A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part I: The hardening law

• Autorzy: Petryk H. , Stupkiewicz S.
• Języki publikacji: EN

Abstrakty

EN

A simple gradient-enhancement of the classical continuum theory of plasticity of single crystals deformed by multislip is proposed for incorporating size effects in a manner consistent with phenomenological laws established in materials science. Despite considerable efforts in developing gradient theories, there is no consensus regarding the minimal set of physically based assumptions needed to capture the slip-gradient effects in metal single crystals and to provide a benchmark for more refined approaches. In order to make a step towards such a reference model, the concept of the tensorial density of geometrically necessary dislocations generated by slip-rate gradients is combined with a generalized form of the classical Taylor formula for the flow stress. In the governing equations in the rate form, the derived internal length scale is expressed through the current flow stress and standard parameters so that no further assumption is needed to define a characteristic length. It is shown that this internal length scale is directly related to the mean free path of dislocations and possesses physical interpretation which is frequently missing in other gradient-plasticity models.

Słowa kluczowe

EN

gradient plasticity; geometrically necessary dislocation; single crystal; strain hardening; internal length scale; size effect

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2016
• Tom: Vol. 68, nr 6
• Strony: 459--485
• Opis fizyczny: Bibliogr. 68 poz.

Twórcy

autor

Petryk H.

• Institute of Fundamental Technological Research Polish Academy of Sciences Pawińskiego 5B 02-106 Warsaw, Poland

autor

Stupkiewicz S.

• Institute of Fundamental Technological Research Polish Academy of Sciences Pawińskiego 5B 02-106 Warsaw, Poland

