, Volume 47, Issue 2, pp 423–436 | Cite as

Stresses distributions in a rotating functionally graded piezoelectric hollow cylinder

  • Hong-Liang Dai
  • Ting Dai
  • Hong-Yan Zheng


An analytic solution to the axisymmetric problem of a long, radially polarized, hollow cylinder composed of functionally graded piezoelectric material (FGPM) rotating about its axis at a constant angular velocity is presented. For the case that electric, thermal and mechanical properties of the material obey different power laws in the thickness direction, distributions for radial displacement, stresses and electric potential in the FGPM hollow cylinder are determined by using the theory of electrothermoelasticity. Some useful discussions and numerical examples are presented to show the significant influence of material nonhomogeneity, and adopting suitable graded indexes and applying suitable geometric size and rotating velocity ω may optimize the rotating FGPM hollow cylindrical structures. This will be of particular importance in modern engineering application.


Stress distribution FGPM hollow cylinder Rotating Electrothermoelastic 



radial displacement [m]


radial variable [m]


inner and outer radii of the FGPM hollow cylinder [m]

σi (i=r,θ,z)

components of stresses [N/m2]


temperature distribution [K]


electric potential [W/A]


radial electric displacement [C/m2]

cij (i=1,2;j=1,2,3)

elastic constant [N/m2]

e1i (i=1,2,3)

piezoelectric constants [C/m2]


dielectric constant [C2/N m2]


pyroelectric coefficient [C/m2 K]

αi (i=1,2,3)

thermal stress modulus [Pa/K]


mass density [kg/m3]


constant angular velocity of rotation [rad/s]


thermal conduction coefficient [W/m K]


ratio of the convective heat-transfer coefficients [W/K]

