Skip to main content
Log in

Recent Progress in Discrete Dislocation Dynamics and Its Applications to Micro Plasticity

  • Published:
JOM Aims and scope Submit manuscript

Abstract

We present a self-contained review of the discrete dislocation dynamics (DDD) method for the numerical investigation of plasticity in crystals, focusing on recent development and implementation progress. The review covers the theoretical foundations of DDD within the framework of incompatible elasticity, its numerical implementation via the nodal method, the extension of the method to finite domains and several implementation details. Applications of the method to current topics in micro-plasticity are presented, including the size effects in nano-indentation, the evolution of the dislocation microstructure in persistent slip bands, and the phenomenon of dislocation avalanches in micro-pillar compression.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. The more general case involving thermal and vacancy diffusion causing dislocation climb is treated in Ref. 24.

  2. At higher dislocation velocity, the quadratic form of π cannot be used and non-linear dislocation mobility laws are obtained. Moreover, the quadratic form of π does not capture lattice-controlled mobility typical of bcc metals at low temperature.

  3. The arbitrariness of \( \beta_{ij}^{P*} \) implies that its bounding curve (or virtual dislocation line) is also arbitrary, in agreement with the "mirror image construction" of 22 and the "independence of the total stress on the choice of virtual segments" stated in Ref. 53.

References

  1. A.H. Cottrell, Dislocations in Solids V11, ed. F.R.N. Nabarro and M. Duesbery (Amsterdam: North-Holland, 2002), pp. vii–xvii.

    Google Scholar 

  2. W.K. Liu, E.G. Karpov, S. Zhang, and H.S. Park, Comput. Method Appl. M 193, 1529 (2004).

    MathSciNet  MATH  Google Scholar 

  3. W.A. Curtin and R.E. Miller, Model. Simul. Mater. Sci. 11, R33 (2003).

    Google Scholar 

  4. S. Groh and H.M. Zbib, J. Eng. Mater. Technol 131, 041209 (2009).

    Google Scholar 

  5. J.M. Burgers, P K Ned Akad Wetensc 42, 293 (1939).

    Google Scholar 

  6. M. Peach and J. Koehler, Phys. Rev. 80, 436 (1950).

    MathSciNet  MATH  Google Scholar 

  7. J. Friedel, Dislocations (Reading, MA: Addison-Wesley, 1967).

    Google Scholar 

  8. F.R.N. Nabarro, Theory of Crystal Dislocations (New York: Dover Publications, 1987).

    Google Scholar 

  9. J.P. Hirth and J. Lothe, Theory of Dislocations, 2nd ed. (Malabar, FL: Krieger Publishing Company, 1992).

    Google Scholar 

  10. L.M. Brown, Philos. Mag. 10, 441 (1964).

    MATH  Google Scholar 

  11. D.J. Bacon, Phys. Status Solidi B 23, 527 (1967).

    Google Scholar 

  12. A.J.E. Foreman and M.J. Makin, Philos. Mag. 14, 911 (1966).

    Google Scholar 

  13. A.J.E. Foreman, Philos. Mag. 15, 1011 (1967).

    Google Scholar 

  14. D.J. Bacon, U.F. Kocks, and R.O. Scattergood, Philos. Mag. 28, 1241 (1973).

    Google Scholar 

  15. J. Lepinoux and L.P. Kubin, Scr. Metall. Mater. 21, 833 (1987).

    Google Scholar 

  16. N.M. Ghoniem and R. Amodeo, Solid State Phenom. 3&4, 377 (1988).

    Google Scholar 

  17. A.N. Gulluoglu, D.J. Srolovitz, R. LeSar, and P.S. Lomdahl, Scr. Metall. 23, 1347 (1989).

    Google Scholar 

  18. L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet, Solid State Phenom. 23, 455 (1992).

    Google Scholar 

  19. K.W. Schwarz, Phys. Rev. Lett. 78, 4785 (1997).

    Google Scholar 

  20. H. Zbib, M. Rhee, and J. Hirth, Int. J. Mech. Sci. 40, 113 (1998).

    MATH  Google Scholar 

  21. N.M. Ghoniem, S. Tong, and L. Sun, Phys. Rev. B 61, 913 (2000).

    Google Scholar 

  22. D. Weygand, L. Friedman, E. Van der Giessen, and A. Needleman, Model. Simul. Mater. Sci. 10, 437 (2002).

    Google Scholar 

  23. V.V. Bulatov, W. Cai, J. Fier, M. Hiratani, G. Hommes, T. Pierce, M. Tang, M. Rhee, K. Yates, and T. Arsenlis, Supercomputing 2004. Proc. ACM/IEEE SC2004 Conf. (Piscataway, NJ: IEEE, 2014). doi:10.1109/SC.2004.53.

