Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Expansion/contraction of a spherical elastic/plastic shell revisited

  • 208 Accesses


A semi-analytic solution for the elastic/plastic distribution of stress and strain in a spherical shell subject to pressure over its inner and outer radii and subsequent unloading is presented. The Bauschinger effect is taken into account. The flow theory of plasticity is adopted in conjunction with quite an arbitrary yield criterion and its associated flow rule. The yield stress is an arbitrary function of the equivalent strain. It is shown that the boundary value problem is significantly simplified if the equivalent strain is used as an independent variable instead of the radial coordinate. In particular, numerical methods are only necessary to evaluate ordinary integrals and solve simple transcendental equations. An illustrative example is provided to demonstrate the distribution of residual stresses and strains.

This is a preview of subscription content, log in to check access.


  1. 1

    Hill R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)

  2. 2

    Mendelson A.: Plasticity: Theory and Application. The Macmillan Company, New York (1986)

  3. 3

    Chakrabarty J.: Theory of Plasticity. McGraw-Hill, New York (1987)

  4. 4

    Durban D., Baruch M.: Behaviour of an incrementally elastic thick walled sphere under internal and external pressure. Int. J. Non-linear Mech. 9, 105–119 (1974)

  5. 5

    Cowper G.R.: The elastoplastic thick-walled sphere subjected to a radial temperature gradient. ASME J. Appl. Mech. 27, 496–500 (1960)

  6. 6

    Johnson W., Mellor P.B.: Elastic-plastic behaviour of thick-walled spheres of non-work-hardening material subject to a steady state radial temperature gradient. Int. J. Mech. Sci. 4, 147–158 (1962)

  7. 7

    Akis T.: Elastoplastic analysis of functionally graded spherical pressure vessels. Comp. Mater. Sci. 46, 545–554 (2009)

  8. 8

    Sadeghian M., Toussi H.E.: Axisymmetric yielding of functionally graded spherical vessel under thermo-mechanical loading. Comp. Mater. Sci. 50, 975–981 (2011)

  9. 9

    Chadwick P.: Compression of a spherical shell of work-hardening material. Int. J. Mech. Sci. 5, 165–182 (1963)

  10. 10

    Dastidar D.G., Ghosh P.: A transient thermal problem: a hollow sphere of strain-hardening material with temperature-dependent properties. Int. J. Mech. Sci. 16, 359–371 (1974)

  11. 11

    Ishikawa H.: Transient thermoelastoplastic stress analysis for a hollow sphere using the incremental theory of plasticity. Int. J. Solids Struct. 13, 645–655 (1977)

  12. 12

    Durban D.: Thermo-elastic/plastic behaviour of a strain-hardening thick-walled sphere. Int. J. Solids Struct. 19, 643–652 (1983)

  13. 13

    Durban D.: Thermoplastic behavior of a thick-walled sphere. AIAA J. 19, 826–828 (1981)

  14. 14

    Durban D., Baruch M.: Analysis of an elasto-plastic thick walled sphere loaded by internal and external pressure. Int. J. Non-linear Mech. 12, 9–21 (1977)

  15. 15

    Katzir Z., Rubin M.B.: A simple formula for dynamic spherical cavity expansion in a compressible elastic perfectly plastic material with large deformations. Math. Mech. Solids 16, 665–681 (2011)

  16. 16

    Durban D., Fleck N.A.: Spherical cavity expansion in a Drucker–Prager solid. ASME J. Appl. Mech. 64, 743–750 (1997)

  17. 17

    Rees D.W.A.: Description of reversed yielding in bending. Proc. IMechE Part C: J. Mech. Eng. Sci. 221, 981–991 (2007)

Download references

Author information

Correspondence to Yeau-Ren Jeng.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alexandrov, S., Pirumov, A. & Jeng, Y. Expansion/contraction of a spherical elastic/plastic shell revisited. Continuum Mech. Thermodyn. 27, 483–494 (2015). https://doi.org/10.1007/s00161-014-0365-6

Download citation


  • Spherical shell
  • Bauschinger effect
  • Semi-analytic solution