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Strain Gradient Crystal Plasticity: Thermodynamics and Implementation

  • Tuncay Yalçinkaya
Reference work entry

Abstract

This chapter studies the thermodynamical consistency and the finite element implementation aspects of a rate-dependent nonlocal (strain gradient) crystal plasticity model, which is used to address the modeling of the size-dependent behavior of polycrystalline metallic materials. The possibilities and required updates for the simulation of dislocation microstructure evolution, grain boundary-dislocation interaction mechanisms, and localization leading to necking and fracture phenomena are shortly discussed as well. The development of the model is conducted in terms of the displacement and the plastic slip, where the coupled fields are updated incrementally through finite element method. Numerical examples illustrate the size effect predictions in polycrystalline materials through Voronoi tessellation.

Keywords

Size effect Strain gradient Crystal plasticity Finite element Polycrystalline plasticity Thermodynamics 

Notes

Acknowledgements

Tuncay Yalçinkaya gratefully acknowledges the support by the Scientific and Technological Research Council of Turkey (TÜBİTAK) under the 3001 Programme (Grant No. 215M381).

References

  1. A. Acharya, J.L. Bassani, Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys. Solids 48, 1565–1595 (2000)MathSciNetCrossRefGoogle Scholar
  2. E.C. Aifantis, On the microstructural origin of certain inelastic models. J. Eng. Mater. Technol. 106, 326–330 (1984)CrossRefGoogle Scholar
  3. E.C. Aifantis, Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)Google Scholar
  4. K. Aifantis, J. Senger, D. Weygand, M. Zaiser, Discrete dislocation dynamics simulation and continuum modeling of plastic boundary layers in tricrystal micropillars. IOP Conf. Ser. Mater. Sci. Eng. 3, 012025 (2009)CrossRefGoogle Scholar
  5. A. Arsenlis, D.M. Parks, R. Becker, V.V. Bulatov, On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals. J. Mech. Phys. Solids 52, 1213–1246 (2004)MathSciNetCrossRefGoogle Scholar
  6. M.F. Ashby, The deformation of plastically non-homogeneous materials. Philos. Mag. 21, 399–424 (1970)CrossRefGoogle Scholar
  7. ASTM, Annual Book of ASTM Standards (ASTM International, West Conshohocken, 2009)Google Scholar
  8. F. Aurenhammer, Voronoi diagrams – a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)CrossRefGoogle Scholar
  9. J.L. Bassani, Incompatibility and a simple gradient theory. J. Mech. Phys. Solids 49, 1983–1996 (2001)CrossRefGoogle Scholar
  10. E. Bayerschen, A.T. McBride, B.D. Reddy, T. Böhlke, Review on slip transmission criteria in experiments and crystal plasticity models. J. Mater. Sci. 51(5), 2243–2258 (2016)CrossRefGoogle Scholar
  11. C.J. Bayley, W.A.M. Brekelmans, M.G.D. Geers, A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. Int. J. Solids Struct. 43, 7268–7286 (2006)CrossRefGoogle Scholar
  12. P. van Beers, V. Kouznetsova, M. Geers, Defect redistribution within a continuum grain boundary plasticity model. J. Mech. Phys. Solids 83, 243–262 (2015a)MathSciNetCrossRefGoogle Scholar
  13. P. van Beers, V. Kouznetsova, M. Geers, Grain boundary interfacial plasticity with incorporation of internal structure and energy. Mech. Mater. 90, 69–82 (2015b). Proceedings of the IUTAM Symposium on Micromechanics of Defects in SolidsGoogle Scholar
  14. U. Borg, A strain gradient crystal plasticity analysis of grain size effects in polycrystals. Eur. J. Mech. A-Solid. 26, 313–324 (2007)CrossRefGoogle Scholar
  15. S.H. Chen, T.C. Wang, A new hardening law for strain gradient plasticity. Acta Mater. 48, 3997–4005 (2000)CrossRefGoogle Scholar
  16. A. Di Schino, J. Kenny, Grain size dependence of the fatigue behaviour of a ultrafine-grained AISI 304 stainless steel. Mater. Lett. 57(21), 3182–3185 (2003)CrossRefGoogle Scholar
  17. F.P.E. Dunne, D. Rugg, A. Walker, Lengthscale-dependent, elastically anisotropic, physically-based HCP crystal plasticity: application to cold-dwell fatigue in Ti alloys. Int. J. Plast. 23, 1061–1083 (2007)CrossRefGoogle Scholar
  18. L.P. Evers, W.A.M. Brekelmans, M.G.D. Geers, Non-local crystal plasticity model with intrinsic SSD and GND effects. J. Mech. Phys. Solids 52, 2379–2401 (2004)CrossRefGoogle Scholar