Bibliografia

• 1. R. Hill, Generalized constitutive relations for incremental deformation of metal crystals by multislip, J. Mech. Phys. Solids 14 (1966), 95–102.
• 2. J.R. Rice, Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity, J. Mech. Phys. Solids 19 (1971), 433–455.
• 3. R. Hill, J.R. Rice, Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids 20 (1972), 401–413.
• 4. R.J. Asaro, Micromechanics of crystals and polycrystals, Adv. Appl. Mech. 23 (1983), 1–115.
• 5. N.A. Fleck, J.W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech. 33 (1997), 295–361.
• 6. W.D. Nix, H. Gao, Indentation size effects in crystalline materials: a law for strain gradient plasticity, J. Mech. Phys. Solids 46 (1998), 411–425.
• 7. H. Gao, Y. Huang, W.D. Nix, J.W. Hutchinson, Mechanism-based strain gradient plasticity—I. Theory, J. Mech. Phys. Solids 47 (1999), 1239–1263.
• 8. A. Arsenlis, D.M. Parks, Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density, Acta Mater. 47 (1999), 1597–1611.
• 9. A. Acharya, J.L. Bassani, Lattice incompatibility and a gradient theory of crystal plasticity, J. Mech. Phys. Solids 48 (2000), 1565–1595.
• 10. A. Menzel, P. Steinmann, On the continuum formulation of higher gradient plasticity for single and polycrystals, J. Mech. Phys. Solids 48 (2000), 1777–1796.
• 11. M.E. Gurtin, On the plasticity of single crystals: free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids 48 (2000), 989–1036.
• 12. M.E. Gurtin, A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations, J. Mech. Phys. Solids 50 (2002), 5–32.
• 13. H. Gao, Y. Huang, Taylor-based nonlocal theory of plasticity, Int. J. Sol. Struct. 38 (2001), 2615–2637.
• 14. B. Svendsen, Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations, J. Mech. Phys. Solids 50 (2002), 1297–1329.
• 15. S. Forest, R. Sievert, E.C. Aifantis, Strain gradient crystal plasticity: thermomechanical formulations and applications, J. Mech. Beh. Mat. 13 (2002), 219–232.
• 16. I. Groma, F.F. Csikor, M. Zaiser, Spatial correlations and higher-order gradient terms in a continuum description of dislocation dynamics, Acta Mater. 51 (2003), 1271–1281.
• 17. 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 (2004), 2379–2401.
• 18. C.S. Han, H. Gao, Y. Huang, W.D. Nix, Mechanism-based strain gradient crystal plasticity—I. Theory, J. Mech. Phys. Solids 53 (2005), 1188–1203.
• 19. G.Z. Voyiadjis, R.K. Abu Al-Rub, Gradient plasticity theory with a variable length scale parameter, Int. J. Sol. Struct. 42 (2005), 3998–4029.
• 20. 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. Sol. Struct. 43 (2006), 7268–7286.
• 21. A. Ma, F. Roters, D. Raabe, A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations, Acta Mater. 54 (2006), 2169–2179.
• 22. M.E. Gurtin, L. Anand, S.P. Lele, Gradient single-crystal plasticity with free energy dependent on dislocation densities, J. Mech. Phys. Solids 55 (2007), 1853–1878.
• 23. J. Kratochvil, M. Kruzik, R. Sedlacek, Statistically based continuum model of misoriented dislocation cell structure formation, Phys. Rev. B 75 (2007), 064104.
• 24. N. Ohno, D. Okumura, Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations, J. Mech. Phys. Solids 55 (2007), 1879–1898.
• 25. M. Kuroda, V. Tvergaard, A finite deformation theory of higher-order gradient crystal plasticity, J. Mech. Phys. Solids 56 (2008), 2573–2584.
• 26. E. Kroner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen, Arch. Ration. Mech. Anal. 4 (1960), 273–334.
• 27. O.W. Dillon, J. Kratochvil, A strain gradient theory of plasticity, Int. J. Sol. Struct. 6 (1970), 1513–1533.
• 28. E.C. Aifantis, On the microstructural origin of certain inelastic models, Trans. ASME J. Eng. Mat. Tech. 106 (1984), 326–330.
• 29. N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metall. Mater. 42 (1994), 475–487.
• 30. P. Steinmann, Views on multiplicative elastoplasticity and the continuum theory of dislocations , Int. J. Engng. Sci. 34 (1996), 1717–1735.
• 31. M.E. Gurtin, A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on the accumulation of geometrically necessary dislocations, Int. J. Plasticity 26 (2010), 1073–1096.
• 32. B.D. Reddy, The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity, Continuum Mech. Thermodyn. 23 (2011), 551–572.
• 33. S. Bargmann, B. Svendsen, M. Ekh, An extended crystal plasticity model for latent hardening in polycrystals, Comp. Mech. 48 (2011), 631–645.
• 34. J.R. Mayeur, D.L. McDowell, D.J. Bammann, Dislocation-based micropolar single crystal plasticity: Comparison of multi- and single criterion theories, J. Mech. Phys. Solids 59 (2011), 398–422.
• 35. N.M. Cordero, S. Forest, E.P. Busso, Generalized continuum modelling of grain size effects in polycrystals, C. R. Mecanique 340 (2012), 261–274.
• 36. S. Wulfinghoff, T. Boehlke, Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics, Proc. R. Soc. A 468 (2012), 2682–2703.
