Skip to main content
SpringerLink
Log in

Scalar-based strain gradient plasticity theory to model size-dependent kinematic hardening effects

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

A common belief in phenomenological strain gradient plasticity modeling is that including the gradient of scalar variables in the constitutive setting leads to size-dependent isotropic hardening, whereas the gradient of second-order tensors induces size-dependent kinematic hardening. The present paper shows that it is also possible to produce size-dependent kinematic hardening using scalar-based gradient theory. For this purpose, a new model involving the gradient of the equivalent plastic strain is developed and compared with two reference scalar-based and tensor-based theories. Theoretical investigations using simple monotonic loading conditions are first presented to assess the ability of the proposed model to solve some issues related to existing scalar-based gradient theories. Simulations under cyclic loading conditions are then provided to investigate the nature of the resulting hardening. These simulations show that the proposed model is capable of producing size-dependent kinematic hardening effects at more affordable costs, compared to existing tensor-based strain gradient plasticity theories.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. In general, \(\varepsilon _\mathrm{eq}\) is not considered as a physically relevant hardening variable.

References

  1. Acharya, A., Bassani, J.L.: Lattice incompatibility and a gradient theory of crystal plasticity. J. Mech. Phys. Solids 48(8), 1565–1595 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Aifantis, E.C.: On the microstructural origin of certain inelastic models. J. Eng. Mater. Technol. 106(4), 326–330 (1984)

    Article  Google Scholar 

  3. Aifantis, E.C.: Section 4.12—gradient plasticity. In Lemaitre J (ed.) Handbook of Materials Behavior Models, pp. 281–297. Academic Press, Burlington (2001)

  4. Bardella, L., Panteghini, A.: Modelling the torsion of thin metal wires by distortion gradient plasticity. J. Mech. Phys. Solids 78, 467–492 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Dahlberg, C.F.O., Boåsen, M.: Evolution of the length scale in strain gradient plasticity. Int. J. Plast 112, 220–241 (2019)

    Article  Google Scholar 

  6. Dahlberg, C.F.O., Saito, Y., Öztop, M.S., Kysar, J.W.: Geometrically necessary dislocation density measurements at a grain boundary due to wedge indentation into an aluminum bicrystal. J. Mech. Phys. Solids 105, 131–149 (2017)

    Article  ADS  Google Scholar 

  7. de Borst, R., Mühlhaus, H.-B.: Gradient-dependent plasticity: formulation and algorithmic aspects. Int. J. Numer. Methods Eng. 35(3), 521–539 (1992)

    Article  MATH  Google Scholar 

  8. de Borst, R., Pamin, J.: Some novel developments in finite element procedures for gradient-dependent plasticity. Int. J. Numer. Methods Eng. 39(14), 2477–2505 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, volume 69 of Mathématiques et Applications. Springer, Berlin (2012)

  10. Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. El-Naaman, S.A., Nielsen, K.L., Niordson, C.F.: An investigation of back stress formulations under cyclic loading. Mech. Mater. 130, 76–87 (2019)

    Article  Google Scholar 

  12. Engelen, R.A., Fleck, N.A., Peerlings, R.H., Geers, M.G.: An evaluation of higher-order plasticity theories for predicting size effects and localisation. Int. J. Solids Struct. 43(7–8), 1857–1877 (2006)

    Article  MATH  Google Scholar 

  13. Eymard, R., Guichard, C.: Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form. Computational and Applied Mathematics 37(4), 4023–4054 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fleck, N., Willis, J.: A mathematical basis for strain-gradient plasticity theory. Part II: Tensorial plastic multiplier. J. Mech. Phys. Solids 57, 1045–1057 (2009a)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Fleck, N., Willis, J.: A mathematical basis for strain-gradient plasticity theory. Part I: Scalar plastic multiplier. J. Mech. Phys. Solids 57, 161–177 (2009b)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49(10), 2245–2271 (2001)

