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Micromorphic Crystal Plasticity

  • Samuel ForestEmail author
  • J. R. Mayeur
  • D. L. McDowell
Reference work entry

Abstract

The micromorphic approach to crystal plasticity represents an extension of the micropolar (Cosserat) framework, which is presented in a separate chapter. Cosserat crystal plasticity is contained as a special constrained case in the same way as the Cosserat theory is a special restricted case of Eringen's micromorphic model, as explained also in a separate chapter. The micromorphic theory is presented along the lines of Aslan et al. (Int J Eng Sci 49:1311–1325, 2011) and Forest et al. (Micromorphic approach to crystal plasticity and phase transformation. In: Schroeder J, Hackl K (eds) Plasticity and beyond. CISM international centre for mechanical sciences, courses and lectures, vol 550, Springer, pp 131–198, 2014) and compared to the micropolar model in some applications. These extensions of conventional crystal plasticity aim at incorporating the dislocation density tensor introduced by Kröner (Initial studies of a plasticity theory based upon statistical mechanics. In: Kanninen M, Adler W, Rosenfield A, Jaffee R (eds) Inelastic behaviour of solids. McGraw-Hill, pp 137–147, 1969). and Cermelli and Gurtin (J Mech Phys Solids 49:1539–1568, 2001) into the constitutive framework. The concept of dislocation density tensor is equivalent to that of the so-called geometrically necessary dislocations (GND) introduced by Ashby (The deformation of plastically non-homogeneous alloys. In: Kelly A, Nicholson R (eds) Strengthening methods in crystals. Applied Science Publishers, London, pp 137–192, 1971). The applications presented in this chapter deal with pile-up formation in laminate microstructures and strain localization phenomena in polycrystals.

Keywords

Micromorphic medium Crystal plasticity Dislocation density tensor Geometrically necessary dislocations Strain gradient plasticity Size effect 

Notes

Acknowledgments

The first author is indebted to Dr. N.M. Cordero, Dr. S. Wulfinghoff, and Prof. E.B. Busso for their contribution to the presented micromorphic crystal plasticity theory. These contributions are duly cited in the references quoted in the text and listed below.

