Finite Element Analysis of Thermodynamically Consistent Strain Gradient Plasticity Theory and Applications

  • George Z. VoyiadjisEmail author
  • Yooseob Song
Living reference work entry


In this chapter, a coupled thermomechanical gradient-enhanced continuum plasticity theory containing the flow rules of the grain interior and grain boundary areas is developed within the thermodynamically consistent framework. Two-dimensional finite element implementation for the proposed gradient plasticity theory is carried out to examine the micro-mechanical and thermal characteristics of small-scale metallic volumes. The proposed model is conceptually based on the dislocation interaction mechanisms and thermal activation energy. The thermodynamic conjugate microstresses are decomposed into dissipative and energetic components; correspondingly, the dissipative and energetic length scales for both the grain interior and grain boundary are incorporated in the proposed model, and an additional length scale related to the geometrically necessary dislocation-induced strengthening is also included. Not only the partial heat dissipation caused by the fast transient time but also the distribution of temperature caused by the transition from the plastic work to the heat is included into the coupled thermomechanical model by deriving a generalized heat equation. The derived constitutive framework and two-dimensional finite element model are validated through the comparison with the experimental observations conducted on microscale thin films. The proposed enhanced model is examined by solving the simple shear problem and the square plate problem to explore the thermomechanical characteristics of small-scale metallic materials. Finally, some significant conclusions are presented.


Strain gradient plasticity Thermomechanical coupling Grain boundary 2D FEM Validation Size effect 


