Journal of Elasticity

, Volume 96, Issue 1, pp 81–95 | Cite as

A Pressurized Functionally Graded Hollow Cylinder with Arbitrarily Varying Material Properties

  • Xian-Fang Li
  • Xu-Long Peng


The elastic analysis of a pressurized functionally graded material (FGM) annulus or tube is made in this paper. Different from existing studies, this study deals with an axisymmetrical FGM hollow cylinder or disk with arbitrarily varying material properties. A simple and efficient approach is suggested, which reduces the associated problem to solving a Fredholm integral equation. The resulting equation is approximately solved by expanding the solution as series of Legendre polynomials. The stresses and displacements can be represented in terms of the solution to the equation. For radius-dependent Young’s modulus, numerical results of the distribution of the radial and circumferential stresses are presented graphically. Our results indicate that change in the gradient of the FGM tube does not produce a substantial variation of the radial stress, but strongly affects the distribution of the hoop stress. In particular, the hoop stress may reach its maximum at an internal position or at the outer surface when the tube is internally pressurized. The results obtained are helpful in designing FGM cylindrical vessels to prevent failure.


Functionally graded materials Pressurized cylindrical tube Arbitrarily varying gradients 

Mathematics Subject Classification (2000)

74E05 74B05 45B05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970) MATHGoogle Scholar
  2. 2.
    Shi, Z.F., Zhang, T.T., Xiang, H.J.: Exact solutions of heterogeneous elastic hollow cylinders. Compos. Struct. 79, 140–147 (2007) CrossRefGoogle Scholar
  3. 3.
    Lutz, M.P., Zimmerman, R.W.: Effect of the interphase zone on the bulk modulus of a particular composite. ASME J. Appl. Mech. 63, 855–861 (1996) MATHCrossRefGoogle Scholar
  4. 4.
    Tanaka, K., Watanabe, H., Sugano, Y., Poterasu, V.F.: A multicriterial material tailoring of a hollow cylinder in functionally gradient materials: Scheme to global reduction of thermoelastic stresses. Comput. Methods Appl. Mech. Eng. 135, 369–380 (1996) MATHCrossRefGoogle Scholar
  5. 5.
    Horgan, C.O., Chan, A.M.: The stress response of functionally graded isotropic linearly elastic rotating disks. J. Elasticity 55, 219–230 (1999) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Tarn, J.Q.: Exact solutions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads. Int. J. Solids Struct. 38, 8189–8206 (2001) MATHCrossRefGoogle Scholar
  7. 7.
    Jaabbari, M., Sohrabpour, S., Elsam, M.R.: Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int. J. Press. Vessels Piping 79, 493–497 (2002) CrossRefGoogle Scholar
  8. 8.
    Kim, K.S., Noda, N.: Green’s function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material. Acta Mech. 156, 145–161 (2002) MATHCrossRefGoogle Scholar
  9. 9.
    Oral, A., Anlas, G.: Effects of radially varying moduli on stress distribution of nonhomogeneous anisotropic cylindrical bodies. Int. J. Solids Struct. 42, 5568–5588 (2005) MATHCrossRefGoogle Scholar
  10. 10.
    Batra, R.C., Iaccarino, G.L.: Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder. Int. J. Non-Linear Mech. 43, 383–398 (2008) CrossRefADSGoogle Scholar
  11. 11.
    Ting, T.C.T.: Pressuring, shearing, torsion and extension of a circular tube or bar of cylindrically anisotropic material. Proc. R. Soc. Lond. A 452, 2397–2421 (1996) MATHCrossRefADSGoogle Scholar
  12. 12.
    Ting, T.C.T.: New solutions to pressuring, shearing, torsion and extension of a circular tube or bar. Proc. R. Soc. Lond. A 455, 3527–3542 (1999) MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Alshits, V.I., Kirchner, H.O.K.: Cylindrically anisotropic, radially inhomogeneous elastic materials. Proc. R. Soc. Lond. A 457, 671–693 (2001) MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Tarn, J.Q., Chang, H.H.: Torsion of cylindrical orthotropic elastic circular bars with radial inhomogeneity: some exact solutions and end effects. Int. J. Solids Struct. 45, 303–319 (2008) MATHGoogle Scholar
  15. 15.
    Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Body. Mir, Moscow (1981) MATHGoogle Scholar
  16. 16.
    Horgan, C.O., Chan, A.M.: The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. J. Elasticity 55, 43–59 (1999) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Yang, Y.Y.: Time-dependent stress analysis in functionally graded materials. Int. J. Solids Struct. 37, 7593–7608 (2000) MATHCrossRefGoogle Scholar
  18. 18.
    Theotokoglou, E.E., Stampouloglou, I.H.: The radially nonhomogeneous elastic axisymmetric problem. Int. J. Solids Struct. 45, 6535–6552 (2008) CrossRefMATHGoogle Scholar
  19. 19.
    Tutuncu, N., Ozturk, M.: Exact solutions for stresses in functionally graded pressure vessels. Compos., Part B Eng. 32, 683–686 (2001) CrossRefGoogle Scholar
  20. 20.
    Xiang, H.J., Shi, Z.F., Zhang, T.T.: Elastic analyses of heterogeneous hollow cylinders. Mech. Res. Commun. 33, 681–691 (2006) CrossRefGoogle Scholar
  21. 21.
    Tutuncu, N.: Stresses in thick-walled FGM cylinders with exponentially-varying properties. Eng. Struct. 29, 2032–2035 (2007) CrossRefGoogle Scholar
  22. 22.
    Zhang, X., Hasebe, N.: Elasticity solution for a radially nonhomogeneous hollow circular cylinder. J. Appl. Mech. ASME 66, 598–606 (1999) MathSciNetGoogle Scholar
  23. 23.
    Dryden, J., Jayaraman, K.: Effect of inhomogeneity on the stress in pipes. J. Elasticity 83, 179–189 (2006) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Baker, C.T.H.: Expansion methods. In: Delves, L.M., Walsh, J. (eds.) Numerical Solution of Integral Equations, pp. 80–96. Clarendon, Oxford (1974) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.IMST, School of Civil Engineering and ArchitectureCentral South UniversityChangshaChina

Personalised recommendations