    Article  ADS  MATH  Google Scholar 

  17. Fleck, N.A., Hutchinson, J.W., Willis, J.R.: Strain gradient plasticity under non-proportional loading. Proc. Math. Phys. Eng. Sci. 470(2170), 0267 (2014)

    MathSciNet  Google Scholar 

  18. Fleck, N.A., Hutchinson, J.W., Willis, J.R.: Guidelines for constructing strain gradient plasticity theories. J. Appl. Mech. 82(7), 071002 (2015)

    Article  ADS  Google Scholar 

  19. Fleck, N.A., Willis, J.R.: Strain gradient plasticity: energetic or dissipative? Acta. Mech. Sin. 31(4), 465–472 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)

    Google Scholar 

  21. Forest, S.: Questioning size effects as predicted by strain gradient plasticity. J. Mech. Behav. Mater. 22, 101–110 (2013)

    Article  Google Scholar 

  22. Forest, S., Aifantis, E.C.: Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int. J. Solids Struct. 47, 3367–3376 (2010)

    Article  MATH  Google Scholar 

  23. Forest, S., Bertram, A.: Formulations of strain gradient plasticity. In Altenbach, H., Maugin, G.A., Erofeev, V., (eds.), Mechanics of Generalized Continua, pp. 137–150. Advanced Structured Materials vol. 7, Springer (2011)

  24. Forest, S., Mayeur, J.R., McDowell, D.L.: Micromorphic crystal plasticity. In: Voyiadjis, G.Z. (ed.) Handbook of Nonlocal Continuum Mechanics for Materials and Structures, pp. 1–44. Springer International Publishing, New York (2018)

    Google Scholar 

  25. Forest, S., Sievert, R.: Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160(1–2), 71–111 (2003)

    Article  MATH  Google Scholar 

  26. Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus, première partie : théorie du second gradient. J. de Mécanique 12, 235–274 (1973)

    MathSciNet  MATH  Google Scholar 

  27. Gudmundson, P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52(6), 1379–1406 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Gurtin, M.E.: On a framework for small-deformation viscoplasticity: Free energy, microforces, strain gradients. Int. J. Plast 19(1), 47–90 (2003)

    Article  MATH  Google Scholar 

  29. 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(11), 2545–2568 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. 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(7), 1624–1649 (2005a)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Gurtin, M.E., Anand, L.: A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part II: Finite deformations. Int. J. Plast 21(12), 2297–2318 (2005b)

    Article  MATH  Google Scholar 

  32. 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(3), 405–421 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Gurtin, M.E., Anand, L., Lele, S.P.: Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55(9), 1853–1878 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Hayashi, I., Sato, M., Kuroda, M.: Strain hardening in bent copper foils. J. Mech. Phys. Solids 59(9), 1731–1751 (2011)

    Article  ADS  Google Scholar 

  35. Hutchinson, J.W.: Generalizing J 2 flow theory: Fundamental issues in strain gradient plasticity. Acta. Mech. Sin. 28(4), 1078–1086 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. Idiart, M.I., Deshpande, V.S., Fleck, N.A., Willis, J.R.: Size effects in the bending of thin foils. Int. J. Eng. Sci. 47(11–12), 1251–1264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Jebahi, M., Cai, L., Abed-Meraim, F.: Strain gradient crystal plasticity model based on generalized non-quadratic defect energy and uncoupled dissipation. Int. J. Plast 126, 102617 (2020)

    Article  Google Scholar 

  38. Liu, D., He, Y., Dunstan, D.J., Zhang, B., Gan, Z., Hu, P., Ding, H.: Toward a further understanding of size effects in the torsion of thin metal wires: An experimental and theoretical assessment. Int. J. Plast 41, 30–52 (2013)

