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A canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin accounting for the Burgers vector

  • François EbobisseEmail author
  • Klaus Hackl
  • Patrizio Neff
Original Article
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Abstract

In this paper, we propose a canonical variational framework for rate-independent phenomenological geometrically linear gradient plasticity with plastic spin. The model combines the additive decomposition of the total distortion into non-symmetric elastic and plastic distortions, with a defect energy contribution taking account of the Burgers vector through a dependence only on the dislocation density tensor \({{\,\mathrm{Curl}\,}}p\) giving rise to a non-symmetric nonlocal backstress, and isotropic hardening response only depending on the accumulated equivalent plastic strain. The model is fully isotropic and satisfies linearized gauge invariance conditions, i.e., only true state variables appear. The model satisfies also the principle of maximum dissipation which allows to show existence for the weak formulation. For this result, a recently introduced Korn’s inequality for incompatible tensor fields is necessary. Uniqueness is shown in the class of strong solutions. For vanishing energetic length scale, the model reduces to classical elasto-plasticity with symmetric plastic strain \(\mathbf \varepsilon _p\) and standard isotropic hardening.

Keywords

Plasticity Gradient plasticity Variational modeling Dissipation function Geometrically necessary dislocations Incompatible distortions Rate-independent models Isotropic hardening Generalized standard material Variational inequality Convex analysis Associated flow rule Defect energy Dislocation density Plastic rotation Global dissipation inequality Burgers vector Plastic spin 

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Notes

Acknowledgements

The research of Francois Ebobisse has been supported by the National Research Foundation (NRF) of South Africa through the Incentive Grant for Rated Researchers and the International Centre for Theoretical Physics (ICTP) through the Associateship Scheme. The first draft of this work was written at Essen (Germany) while Francois Ebobisse was visiting the Faculty of Mathematics of the University of Duisburg-Essen.

References

  1. 1.
    Aifantis, E.C.: On the microstructural origin of certain inelastic models. ASME J. Eng. Mater. Technol. 106, 326–330 (1984)CrossRefGoogle Scholar
  2. 2.
    Aifantis, E.C.: The physics of plastic deformation. Int. J. Plasticity 3, 211–247 (1987)zbMATHCrossRefGoogle Scholar
  3. 3.
    Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)zbMATHCrossRefGoogle Scholar
  4. 4.
    Aifantis, E.C.: Gradient plasticity. In: Lemaitre, J. (ed.) Handbook of Materials Behavior Models, pp. 281–297. Academic Press, New York (2001)CrossRefGoogle Scholar
  5. 5.
    Aifantis, E.C.: Update on a class of gradient theories. Mech. Mater. 35, 259–280 (2003)CrossRefGoogle Scholar
  6. 6.
    Aifantis, E.C.: Gradient material mechanics: perspectives and prospects. Acta Mech. 225, 999–1012 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Alber, H.D.: Materials with Memory. Initial-Boundary Value Problems for Constitutive Equations with Internal Variables. volume 1682 of Lecture Notes in Mathematics. Springer, Berlin (1998)Google Scholar
  8. 8.
    Anand, L., Gurtin, M.E., Reddy, B.D.: The stored energy of cold work, thermal annealing, and other thermodynamic issues in single crystal plasticity at small length scales. Int. J. Plasticity 64, 1–25 (2015)CrossRefGoogle Scholar
  9. 9.
    Bardella, L.: A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bardella, L.: Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scale involved. Int. J. Plasticity 23, 296–322 (2007)zbMATHCrossRefGoogle Scholar
  11. 11.
    Bardella, L.: A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin. Eur. J. Mech. A Solids 28(3), 638–646 (2009)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bardella, L.: Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin. Int. J. Eng. Sci. 48(5), 550–568 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bardella, L., Giacomini, A.: Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity. J. Mech. Phys. Solids 56, 2906–2934 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bardella, L., Panteghini, A.: Modelling the torsion of thin metal wires by distortion gradient plasticity. J. Mech. Phys. Solids 78, 467–492 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Bargmann, S., Reddy, B.D., Klusemann, B.: A computational study of a model of single-crystal strain gradient viscoplasticity with a fully-interactive hardening relation. Int. J. Solids Struct. 51(15–16), 2754–2764 (2014)CrossRefGoogle Scholar
  16. 16.
