# Elasticity Solution of Functionally Graded Carbon Nanotube Reinforced Composite Cylindrical Panel

Author

Department of Mechanical Engineering, Tarbiat Modares University, Tehran, 14115-143, Iran

Abstract

Based on three-dimensional theory of elasticity, static analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) 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, stress-displacement 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 non-simply 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

### Full Text

Elasticity Solution of Functionally Graded Carbon Nanotube Reinforced Composite Cylindrical Panel

A. Alibeigloo*

Department of Mechanical Engineering, Tarbiat Modares University, Tehran, 14115-143, Iran

1. 1.     Introduction

1. 2.     Basic Equations

2.1 .  FG-CNTRC 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, FG-V, ,  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 

(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, 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  is mid-radius 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 in-plane stresses in terms of state variables are expressed as:

(9)

The relations for simply supported edges boundary conditions are:

at

at                          (10)

1. 3.     Solution Procedure

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 state-space equations for the FG-CNTRC 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 .

Non-dimensional in-plane 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 in-plane stresses can be determined.

1. 4.     Numerical Results and Discussion

In this section a simply-supported FG-CNT 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.2-8. Figs. 2a-2e depict the effect of five cases of CNT distributions, UD, FG-V, ,  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 FG-V 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.3a-3f. 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 in-plane 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. 4a-4c. 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.5a-5g. 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.

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.

1. 5.     Conclusions

Bending behaviour of FG-CNTRC cylindrical panel with simply supported edges and various cases of CNT distribution was examined. The governing differential equations are based on 3-D 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;

• Radial normal, transverse shear stresses and axial displacement in the case of  at a point are always smaller in magnitude than those at the corresponding points in the other two cases of CNT distribution.
• The existence of CNT in cylindrical panel decreases the axial and circumferential displacement components as well as the normal and shear stresses in radial and circumferential directions.
• The effect of CNT volume fraction on the axial displacement in contrast with the circumferential displacement at the outer radius is more significant while it is negligible at the inner radius
• The influence of external load on the axial displacement In contrast with the circumferential displacement at outer radius is more noticeable.
• The effect of CNT volume fraction on the bending behaviour of the thin FG-CNTRC cylindrical panel is more significant than that for the thick panel.

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

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,

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 FG-V CNTRC hybrid beam, with L/h = 50,

b. Circumferential normal stress

c. Axial normal stress

d. Transverse shear stress

e. In-plane 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,

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 FG-V 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 y-directions displacements in the r- , -  and z-directions, respectively shear strains normal strains

Appendix

,

where

References

      Thostenson, ET, Ren,ZF, Chou, TW. Advances in the science and technology of carbon nanotubes and their composite: a review, Compos Sci Thecnol 2001; 61:1899-912.

      Gou J, Minaie B, Wang B, Liang Z, Zhang C.  Computational and experimental study of interfacial bonding of single-walled nanotube reinforced composites stiffness, Comp Mater Sci 2004;31:225–236.

      Wuite J, Adali S. Deflection and stress behavior of nanocomposite reinforced beams using a multi scale analysis, Compos Struct 2005;71:388–396.

      Vodenitcharova T, Zhang C. Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube, Int J Sol Struct 2006; 43:3006–3024.

      Shen HS. Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments, Compos Struct 2009;91:9–19.

      Formica G, Lacarbonara W, Alessi R. Vibrations of carbon nanotube-reinforced composites, Sound Vib 2010;329:1875–1889.

      Shen HS, Zhang C. Thermal buckling and post buckling behavior of functionally grade carbon nanotube-reinforced composite plates, Mater Design 2010;31:3403–3411.

      Ke LL, Yang J, Kitipornchai S. Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams, Compos Struct 2010;92:676-683.

      Shen HS. Post buckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells, Compos Struct  2011; 93: 2096–2108.

   Shen HS. Post buckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part II: Pressure-loaded shells, Compos Struct 2011;93: 2496–2503.

  Wang  ZX, Shen HS. Nonlinear vibration of nanotube-reinforced composite plates in thermal environments, Comp Mater Sci 2011;50: 2319–2330.

  Mehrabadi  SJ, Sobhani Aragh B, Khoshkhahesh V, Taherpour A. Mechanical buckling of nanocomposite rectangular plate reinforced by aligned and strait single-walled carbon nanotubes, Compos Part B-Eng 2012;43(4):2031-2040.

  Zhu P, Lei ZX,  Liew KM. Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory, Compos Struct 2012;94:1450–1460.

  Wang ZX,  Shen HS. Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets, Compos Part B-Eng 2012;43:411–421.

  Yas MH, Heshmati M. Dynamic analysis of functionally graded nanocomposite beam reinforced by randomly oriented carbon nanotube under the action of moving load, Appl Math Model 2012;36:1371–1394.

   Alibeigloo A. Static analysis of functionally graded carbon nanotube-reinforced composite plate embedded in piezoelectric layers by using theory of elasticity, Compos Struct 2013;95:612-622.