References

  1. E. Aifantis, The physics of plastic deformation. Int. J. Plast. 3, 211–248 (1987)CrossRefGoogle Scholar
  2. R.J. Asaro, Elastic–plastic memory and kinematic hardening. Acta Metall. 23, 1255–1265 (1975)CrossRefGoogle Scholar
  3. R. Asaro, Crystal plasticity. J. Appl. Mech. 50, 921–934 (1983)CrossRefGoogle Scholar
  4. Ashby, M., 1971. The deformation of plastically non-homogeneous alloys, in Strengthening Methods in Crystals, ed. by A. Kelly, R. Nicholson (Applied Science Publishers, London), pp. 137–192Google Scholar
  5. O. Aslan, N.M. Cordero, A. Gaubert, S. Forest, Micromorphic approach to single crystal plasticity and damage. Int. J. Eng. Sci. 49, 1311–1325 (2011)MathSciNetCrossRefGoogle Scholar
  6. V. Bennett, D. McDowell, Crack tip displacements of microstructurally small surface cracks in single phase ductile polycrystals. Eng. Fract. Mech. 70(2), 185–207 (2003)CrossRefGoogle Scholar
  7. V. Berdichevsky, On thermodynamics of crystal plasticity. Scripta Mat. 54, 711–716 (2006a)CrossRefGoogle Scholar
  8. V. Berdichevsky, On thermodynamics of crystal plasticity. Scr. Mater. 54, 711–716 (2006b)CrossRefGoogle Scholar
  9. P. Cermelli, M. Gurtin, On the characterization of geometrically necessary dislocations in finite plasticity. J. Mech. Phys. Solids 49, 1539–1568 (2001)CrossRefGoogle Scholar
  10. H.J. Chang, N.M. Cordero, C. Déprés, M. Fivel, S. Forest, Micromorphic crystal plasticity versus discrete dislocation dynamics analysis of multilayer pile-up hardening in a narrow channel. Arch. Appl. Mech. 86, 21–38 (2016)CrossRefGoogle Scholar
  11. W. Claus, A. Eringen, Three dislocation concepts and micromorphic mechanics, in Developments in Mechanics. Proceedings of the 12th Midwestern Mechanics Conference, vol. 6, (1969), pp. 349–358Google Scholar
  12. S. Conti, M. Ortiz, Dislocation microstructures and the effective behavior of single crystals. Arch. Ration. Mech. Anal. 176, 103–147 (2005)MathSciNetCrossRefGoogle Scholar
  13. N. Cordero, A. Gaubert, S. Forest, E. Busso, F. Gallerneau, S. Kruch, Size effects in generalised continuum crystal plasticity for two–phase laminates. J. Mech. Phys. Solids 58, 1963–1994 (2010a)MathSciNetCrossRefGoogle Scholar
  14. N.M. Cordero, A. Gaubert, S. Forest, E. Busso, F. Gallerneau, S. Kruch, Size effects in generalised continuum crystal plasticity for two-phase laminates. J. Mech. Phys. Solids 58, 1963–1994 (2010b)MathSciNetCrossRefGoogle Scholar
  15. N.M. Cordero, S. Forest, E. Busso, S. Berbenni, M. Cherkaoui, Grain size effects on plastic strain and dislocation density tensor fields in metal polycrystals. Comput. Mater. Sci. 52, 7–13 (2012)CrossRefGoogle Scholar
  16. L. De Luca, A. Garroni, M. Ponsiglione, Gamma-convergence analysis of Systems of Edge Dislocations: the self energy regime. Arch. Ration. Mech. Anal. 206, 885–910 (2012)MathSciNetCrossRefGoogle Scholar
  17. C. Déprés, C.F. Robertson, M.C. Fivel, Low-strain fatigue in aisi 316l steel surface grains: a three-dimensional discrete dislocation dynamics modelling of the early cycles i. Dislocation microstructures and mechanical behaviour. Philos. Mag. 84(22), 2257–2275 (2004)CrossRefGoogle Scholar
  18. Eringen, A., Claus, W., 1970. A micromorphic approach to dislocation theory and its relation to several existing theories, in Fundamental Aspects of Dislocation Theory, ed. by J. Simmons, R. de Wit, R. Bullough. National Bureau of Standards (US) Special Publication 317, vol. II (U.S. Government Printing Office, Washington, DC), pp. 1023–1062Google Scholar
  19. B. Fedelich, A microstructural model for the monotonic and the cyclic mechanical behavior of single crystals of superalloys at high temperatures. Int. J. Mech. Sci. 18, 1–49 (2002)zbMATHGoogle Scholar
  20. S. Forest, Some links between cosserat, strain gradient crystal plasticity and the statistical theory of dislocations. Philos. Mag. 88, 3549–3563 (2008)CrossRefGoogle Scholar
  21. S. Forest, The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE J. Eng. Mech. 135, 117–131 (2009)CrossRefGoogle Scholar
  22. S. Forest, Nonlinear regularisation operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage. Proc. R. Soc. A 472, 20150755 (2016)CrossRefGoogle Scholar