    Bauer, S., Neff, P., Pauly, D., Starke, G.: New Poincaré-type inequalities. Comptes Rendus Math. 352(4), 163–166 (2014)CrossRefGoogle Scholar
  17. 17.
    Bauer, S., Neff, P., Pauly, D., Starke, G.: Dev-Div-and DevSym-devCurl-inequalities for incompatible square tensor fields with mixed boundary conditions. ESAIM Control Optim. Calc. Var. 22(1), 112–133 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Berdichevsky, V.L., Sedov, L.I.: Dynamic theory of continuously distributed dislocations. Its relation to plasticity theory. PMM 31(6), 981–1000 (1967) (English translation: J. Appl. Math. Mech. (PMM), 989–1006, (1967))Google Scholar
  19. 19.
    Berdichevsky, V.L.: Continuum theory of dislocations revisited. Cont. Mech. Thermod. 18, 195–222 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. Lond. Ser. A. 458, 299–317 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chiricotto, M., Giacomelli, L., Tomassetti, G.: Dissipative scale effects in strain-gradient plasticity: the case of simple shear. SIAM J. Appl. Math. 76(2), 688–704 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dafalias, Y.F.: The plastic spin. J. Appl. Mech. 52, 865–871 (1985)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Dal Maso, G., De Simone, A., Mora, M.G.: Quasistatic evolution problems for linearly elastic–perfectly plastic material. Arch. Ration. Mech. Anal. 180, 237–291 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    De Wit, R.: A view of the relation between the continuum theory of lattice defects and non-Euclidean geometry in the linear approximation. Int. J. Eng. Sci. 19, 1475–1506 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity. Part 1: formulation and analysis. Comput. Methods Appl. Mech. Eng. 196, 3881–3897 (2007)Google Scholar
  26. 26.
    Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: algorithms and numerical analysis. Comput. Methods Appl. Mech. Eng. 197, 1–22 (2007)Google Scholar
  27. 27.
    Ebobisse, F., McBride, A.T., Reddy, B.D.: On the mathematical formulations of a model of gradient plasticity. In: Reddy, B.D. (ed.) IUTAM-Symposium on Theoretical, Modelling and Computational Aspects of Inelastic Media in Cape Town, pp. 117–128. Springer, Berlin (2008)CrossRefGoogle Scholar
  28. 28.
    Ebobisse, F., Neff, P.: Existence and uniqueness in rate-independent infinitesimal gradient plasticity with isotropic hardening and plastic spin. Math. Mech. Solids 15, 691–703 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Ebobisse, F., Neff, P., Aifantis, E.C.: Existence result for a dislocation based model of single crystal gradient plasticity with isotropic or linear kinematic hardening. Quart. J. Mech. Appl. Math. 71, 99–124 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ebobisse, F., Neff, P., Forest, S.: Well-posedness for the microcurl model in both single and polycrystal gradient plasticity. Int. J. Plasticity 107, 1–26 (2018)CrossRefGoogle Scholar
  31. 31.
    Ebobisse, F., Neff, P., Reddy, B.D.: Existence results in dislocation based rate-independent isotropic gradient plasticity with kinematical hardening and plastic spin: the case with symmetric local backstress. http://arxiv.org/pdf/1504.01973.pdf (in review)
  32. 32.
    Ebobisse, F., Neff, P.: A fourth order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor. http://arxiv.org/pdf/1706.08770.pdf (in review)
  33. 33.
    Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (eds.) Advances in Applied Mechanics, vol. 33, pp. 295–361. Academic Press, New York (1997)Google Scholar
  34. 34.
    Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49, 2245–2271 (2001)ADSzbMATHCrossRefGoogle Scholar
  35. 35.
    Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity theory. Part I: scalar plastic multiplier. J. Mech. Phys. Solids 57, 161–177 (2009)Google Scholar
  36. 36.
    Fleck, N.A., Willis, J.R.: A mathematical basis for strain-gradient plasticity theory. Part II: tensorial plastic multiplier. J. Mech. Phys. Solids 57, 1045–1057 (2009)Google Scholar
  37. 37.