  24. G. Po and N.M. Ghoniem, J. Mech. Phys. Solids 66, 103 (2014).

    MathSciNet  Google Scholar 

  25. P. Pant, K. Schwarz, and S. Baker, Acta Mater. 51, 3243 (2003).

    Google Scholar 

  26. R. Miller, L. Shilkrot, and W. Curtin, Acta Mater. 52, 271 (2004).

    MathSciNet  Google Scholar 

  27. H.G.M. Kreuzer and R. Pippan, Mater. Sci. Eng. A 387–389, 254 (2004).

    Google Scholar 

  28. M. Fivel, M. Verdier, and G. Canova, Mater. Sci. Eng. A 234, 923 (1997).

    Google Scholar 

  29. H. Tang, K. Schwarz, and H. Espinosa, Acta Mater. 55, 1607 (2007).

    Google Scholar 

  30. J. El-Awady, S. Bulent Biner, and N. Ghoniem, Mech. Phys. Solids 56, 2019 (2008).

    MATH  Google Scholar 

  31. S. Akarapu, H. Zbib, and D. Bahr, Int. J. Plast. 26, 239 (2010).

    MATH  Google Scholar 

  32. H. Ziegler, Progress in Solid Mechanics, ed. I.N. Sneddon and R. Hill (Amsterdam: North Holland, 1963), pp. 91–193.

    Google Scholar 

  33. T. Mura, Micromechanics of Defects in Solids, 2nd ed. (Dordrecht: Kluwer Academic Publishers, 1987).

    Google Scholar 

  34. G.A. Maugin, Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics, 1st edn. (Chapman and all/CRC, 2010).

  35. L. Onsager, Phys. Rev. 37, 405 (1931).

    Google Scholar 

  36. L. Onsager, Ann. NY Acad. Sci. 46, 241 (1945).

    Google Scholar 

  37. L. Wickham, K.W. Schwarz, and J. Stolken, Phys. Rev. Lett. 83, 4574 (1999).

    Google Scholar 

  38. L.P. Kubin, R. Madec, and B. Devincre, Mater. Res. Soc. Symp. Proc. 779, W1.6.1 (2003).

    Google Scholar 

  39. D. Manocha and J. Demmel, ACM T Graphic 13, 73 (1994).

    MATH  Google Scholar 

  40. M. Duesbery, N. Louat, and K. Sadananda, Acta Metall. Mater. 40, 149 (1992).

    Google Scholar 

  41. B. Ramirez, N.M. Ghoniem, and G. Po, Phys. Rev. B 86, 094115 (2012).

    Google Scholar 

  42. K. Kang, J. Yin, and W. Cai, J. Mech. Phys. Solids 62, 181 (2014).

    Google Scholar 

  43. P. PauA, J. Kratochvil, and M. Benes, Acta Mater. 61, 7917 (2013).

    Google Scholar 

  44. G.E. Beltz and L.B. Freund, Phys. Status Solidi B 180, 303 (1993).

    Google Scholar 

  45. S. Aubry, K. Kang, S. Ryu, and W. Cai, Scr. Mater. 64, 1043 (2011).

    Google Scholar 

  46. C.R. Weinberger, A.T. Jennings, K. Kang, and J.R. Greer, J. Mech. Phys. Solids 60, 84 (2012).

    Google Scholar 

  47. G. Po, M. Lazar, D. Seif, and N.M. Ghoniem, J. Mech. Phys. Solids 68, 161 (2014).

    MathSciNet  Google Scholar 

  48. R.E. Miller and A. Acharya, J. Mech. Phys. Solids 52, 1507 (2004).

    MathSciNet  MATH  Google Scholar 

  49. J. Eshelby, Dislocations in Solids V1, ed. F.R.N. Nabarro (Amsterdam: North-Holland, 1979), pp. 167–221.

    Google Scholar 

  50. V.A. Lubarda, J.A. Blume, and A. Needleman, Acta Metall. Mater. 41, 625 (1993).

    Google Scholar 

  51. E. Van der Giessen and A. Needleman, Model. Simul. Mater. Sci. 3, 689 (1995).

    Google Scholar 

