Skip to main content
SpringerLink
Log in

Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

Following Ogden, a class of objective (Lagrangian and Eulerian) tensors is identified among the second-rank tensors characterizing continuum deformation, but a more general definition of objectivity than that used by Ogden is introduced. Time rates of tensors are determined using convective rates. Sufficient conditions of objectivity are obtained for convective rates of objective tensors. Objective convective rates of strain tensors are used to introduce pairs of symmetric stress and strain tensors conjugate in a generalized sense. The classical definitions of conjugate Lagrangian (after Hill) and Eulerian (after Xiao et al.) stress and strain tensors are particular cases of the definition of conjugacy of stress and strain tensors in the generalized sense used in the present paper. Pairs of objective stress and strain tensors conjugate in the generalized sense are used to formulate constitutive relations for a hyperelastic medium. A family of objective generalized strain tensors is introduced, which is broader than Hill’s family of strain tensors. The basic forms of the hyperelastic constitutive relations are obtained with the aid of pairs of Lagrangian stress and strain tensors conjugate after Hill (the strain tensors in these pairs belong to the family of generalized strain tensors). A method is presented for generating reduced forms of the constitutive relations with the aid of pairs of Lagrangian and Eulerian stress and strain tensors conjugate in the generalized sense which are obtained from pairs of Lagrangian tensors conjugate after Hill by mapping tensor fields on one configuration of a deformable body to tensor fields on another configuration.

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. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3. Springer, Berlin (1965)

    Google Scholar 

  2. Truesdell, C.: A First Course in Rational Continuum Mechanics. The John Hopkins University, Baltimore (1972)

    Google Scholar 

  3. Hill, R.: On constitutive inequalities for simple materials—1. J. Mech. Phys. Solids 16(4), 229–242 (1968)

    Article  MATH  ADS  Google Scholar 

  4. Hill, R.: Aspects of invariance in solid mechanics In: Yih, C.-S. (ed.) Advances in Applied Mechanics, vol. 18, pp. 1–75. Academic Press, New York (1978)

    Chapter  Google Scholar 

  5. Lur’e, A.I.: Nonlinear Theory of Elasticity. Nauka, Moscow (1980) (in Russian)

    MATH  Google Scholar 

  6. Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs (1983)

    MATH  Google Scholar 

  7. Ogden, R.W.: Non-linear Elastic Deformations. Ellis Horwood, Chichester (1984)

    Google Scholar 

  8. Chernykh, K.Ph.: Nonlinear Theory of Elasticity in Engineering Analysis. Mashinostroenie, Leningrad (1986) (in Russian)

    Google Scholar 

  9. Pozdeev, A.A., Trusov, P.V., Nyashin, Yi.I.: Large Elastoplastic Strains. Nauka, Moscow (1986) (in Russian)

    Google Scholar 

  10. Curnier, A., Rakotomanana, L.: Generalized strain and stress measures: critical survey and new results. Eng. Trans. 39(3–4), 461–538 (1991)

    MathSciNet  Google Scholar 

  11. Curnier, A.: Computational Methods in Solid Mechanics. Kluwer Academic, Dordrecht (1994)

    MATH  Google Scholar 

  12. Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Egineering. Wiley, Chichester (2000)

    Google Scholar 

  13. Korobeynikov, S.N.: Nonlinear Strain Analysis of Solids. Sib. Div. Russ. Acad. Sci., Novosibirsk (2000) (in Russian)

    Google Scholar 

  14. Kondaurov, V.I., Fortov, V.E.: Thermomechanical Foundations of Condensed Media. Moscow Phys.-Tech. Inst., Moscow (2002) (in Russian)

    Google Scholar 

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

    MATH  Google Scholar 

  16. Murdoch, A.I.: On objectivity and material symmetry for simple elastic solids. J. Elast. 60, 233–242 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Korobeynikov, S.N.: Objective Lie derivatives of tensors in continuum mechanics. In: Andreev, V.K. (ed.) Proceedings of the 3rd Int. Conf. on Symmetry and Differential Equations, pp. 133–139. Institute of Numerical Simulation, Krasnoyarsk (2002) (in Russian)

    Google Scholar 

  18. Liu, I.S.: On the transformation property of the deformation gradient under a change of frame. J. Elast. 71, 73–80 (2003)