    Forest, S., Guéninchault, N.: Inspection of free-energy functions in gradient crystal plasticity. Acta Mech. Sinica 29, 763–772 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Francfort, G., Giacomini, A.: Small strain heterogeneous elastoplasticity revisited. Commun. Pure Appl. Math. 65(9), 1185–1241 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Francfort, G., Giacomini, A., Marigo, J.: The elasto-plasticity exquisite corpse: a Suquet legacy. J. Mech. Phys. Solids i97, 125–139 (2016)Google Scholar
  40. 40.
    Gao, H., Huang, Y., Nix, W.D., Hutchinson, J.W.: Mechanism-based strain gradient plasticity-I. Theory J. Mech. Phys. Solids 47, 1239–1263 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Ghiba, I.-D., Neff, P., Madeo, A., Münch, I.: A variant of the linear isotropic indeterminate couple stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and complete traction boundary conditions. Math. Mech. Solids 22(6), 1221–1266 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Giacomini, A.: On the energetic formulation of the Gurtin and Anand model in strain gradient plasticity. Discrete Contin. Dyn. Syst. Ser. B 17, 527–552 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Giacomini, A., Lussardi, L.: A quasistatic evolution for a model in strain gradient plasticity. SIAM J. Math. Anal. 40(3), 1201–1245 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Gudmundson, P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Gurtin, M.E.: A gradient theory of single-crystal visco-plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Gurtin, M.E.: A gradient theory of small deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568 (2004)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Gurtin, M.E., Needleman, A.: Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector. J. Mech. Phys. Solids 53, 1–31 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Gurtin, M.E., Anand, L.: A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: small deformations. J. Mech. Phys. Solids 53, 1624–1649 (2005)Google Scholar
  49. 49.
    Gurtin, M.E., Anand, L.: A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part II: finite deformation. Int. J. Plasticity 21(12), 2297–2318 (2005)Google Scholar
  50. 50.
    Gurtin, M.E., Anand, L.: 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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  52. 52.
    Gurtin, M.E., Reddy, B.D.: Gradient single-crystal plasticity within a von Mises-Hill framework based on a new formulation of self- and latent-hardening relations. J. Mech. Phys. Solids 68, 134–160 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Hackl, K.: Generalized standard media and variational principles in classical and finite strain elastoplasticity. J. Mech. Phys. Solids 45, 667–688 (1997)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Hackl, K., Fischer, F.D.: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials. Proc. R. Soc. A 464(2089), 117–132 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Hackl, K., Heinz, S., Mielke, A.: A model for the evolution of laminates in finite-strain elastoplasticity. Zeit. f. Angew. Math. Mech. 92(11–12), 888–909 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Hackl, K., Hoppe, U., Kochmann, D.M.: Variational modeling of microstructures in plasticity. In: Schröder, J., Hackl, K. (eds.) Plasticity and Beyond, pp. 65–129. Springer, Vienna (2014)CrossRefzbMATHGoogle Scholar
  57. 57.
    Hackl, K., Kochmann, D.M.: Relaxed potentials and evolution equations for inelastic microstructures. In: Reddy, B.D. (ed.) IUTAM-Symposium on Theoretical, Modelling and Computational Aspects of Inelastic Media in Cape Town, pp. 27–39. Springer, Berlin (2008)CrossRefGoogle Scholar
  58. 58.
    Hackl, K., Fischer, F., Svoboda, J.: A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials. Proc. R. Soc. A 467(2128), 1186–1196 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Hackl, K., Fischer, F., Svoboda, J.: A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials, addendum. Proc. R. Soc. A 467(2132), 2422–2426 (2011)ADSzbMATHCrossRefGoogle Scholar
  60. 60.
    Han, W., Reddy, B.D.: Plasticity: Mathematical Theory and Numerical Analysis. Springer, New York (1999)zbMATHGoogle Scholar
  61. 61.
    He, Q.-C., Vallée, C., Lerintiu, C.: Explicit expressions for the plastic normality-flow rule associated to the Tresca yield criterion. Z. Angew. Math. Phys. 56, 357–366 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Hill, R.: The Mathematical Theory of Plasticity. Oxford University Press, New York (1950)zbMATHGoogle Scholar
  63. 63.
