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Strain Gradient Crystal Plasticity: Intergranular Microstructure Formation

  • İzzet Özdemir
  • Tuncay Yalçinkaya
Reference work entry

Abstract

This chapter addresses the formation and evolution of inhomogeneous plastic deformation field between grains in polycrystalline metals by focusing on continuum scale modeling of dislocation-grain boundary interactions within a strain gradient crystal plasticity (SGCP) framework. Thermodynamically consistent extension of a particular strain gradient plasticity model, addressed previously (see also, e.g., Yalcinkaya et al, J Mech Phys Solids 59:1–17, 2011), is presented which incorporates the effect of grain boundaries on plastic slip evolution explicitly. Among various choices, a potential-type non-dissipative grain boundary description in terms of grain boundary Burgers tensor (see, e.g., Gurtin, J Mech Phys Solids 56:640–662, 2008) is preferred since this is the essential descriptor to capture both the misorientation and grain boundary orientation effects. A mixed finite element formulation is used to discretize the problem in which both displacements and plastic slips are considered as primary variables. For the treatment of grain boundaries within the solution algorithm, an interface element is formulated. The capabilities of the framework is demonstrated through 3D bi-crystal and polycrystal examples, and potential extensions and currently pursued multi-scale modeling efforts are briefly discussed in the closure.

Keywords

Strain gradient plasticity Grain boundary Grain boundary-dislocation interaction Misorientation Grain boundary Burgers tensor 

