Skip to main content
SpringerLink
Log in

From Atomistic Model to the Peierls–Nabarro Model with \({\gamma}\)-surface for Dislocations

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

The Peierls–Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this paper, we study the convergence from a full atomistic model to the PN model with \({\gamma}\)-surface for the dislocation in a bilayer system. We prove that the displacement field and the total energy of the dislocation solution of the PN model are asymptotically close to those of the full atomistic model. Our work can be considered as a generalization of the analysis of the convergence from atomistic model to Cauchy–Born rule for crystals without defects.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ariza, M.P., Ortiz, M.: Discrete dislocations in graphene. J. Mech. Phys. Solids 58, 710–734 (2010)

    Article  ADS  Google Scholar 

  2. Ariza, M.P., Serrano, R., Mendez, J.P., Ortiz, M.: Stacking faults and partial dislocations in graphene. Philos. Mag. 92, 2004–2021 (2012)

    Article  ADS  Google Scholar 

  3. Le Bris, C., Lions, P.-L., Blanc, X.: From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. 164, 341–381 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Oxford University Press, Oxford (1954)

    MATH  Google Scholar 

  5. Braides, A., Dal Maso, G., Garroni, A.: Variational formulation of softening phenomena in fracture mechanics: The one-dimensional case. Arch. Ration. Mech. Anal. 146, 23–58 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bulatov, V.V., Kaxiras, E.: Semidiscrete variational Peierls framework for dislocation core properties. Phys. Rev. Lett. 78, 4221–4223 (1997)

    Article  ADS  Google Scholar 

  7. Conti, S., Dolzmann, G., Kirchheim, B., Müller, S.: Sufficient conditions for the validity of the Cauchy=-Born rule close to SO(n). J. Eur. Math. Soc. 8, 515–530 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Conti, S., Garroni, A., Müller, S.: Singular kernels, multiscale decomposition of microstructure, and dislocation models. Arch. Ration. Mech. Anal. 199, 779–819 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conti, S., Garroni, A., Müller, S.: Dislocation microstructures and strain-gradient plasticity with one active slip plane. J. Mech. Phys. Solids 93, 240–251 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  10. Conti, S., Garroni, A., Ortiz, M.: The line-tension approximation as the dilute limit of linear-elastic dislocations. Arch. Ration. Mech. Anal. 218(2), 699–755 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dai, S., Xiang, Y., Srolovitz, D.J.: Structure and energy of (111) low-angle twist boundaries in Al. Cu and Ni. Acta Mater. 61(4), 1327–1337 (2013)

    Article  Google Scholar 

  12. Dai, S., Xiang, Y., Srolovitz, D.J.: Atomistic, generalized Peierls-Nabarro and analytical models for (111) twist boundaries in Al, Cu and Ni for all twist angles. Acta Mater. 69, 162–174 (2014)

    Article  Google Scholar 

  13. Dai, S., Xiang, Y., Srolovitz, D.J.: Structure and energetics of interlayer dislocations in bilayer graphene. Phys. Rev. B 93, 085410 (2016)

    Article  ADS  Google Scholar 

  14. Dai, S., Xiang, Y., Srolovitz, D.J.: Twisted bilayer graphene: Moire with a twist. Nano Lett. 16, 5923–5927 (2016)

    Article  ADS  Google Scholar 

  15. Dai, S., Xiang, Y., Zhang, T.-Y.: A continuum model for core relaxation of incoherent twin boundaries based on the Peierls-Nabarro framework. Scr. Mater. 64, 438–441 (2011)

    Article  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dipierro, S., Palatucci, G., Valdinoci, E.: Dislocation dynamics in crystals: a macroscopic theory in a fractional laplace setting. Commun. Math. Phys. 333, 1061–1105 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Dipierro, S., Patrizi, S., Valdinoci, E.: Chaotic orbits for systems of nonlocal equations. Commun. Math. Phys. 349, 583–626 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. E, W., Lu, J.: Electronic structure of smoothly deformed crystals: Cauchy–Born rule for the nonlinear tight-binding model. Commun. Pure Appl. Math. 63, 1432–1468 (2010)

  20. E, W., Ming, P.: Cauchy–Born rule and the stability of crystalline solids: static problems. Arch. Ration. Mech. Anal. 183, 241–297 (2007)

  21. El Hajj, A., Ibrahim, H., Monneau, R.: Dislocation dynamics: from microscopic models to macroscopic crystal plasticity. Continuum Mech. Thermodyn. 21, 109–123 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Eshelby, J.D.: Edge dislocations in anisotropic materials. Philos. Mag. 40, 903–912 (1949)