    Article  MATH  Google Scholar 

  19. Annin, B.D., Korobeynikov, S.N.: Generalized conjugate stress and strain tensors. J. Ind. Math. 7(3), 21–43 (2004) (in Russian)

    MATH  Google Scholar 

  20. Guo, Z.-H.: Time derivatives of tensor fields in non-linear continuum mechanics. Arch. Mech. (Arch. Mech. Stosow.) 15(1), 131–163 (1963)

    MATH  Google Scholar 

  21. Meyers, A., Schieße, P., Bruhns, O.T.: Some comments on objective rates of symmetric Eulerian tensors with application to Eulerian strain rates. Acta Mech. 139(1–4), 91–103 (2000)

    Article  MATH  Google Scholar 

  22. Xiao, H., Bruhns, O.T., Meyers, A.: Objective corotational rates and unified work-conjugacy relation between Eulerian and Lagrangean strain and stress measures. Arch. Mech. 50(6), 1015–1045 (1998)

    MATH  MathSciNet  Google Scholar 

  23. Green, A.E., Naghdi, P.M.: A general theory of an elastic-plastic continuum. Arch. Ration. Mech. Anal. 18(4), 251–281 (1965)

    Article  MathSciNet  Google Scholar 

  24. Novozhilov, V.V.: Principles of processing of the results of static tests of isotropic materials. Prikl. Mat. Mekh. 15(6), 709–722 (1951) (in Russian)

    MATH  Google Scholar 

  25. Ziegler, H., McVean, D.: On the notion of an elastic solid. In: Broberg, B. et al. (eds.) Recent Progress in Applied Mechanics (The Folke Odqvist Volume), pp. 561–572. Almquist & Wiksell/Wiley, Stockholm/New York (1967)

    Google Scholar 

  26. Doyle, T.C., Ericksen, J.L.: Nonlinear elasticity. In: Dryden, H.L., von Karman, Th. (eds.) Advances in Applied Mechanics, vol. 4, pp. 53–115. Academic Press, New York (1956)

    Google Scholar 

  27. Atluri, S.N.: Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analysis of finitely deformed solids, with application to plates and shells—I. Theory. Comput. Struct. 18(1), 93–116 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lehmann, Th., Guo, Z.-H., Liang, H.: The conjugacy between Cauchy stress and logarithm of the left stretch tensor. Eur. J. Mech. A/Solids 10(4), 395–404 (1991)

    MATH  MathSciNet  Google Scholar 

  29. Haupt, P., Tsakmakis, Ch.: On the application of dual variables in continuum mechanics. Contin. Mech. Thermodyn. 1, 165–196 (1989)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Svendsen, B., Tsakmakis, Ch.: A local differential geometric formulation of dual stress-strain pairs and time derivatives. Arch. Mech. 46(1–2), 49–91 (1994)

    MATH  MathSciNet  Google Scholar 

  31. Haupt, P., Tsakmakis, Ch.: Stress tensors associated with deformation tensors via duality. Arch. Mech. 48(2), 347–384 (1996)

    MATH  MathSciNet  Google Scholar 

  32. Xiao, H., Bruhns, O.T., Meyers, A.: Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mech. 124, 89–105 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Xiao, H., Bruhns, O.T., Meyers, A.: Direct relationship between the Lagrangean logarithmic strain and the Lagrangean stretching and the Lagrangean Kirchhoff stress. Mech. Res. Commun. 25(1), 59–67 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Xiao, H., Bruhns, O.T., Meyers, A.: Existence and uniqueness of the integrable-exactly hypoelastic equation \(\stackrel{\circ}{\boldsymbol{\tau}}{}^{\ast}=\lambda(\mathrm{tr}\,\mathbf{D})\mathbf{I}+2\mu\mathbf{D}\) and its significance to finite inelasticity. Acta Mech. 138, 31–50 (1999)

    Article  MATH  Google Scholar 

  35. Steinmann, P.: On spatial and material settings of thermo-hyperelastodynamics. J. Elast. 66, 109–157 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Xiao, H., Bruhns, O.T., Meyers, A.: Hypo-elasticity model based upon the logarithmic stress rate. J. Elast. 47, 51–68 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  37. Annin, B.D., Korobeynikov, S.N.: Allowable forms of elastic deformation laws in elaso-plastic constitutive relations. J. Ind. Math. 1(1), 21–34 (1998) (in Russian)