  52. J. Deng, A. El-Azab, and B.C. Larson, Philos. Mag. 88, 3527 (2008).

    Google Scholar 

  53. C.R. Weinberger, S. Aubry, S.W. Lee, W.D. Nix, and W. Cai, Model. Simul. Mater. Sci. 17, 7 (2009).

    Google Scholar 

  54. A. Gouldstone, N. Chollacoop, M. Dao, J. Li, A.M. Minor, and Y.L. Shen, Acta Mater. 55, 4015 (2007).

    Google Scholar 

  55. D. Tabor, The Hardness of Metals (Oxford: Oxford University Press, 2000).

    Google Scholar 

  56. N. Gane, Proc. R Soc. Lond. A 317, 367 (1970).

    Google Scholar 

  57. G. Upit and S. Varchenya, The Science of Hardness Testing and its Research Applications, ed. J.H. Westbrook (Metals Park: ASM, 1973), pp. 135–146.

    Google Scholar 

  58. G.M. Pharr, E.G. Herbert, and Y. Gao, Annu. Rev. Mater. Res. 40, 271 (2010).

    Google Scholar 

  59. W.D. Nix and H. Gao, J. Mech. Phys. Solids 46, 411 (1998).

    MATH  Google Scholar 

  60. Q. Ma and D.R. Clarke, J. Mater. Res. 10, 853 (1995).

    Google Scholar 

  61. K.W. McElhaney, J.J. Vlassak, and W.D. Nix, J. Mater. Res. 13, 1300 (1998).

    Google Scholar 

  62. J.G. Swadener, E.P. George, and G.M. Pharr, J. Mech. Phys. Solids 50, 681 (2002).

    MATH  Google Scholar 

  63. E. Demir, D. Raabe, N. Zaafarani, and S. Zaefferer, Acta Mater. 57, 559 (2009).

    Google Scholar 

  64. M. Fivel, C.F. Robertson, G.R. Canova, and L. Boulanger, Acta Mater. 46, 6183 (1998).

    Google Scholar 

  65. H.G.M. Kreuzer and R. Pippan, Mater. Sci. Eng. A 400, 460 (2005).

    Google Scholar 

  66. A. Widjaja, E. Van der Giessen, and A. Needleman, Mater. Sci. Eng. A 400, 456 (2005).

    Google Scholar 

  67. A. Widjaja, E. Van der Giessen, V. Deshpande, and A. Needleman, J. Mater. Res. 22, 655 (2007).

    Google Scholar 

  68. M.S. Mohamed, Ph.D. Dissertation, Florida State University (2012).

  69. N. Thomson, N. Wadsworth, and N. Louat, Philos. Mag. 1, 113 (1956).

    Google Scholar 

  70. A.T. Winter, Philos. Mag. 30, 719 (1974).

    Google Scholar 

  71. Z.S. Basinski and S.J. Basinski, Prog. Mater Sci. 36, 89 (1992).

    Google Scholar 

  72. T.H. Alden and W.A. Backofen, Acta Metall. 9, 352 (1961).

    Google Scholar 

  73. B. Ma and C. Laird, Acta Metall. 37, 325 (1989).

    Google Scholar 

  74. L.M. Brown, Mater. Sci. Eng. A A285, 35 (2000).

    Google Scholar 

  75. L.M. Brown, Philos. Mag. 84, 2501 (2004).

    Google Scholar 

  76. H. Mughrabi and F. Pschenitzka, Philos. Mag. 85, 3029 (2005).

    Google Scholar 

  77. L.M. Brown, Philos. Mag. 86, 4055 (2006).

    Google Scholar 

  78. K.W. Schwarz and H. Mughrabi, Philos. Mag. Lett. 86, 773 (2006).

    Google Scholar 

  79. J. Kristan and J. Kratochvil, Philos. Mag. 87, 4593 (2007).

    Google Scholar 

  80. J.A. El-Awady, N.M. Ghoniem, and H. Mughrabi, 8th Global Innovations Symposium: Trends in Materials and Manufacturing Technologies for Energy Production, ed. Brent L. Adams and Hamid Garmestani (Warrendale, PA: TMS, 2007), pp. 23–35.