    Hill, R.: A variational principle of maximum plastic work in classical plasticity. Q. J. Mech. Appl. Math. 1(1), 18–28 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Kochmann, D.M., Hackl, K.: The evolution of laminates in finite crystal plasticity: a variational approach. Contin. Mech. Thermodyn. 23, 63–85 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Kraynyukova, N., Neff, P., Nesenenko, S., Chełmiński, K.: Well-posedness for dislocation based gradient visco-plasticity with isotropic hardening. Nonlinear Anal. Real World Appl. 25, 96–111 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Krishnan, J., Steigmann, D.J.: A polyconvex formulation of isotropic elastoplasticity. IMA J. Appl. Math. 79, 722–738 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Kröner, E.: Continuum theory of defects. In: Balian, R., et al. (eds.) Les Houches, Session 35, 1980 - Physique des defauts, pp. 215–315. North-Holland, New York (1981)Google Scholar
  68. 68.
    Lazar, M.: Dislocation theory as a 3-dimensional translation gauge theory. Ann. Phys. (Leipzig) 9, 461–473 (2000)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Lazar, M.: An elastoplastic theory of dislocations as a physical field theory with torsion. J. Phys. A Math. Gen. 35, 1983–2004 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Lazar, M., Anastassiadis, C.: The gauge theory of dislocations: conservation and balance laws. Philos. Mag. 88, 1673–1699 (2008)ADSCrossRefGoogle Scholar
  71. 71.
    Lubliner, J.: Plasticity Theory. Dover Publications, Mineola (2008)zbMATHGoogle Scholar
  72. 72.
    Lucchesi, M., Šilhavý, M.: Il’yushin’s conditions in non-isothermal plasticity. Arch. Ration. Mech. Anal. 113, 121–163 (1991)zbMATHCrossRefGoogle Scholar
  73. 73.
    Mainik, A., Mielke, A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial. Differ. Equ. 22(1), 72–99 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Mandel, J.: Plasticité Classique et Viscoplasticité. Courses and Lectures, No 97, International Center for Mechanical Sciences, Udine. Springer, Berlin (1971)Google Scholar
  75. 75.
    Mandel, J.: Equations constitutives et directeurs dans les milieux plastiques et viscoplasticques. Int. J. Solids Struct. 9, 725–740 (1973)zbMATHCrossRefGoogle Scholar
  76. 76.
    Martin, J.: Plasticity: Fundamental and General Results. MIT Press, Cambridge (1975)Google Scholar
  77. 77.
    Menzel, M., Steinmann, P.: On the formulation of higher gradient plasticity for single and polycrystals. J. Phys. Fr. 8, 239–247 (1998)Google Scholar
  78. 78.
    Menzel, M., Steinmann, P.: On the continuum formulation of higher gradient plasticity for single and polycrystals. J. Mech. Phys. Solids, 48:1777–1796, 2000. Erratum: 49:1179-1180, 2001Google Scholar
  79. 79.
    Mielke, A.: Analysis of energetic models for rate-independent materials. In Li, T. (ed.) Proceedings of the International Congress of Mathematicians 2002, Beijing, III, pp. 817–828. Higher Education Press (2002)Google Scholar
  80. 80.
    Mielke, A.: Evolution of rate-independent systems. In: Dafermos, A., Feireisl, E. (eds.) Evolution equations, vol. II, Handbook of Differential Equations, pp. 461–559. Elsevier/North-Holland, Amsterdam (2005)Google Scholar
  81. 81.
    von Mises, R.: Mechanik der plastischen Formänderung von Kristallen Zeit. Angew. Math. Mech. 8, 161 (1928)zbMATHCrossRefGoogle Scholar
  82. 82.
    Moreau, J.J.: Application of convex analysis to the treatment of elastoplastic systems. In: Germain, P., Nayroles, B. (eds.) Applications of Methods of Functional Analysis to Problems in Mechanics. Springer, Berlin (1976)Google Scholar
  83. 83.
    Mühlhaus, H.B., Aifantis, E.C.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28(7), 845–853 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Neff, P.: On Korn’s first inequality with non-constant coefficients. Proc. R. Soc. Edinb. Sect. A 132(1), 221–243 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Neff, P.: Remarks on invariant modelling in finite strain gradient plasticity. Tech. Mech. 28(1), 13–21 (2008)Google Scholar
  86. 86.
