Advertisement

Continuum Mechanics and Thermodynamics

, Volume 27, Issue 6, pp 1039–1058 | Cite as

Finite gradient elasticity and plasticity: a constitutive mechanical framework

  • Albrecht BertramEmail author
Original Article

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.

Keywords

Gradient plasticity Gradient elasticity Finite deformations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)Google Scholar
  2. 2.
    Auffray N., Le Quang H., He Q.C.: Matrix representations for 3D strain-gradient elasticity. J. Mech. Phys. Solids 61, 1202–1223 (2013)zbMATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Ashby M.F.: The deformation of plastically nonhomogeneous materials. Philos. Mag. 21(170), 399–424 (1970)CrossRefADSGoogle Scholar
  4. 4.
    Bammann, D.J.: A model of crystal plasticity containing a natural length scale. Mater. Sci. Eng. A309–310, 406–410 (2001)Google Scholar
  5. 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)zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Bertram A.: Material systems: a framework for the description of material behavior. Arch. Ration. Mech. Anal. 80(2), 99–133 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bertram A.: Axiomatische Einführung in die Kontinuumsmechanik. BI Wissenschaftsverlag, Mannheim (1989)zbMATHGoogle Scholar
  8. 8.
    Bertram A.: An alternative approach to finite plasticity based on material isomorphisms. Int. J. Plast. 52, 353–374 (1998)Google Scholar
  9. 9.
    Bertram A., Svendsen B.: On material objectivity and reduced constitutive equations. Arch. Mech. 536, 653–675 (2001)MathSciNetGoogle Scholar
  10. 10.
    Bertram, A.: Elasticity and Plasticity of Large Deformations: An Introduction. Springer, Berlin (2005, 2008, 2012)Google Scholar
  11. 11.
    Bertram A., Forest S.: Mechanics based on an objective power functional. Techn. Mech. 27(1), 1–17 (2007)Google Scholar
  12. 12.
    Bertram A., Krawietz K.: On the introduction of thermoplasticity. Acta Mech. 223(10), 2257–2268 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bertram A., Forest S.: The thermodynamics of gradient elastoplasticity. Continuum Mech. Thermodyn. 26, 269–286 (2014)MathSciNetCrossRefADSGoogle Scholar
  14. 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. 15.
    Bleustein J.L.: A note on the boundary conditions of Toupin’s strain-gradient theory. Int. J. Solids Struct. 3, 1053–1057 (1967)CrossRefGoogle Scholar
  16. 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)ADSGoogle Scholar
  17. 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)CrossRefADSGoogle Scholar
  18. 18.
    Cleja-Ţigoiu S.: Couple stresses and non-Riemannian plastic connection in finite elasto-plasticity. Z. Angew. Math. Phys. 53, 996–1013 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 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)zbMATHCrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 21.
    Cross J.J.: Mixtures of fluids and isotropic solids. Arch. Mech. 25(6), 1025–1039 (1973)zbMATHMathSciNetGoogle Scholar
  22. 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)zbMATHMathSciNetCrossRefADSGoogle Scholar
  23. 23.
    Del Piero G.: On the method of virtual power in continuum mechanics. J. Mech. Mater. Struct. 4, 281–292 (2009)CrossRefGoogle Scholar
  24. 24.
    Elzanowski M., Epstein M.: The symmetry group of second-grade materials. Int. J. Non-Linear Mech. 27(4), 635–638 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 27.
    Forest S., Sievert R.: Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160, 71–111 (2003)zbMATHCrossRefGoogle Scholar
  28. 28.
    Germain P.: La méthode des puissances virtuelles en mécanique des milieux continus. J. Mécanique 12(2), 235–274 (1973)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Gudmundson P.: A unified treatment of strain gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406 (2004)zbMATHMathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Gurtin M.E.: On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48, 989–1036 (2000)zbMATHMathSciNetCrossRefADSGoogle Scholar
  31. 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)zbMATHCrossRefGoogle Scholar
  32. 32.
    Gurtin M.E., Fried E., Anand L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2009)Google Scholar
  33. 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)zbMATHMathSciNetCrossRefADSGoogle Scholar
  34. 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)zbMATHGoogle Scholar
  35. 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)Google Scholar
  36. 36.
    de Leon M., Epstein M.: The geometry of uniformity in second-grade elasticity. Acta Mech. 114, 217–224 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 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)zbMATHMathSciNetCrossRefADSGoogle Scholar
  38. 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)zbMATHCrossRefGoogle Scholar
  39. 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)zbMATHMathSciNetCrossRefADSGoogle Scholar
  40. 40.
    Mindlin R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)CrossRefGoogle Scholar
  41. 41.
    Mindlin R.D., Eshel N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)zbMATHCrossRefGoogle Scholar
  42. 42.
    Müller Ch., Bruhns O.T.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22, 1658–1682 (2006)zbMATHCrossRefGoogle Scholar
  43. 43.
    Murdoch A.I.: Symmetry considerations for materials of second grade. J. Elast. 91, 43–50 (1979)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Neff P.: Remarks on invariant modelling in finite strain gradient plasticity. Technische Mechanik 28(1), 13–21 (2008)Google Scholar
  45. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Noll W.: Materially uniform simple bodies with inhomogeneities. Arch. Rat. Mech. Anal. 27, 1–32 (1967)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Podio-Guidugli P., Vianello M.: On a stress-power-based characterization of second-gradient elastic fluids. Continuum Mech. Thermodyn. 25, 399–421 (2013)MathSciNetCrossRefADSGoogle Scholar
  48. 48.
    Polizzotto C.: A nonlocal strain gradient plasticity theroy for finite deformations. Int. J. Plast. 25, 1280–1300 (2009)zbMATHCrossRefGoogle Scholar
  49. 49.
    Sievert R.: A geometrically nonlinear elasto-viscoplasticity theory of second grade. Technische Mechanik 31(2), 83–111 (2011)Google Scholar
  50. 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)zbMATHCrossRefGoogle Scholar
  51. 51.
    Svendsen B.: Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. J. Mech. Phys. Solids 50, 1297–1329 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  52. 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)zbMATHMathSciNetCrossRefADSGoogle Scholar
  53. 53.
    Testa V., Vianello M.: The symmetry group of gradient sensitive fluids. Int. J. Non-linear Mech. 40, 621–631 (2005)zbMATHMathSciNetCrossRefADSGoogle Scholar
  54. 54.
    Toupin R.A.: Elastic materials with couple stresses. Arch. Rat. Mech. Anal. 11, 385–414 (1962)zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Triantafyllidis N., Aifantis E.C.: A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elast. 16, 225–237 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Trostel, R.: Gedanken zur Konstruktion mechanischer Theorien. In: Trostel, R. (ed.) Beiträge zu den Ingenieurwissenschaften. Univ.-Bibl. Techn. Univ. Berlin, 96–134 (1985)Google Scholar
  57. 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)Google Scholar
  58. 58.
    Wang C.-C.: Inhomogeneities in second-grade fluid bodies and istotropic solid bodies. Arch. Mech. 25(5), 765–780 (1973)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Otto-von-Guericke Universität MagdeburgMagdeburgGermany

Personalised recommendations