Internal State Variable Theory

  • D. L. McDowell

Abstract

Many practical problems of interest deal with irreversible, path dependent aspects of material behavior, such as hysteresis due to plastic deformation or phase transition, fatigue and fracture, or diffusive rearrangement. Some of these processes occur so slowly and so near equilibrium that attendant models forego description of nonequilibrium aspects of dissipation (e.g., grain growth). On the other hand, some irreversible behaviors such as thermally activated dislocation glide can occur farther from equilibrium with a spectrum of relaxation times. The fact that quasi-stable, nonequilibrium configurations of defects can exist in lattices at multiple length scales, combined with the long range nature of interaction forces, presents an enormous challenge to the utility of high fidelity, high degree of freedom (DoF) dynamical models that employ atomistic or molecular modeling methods. For example, analyses of simple crystal structures using molecular dynamics have now reached scales on the order of microns, but are limited to rather idealized systems such as pure metals and to small time durations of the order of nanoseconds. High fidelity analyses of generation, motion and interaction of line defects in lattices based on discrete dislocation dynamics, making use of interactions based on linear elastic solutions, cover somewhat higher length scales and longer time scales, but are also limited in considering realistic multiphase, hierarchical microstructures. Crystal plasticity as well cannot be used for large scale finite element simulations, for example, crash simulations of a vehicle into a barrier.

