Author
Department of Mechanical Engineering, Tarbiat Modares University, Tehran, 14115143, Iran
Abstract
Keywords
Elasticity Solution of Functionally Graded Carbon Nanotube Reinforced Composite Cylindrical Panel
A. Alibeigloo^{*}
Department of Mechanical Engineering, Tarbiat Modares University, Tehran, 14115143, Iran
paper INFO 

ABSTRACT 
Paper history: Received 7 August 2014 Received in revised form 7 September 2014 Accepted 24 September 2014 
Based on threedimensional theory of elasticity, static analysis of functionally graded carbon nanotube reinforced composite (FGCNTRC) cylindrical panel subjected to mechanical uniformed load with simply supported boundary conditions is carried out. In the process, stress and displacement fields are expanded according to the Fourier series along the axial and circumferential coordinates. From constitutive law, stressdisplacement relations and equilibrium equations, state space equation is obtained. The obtained first order governing differential equations can be solved analytically. The effects of CNT distribution cases, the volume fraction of CNT, length to mid radius ratio, span of the cylindrical panel, variation of mechanical load and radius to thickness ratio on the bending behaviour of the cylindrical panel are examined. It should be noted that by using Fourier series solution it is possible only to solve the static behaviour of cylindrical panel with simply supported for all of edges and for the nonsimply supported boundary conditions it is possible to solve numerically. The obtained analytical solution can be used to validate the results of approximate two dimensional conventional theories. 