  23. S. Forest, N. Guéninchault, Inspection of free energy functions in gradient crystal plasticity. Acta. Mech. Sinica. 29, 763–772 (2013)  https://doi.org/10.1007/s10409-013-0088-0MathSciNetCrossRefGoogle Scholar
  24. S. Forest, R. Sedláček, Plastic slip distribution in two–phase laminate microstructures: Dislocation–based vs. generalized–continuum approaches. Philos. Mag. A 83, 245–276 (2003a)CrossRefGoogle Scholar
  25. S. Forest, R. Sedláček, Plastic slip distribution in two–phase laminate microstructures: Dislocation–based vs. generalized–continuum approaches. Philos. Mag. A 83, 245–276 (2003b)CrossRefGoogle Scholar
  26. S. Forest, R. Sievert, Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160, 71–111 (2003)CrossRefGoogle Scholar
  27. S. Forest, R. Sievert, Nonlinear microstrain theories. Int. J. Solids Struct. 43, 7224–7245 (2006)MathSciNetCrossRefGoogle Scholar
  28. S. Forest, F. Pradel, K. Sab, Asymptotic analysis of heterogeneous Cosserat media. Int. J. Solids Struct. 38, 4585–4608 (2001)MathSciNetCrossRefGoogle Scholar
  29. Forest, S., Ammar, K., Appolaire, B., Cordero, N., Gaubert, A., 2014. Micromorphic approach to crystal plasticity and phase transformation, in Plasticity and Beyond, ed. by J. Schroeder, K. Hackl. CISM International Centre for Mechanical Sciences, Courses and Lectures, no. 550 (Springer, Vienna), pp. 131–198CrossRefGoogle Scholar
  30. M. Geers, R. Peerlings, M. Peletier, L. Scardia, Asymptotic behaviour of a pile–up of infinite walls of edge dislocations. Arch. Ration. Mech. Anal. 209, 495–539 (2013)MathSciNetCrossRefGoogle Scholar
  31. P. Germain, The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)CrossRefGoogle Scholar
  32. P. Grammenoudis, C. Tsakmakis, Micromorphic continuum part I: strain and stress tensors and their associated rates. Int. J. Non–Linear Mech. 44, 943–956 (2009)CrossRefGoogle Scholar
  33. I. Groma, F. Csikor, M. Zaiser, Spatial correlations and higher–order gradient terms in a continuum description of dislocation dynamics. Acta Mater. 51, 1271–1281 (2003)CrossRefGoogle Scholar
  34. I. Groma, G. Györgyi, B. Kocsis, Dynamics of coarse grain grained dislocation densities from an effective free energy. Philos. Mag. 87, 1185–1199 (2007)CrossRefGoogle Scholar
  35. M. Gurtin, A gradient theory of single–crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32 (2002)MathSciNetCrossRefGoogle Scholar
  36. M. Gurtin, L. Anand, Nanocrystalline grain boundaries that slip and separate: a gradient theory that accounts for grain-boundary stress and conditions at a triple-junction. J. Mech. Phys. Solids 56, 184–199 (2008)MathSciNetCrossRefGoogle Scholar
  37. M. 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
  38. W. Han, B. Reddy, Plasticity: Mathematical Theory and Numerical Analysis (Springer, New York, 2013)CrossRefGoogle Scholar
  39. C. Hirschberger, P. Steinmann, Classification of concepts in thermodynamically consistent generalized plasticity. ASCE J. Eng.Mech. 135, 156–170 (2009)CrossRefGoogle Scholar
  40. D.E. Hurtado, M. Ortiz, Surface effects and the size-dependent hardening and strengthening of nickel micropillars. J. Mech. Phys. Solids 60(8), 1432–1446 (2012)MathSciNetCrossRefGoogle Scholar
  41. D.E. Hurtado, M. Ortiz, Finite element analysis of geometrically necessary dislocations in crystal plasticity. Int. J. Numer. Methods Eng. 93(1), 66–79 (2013)MathSciNetCrossRefGoogle Scholar
  42. R. Kametani, K. Kodera, D. Okumura, N. Ohno, Implicit iterative finite element scheme for a strain gradient crystal plasticity model based on self-energy of geometrically necessary dislocations. Comput. Mater. Sci. 53(1), 53–59 (2012)CrossRefGoogle Scholar
  43. Kröner, E., 1969. Initial studies of a plasticity theory based upon statistical mechanics, in Inelastic Behaviour of Solids, ed. by M. Kanninen, W. Adler, A. Rosenfield, R. Jaffee (McGraw-Hill, New York/London), pp. 137–147Google Scholar
  44. J. Lee, Y. Chen, Constitutive relations of micromorphic thermoplasticity. Int. J. Eng. Sci. 41, 387–399 (2003)MathSciNetCrossRefGoogle Scholar
  45. J. Mandel, Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Struct. 9, 725–740 (1973)CrossRefGoogle Scholar