    Google Scholar 

  38. Trusov, P.V., Dudar’, O.I., Keller, I.A.: Tensor Algebra and Analysis. Perm’ State Technical University, Perm’ (1998) (in Russian)

    Google Scholar 

  39. Sedov, L.I.: Introduction to the Mechanics of a Continuous Medium. Addison-Wesley, Reading (1965)

    MATH  Google Scholar 

  40. Bruhns, O.T., Meyers, A., Xiao, H.: On non-corotational rates of Oldroyd’s type and relevant issues in rate constitutive formulations. Proc. R. Soc. Lond. A 460, 909–928 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Korobeinikov, S.N.: Natural stress tensors. J. Appl. Mech. Tech. Phys. 42(6), 1051–1056 (2001)

    Article  MathSciNet  Google Scholar 

  42. Simo, J.C., Marsden, J.E.: On the rotated stress tensor and the material version of the Doyle–Ericksen formula. Arch. Ration. Mech. Anal. 86, 213–231 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  43. Korobeinikov, S.N.: Strictly conjugate stress and strain tensors. J. Appl. Mech. Tech. Phys. 41(3), 513–518 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  44. Korobeinikov, S.N.: Conjugate nonsymmetric stress and strain tensors. In: Rastorguev, G.I. (ed.) Proceedings of the 1st Russian–Korean Int. Symp. on Applied Mechanics (RUSKO-AM-2001), pp. 69–73. Novosibirsk State Technical University, Novosibirsk (2001)

    Google Scholar 

  45. Curnier, A., Zysset, Ph.: A family of metric strains and conjugate stresses, prolonging usual material laws from small to large transformations. Int. J. Solids Struct. 43, 3057–3086 (2006)

    Article  MATH  Google Scholar 

  46. Reinhardt, W.B., Dubey, R.N.: Application of objective rates in mechanical modeling of solids. Trans. ASME J. Appl. Mech. 63(3), 692–698 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  47. Murnaghan, F.D.: Finite deformations of an elastic solid. Am. J. Math. 59(2), 235–260 (1937)

    Article  MATH  MathSciNet  Google Scholar 

  48. Murnaghan, F.D.: Finite Deformation of an Elastic Solid. Wiley, New York (1951)

    MATH  Google Scholar 

  49. Dluzewski, P.: Anisotropic hyperelasticity based upon general strain measures. J. Elast. 60, 119–129 (2000)

    Article  MATH  Google Scholar 

  50. Xiao, H., Bruhns, O.T., Meyers, A.: A natural generalization of hypoelasticity and Eulerian rate type formulation of hyperelasticity. J. Elast. 56, 59–93 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  51. Anand, L.: On Hencky’s approximate strain-energy function for moderate deformations. Trans. ASME J. Appl. Mech. 46(1), 78–82 (1979)

    MATH  MathSciNet  Google Scholar 

  52. Anand, L.: Moderate deformations in extension-torsion of incompressible isotropic elastic materials. J. Mech. Phys. Solids 34, 293–304 (1986)

    Article  ADS  Google Scholar 

  53. Bruhns, O.T., Xiao, H., Meyers, A.: Henky’s elasticity model with the logarithmic strain measure: a study on Poynting effect and stress response in torsion of tubes and rods. Arch. Mech. 52(4–5), 489–509 (2000)

    MATH  MathSciNet  Google Scholar 

  54. Xiao, H., Bruhns, O.T., Meyers, A.: Explicit dual stress-strain and strain-stress relations of incompressible isotropic hyperelastic solids via deviatoric Hencky strain and Cauchy stress. Acta Mech. 168, 21–33 (2004)

    Article  MATH  Google Scholar 

  55. Xiao, H., Bruhns, O.T., Meyers, A.: Objective stress rates, path-dependence properties and non-integrability problems. Acta Mech. 176, 135–151 (2005)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Novosibirsk, 630090, Russia

    S. N. Korobeynikov

Authors
  1. S. N. Korobeynikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. N. Korobeynikov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korobeynikov, S.N. Objective Tensor Rates and Applications in Formulation of Hyperelastic Relations. J Elasticity 93, 105–140 (2008). https://doi.org/10.1007/s10659-008-9166-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-008-9166-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation