Continuum theory of dislocations revisited

Original Article

Abstract

The paper continues the discussion of continuum theory of dislocations suggested by Berdichevsky and Sedov (PMM 31(6): 981–1000, 1967). The major new points are: the choice of energy, the variational form of the governing dynamical equations, the variational principle for the final plastic state.

Keywords

Continuum theory of dislocations Plastic spin Strain gradient plasticity 

References

  1. 1.
    Berdichevsky, V.L, Sedov, L.I.: Dynamic theory of continuously distributed dislocations. Its relation to plasticity theory. PMM, 31(6), 981–1000, (1967) (English translation: J. Appl. Math. Mech. (PMM), 989–1006 (1967))Google Scholar
  2. 2.
    Nye J.F. (1953). Some geometrical relations in dislocated crystals. Acta Metall. 1:153–162CrossRefGoogle Scholar
  3. 3.
    Kunin, I.A.: Methods of tensor analysis in the theory of dislocations. A supplement to the Russian edition of J.A. Schouten’s Tensor analysis for physicists, English translation (1967) is available from the US Department of Commerce, Clearinghouse for Federal Scientific and Technical Information, TT 67/514 26, Springfield, VA 22151 (1965)Google Scholar
  4. 4.
    Kunin I.A. (1983). Elastic Media with Microstructure, vol 2. Springer, Berlin Heidelberg New YorkGoogle Scholar
  5. 5.
    Kröner E. (1958). Kontinuumtheorie der Versetzungen und Eigenspannungen. Springer, Berlin Heidelberg New YorkGoogle Scholar
  6. 6.
    Kondo, K.: On the geometrical and physical foundations of the theory of yielding. In: Proceedings of the 2nd Japan Congress on Applied Mechanics, (1952), Tokyo, (1953)Google Scholar
  7. 7.
    Kondo, K.: Memoirs of the Unified Study of the Basic Problems in Engineering by Means of Geometry, vol. 1–3. pp 1955–1962. Gakujutsubunken-fukyukai, Tokyo (1955)Google Scholar
  8. 8.
    Bilby, B.A, Bullough, R., Smith, E.: Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc R Soc A231 (1955)Google Scholar
  9. 9.
    Bilby, B.A, Smith, E.: Continuous distribution of dislocations. Proc R Soc A236, (1956)Google Scholar
  10. 10.
    Bilby, B.A: Continuous distributions of dislocations. In: Sneddon I. (ed.) Progress. Solid Mechanics, vol. 1, Amsterdam (1960)Google Scholar
  11. 11.
    Le K.C., Stumpf H. (1996). On the determination of the crystal reference in nonlinear continuum theory of dislocations. Proc. R. Soc. A452:359–371Google Scholar
  12. 12.
    Sedov L.I., Berdichevsky V.L (1968). A dynamic theory of continual dislocations. In: Kröner E. (eds) Mechanics of Generalized Continua. Springer, Berlin Heidelberg New York, pp 214–238Google Scholar
  13. 13.
    Sedov L.I. (1965). Mathematical methods of constructing models of continuum media. Usp. Matem. Nauk 20(5):123–182MathSciNetGoogle Scholar
  14. 14.
    Aifantis E.C. (1984). On the microstructural origin of certain inelastic models. Trans. ASME J. Eng. Mater. Technol. 106:326–330CrossRefGoogle Scholar
  15. 15.
    Aifantis E.C. (1987). Physics of plastic deformation. Int. J. Plasticity 3:211–247MATHCrossRefGoogle Scholar
  16. 16.
    Fleck N.A., Hutchinson J.W. (1993). A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41:1825–1857MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Fleck N.A., Muller G.M., Ashby M.F., Hutchinson J.W. (1994). Strain gradient plasticity: theory and experiment. Acta Metall. Material. 42:475–487CrossRefGoogle Scholar
  18. 18.
    Acharia, A., Bassani, J.L.: Incompatible lattice deformations and crystal plasticity. In: Ghoneim, N., (ed.) Plastic and Fracture Instabilities in Materials, AMD-Vol. 200/MD-Vol 57, pp 75–80. ASME, New York, (1995)Google Scholar
  19. 19.