Keywords

Entropy Fatigue Manifold Cavitation Cold Work 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B.D. Coleman and M.E. Gurtin, “Thermodynamics with internal state variables,” J. Chem. Phys., 47, 597–613, 1967.CrossRefADSGoogle Scholar
  2. [2]
    J. Kestin and J.R. Rice, “Paradoxes in the application of thermo-dynamics to strained solids,” E.B. Stuart, B. Gal Or and A.J. Brainard (eds.), A Critical Review of Thermodynamics, Mono-Book Corp., Baltimore, pp. 275–298, 1970.Google Scholar
  3. [3]
    P. Germain, Q.S. Nguyen, and P. Suquet, “Continuum thermodynamics,” J. Appl. Mech. Trans. ASME, 50, 1010, 1983.MATHCrossRefADSGoogle Scholar
  4. [4]
    J. Kestin, “Local equilibrium formalism applied to mechanics of solids,” Int. J. Sol. Struct., 29(14-15), 1827–1836, 1992.MATHCrossRefGoogle Scholar
  5. [5]
    W. Muschik, “Fundamentals of nonequilibrium thermodynamics,” In: W. Mushik, (ed.), Non-Equilibrium Thermodynamics with Applications to Solids, CISM Courses and Lectures No. 336, International Centre for Mechanical Sciences, Springer-Verlag, New York, pp. 1–63, 1993.Google Scholar
  6. [6]
    J. Bataille and J. Kestin, “L’interprétation physique de la thermodynamique rationnelle,” J. de Méchanique, 14, 365–384, 1975.MATHMathSciNetADSGoogle Scholar
  7. [7]
    J. Lemaitre and J.L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge, 1990.MATHGoogle Scholar
  8. [8]
    D.L. McDowell, “Multiaxial effects in metallic materials. Symposium on Durability and Damage Tolerance,” ASME AD-Vol. 43, ASME Winter Annual Meeting, Chicago, IL, Nov. 6–11, pp. 213–267, 1994.Google Scholar
  9. [9]
    Z.P Bazant and E.-P Chen, “Scaling of structural failure,” Appl. Mech. Rev., 50(10), 593–627, 1997.CrossRefGoogle Scholar
  10. [10]
    A.C. Eringen, “Non-local polar elastic continua,” Int. J. Engrg. Sci., 10, 1–16, 1972.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    H.M. Zbib and E.C. Aifantis, “On the gradient-dependent theory of plasticity and shear banding,” Acta Mech., 92, 209–225, 1992.MATHCrossRefGoogle Scholar
  12. [12]
    H.M. Zbib and E.C. Aifantis, “Size effects and length scales in gradient plasticity and dislocation dynamics,” Scripta Mater., 48, 155–160, 2003.CrossRefGoogle Scholar
  13. [13]
    E.C. Aifantis, “Pattern formation in plasticity,” Int. J. Engrg. Sci., 33(15), 2161–2178, 1995.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    T.E. Lacy, D.L. McDowell, and R. Talreja, “Gradient concepts for evolution of damage,” Mech. Mater, 31, 831–860, 1999.CrossRefGoogle Scholar
  15. [15]
    S. Nemat-Nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, Amsterdam, 1993.MATHGoogle Scholar
  16. [16]
    M. Zhou and D.L. McDowell, “Equivalent continuum for dynamically deforming atomistic particle systems,” Phil. Mag. A, 82(13), 2547–2574, 2002.ADSCrossRefGoogle Scholar
  17. [17]
    H. Ziegler, “An Introduction to Thermomechanics,” In: E. Becker, B. Budiansky, W.T. Koiter, and H.A. Lauwerier (eds.), North Holland Series in Applied Mathematics and Mechanics, 2nd edn., vol. 21, North Holland, Amsterdam, New York, 1983.Google Scholar
  18. [18]
    G. Lebon, “Fundamentals of nonequilibrium thermodynamics,” In: W. Mushik (ed.), Non-Equilibrium Thermodynamics with Applications to Solids, CISM Courses and Lectures No. 336, International Centre for Mechanical Sciences, Springer-Verlag, New York, pp. 139–204, 1993.Google Scholar
  19. [19]
    D. Krajcinovic, Damage Mechanics, Elsevier, Amsterdam, 1996.Google Scholar
  20. [20]
    R.S. Kumar and R. Talreja, “A continuum damage model for linear viscoelastic composite materials,” Mech. Mater., 35(3-6), 463–480, 2003.CrossRefGoogle Scholar
  21. [21]
    J.R. Rice, “Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity,” J. Mech. Phys. Sol., 19, 433–455, 1971.MATHCrossRefADSGoogle Scholar
  22. [22]
    R.J. Asaro, “Crystal plasticity,” J. Appl. Mech., Trans. ASME, 50, 921–934, 1983.MATHCrossRefGoogle Scholar
  23. [23]
    A. Nadai, Theory of flow and fracture of solids, McGraw-Hill, New York, 1963.Google Scholar
  24. [24]
    A. Needleman and J.R. Rice, “Plastic creep flow effects in the diffusive cavitation of grain boundaries,” Acta Met., 28(10), 1315–1332, 1980.CrossRefGoogle Scholar
  25. [25]
    D. Moldovan, D. Wolf, S.R. Phillpot, and A.J. Haslam, “Role of grain rotation during grain growth in a columnar micro structure by mesoscale simulation,” Acta Mater., 50, 3397–3414, 2002.CrossRefGoogle Scholar
  26. [26]
    F. Cleri, G. D’Agostino, A. Satta, and L. Colombo, “Microstructure evolution from the atomic scale up,” Comput. Mater. Sci., 24, 21–27, 2002.CrossRefGoogle Scholar
  27. [27]
    J.C. Moosbrugger and D.L. McDowell, “A rate dependent bounding surface model with a generalized image point for cyclic nonproportional viscoplasticity,” J. Mech. Phys. Sol., 38(5), 627–656, 1990.CrossRefADSGoogle Scholar
  28. [28]
    J.-L. Chaboche, “Constitutive equations for cyclic plasticity and cyclic viscoplasticity,” Int. J. Plasticity, 5(3), 247–302, 1989.MATHCrossRefGoogle Scholar
  29. [29]
    D.L. McDowell, “An experimental study of the structure of constitutive equations for nonproportional cyclic plasticity,” J. Engrg. Mater. Techn., Trans. ASME, 107, 307–315, 1985.CrossRefGoogle Scholar
  30. [30]
    P.J. Armstrong and C.O. Frederick, “A mathematical representation of the multiaxial Bauschinger effect,” CEGB Report RD/B/N731, Berkeley Nuclear Laboratories, 1966.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • D. L. McDowell
    • 1
  1. 1.Georgia Institute of TechnologyAtlantaUSA

Personalised recommendations