Dissipative Systems in Contact with a Heat Bath: Application to Andrade Creep

  • Florian Theil
  • Tim Sullivan
  • Marisol Koslovski
  • Michael Ortiz
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 21)

Abstract

We develop a theory of statistical mechanics for dissipative systems governed by equations of evolution that assigns probabilities to individual trajectories of the system. The theory is made mathematically rigorous and leads to precise predictions regarding the behavior of dissipative systems at finite temperature. Such predictions include the effect of temperature on yield phenomena and rheological time exponents. The particular case of an ensemble of dislocations moving in a slip plane through a random array of obstacles is studied numerically in detail. The numerical results bear out the analytical predictions regarding the mean response of the system, which exhibits Andrade creep.

Keywords

Slip Plane Dissipative System Heat Bath Physical Review Letter Dislocation Dynamic 
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References

  1. 1.
    Edward N. da C. Andrade, Proceedings of the Royal Society of London A84, 1910, 1.Google Scholar
  2. 2.
    Edward N. da C. Andrade, Proceedings of the Royal Society of London A90, 1914, 329.Google Scholar
  3. 3.
    Daryl C. Chrzan and M.J. Mills, Criticality in the plastic deformation of Ni3Al. Physical Review Letters 69(19), 1992, 2795–2798.Google Scholar
  4. 4.
    S. Conti and M. Ortiz, Minimum principles for the trajectories of systems governed by rate problems. Journal of the Mechanics and Physics of Solids 56(5), 2008, 1885–1904.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A.H. Cottrell, Criticality in andrade creep. Philosophical Magazine A 74(4), 1996, 1041–1046.CrossRefGoogle Scholar
  6. 6.
    A.H. Cottrell, Strain hardening in andrade creep. Philosophical Magazine Letters 74(5), 1996, 375–379.CrossRefGoogle Scholar
  7. 7.
    G.S. Daehn, Primary creep transients due to non-uniform obstacle sizes. Materials Science and Engineering A 319–321, 2001, 765–769.CrossRefGoogle Scholar
  8. 8.
    A. Garroni and S. Müller, Γ-limit of a phase-field model of dislocations. SIAM Journal on Mathematical Analysis 36(6), 2005, 1943–1964 (electronic).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Garroni and S. Müller. A variational model for dislocations in the line tension limit. Archive for Rationional Mechanics and Analysis 181(3), 2006, 535–578.MATHCrossRefGoogle Scholar
  10. 10.
    M. Koslowski, A.M. Cuitiño and M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals. Journal of the Mechanics and Physics in Solids 50(12), 2002, 2597–2635.MATHCrossRefGoogle Scholar
  11. 11.
    A. Mielke and M. Ortiz, A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems. ESAIM: Control, Optimisation and Calculus of Variations, 14, 2008, 494–516.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Mielke and F. Theil, On rate-independent hysteresis models. NoDEA Nonlinear Differential Equations Appl. 11(2), 2004, 151–189.MATHMathSciNetGoogle Scholar
  13. 13.
    A. Mielke, F. Theil and V.I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle. Archive for Rationional Mechanics and Analysis 162(2), 2002, 137–177.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M.C. Miguel, A. Vespignani, M. Zaiser and S. Zapperi, Dislocation jamming and Andrade creep. Physical Review Letters 89(16), 2002, 165501–1–4.CrossRefGoogle Scholar
  15. 15.
    M. Ortiz and E.A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. Journal of the Mechanics and Physics of Solids 47(2), 1999, 397–462.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering 171(3–4), 1999, 419–444.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    R.A. Radovitzky and M. Ortiz, Error estimation and adaptive meshing in strongly nonlinear dynamic problems. Computer Methods in Applied Mechanics and Engineering 172(1–4), 1999, 203–240.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    T. Sullivan, M. Koslowski, F. Theil and M. Ortiz, Dissipative systems in contact with a heat bath: Application to andrade creep. Journal of the Mechanics and Physics of Solids, 2009, to appear.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Florian Theil
    • 1
  • Tim Sullivan
    • 1
  • Marisol Koslovski
    • 2
  • Michael Ortiz
    • 3
  1. 1.Mathematics InstituteWarwick UniversityCoventryUnited Kingdom
  2. 2.School of Mechanical EngineeringPurdue UniversityWest LafayetteU.S.A.
  3. 3.Engineering and Applied Sciences DivisionCalifornia Institute of TechnologyPasadenaU.S.A.

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