Continuum Mechanics and Thermodynamics

, Volume 22, Issue 4, pp 291–298 | Cite as

Polygonization as low energy dislocation structure

  • K. C. Le
  • Q. S. Nguyen
Original Article


Within continuum dislocation theory, one-dimensional energy functional of a bent beam, made of a single crystal, is derived. By relaxing the continuously differentiable minimizer of this energy functional, we construct a sequence of piecewise smooth deflections and piecewise constant plastic distortions reducing the energy and exhibiting polygonization. The number of polygons can be estimated by comparing the surface energy of small angle tilt boundaries with the contribution of the gradient terms from the weak minimizer in the bulk energy.


Crystal Bending Tilt boundaries Dislocations Polygon 


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  1. 1.
    Berdichevsky V.L.: Variational Principles of Continuum Mechanics. Nauka, Moskva (1983)Google Scholar
  2. 2.
    Berdichevsky V.L.: Continuum theory of dislocations revisited. Continuum Mech. Thermodyn. 18, 195–222 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Berdichevsky V.L.: On thermodynamics of crystal plasticity. Scripta Mater. 54, 711–716 (2006)CrossRefGoogle Scholar
  4. 4.
    Bilby B.A., Gardner L.R.T., Smith E.: The relation between dislocation density and stress. Acta Metall. 6, 29–33 (1958)CrossRefGoogle Scholar
  5. 5.
    Cahn R.W.: Recrystalization of single crystals after plastic bending. J. Inst. Met. 76, 121–125 (1949)Google Scholar
  6. 6.
    Gilman J.J.: Structure and polygonization of bent zinc monocrystals. Acta Metall. 3, 277–288 (1955)CrossRefGoogle Scholar
  7. 7.
    Kuhlmann-Wilsdorf D.: Q: Dislocation structures—how far from equilibrium? A: very closed indeed. Mater. Sci. Eng. A 315, 211–216 (2001)CrossRefGoogle Scholar
  8. 8.
    Le K.C.: Vibrations of Shells and Rods. Springer, Berlin (1999)MATHGoogle Scholar
  9. 9.
    Le K.C., Nguyen Q.S.: Plastic yielding and work hardening of single crystals in a soft device. Comptes Rendus Mecanique 337, 709–715 (2009)CrossRefGoogle Scholar
  10. 10.
    Mura T.: Micromechanics of Defects in Solids. Kluwer Academic Publishers, Oxford (1987)Google Scholar
  11. 11.
    Nye J.F.: Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162 (1953)CrossRefGoogle Scholar
  12. 12.
    Read W.T., Shockley W.: Dislocation models of crystal grain boundaries. Phys. Rev. 78, 275–289 (1950)MATHCrossRefADSGoogle Scholar
  13. 13.
    Read W.T.: Dislocation theory of plastic bending. Acta Metall. 5, 83–88 (1957)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Ruhr-Universität BochumBochumGermany
  2. 2.LMS, Ecole PolytechniquePalaiseauFrance

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