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Existence and Stability of a Screw Dislocation under Anti-Plane Deformation

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Abstract

We formulate a variational model for a geometrically necessary screw dislocation in an anti-plane lattice model at zero temperature. Invariance of the energy functional under lattice symmetries renders the problem non-coercive. Nevertheless, by establishing coercivity with respect to the elastic strain and a concentration compactness principle, we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model.

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References

  1. Alicandro R., Cicalese M.: Variational analysis of the asymptotics of the XY model. Arch. Ration. Mech. Anal. 192(3), 501–536 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alicandro R., Cicalese M., Ponsiglione M.: Variational equivalence between Ginzburg–Landau, XY spin systems and screw dislocations energies. Indiana Univ. Math. J. 60(1), 171–208 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Amodeo R.J., Ghoniem N.M.: Dislocation dynamics. i. a proposed methodology for deformation micromechanics. Phys. Rev. B, 41, 6958–6967 (1990)

    Article  ADS  Google Scholar 

  4. Ariza, M.P., Ortiz, M.: Discrete crystal elasticity and discrete dislocations in crystals. Arch. Ration. Mech. Anal. 178(2) (2005)

  5. Brezis, H., Coron, J.-M., Lieb, E.H.: Harmonic maps with defects. Commun. Math. Phys. 107(4) (1986)

  6. Bulatov, V.V., Cai, W.: Computer Simulations of Dislocations, Oxford Series on Materials Modelling, Vol. 3. Oxford University Press, Oxford (2006)

  7. Cermelli P., Leoni G.: Renormalized energy and forces on dislocations. SIAM J. Math. Anal., 37(4), 1131–1160 (2005) (electronic)

    MATH  MathSciNet  Google Scholar 

  8. Conti S., Garroni A., Müller S.: Singular kernels, multiscale decomposition of microstructure, and dislocation models. Arch. Ration. Mech. Anal. 199(3), 779– (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Diestel, R.: Graph theory, Graduate Texts in Mathematics, 4th edn, Vol. 173. Springer, Heidelberg, (2010)

  10. Ehrlacher, V., Ortner, C., Shapeev, A.: in preparation

  11. Föll, H.: Defects in crystals (2013). http://www.tf.uni-kiel.de/matwis/amat/def_en/

  12. Garroni A., Müller S.: Γ-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36(6), 1943–1964 (2005) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  13. Garroni, A., Leoni, G., Ponsiglione, M.: Gradient theory for plasticity via homogenization of discrete dislocations. J. Eur. Math. Soc. (JEMS) 12(5), 1231–1266 (2010)

    Google Scholar 

  14. Garroni A., Müller S.: A variational model for dislocations in the line tension limit. Arch. Ration. Mech. Anal. 181(3), 535–578 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hirth J.P., Lothe J.: Theory of Dislocations. Krieger Publishing Company, Malabar (1982)

    Google Scholar 

  16. Hudson, T., Mason, J., Ortner, C.: in preparation

  17. Hudson, T., Ortner, C.: Manuscript

  18. Hull, D., Bacon, D.J.: Introduction to Dislocations, vol. 37. Butterworth-Heinemann, 2011

  19. Koslowski M., Cuitiño A.M., Ortiz M.: A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals. J. Mech. Phys. Solids 50(12), 2597–2635 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. Luskin, M., Ortner, C., Van Koten, B.: Formulation and optimization of the energy-based blended quasicontinuum method. Comput. Methods Appl. Mech. Eng. 253 (2013)

  21. Peletier, M.A., Geers, M.G.D., Peerlings, R.H.J., Scardia, L.: Asymptotic behaviour of a pile-up of infinite walls of edge dislocations. ArXiv e-prints (2012)

  22. Miller, R., Tadmor, E.: A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17 (2009)

  23. Orowan E.: Zur kristallplastizität. III. Zeitschrift für Physik 89, 634–659 (1934)

    Article  ADS  Google Scholar 

  24. Ortner, C., Shapeev, A.: Interpolants of lattice functions for the analysis of atomistic/continuum multiscale methods. ArXiv e-prints, 1204.3705 (2012)

  25. Ortner, C., Shapeev, A.V.: Analysis of an energy-based atomistic/continuum coupling approximation of a vacancy in the 2D triangular lattice. Math. Comput. (2014, in press)

  26. Ortner, C., Theil, F.: Nonlinear elasticity from atomistic mechanics (2012). arXiv:1202.3858v3

  27. Polanyi M.: Über eine art gitterstörung, die einen kristall plastisch machen könnte. Zeitschrift für Physik 89, 660–664 (1934)

    Article  ADS  Google Scholar 

  28. Ponsiglione, M.: Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuous. SIAM J. Math. Anal. 39(2) (2007)

  29. Rodney D., Le Bouar Y., Finel A.: Phase field methods and dislocations. Acta Mater. 51(1), 17–30 (2003)

    Article  Google Scholar 

  30. Scardia L., Zeppieri C.I.: Line-tension model for plasticity as the Γ-limit of a nonlinear dislocation energy. SIAM J. Math. Anal. 44(4), 2372–2400 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Shilkrot L.E., Miller R.E., Curtin W.A.: Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. J. Mech. Phys. Solids 52, 755–787 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. Sinclair, J.E.: Improved atomistic model of a BCC dislocation core. J. Appl. Phys. 42, 5231 (1971)

    Google Scholar 

  33. Taylor, G.I.: The mechanism of plastic deformation of crystals. Part I. Theoretical. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 145(855) (1934)

  34. Vítek V., Perrin R.C., Bowen D.K.: The core structure of \({\frac12(111)}\) screw dislocations in BCC crystals. Philos. Mag. 21(173), 1049–1073 (1970)

    Article  ADS  Google Scholar 

  35. Volterra, V.: Sur l’équilibre des corps élastiques multiplement connexes. Ann. Sci. École Norm. Sup. (3), 24

  36. Voskoboinikov R.E., Chapman S.J., Ockendon J.R., Allwright D.J.: Continuum and discrete models of dislocation pile-ups. I. Pile-up at a lock. J. Mech. Phys. Solids 55(9), 2007–2025 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK

    Thomas Hudson

  2. Mathematics Institute, Zeeman Building University of Warwick, Coventry, CV4 7AL, UK

    Christoph Ortner

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  1. Thomas Hudson
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  2. Christoph Ortner
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Correspondence to Thomas Hudson.

Additional information

Communicated by S. Müller

C. Ortner was supported by the EPSRC grant EP/H003096 “Analysis of atomistic-to-continuum coupling methods”. T. Hudson was supported by the UK EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1).

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Hudson, T., Ortner, C. Existence and Stability of a Screw Dislocation under Anti-Plane Deformation. Arch Rational Mech Anal 213, 887–929 (2014). https://doi.org/10.1007/s00205-014-0746-9

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  • DOI: https://doi.org/10.1007/s00205-014-0746-9

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