    Article  MATH  Google Scholar 

  23. Fino, A.Z., Ibrahim, H., Monneau, R.: The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model. J. Differ. Equ. 252, 258–293 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Frenkel, Y.I., Kontorova, T.: The model of dislocation in solid body. Zh. Eksp. Teor. Fiz 8, 1340–1348 (1938)

    MATH  Google Scholar 

  25. Friesecke, G., Theil, F.: On the validity and failure of the Cauchy-Born rule in a two-dimensional mass-spring model. J. Nonlinear Sci. 12, 445–478 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Garroni, A., Müller, S.: \(\Gamma \)-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36, 1943–1964 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, New York (2001)

    MATH  Google Scholar 

  30. Gonzalez, M.d.M., Monneau, R.: Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete Contin. Dyn. Syst. 32, 1255–1286 (2010)

  31. Hartford, J., von Sydow, B., Wahnström, G., Lundqvist, B.I.: Peierls barriers and stresses for edge dislocations in Pd and Al calculated from first principles. Phys. Rev. B 58, 2487–2496 (1998)

    Article  ADS  Google Scholar 

  32. Hirth, J.P., Lothe, J.: Theory of Dislocations, 2nd edn. Wiley, New York (1982)

    MATH  Google Scholar 

  33. Hudson, T., Ortner, C.: Existence and stability of a screw dislocation under anti-plane deformation. Arch. Ration. Mech. Anal. 213, 887–929 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kaxiras, E., Duesbery, M.S.: Free energies of generalized stacking faults in Si and implications for the brittle-ductile transition. Phys. Rev. Lett. 70, 3752–3755 (1993)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Lu, G., Bulatov, V.V., Kioussis, N.: A nonplanar Peierls-Nabarro model and its application to dislocation cross-slip. Philos. Mag. 83, 3539–3548 (2003)

    Article  ADS  Google Scholar 

  37. Lu, G., Kioussis, N., Bulatov, V.V., Kaxiras, E.: Generalized stacking fault energy surface and dislocation properties of aluminum. Phys. Rev. B 62, 3099–3108 (2000)

    Article  ADS  Google Scholar 

  38. Lu, J., Ming, P.: Convergence of a force-based hybrid method in three dimensions. Commun. Pure Appl. Math. 66, 83–108 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Makridakis, C., Suli, E.: Finite element analysis of Cauchy-Born approximations to atomistic models. Arch. Ration. Mech. Anal. 207, 813–843 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mianroodi, J.R., Svendsen, B.: Atomistically determined phase-field modeling of dislocation dissociation, stacking fault formation, dislocation slip, and reactions in fcc systems. J. Mech. Phys. Solids 77, 109–122 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  41. Miller, R., Phillips, R., Beltz, G., Ortiz, M.: A non-local formulation of the Peierls dislocation model. J. Mech. Phys. Solids 46, 1845–1867 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Monneau, R., Patrizi, S.: Derivation of Orowan's law from the Peierls-Nabarro model. Comm. Partial Differential Equations 37, 1887–1911 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Movchan, A.B., Bullough, R., Willis, J.R.: Stability of a dislocation: Discrete model. Eur. J. Appl. Math. 9, 373–396 (1998)

    Article  MATH  Google Scholar 

  44. Movchan, A.B., Bullough, R., Willis, J.R.: Two-dimensional lattice models of the Peierls type. Philos. Mag. 83, 569–587 (2003)

    Article  ADS  Google Scholar 

  45. Nabarro, F.R.N.: Dislocations in a simple cubic lattice. Proc. Phys. Soc. 59, 256–272 (1947)

    Article  ADS  Google Scholar 

  46. Nabarro, F.R.N.: Fifty-year study of the Peierls-Nabarro stress. Mater. Sci. Eng. A 234, 67–76 (1997)

    Article  Google Scholar 

  47. Ortner, C., Theil, F.: Justification of the Cauchy-Born approximation of elastodynamics. Arch. Ration. Mech. Anal. 207, 1025–1073 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Patrizi, S., Valdinoci, E.: Crystal dislocations with different orientations and collisions. Arch. Ration. Mech. Anal. 217, 231–261 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Patrizi, S., Valdinoci, E.: Homogenization and Orowans law for anisotropic fractional operators of any order. Nonlinear Anal. 119, 3–36 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Patrizi, S., Valdinoci, E.: Long-time behavior for crystal dislocation dynamics. arXiv:1609.04441 (2016)

