The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity

  • B. D. ReddyEmail author
Original Article


Variational formulations are constructed for rate-independent problems in small-deformation single-crystal strain-gradient plasticity. The framework, based on that of Gurtin (J Mech Phys Solids 50: 5–32, 2002), makes use of the flow rule expressed in terms of the dissipation function. Provision is made for energetic and dissipative microstresses. Both recoverable and non-recoverable defect energies are incorporated into the variational framework. The recoverable energies include those that depend smoothly on the slip gradients, the Burgers tensor, or on the dislocation densities (Gurtin et al. J Mech Phys Solids 55:1853–1878, 2007), as well as an energy proposed by Ohno and Okumura (J Mech Phys Solids 55:1879–1898, 2007), which leads to excellent agreement with experimental results, and which is positively homogeneous and therefore not differentiable at zero slip gradient. Furthermore, the variational formulation accommodates a non-recoverable energy due to Ohno et al. (Int J Mod Phys B 22:5937–5942, 2008), which is also positively homogeneous, and a function of the accumulated dislocation density. Conditions for the existence and uniqueness of solutions are established for the various examples of defect energy, with or without the presence of hardening or slip resistance.


Strain-gradient plasticity Dissipation function Single-crystal plasticity Variational problem Defect energy Recoverable energy Non-recoverable energy 


  1. 1.
    Reddy, B.D.: The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity. Continuum Mech. Thermodyn. doi: 10.1007/s00161-011-0194-9 (2011)
  2. 2.
    Gurtin M.E., Anand L.: A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: small deformations. J. Mech. Phys. Solids 53, 1624–1649 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Gurtin M.E.: On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48, 989–1036 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Gurtin M.E.: A theory of viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Gurtin M.E., Anand L., Lele S.P.: Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Gurtin M.E., Needleman A.: Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector. J. Mech. Phys. Solids 53, 1–31 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Gurtin M.E., Fried E., Anand L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)Google Scholar
  8. 8.
    Ertürk I., van Dommelen J.A.W., Geers M.G.D.: Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories. J. Mech. Phys. Solids 57, 1801–1814 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Bayley C.J., Brekelmans W.A.M., Geers M.G.D.: A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. Int. J. Solids Struct. 43, 7268–7286 (2006)zbMATHCrossRefGoogle Scholar
  10. 10.
    Evers L.P., Brekelmans W.A.M., Geers M.G.D.: Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. Int. J. Solids Struct. 41, 5209–5230 (2004)zbMATHCrossRefGoogle Scholar
  11. 11.
    Evers L.P., Brekelmanns W.A.M., Geers M.G.D.: Non-local crystal plasticity model with intrinsic SSD and GND effects. J. Mech. Phys. Solids 52, 2379–2401 (2004)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Bayley C.J., Brekelmans W.A.M., Geers M.G.D.: A three-dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philos. Mag. 87, 1361–1378 (2007)ADSCrossRefGoogle Scholar
  13. 13.
    Kuroda M., Tvergaard V.: Studies of scale dependent crystal viscoplasticity models. J. Mech. Phys. Solids 54, 1789–1810 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Kuroda M., Tvergaard V.: On the formulations of higher-order strain gradient crystal plasticity models. J. Mech. Phys. Solids 56, 1591–1608 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    Svendsen B., Bargmann S.: On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. J. Mech. Phys. Solids 58, 1253–1271 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    Ohno N., Okumura D.: Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations. J. Mech. Phys. Solids 55, 1879–1898 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Ohno N., Okumura D., Shibata T.: Grain-size dependent yield behavior under loading, unloading and reverse loading. Int. J. Mod. Phys. B 22, 5937–5942 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    Gurtin M.E.: A finite deformation, gradient theory of single-crystal plasticity dependent on the accumulation of geometrically necessary dislocations. Int. J. Plast. 26, 1073–1096 (2010)CrossRefGoogle Scholar
  19. 19.
    Gurtin M.E., Reddy B.D.: Alternative formulations of isotropic hardening for Mises materials, and associated variational inequalities. Continuum Mech Thermodyn 21, 237–250 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Voyiadjis G.Z., Deliktas B.: Mechanics of strain gradient plasticity with particular reference to decomposition of the state variables into energetic and dissipative components. Int. J. Eng. Sci. 47, 1405–1423 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Bittencourt E., Needleman A., Gurtin M.E., van der Giessen E.: A comparison of nonlocal continuum and discrete dislocation plasticity predictions. J. Mech. Phys. Solids 51, 281–310 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Han W., Reddy B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer, New York (1999)zbMATHGoogle Scholar
  23. 23.
    Friâa A.: Le matériau de Norton-Hoff generalisé et ses applications à l’analyse limite. Comptes Rendus de l’Académie des Sciences A 286, 953–956 (1978)zbMATHGoogle Scholar
  24. 24.
    Hoff, N.J. (ed): First IUTAM Colloquium on Creep in Structures (Stanford, California, 11–15 July 1960). Springer, Berlin (1962)Google Scholar
  25. 25.
    Norton F.H.: Creep of Steel at High Temperatures. McGraw-Hill, New York (1929)Google Scholar
  26. 26.
    Gurtin M.E., Ohno N.: A gradient theory of small-deformation, single-crystal plasticity that accounts for GND-induced interactions between slip systems. J. Mech. Phys. Solids 59, 320–343 (2010)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Carstensen C., Hackl K., Mielke A.: Nonconvex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. A 458, 299–317 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Mielke A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Continuum Mech. Thermodyn. 15, 351–382 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Yalcinkaya T., Brekelmans W.A.M., Geers M.G.D.: Deformation patterning driven by rate dependent non-convex strain gradient plasticity. J. Mech. Phys. Solids 59, 1–17 (2010)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Applied Mathematics, Centre for Research in Computational and Applied MechanicsUniversity of Cape TownRondeboschSouth Africa

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