d/0 J7< ℎ − 4d// x7
− 0 4/0 0 − 8x7 d// 4Y=j + Y=y 6 + 8x7 d/0 Y
Comparison of Eqs. (38) and (40) yields
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(39)
(40)
ACCEPTED MANUSCRIPT J7= =
ℎ
(41)
While Eq. (41) is satisfied, the close condition and
ªП«¬ ª®¯
= 0 hold simultaneously. From the previous
equation, it is found that the pre-buckling circumferential stress depends on only the radial pressure when
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the shell can freely move in the radial direction. Accordingly, if the lateral pressure vanishes, the (39)) and regarding relationship (41) and / ≠ 0 we obtain
circumferential stress becomes zero, too. Using the Ritz method (the last two partial derivatives of Eq. /0
=−
1 7/ + 7~00 + 7 0 − 2 3 0 J7< ℎ 71
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= 7N 0 + 7 /0 + 7 /0 0 − 3 0 0 J7< ℎ where
(42) (43)
7/ = 40 0 + N 63 ~ + 41 0 + ~ 0 + + 63 0 0 + 4/ 0 + 6~
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7~ = 0 ~0 + 810 N0 3 ~ + 1 ~0 + ~ ~0 + 91 N0 + 9~ N0 3 0 0 + / ~0 + / N0 ~ 7 = 640 / 0 + 20 ~ 3 ~ + 2 1 ~ + ~ ~ 3 0 0 + 2/ ~ ~ 71 = 64 0 00 3 ~ + / 10 ~ 7N = 83 ~ 40 /0 + N
(44)
7 = 320 / 0 + 0 ~ 3 ~ + 1 ~ + ~ ~ 3 0 0 + /~ ~
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7 = 0 ~0 + 810 N0 3 ~ + 1 ~0 + ~ ~0 + 91 N0 + 9~ N0 3 0 0 + / ~0 + / N0 ~ 2 [ + 71 + 7/ 7 + 7 7 − 71 7N 0 [7 + 7 − 2710 ]3 0 7/ 7
Also, from Eqs. (40) and (41), one can achieve J7< ℎ =
+
7 7 + 7~ 7 00
+
7~ 7 01 ]
(45)
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It is noticeable that by setting the constant 0 to zero i.e., dropping the nonlinear buckling shape, the
linear critical load of the nanocomposite shell is attained. By inserting Eqs. (30) and (33) into Eq. (29) and adopting Eq. (41), the following expression for the end shortening ratio of the shell is obtained 30 0
+ 200 + x7 ¢d00 J7< ℎ − d/0 J7= ℎ − d00 Y
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< =
=
30 1 1 ±200 − $ + 7~ 00 + 7 0 − 3 0 J7< ℎ(² 8 71 7/ 2
+ x7 ¢d00J7< ℎ − d/0 J7= ℎ − d00 Y
4. Results and discussion
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(46)
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In what follows, numerical results are provided to investigate the nonlinear postbuckling analysis of piezoelectric FG-CNTRC shells under combined electro-thermal loadings, axial compression and lateral loads. The CNT efficiency parameters *X can be determined by matching the Young’s modulus of
CNTRCs .//& and .00& obtained from the rule of mixture with their counterparts from MD simulations ∗ ³´ = 0.12 ∶
*/ = 0.13l7, *0 = 1.022, *1 = 0.715,
∗ ³´ = 0.17 ∶ ∗ ³´
= 0.28 ∶
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given in [42]. The values of these parameters for three magnitudes of CNTs volume fraction are given by */ = 0.142, *0 = 1.626, *1 = 1.138, */ = 0.141, *0 = 1.585, *1 = 1.109
The matrix of the nanocomposite is selected to be PmPV whose material properties are as %&' =
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1150 ¹º/1 , 5&' = 0.34, .&' = 2.52», 3&' = 45 × 10]N . Temperature dependent material properties of the reinforcement materials, say SWCNTs, are listed in Table 1. For the piezoelectric layers, the
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material properties are taken to be % = 7600 ¹º/1 , »1 = 0.9 × 10]N , 5 = 0.3, .// = 632».
Table 2 indicates the influences of the applied constant voltage on the critical buckling load of CNTRC shell. The results are given for different CNT volume fractions and distribution-types of the reinforcement material through the thickness. As seen, when the voltage is selected a negative value, the liner buckling load is obtained larger than that in the case of positive applied voltage as well as in the absence of excitation voltage. It is also noted that an increase in the carbon nanotube volume fraction leads to the
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increase of the buckling load i.e., the stiffness of the nanocomposite shell.
parameter 0 /ℎ for various mode numbers. The nonlinear critical condition is defined as the lowest point Figs. 1a and 1b separately show variation of the nonlinear critical axial stress and lateral pressure with the
of the external force that is calculated from Eq. (45). In Fig. 2a, the mode numbers are selected among
10 − 5 and in Fig. 2b those are in the range of 13,12, … ,8. From the lowest point of these curves, an
condition with the nonlinear critical axial stress J³¾ or lateral pressure ³¾ and the associated nonlinear
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envelope curve is obtained. The lowest point of the envelope curve is viewed as the nonlinear critical
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buckling mode (, ).
Depicted in Fig. 2 is the postbuckling path of piezoelectric FGX-CNTRC cylindrical shell. The shell is
taken to be axially stressed and no radial pressure is applied. The solid lines indicate the postbuckling equilibrium path. It is observed from this figure that the system first follows a prebuckling path until it reaches the linear bifurcation point at which the axial stress is maximum i.e., it is the linear critical load called the upper critical load. Afterward, the system follows the postbuckling path and the stress considerably decreases until it obtains its own minimum value at the lowest point. This point is called the region a continuous modes jump (i.e. = 1; = 10, 9, . . . ; 6) is observed.
lower critical load after which the stress slightly increases. Moreover, it is noted that in the postbuckling
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Variation of the axial stress of the pre-pressured piezoelectric FG-CNTRC cylindrical shell with the loadshortening ratio under various combinations of the mode is depicted in Fig. 3. The results are provided for different CNT volume fractions. It is apparent that the higher volume fraction, the higher values of the upper and lower critical loads. In other words, an increase in the volume fraction of the reinforcement
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makes the carrying capacity of the structure. Additionally, a uniform postbuckling modes jump is noticeable.
Postbuckling responses of the non-pressured and pre-pressured piezoelectric FG-CNTRC cylindrical shells for different values of lateral pressure are plotted in Fig. 4. It is seen that when the shell is subjected negligible. Furthermore, the postbuckling modes keep jumping uniformly in the range of = 10,9, … ,6.
to radial pressure, the lower critical load diminishes, however, the difference among the curves are
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Fig. 5 depicts the effects of piezoelectric layer thickness-to-nanocomposite thickness ratio ℎ /ℎ on the
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postbuckling behaviors of the non-pressured piezoelectric FG-CNTRC shell under various combinations
of the mode. From this figure, one can see that the thicknesses ratio affects the postbuckling path of the shell so that as the thickness of the piezoelectric actuator layer goes up, the lower critical load increases, too.
