Skip to main content
SpringerLink
Log in

The thermodynamics of gradient elastoplasticity

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

Abstract

A thermomechanical framework for the modelling of gradient plasticity is developed within the range of linear strains. Full anisotropy is considered. Special focus is given to the restrictions imposed by the Clausius–Duhem inequality. A rather general example gives a complete anisotropic model and shows its thermodynamic consistency. This is finally particularized for the isotropic case by using isotropic tensor-function representations.

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

Similar content being viewed by others

References

  1. Abu Al-Rub R.K.: Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals. Int. J. Plast. 24, 1277–1306 (2008)

    Article  MATH  Google Scholar 

  2. Aifantis E.C.: The physics of plastic deformation. Int. J. Plast. 3, 211–248 (1987)

    Article  MATH  Google Scholar 

  3. Altenbach H., Maugin G.A., Erofeev V.: Mechanics of Generalized Continua. Advanced Structured Materials vol. 7. Springer, Berlin (2011)

    Book  Google Scholar 

  4. Bertram A., Forest S.: Mechanics based on an objective power functional. Techn. Mechanik 27(1), 1–17 (2007)

    Google Scholar 

  5. Bertram A., Krawietz A.: On the introduction of thermoplasticity. Acta Mech. 223(10), 2257–2268 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bertram A.: An alternative approach to finite plasticity based on material isomorphisms. Int. J. Plast. 52, 353–374 (1998)

    Google Scholar 

  7. Bertram, A.: Elasticity and Plasticity of Large Deformations—an Introduction. Springer (2005, 2008, 2012)

  8. Boutin C.: Microstructural effects in elastic composites. Int. J. Solids Struct. 33, 1023–1051 (1996)

    Article  MATH  Google Scholar 

  9. Chambon R., Caillerie D., Matsuchima T.: Plastic continuum with microstructure, local second gradient theories for geomaterials. Int. J. Solids Structures 38, 8503–8527 (2001a)

    Article  MATH  Google Scholar 

  10. Chambon R., Caillerie D., Tamagnini C.: A finite deformation second gradient theory of plasticity. Comptes Rendus de l’Académie des Sciences - Series IIB - Mechanics 329, 797–802 (2001b)

    Article  ADS  Google Scholar 

  11. Dillon O.W., Kratochvil J.: A strain gradient theory of plasticity. Int. J. Solids Struct. 6, 1513–1533 (1970)

    Article  MATH  Google Scholar 

  12. Enakoutsa K., Leblond J.B.: Numerical implementation and assessment of the GLPD micromorphic model of ductile rupture. Eur. J. Mech. A/solids 28, 445–460 (2009)

    Article  ADS  MATH  Google Scholar 

  13. Faciu C., Molinari A.: A non-local rate-type viscoplastic approach to patterning of deformation. Acta Mech. 126, 71–99 (1998)

    Article  MATH  Google Scholar 

  14. Fleck N.A., Hutchinson J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997)

    Article  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  16. Forest, S., Cardona, J.M., Sievert, R.: Thermoelasticity of Second-Grade Media, Continuum Thermomechanics, The Art and Science of Modelling Material Behaviour, Paul Germain’s Anniversary Volume, G.A. Maugin and R. Drouot and F. Sidoroff, Kluwer, 163–176 (2000)

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

    Article  MATH  Google Scholar 

  18. Forest S., Amestoy M.: Hypertemperature in thermoelastic solids. Comptes Rendus Mécanique 336, 347–353 (2008)

    Article  ADS  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Forest S., Trinh D.K.: Generalized continua and non-homogeneous boundary conditions in homogenisation methods. Z. angew. Math. Mech. 91, 90–109 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  23. Germain P., Nguyen Q.S., Suquet P.: Continuum thermodynamics. J Appl. Mech. 50, 1010–1020 (1983)

    Article  MATH  Google Scholar 

  24. Gologanu, M., Leblond, J.B., Devaux, J.: Recent extensions of Gurson’s model for porous ductile metals. Continuum micromechanics, CISM Courses and Lectures No. 377, 61–130, Springer (1997)

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

    Article  MATH  Google Scholar 

  26. Hirschberger C.B., Steinmann P.: Classification of concepts in thermodynamically consistent generalized plasticity. ASCE J. Eng. Mech. 135, 156–170 (2009)