  1. Abaqus, User’s Manual (Version 6.12) (Dassault Systemes Simulia Corporation, Providence, 2012)Google Scholar
  2. E.C. Aifantis, J. Eng. Mater-T Asme. 106, 4 (1984)CrossRefGoogle Scholar
  3. E.C. Aifantis, Int. J. Plast. 3, 3 (1987)CrossRefGoogle Scholar
  4. K.E. Aifantis, J.R. Willis, J. Mech. Phys. Solids 53, 5 (2005)CrossRefGoogle Scholar
  5. A.H. Almasri, G.Z. Voyiadjis, Acta Mech. 209, 1–2 (2010)CrossRefGoogle Scholar
  6. L. Anand, M.E. Gurtin, S.P. Lele, C. Gething, J. Mech. Phys. Solids 53, 8 (2005)Google Scholar
  7. A. Arsenlis, D.M. Parks, Acta Mater. 47, 5 (1999)CrossRefGoogle Scholar
  8. L. Bardella, J. Mech. Phys. Solids 54, 1 (2006)MathSciNetCrossRefGoogle Scholar
  9. D.J. Benson, H.H. Fu, M.A. Meyers, Mater. Sci. Eng. A 319, 319–321 (2001)Google Scholar
  10. T. Bergeman, J. Qi, D. Wang, Y. Huang, H.K. Pechkis, E.E. Eyler, P.L. Gould, W.C. Stwalley, R.A. Cline, J.D. Miller, D.J. Heinzen, J. Phys. B-Atom. Mol. Opt. Phys. 39, 19 (2006)CrossRefGoogle Scholar
  11. S.D. Brorson, A. Kazeroonian, J.S. Moodera, D.W. Face, T.K. Cheng, E.P. Ippen, M.S. Dresselhaus, G. Dresselhaus, Phys. Rev. Lett. 64, 18 (1990)CrossRefGoogle Scholar
  12. H.B. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, Hoboken, 1985)zbMATHGoogle Scholar
  13. P. Cermelli, M.E. Gurtin, Int. J. Solids Struct. 39, 26 (2002)CrossRefGoogle Scholar
  14. W.A.T. Clark, R.H. Wagoner, Z.Y. Shen, T.C. Lee, I.M. Robertson, H.K. Birnbaum, Scr. Met. Mater. 26, 2 (1992)CrossRefGoogle Scholar
  15. B.D. Coleman, W. Noll, Arch. Ration. Mech. Anal. 13, 3 (1963)Google Scholar
  16. D. Faghihi, G.Z. Voyiadjis, J. Eng. Mater-T Asme. 136, 1 (2014)Google Scholar
  17. N.A. Fleck, J.W. Hutchinson, Adv. Appl. Mech. 33, 295 (1997)CrossRefGoogle Scholar
  18. N.A. Fleck, J.W. Hutchinson, J. Mech. Phys. Solids 49, 10 (2001)Google Scholar
  19. N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Acta Metall. Mater. 42, 2 (1994)Google Scholar
  20. N.A. Fleck, J.W. Hutchinson, J.R. Willis, P. Roy. Soc. A-Math. Phy. 470, 2170 (2014)CrossRefGoogle Scholar
  21. N.A. Fleck, J.W. Hutchinson, J.R. Willis, J. Appl. Mech-T Asme. 82, 7 (2015)CrossRefGoogle Scholar
  22. S. Forest, J. Eng. Mech-Asce. 135, 3 (2009)CrossRefGoogle Scholar
  23. S. Forest, M. Amestoy, Cr. Mecanique 336, 4 (2008)CrossRefGoogle Scholar
  24. P. Fredriksson, P. Gudmundson, Model. Simul. Mater. Sci. 15, 1 (2007)CrossRefGoogle Scholar
  25. A. Garroni, G. Leoni, M. Ponsiglione, J. Eur. Math. Soc. 12, 5 (2010)Google Scholar
  26. S. Giacomazzi, F. Leroi, C. L'Henaff, J.J. Joffraud, Lett. Appl. Microbiol. 38, 2 (2004)CrossRefGoogle Scholar
  27. H. Gleiter, Acta Mater. 48, 1 (2000)CrossRefGoogle Scholar
  28. P. Gudmundson, J. Mech. Phys. Solids 52, 6 (2004)MathSciNetCrossRefGoogle Scholar
  29. M.E. Gurtin, Int. J. Plast. 19, 1 (2003)CrossRefGoogle Scholar
  30. M.E. Gurtin, J. Mech. Phys. Solids 52, 11 (2004)CrossRefGoogle Scholar
  31. M.E. Gurtin, J. Mech. Phys. Solids 56, 2 (2008)Google Scholar
  32. M.E. Gurtin, L. Anand, J. Mech. Phys. Solids 53, 7 (2005)Google Scholar
  33. M.E. Gurtin, L. Anand, J. Mech. Phys. Solids 57, 3 (2009)CrossRefGoogle Scholar
  34. M.E. Gurtin, E. Fried, L. Anand, The Mechanics and Thermodynamics of Continua (Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar
  35. S. Han, T. Kim, H. Lee, H. Lee, Electronics System-Integration Technology Conference, 2008. ESTC 2008. 2nd (2008)Google Scholar
  36. M.A. Haque, M.T.A. Saif, Acta Mater. 51, 11 (2003)CrossRefGoogle Scholar
  37. J.P. Hirth, J. Lothe, Theory of Dislocations (Krieger Publishing Company, 1982). ISBN: 0894646176, 9780894646171Google Scholar
  38. Y. Huang, J. Qi, H.K. Pechkis, D. Wang, E.E. Eyler, P.L. Gould, W.C. Stwalley, J. Phys. B-Atom. Mol. Opt. Phys. 39, 19 (2006)CrossRefGoogle Scholar
  39. J.W. Hutchinson, Acta. Mech. Sinica-Prc. 28, 4 (2012)Google Scholar
  40. J.S. Hwang, H.L. Park, T.W. Kim, H.J. Lee, Phys. Status Solidi a-Appl. Res. 148, 2 (1995)CrossRefGoogle Scholar