Keywords: Carbon nanotube Cylindrical panel Boundary layer Static Elasticity



© 2014 Published by Semnan University Press. All rights reserved. 
The high strength and stiffness of carbon nanotubes cause to use them as reinforcing constituents instead of conventional fibers in composite structures such as beam, plate and shell. The introduction of CNT into polymer matrix increases the application of reinforcing composite. Study on the mechanical and thermal properties of CNTRC structures has increased by many researchers in recent years. Thostenson et al. [1] presented a review on the researches and application of CNT and CNTRC. Gou et al. [2] used the molecular dynamics (MD) simulations and experimental method to investigate the interfacial bonding of singlewalled nanotube (SWNT) reinforced epoxy composites. Wuite and Adali [3] carried out a multi scale analysis of the deflection and stress behaviour of carbon nanotube (CNT) reinforced polymer composite beams. Vodenitcharova and Zhang [4] investigated pure bending and bendinginduced local buckling of a nanocomposite beam reinforced by a SWNT computationally as well as experimentally using Airy stressfunction approach. Shen [5] discussed nonlinear bending behaviour of simply supported and functionally graded composite plates reinforced by SWCNTs subjected to transverse uniformed or sinusoidal load in thermal environments. Formica et al. [6] used an equivalent continuum model and Eshelby–Mori–Tanaka approach to study the vibrational behaviour of CNTRC. By using multiscale approach, Shen and Zhang [7] discussed thermal buckling and post buckling behaviour of functionally graded nanocomposite plates reinforced by SWCNTs subjected to inplane temperature variation. Based on Timoshenko beam theory and von Karman geometric nonlinearity, Ke et al. [8] discussed nonlinear free vibration of FG nanocomposite beams reinforced by SWCNTs by using Ritz method. Shen [9] used higher order shear deformation theory as well as a von Kármántype of kinematic nonlinearity to investigate the post buckling behaviour of nanocomposite cylindrical shells reinforced by SWCNTs and subjected to axial environments. Based on a micromechanical model and multiscale approach, Shen [10] discussed post buckling behaviour of FGCNTRC cylindrical shells subjected to mechanical load in thermal environments. Based on a higher order shear deformation plate theory, Wanga and Shen [11] investigated the nonlinear vibration of FGSWCNT plates rested on elastic foundation in thermal environments by using an improved perturbation technique. Mehrabadi et al. [12] discussed mechanical buckling behaviour of FG nanocomposite plate reinforced by SWCNTs by using Mindlin plate theory based on firstorder shear deformation (FSDT). Zhu et al. [13] carried out bending and free vibration analysis of composite plates reinforced by SWCNTs by using the finite element method based on the first order shear deformation plate theory. Wang and Shen [14] investigated nonlinear bending and vibration behaviour of sandwich plate with CNTRC face sheets by using multiscale approach and twostep perturbation technique. Yas and Heshmati [15] used Timoshenko beam theory to analysis the vibration of FG nanocomposite beams reinforced by randomly oriented straight SWCNTs subjected to moving load. Recently the author [16] presented an analytical solution for bending behaviour of FGCNT composite plate integrated with piezoelectric actuator and sensor under an applied electric field and mechanical load. Bhardwaj et al. [17] investigated the nonlinear static and dynamic behaviour of crossply CNTRC laminated plate by using the double Chebyshev series. By using higher order shear deformation theory and Von Karman type of kinematic nonlinearity, Shen [18] discussed post buckling of FGCNTRC cylindrical shell in thermal environment. Nonlinear vibration of FGCNTRC cylindrical shell was investigated by Shen and Xiang [19] using the equation of the motion based on higherorder shear deformation theory with a Von Karmantype of kinematic nonlinearity. MoradiDastjerdi et al. [20] analysed the dynamic behaviour of FGCNTRC cylindrical shell subjected to impact load by making the use of mesh free method. By using Eshelby–Mori–Tanaka approach and twodimensional differential quadrature method, free vibration analysis of CNTRC cylindrical panel was presented by Sobhani Aragh [21]. According to the above mentioned research it is seen that static analysis of FGCNTRC cylindrical panel has not been yet considered. In this paper, based on the theory of elasticity, bending behaviour of FGCNTRC cylindrical panel subjected to uniformed internal pressure is studied.
2.1 . FGCNTRC Layer
A CNTRC cylindrical panel with geometry and dimensions according to the Fig.1 is considered. The SWCNT reinforcement is either uniformly distributed (UD) or functionally graded (FG) in four cases, FGV, , and in the thickness direction. Displacements component along the r, and z directions are denoted by , and , respectively. According to the rule of mixture and considering the CNT efficiency parameters, the effective mechanical properties of mixture of CNTs and isotropic polymer matrix can be written as the following [6]
(1.1)
(1.2)
(1.3)
The relation between the CNT and matrix volume fractions is stated as:
(2)
The volume fraction of CNT for five cases UD, FGV, , and distribution along the thickness according to the Fig.1 has the following relations, respectively:
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
where is midradius of the panel and
(3.6)
The Poisson’s ratio, and the density of the nanocomposite panel is assumed as:
(4.1)
(4.2)
And the other effective mechanical properties of mixture of CNTs and isotropic polymer matrix are:
, , ,
,
The constitutive equations for CNTRC panel layer are written as:
(5)
where
And the relation between the stiffness elements, and engineering constants , and are described in the appendix. In the absence of body forces, the governing equilibrium equations in three dimensions are:
(6)
The linear relations between the strain and displacements are:
,
(7)
By using Eqs. (5)(7), the following state space equations can be derived:
(8)
where is the state variable vector, and is the coefficients matrix (see Appendix).
