The Anderson-Bishop Problem—Thermal Ratchetting of a Polycrystalline Metals

Abstract

The paper readdresses a theory of super-plastic behaviour induced by cyclic temperature for certain classes of polycrystalline metals and composite materials, originally analyses by Anderson and Bishop in the 1950’s for polycrystalline alpha uranium. The background to the original analysis and its subsequent history in the literature is discussed. Two distinct phenomena are involved. The first is ratchetting due to a fluctuating residual stress field and commonly found in structural analysis. The second form of ratchetting is due to the severe kinematic constraint on the deformation of each crystal within a polycrystal. The Anderson Bishop analysis did not take into account the former. Adopting the same kinematic assumptions as these authors, new solutions are discussed for simplified polycrystalline models and an isotropic polycrystal. These new solutions provide functional forms for material behaviour that need to be taken into account in discussions of the phenomena by material scientists.

Keywords

Plastic Strain Strain Growth Residual Stress Field Helical Spring Reverse Plasticity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The first author first became aware of the Anderson and Bishop solutions when working with Roger Anderson on the modelling of a similar phenomena in the irradiation induced swelling of structural steels [20]. In recent years, following an interest in the thermal properties of metal matrix composites it became clear that the solutions have been extensively applied to problems in material science.

The authors wish to thanks Roger Anderson for assistance in this work. The work for this paper was part support by a Leverhulme Emeritus Fellowship to the first named author, which is gratefully acknowledges.

References

  1. 1.
    Anderson RG, Bishop JFW (1957) United Kingdom Atomic Energy Authority Industrial Group report TN/C681, UK National Archive reference AB 7/7441. Copy is available with the permission of the National Archive for downloading from http://www.le.ac.uk/departments/engineering/people/academic-staff/alan-ponter/papers
  2. 2.
    Anderson RG (1958) United Kingdom Atomic Energy Authority Industrial Group report TN/C681, UK National Archive reference AB 7/ 6213. Copy is available with the permission of the National Archive for downloading from http://www.le.ac.uk/departments/engineering/people/academic-staff/alan-ponter/papers
  3. 3.
    Anderson RG, Bishop JFW (1962) The effect of neutron irradiation and thermal cycling on permanent deformations in uranium under load. In: Proceedings of the institute of metals symposium on uranium and graphite, Institute of Metals, London, paper 3. Copy is available with the permission of the National Archive for downloading from http://www.le.ac.uk/departments/engineering/people/academic-staff/alan-ponter/papers
  4. 4.
    Greenwood GW, Johnson RH (1964) The deformation of metals under stresses during phase transformation. Proc R Soc A 283:403–422 CrossRefGoogle Scholar
  5. 5.
    Greenwood GW (2010) Generation of internal stress and its effects. Mater Sci Technol 26(4):398–403 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Leblond JB, Devaux J, Devaux JC (1989) Mathematical modelling of transformation plasticity in steels, I: case of ideal-plastic phases. Int J Plast 5:551–572 CrossRefGoogle Scholar
  7. 7.
    Leblond JB (1989) Mathematical modelling of transformation plasticity in steels, II: coupling with strain hardening phenomena. Int J Plast 5:573–591 CrossRefGoogle Scholar
  8. 8.
    Taleb L, Sidoroff F (2003) A micromechanical modeling of the Greenwood–Johnson mechanism in transformation induced plasticity. Int J Plast 19(10):1821–1842 MATHCrossRefGoogle Scholar
  9. 9.
    Bishop JFW, Hill R (1951) A theory of the plastic distortion of a polycrystalline aggregate under combined stress. Philos Mag 42:1298 MathSciNetMATHGoogle Scholar
  10. 10.
    Hutchinson JM, McCrum NG (1974) Microstress mechanisms for the time dependence of the modulus of crystalline polymers following an imposed change in volume. Nat Phys Sci 270:295–296 Google Scholar
  11. 11.
    Daehn GS, Gonzalez-Doncel G (1989) Deformation of whisker-reinforced metal matrix composites under changing temperature conditions. Metall Trans 20A:2355–2368 Google Scholar
  12. 12.
    Clyne TW, Withers PJ (1993) An introduction to metal matrix composites. Cambridge University Press, Cambridge, Chap 5 CrossRefGoogle Scholar
  13. 13.
    Bree J (1967) Elasto-plastic behaviour of thin tubes subjected to internal pressure and intermittent high-heat fluxes with applications to fast reactor fuel elements. J Strain Anal 2(3):226–238 CrossRefGoogle Scholar
  14. 14.
    Ponter ARS, Cocks ACF (1984) The incremental strain growth of an elastic-plastic body loaded in excess of the shakedown limit. J Appl Mech 51(3):465–469 MATHCrossRefGoogle Scholar
  15. 15.
    Ponter ARS, Cocks ACF (1984) The incremental strain growth of elastic-plastic bodies subjected to high levels of cyclic thermal loading. J Appl Mech 51(3):470–474 MATHCrossRefGoogle Scholar
  16. 16.
    Goodall IW, Goodman AM, Chell GC, Ainsworth RA, Williams JA (1991) R5: an assessment procedure for the high temperature response of structures. Report, Nuclear Electric Ltd., Barnwood, Gloucester Google Scholar
  17. 17.
    Chen HF, Ponter ARS, Ainsworth RA (2006) The linear matching method applied to the high temperature life assessment of structures, part 1: assessments involving constant residual stress fields. Int J Press Vessels Piping 83:123–135 CrossRefGoogle Scholar
  18. 18.
    Chen HF, Ponter ARS, Ainsworth RA (2006) The linear matching method applied to the high temperature life assessment of structures, part 2: assessments beyond shakedown involving changing residual stress fields. Int J Press Vessels Piping 83:136–147 CrossRefGoogle Scholar
  19. 19.
    Ponter ARS, Cocks ACF (2013) Thermal ratchetting of polycrystalline metals with inhomogeneous thermal properties. Philos Mag, accepted May 2013, available online Google Scholar
  20. 20.
    Anderson RG, Ponter ARS (1972) An estimate of the Cottrell creep of a metal during swelling. In: British nuclear energy society conference on irradiation embrittlement and creep in fuel cladding and core components, London Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of EngineeringUniversity of LeicesterLeicesterUK
  2. 2.Department of Engineering ScienceUniversity of OxfordOxfordUK

Personalised recommendations