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Journal of Nonlinear Science

, Volume 29, Issue 1, pp 319–344 | Cite as

Variational Evolution of Dislocations in Single Crystals

  • Riccardo Scala
  • Nicolas Van GoethemEmail author
Article
  • 51 Downloads

Abstract

In this paper, we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents. Moreover, we discuss a novel dissipation structure for such currents, namely the flat distance, that will serve to drive the evolution of the dislocation clusters.

Keywords

Dislocation clusters Finite elasticity Time evolution Energetic solutions Variational method 

Mathematics Subject Classification

35Q74 74B20 35A15 74G65 49Q15 

Notes

Acknowledgements

We acknowledge the support of the FCT Starting Grant “ Mathematical theory of dislocations: geometry, analysis, and modeling” (IF/00734/2013). We thank the anonymous referees for their careful reading and interesting suggestions, which allow us to deeply improve our discussion.

References

  1. Alicandro, R., De Luca, L., Garroni, A., Ponsiglione, M.: Metastability and dynamics of discrete topological singularities in two dimensions: a \(\Gamma \)-convergence approach. Arch. Ration. Mech. Anal. 214(1), 269–330 (2014)MathSciNetzbMATHGoogle Scholar
  2. Alicandro, R., De Luca, L., Garroni, A., Ponsiglione, M.: Minimising movements for the motion of discrete screw dislocations along glide directions. Calc. Var. Partial Differ. Equ. 56(5), 19 (2017)MathSciNetzbMATHGoogle Scholar
  3. Amstutz, S., Van Goethem, N.: Incompatibility-governed elasto-plasticity for continua with dislocations. Proc. R. Soc. A 473(2199) (2017)Google Scholar
  4. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)MathSciNetzbMATHGoogle Scholar
  5. Ball, J.M.: Some open problems in elasticity. In: Geometry, Mechanics, and Dynamics. Volume in Honor of the 60th Birthday of J.E. Marsden, pp. 3–59. Springer, New York (2002)Google Scholar
  6. Bauman, P., Phillips, D., Owen, N.C.: Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity. Proc. R. Soc. Edinb. Sect. A Math. 119(3–4), 241–263 (1991)Google Scholar
  7. Berdichevsky, V.: Continuum theory of dislocations revisited. Cont. Mech. Therm. 18(9), 195–222 (2006)MathSciNetzbMATHGoogle Scholar
  8. Blass, T., Fonseca, I., Leoni, G., Morandotti, M.: Dynamics for systems of screw dislocations. SIAM J. Appl. Math. 75(2), 393–419 (2015)MathSciNetzbMATHGoogle Scholar
  9. Bonaschi, G.A., Van Meurs, P., Morandotti, M.: Dynamics of screw dislocations: a generalised minimising-movements scheme approach. Eur. J. Appl. Math. 28(4), 636–655 (2017)MathSciNetzbMATHGoogle Scholar
  10. Bonet, J., Gil, A.J., Ortigosa, R.: A computational framework for polyconvex large strain elasticity. Comput. Methods Appl. Mech. Eng. 283, 1061–1094 (2015)MathSciNetzbMATHGoogle Scholar
  11. Bulatov, V.V., Chang, J., Cai, W., Yip, S.: Molecular dynamics simulations of motion of edge and screw dislocations in a metal. Comput. Mater. Sci. 23(1), 111–115 (2002)Google Scholar
  12. Cleveringa, H.H.M., van der Giessen, E., Needleman, A.: A discrete dislocation analysis of bending. Int. J. Plast. 15(8), 837–868 (1999)zbMATHGoogle Scholar
  13. Conti, S., Garroni, A., Massaccesi, A.: Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity. Calc. Var. Partial Differ. Equ. 54(2), 1847–1874 (2015)MathSciNetzbMATHGoogle Scholar
  14. Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences. Springer, New York (2008)zbMATHGoogle Scholar
  15. Francfort, G., Mielke, A.: Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math. 595, 55–91 (2006)MathSciNetzbMATHGoogle Scholar
  16. Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations I. Cartesian currents. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 37. Springer, Berlin (1998)Google Scholar
  17. Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Applied Mathematical Sciences, vol. 137. Springer, New York (2000)Google Scholar