  19. X. Feaugas, H. Haddou, Grain-size effects on tensile behavior of nickel and AISI 316l stainless steel. Metall. Mater. Trans. A 34A, 2329–2340 (2003)CrossRefGoogle Scholar
  20. N.A. Fleck, J.W. Hutchinson, Strain gradient plasticity. Adv. Appl. Mech. 33, 184–251 (1997)zbMATHGoogle Scholar
  21. N.A. Fleck, J.W. Hutchinson, A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001)CrossRefGoogle Scholar
  22. N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487 (1994)CrossRefGoogle Scholar
  23. N.A. Fleck, J.W. Hutchinson, J.R. Willis, Strain gradient plasticity under non-proportional loading. Proc. R. Soc. A 470, 20140267 (2014)MathSciNetCrossRefGoogle Scholar
  24. M.G.D. Geers, W.A.M. Brekelmans, C.J. Bayley, Second-order crystal plasticity: internal stress effects and cyclic loading. Modell. Simul. Mater. Sci. Eng. 15, 133–145 (2007)CrossRefGoogle Scholar
  25. D. Gottschalk, A. McBride, B. Reddy, A. Javili, P. Wriggers, C. Hirschberger, Computational and theoretical aspects of a grain-boundary model that accounts for grain misorientation and grain-boundary orientation. Comput. Mater. Sci. 111, 443–459 (2016)CrossRefGoogle Scholar
  26. P. Gudmundson, A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004)MathSciNetCrossRefGoogle Scholar
  27. M.E. Gurtin, On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48, 989–1036 (2000)MathSciNetCrossRefGoogle Scholar
  28. M.E. Gurtin, A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32 (2002)MathSciNetCrossRefGoogle Scholar
  29. M.E. Gurtin, A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations. Int. J. Plast. 24, 702–725 (2008)CrossRefGoogle Scholar
  30. M.E. Gurtin, L. Anand, Thermodynamics applied to gradient theories involving the accumulated plastic strain: the theories of Aifantis and Fleck and Hutchinson and their generalization. J. Mech. Phys. Solids 57, 405–421 (2009)MathSciNetCrossRefGoogle Scholar
  31. C.S. Han, H. Gao, Y. Huang, W.D. Nix, Mechanism-based strain gradient crystal plasticity – I. Theory. J. Mech. Phys. Solids 53, 1188–1203 (2005a)MathSciNetCrossRefGoogle Scholar
  32. C.S. Han, H. Gao, Y. Huang, W.D. Nix, Mechanism-based strain gradient crystal plasticity – II. Analysis. J. Mech. Phys. Solids 53, 1204–1222 (2005b)MathSciNetCrossRefGoogle Scholar
  33. M.A. Haque, M.T.A. Saif, Strain gradient effect in nanoscale thin films. Acta Mater. 51, 3053–3061 (2003)CrossRefGoogle Scholar
  34. Y. Huang, S. Qu, K.C. Hwang, M. Li, H. Gao, A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plast. 20, 753–782 (2004)CrossRefGoogle Scholar
  35. J.W. Hutchinson, Generalizing j2 flow theory: fundamental issues in strain gradient plasticity. Acta Mech. Sinica 28, 1078–1086 (2012)MathSciNetCrossRefGoogle Scholar
  36. B. Klusemann, T. Yalçinkaya, Plastic deformation induced microstructure evolution through gradient enhanced crystal plasticity based on a non-convex helmholtz energy. Int. J. Plast. 48, 168–188 (2013)CrossRefGoogle Scholar
  37. B. Klusemann, T. Yalçinkaya, M.G.D. Geers, B. Svendsen, Application of non-convex rate dependent gradient plasticity to the modeling and simulation of inelastic microstructure development and inhomogeneous material behavior. Comput. Mater. Sci. 80, 51–60 (2013)CrossRefGoogle Scholar
  38. M. Kuroda, V. Tvergaard, On the formulations of higher-order strain gradient crystal plasticity models. J. Mech. Phys. Solids 56, 1591–1608 (2008)MathSciNetCrossRefGoogle Scholar
  39. G. Lancioni, T. Yalçinkaya, A. Cocks, Energy-based non-local plasticity models for deformation patterning, localization and fracture. Proc. R. Soc. A 471: 20150275 (2015a)CrossRefGoogle Scholar
  40. G. Lancioni, G. Zitti, T. Yalcinkaya, Rate-independent deformation patterning in crystal plasticity. Key Eng. Mater. 651–653, 944–949 (2015b)CrossRefGoogle Scholar
  41. V. Levkovitch, B. Svendsen, On the large-deformation- and continuum-based formulation of models for extended crystal plasticity. Int. J. Solids Struct. 43, 7246–7267 (2006)MathSciNetCrossRefGoogle Scholar
  42. L. Liang, F.P.E. Dunne, GND accumulation in bi-crystal deformation: crystal plasticity analysis and comparison with experiments. Int. J. Mech. Sci. 51, 326–333 (2009)CrossRefGoogle Scholar
  43. A. Ma, F. Roters, D. Raabe, A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations. Acta Mater. 54, 2169–2179 (2006)CrossRefGoogle Scholar
  44. H.B. Mühlhaus, E.C. Aifantis, A variational principle for gradient plasticity. Int. J. Solids Struct. 28, 845–857 (1991)MathSciNetCrossRefGoogle Scholar