    Neff, P.: Uniqueness of strong solutions in infinitesimal perfect gradient plasticity. In: Reddy, B.D. (ed.) IUTAM-Symposium on Theoretical, Modelling and Computational Aspects of Inelastic Media in Cape Town, pp. 129–140. Springer, Berlin (2008)CrossRefGoogle Scholar
  87. 87.
    Neff, P., Münch, I.: Curl bounds Grad on SO(3). ESAIM Control Optim. Calc. Var. 14(1), 148–159 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Neff, P., Chełmiński, K., Alber, H.D.: Notes on strain gradient plasticity. Finite strain covariant modelling and global existence in the infinitesimal rate-independent case. Math. Mod. Methods Appl. Sci. 19(2), 1–40 (2009)Google Scholar
  89. 89.
    Neff, P., Ghiba, I.-D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Cont. Mech. Therm. 26, 639–681 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Neff, P., Ghiba, I.-D., Lazar, M., Madeo, A.: The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations. Q. J. Mech. Appl. Math. 68, 53–84 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Neff, P., Pauly, D., Witsch, K.J.: On a canonical extension of Korn’s first and Poincaré’s inequalities to H(Curl) motivated by gradient plasticity with plastic spin. Comp. Rend. Math. 349(23–24), 1251–1254 (2011)zbMATHCrossRefGoogle Scholar
  92. 92.
    Neff, P., Pauly, D., Witsch, K.J.: On a canonical extension of Korn’s first and Poincaré’s inequalities to H(Curl). J. Math. Sci. (NY) 185(5), 721–727 (2012)zbMATHCrossRefGoogle Scholar
  93. 93.
    Neff, P., Pauly, D., Witsch, K.J.: Maxwell meets Korn: a new coercive inequality for tensor fields with square integrable exterior derivatives. Math. Methods Appl. Sci. 35(1), 65–71 (2012)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    Neff, P., Pauly, D., Witsch, K.J.: Poincaré meets Korn via Maxwell: extending Korn’s first inequality to incompatible tensor fields. J. Differ. Equ. 258(4), 1267–1302 (2014)ADSzbMATHCrossRefGoogle Scholar
  95. 95.
    Neff, P., Sydow, A., Wieners, C.: Numerical approximation of incremental infinitesimal gradient plasticity. Int. J. Numer. Methods Eng. 77(3), 414–436 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Nesenenko, S., Neff, P.: Well-posedness for dislocation based gradient visco-plasticity I: subdifferential case. SIAM J. Math. Anal. 44(3), 1695–1712 (2012)zbMATHCrossRefGoogle Scholar
  97. 97.
    Nesenenko, S., Neff, P.: Well-posedness for dislocation based gradient visco-plasticity II: general non-associative monotone plastic flow. Math. Mech. Complex Syst. 1(2), 149–176 (2013)zbMATHCrossRefGoogle Scholar
  98. 98.
    Nguyen, Q.-S.: Variational principles in the theory of gradient plasticity. C. R. Mecanique 339, 743–750 (2011)ADSCrossRefGoogle Scholar
  99. 99.
    Nye, J.F.: Some geometrical relations in dislocated solids. Acta Metall. 1, 153–162 (1953)CrossRefGoogle Scholar
  100. 100.
    Onat, E.T.: The notion of state and its implications in thermodynamics of inelastic solids. In: Parkus, H., Sedov, L.I. (eds.) Proceedings of the IUTAM Symposium on Irreversible Aspects of the Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids, Vienna, pp. 292–314. Springer, Wien (1996)Google Scholar
  101. 101.
    Ohno, N., Okumura, D.: Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations. J. Mech. Phys. Solids 55, 1879–1898 (2007)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    Ortiz, M., Repetto, E.: Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397–462 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Panteghini, A., Bardella, L.: On the finite element implementation of higher-order gradient plasticity, with focus on theories based on plastic distortion incompatibility. Comput. Methods Appl. Mech. Eng. 310, 840–865 (2016)ADSMathSciNetCrossRefGoogle Scholar
  104. 104.
    Panteghini, A., Bardella, L.: On the role of higher-order conditions in distortion gradient plasticity. J. Mech. Phys. Solids 118, 293–321 (2018)ADSMathSciNetCrossRefGoogle Scholar
  105. 105.
    Poh, L.H.: Scale transition of a higher order plasticity model–a consistent homogenization theory from meso to macro. J. Mech. Phys. Solids 61, 2692–2710 (2013)ADSMathSciNetCrossRefGoogle Scholar
  106. 106.
