Skip to main content
SpringerLink
Log in

Finite gradient elasticity and plasticity: a constitutive mechanical framework

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

Abstract

Following a suggestion by Forest and Sievert (Acta Mech 160:71–111, 2003), a constitutive frame for a general gradient elastoplasticity for finite deformations is established. The basic assumptions are the principle of Euclidean invariance and the isomorphy of the elastic ranges. Both the elastic and the plastic laws include the first and the second deformation gradient. The starting point is an objective expression for the stress power.

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. Anand, L., Aslan, O., Chester, S.A.: A large-deformation gradient theory for elastic–plastic materials: strain softening and regularization of shear bands. Int. J. Plast. 30–31, 116–143 (2012)

  2. Auffray N., Le Quang H., He Q.C.: Matrix representations for 3D strain-gradient elasticity. J. Mech. Phys. Solids 61, 1202–1223 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Ashby M.F.: The deformation of plastically nonhomogeneous materials. Philos. Mag. 21(170), 399–424 (1970)

    Article  ADS  Google Scholar 

  4. Bammann, D.J.: A model of crystal plasticity containing a natural length scale. Mater. Sci. Eng. A309–310, 406–410 (2001)

  5. Baek S., Srinivasa A.R.: A variational procedure utilizing the assumption of maximum dissipation rate for gradient-dependent elastic–plastic materials. Int. J. Non-Linear Mech. 38, 659–662 (2003)

    Article  MATH  ADS  Google Scholar 

  6. Bertram A.: Material systems: a framework for the description of material behavior. Arch. Ration. Mech. Anal. 80(2), 99–133 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bertram A.: Axiomatische Einführung in die Kontinuumsmechanik. BI Wissenschaftsverlag, Mannheim (1989)

    MATH  Google Scholar 

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

    Google Scholar 

  9. Bertram A., Svendsen B.: On material objectivity and reduced constitutive equations. Arch. Mech. 536, 653–675 (2001)

    MathSciNet  Google Scholar 

  10. Bertram, A.: Elasticity and Plasticity of Large Deformations: An Introduction. Springer, Berlin (2005, 2008, 2012)

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Bertram A., Forest S.: The thermodynamics of gradient elastoplasticity. Continuum Mech. Thermodyn. 26, 269–286 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  14. Bertram, A.: The Mechanics and Thermodynamics of Finite Gradient Elasticity and Plasticity. Preprint Otto-von-Guericke Universität Magdeburg (2013). http://www.uni-magdeburg.de/ifme/l-festigkeit/pdf/1/Preprint_Gradientenplasti_finite_16.10.12.pdf

  15. Bleustein J.L.: A note on the boundary conditions of Toupin’s strain-gradient theory. Int. J. Solids Struct. 3, 1053–1057 (1967)

    Article  Google Scholar 

  16. 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 (2001)

    ADS  Google Scholar 

  17. Ciarletta P., Maugin G.A.: Elements of a finite strain-gradient thermomechanical theory for material growth and remodeling. Int. J. Nonlinear Mech. 46, 1341–1346 (2011)

    Article  ADS  Google Scholar 

  18. Cleja-Ţigoiu S.: Couple stresses and non-Riemannian plastic connection in finite elasto-plasticity. Z. Angew. Math. Phys. 53, 996–1013 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cleja-Ţigoiu S.: Elasto-plastic materials with lattice defects modeled by seond order deformations with non-zero curvature. Int. J. Fract. 166, 61–75 (2010)

    Article  MATH  Google Scholar 

  20. Cleja-Ţigoiu S.: Non-local elasto-viscoplastic models with dislocations in finite elasto-plasticity. Part I: constitutive framework. Math. Mech. Solids 18(4), 349–372 (2011)

    Article  Google Scholar 

  21. Cross J.J.: Mixtures of fluids and isotropic solids. Arch. Mech. 25(6), 1025–1039 (1973)

    MATH  MathSciNet  Google Scholar 

  22. Dell’Isola F., Sciarra G., Vidoli S.: Generalized Hooke’s law for isotropic second gradient materials. Proc. R. Soc. A. 465, 2177–2196 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Del Piero G.: On the method of virtual power in continuum mechanics. J. Mech. Mater. Struct. 4, 281–292 (2009)

    Article  Google Scholar 

  24. Elzanowski M., Epstein M.: The symmetry group of second-grade materials. Int. J. Non-Linear Mech. 27(4), 635–638 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Ekh M., Grymer M., Runesson K., Svedberg T.: Gradient crystal plasticity as part of the computational modelling of polycrystals. Int. J. Numer. Meth. Eng. 72, 197–220 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Fleck N.A., Muller G.M., Ashby M.F., Hutchinson J.W.: Strain gradient plasticity: theory and experiments. Acta Metall. Mater. 42(2), 475–487 (1994)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  28. Germain P.: La méthode des puissances virtuelles en mécanique des milieux continus. J. Mécanique 12(2), 235–274 (1973)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  ADS  Google Scholar 