Depicted in Fig. 6 are the postbuckling equilibrium paths of the piezoelectric FG-CNTRC cylindrical shell for different distribution-types of the carbon nanotubes along the thickness. As observed, the nanocomposite shell with the CNT distribution of FGX-type has the greatest lower critical load and the is also observed in the range of = 10,9, … ,6.
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FGO nanocomposite shell has the smallest one and lower postbuckling equilibrium path. The modes jump
Fig. 7 represents the influence of the radius-to-thickness ratio on postbuckling behavior of the nonfrom this figure that by increasing the non-dimensional parameter /ℎ, the lower critical load reduces
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pressured piezoelectric FG-CNTRC shell along with the corresponding buckling modes. It is observed
and the postbuckling equilibrium path moves down. Also, this parameter has considerable effect on the
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linear critical buckling load.
In the end, it is worth mentioning that the type of matrix may affect the concentration ranges (volume fraction) of CNTs and consequently affects the mechanical enhancement of the composite. In our case for the selected matrix, higher volume fractions results in higher values of the upper and lower critical loads and uniform postbuckling modes lead to more values. Also, different distributions of CNT in epoxy resin matrices cause different critical loads. But, it is clear that the influence of variation of volume fraction due to the inclusion of CNT with higher mass fractions is noticeable. Although an increase in the volume fraction leads to increasing the carrying load capacity of FG-CNTRC shells, more increase in the volume fraction as well as the weight fractions may lead to reducing the mechanical properties of nanocomposites due to the agglomeration of nanotubes, as reported by Muthu et al. [27] and Rousakis et al. [28].
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Therefore, different optimum dispersions of CNTs may be obtained by choosing various epoxy resin matrices to achieve the best mechanical properties for the load-carrying capacities of nanocomposites.
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4. Conclusion The nonlinear buckling and postbuckling problems of axially and laterally combine-loaded piezoelectric CNTRC cylindrical shells were solved based on the classic theory and von Kármán geometric nonlinearity. The Ritz energy method was employed to attain analytical solution for linear and nonlinear critical buckling load. The effects of CNT volume fraction, distribution-type of the reinforcement
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material, geometric parameter of the structure and piezoelectric layer on postbuckling response of the nanocomposite shell were investigated. It was found that a postbuckling modes jump occurs after the system passes the prebuckling equilibrium path. Additionally, it was revealed that the nonlinear critical
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load of the shell decreases by increasing the lateral pressure and radius-to-thickness ratio. It was concluded that a nanocomposite shell with piezoelectric layers and CNT distribution of FGX -type is of larger nonlinear critical buckling load. Also, it was found that for the selected matrix, if the CNT volume fraction is higher, the resulting nanocomposite shows higher values of upper and lower critical loads.
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TECHNOLOGY DIV WRIGHT-PATTERSON AFB OHIO, 1965. [42] Han, Y., Elliott, J. Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Computational Materials Science 39 (2007), 315–23.
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Table 1: Temperature-dependent material properties for (10, 10) SWCNT. = 9.26 , = List of captions
j 0.68 , ℎ = 0.067 , 5/0 = 0.175, % = 1400 ¹º⁄1 . / , 17
= 0 , = 10.
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Table 2: Effects of the applied constant voltage on the critical buckling load of CNTRC shell.
&_ &
=
Figure 1: Diagrammatic sketch of solving the non-linear critical load and the buckling mode: (a) the non∗ linear critical axial stress = 0.12, &_ &
=
/ ,J 17 7<
&
=
/ , 17
= 4» , = 10 − 5; (b) the non-linear critical lateral
= 5`» , = 13 − 8.
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∗ = 0.12, pressure
&_
∗ Figure 2: Diagrammatic sketch of the load-shortening ratio response curve. = 0.12,
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0 , = 10 − 6.
&_ &
=
/ , 17
=
Figure 3: Postbuckling responses of pre-pressured piezoelectric FG-CNTRC cylindrical shell under ∗ various volume fraction of carbon nanotubes .
&_ &
=
/ , 17
= 0 .
Figure 4: Postbuckling responses of non-pressured and pre-pressured piezoelectric FG-CNTRC ∗ = 0.12, cylindrical shells for different values of
&_ &
=
/ , 17
= 10 − 6.
Figure 5: Postbuckling responses of non-pressured piezoelectric FG-CNTRC cylindrical shell under &_ &
∗
= 0.12, , = 0 .
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different values of
Figure 6: Postbuckling responses of piezoelectric FG-CNTRC cylindrical shell for different types of ∗ CNTRC shell. = 0.12,
&_ &
=
/
17
, = 0, = 10 − 6.
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Figure 7: Postbuckling responses of non-pressured piezoelectric FG-CNTRC cylindrical shell for ∗ different values of = 0.12,
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&
&_ &
=
/
17
, = 0.
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Table 1: Temperature-dependent material properties for (10, 10) SWCNT. = 9.26 , =
500
j 2/0
T»
5.5308
6.9348
1.9643
5.6466 5.4744
7.08
6.8641
1.9445 1.9644
j 3//
× 10]N /
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700
j .00
T»
20
j 300
× 10]N /
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300
j .//
T»
3.4584 4.5361 4.6677
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TVkVÀuÀV
j = 0.175, % = 1400 ¹º⁄1 0.68 , ℎ = 0.067 , 5/0
5.1682 5.0189 4.8943
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Table 2: Effects of the applied constant voltage on the critical buckling load of CNTRC shell. & =
0.12
15.0120
14.9833
FGO
14.8237
14.7947
14.7655
UD
FGX
FGO UD
FGX
15.2117 16.1788 16.4877 15.9318 16.7864 17.3653 16.4675
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FGO
−500
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0.28
UD
FGX
0.17
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v7
0
15.1888
21
500
14.9546 15.1604
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ÁYTÁ − u kV
= 0 , = 10
16.1523 16.4617
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∗
/ , 17
&_
15.9040 16.7613 17.3410 16.4418
16.1258 16.4357 15.8780 16.7361
17. .3166 16.4161
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(a)
(b)
Figure 1: Diagrammatic sketch of solving the non-linear critical load and the buckling mode: (a) the nonlinear critical axial stress Â∗ÃÄ = Å. ÆÇ,
ÈÉ È
=
Æ ,Ë ÊÅ
lateral pressure Â∗ÃÄ = Å. ÆÇ,
ÈÉ È
= ÌÍÎÏ , Ð = ÆÅ − Ñ; (b) the non-linear critical
=
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Æ , ÒÅÓ ÊÅ
= ÑÔÎÏ , Ð = ÆÊ − Õ
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Figure 2: Diagrammatic sketch of the load-shortening ratio response curve. Â∗ÃÄ = Å. ÆÇ,
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Å , Ð = ÆÅ − Ö, ÂÅ = ÑÅÅ
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ÈÉ È
=
Æ
ÊÅ
,Ë =
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Figure 3: Postbuckling responses of non-pressured piezoelectric FGX-CNTRC cylindrical shell under
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various volume fraction of carbon nanotubes Â∗ÃÄ .