  18. Knees, D., Mielke, A.: Energy release rate for cracks in finite-strain elasticity. Math. Methods Appl. Sci. 31(5), 501–528 (2008)MathSciNetzbMATHGoogle Scholar
  19. Krantz, S.G., Parks, H.R.: Geometric Integration Theory. Cornerstones, Birkhäuser (2008)zbMATHGoogle Scholar
  20. Mainik, A., Mielke, A.: Global existence for rate-independent gradient plasticity at finite strain. J. Nonlinear Sci. 19(3), 221–248 (2009)MathSciNetzbMATHGoogle Scholar
  21. Mielke, A.: Evolution of rate-independent systems. In: Handbook of Differential Equations: Evolutionary Equations, vol. II, pp. 461–559. Elsevier/North-Holland, Amsterdam (2005)Google Scholar
  22. Mielke, A., Roubicek, T.: Rate-Independet Systems: Theory and Applications. Springer, New York (2015)zbMATHGoogle Scholar
  23. Mielke, A., Roubicek, T.: Rate-independent damage processes in nonlinear elasticity. Math. Models Methods Appl. Sci. 16, 81–123 (2006)MathSciNetzbMATHGoogle Scholar
  24. Monnet, G., Terentyev, D.: Structure and mobility of the edge dislocation in bcc iron studied by molecular dynamics. Acta Mater. 57(5), 1416–1426 (2009)Google Scholar
  25. Mora, M.G., Peletier, M., Scardia, L.: Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement. SIAM J. Math. Anal. 49, 4149–4205 (2017)MathSciNetzbMATHGoogle Scholar
  26. Müller, G., Friedrich, J.: Challenges in modeling of bulk crystal growth. J. Cryst. Growth 266(1–3), 1–19 (2004)Google Scholar
  27. 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)MathSciNetzbMATHGoogle Scholar
  28. Patrizi, S., Valdinoci, E.: Long-time behavior for crystal dislocation dynamics. Math. Models Methods Appl. Sci. 27(12), 2185–2228 (2017)MathSciNetzbMATHGoogle Scholar
  29. Scala, R., Van Goethem, N.: Constraint reaction and the Peach-Koehler force for dislocation networks. Math. Mech. Complex Syst. 4(2), 105–138 (2016a)MathSciNetzbMATHGoogle Scholar
  30. Scala, R., Van Goethem, N.: Currents and dislocations at the continuum scale. Methods Appl. Anal. 23(1), 1–34 (2016b)MathSciNetzbMATHGoogle Scholar
  31. Scala, R., Van Goethem, N.: A variational approach to single crystals with dislocations. Preprint (2018)Google Scholar
  32. Scala, R., Van Goethem, N.: Geometric and analytic properties of dislocation singularities. Proc. R. Soc. Edinb. Sect. A 149(4) (2019)Google Scholar
  33. Scala, R., Melching, D., Zeman, J.: Damage model for plastic materials at finite strains. Preprint (2017)Google Scholar
  34. Thomas, M.: Quasistatic damage evolution with spatial BV-regularization. Discret. Contin. Dyn. Syst. S 6(1), 235–255 (2013)MathSciNetzbMATHGoogle Scholar
  35. Thomas, M., Mielke, A.: Damage of nonlinearly elastic materials at small strain: existence and regularity results. ZAMM Z. Angew. Math. Mech. 90(2), 88–112 (2010)MathSciNetzbMATHGoogle Scholar
  36. van der Giessen, E., Needleman, A.: Discrete dislocation plasticity: a simple planar model. Model. Simul. Mater. Sci. Eng. 3, 689–735 (1995)Google Scholar
  37. Van Goethem, N., Dupret, F.: A distributional approach to \(2{D}\) Volterra dislocations at the continuum scale. Eur. J. Appl. Math. 23(3), 417–439 (2012a)Google Scholar
  38. Van Goethem, N., Dupret, F.: A distributional approach to the geometry of 2D dislocations at the continuum scale. Ann. Univ. Ferrara 58(2), 407–434 (2012b).  https://doi.org/10.1007/s11565-012-0149-5
  39. van Meurs, P.J.P.: Discrete-to-continuum limits of interacting dislocations. Ph.D. thesis, Technische Universiteit, Eindhoven (2015)Google Scholar
  40. Zbib, H.M.: Introduction to discrete dislocation dynamics. In: Sansour, C., Skatulla, S. (eds.) Springer (2012)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universidade de Lisboa, Faculdade de Ciências, CMAF_CIOAlameda da UniversidadeLisboaPortugal

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