    Gao H., Huang Y., Nix W.D., Hutchinson J.W. (1999). Mechanism-based strain gradient plasticity-I. Theory. J. Mech. Phys. Solids 47:1239–1263MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Aifantis E.C. (1999). Strain gradient interpretation of size effects. Int. J. Fracture 95:299–314CrossRefGoogle Scholar
  21. 21.
    Shu J.Y., Fleck N.A. (1999). Strain gradient crystal plasticity: size-dependent deformation of bicrystals. J. Mech. Phys. Solids 47:297–324MATHCrossRefGoogle Scholar
  22. 22.
    Bassani J.L. (2001). Incompatibility and a simple gradient theory of plasticity. J. Mech. Phys. Solids 49:1983–1996MATHCrossRefGoogle Scholar
  23. 23.
    Fleck N.A., Hutchinson J.W. (2001). A reformulation of strain- gradient plasticity. J. Mech. Phys Solids 49:2245–2271MATHCrossRefGoogle Scholar
  24. 24.
    Gudmundson P. (2004). A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52:1379–1406MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Gurtin M. (2004). A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52:2545–2568MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Huang Y., Qu S., Hwang K.C., Li M., Gao H. (2004). A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plasticity 20:753–782MATHCrossRefGoogle Scholar
  27. 27.
    Aifantis K.E., Willis J.R. (2005). The role of interfaces in enhancing the yield strength of composites and polycrystals. J. Mech. Phys. Solids 53:1047–1070MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Han C.S., Gao H., Huang Y., Nix W.D. (2005). Mechanism-based strain gradient crystal plasticity- I. Theory. J. Mech. Phys. Solids 53:1188–1203MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Berdichevsky V., Dimiduk D. (2005). On failure of continuum plasticity theories on small scales. Scripta Material. 52:1017–1019CrossRefGoogle Scholar
  30. 30.
    Berdichevsky V. (2005). Homogenization in micro-plasticity. J. Mech. Phys. Solids 53:2457–2469MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Popov V.L., Kröner E. (2001). Theory of elastoplastic media with mesostructure. Theor Appl Fracture Mech. 37:299–310CrossRefGoogle Scholar
  32. 32.
    Berdichevsky V. (2006). On thermodynamics of crystal plasticity. Scripta Material. 54:711–716CrossRefGoogle Scholar
  33. 33.
    Ortiz M., Repetto E.A. (1999). Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47:397–462MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Berdichevsky, V.L.: Structure of equations of macrophysics. Phys. Rev. E68:066126 (2003)Google Scholar
  35. 35.
    Berdichevsky V.L. (1983). Variational principles of continuum mechanics. Nauka, MascowGoogle Scholar
  36. 36.
    Puglisi G., Truskinovsky L. (2005). Thermodynamics of rate independent plasticity. J. Mech. Phys. Solids 53:655–679MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Berdichevsky V.L. (1979). Variational-asymptotic method of constructing shell theory. J. Appl. Math. Mech. (PMM) 44(4):664–687Google Scholar
  38. 38.
    Dafalias Y.F. (1985). The plastic spin. J. Appl. Mech. 52:865–871MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Bilby B.A., Gardner L.R.T., Stroh A.N. (1957). Extrait des Actes du IX Congrès International de Mechanique Appliqueè, Bruxelles, 35–43Google Scholar
  40. 40.
    Kröner E. (1960). Arch. Ration. Mech. Anal. 4:273MATHCrossRefGoogle Scholar
  41. 41.
    Lee E.H. (1969). ASME J. Appl. Mech. 36:1MATHGoogle Scholar
  42. 42.
    Mandel J. (1973). Thermodynamics and plasticity. In: Delgado Domingos J.J., Nina M.N.R., Whitelaw J.H. (eds) Foundations of Thermodynamics. Wiley, New York, pp 283–315Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mechanical EngineeringWayne State UniversityDetroitUSA

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