   Bhardwaj G, Upadhyay AK, Pandey R, Shukla KK. Non-linear flexural and dynamic response of CNT reinforced laminated composite plate, Compos Part B-Eng 2013;45:89-100.

   Shen HS. Thermal buckling and post buckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells, Compos Part B-Eng 2012; 43:1030-1038.

  Shen HS, Xiang Y. Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments, Comput. Methods Appl. Mech. Eng 2012; 213-216:196–205.

  Moradi-Dastjerdi R, Foroutan M, Pourasgha A.  Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method, Mater Design 2013;44:256–266.

  Moradi-Dastjerdi  R,  Foroutan  A,  Pourasgha A. Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels. Compos Part B-Eng 2012;43:1943–1954.

### References

          Thostenson, ET, Ren,ZF, Chou, TW. Advances in the science and technology of carbon nanotubes and their composite: a review, Compos Sci Thecnol 2001; 61:1899-912.
          Gou J, Minaie B, Wang B, Liang Z, Zhang C.  Com-putational and experimental study of interfacial bonding of single-walled nanotube reinforced composites stiffness, Comp Mater Sci 2004;31:225–236.
          Wuite J, Adali S. Deflection and stress behavior of nanocomposite reinforced beams using a mul-ti scale analysis, Compos Struct 2005;71:388–396.
          Vodenitcharova T, Zhang C. Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube, Int J Sol Struct 2006; 43:3006–3024.
          Shen HS. Nonlinear bending of functionally grad-ed carbon nanotube-reinforced composite plates in thermal environments, Compos Struct 2009;91:9–19.
          Formica G, Lacarbonara W, Alessi R. Vibrations of carbon nanotube-reinforced composites, Sound Vib 2010;329:1875–1889.
          Shen HS, Zhang C. Thermal buckling and post buckling behavior of functionally grade carbon nanotube-reinforced composite plates, Mater De-sign 2010;31:3403–3411.
          Ke LL, Yang J, Kitipornchai S. Nonlinear free vi-bration of functionally graded carbon nanotube-reinforced composite beams, Compos Struct 2010;92:676-683.
          Shen HS. Post buckling of nanotube-reinforced composite cylindrical shells in thermal envi-ronments, Part I: Axially-loaded shells, Compos Struct  2011; 93: 2096–2108.
        Shen HS. Post buckling of nanotube-reinforced composite cylindrical shells in thermal envi-ronments, Part II: Pressure-loaded shells, Com-pos Struct 2011;93: 2496–2503.
        Wang  ZX, Shen HS. Nonlinear vibration of nano-tube-reinforced composite plates in thermal en-vironments, Comp Mater Sci 2011;50: 2319–2330.
        Mehrabadi  SJ, Sobhani Aragh B, Khoshkhahesh V, Taherpour A. Mechanical buckling of nano-composite rectangular plate reinforced by aligned and strait single-walled carbon nano-tubes, Compos Part B-Eng 2012;43(4):2031-2040.
        Zhu P, Lei ZX,  Liew KM. Static and free vibration analyses of carbon nanotube-reinforced compo-site plates using finite element method with first order shear deformation plate theory, Compos Struct 2012;94:1450–1460.
        Wang ZX,  Shen HS. Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets, Compos Part B-Eng 2012;43:411–421.
        Yas MH, Heshmati M. Dynamic analysis of func-tionally graded nanocomposite beam reinforced by randomly oriented carbon nanotube under the action of moving load, Appl Math Model 2012;36:1371–1394.
        Alibeigloo A. Static analysis of functionally grad-ed carbon nanotube-reinforced composite plate embedded in piezoelectric layers by using theo-ry of elasticity, Compos Struct 2013;95:612-622.
        Bhardwaj G, Upadhyay AK, Pandey R, Shukla KK. Non-linear flexural and dynamic response of CNT reinforced laminated composite plate, Compos Part B-Eng 2013;45:89-100.
        Shen HS. Thermal buckling and post buckling behavior of functionally graded carbon nano-tube-reinforced composite cylindrical shells, Compos Part B-Eng 2012; 43:1030-1038.
        Shen HS, Xiang Y. Nonlinear vibration of nano-tube-reinforced composite cylindrical shells in thermal environments, Comput. Methods Appl. Mech. Eng 2012; 213-216:196–205.
        Moradi-Dastjerdi R, Foroutan M, Pourasgha A.  Dynamic analysis of functionally graded nano-composite cylinders reinforced by carbon nano-tube by a mesh-free method, Mater Design 2013;44:256–266.
        Moradi-Dastjerdi  R,  Foroutan  A,  Pourasgha A. Eshelby–Mori–Tanaka approach for vibrational behavior of continuously graded carbon nano-tube-reinforced cylindrical panels. Compos Part B-Eng 2012;43:1943–1954.