    Poh, L.H., Peerlings, R.H.J., Geers, M.G.D., Swaddiwudhipong, S.: An implicit tensorial gradient plasticity model–formulation and comparison with a scalar gradient model. Int. J. Solids Struct. 48(18), 2595–2604 (2011)CrossRefGoogle Scholar
  107. 107.
    Poh, L.H., Peerlings, R.H.J.: The plastic rotation effect in an isotropic gradient plasticity model for applications at the meso scale. Int. J. Solids Struct. 78–79, 57–69 (2016)CrossRefGoogle Scholar
  108. 108.
    Polizzotto, C.: A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle. Int. J. Plasticity 25, 2169–2180 (2009)CrossRefGoogle Scholar
  109. 109.
    Reddy, B.D.: The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity. Cont. Mech. Therm. 23, 527–549 (2011)Google Scholar
  110. 110.
    Reddy, B.D.: The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity. Cont. Mech. Therm. 23, 551–572 (2011)Google Scholar
  111. 111.
    Reddy, B.D., Ebobisse, F., McBride, A.: Well-posedness of a model of strain gradient plasticity for plastically irrotational materials. Int. J. Plasticity 24, 55–73 (2008)zbMATHCrossRefGoogle Scholar
  112. 112.
    Reddy, B.D., Wieners, C., Wohlmuth, B.: Finite element analysis and algorithms for single-crystal strain-gradient plasticity. Int. J. Numer. Methods Eng. 90, 784–804 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Rubin, M.B.: Physical reasons for abandoning plastic deformation measures in plasticity and visco-plasticity. Arch. Mech. 53, 519–539 (2001)zbMATHGoogle Scholar
  114. 114.
    Simo, J.C., Hughes, T.J.R.: Computational Inelasticity. Springer, New York (1998)zbMATHGoogle Scholar
  115. 115.
    Steigmann, D.J., Gupta, A.: Mechanically equivalent elastic-plastic deformations and the problem of plastic spin. Theor. Appl. Mech. (Belgrade) 38, 397–417 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Stölken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46, 5109–5115 (1998)CrossRefGoogle Scholar
  117. 117.
    Suquet, P.-M.: Sur un espace fonctionel pour les équations de la plasticité. Ann. Fac. Sci. Toulouse Math. (5), 1(1), 77–87 (1979)Google Scholar
  118. 118.
    Suquet, P.-M.: Sur les équations de la plasticité: existence et regularité des solutions. J. Mécanique 20, 3–39 (1981)MathSciNetzbMATHGoogle Scholar
  119. 119.
    Svendsen, B.: Continuum thermodynamic models for crystal plasticity including the effects of geometrically necessary dislocations. J. Mech. Phys. Solids 50(25), 1297–1329 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Svendsen, B., Neff, P., Menzel, A.: On constitutive and configurational aspects of models for gradient continua with microstructure. Z. Angew. Math. Mech. 89(8), 687–697 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 121.
    Tresca, H.E.: Mémoires sur l’écoulement des corps solides. Mém. Sav. Acad. Sci., Paris, (Sciences Mathématiques et physiques) 10, 75–135 (1872)Google Scholar
  122. 122.
    Tsagrakis, I., Aifantis, E.C.: Recent developments in gradient plasticity–Part I: formulation and size effects. J. Eng. Mater. Technol. 124(3), 352–357 (2002)CrossRefGoogle Scholar
  123. 123.
    Tsagrakis, I., Efremidis, G., Konstantinidis, A., Aifantis, E.C.: Deformation vs. flow and wavelet-based models of gradient plasticity. Int. J. Plasticity 22, 1456–1485 (2006)Google Scholar
  124. 124.
    Zbib, H.M., Aifantis, E.C.: On the gradient-dependent theory of plasticity and shear banding. Acta Mechanica 92, 209–225 (1992)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • François Ebobisse
    • 1
    Email author
  • Klaus Hackl
    • 2
  • Patrizio Neff
    • 3
  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa
  2. 2.Lehrstuhl für Mechanik-MaterialtheorieRuhr-Universität BochumBochumGermany
  3. 3.Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für MathematikUniversität Duisburg-EssenEssenGermany

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