  45. W. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46(3), 411–425 (1998)CrossRefGoogle Scholar
  46. T. Ohashi, Crystal plasticity analysis of dislocation emission from micro voids. Int. J. Plast. 21, 2071–2088 (2005)CrossRefGoogle Scholar
  47. I. Özdemir, T. Yalçinkaya, Modeling of dislocation-grain boundary interactions in a strain gradient crystal plasticity framework. Comput. Mech. 54, 255–268 (2014)MathSciNetCrossRefGoogle Scholar
  48. A. Panteghini, L. Bardella, On the finite element implementation of higher-order gradient plasticity, with focus on theories based on plastic distortion incompatibility. Comput. Methods Appl. Mech. Eng. (2016). https://doi.org/10.1016/j.cma.2016.07.045 CrossRefGoogle Scholar
  49. P. Perzyna, Temperature and rate dependent theory of plasticity of crystalline solids. Revue Phys. Appl. 23, 445–459 (1988)CrossRefGoogle Scholar
  50. B.D. Reddy, The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity. Contin. Mech. Thermodyn. 23, 527–549 (2011a)MathSciNetCrossRefGoogle Scholar
  51. B.D. Reddy, The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity. Contin. Mech. Thermodyn. 23, 551–572 (2011b)MathSciNetCrossRefGoogle Scholar
  52. J.R. Rice, Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)CrossRefGoogle Scholar
  53. J.Y. Shu, N.A. Fleck, Strain gradient crystal plasticity: size-dependent deformation of bicrystals. J. Mech. Phys. Solids 47, 297–324 (1999)CrossRefGoogle Scholar
  54. M. Silhavy, The Mechanics and Thermodynamics of Continuous Media, 1st edn. (Springer, Berlin, 1997)CrossRefGoogle Scholar
  55. J.S. Stölken, A.G. Evans, A microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109–5115 (1998)CrossRefGoogle Scholar
  56. B. Svendsen, On thermodynamic- and variational-based formulations of models for inelastic continua with internal length scales. Comput. Methods Appl. Mech. Eng. 193, 5429–5452 (2004)MathSciNetCrossRefGoogle Scholar
  57. B. Svendsen, S. Bargmann, On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. J. Mech. Phys. Solids 58, 1253–1271 (2010)MathSciNetCrossRefGoogle Scholar
  58. J.G. Swadenera, E.P. Georgea, G.M. Pharra, The correlation of the indentation size effect measured with indenters of various shapes. J. Mech. Phys. Solids 50, 681–694 (2002)CrossRefGoogle Scholar
  59. G.I. Taylor, Plastic strain in metals. J. Inst. Met. 62, 307–325 (1938)Google Scholar
  60. C.A. Volkert, E.T. Lilleodden, Size effects in the deformation of sub-micron au columns. Philos. Mag. 86, 5567–5579 (2006)CrossRefGoogle Scholar
  61. G. Voyiadjis, R. Abu Al-Rub, Gradient plasticity theory with a variable length scale parameter. Int. J. Solids Struct. 42(14), 3998–4029 (2005)CrossRefGoogle Scholar
  62. J. Wang, J. Lian, J.R. Greer, W.D. Nix, K.S. Kim, Size effect in contact compression of nano- and microscale pyramid structures. Acta Mater. 54, 3973–3982 (2006)CrossRefGoogle Scholar
  63. T. Yalcinkaya, Microstructure evolution in crystal plasticity : strain path effects and dislocation slip patterning. PhD Thesis, Eindhoven University of Technology, 2011Google Scholar
  64. T. Yalçinkaya, Multi-scale modeling of microstructure evolution induced anisotropy in metals. Key Eng. Mater. 554–557, 2388–2399 (2013)CrossRefGoogle Scholar
  65. T. Yalcinkaya, G. Lancioni, Energy-based modeling of localization and necking in plasticity. Proc. Mater. Sci. 3, 1618–1625 (2014)CrossRefGoogle Scholar
  66. T. Yalcinkaya, W.A.M. Brekelmans, M.G.D. Geers, Deformation patterning driven by rate dependent nonconvex strain gradient plasticity. J. Mech. Phys. Solids 59, 1–17 (2011)MathSciNetCrossRefGoogle Scholar
  67. T. Yalcinkaya, W.A.M. Brekelmans, M.G.D. Geers, Non-convex rate dependent strain gradient crystal plasticity and deformation patterning. Int. J. Solids Struct. 49, 2625–2636 (2012)CrossRefGoogle Scholar
  68. S. Yefimov, I. Groma, E. van der Giessena, A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations. J. Mech. Phys. Solids 52, 279–300 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Tuncay Yalçinkaya
    • 1
    • 2
  1. 1.Aerospace Engineering ProgramMiddle East Technical University Northern Cyprus CampusGuzelyurtTurkey
  2. 2.Department of Aerospace EngineeringMiddle East Technical UniversityAnkaraTurkey

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