  51. Patrizi, S., Valdinoci, E.: Relaxation times for atom dislocations in crystals. Calc. Var. Partial Differ. Equ. 55, 71 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. Peierls, R.: The size of a dislocation. Proc. Phys. Soc. 52, 34–37 (1940)

    Article  ADS  Google Scholar 

  53. Ponsiglione, M.: Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuous. SIAM J. Math. Anal. 39(2), 449–469 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Schoeck, G.: The generalized Peierls-Nabarro model. Philos. Mag. A 69, 1085–1095 (1994)

    Article  ADS  Google Scholar 

  55. Schoeck, G.: Peierls energy of dislocations: a critical assessment. Phys. Rev. Lett. 82, 2310–2313 (1999)

    Article  ADS  Google Scholar 

  56. Schoeck, G.: The Peierls energy revisited. Philos. Mag. A 79, 2629–2636 (1999)

    Article  ADS  Google Scholar 

  57. Shen, C., Li, J., Wang, Y.: Predicting structure and energy of dislocations and grain boundaries. Acta Mater. 74, 125–131 (2014)

    Article  Google Scholar 

  58. Shen, C., Wang, Y.: Incorporation of \(\gamma \)-surface to phase field model of dislocations: simulating dislocation dissociation in fcc crystals. Acta Mater. 52, 683–691 (2004)

    Article  Google Scholar 

  59. Vitek, V.: Intrinsic stacking faults in body-centred cubic crystals. Philos. Mag. 18, 773–786 (1968)

    Article  ADS  Google Scholar 

  60. Vitek, V., Lejček, L., Bowen, D.K.: On the factors controlling the structure of dislocation cores in b.c.c. crystals. In: Gehlen P.C., Jr. Beeler J.R., Jaffee R.I. (eds.) Interatomic Potentials and Simulation of Lattice Defects, pp. 493–508. Plenum Press, New York, 1971

  61. Wang, S.: The dislocation equation as a generalization of Peierls equation. Philos. Mag. 95, 3768–3784 (2015)

    Article  ADS  Google Scholar 

  62. Wang, S., Dai, S., Li, X., Yang, J., Srolovitz, D.J., Zheng, Q.S.: Measurement of the cleavage energy of graphite. Nat. Commun. 6, 7853 (2015)

    Article  ADS  Google Scholar 

  63. Wei, H., Xiang, Y.: A generalized Peierls-Nabarro model for kinked dislocations. Philos. Mag. 89, 2333–2354 (2009)

    Article  ADS  Google Scholar 

  64. Wei, H., Xiang, Y., Ming, P.: A generalized peierls-nabarro model for curved dislocations using discrete fourier transform. Commun. Comput. Phys. 4, 275–293 (2008)

    MathSciNet  MATH  Google Scholar 

  65. Wu, Z.X., Curtin, W.A.: Mechanism and energetics of c+ał dislocation cross-slip in hcp metals. Proc. Natl. Acad. Sci. 113, 11137–11142 (2016)

    Article  ADS  Google Scholar 

  66. Xiang, Y.: Modeling dislocations at different scales. Commun. Comput. Phys. 1, 383–424 (2006)

    MATH  Google Scholar 

  67. Xiang, Y.: Continuum approximation of the Peach-Koehler force on dislocations in a slip plane. J. Mech. Phys. Solids 57, 728–743 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  68. Xiang, Y., Wei, H., Ming, P.: E, W.: A generalized Peierls-Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu. Acta Mater. 56, 1447–1460 (2008)

    Article  Google Scholar 

  69. Xu, G., Argon, A.S.: Homogeneous nucleation of dislocation loops under stress in perfect crystals. Philos. Mag. Lett. 80, 605–611 (2000)

    Article  ADS  Google Scholar 

  70. Zhou, S., Han, J., Dai, S., Sun, J., Srolovitz, D.J.: van der Waals bilayer energetics: Generalized stacking-fault energy of graphene, boron nitride, and graphene/boron nitride bilayers. Phys. Rev. B 92, 155438 (2015)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Tao Luo & Yang Xiang

  2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, No. 55, East Road Zhong-Guan-Cun, Beijing, 100190, China

    Pingbing Ming

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Pingbing Ming

Authors
  1. Tao Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pingbing Ming
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Xiang.

Additional information

Communicated by I. Fonseca

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, T., Ming, P. & Xiang, Y. From Atomistic Model to the Peierls–Nabarro Model with \({\gamma}\)-surface for Dislocations. Arch Rational Mech Anal 230, 735–781 (2018). https://doi.org/10.1007/s00205-018-1257-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-018-1257-x

Search

Navigation