Thermo-elastic behavior of a thick-walled composite cylinder reinforced with functionally graded SWCNTs

Document Type : Research Paper


1 Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, I. R. Iran , Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, I.R.Iran.

2 Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, I. R. Iran

3 Department of Chemical Engineering, Faculty of Engineering, University of Kashan, Kashan, I. R. Iran.



In this article, thermo-elastic-behavior of a thick-walled cylinder made from a polystyrene nanocomposite reinforced with functionally graded (FG) single-walled carbon nanotubes (SWCNTs) was carried out in radial direction while subjected to a steady state thermal field. The SWCNTs were assumed aligned, straight with infinite length and a uniform layout. Two types of variations in the volume fraction of SWCNTs were considered in the structure of the FG cylinder along the radius from inner to outer surface, namely: incrementally increasing (Inc Inc) and incrementally decreasing (Inc Dec). These are compared with uniformly distributed (UD) structure. Mori-Tanaka method was used for stress-strain analysis. Using equations of motion, stress-strain and their corresponding constitutive correlations of a polystyrene vessel, a second order ordinary differential equation was proposed based on the radial displacement which was solved in order to obtain the distribution of displacement and radial, circumferential and axial stresses. For constant temperatures at the inner and outer surfaces of the FG cylinder considered here, results in this work indicate that radial and circumferential stresses and displacement are lower for the Inc Inc FG cylinder, and the axial stresses are higher irrespective of the structure of the FG material.


[1] Saito R, DresselhausG, Dresselhaus M S. Physical Properties of Carbon Nanotubes. Imperial College Press: London; 1998.
[2] Qian D, Wagner G J, Liu W K, Yu M F and Ruoff R S. Mechanics of Carbon Nanotubes. Appl Mech Rev 2002; 55(6): 495–533.
[3] Ajayan PM, Stephan O, Colliex C, Trauth D. Aligned carbon nanotube arrays formed by cutting a polymer resin—nanotube composite. Science 1994; 256: 1212–4.
[4] LourieO, Cox D M, Wagner H D. Buckling and Collapse of Embedded Carbon Nanotube.Phys Rev Lett 1998; 81(8): 1638–41.
[5] Haggenmueller R, Gommans H H, Rinzler A G, Fischer J E, Winey K I. AlignedSingle-Wall Carbon Nanotubes In Composites by Melt Processing Methods. Chem Phys Lett 2000; 330: 219–25.
[6] Fidelus J D, Wiesel E, Gojny F H, Schulte K,Wagner H D. Thermo-mechanical properties of randomly oriented   Carbon/epoxy nanocomposites. Compos Part A: Appl Sci Manufact 2005; 36:1555–61.
[7] Bonnet P, Sireude D, Garnier B, Chauvet O. Thermal  properties and percolation in carbon nanotube–polymer composites. J Appl Phys 2007; 91: 201910.
[8] Qian D, Dickey E C, AndrewsR, Rantell T. Load Transferand  and Deformation Mechanisms in Carbon Nanotube-Polystyrene Composites. Appl Phys Lett 2000; 76: 2868–70.
[9] Odegard GM, Gates T S, Wise K E, Park C, Siochi E J. Constitutive Modeling of Nanotube-Reinforced Polymer Composites. Compos Sci Technol 2002; 63(11): 1671–87.
[10] Wuite J, Adali S. Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis. Comps Struct 2005; 71:388–96.
[11] Vodenitcharova T, Zhang L C. Bending and local buckling of a nanocomposite beam reinforced by a single-walled carbon nanotube. Int J Solids Struct 2006; 43:3006–24.
[12] Han Y, Elliott J. Molecular dynamics simulations of the elastic properties of polymer/ carbon nanotube composites. Comput Mater Sci 2007; 39:315–23.
[13] Zhu R, Pan E, Roy AK. Molecular dynamics study of the stress–strain behavior of carbon-nanotube reinforced Epon 862 composites. Mater Sci Eng A 2007; 447: 51–7.
[14] Shen HS. Nonlinear bending of functionally graded carbon nanotubereinforced composite plates in thermal environments. Compos Struct 2009; 91:9–19.
[15] Ke L. L., Yang J., Kitipornchai S. Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos Struct 2010; 92(3): 676-83.
[16] Wang X. Thermal shock in a hollow cylinder caused by rapid arbitrary heating. J Sound Vib 1995; 183: 899–906.
[17] Cho H, Kardomateas G A, Valle C S. Elastodynamic solution for the thermal shock stresses in an orthotropic thick cylindrical shell. J Appl Mech 1998; 65: 184–192.
[18] Ding H J, Wang H M, Chen W Q. A theoretical solution of cylindrically isotropic cylindrical tube for axisymmetric plane strain dynamic thermoelastic problem. Acta Mech Solida Sinica 2001; 14: 357–63.
[19] Pelletier J L, Vel S S. An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells. Int J Solids Struct 2006; 43: 1131–58.
[20] Horgan C O, Chan A M. The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. J Elasticity 1999; 55: 43–59.
[21] Tarn J Q. Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads. Int J Solids Struct 2001; 38: 8189–206.
[22] Abd-Alla A M, Farhan A M. Effect of the non-homogenity on the composite infinite cylinder of orthotropic material. Phys Lett A 2008; 372: 756–60.
[23] Shi D L, Feng X Q, Huang Y Y, Hwang K C, Gao H. The effect of nanotube waviness and agglomeration on the elastic property of carbon nanotube –reinforced composites. J Eng Mater Technol 2004; 126: 250-7.
[24]Hill R. A Self- Consistent Mechanics of Composite Materials J Mech Phys Solids 1965; 13: 213–22.
[25] Popov VN, Van Doren V E, Balkanski M. Elastic Properties of Crystals of Single-Walled Carbon Nanotubes. Solid State Commun 2000; 114: 395–9.
[26] Mark J E. Polymer data handbook, Oxford University Press, New York. Oxford: 828-37, 1999.
[27] Hetnarski R B, Eslami M R. Thermal stresses advanced theory and application. Solid Mechanics; 2008.