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ÈÉ È
=
Æ ,Ë ÊÅ
= Å, ÂÅ = ÑÅÅ
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Figure 4: Postbuckling responses of non-pressured and pre-pressured piezoelectric FGX-CNTRC
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cylindrical shells for different values of Ë Â∗ÃÄ = Å. ÆÇ,
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ÈÉ È
= ÊÅ , Ð = ÆÅ − Ö, ÂÅ = ÑÅÅ Æ
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Figure 5: Postbuckling responses of non-pressured piezoelectric FGX-CNTRC cylindrical shell under
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different values of
ÈÉ È
Â∗ÃÄ = Å. ÆÇ, , Ë = Å , ÂÅ = ÑÅÅ
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Figure 6: postbuckling responses of piezoelectric FG-CNTRC cylindrical shell for different types of
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CNTRC shell. Â∗ÃÄ = Å. ÆÇ,
ÈÉ È
=
Æ ,Ë ÊÅ
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= Å, Ð = ÆÅ − Ö, ÂÅ = ÑÅÅ
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Figure 7: Postbuckling responses of non-pressured piezoelectric FGX-CNTRC cylindrical shell for × È
Â∗ÃÄ = Å. ÆÇ,
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different values of
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ÈÉ È
= ÊÅ , Ë = Å, ÂÅ = ÑÅÅ Æ
S1359-8368(15)00744-1
DOI:
10.1016/j.compositesb.2015.12.012
Reference:
JCOMB 3931
To appear in:
Composites Part B
Received Date: 31 August 2015 Revised Date:
2 December 2015
Accepted Date: 24 December 2015
Please cite this article as: Ansari R, Pourashraf T, Gholami R, Shahabodini A, Analytical solution for nonlinear postbuckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers, Composites Part B (2016), doi: 10.1016/j.compositesb.2015.12.012. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Analytical solution for nonlinear postbuckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers R. Ansari*,a, T. Pourashrafa, R. Gholami*,b, A. Shahabodinia b
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran
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a
Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran
Abstract. This paper presents an analytical solution procedure for the nonlinear postbuckling analysis of piezoelectric functionally graded carbon nanotubes reinforced composite (FG-CNTRC) cylindrical shells
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subjected to combined electro-thermal loadings, axial compression and lateral loads. The carbon nanotubes are assumed to be aligned and straight with uniform and functionally graded distributions in
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the thickness direction. The kinematics and constitutive relations are written on the basis of the classic theory and the von Kármán nonlinear strain–displacement relations of large deformation. Applying the Ritz energy approach, analytical solutions are proposed for the nonlinear critical axial load, lateral pressure as well as the load-shortening ratio of the piezoelectric FG-CNTRC shell. Numerical results are presented to study the effects of dimensional parameters, CNT volume fraction, distribution type of the reinforcement and piezoelectric thickness on the nonlinear buckling behavior of the piezoelectric nanocomposite shell. It is revealed that the carrying capacity of the structure increases as the shell is
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integrated by the piezoelectric layers and reinforced by higher CNT volume fraction. Furthermore, FGXand FGO- CNTRC piezoelectric shells are indicated to have higher and lower carrying capacities compared to UD-CNTRC piezoelectric shells, respectively. Keywords: A. Carbon-carbon composites; A. Smart materials; A. Nano-structures; B. Buckling;
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1. Introduction
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C. Analytical modelling.
In recent years, carbon nanotubes as the reinforcement materials in the composite structures have attracted a lot of attention [1, 2]. This is due to the promising thermo-mechanical properties of CNTs such as low density, high strength and stiffness-to-weight ratio. So, replacing the traditional carbon fibers by carbon nanotubes in a matrix can effectively improve composite properties and consequently brings about their application in reinforcing the composites, electronic devices and etc [3-6]. However, the volume fraction of carbon nanotubes in a polymer is limited so that, a more amount may lead to deterioration of their mechanical properties [7]. To overcome this shortcoming, Shen [8] combined the concept of the functionally graded material with CNTs through a nonuniform distribution of CNTs along the thickness *
Corresponding authors. [email protected] (R. Ansari) and [email protected] (R. Gholami).
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of a plate structure. He used a functionally graded distribution of carbon nanotubes within the matrix and improved the load bending moment curves of the plates. Various studies were further carried out on the linear and nonlinear behaviors of the CNT-based functionally graded beam, plate and shell structures. Concerning works are cited below.
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Thermal buckling and postbuckling response of perfect and imperfect functionally graded CNTRC plate [9] and shell [10] were investigated based on a multiscale approach. Shen [11] discussed the postbuckling behavior of CNT-reinforced nanocomposite cylindrical shells under axial compression along with thermal environments based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. They obtained the buckling loads and postbuckling equilibrium paths using a perturbation
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technique. Jafari et al. [12] adopted the first order shear deformation theory to account for the transverse shear deformation in the buckling analysis of the FG-CNTRC plates under uniaxial or biaxial loadings.
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Based on the same theory and solution method, postbuckling of CNTRC cylindrical shells subjected to combined axial and radial mechanical loads in thermal environments was analyzed by Shen and Xiang [13]. Arani et al. [14] discussed static stress analysis of CNTRC cylinder under non-axisymmetric thermo-mechanical loads and uniform electro-magnetic fields. Lei et al. [15] employed the element free kp Ritz method to describe the buckling response of the FG-CNTRC plates under various in-plane loading conditions. Van Dung [16] studied torsional buckling and postbuckling of FG shells in thermal environment. Rafiee et al. [17] proposed an exact closed form solution for the thermal buckling and
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postbuckling of piezoelectric FG-CNTRC Timoshenko beam. They found that a functionally graded distribution of the reinforcement material can increase the bucking load capacity. With the use of Ritz method, Liew et al. [18] analyzed the postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels. A three dimensional solution approach was proposed by Wu and Chang [19] to the
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determination of the critical buckling load of a plate integrated by two piezoelectric layers. Based upon the large deformation theory with the Von Karman–Donnell-type of kinematic non-linearity, Sofiyev [20] investigated non-linear buckling behavior of truncated FG conical shells. Shen and Xiang [21] analyzed
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the postbuckling of axially compressed cylindrical panels reinforced by aligned single-walled carbon nanotubes resting on elastic foundation in thermal environment according to a higher-order shear deformation theory and von Kármán assumptions. Based on the similar theory, they [22] also investigated the postbuckling behavior of nanocomposite cylindrical panels resting on elastic foundations and subjected to uniform temperature rise. The nonlinear bending and postbuckling of a plate composed of conventional fiber reinforced composite (FRC) layers and CNTRC layers were investigated by Fan et al. [23]. In addition to the mentioned works above, another research concerning the linear and nonlinear buckling of the functionally graded carbon nanotube-reinforced composite cylindrical shells under different thermal and mechanical loadings namely radial pressure, combined radial and axial loads and
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torsional load can be found in the works of Shen and his coworkers [24-26]. The results of these papers showed that a symmetric distribution of the reinforcement material with respect to the mid-surface of the shell may increase the strength of it against the buckling. He also found that the CNT volume fraction has a profound effect on the buckling load and postbuckling strength of CNTRC shells. Muthu et al. [27]
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studied the characterization of the glass reinforced fiber composites (GRPs) hybridized with functionalized multiwall carbon nanotube. Tensile testing and mechanical characterization of three different epoxy resins reinforced by different concentrations of multiwall carbon nanotubes were investigated by Rousakis et al. [28].