Non-dimensional quantities

\(R = \frac{r - a}{b - a}\),


\(u^{*} = \frac{u}{a}\),


\(T^{*} = \frac{T(r)}{T_{0}}\),


\(\sigma_{i}^{*} = \frac{\sigma _{i}}{P_{a}}\quad (i =r,\theta,z)\),


\(\phi^{*} = \frac{\phi}{\phi _{a}}\)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ueda S (2008) A cracked functionally graded piezoelectric material strip under transient thermal loading. Acta Mech 199:53–70 CrossRefMATHGoogle Scholar
  2. 2.
    Chue CH, Hsu WH (2008) Antiplane internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane. Meccanica 43(3):307–325 CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Zhong Z, Shang ET (2003) Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate. Int J Solids Struct 40:5335–5352 CrossRefMATHGoogle Scholar
  4. 4.
    Wu XH, Shen YP, Chen CQ (2003) An exact solution for functionally graded piezothermoelastic cylindrical shell as sensors or actuators. Mater Lett 57(22–23):3532–3542 CrossRefGoogle Scholar
  5. 5.
    Ootao Y, Tanigawa Y (2005) The transient piezothermoelastic problem of a thick functionally graded thermopiezoelectric strip due to nonuniform heat supply. Arch Appl Mech 74:449–465 CrossRefMATHGoogle Scholar
  6. 6.
    Ootao Y, Tanigawa Y (2007) Transient piezothermoelastic analysis for a functionally graded thermopiezo-electric hollow sphere. Compos Struct 81:540–549 CrossRefGoogle Scholar
  7. 7.
    Dai HL, Fu YM, Yang JH (2007) Electromagnetoelastic solutions for functionally graded piezoelectric solid cylinder and sphere. Acta Mech Sin 23:55–63 CrossRefMATHADSGoogle Scholar
  8. 8.
    Dai HL, Fu YM (2007) Magnetothermoelastic interactions in hollow structures of functionally graded material subjected to mechanical loads. Int J Press Vessels Piping 84:132–138 CrossRefGoogle Scholar
  9. 9.
    Singh BM, Rokne J, Dhaliwal RS (2006) Torsional vibrations of functionally graded finite cylinders. Meccanica 41(4):459–470 CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Galic D, Horgan CO (2003) The stress response of radially polarized rotating piezoelectric cylinders. J Appl Mech 70(3):426–435 CrossRefMATHGoogle Scholar
  11. 11.
    Akis T, Eraslan AN (2006) The stress response and onset of yield of rotating FGM hollow shafts. Acta Mech 187(1–4):169–187 CrossRefMATHGoogle Scholar
  12. 12.
    Argeso H, Eraslan AN (2007) A computational study on functionally graded rotating solid shafts. Int J Comput Methods Eng Sci Mech 8(6):391–399 CrossRefMATHGoogle Scholar
  13. 13.
    Wang HM, Ding HJ (2007) Control of stress response in a rotating infinite hollow multilayered piezoelectric cylinder. Arch Appl Mech 77(1):11–20 CrossRefMATHGoogle Scholar
  14. 14.
    Zenkour AM, Elsibai KA, Mashat DS (2008) Elastic and viscoelastic solutions to rotating functionally graded hollow and solid cylinders. Appl Math Mech 29(12):1601–1616 CrossRefMATHGoogle Scholar
  15. 15.
    Babaei MH, Chen ZT (2008) Analytical solution for the electromechanical behavior of a rotating functionally graded piezoelectric hollow shaft. Arch Appl Mech 78(7):489–500 CrossRefMATHGoogle Scholar
  16. 16.
    Chang SW, Yang TL, Shi DW (2009) Jet-array impingement heat transfer in a concentric annular channel with rotating inner cylinder. Int J Heat Mass Transf 52(5–6):1254–1267 CrossRefGoogle Scholar
  17. 17.
    Bayat M, Sahari BB, Saleem M et al. (2009) Thermoelastic solution of a functionally graded variable thickness rotating disk with bending based on the first-order shear deformation theory. Thin-Walled Struct 47(5):568–582 CrossRefGoogle Scholar
  18. 18.
    Abd-Alla AM, Mahmoud SR (2010) Magnetothermoelastic problem in rotating non-homogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model. Meccanica 45(4):451–462 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Tarn JQ (2001) Exact solutions for functions for functionally graded anisotropic cylinders subjected to thermal and mechanical loads. Int J Solids Struct 38:8189–8206 CrossRefMATHGoogle Scholar
  20. 20.
    Dai HL, Wang X (2006) Dynamic focusing effect of piezoelectric fibers subjected to thermal shock. Arch Appl Mech 75:379–394 CrossRefMATHGoogle Scholar
  21. 21.
    Heyliger P (1996) A note on the static behavior of simply-supported laminated piezoelectric cylinders. Int J Solids Struct 34:3781–3794 CrossRefGoogle Scholar
  22. 22.
    Hosseini SM, Akhlaghi M, Shakeri M (2007) Transient heat conduction in functionally graded thick hollow cylinders by analytical method. Int J Heat Mass Transf 43:669–675 CrossRefGoogle Scholar
  23. 23.
    Rahimi GH, Zamani Nejad M (2008) Exact solutions for thermal stresses in a rotating thick-walled cylinder of functionally graded materials. J Appl Sci 8(18):3267–3272 CrossRefGoogle Scholar
  24. 24.
    Dunn ML, Taya M (1994) Electroelastic field concentrations in and around inhomogeneities in piezo-electric solids. J Appl Mech 61:474–475 CrossRefGoogle Scholar
  25. 25.
    Jabbari M, Sohrabpour S, Eslami MR (2002) Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int J Press Vessels Piping 79:493–497 CrossRefGoogle Scholar
  26. 26.
    Abd-All AM, Abd-alla AN, Zeidan NA (1999) Transient thermal stresses in a rotation non-homogeneous cylindrically orthotropic composite tubes. Appl Math Comput 105:253–269 CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyHunan UniversityChangshaChina
  2. 2.Department of Engineering Mechanics, College of Mechanical & Vehicle EngineeringHunan UniversityChangshaChina
  3. 3.Department of Vehicle Engineering, College of Mechanical & Vehicle EngineeringHunan UniversityChangshaChina

Personalised recommendations