  30. Gurtin M.E.: On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48, 989–1036 (2000)

    Article  MATH  MathSciNet  ADS  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, 2297–2318 (2005)

    Article  MATH  Google Scholar 

  32. Gurtin M.E., Fried E., Anand L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  33. Hwang K.C., Jiang H., Huang Y., Gao H., Hu N.: A finite deformation theory of strain gradient plasticity. J. Mech. Phys. Solids 50, 81–99 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Korteweg D.J.: Sur la forme que presennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par de variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothése d’une variation continue de la densité. Archives Néerlandaises Sci. Exactes Naturelles. Ser. II 6, 1–24 (1901)

    MATH  Google Scholar 

  35. Krawietz, A.: Zur Elimination der Verschiebungen bei großen Verformungen. In: Alexandru, C., Gödert, G., Görn, W., Parchem, R., Villwock, J. (eds.) Beiträge zu Mechanik, pp. 133–147. TU, Berlin (1993)

  36. de Leon M., Epstein M.: The geometry of uniformity in second-grade elasticity. Acta Mech. 114, 217–224 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  37. Leroy Y.M., Molinari A.: Spatial patterns and size effects in shear zones: A hyperelastic model with higher-order gradients. J. Mech. Phys. Sol. 41(4), 631–663 (1993)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  38. Luscher D.J., McDowell D.L., Bronkhorst C.A.: A second gradient theoretical framework for hierarchical multiscale modeling of materials. Int. J. Plast. 26, 1248–1275 (2010)

    Article  MATH  Google Scholar 

  39. Miehe C.: Variational gradient plasticity at finite strains. Part I: Mixed potentials for the evolution and update problems of gradient-extended dissipative solids. Comput. Methods Appl. Mech. Eng. 268, 677–703 (2014)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  40. Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)

    Article  Google Scholar 

  41. 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 

  42. Müller Ch., Bruhns O.T.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22, 1658–1682 (2006)

    Article  MATH  Google Scholar 

  43. Murdoch A.I.: Symmetry considerations for materials of second grade. J. Elast. 91, 43–50 (1979)

    Article  MathSciNet  Google Scholar 

  44. Neff P.: Remarks on invariant modelling in finite strain gradient plasticity. Technische Mechanik 28(1), 13–21 (2008)

    Google Scholar 

  45. 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. Models Methods Appl. Sci. 19(2), 307–346 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  46. Noll W.: Materially uniform simple bodies with inhomogeneities. Arch. Rat. Mech. Anal. 27, 1–32 (1967)

    Article  MathSciNet  Google Scholar 

  47. Podio-Guidugli P., Vianello M.: On a stress-power-based characterization of second-gradient elastic fluids. Continuum Mech. Thermodyn. 25, 399–421 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  48. Polizzotto C.: A nonlocal strain gradient plasticity theroy for finite deformations. Int. J. Plast. 25, 1280–1300 (2009)

    Article  MATH  Google Scholar 

  49. Sievert R.: A geometrically nonlinear elasto-viscoplasticity theory of second grade. Technische Mechanik 31(2), 83–111 (2011)

    Google Scholar 

  50. Suiker A.S.J., Chang C.S.: Application of higher-order tensor theory for formulating enhanced continuum models. Acta Mech. 142, 223–234 (2000)

    Article  MATH  Google Scholar 

  51. Svendsen B.: Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. J. Mech. Phys. Solids 50, 1297–1329 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  52. Svendsen B., Neff P., Menzel A.: On constitutive and configurational aspects of models for gradient continua with microstructure. ZAMM 89(8), 687–697 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  53. Testa V., Vianello M.: The symmetry group of gradient sensitive fluids. Int. J. Non-linear Mech. 40, 621–631 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  55. Triantafyllidis N., Aifantis E.C.: A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elast. 16, 225–237 (1986)

    Article  MATH  MathSciNet  Google Scholar 

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

  57. Truesdell, C.A., Noll, W.: The non-linear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3. Springer, Berlin (1965). 2nd edn. (1992), 3rd edn. by S. Antman (2004)

  58. Wang C.-C.: Inhomogeneities in second-grade fluid bodies and istotropic solid bodies. Arch. Mech. 25(5), 765–780 (1973)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany

    Albrecht Bertram

Authors
  1. Albrecht Bertram
    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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bertram, A. Finite gradient elasticity and plasticity: a constitutive mechanical framework. Continuum Mech. Thermodyn. 27, 1039–1058 (2015). https://doi.org/10.1007/s00161-014-0387-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-014-0387-0

Keywords

Search

Navigation