  41. A.M. Ivanitsky, D.A. Kadakov, Izv. Vyssh. Uchebn. Zaved Radiofiz. 26, 9 (1983)Google Scholar
  42. T.W. Kim, H.L. Park, J. Cryst. Growth 159, 1–4 (1996)CrossRefGoogle Scholar
  43. T.W. Kim, H.L. Park, J.Y. Lee, Appl. Phys. Lett. 64, 19 (1994)CrossRefGoogle Scholar
  44. T.C. Lee, I.M. Robertson, H.K. Birnbaum, Scr. Metall. 23, 5 (1989)Google Scholar
  45. S.P. Lele, L. Anand, Philos. Mag. 88, 30–32 (2008)CrossRefGoogle Scholar
  46. J.M. Lim, K. Cho, M. Cho, Appl. Phys. Lett. 110, 1 (2017)CrossRefGoogle Scholar
  47. W.J. Liu, K. Saanouni, S. Forest, P. Hu, J. Non-Equil. Thermody. 42, 4 (2017)CrossRefGoogle Scholar
  48. V.A. Lubarda, Int. J. Solids Struct. 45, 1 (2008)CrossRefGoogle Scholar
  49. D.L. McDowell, Int. J. Plast. 26, 9 (2010)CrossRefGoogle Scholar
  50. H. Mecking, U.F. Kocks, Acta Metall. 29, 1865–1875 (1981)Google Scholar
  51. M.A. Meyers, A. Mishra, D.J. Benson, Prog. Mater. Sci. 51, 427 (2006)CrossRefGoogle Scholar
  52. H.B. Muhlhaus, E.C. Aifantis, Int. J. Solids Struct. 28, 7 (1991)CrossRefGoogle Scholar
  53. L. Nicola, Y. Xiang, J.J. Vlassak, E. Van der Giessen, A. Needleman, J. Mech. Phys. Solids 54, 10 (2006)CrossRefGoogle Scholar
  54. W.D. Nix, H.J. Gao, J. Mech. Phys. Solids 46, 3 (1998)CrossRefGoogle Scholar
  55. T. Ohmura, A.M. Minor, E.A. Stach, J.W. Morris, J. Mater. Res. 19, 12 (2004)Google Scholar
  56. N. Ohno, D. Okumura, J. Mech. Phys. Solids 55, 9 (2007)CrossRefGoogle Scholar
  57. P.A. Parilla, M.F. Hundley, A. Zettl, Solid State Commun. 87, 6 (1993)Google Scholar
  58. H.L. Park, S.H. Lee, T.W. Kim, Compd. Semicond. 1995, 145 (1996)Google Scholar
  59. J.M. Pipard, N. Nicaise, S. Berbenni, O. Bouaziz, M. Berveiller, Comput. Mater. Sci. 45, 604–610 (2009)CrossRefGoogle Scholar
  60. X. Qing, G. Xingming, Int. J. Solids Struct. 43, 25–26 (2006)Google Scholar
  61. W.A. Soer, K.E. Aifantis, J.T.M. De Hosson, Acta Mater. 53, 17 (2005)CrossRefGoogle Scholar
  62. Y. Song, G.Z. Voyiadjis, Int. J. Solids Struct. 134, 195–215 (2018a)Google Scholar
  63. Y. Song, G.Z. Voyiadjis, J. Theor. App. Mech-Pol. 56, 2 (2018b)Google Scholar
  64. S. Sun, B.L. Adams, W.E. King, Philos. Mag. A 80, 1 (2000)Google Scholar
  65. D.Y. Tzou, Y.S. Zhang, Int. J. Eng. Sci. 33, 10 (1995)CrossRefGoogle Scholar
  66. R. Venkatraman, P.R. Besser, J.C. Bravman, S. Brennan, J. Mater. Res. 9, 2 (1994)CrossRefGoogle Scholar
  67. E. Voce, Meta 51, 219 (1955)Google Scholar
  68. G.Z. Voyiadjis, B. Deliktas, Int. J. Plast. 25, 10 (2009a)CrossRefGoogle Scholar
  69. G.Z. Voyiadjis, B. Deliktas, Int. J. Eng. Sci. 47, 11–12 (2009b)CrossRefGoogle Scholar
  70. G.Z. Voyiadjis, B. Deliktas, Acta Mech. 213, 1–2 (2010)CrossRefGoogle Scholar
  71. G.Z. Voyiadjis, D. Faghihi, Int. J. Plast. 30–31, 218 (2012)CrossRefGoogle Scholar
  72. G.Z. Voyiadjis, R. Peters, Acta Mech. 211, 1–2 (2010)CrossRefGoogle Scholar
  73. G.Z. Voyiadjis, Y. Song, Philos. Mag. 97, 5 (2017)CrossRefGoogle Scholar
  74. G.Z. Voyiadjis, C. Zhang, Mat. Sci. Eng. A-Struct. 621, 218 (2015)CrossRefGoogle Scholar
  75. G.Z. Voyiadjis, A.H. Almasri, T. Park, Mech. Res. Commun. 37, 3 (2010)CrossRefGoogle Scholar
  76. G.Z. Voyiadjis, D. Faghihi, Y.D. Zhang, Int. J. Solids Struct. 51, 10 (2014)CrossRefGoogle Scholar
  77. G.Z. Voyiadjis, Y. Song, T. Park, J. Eng. Mater. Technol. 139, 2 (2017)Google Scholar
  78. Y. Xiang, J.J. Vlassak, Acta Mater. 54, 20 (2006)Google Scholar
  79. M. Yaghoobi, G.Z. Voyiadjis, Acta Mater. 121, 190 (2016)CrossRefGoogle Scholar
  80. C. Zhang, G.Z. Voyiadjis, Mat. Sci. Eng. A-Struct. 659, 55 (2016)CrossRefGoogle Scholar
  81. B. Zhang, Y. Song, G.Z. Voyiadjis, W.J. Meng, J. Mater. Res. 361(1–2), 160–164 (2018)Google Scholar
  82. J.R. Zhao, J.C. Li, Q. Jiang, J. Alloys Compd. 361, 160 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringLouisiana State UniversityBaton RougeUSA

Personalised recommendations