    Article  Google Scholar 

  39. Martínez-Pañeda, E., Deshpande, V.S., Niordson, C.F., Fleck, N.A.: The role of plastic strain gradients in the crack growth resistance of metals. J. Mech. Phys. Solids 126, 136–150 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  40. Mühlhaus, H.B., Alfantis, E.C.: A variational principle for gradient plasticity. Int. J. Solids Struct. 28(7), 845–857 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. Niordson, C.F., Hutchinson, J.W.: On lower order strain gradient plasticity theories. Eur. J. Mech. A/Solids 22(6), 771–778 (2003)

    Article  ADS  MATH  Google Scholar 

  42. Panteghini, A., Bardella, L.: On the role of higher-order conditions in distortion gradient plasticity. J. Mech. Phys. Solids 118, 293–321 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  43. Panteghini, A., Bardella, L., Niordson, C.F.: A potential for higher-order phenomenological strain gradient plasticity to predict reliable response under non-proportional loading. Proc. Math. Phys. Eng. Sci. 475(2229), 20190258 (2019)

    MathSciNet  Google Scholar 

  44. Peerlings, R.: On the role of moving elastic-plastic boundaries in strain gradient plasticity. Modell. Simul. Mater. Sci. Eng. 15, S109–S120 (2007)

    Article  ADS  Google Scholar 

  45. Peerlings, R., Geers, M., de Borst, R., Brekelmans, W.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38(44–45), 7723–7746 (2001)

    Article  MATH  Google Scholar 

  46. Poh, L., Peerlings, R., Geers, M., Swaddiwudhipong, S.: An implicit tensorial gradient plasticity model: formulation and comparison with a scalar gradient model. Int. J. Solids Struct. 48, 2595–2604 (2011)

    Article  Google Scholar 

  47. Rys̀, M., Forest, S., Petryk, H.: A micromorphic crystal plasticity model with the gradient-enhanced incremental hardening law. Int. J. Plast. 128, 102655 (2020)

  48. Sarac, A., Oztop, M.S., Dahlberg, C.F.O., Kysar, J.W.: Spatial distribution of the net burgers vector density in a deformed single crystal. Int. J. Plast 85, 110–129 (2016)

    Article  Google Scholar 

  49. Steinmann, P.: Views on multiplicative elastoplasticity and the continuum theory of dislocations. Int. J. Eng. Sci. 34, 1717–1735 (1996)

    Article  MATH  Google Scholar 

  50. Stupkiewicz, S., Petryk, H.: A minimal gradient-enhancement of the classical continuum theory of crystal plasticity. Part II: Size effects. Arch. Mech. 68(6), 487–513 (2016)

    MathSciNet  MATH  Google Scholar 

  51. Voyiadjis, G.Z., Song, Y.: Strain gradient continuum plasticity theories: theoretical, numerical and experimental investigations. Int. J. Plast 121, 21–75 (2019)

    Article  Google Scholar 

  52. Wulfinghoff, S., Bayerschen, E., Böhlke, T.: Conceptual difficulties in plasticity including the gradient of one scalar plastic field variable. PAMM Proc. Appl. Math. Mech. 14, 317–318 (2014)

    Article  Google Scholar 

  53. Wulfinghoff, S., Böhlke, T.: Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics. Proc. Math. Phys. Eng. Sci. 468(2145), 2682–2703 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Arts et Metiers Institute of Technology, CNRS, Université de Lorraine, LEM3, 57000, Metz, France

    Mohamed Jebahi

  2. MINES ParisTech, PSL University, Centre des matériaux (CMAT), CNRS UMR 7633, BP 87, 91003, Evry, France

    Samuel Forest

Authors
  1. Mohamed Jebahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel Forest
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel Forest.

Additional information

Communicated by Marcus Aßmus, Victor A. Eremeyev and Andreas Öchsner.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jebahi, M., Forest, S. Scalar-based strain gradient plasticity theory to model size-dependent kinematic hardening effects. Continuum Mech. Thermodyn. 33, 1223–1245 (2021). https://doi.org/10.1007/s00161-020-00967-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-020-00967-0

Keywords

Search

Navigation