Based on the first order theory of shells, Jam and Kiani [29] studied the buckling of FG-CNTRC conical
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shells under pressure loading. Sofiyev [30] studied the non-linear dynamic buckling response of the FG coated truncated conical shells to predict the effects of compositional profiles of coatings, variation of
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truncated conical shell parameters and loading speed on the dimensionless linear and non-linear critical time parameters. Alizada et al. [31] presented a stress analysis of a substrate coated by nanomaterials with a vacancy under uniform extension load to determine the contact pressure, stresses and damage conditions. Based on the Timoshenko beam theory and von Kármán geometric nonlinearity, Ansari et al. [32] analyzed the forced vibration of nanocomposite beams reinforced by single-walled carbon nanotubes. They used a numerical strategy including the differential quadrature and Galerkin approaches and pseudo-arc length continuum technique to solve the nonlinear governing equations. Ansari et al. [33]
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employed the same strategy to investigate the forced vibration behavior of carbon nanotube-reinforced composite plates. Their results revealed that when the mid-plane of plate is CNT-rich the hardening behavior of the plate is intensified.
Integrating the engineering structures with piezoelectric layers is practically of much importance. These
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structures are smart and have application in different fields such as structural vibration control, thermal stress control and structural health monitoring [34]. Prompted by this, the present paper deals with the postbuckling of functionally graded carbon nanotube-reinforced composite cylindrical shells with surface-
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bonded piezoelectric layers under combined action of axial compression and lateral loads. The geometrical nonlinearity of von-Kármán and Donnell kinematic assumptions are used to derive the governing equation. Using Airy’s stress function and Ritz energy approach, analytical relationships are presented to study the effects of various model parameters on the postbuckling strength of piezoelectric FG-CNTRC shell.
Consider a carbon nanotube-reinforced composite cylindrical shell of length , mean radius and 2. Mathematical formulation
thickness ℎ whose inner and outer surfaces are perfectly bonded with piezoelectric actuator layers of
3
ACCEPTED MANUSCRIPT thickness ℎ . In this way, the total thickness of the shell is equal to = ℎ + 2ℎ . A coordinate system
, , with the origin located at the left end on the middle plane is considered in which the axes and
are selected in the longitudinal and circumferential directions of the shell and is in the direction of the
CNTRC shell along , and axes are denoted by , and , respectively and the associated
inward normal to the middle surface. Displacement components of an arbitrary point of piezoelectric FG-
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components on the middle plane (physical neutral surface) are represented by , and . 2.1 Material properties of CNTRC
The present nanocomposite is assumed to be made of a mixture of single wall carbon nanotube (SWCNT)
SC
as the reinforcement material and an isotropic matrix. The distribution profile of carbon nanotubes through the shell thickness is taken to be either uniformly distributed (referred to UD) or functionally
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graded (referred to FG). In the case of functionally graded-CNTRC, the reinforcement has a linear distribution in the thickness direction of two forms FGX and FGO. For type FGX a mid-plane symmetric graded distribution of carbon nanotube is achieved and both outer and inner surfaces are CNT-rich and for type FGO the middle plane is CNT-rich, only. Accordingly, the CNT volume fraction for the considered nanocomposite shell takes the following forms ∗ = UD
= 2 1 −
∗ =
% % + $ ( − $ ( %&' %&'
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where
2|| ∗ FGX ℎ
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= 2
2|| ∗ FGO ℎ
(1a) (1b) (1c)
(2)
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and indicates the mass fraction of the carbon nanotubes, % and %&' are the mass densities of the
carbon nanotube and matrix, respectively. The structure of CNT considerably affects the properties of the resulting composite. The effective material properties of the two-phase nanocomposites, combination of
CNT and a matrix, are estimated through the well-known Mori-Tanaka scheme [35] or the rule of mixtures [36]. Predicting the overall material properties and consequently accounting for the impact of carbon nanotubes is simple and convenient by applying the rule of mixtures. To take the small scale efficiency parameters *+ , = 1,2,3 calculated by matching the effective properties of CNTRC effects into account, this rule needs to be extended. This is accomplished by introducing the CNT
determined from the MD simulations with those from the rule of mixture. Based on the extended version of this rule, the effective Young’s modulus and shear modulus of CNTRCs can be expressed as [11]
4
ACCEPTED MANUSCRIPT .//& = */ .// + '& .&'
(3a)
*1 '& = + ' 2/0& 2/0 2&
(3c)
*0 '& = + ' .00& .00 .&
(3b)
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in which .// , .00 and 2/0 represent Young’s modulus and shear modulus of SWCNTs , respectively
and .&' and 2&' stand for the relevant properties of the isotropic matrix of the host. Moreover, in Eq. (3),
the CNT and matrix volume fractions are related by + '& = 1. The thermal expansion coefficients in 3//& = 3// + '& 3&'
& 300& = 41 + 5/0 6 300 + 1 + 5&' '& 3&' − 5/0 3//&
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the longitudinal and transverse directions are given by
(4a) (4b)
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where 3// , 300 and 3&' are thermal expansion coefficients and 5/0 , 5&' denote Poisson’s ratios,
respectively, of the carbon nanotube and matrix.
2.2 Fundamental equations
In the current paper, the classic theory including some assumptions such as the thin shell assumption, the Kirchhoff–Love hypotheses, the shallow shell approximations and etc [37] is adopted to describe the dimensional displacement components , and take the form [38]
deformation of piezoelectric FG-CNTRC cylindrical shells. According to this theory, the three-
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8 , , 8 8 , , , = , − − 7 , 8 , , = , − − 7
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, , = , .