References

  1. R.K. Abu Al-Rub, Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals. Int. J. Plast. 24, 1277–1306 (2008)CrossRefGoogle Scholar
  2. K.E. Aifantis, J.R. Willis, The role of interfaces in enhancing the yield strength of composites and polycrystals. J. Mech. Phys. Solids 53, 1047–1070 (2005)MathSciNetCrossRefGoogle Scholar
  3. K.E. Aifantis, W.A. Soer, J.T. de Hosson, J. Willis, Interfaces within strain gradient plasticity: theory and experiments. Acta Mater. 54, 5077–5085 (2006)CrossRefGoogle Scholar
  4. E. Bayerschen, A.T. McBride, B.D. Reddy, T. Böhlke, Review of slip transmission criteria in experiments and crystal plasticity models. J. Mater. Sci. 51, 2243–2258 (2016)CrossRefGoogle Scholar
  5. 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
  6. U. Borg, A strain gradient crystal plasticity analysis of grain size effects in polycrystals. Eur. J. Mech. A. Solids 26, 313–324 (2007)CrossRefGoogle Scholar
  7. T. Borg, N.A. Fleck, Strain gradient effects in surface roughening. Model. Simul. Mater. Sci. Eng. 15, S1–S12 (2007)CrossRefGoogle Scholar
  8. M. de Koning, R. Miller, V.V. Bulatov, F.F. Abraham, Modelling grain boundary resistence in intergranular dislocation slip transmission. Philos. Mag. A 82, 2511–2527 (2002)CrossRefGoogle Scholar
  9. M. de Koning, R.J. Kurtz, V.V. Bulatov, C.H. Henager, R.G. Hoagland, W. Cai, M. Nomura, Modeling of dislocation-grain boundary interactions. J. Nucl. Mater. 323, 281–289 (2003)CrossRefGoogle Scholar
  10. M. Ekh, S. Bargmann, M. Grymer, Influence of grain boundary conditions on modeling of size-dependence in polycrystals. Acta Mech. 218, 103–113 (2011)CrossRefGoogle Scholar
  11. N.A. Fleck, J.W. Hutchinson, A formulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001)CrossRefGoogle Scholar
  12. P. Fredriksson, P. Gudmundson, Size-dependent yield strength of thin films. Int. J. Plast. 21, 1834–1854 (2005)CrossRefGoogle Scholar
  13. C. Fressengeas, V. Taupin, L. Capolunga, Continuous modeling of the structure f symmetric tilt boundaries. Int. J. Solids Struct. 51, 1434–1441 (2014)CrossRefGoogle Scholar
  14. M.G.D. Geers, W.A.M. Brekelmans, C.J. Bayley, Second-order crystal plasticity: internal stress effects and cyclic loading. Model. Simul. Mater. Sci. Eng. 15, 133–145 (2007)CrossRefGoogle Scholar
  15. D. Gottschalk, A. McBride, B.D. Reddy, A. Javili, P. Wriggers, C.B. 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
  16. P. Gudmundson, A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004)MathSciNetCrossRefGoogle Scholar
  17. 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
  18. 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
  19. M.E. Gurtin, A theory of grain boundaries that accounts automatically for grain misorientation and grain-boundary orientation. J. Mech. Phys. Solids 56, 640–662 (2008)MathSciNetCrossRefGoogle Scholar
  20. M.E. Gurtin, L. Anand, S.P. Lele, Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878 (2007)MathSciNetCrossRefGoogle Scholar
  21. R. Kumar, F. Szekely, E. Van der Giessen, Modelling dislocation transmission across tilt grain boundaries in 2D. Comput. Mater. Sci. 49, 46–54 (2010)CrossRefGoogle Scholar
  22. G. Lancioni, T. Yalçinkaya, A. Cocks, Energy-based non-local plasticity models for deformation patterning, localization and fracture. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 471(2180) (2015a)CrossRefGoogle Scholar
  23. G. Lancioni, G. Zitti, T. Yalcinkaya, Rate-independent deformation patterning in crystal plasticity. Key Eng. Mater. 651–653, 944–949 (2015b)CrossRefGoogle Scholar
  24. T.C. Lee, I.M. Robertson, H.K. Birnbaum, Prediction of slip transfer mechanisms across grain boundaries. Scr. Metall. 23(5), 799–803 (1989)CrossRefGoogle Scholar
  25. Z. Li, C. Hou, M. Huang, C. Ouyang, Strengthening mechanism in micro-polycrystals with penetrable grain boundaries by discrete dislocation dynamics simulation and Hall-Petch effect. Comput. Mater. Sci. 46, 1124–1134 (2009)CrossRefGoogle Scholar
  26. A. Ma, F. Roters, D. Raabe, On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modeling – theory, experiments, and simulations. Acta Mater. 54, 2181–2194 (2006)CrossRefGoogle Scholar
  27. T.J. Massart, T. Pardoen, Strain gradient plasticity analysis of the grain-size-dependent strength and ductility of polycrystals with evolving grain boundary confinement. Acta Mater. 58, 5768–5781 (2010)CrossRefGoogle Scholar
  28. A.T. McBride, D. Gottschalk, B.D. Reddy, P. Wriggers, A. Javili, Computational and theoretical aspects of a grain-boundary model at finite deformations. Tech. Mech. 36, 102–119 (2016)Google Scholar
  29. D.L. McDowell, Viscoplasticity of heterogeneous metallic materials. Mater. Sci. Eng. R 62, 67–123 (2008)CrossRefGoogle Scholar
  30. J. Mosler, I. Scheider, A thermodynamically and variationally consistent class of damage-type cohesive models. J. Mech. Phys. Solids 59(8), 1647–1668 (2011)MathSciNetCrossRefGoogle Scholar
  31. 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
  32. Z. Shen, R.H. Wagoner, W.A.T. Clark, Dislocation pile-up and grain boundary interactions in 304 stainless steel. Scr. Metall. 20(6), 921–926 (1986)CrossRefGoogle Scholar
  33. D.E. Spearot, D.L. McDowell, Atomistic modeling of grain boundaries and dislocation processes in metallic polycrystalline materials. J. Eng. Mater. Tech. 131, 041,204 (2009)CrossRefGoogle Scholar
  34. M. Stricker, J. Gagel, S. Schmitt, K. Schulz, D. Weygand, P.B. Gumbsch, On slip transmission and grain boundary yielding. Meccanica 51, 271–278 (2016)MathSciNetCrossRefGoogle Scholar
  35. M.A. Tschopp, D.L. McDowell, Asymmetric tilt grain boundary structure and energy in copper and aluminium. Philos. Mag. 87, 3871–3892 (2007)CrossRefGoogle Scholar
  36. P.R.M. van Beers, G.J. McShane, V.G. Kouznetsova, M.G.D. Geers, Grain boundary interface mechanics in strain gradient crystal plasticity. J. Mech. Phys. Solids 61, 2659–2679 (2013)MathSciNetCrossRefGoogle Scholar
  37. P.R.M. van Beers, V.G. Kouznetsova, M.G.D. Geers, Defect redistribution within continuum grain boundary plasticity model. J. Mech. Phys. Solids 83, 243–262 (2015a)MathSciNetCrossRefGoogle Scholar
  38. P.R.M. van Beers, V.G. Kouznetsova, M.G.D. Geers, M.A. Tschopp, D.L. McDowell, A multiscale model to grain boundary structure and energy: from atomistics to a continuum description. Acta Mater. 82, 513–529 (2015b)CrossRefGoogle Scholar
  39. E. Van der Giessen, A. Needleman, Discrete dislocation plasticity: a simple planar model. Model. Simul. Mater. Sci. Eng. 3, 689–735 (1995)CrossRefGoogle Scholar
  40. D. Wolf, S Yip, Material Interfaces: Atomic-Level Structure and Properties (Chapman and Hall, London, 1992)Google Scholar
  41. T. Yalçinkaya, Multi-scale modeling of microstructure evolution induced anisotropy in metals. Key Eng. Mater. 554–557, 2388–2399 (2013)CrossRefGoogle Scholar
  42. T. Yalçinkaya, 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
  43. T. Yalcinkaya, G. Lancioni, Energy-based modeling of localization and necking in plasticity. Procedia Mater. Sci. 3, 1618–1625 (2014)CrossRefGoogle Scholar
  44. T. Yalcinkaya, W.A.M. Brekelmans, M.G.D. Geers, Deformation patterning driven by rate dependent non-convex strain gradient plasticity. J. Mech. Phys. Solids 59, 1–17 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Civil Engineeringİzmir Institute of TechnologyİzmirTurkey
  2. 2.Aerospace Engineering ProgramMiddle East Technical University Northern Cyprus CampusGuzelyurtTurkey
  3. 3.Department of Aerospace EngineeringMiddle East Technical UniversityAnkaraTurkey

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