Abstract

We present a variational framework for studying screw dislocations subject to antiplane shear. Using a classical model developed by P. Cermelli and M. E. Gurtin [The motion of screw dislocations in crystalline materials undergoing antiplane shear: glide, cross-slip, fine cross-slip, Arch. Ration. Mech. Anal. 148 (1999) 3–52], methods of Calculus of Variations are exploited to prove existence of solutions, and to derive a useful expression of the Peach-Köhler forces acting on a system of dislocation. This provides a setting for studying the dynamics of the dislocations, which is done in another paper of T. Blass, I. Fonseca, G. Leoni and M. Morandotti [Dynamics for systems of screw dislocations, SIAM J. Appl. Math. 75 (2015) 393–419].

Author information

Contact details are reproduced from the original publication and may be historical.

Marco Morandotti

SISSA - International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy

marco.morandotti@sissa.it

Keywords

  • Dislocations
  • variational methods
  • renormalized energy
  • Peach-Köhler force

Mathematics Subject Classification

70G75; 74B05, 49J99, 70F99

References

  1. R. Alicandro, M. Ponsiglione. Ginzburg–Landau functionals and renormalized energy: a revised Γ-convergence approach, J. Funct. Anal. 266(8) (2014) 4890–4907.https://doi.org/10.1016/j.jfa.2014.01.024Google Scholar
  2. F. Bethuel, H. Brezis, F. Hélein: Tourbillons de Ginzburg-Landau et énergie renormalisée, C. R. Acad. Sci., Paris, Sér. I 317(2) (1993) 165–171.https://doi.org/10.1007/978-1-4612-0287-5Google Scholar
  3. F. Bethuel, H. Brezis, F. Hélein: Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications 13, Birkhäuser, Boston (1994).https://doi.org/10.1007/978-1-4612-0287-5Google Scholar
  4. T. Blass, I. Fonseca, G. Leoni, M. Morandotti: Dynamics for systems of screw dislocations, SIAM J. Appl. Math. 75(2) (2015) 393–419.https://doi.org/10.1137/140980065Google Scholar
  5. E. Buzano, P. Cermelli: A singular variational problem in dislocation theory, Z. Angew. Math. Phys. 51(6) (2000) 968–983.https://doi.org/10.1007/PL00001532Google Scholar
  6. P. Cermelli, M. E. Gurtin: The motion of screw dislocations in crystalline materials undergoing antiplane shear: glide, cross-slip, fine cross-slip, Arch. Ration. Mech. Anal. 148(1) (1999) 3–52.https://doi.org/10.1007/s002050050155Google Scholar
  7. P. Cermelli, G. Leoni: Renormalized energy and forces on dislocations, SIAM J. Math. Anal. 37(4) (2005) 1131–1160.https://doi.org/10.1137/040621636Google Scholar
  8. L. De Luca, A. Garroni, M. Ponsiglione: Γ-convergence analysis of systems of edge dislocations: the self energy regime, Arch. Ration. Mech. Anal. 206(3) (2012) 885–910.https://doi.org/10.1007/s00205-012-0546-zGoogle Scholar
  9. J. D. Eshelby: The force on an elastic singularity, Philos. Trans. Roy. Soc. London, Ser. A 244 (1951) 84–112.https://doi.org/10.1098/rsta.1951.0016Google Scholar
  10. J. D. Eshelby: The force on a disclination in a liquid crystal, Phil. Mag. A 42(42) (1980) 359–367.https://doi.org/10.1080/01418618008239363Google Scholar
  11. L. C. Evans: Weak Convergence Methods for Nonlinear Partial Differential Equations (Chicago, 1988), CBMS Regional Conference Series in Mathematics 74, American Mathematical Society, Providence (1990).https://doi.org/10.1090/cbms/074Google Scholar
  12. A. Garroni, G. Leoni, M. Ponsiglione: Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc. (JEMS) 12(5) (2010) 1231–1266.https://doi.org/10.4171/jems/228Google Scholar
  13. M. G. D. Geers, R. H. J. Peerlings, M. A. Peletier, L. Scardia: Asymptotic behaviour of a pile-up of infinite walls of edge dislocations, Arch. Ration. Mech. Anal. 209(2) (2013) 495–539.https://doi.org/10.1007/s00205-013-0635-7Google Scholar
  14. M. E. Gurtin: The nature of configurational forces, Arch. Rational Mech. Anal. 131(1) (1995) 67–100.https://doi.org/10.1007/BF00386071Google Scholar
  15. J. P. Hirth, J. Lothe: Theory of Dislocations, Krieger, Malabar (1982).Google Scholar
  16. F. R. N. Nabarro: Theory of Crystal Dislocations, International Series of Monographs on Physics, Clarendon Press, Oxford (1967).Google Scholar
  17. W. T. Read: Dislocations in Crystals, International Series in Pure and Applied Physics, McGraw-Hill, New York (1953).Google Scholar
  18. A. L. Roitburd: Equilibrium of crystals formed in the solid phase, Sov. Phys., Dokl. 16 (1971) 305–308.Google Scholar
  19. E. Sandier, S. Serfaty: Limiting vorticities for the Ginzburg-Landau equations, Duke Math. J. 117(3) (2003) 403–446.https://doi.org/10.1215/S0012-7094-03-11732-9Google Scholar
  20. E. B. Tadmor, M. Ortiz, R. Phillips: Quasicontinuum analysis of defects in solids, Phil. Mag. A 73(6) (1996) 1529–1563.https://doi.org/10.1080/01418619608243000Google Scholar
  21. B. Van Koten, X. Helen Li, M. Luskin, C. Ortner: A computational and theoretical investigation of the accuracy of quasicontinuum methods, in: Numerical Analysis of Multiscale Problems (Durham, 2010), I. G. Graham et al. (ed.), Lect. Notes Comput. Sci. Eng. 83, Springer, Berlin (2012) 67–96.https://doi.org/10.1007/978-3-642-22061-6_3Google Scholar

Suggested citation

T. Blass, M. Morandotti. “Renormalized Energy and Peach-Köhler Forces for Screw Dislocations with Antiplane Shear.” Journal of Convex Analysis 24 (2017), No. 2, 547–570. https://doi.org/10.68381/jca24036

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.