(5a) (5b) (5c)
in which 7 denotes the coordinate associated with the physical neutral surface. From the nonlinearity of von-Kármán, the strain–displacement relationships are stated as follows
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;< ;<7 B< 7 ; B ; : = ? = @ = A + − 7 : = ? ><= B<= >7 in which
(6)
<
5
ACCEPTED MANUSCRIPT 8 0 8 1 8 0 F − 0 I F + I 8 8 2 8 D D ;<7 B< D 8 0 D D D8 1 8 D D 0 7 B − +
; − , : = ? = @ =A = 0 E8 2 8 H B<= E 8 H ><7 0 D 8 8 8 8 D D D D D−2 8 D D + + C 8 8 8 8 G C 8 8 G
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(7)
In the presence of thermal and electrical loadings, the constitutive relations for matrix and piezoelectric J< L// : J= ? = KL/0 J<= 0
;< 0 3// 0 0 V1/ .< 0 O P: ;= ? − Q300 R ∆TU − K0 0 V10O @.= A ><= 0 LNN .W 0 0 0
layers are given by
L/0 L00 0
.//& .00& 5/0& ./0& , L00& = , L/0& = ,L = 2/0& 1 − 5/0& 50/& 1 − 5/0& 50/& 1 − 5/0& 50/& NN& .// , 1 − 5 0
for the CNTRC host and L// = L00 = 3// = 300
L/0 =
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L//& =
SC
where L+X are the reduced elastic modulus given by
5 .// .// , LNN = 2/0 = 0 1 − 5 2 1 + 5
(8)
(9)
(10)
for piezoelectric layer. ∆T = T − T7 is the temperature difference measured from the reference
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temperature T7 . Moreover, V1/ and V10 are piezoelectric constants and .< , .= and .W stand for the
displacement electric field. The electric potential is assumed to linearly distribute in the thickness direction. Since the piezoelectric layer is very thin the self-induced electric potential is much smaller than between applied voltage and electric field intensity [39] within a piezoelectric actuator can be stated as ℎ
.W =
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follows [40]
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the applied voltage and the voltage is applied to the actuator along the thickness only, the relationship
(11)
The force, moment and transverse shear force resultants per unit length are related to stress components as Y< =
&/0
Z J<& [ ]&/0
Y= = Z
&/0
]&/0
J=& [
+Z
]&/0
& ] ^&_ 0
+Z
]&/0
& ] ^&_ 0
J< [
J= [
+Z
&
^&_ 0
&/0
+Z
&
^&_ 0
&/0
J< [
(12a)
J= [
(12b)
6
ACCEPTED MANUSCRIPT
&/0
Z J<& ]&/0
`= = Z
&/0
]&/0
`<= = Z
J=&
&/0
+Z
]&/0
& ] ^&_ 0
− 7 [ + Z
J<= [ ]&/0
& ] ^&_ 0 ]&/0
− 7 [ + Z
& ] ^&_ 0
& J<=
− 7 [ + Z
]&/0
+Z
&/0
J<
J=
& ] ^&_ 0
]&/0
&
^&_ 0
J<= [
(12c)
− 7[ + Z
&
^&_ 0
&/0
− 7 [ + Z
J<=
&
^&_ 0
&/0
− 7 [ + Z
d// cd b /0 0 <= =b ` e E < H b // D D `= D D be/0 C`<= G a 0
d/0 d00 0 e/0 e00 0
0 0 dNN 0 0 eNN
e// e/0 0 f// f/0 0
e/0 e00 0 f/0 f00 0
J= − 7 [
J<= − 7 [
Y< Y
(12d)
(12e)
(12f)
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Y< FY I = D DY D D
&
^&_ 0
&/0
Substituting Eqs. (6) and (8) into Eqs. (12) gives
J< − 7 [
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`< =
& Z J<= [ ]&/0
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Y<= =
&/0
(13)
where the subscripts ‘T’ and ‘k’ signal the thermal and electric loads, respectively and the coefficients of stretching stiffness d+X , stretching-bending coupling stiffness e+X and bending stiffness f+X are given by l 0
l ] 0
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4d+X , e+X , f+X 6 = Z L+X 1, − 7 , − 7 0 [ ; ,, n = 1,2,6
(14)
It is remarkable that for symmetric distribution of the carbon nanotubes through thickness direction of the nano-composite shells, the stretching–bending coupling effect will not be appeared. From the physical = 7 [40] &/0
p]&/0 .// [ &/0
(15)
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7 =
p]&/0 .// [
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neutral surface concept and constant Poisson’s ratio, the physical neutral surface of a shell is obtained by
Thermal force and piezoelectric resultants are calculated as
7
ACCEPTED MANUSCRIPT &/0 `
L// + Z qL/0 0 &/0
L/0& L00& 0
l/0
L// + Z qL/0 0 ]l/0 ]&/0
Y<=
L/0 L00 0
0 3// 0 r :300 ? 41, − 7 6∆T[ 0 LNN
0 3// 0 r :300 ? 41, − 7 6∆T[ 0 LNN
(16)
l/0 ]&/0 `< V1/ V1/ u `= t = Z QV10 R 41, − 7 6 [ + Z QV10 R 41, − 7 6 [ ℎ ℎ 0 0 &/0 ]l/0 `<=
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Y< s Y=
L/0 L00 0
0 3//& 0 O Q300& R 41, − 7 6∆T[ 0 LNN&
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Y
(17)
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Let the same constant excitation voltage = v7 be applied to both the top and down piezoelectric
actuators. In this manner, by means of Eq. (17), one can obtain the force per unit length of the shell as Y< = Y= = Yv7 = 2v7 V1/
(18)
Using Eq. (7), the compatibility equation is attained as
7 8 0 ;<7 8 0 ;=7 8 0 ><= 80 0 80 80 1 8 0 + − =
− 0 − 8 0 8 0 8 8 8 8 8 8 0 8 0
Airy’s stress function w is defined such that 8 0w , 8 0
Y= =
80 w , 8 0
Y<= = −
From Eqs. (13) and (20), one has
80 w 80w − d + d00 Y
;=7 = x7 P−d/0
80 w 80w + d + d// 4Y=j + Y=y 6 − d/0 Y
80w 8 8
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7 ><= = x/
(20)
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;<7 = x7 Pd00
80 w 8 8
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Y< =
in which x7 =
/ , z{{ z|| ]z|{|
relation becomes
x/ = −
(19)
(21a)
(21b)
(21c) / . z}}
By inserting relationships (21) into Eq. (19), the compatibility
8 ~ w −2d/0 x7 − x/ 8 ~ w d00 8 ~ w 1 80 8 0 8 0 1 80 + + = [ − − ] 8 ~ x7 d// 8 0 8 0 d// 8 ~ x7 d// 8 8 8 0 8 0 8 0 0
The strain energy of the FG-CNTRC shell is evaluated as
8
(22)
ACCEPTED MANUSCRIPT
П+
1 = 4J< ;< + J= ;= + <= ><= 6[ [ [ 2
(23)
where is the volume of body. With regard to Eqs. (6), (8) and (21), the strain energy is rewritten as 8 0w 80 w 80w 80w 80w = Z Z[/ 0 + 0 0 + 1 0 0 + ~ − / Y
0
− 0 4Y=j + Y=y 6 − 1 Y
80 80 80 + 0 + ] [ [ 8 8 0 8 8
0
0
f00 f// = , N = , = f/0 , = fNN 2 2
(24)
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0
80 80 + N 8 0 8 0
d00 d// d/0 1 0 , 0 = 0 , 1 = − 0 , ~ = d , 2 d// d00 − d/0 2 d// d00 − d/0 d// d00 − d/0 NN
where / =
0
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7
0
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П+
0
(25)
Assume that the shell is synchronously subjected to average axial stress J7< on two ends and uniform radial pressure . Using Eqs. (7) and (21), the work done by the external forces is calculated as 0
0
П< = Z Z [ [ − J7< ℎ Z Z P 7
0
7
7
0
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7
8 − x7 d00 Y
= Z Z [ [ − J7< ℎ Z Z[x7 d00 7
7
7
П j = П+ − П<
0
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Then, the total potential energy of the system is
7
8 w 8 w 1 8 − d/0 0 − 0 ] [ [ 0 8 8 2 8 0
(26)
(27)
Regarding relationships (7) and (21), the following circumferential closed condition should be satisfied for a cylindrical shell 0 7
7
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8 Z Z $ ( [ [ 8 0
= Z Z[x7 P−d/0 7
7
8 w 8 w 1 8 + d// 0 + d// 4Y=j + Y=y 6 − d/0 Y
0
In like manner, the average end-shortening ratio of the shell < may be stated as
9
(28)
ACCEPTED MANUSCRIPT 0
1 8 < = − Z Z $ ( [ [ 2 8 7
7
0
80 w 80w 1 =− Z Z[x7 Pd00 0 − d/0 0 + d00 Y
7
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1 8 − 0 ] [ [ 2 8
(29)
3. Analytical solution procedure
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Herein, According to [41], the deflection of the piezoelectric FG-CNTRC shell can be expanded into a series of multi-items
, = 7 + / sin 3 sin + 0 sin 3 0 in which 3 =
'
(30)
and = with , as the axial half-wave numbers in the direction and the wave
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numbers in the direction. Also, 7 , / and 0 are unknown amplitudes which are associated with the pre-
buckling, linear buckling and nonlinear buckling states, respectively. On substitution of Eq. (30) into Eq.
(22), we get
8 ~w 8 ~w 8~w 0 0 + + 8 ~ 8 0 8 0 8 ~
= [7/ cos 23 + [70 cos 2 + [71 sin 3 sin 0 =
d00 , d//
[7/ =
1 −4 $ 0 3 0 + /0 3 0 0 (, 2x7 d//
1 1 1
/0 3 0 0 , [71 =
/ 3 0 − / 0 3 0 0 , [7~ =
3 0 0 . = 2x7 d// x7 d// x7 d// / 0
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[70
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−2d/0x7 − x/ , x7 d//
where 0 =
+ [7~ sin 33 sin
(31)
The general solution of the stress function w is taken to be of the form
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w = [/ cos 23 + [0 cos 2 + [1 sin 3 sin + [~ sin 33 sin − J7< ℎ 0 − J7= ℎ 2
0 2
(32)
(33)
in which J7= denotes the average circumferential stress, positive when the shell is circumferentially
compressed, and
10
ACCEPTED MANUSCRIPT [/ =
[7/ = / 0 + 0 /0 , 163 ~
[70 = 1 /0 , 16 0 ~
[0 =
/ = −
~ = − N =
1
8x7 d// 3 0
,
0 =
0 , 32x7 d// 3 0
3 0 0 , x7 d// 3 ~ + 0 3 0 0 + 0 ~
3 0 0 x7 d// 813 ~ + 9 0 3 0 0 + 0 ~
1 =
=
30 , 32x7 d// 0 0
30 , x7 d// 3 ~ + 0 3 0 0 + 0 ~
(34)
(35)
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in which
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[71 [7~ [1 = ~ = ~ / 0 + / , [~ = = N / 0 . 0 0 0 0 ~ ~
3 + 3 +
813 + 9 0 3 0 0 + 0 ~
With the aid of Eqs. (30) and (33), the relationships of strain energy in Eq. (24) and external work in Eq. 1 П+ = ¡/ −4 Y
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(26) become
+ 0 ¢−44Y=j + Y=y 6 + 32[/0 3 ~ + [10 3 ~ + 81[~0 3 ~ £ 0
+ 1 4−4 Y
+ /0 ~ + N 3 ~ /0 + 800 + /0 3 0 0 + /0 3 0 0
(36)
0 0 + 4ℎ0 4/ J7< + 0 J7= + 1 J7< J7= 6¤
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1 П< = ¥ 27 + 0 + J7< ℎ¦ /0 + 200 3 0 + 8x7 ℎ4d00 J7< − d/0 J7= 6§¨ 4
(37)
Eqs. (36) and (37) are inserted into Eq. (27) to achieve the total potential energy. From Eqs. (28), (30) and (33), the average circumferential stress is obtained by 1
27 + 0 /0 0 + d// 4Y=j + Y=y 6 − d/0 Y
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J7= =
8П jy© 8П jy© 8П jy© = = =0 87 8/ 80
(38)
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By applying the Ritz energy approach one has
From the preceding equation and considering the relationship (38) for J7= , we get 8П jy© = 0 0 ¡40 27 + 0 87 2d// x7
0 0 + ¦4x7d// 1 + x7 d/0 J7< ℎ − 4d// x7
− 0 4/0 0 − 8x7 d// 4Y=j + Y=y 6 + 8x7 d/0 Y
Comparison of Eqs. (38) and (40) yields
11
(39)
(40)
ACCEPTED MANUSCRIPT J7= =
ℎ
(41)
While Eq. (41) is satisfied, the close condition and
ªП«¬ ª®¯
= 0 hold simultaneously. From the previous
equation, it is found that the pre-buckling circumferential stress depends on only the radial pressure when
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the shell can freely move in the radial direction. Accordingly, if the lateral pressure vanishes, the (39)) and regarding relationship (41) and / ≠ 0 we obtain
circumferential stress becomes zero, too. Using the Ritz method (the last two partial derivatives of Eq. /0
=−
1 7/ + 7~00 + 7 0 − 2 3 0 J7< ℎ 71
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= 7N 0 + 7 /0 + 7 /0 0 − 3 0 0 J7< ℎ where
(42) (43)
7/ = 40 0 + N 63 ~ + 41 0 + ~ 0 + + 63 0 0 + 4/ 0 + 6~
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7~ = 0 ~0 + 810 N0 3 ~ + 1 ~0 + ~ ~0 + 91 N0 + 9~ N0 3 0 0 + / ~0 + / N0 ~ 7 = 640 / 0 + 20 ~ 3 ~ + 2 1 ~ + ~ ~ 3 0 0 + 2/ ~ ~ 71 = 64 0 00 3 ~ + / 10 ~ 7N = 83 ~ 40 /0 + N
(44)
7 = 320 / 0 + 0 ~ 3 ~ + 1 ~ + ~ ~ 3 0 0 + /~ ~
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7 = 0 ~0 + 810 N0 3 ~ + 1 ~0 + ~ ~0 + 91 N0 + 9~ N0 3 0 0 + / ~0 + / N0 ~ 2 [ + 71 + 7/ 7 + 7 7 − 71 7N 0 [7 + 7 − 2710 ]3 0 7/ 7
Also, from Eqs. (40) and (41), one can achieve J7< ℎ =
+
7 7 + 7~ 7 00
+
7~ 7 01 ]
(45)
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It is noticeable that by setting the constant 0 to zero i.e., dropping the nonlinear buckling shape, the
linear critical load of the nanocomposite shell is attained. By inserting Eqs. (30) and (33) into Eq. (29) and adopting Eq. (41), the following expression for the end shortening ratio of the shell is obtained 30 0
+ 200 + x7 ¢d00 J7< ℎ − d/0 J7= ℎ − d00 Y
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< =
=
30 1 1 ±200 − $ + 7~ 00 + 7 0 − 3 0 J7< ℎ(² 8 71 7/ 2
+ x7 ¢d00J7< ℎ − d/0 J7= ℎ − d00 Y
4. Results and discussion
12
(46)
ACCEPTED MANUSCRIPT
In what follows, numerical results are provided to investigate the nonlinear postbuckling analysis of piezoelectric FG-CNTRC shells under combined electro-thermal loadings, axial compression and lateral loads. The CNT efficiency parameters *X can be determined by matching the Young’s modulus of
CNTRCs .//& and .00& obtained from the rule of mixture with their counterparts from MD simulations ∗ ³´ = 0.12 ∶
*/ = 0.13l7, *0 = 1.022, *1 = 0.715,
∗ ³´ = 0.17 ∶ ∗ ³´
= 0.28 ∶
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given in [42]. The values of these parameters for three magnitudes of CNTs volume fraction are given by */ = 0.142, *0 = 1.626, *1 = 1.138, */ = 0.141, *0 = 1.585, *1 = 1.109
The matrix of the nanocomposite is selected to be PmPV whose material properties are as %&' =
SC
1150 ¹º/1 , 5&' = 0.34, .&' = 2.52», 3&' = 45 × 10]N . Temperature dependent material properties of the reinforcement materials, say SWCNTs, are listed in Table 1. For the piezoelectric layers, the
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material properties are taken to be % = 7600 ¹º/1 , »1 = 0.9 × 10]N , 5 = 0.3, .// = 632».