  81. C. Depres, C.F. Robertson, and M.C. Fivel, Philos. Mag. 84, 2257 (2004).

    Google Scholar 

  82. H. Mughrabi, Mater. Sci. Eng. 33, 207 (1978).

    Google Scholar 

  83. A.S. Tetelman, Acta Metall. 10, 813 (1962).

    Google Scholar 

  84. W.G. Johnston and J.J. Gilman, J. Appl. Phys. 31, 632 (1960).

    Google Scholar 

  85. A.S. Tetelman, Philos. Mag. 7, 1801 (1962).

    Google Scholar 

  86. P. Veyssiere and F. Gregori, Philos. Mag. Lett. 81, 795 (2001).

    Google Scholar 

  87. P. Veyssiere and F. Gregori, Philos. Mag. A 82, 567 (2002).

    Google Scholar 

  88. P. Veyssiere, Y.L. Chiu, and M. Niewczas, Z. Metallkd. 97, 189 (2006).

    Google Scholar 

  89. M.D. Uchic, D.M. Dimiduk, J.N. Florando, and W.D. Nix, Science 305, 986 (2004).

    Google Scholar 

  90. D.M. Dimiduk, C. Woodward, R. LeSar, and M.D. Uchic, Science 312, 1188 (2006).

    Google Scholar 

  91. A.T. Jennings, M.J. Burek, and J.R. Greer, Phys. Rev. Lett. 104, 135503 (2010).

    Google Scholar 

  92. J.R. Greer, J.Y. Kim, and M.J. Burek, Jom-J. Min. Met. Mater. 61, 19 (2009).

    Google Scholar 

  93. M.D. Uchic, P.A. Shade, and D.M. Dimiduk, Ann. Rev. Mater. Res. 39, 361 (2009).

    Google Scholar 

  94. S.W. Lee, S.M. Han, and W.D. Nix, Acta Mater. 57, 4404 (2009).

    Google Scholar 

  95. A.T. Jennings, J. Li, and J.R. Greer, Acta Mater. 59, 5627 (2011).

    Google Scholar 

  96. W.D. Nix, J.R. Greer, G. Feng, and E.T. Lilleodden, Thin Solid Films 515, 3152 (2007).

    Google Scholar 

  97. J.R. Greer, C.R. Weinberger, and W. Cai, Mater. Sci. Eng. A 493, 21 (2008).

    Google Scholar 

  98. A. Van den Beukel, Phys. Status Solidi A 30, 197 (1975).

    Google Scholar 

  99. S.D. Mesarovic, J. Mech. Phys. Solids 43, 671 (1995).

    MathSciNet  MATH  Google Scholar 

  100. M. Koslowski, R. LeSar, and R. Thomson, Phys. Rev. Lett. 93, 125502 (2004).

    Google Scholar 

  101. F.F. Csikor, C. Motz, D. Weygand, M. Zaiser, and S. Zapperi, Science 318, 251 (2007).

    Google Scholar 

  102. P.D. Ispanovity, I. Groma, G. Gyorgyi, F.F. Csikor, and D. Weygand, Phys. Rev. Lett. 105, 085503 (2010).

    Google Scholar 

  103. J.A. El-Awady, M. Wen, and N.M. Ghoniem, J. Mech. Phys. Solids 57, 32 (2009).

    MATH  Google Scholar 

  104. J.R. Greer, W.C. Oliver, and W.D. Nix, Acta Mater. 53, 1821 (2005).

    Google Scholar 

  105. Z.W. Shan, R.K. Mishra, S.A. Syed Asif, O.L. Warren, and A.M. Minor, Nat. Mater. 7, 115 (2007).

    Google Scholar 

  106. J.T. Fourie and H.G.F. Wilsdorf, J. Appl. Phys. 31, 2219 (1960).

    Google Scholar 

  107. J.T. Fourie and R.J. Murphy, Philos. Mag. 7, 1617 (1962).

    Google Scholar 

  108. J.T. Fourie, Philos. Mag. 10, 1027 (1964).

    Google Scholar 

  109. S.I. Rao, D.M. Dimiduk, M. Tang, M.D. Uchic, T.A. Parthasarathy, and C. Woodward, Philos. Mag. 87, 4777 (2007).

    Google Scholar 

  110. S.I. Rao, D.M. Dimiduk, T.A. Parthasarathy, M.D. Uchic, M. Tang, and C. Woodward, Acta Mater. 56, 3245 (2008).