The inplane stresses in terms of state variables are expressed as:
(9)
The relations for simply supported edges boundary conditions are:
at
at (10)
In order to satisfy the simply supported boundary conditions, Eq. (10), displacement and stress components are assumed as:
(11)
Where , , n and m are the wave numbers along the axial and circumferential directions, respectively.
For convenience, the dimensionless physical quantities are defined by:
, , ,
(12)
By using Eqs. (8), (11) and (12), the following statespace equations for the FGCNTRC layer is derived
(13)
where and is defined in the appendix.
The general solution for Eq. (13) can be explicitly expressed as:
(14)
where is at
Eq. (14) at yields:
(15)
where .
Nondimensional inplane stresses can be derived from Eqs. (9), (11) and (12) as:
(16)
Inner and outer Surfaces boundary conditions are assumed as:
at
at (17)
Applying surfaces tractions, Eqs. 17 to the Eq. 15, displacement components at the inner surface of the panel are obtained:
(18)
where is the element of matrix M .
By using Eqs. (18), (17.1) and (14) state variables in three dimensions can be derived. Finally by substituting the obtained state variables into the induced variable, Eq. (16) the inplane stresses can be determined.
In this section a simplysupported FGCNT cylindrical panel with the following material properties for the CNT and polymer matrix is considered to illustrate the foregoing analysis.
, , , , , , , , ,
To show the effect of CNT on the bending behaviour of nanocomposite, numerical illustration is made. The effect of CNT volume fraction on the stress and displacement field at mid radius of cylindrical panel for various spans and lengths to mid radius ratio is depicted in table1. According to the table CNT inclusion in the cylindrical panel affects the mechanical entities in smaller spans as well as the smaller lengths to mid radius ratio. For further discussion, numerical investigations were carried out and presented in Figs.28. Figs. 2a2e depict the effect of five cases of CNT distributions, UD, FGV, , and on the stress and displacement field for the CNTRC cylindrical panel. According to the Figures, the transverse normal and shear stresses, , and axial displacement, transverse displacement, has minimum value in case and maximum value in FGV case where as circumferential displacement, is not affected by the case of CNT distribution. Also it is observed that the effect of case of CNT distribution on the transverse shear stress, is insignificant. Henceforth, all of the numerical results are presented for the case. The effect of CNT volume fraction on stress and displacement fields is presented in Figs.3a3f. From the figures it is observed that the effect of increasing the CNT volume fraction on the transverse normal and shear stresses is not significant in comparison with the inplane stresses. Also, from Fig.3b, it is seen that the circumferential stress decreases linearly with nearly constant value at the inner radius when the CNT volume fraction increases.
According to Fig.3c increase in the CNT volume fraction leads to increase in the axial stress nonlinearly with remaining almost constant at the outer radius. As the Figs. 3e and 3f depict, increase in the CNT volume fraction leads to decrease in the circumferential and axial displacements. Moreover, it can be observed that this effect in the circumferential displacement is more significant near the inner region whereas this effect for the axial displacement is noticeable near the outer surface.
Through the thickness distribution of stress and displacement components for various spans of the panel are depicted in Figs. 4a4c. From the figures it is seen that increasing the span of the panel leads to change gradually the slope of the stresses and displacement distribution curves. Furthermore it can be observed that the span effect for the radial normal and transverse shear stresses at mid radius and for the axial displacement at the outer radius is more significant. The influence of internal uniformed pressure on the mechanical behaviour of nanocomposite cylindrical panel is presented in Figs.5a5g. From the figures it is seen that the stresses and displacement increase when the applied load increases.
From the Figs.5b and 5c, it is revealed that the neutral axis along the axial and circumferential direction does not coincide with the mid surface of the cylindrical panel. Moreover it is seen that the effect of mechanical load on the axial normal stress (Fig.5b) at the outer surface is more significant while it is almost negligible at the inner surface.
As the Fig. 5c shows the influence of applied mechanical load on the circumferential stress at the inner surface is greater than that at the outer surface. Transverse shear stress, in FGCNTRC cylindrical panel, in contrast with the isotropic cylindrical panel is not axisymmetry with respect to the mid radius (Fig.4d). Fig.5e depicts that the effect of external applied load on the inplane shear stress at the outer surface is greater than that at the inner surface, also it is seen that this shear stress at the mid surface, without depending on the external mechanical load, is always zero. As the Figs. 5f and 5g show, increasing the external load leads to increase in the slope through the thickness distribution of axial and circumferential displacement. Also it can be concluded that the effect of external load in axial displacement at the outer radius is greater than that at the inner radius and it is converse for the circumferential displacement. Figs.6a and 6b show the influence of the CNT volume fraction on the radial and axial stresses at mid radius of thin and thick FGCNTRC panels, respectively.
From these figures it is evident that the effect of increasing the CNT volume fraction on both radial and axial stresses in the thin panel is more considerable than that for the thick panel. Radial stress distribution along the thickness direction for the thin and thick cylindrical panels with and without containing the CNT is presented in Fig. 7.
Table1. Effect of CNT volume fraction on the stress and displacement field at mid radius of cylindrical panel with various span and length to mid radius ratio and S = 10,
0.11 
2 
0.390 
2.285 
10.091 
0.874 
0.986 
1.328 
2.418 