Table 2 indicates the influences of the applied constant voltage on the critical buckling load of CNTRC shell. The results are given for different CNT volume fractions and distribution-types of the reinforcement material through the thickness. As seen, when the voltage is selected a negative value, the liner buckling load is obtained larger than that in the case of positive applied voltage as well as in the absence of excitation voltage. It is also noted that an increase in the carbon nanotube volume fraction leads to the
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increase of the buckling load i.e., the stiffness of the nanocomposite shell.
parameter 0 /ℎ for various mode numbers. The nonlinear critical condition is defined as the lowest point Figs. 1a and 1b separately show variation of the nonlinear critical axial stress and lateral pressure with the
of the external force that is calculated from Eq. (45). In Fig. 2a, the mode numbers are selected among
10 − 5 and in Fig. 2b those are in the range of 13,12, … ,8. From the lowest point of these curves, an
condition with the nonlinear critical axial stress J³¾ or lateral pressure ³¾ and the associated nonlinear
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envelope curve is obtained. The lowest point of the envelope curve is viewed as the nonlinear critical
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buckling mode (, ).
Depicted in Fig. 2 is the postbuckling path of piezoelectric FGX-CNTRC cylindrical shell. The shell is
taken to be axially stressed and no radial pressure is applied. The solid lines indicate the postbuckling equilibrium path. It is observed from this figure that the system first follows a prebuckling path until it reaches the linear bifurcation point at which the axial stress is maximum i.e., it is the linear critical load called the upper critical load. Afterward, the system follows the postbuckling path and the stress considerably decreases until it obtains its own minimum value at the lowest point. This point is called the region a continuous modes jump (i.e. = 1; = 10, 9, . . . ; 6) is observed.
lower critical load after which the stress slightly increases. Moreover, it is noted that in the postbuckling
13
ACCEPTED MANUSCRIPT
Variation of the axial stress of the pre-pressured piezoelectric FG-CNTRC cylindrical shell with the loadshortening ratio under various combinations of the mode is depicted in Fig. 3. The results are provided for different CNT volume fractions. It is apparent that the higher volume fraction, the higher values of the upper and lower critical loads. In other words, an increase in the volume fraction of the reinforcement
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makes the carrying capacity of the structure. Additionally, a uniform postbuckling modes jump is noticeable.
Postbuckling responses of the non-pressured and pre-pressured piezoelectric FG-CNTRC cylindrical shells for different values of lateral pressure are plotted in Fig. 4. It is seen that when the shell is subjected negligible. Furthermore, the postbuckling modes keep jumping uniformly in the range of = 10,9, … ,6.
to radial pressure, the lower critical load diminishes, however, the difference among the curves are
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Fig. 5 depicts the effects of piezoelectric layer thickness-to-nanocomposite thickness ratio ℎ /ℎ on the
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postbuckling behaviors of the non-pressured piezoelectric FG-CNTRC shell under various combinations
of the mode. From this figure, one can see that the thicknesses ratio affects the postbuckling path of the shell so that as the thickness of the piezoelectric actuator layer goes up, the lower critical load increases, too.
Depicted in Fig. 6 are the postbuckling equilibrium paths of the piezoelectric FG-CNTRC cylindrical shell for different distribution-types of the carbon nanotubes along the thickness. As observed, the nanocomposite shell with the CNT distribution of FGX-type has the greatest lower critical load and the is also observed in the range of = 10,9, … ,6.
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FGO nanocomposite shell has the smallest one and lower postbuckling equilibrium path. The modes jump
Fig. 7 represents the influence of the radius-to-thickness ratio on postbuckling behavior of the nonfrom this figure that by increasing the non-dimensional parameter /ℎ, the lower critical load reduces
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pressured piezoelectric FG-CNTRC shell along with the corresponding buckling modes. It is observed
and the postbuckling equilibrium path moves down. Also, this parameter has considerable effect on the
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linear critical buckling load.
In the end, it is worth mentioning that the type of matrix may affect the concentration ranges (volume fraction) of CNTs and consequently affects the mechanical enhancement of the composite. In our case for the selected matrix, higher volume fractions results in higher values of the upper and lower critical loads and uniform postbuckling modes lead to more values. Also, different distributions of CNT in epoxy resin matrices cause different critical loads. But, it is clear that the influence of variation of volume fraction due to the inclusion of CNT with higher mass fractions is noticeable. Although an increase in the volume fraction leads to increasing the carrying load capacity of FG-CNTRC shells, more increase in the volume fraction as well as the weight fractions may lead to reducing the mechanical properties of nanocomposites due to the agglomeration of nanotubes, as reported by Muthu et al. [27] and Rousakis et al. [28].
14
ACCEPTED MANUSCRIPT
Therefore, different optimum dispersions of CNTs may be obtained by choosing various epoxy resin matrices to achieve the best mechanical properties for the load-carrying capacities of nanocomposites.
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4. Conclusion The nonlinear buckling and postbuckling problems of axially and laterally combine-loaded piezoelectric CNTRC cylindrical shells were solved based on the classic theory and von Kármán geometric nonlinearity. The Ritz energy method was employed to attain analytical solution for linear and nonlinear critical buckling load. The effects of CNT volume fraction, distribution-type of the reinforcement
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material, geometric parameter of the structure and piezoelectric layer on postbuckling response of the nanocomposite shell were investigated. It was found that a postbuckling modes jump occurs after the system passes the prebuckling equilibrium path. Additionally, it was revealed that the nonlinear critical
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load of the shell decreases by increasing the lateral pressure and radius-to-thickness ratio. It was concluded that a nanocomposite shell with piezoelectric layers and CNT distribution of FGX -type is of larger nonlinear critical buckling load. Also, it was found that for the selected matrix, if the CNT volume fraction is higher, the resulting nanocomposite shows higher values of upper and lower critical loads.
