Continuum Mechanics and Thermodynamics

, Volume 17, Issue 1, pp 83–99 | Cite as

Linear stability analysis for a thermoviscoplastic material under cyclic axial loading

  • F. DinzartEmail author
  • A. Molinari
Original article


Some mechanical properties exhibit a very strong dependence upon temperature; these evolutions can be properly analyzed by the steady state response in cyclic loading. To relate experimental conditions to thermomechanical characteristics, the existence and the stability of steady state solutions are studied for cylinders submitted to cyclic compression. The material, considered as rigid viscoplastic, is modeled by a non-Newtonian temperature dependent viscous law. Closed form solutions are obtained in the framework of a large deformation theory by neglecting thermal expansion and inertia effects. Steady state regime is analyzed. The stress versus strain rate response and the temperature distribution are established as functions of the geometry of the cylinder, the loading characteristics and the material parameters. The stability of steady state solutions is analyzed with use of a linear perturbation scheme.


cyclic loading thermoviscoplastic behavior steady states thermomechanical instability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Christensen, R.M.: Theory of viscoelasticity: an introduction. Academic Press, 1971Google Scholar
  2. 2.
    Dillon, O.W.: An experimental study of the heat generated during torsional oscillations. J. Mech. Phys. Solids 10, 235-244 (1962)Google Scholar
  3. 3.
    Dinzart, F. Molinari, A.: Cyclic torsion of a polymeric tube: self heating and thermal failure. J. Thermal Stresses 21, 851-879 (1998)Google Scholar
  4. 4.
    Ferry, J.D.: Viscoelastic properties of polymers. Wiley, 1980Google Scholar
  5. 5.
    Gradshteyn, I.S. Ryzhik. I.M.: Table of integrals series and products. Academic Press, New York, 1965Google Scholar
  6. 6.
    Lemaître, J., Chaboche, J.L.: Mécanique des matériaux solides. Dunod, Paris, 1985Google Scholar
  7. 7.
    Leroy, Y.M., Molinari, A. Stability of steady states in shear zones. J. Mech. Phys. Solids 40, 181-212 (1992)Google Scholar
  8. 8.
    Manson, S.S.: Thermal stress and low-cycle fatigue. McGraw-Hill, 1966Google Scholar
  9. 9.
    Molinari, A.: Instabilité thermoviscoplastique en cisaillement simple. J. Theor. Appl. Mech. 4, 659-670 (1985)Google Scholar
  10. 10.
    Molinari, A., Germain, Y.: Self-heating and thermal failure of polymers sustaining a compressive cyclic loading. Int. J. Solids Struct. 33, 3439-3462 (1996)Google Scholar
  11. 11.
    Suresh, S.: Fatigue of materials. Cambridge University Press, 1991Google Scholar
  12. 12.
    Swindeman, R.W.: Cyclic stress-strain response of a 9Cr-1Mo-V-Nb pressure vessel steel at high temperature. Low cycle fatigue ASTM STP 942, 107-122 (1988)Google Scholar
  13. 13.
    Tauchert, T.R.: Heat generation in a viscoelastic solid. Acta. Mech. 3, 385-396 (1966)Google Scholar
  14. 14.
    Tauchert, T.R.: The temperature generated during torsional oscillations of polyethylene rods. Int. J. Engng Sci. 5, 353-365 (1967)Google Scholar
  15. 15.
    Tormey, J.F., Britton, S.C.: Effects of cyclic loading on solid propellant grain structure. AIAA J. 1, 1763-1770 (1963)Google Scholar
  16. 16.
    Tresca, H.: Flow of solids, application in forging. Proc. Inst. Mech. Eng. 30, 301-327 (1878)Google Scholar
  17. 17.
    Watson, G.N. A treatise on the theory of Bessel functions. Cambridge University Press, 1966Google Scholar

Copyright information

© Springer-Verlag Berlin/Heidelberg 2005

Authors and Affiliations

  1. 1.Laboratoire de Physique et Mécanique des MatériauxUMR CNRS 7554, Université de MetzIle du Saulcy, Metz CedexFrance

Personalised recommendations