• 37. C. Miehe, S. Mauthe, F. Hildebrand, Variational gradient plasticity at finite strains. Part III: Local–global updates and regularization techniques in multiplicative plasticity for single crystals, Comp. Meth. Appl. Mech. Engng. 268 (2014), 735–762.
• 38. C.F. Niordson, J.W. Kysar, Computational strain gradient crystal plasticity, J. Mech. Phys. Solids 62 (2014), 31–47.
• 39. L. Anand, M.E. Gurtin, B.D. Reddy, The stored energy of cold work, thermal annealing, and other thermodynamic issues in single crystal plasticity at small length scales, Int. J. Plasticity 64 (2015), 1–25.
• 40. S.D. Mesarovic, S. Forest, J.P. Jaric, Size-dependent energy in crystal plasticity and continuum dislocation models, Proc. R. Soc. A 471 (2015), 20140868.
• 41. E. Kroner, Benefits and shortcomings of the continuous theory of dislocations, Int. J. Sol. Struct. 38 (2001), 1115–1134.
• 42. F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler, D. Raabe, Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications, Acta Mater. 58 (2010), 1152–1211.
• 43. 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 (2010), 1253–1271.
• 44. S. Forest, E.C. Aifantis, Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua, Int. J. Sol. Struct. 47 (2010), 3367–3376.
• 45. J.R. Mayeur, D.L. McDowell, A comparison of Gurtin type and micropolar theories of generalized single crystal plasticity, Int. J. Plasticity 57 (2014), 29–51.
• 46. G.I. Taylor, The mechanism of plastic deformation of crystals. Part I. Theoretical, Proc. R. Soc. Lond. A 145 (1934), 362–387.
• 47. U.F. Kocks, H. Mecking, Physics and phenomenology of strain hardening: the FCC case, Progress Mater. Sci. 48 (2003), 171–273.
• 48. M. Sauzay, L.P. Kubin, Scaling laws for dislocation microstructures in monotonic and cyclic deformation of fcc metals, Progress Mater. Sci. 56 (2011), 725–784.
• 49. M. Niewczas, Intermittent plastic flow of single crystals: central problems in plasticity: a review, Mater. Sci. Technol. 30 (2014) 739–757.
• 50. J.F. Nye, Some geometrical relations in dislocated crystals, Acta Metall. 1 (1953), 153–162.
• 51. M.F. Ashby, The deformation of plastically non-homogeneous materials, Phil. Mag. 21 (1970), 399–424.
• 52. E. Kroner, Continuum theory of defects, [in:] R. Ballian, M. Kleman, J.-P. Poirer (Eds.), Physics of Defects, North-Holland, Amsterdam, 1981.
• 53. P. Cermelli, M.E. Gurtin, On the characterization of geometrically necessary dislocations in finite plasticity, J. Mech. Phys. Solids 49 (2001), 1539–1568.
• 54. M.E. Gurtin, The Burgers vector and the flow of screw and edge dislocations in finitedeformation single-crystal plasticity, J. Mech. Phys. Solids 54 (2006), 1882–1898.
• 55. G. Del Piero, Nonclassical continua, pseudobalance, and the law of action and reaction, Math. Mech. Complex Systems 2 (2014), 71–107.
• 56. G. Del Piero, On the method of virtual power in the mechanics of non-classical continua, [in:] T. Sadowski, P. Trovalusci (Eds.), Multiscale Modelling of Complex Materials, Vol. 556 of CISM International Centre for Mechanical Sciences, Springer, Vienna, 2014, pp. 29–58.
• 57. H. Mughrabi, On the current understanding of strain gradient plasticity, Mater. Sci. Eng. A 387–389 (2004), 209–213.
• 58. H. Petryk, M. Kursa, Incremental work minimization algorithm for rate-independent plasticity of single crystals, Int. J. Num. Meth. Engng. 104 (2015), 157–184.
• 59. A. Acharya, A.J. Beaudoin, Grain-size effect in viscoplastic polycrystals at moderate strains, J. Mech. Phys. Solids 48 (2000), 2213–2230.
• 60. A.J. Beaudoin, A. Acharya, S.R. Chen, D.A. Korzekwa, M.G. Stout, Consideration of grain-size effect and kinetics in the plastic deformation of metal polycrystals, Acta Mater. 48 (2000), 3409–3423.
• 61. S. Stupkiewicz, H. Petryk, A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part II: Size effects, Arch. Mech. 68 (2016), 487-513.
• 62. D. Kuhlmann-Wilsdorf, Theory of plastic deformation – properties of low energy dislocation structures, Mater. Sci. Eng. A 113 (1989), 1–41.
• 63. U. Essmann, H. Mughrabi, Annihilation of dislocations during tensile and cyclic deformation and limits of dislocation densities, Phil. Mag. A 40 (1979), 731–756.
• 64. P. Franciosi, M. Berveiller, A. Zaoui, Latent hardening in copper and aluminium single crystals, Acta Metall. 28 (1980), 273–283.
• 65. R. Madec, B. Devincre, L.P. Kubin, From dislocation junctions to forest hardening, Phys. Rev. Lett. 89 (2002) 255508.
• 66. U.F. Kocks, The relation between polycrystal deformation and single-crystal deformation, Metall. Trans. 1 (1970), 1121–1143.
• 67. P. Franciosi, A. Zaoui, Multislip tests on copper crystals: A junctions hardening effect, Acta Metall. 30 (1982), 2141–2151.
• 68. J. Mandel, Plasticité classique et viscoplasticité, CISM course No. 97, Springer, Wien, 1971.