Paper Title (Use Style: Paper Title) s34

A. Alibeigloo, M. Shaban

Elasticity solution of functionally graded carbon nanotube-reinforced composite cylindrical panel

A. Alibeigloo, M. Shaban

Department of mechanical engineering, Faculty of Engineering, Tarbiat modares University, Tehran, Iran

Email: ,

Received **** 2011

Abstract

Based on three-dimensional theory of elasticity, static analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical panel subjected to mechanical uniform load with simply supported boundary conditions is carried out. By using Fourier series expansion along the axial and circumferential directions and state space technique across the radial direction, closed form solution is derived. The effects of CNT distribution cases as well as the volume fraction of CNT and span angle of the panel on the bending behavior of the cylindrical panel are examined.

Keywords: Carbon nanotube ; cylindrical panel; Static ; Elasticity; Analytical

A. Alibeigloo, M. Shaban

1. Introduction

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. Shen [2] used higher order shear deformation theory as well as a von Kármán-type of kinematic nonlinearity to investigate the postbuckling behavior of nanocomposite cylindrical shells reinforced by SWCNTs and subjected to axial environments. Nonlinear vibration of FG-CNTRC cylindrical shell was investigated by Shen and Xiang [3 using the equation of the motion base on higher-order shear deformation theory with aVon Karman-type of kinematic nonlinearity. Moradi-Dastjerdi et al. [4] analyzed the dynamic behavior of FG-CNTRC cylindrical shell subjected to impact load by making the use of mesh free method. Recently the author [5] presented an analytical solution for bending behavior of FG-CNT composite plate integrated with piezoelectric actuator and sensor under an applied electric field and mechanical load. In the present work bending behavior of FG-CNTRC cylindrical panel subjected to uniform internal pressure is investigated.

2. Basic equations

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, FG-V, FG-Λ, FG-X and FG-O in the thickness direction. Displacements component along the r, θ and z directions are denoted by ur, uθ and uz, respectively. According to the rule of mixture and considering the CNT efficiency parameters, the effective mechanical properties of mixture of CNTs and matrix isotropic polymer can be written as the follow

/ (1.1)
/ (1.2)
/ (1.3)

Relation between the CNT and matrix volume fractions is stated as

(2)

The volume fraction of CNT for five cases UD, FG-V,, 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

, (3.6)

where R is mid-radius of the panel . The Poisson’s ratio,ν12 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 matrix isotropic polymer are

The constitutive equations for CNTRC panel layer are written as

/ (5)

where

and the relation between the stiffness elements, Qij and engineering constants, Eij, Gij and νij are described in 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)

3. Solution procedure

Following displacement and stress components satisfy simply supported boundary condition

/ (8)

By using Eqs. (4)-(8), the following state-space equations for the FG-CNTRC layer is derived

, (9)

where.

4. Numerical results and discussion

In this section a simply-supported FG-CNT cylindrical panel with the following material properties for the CNT and matrix polymer is considered to illustrate the foregoing analysis

, , , , , , , , , ,

To show the effect of CNT in the bending behavior of nanocomposite, numerical illustration is made. Numerical investigation were carried out and presented in Table 1 and Figs.2-3. According to the table, by increasing the span angle of the panel, radial stress decreases whereas the axial stress and transverse shear stress increase. From this figure also it can be concluded that when the CNT volume fraction increase, axial normal and transverse shear stresses increase and radial stress as well as circumferential displacement decrease. Figs. 2a and 2b depict the effect of five cases of CNT distribution, UD, FG-V, FG-Λ, FG-X and FG-O on the stress and displacement field for the CNTRC cylindrical panel. According to the Figures, the transverse normal, and axial displacement, has minimum value in FG-Λ case and maximum value in FG-V case. Effect of CNT volume fraction on stress and displacement fields is presented in Figs.3a and 3b. According to Fig.3a increase the CNT volume fraction causes to increase the axial stress nonlinearly with remaining almost constant at the outer radius. As the Fig. 3b depict, increase the CNT volume fraction causes to decrease axial displacements.

Table 1. Effect of CNT volume fraction on the stress and displacement field at mid radius of cylindrical panel with various span and S = 10, .

0.11 / / -0.390 / 10.091 / -0.986 / -1.328
/ -0.389 / 25.461 / -1.251 / -5.442
/ 0.381 / 31.995 / -1.571 / -34.718
0.14 / / -0.381 / 11.545 / -0.994 / -1.292
/ -0.373 / 28.649 / -1.256 / -5.299
/ 0.382 / 22.977 / -1.559 / -33.721
0.17 / / -0.376 / 12.951 / -1.002 / -1.253
/ -0.365 / 31.891 / -1.264 / -5.152
/ 0.382 / 23.846 / -1.556 / -32.833

(a) UD

(b) FG- Λ

(d) FG-X

(e) FG-O

Fig.1. Geometry of CNTRC.

(a) Radial normal stress

(b) Axial displacement

Fig. 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) Axial normal stress

(b) Axial displacement

Fig. 3. Effect of CNT volume fraction on through the thickness stresses and displacements for the CNTRC cylindrical panel , with L/h = 50, .

5. Conclusion

Bending behavior of FG-CNTRC cylindrical panel with simply supported edges and various cases of CNT distribution was examined. From numerical illustration the following conclusion are derived;

· Radial normal and axial displacement in the case of FG-Λ, at a point is always smaller in magnitude than those at the corresponding points in the other two cases of CNT distribution.

· Existence of CNT in cylindrical panel decreases the axial displacement component and increases the axial stress.

6. Appendix

where

7. References

[1] ET Thostenson, ZF Ren, TWChou. “Advances in the science and technology of carbon nanotubes and their composite: a review”. Composite Science Thecnology, Vol. 61, pp.1899-912, 2001.

[2] HS Shen. “Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells”, Composite Structures. Vol 93, pp.2096–2108, 2011.

[3] HS Shen, Y Xiang. “Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal invironments” . Computer Methods in Applied Mechanic Engineering, Vol. 213–216 pp. 196–205, 2012.

[4] R Moradi-Dastjerdi, M Foroutan, “A Pourasgha. Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method”. Materials and Design Vol. 44, pp. 256–266, 2013.

[5] Alibeigloo A. “Static analysis of functionally graded carbon nanotube-reinforced composite plate embedded in piezoelectric layers by using theory of elasticity”. Composite Structures Vol. 95, pp. 612-622, 2013.