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Journal of Nonlinear Science

, Volume 27, Issue 5, pp 1435–1461 | Cite as

A Variational Model for Dislocations at Semi-coherent Interfaces

  • Silvio Fanzon
  • Mariapia Palombaro
  • Marcello PonsiglioneEmail author
Article

Abstract

We propose and analyze a simple variational model for dislocations at semi-coherent interfaces. The energy functional describes the competition between two terms: a surface energy induced by dislocations and a bulk elastic energy, spent to decrease the amount of dislocations needed to compensate the lattice misfit. We prove that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter. The proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations. Even if we deal with finite elasticity, linearized elasticity naturally emerges in our analysis since the far-field strain vanishes as the interface size increases.

Keywords

Nonlinear elasticity Geometric rigidity Linearization Crystals Dislocations Heterostructures 

Mathematics Subject Classification

74B20 74K10 74N05 49J45 

References

  1. Agostiniani, V., Dal Maso, G., DeSimone, A.: Linear elasticity obtained from finite elasticity by \({\Gamma }\)-convergence under weak coerciveness conditions. Ann. Inst. H. Poincaré Anal. nonlinéaire. 29(5), 715–735 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alicandro, R., Palombaro, M., Lazzaroni, G.: Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. http://cvgmt.sns.it/paper/3248/, Preprint (2016)
  3. Conti, S., Garroni, A., Müller, S.: The line-tension approximation as the dilute limit of linear-elastic dislocations. Arch. Ration. Mech. Anal. 218, 699–755 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dal Maso, G., Negri, M., Percivale, D.: Linearized elasticity as \(\Gamma \)-limit of finite elasticity. Set-Valued Anal. 10(2–3), 165–183 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. De Luca, L., Garroni, A., Ponsiglione, M.: \(\Gamma \)-convergence analysis of systems of edge dislocations: the self energy regime. Arch. Ration. Mech. Anal. 206(3), 885–910 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Ernst, F.: Metal-oxide interfaces. Mat. Sci. Eng. R. 14, 97–156 (1995)CrossRefGoogle Scholar
  7. Fonseca, I., Fusco, N., Leoni, G., Morini, M.: A model for dislocations in epitaxially strained elastic films. Preprint (2016)Google Scholar
  8. Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55(11), 1461–1506 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hirsch, P.B.: Nucleation and propagation of misfits dislocations in strained epitaxial layer systems. In: Proceedings of the Second International Conference Schwäbisch Hall, Federal Republic of Germany, 30 July–3 August 1990Google Scholar
  10. Hirth, J.P., Lothe, J.: Theory of Dislocations, 2nd edn. Wiley, Hoboken (1982)zbMATHGoogle Scholar
  11. Lauteri, G. and Luckhaus, S.: An energy estimate for dislocation configurations and the emergence of cosserat-type structures in metal plasticity. https://arxiv.org/pdf/1608.06155.pdf, Preprint (2016)
  12. Lazzaroni, G., Palombaro, M., Schlömerkemper, A.: A discrete to continuum analysis of dislocations in nanowires heterostructures. Commun. Math. Sci. 13, 1105–1133 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lazzaroni, G., Palombaro, M., Schlömerkemper, A.: Rigidity of three-dimensional lattices and dimension reduction in heterogeneous nanowires. Discrete Contin. Dyn. Syst. Ser. S 10(1), 119–139 (2017)Google Scholar
  14. Leoni, G.: Lecture Notes on Epitaxy. CRM Series, Edizioni della Scuola Normale Superiore, Springer (to appear)Google Scholar
  15. Müller, S., Palombaro, M.: Existence of minimizers for a polyconvex energy in a crystal with dislocations. Calc. Var. Partial Differ. Equ. 31(4), 473–482 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Müller, S., Palombaro, M.: Derivation of a rod theory for biphase materials with dislocations at the interface. Calc. Var. Partial Differ. Equ. 48(3–4), 315–335 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Nabarro, F.R.N.: Theory of Crystal Dislocations. Clarendon Press, Oxford (1967)Google Scholar
  18. Ortiz, M.: Lectures at the Vienna summer school on microstructures. Vienna, 25–29 September 2000Google Scholar
  19. Read, W.T., Shockley, W.: Dislocation models of crystal grain boundaries. Phys. Rev. 78, 275–289 (1950)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SussexBrightonUK
  2. 2.Dipartimento di MatematicaSapienza Università di RomaRomeItaly

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