  46. L. Méric, P. Poubanne, G. Cailletaud, Single crystal modeling for structural calculations. Part 1: Model presentation. J. Eng. Mat. Technol. 113, 162–170 (1991)CrossRefGoogle Scholar
  47. S.D. Mesarovic, R. Baskaran, A. Panchenko, Thermodynamic coarsening of dislocation mechanics and the size-dependent continuum crystal plasticity. J. Mech. Phys. Solids 58(3), 311–329 (2010)MathSciNetCrossRefGoogle Scholar
  48. S. Mesarovic, S. Forest, J. Jaric, Size-dependent energy in crystal plasticity and continuum dislocation models. Proc. R. Soc. A 471, 20140868 (2015)MathSciNetCrossRefGoogle Scholar
  49. C. Miehe, S. Mauthe, F.E. Hildebrand, Variational gradient plasticity at finite strains. Part III: local-global updates and regularization techniques in multiplicative plasticity for single crystals. Comput. Methods Appl. Mech. Eng. 268, 735–762 (2014)MathSciNetCrossRefGoogle Scholar
  50. J. Nye, Some geometrical relations in dislocated crystals. Acta Metall. 1, 153–162 (1953)CrossRefGoogle Scholar
  51. N. Ohno, D. Okumura, Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations. J. Mech. Phys. Solids 55, 1879–1898 (2007)MathSciNetCrossRefGoogle Scholar
  52. N. Ohno, D. Okumura, Grain–size dependent yield behavior under loading, unloading and reverse loading. Int. J. Mod. Phys. B 22, 5937–5942 (2008)CrossRefGoogle Scholar
  53. M. Ortiz, E. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397–462 (1999)MathSciNetCrossRefGoogle Scholar
  54. H. Proudhon, W. Poole, X. Wang, Y. Bréchet, The role of internal stresses on the plastic deformation of the Al–Mg–Si–Cu alloy AA611. Philos. Mag. 88, 621–640 (2008)CrossRefGoogle Scholar
  55. B.D. Reddy, C. Wieners, B. Wohlmuth, Finite element analysis and algorithms for single-crystal strain-gradient plasticity. Int. J. Numer. Methods Eng. 90(6), 784–804 (2012)MathSciNetCrossRefGoogle Scholar
  56. R. Regueiro, On finite strain micromorphic elastoplasticity. Int. J. Solids Struct. 47, 786–800 (2010)CrossRefGoogle Scholar
  57. C. Sansour, S. Skatulla, H. Zbib, A formulation for the micromorphic continuum at finite inelastic strains. Int. J. Solids Struct. 47, 1546–1554 (2010)CrossRefGoogle Scholar
  58. P. Steinmann, Views on multiplicative elastoplasticity and the continuum theory of dislocations. Int. J. Eng. Sci. 34, 1717–1735 (1996)CrossRefGoogle Scholar
  59. R. Stoltz, R. Pelloux, Cyclic deformation and Bauschinger effect in Al–Cu–Mg alloys. Scr. Metall. 8, 269–276 (1974)CrossRefGoogle Scholar
  60. R. Stoltz, R. Pelloux, The Bauschinger effect in precipitation strengthened aluminum alloys. Metallurgical. Transactions 7A, 1295–1306 (1976)Google Scholar
  61. 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(9), 1253–1271 (2010)MathSciNetCrossRefGoogle Scholar
  62. R. Taillard, A. Pineau, Room temperature tensile properties of Fe-19wt.% Cr alloys precipitation hardened by the intermetallic compound NiAl. Mater. Sci. Eng. 56, 219–231 (1982)CrossRefGoogle Scholar
  63. S. Wulfinghoff, T. Böhlke, Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics. Proc. R. Soc. A: Math. Phys. Eng. Sci. 468(2145), 2682–2703 (2012)MathSciNetCrossRefGoogle Scholar
  64. S. Wulfinghoff, E. Bayerschen, T. Böhlke, A gradient plasticity grain boundary yield theory. Int. J. Plast. 51, 33–46 (2013a)CrossRefGoogle Scholar
  65. S. Wulfinghoff, E. Bayerschen, T. Böhlke, Micromechanical simulation of the hall-petch effect with a crystal gradient theory including a grain boundary yield criterion. PAMM 13, 15–18 (2013b)CrossRefGoogle Scholar
  66. S. Wulfinghoff, S. Forest, T. Böhlke, Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures. J. Mech. Phys. Solids 79, 1–20 (2015)MathSciNetCrossRefGoogle Scholar
  67. A. Zeghadi, S. Forest, A.-F. Gourgues, O. Bouaziz, Ensemble averaging stress–strain fields in polycrystalline aggregates with a constrained surface microstructure–part 2: crystal plasticity. Philos. Mag. 87, 1425–1446 (2007)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre des Materiaux, Mines ParisTech CNRSPSL Research UniversityParisFrance
  2. 2.Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Woodruff School of Mechanical Engineering, School of Materials Science and EngineeringGeorgia Institute of TechnologyAtlantaUSA

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