References
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[1] Thostenson, E.T., Li, W.Z., Wang, D.Z., Ren, Z.F., Chou, T.W. Carbon nanotube/carbon fiber hybrid multiscale composites. Journal of Applied Physics 2002; 91: 6034–6037. [2] Bekyarova, E., Thostenson, E.T., Yu, A., Kim, H., Gao, J., Tang, J., Hahn, H.T., Chou, T.W., Itkis, M.E., Haddon, R.C. Multiscale carbon nanotube–carbon fiber reinforcement for advanced epoxy
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composites. Langmuir 2007; 23: 3970–3974.
[3] Lau KT, Gu C, Gao GH, Ling HY, Reid SR. Stretching process of single- and multi- walled carbon nanotubes for nanocomposite applications. Carbon 2004;42:426–8.
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[8] Shen H-S. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures 91 (2009), 9–19. [9] H.S. Shen, C.L. Zhang, Thermal Buckling and Postbuckling Behavior of Functionally Graded Carbon Nanotube-Reinforced Composite Plates, Materials and Design 31 (2010), 3403-3411.
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Functionally Graded Cylindrical Panels under Axial Compression using a Meshless Approach, Computer Methods in Applied Mechanics and Engineering 268 (2014), 1-17. [19] C.P. Wu, S.K. Chang, Stability of Carbon Nanotube-Reinforced Composite Plates with SurfaceBonded Piezoelectric Layers and under Bi-axial Compression, Composite Structures, 111 (2014), 587601.
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Table 1: Temperature-dependent material properties for (10, 10) SWCNT. = 9.26 , = List of captions
j 0.68 , ℎ = 0.067 , 5/0 = 0.175, % = 1400 ¹º⁄1 . / , 17
= 0 , = 10.
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Table 2: Effects of the applied constant voltage on the critical buckling load of CNTRC shell.
&_ &
=
Figure 1: Diagrammatic sketch of solving the non-linear critical load and the buckling mode: (a) the non∗ linear critical axial stress = 0.12, &_ &
=
/ ,J 17 7<
&
=
/ , 17
= 4» , = 10 − 5; (b) the non-linear critical lateral
= 5`» , = 13 − 8.
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∗ = 0.12, pressure
&_
∗ Figure 2: Diagrammatic sketch of the load-shortening ratio response curve. = 0.12,
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0 , = 10 − 6.
&_ &
=
/ , 17
=
Figure 3: Postbuckling responses of pre-pressured piezoelectric FG-CNTRC cylindrical shell under ∗ various volume fraction of carbon nanotubes .
&_ &
=
/ , 17
= 0 .
Figure 4: Postbuckling responses of non-pressured and pre-pressured piezoelectric FG-CNTRC ∗ = 0.12, cylindrical shells for different values of
&_ &
=
/ , 17
= 10 − 6.
Figure 5: Postbuckling responses of non-pressured piezoelectric FG-CNTRC cylindrical shell under &_ &
∗
= 0.12, , = 0 .
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different values of
Figure 6: Postbuckling responses of piezoelectric FG-CNTRC cylindrical shell for different types of ∗ CNTRC shell. = 0.12,
&_ &
=
/
17
, = 0, = 10 − 6.
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Figure 7: Postbuckling responses of non-pressured piezoelectric FG-CNTRC cylindrical shell for ∗ different values of = 0.12,
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&
&_ &
=
/
17
, = 0.
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Table 1: Temperature-dependent material properties for (10, 10) SWCNT. = 9.26 , =
500
j 2/0
T»
5.5308
6.9348
1.9643
5.6466 5.4744
7.08
6.8641
1.9445 1.9644
j 3//
× 10]N /
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700
j .00
T»
20
j 300
× 10]N /
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300
j .//
T»
3.4584 4.5361 4.6677
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TVkVÀuÀV
j = 0.175, % = 1400 ¹º⁄1 0.68 , ℎ = 0.067 , 5/0
5.1682 5.0189 4.8943
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Table 2: Effects of the applied constant voltage on the critical buckling load of CNTRC shell. & =
0.12
15.0120
14.9833
FGO
14.8237
14.7947
14.7655
UD
FGX
FGO UD
FGX
15.2117 16.1788 16.4877 15.9318 16.7864 17.3653 16.4675
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FGO
−500
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0.28
UD
FGX
0.17
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v7
0
15.1888
21
500
14.9546 15.1604
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ÁYTÁ − u kV
= 0 , = 10
16.1523 16.4617
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∗
/ , 17
&_
15.9040 16.7613 17.3410 16.4418
16.1258 16.4357 15.8780 16.7361
17. .3166 16.4161
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(a)
(b)
Figure 1: Diagrammatic sketch of solving the non-linear critical load and the buckling mode: (a) the nonlinear critical axial stress Â∗ÃÄ = Å. ÆÇ,
ÈÉ È
=
Æ ,Ë ÊÅ
lateral pressure Â∗ÃÄ = Å. ÆÇ,
ÈÉ È
= ÌÍÎÏ , Ð = ÆÅ − Ñ; (b) the non-linear critical
=
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Æ , ÒÅÓ ÊÅ
= ÑÔÎÏ , Ð = ÆÊ − Õ
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Figure 2: Diagrammatic sketch of the load-shortening ratio response curve. Â∗ÃÄ = Å. ÆÇ,
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Å , Ð = ÆÅ − Ö, ÂÅ = ÑÅÅ
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ÈÉ È
=
Æ
ÊÅ
,Ë =
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Figure 3: Postbuckling responses of non-pressured piezoelectric FGX-CNTRC cylindrical shell under
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various volume fraction of carbon nanotubes Â∗ÃÄ .
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ÈÉ È
=
Æ ,Ë ÊÅ
= Å, ÂÅ = ÑÅÅ
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Figure 4: Postbuckling responses of non-pressured and pre-pressured piezoelectric FGX-CNTRC
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cylindrical shells for different values of Ë Â∗ÃÄ = Å. ÆÇ,
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ÈÉ È
= ÊÅ , Ð = ÆÅ − Ö, ÂÅ = ÑÅÅ Æ
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Figure 5: Postbuckling responses of non-pressured piezoelectric FGX-CNTRC cylindrical shell under
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ÈÉ È
Â∗ÃÄ = Å. ÆÇ, , Ë = Å , ÂÅ = ÑÅÅ
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Figure 6: postbuckling responses of piezoelectric FG-CNTRC cylindrical shell for different types of
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CNTRC shell. Â∗ÃÄ = Å. ÆÇ,
ÈÉ È
=
Æ ,Ë ÊÅ
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= Å, Ð = ÆÅ − Ö, ÂÅ = ÑÅÅ
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Figure 7: Postbuckling responses of non-pressured piezoelectric FGX-CNTRC cylindrical shell for × È
Â∗ÃÄ = Å. ÆÇ,
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different values of
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ÈÉ È
= ÊÅ , Ë = Å, ÂÅ = ÑÅÅ Æ