    Article  Google Scholar 

  27. Li J.: A micromechanics-based strain gradient damage model for fracture prediction of brittle materials. Part I: homogenization methodology and constitutive relations. Int. J. Solids Struct. 48, 3336–3345 (2011)

    Article  Google Scholar 

  28. Maugin G.A.: Thermomechanics of nonlinear irreversible behaviors. World Scientific, Singapore (1999)

    MATH  Google Scholar 

  29. Maugin G.A., Metrikine A.V.: Mechanics of generalized continua, one hundred years after the Cosserats. Advances in mechanics and mathematics vol. 21. Springer, Berlin (2010)

    Google Scholar 

  30. Mindlin R.D., Eshel N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)

    Article  MATH  Google Scholar 

  31. Mises R.v.: Mechanik der plastischen Formänderung von Kristallen. Z. angew. Math. Mech. 8(3), 161–185 (1928)

    Article  MATH  Google Scholar 

  32. Mühlich U., Zybell L., Kuna M.: Micromechanical modelling of size effects in failure of porous elastic solids using first order plane strain gradient elasticity. Comput. Mater. Sci. 46, 647–653 (2009)

    Article  Google Scholar 

  33. Papenfuss C., Forest S.: Thermodynamical frameworks for higher grade material theories with internal variables or additional degrees of freedom. J. Non-Equilib. Thermodyn. 31, 319–353 (2006)

    Article  ADS  MATH  Google Scholar 

  34. Polizzotto C.: Unified thermodynamic framework-for nonlocal/gradient continuum theories. Eur. J. Mech. A/Solids 22, 651–668 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Sievert R.: A geometrically nonlinear elasto-viscoplasticity theory of second grade. Technische Mechanik 31, 83–111 (2011)

    Google Scholar 

  36. Shu, J.Y., King, W.E., Fleck, N.A.: Finite elements for materials with strain gradient effects. Int. J. Numer. Meth. Engng. 44, 373–391(1999)

    Google Scholar 

  37. Toupin R.A.: Elastic materials with couple stresses. Arch. Rat. Mech. Anal. 11, 385–414 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  38. Trostel, R.: Gedanken zur Konstruktion mechanischer Theorien. In: Trostel, R. (eds) Beiträge zu den Ingenieurwissenschaften, pp. 96–134. Univ.-Bibl. Techn. Univ., Berlin (1985)

  39. Trostel R.: Mathematische Grundlagen der Technischen Mechanik I- Vektor- und Tensor-Algebra. Vieweg, Wiesbaden (1993)

    Book  MATH  Google Scholar 

  40. Trostel R.: Mathematische Grundlagen der Technischen Mechanik II- Vektor- und Tensor-Analysis. Vieweg, Wiesbaden (1997)

    Book  Google Scholar 

  41. Upadhyay M.V., Capolungo L., Taupin V., Fressengeas C.: Elastic constitutive laws for incompatible crystalline media: the contributions of dislocations, disclinations and G-disclinations. Philos. Mag. 93, 794–832 (2012)

    Article  ADS  Google Scholar 

  42. Voyiadjis G.Z., Faghihi D.: Thermo-mechanical strain gradient plasticity with energetic and dissipative length scales. Int. J. Plast. 30(31), 218–247 (2012)

    Article  Google Scholar 

  43. Voyiadjis, G.Z., Faghihi, D.: Gradient plasticity for thermo-mechanical processes in metals with length and time scales. Philos. Mag., in press, 1–41 (2012)

  44. Zervos A., Papanastasiou P.: Computational post failure analysis with a second gradient theory of elastoplasticity. Eur. J. Environ. Civ. Eng. 14, 1067–1079 (2010)

    Google Scholar 

  45. Zybell L., Mühlich U., Kuna M.: Constitutive equations for porous plane-strain gradient elasticity obtained by homogenization. Arch. Appl. Mech. 79, 359–375 (2009)

    Article  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Magdeburg University, Magdeburg, Germany

    Albrecht Bertram

  2. Centre des Materiaux Mines ParisTech, CNRS UMR 7633, BP 87, 91003, Evry Cedex, France

    Samuel Forest

Authors
  1. Albrecht Bertram
    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 Albrecht Bertram.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bertram, A., Forest, S. The thermodynamics of gradient elastoplasticity. Continuum Mech. Thermodyn. 26, 269–286 (2014). https://doi.org/10.1007/s00161-013-0300-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-013-0300-2

Keywords

Search

Navigation