    Google Scholar 

  111. D. Kiener and A.M. Minor, Nano Lett. 11, 3816 (2011).

    Google Scholar 

Download references

Acknowledgements

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, under Award Number DE-FG02-03ER54708, and the US Air Force Office of Scientific Research (AFOSR), under award number FA9550-11-1-0282.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Giacomo Po.

Appendix A: A Elastic Fields of Dislocation Loops: Vector Notation

Appendix A: A Elastic Fields of Dislocation Loops: Vector Notation

The differential operators U ikl and S ijkl appearing in Eqs. 5 and 6 are, respectively

$$ U_{ikl} = \frac{1}{8\pi}\epsilon_{klm} \left({\delta_{im} \Delta - \frac{1}{1 - \nu}\partial_{i} \partial_{m}} \right) $$
(21)
$$ \begin{gathered} S_{ijkl} = \frac{\mu}{8\pi}\left[{\left({\delta_{il} \epsilon_{jmk} + \delta_{jl} \epsilon_{imk}} \right)\partial_{m} \Delta} \right. \hfill \\ \left. {+ \frac{2}{1 - \nu}\epsilon_{klm} \left({\partial_{i} \partial_{j} \partial_{m} - \delta_{ij} \partial_{m} \Delta} \right)} \right]{\mkern 1mu} \hfill \\ \end{gathered} $$
(22)

where Δ is the Laplace operator, ν is the Poisson ratio, and ji the shear modulus of the material. Using Eq. 21, the displacement field generated by a dislocation becomes

$$ \varvec{u}(\varvec{x}) = - \frac{{\varvec{b}\varOmega (\varvec{x})}}{4\pi } - \frac{1}{8\pi (1 - \nu )}\oint_{{\mathcal{L}}} {\frac{1}{R}\left\{ {\left( {1 - 2\nu } \right)\varvec{b} \times \widehat{\varvec{\xi}}^{\prime } + \left[ {\widehat{\varvec{R}}\cdot\left( {\varvec{b} \times \widehat{\varvec{\xi}}^{\prime } } \right)} \right]\widehat{\varvec{R}}} \right\}} {\text{d}}\ell ' $$
(23)

where Ω(x) is the solid angle subtended by the loop, R = x − x’, \( R = \sqrt {\varvec{R}\cdot\varvec{R}} \), and \( \widehat{\varvec{R}} = \varvec{R}/R \). With the same notation, using Eq. 22 the stress field caused by the loop becomes:

$$ \varvec{\sigma}(\varvec{x}) = \frac{\mu }{4\pi (1 - \nu )}\oint_{{\mathcal{L}}} {\frac{1}{{R^{2} }}\left\{ {(1 - \nu )\left[ {\widehat{\varvec{\xi}}^{\prime } \otimes (\varvec{b} \times \widehat{\varvec{R}}) + (\varvec{b} \times \widehat{\varvec{R}}) \otimes \widehat{\varvec{\xi}}^{\prime } } \right] + \left[ {(\widehat{\varvec{\xi}}^{\prime } \times \varvec{b}) \otimes \widehat{\varvec{R}} + \widehat{\varvec{R}} \otimes (\widehat{\varvec{\xi}}^{\prime } \times \varvec{b})} \right] + \widehat{\varvec{R}}\cdot(\varvec{b} \times \widehat{\varvec{\xi}}^{\prime } )\left[ {3\widehat{\varvec{R}} \otimes \widehat{\varvec{R}} + \varvec{I}} \right]} \right\}} {\text{d}}\ell^{\prime} $$
(24)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Po, G., Mohamed, M.S., Crosby, T. et al. Recent Progress in Discrete Dislocation Dynamics and Its Applications to Micro Plasticity. JOM 66, 2108–2120 (2014). https://doi.org/10.1007/s11837-014-1153-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11837-014-1153-2

Keywords

Navigation