3 
0.604 
1.818 
5.320 
0.850 
1.008 
1.344 
6.621 

4 
0.666 
0.839 
1.920 
0.730 
1.023 
1.369 
8.444 

2 
0.389 
2.890 
25.461 
0.650 
1.251 
5.442 
1.448 

3 
1.106 
6.873 
27.107 
0.735 
1.299 
5.519 
10.166 

4 
1.534 
6.147 
18.879 
0.949 
1.316 
5.561 
21.771 

2 
0.381 
5.393 
31.995 
0.117 
1.571 
34.718 
1.878 

3 
0.002 
5.288 
56.063 
0.062 
1.921 
42.435 
3.567 

4 
0.843 
15.795 
86.937 
0.098 
2.012 
43.794 
17.381 

0.14

2 
0.381 
3.046 
11.545 
0.925 
0.994 
1.292 
2.319 

3 
0.602 
2.250 
6.050 
0.910 
1.017 
1.309 
6.614 

4 
0.670 
1.007 
2.092 
0.810 
1.031 
1.335 
8.446 

2 
0.373 
4.618 
28.649 
0.691 
1.256 
5.299 
1.081 

3 
1.095 
8.901 
30.869 
0.782 
1.310 
5.369 
9.785 

4 
1.534 
7.591 
21.370 
1.030 
1.328 
5.415 
21.595 

2 
0.382 
4.475 
22.977 
0.166 
1.559 
33.721 
2.500 

3 
0.031 
9.050 
63.155 
0.071 
1.926 
41.301 
2.462 

4 
0.808 
22.088 
99.166 
0.091 
2.027 
42.619 
16.200 


2 
0.376 
3.779 
12.951 
0.962 
1.002 
1.253 
2.274 

3 
0.603 
2.640 
6.683 
0.982 
1.026 
1.274 
6.679 

0.17 

4 
0.674 
1.136 
2.182 
0.876 
1.041 
1.300 
8.479 
2 
0.365 
6.300 
31.891 
0.727 
1.264 
5.152 
0.782 

3 
1.093 
10.836 
34.472 
0.808 
1.322 
5.214 
9.583 

4 
1.542 
8.911 
23.587 
1.088 
1.341 
5.266 
21.680 

2 
0.382 
3.736 
23.846 
0.231 
1.556 
32.833 
3.058 

3 
0.053 
12.731 
70.488 
0.086 
1.937 
40.173 
1.500 

4 
0.799 
28.307 
111.647 
0.065 
2.045 
41.409 
15.344 
a. UD
b. FG V
c.
d.
e.
Figure 1. Geometry of CNTRC
The existence of CNT in both thin and thick composite cylindrical panels leads to decrease in the radial stress; in addition, it is seen that the effect of CNT in the thin panel along the radial direction is more important than that for the thick panel.
Bending behaviour of FGCNTRC cylindrical panel with simply supported edges and various cases of CNT distribution was examined. The governing differential equations are based on 3D theory of elasticity. The analysis was carried out by using the Fourier series expansion along the longitudinal and circumferential directions and state space technique in the radial direction. The accuracy of the conventional two dimensional theories can be validated by this closed form solution.
From numerical illustrations the following conclusions are derived;
a. Radial normal stress
b. Transverse shear stress
c. Transverse shear stress
d. Circumferential displacement
e. Axial displacement
Figure 2. Distribution of mechanical entities along the thickness for various cases of CNT distribution for the cylindrical panel with =0.17, m = n=25
a. Radial normal stress
b. Circumferential normal stress
c. Axial normal stress
d. Transverse shear stress
e. Circumferential displacement
f. Axial displacement
Figure 3. Effect of CNT volume fraction on the through the thickness stresses and displacements for the CNTRC cylindrical panel , with L/h = 50,
a. Radial normal stress
b. Transverse shear stress
c. Axial displacement
Figure 4. Effect of span of panel on the through the thickness distribution of stresses and displacements for the FGV CNTRC hybrid beam, with L/h = 50,
a. Radial normal stress
b. Circumferential normal stress
c. Axial normal stress
d. Transverse shear stress
e. Inplane shear stress
f. Axial displacement
g. Circumferential displacement
Figure 5. Effect of internal pressure on the through the thickness distribution of stresses, displacements for the FG CNTRC hybrid beam with L/h = 50,
a. Radial normal stress
b. Axial normal stress
Figure 6. Effect of increasing the CNT volume fraction on radial and axial normal stresses at mid surface of the thick and thin panels for the FGV CNTRC
Figure 7. Distribution of radial stress for the thick and thin panels with and without the CNT
Nomenclature
, , , , 
Young’s modulus, shear modulus, shear modulus of carbon nanotube and matrix respectively

, 
volume fractions of carbon nanotube and matrix, respectively 
h , L 
thickness and axial length of the panel 
, 
mass fraction and density fraction of CNT, respectively 


density fraction of matrix 

normal stresses 

shear stresses 

efficiency parameter of CNT 

state variables 

half wave numbers in the x and ydirections 

displacements in the r ,  and zdirections, respectively 

shear strains 

normal strains 
Appendix
,
where
References
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