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Acta Mechanica

, Volume 227, Issue 3, pp 651–675 | Cite as

Consistent hypo-elastic behavior using the four-dimensional formalism of differential geometry

  • B. Panicaud
  • E. Rouhaud
  • G. Altmeyer
  • M. Wang
  • R. Kerner
  • A. Roos
  • O. Ameline
Original Paper

Abstract

The covariance principle of the theory of relativity within a four-dimensional framework ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space–time and the use of 4D tensors, operations and operators. This 4D formalism enables also to clearly distinguish between the covariance principle (i.e., frame-indifference) and the material objectivity principle (i.e., indifference to any rigid body motion superposition). We propose and apply here a method to build a constitutive relation for elastic materials using such a 4D formalism. The present article is specifically devoted in the application of this methodology to construct hypo-elastic materials with the use of the 4D Lie derivative. It enables thus to obtain consistent non-dissipative models equivalent to (hyper)elastic ones.

Keywords

Constitutive Model Simple Shear Rigid Body Motion Covariant Component Tensor Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Speziale C.G.: A review of material frame-indifference in mechanics. Appl. Mech. Rev. 51, 489 (1998)CrossRefGoogle Scholar
  2. 2.
    Frewer M.: More clarity on the concept of material frame-indifference in classical continuum mechanics. Acta Mech. 202, 213–246 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Romano G., Baretta R.: Covariant hypo-elasticity. Eur. J. Mech. A Solids 30, 1012–1023 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Romano G., Baretta R.: Geometric constitutive theory and frame invariance. Int. J. Non-Linear Mech. 51, 75–86 (2013)CrossRefGoogle Scholar
  5. 5.
    Eringen A.C.: Nonlinear Theory of Continuous Media. McGraw-Hill, New York (1962)Google Scholar
  6. 6.
    Malvern L.E.: Introduction to the Mechanics of Continuous Medium. Prentice Hall, Upper Saddle River (1969)zbMATHGoogle Scholar
  7. 7.
    Truesdell C., Noll W.: The Non-Linear Field Theories of Mechanics, third edition. Springer, New York (2003)zbMATHGoogle Scholar
  8. 8.
    Murdoch A.I.: Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations. Contin. Mech. Thermodyn. 15, 309–320 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu I.S.: Further remarks on euclidean objectivity and the principle of material frame-indifference. Contin. Mech. Thermodyn. 17, 125–133 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Oldroyd J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Stumpf H., Hoppe U.: The application of tensor algebra on manifolds to nonlinear continuum mechanics—invited survey article. Math. Mech. 77, 327–339 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Marsden J.E., Hugues J.R.: Mathematical Foundation of Elasticity. Prentice Hall, Upper Saddle River (1983)Google Scholar
  13. 13.
    Rougée P.: A new Lagrangian intrinsic approach to large deformations in continuous media. Eur. J. Mech. A Solids 10, 15–39 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Rougée P.: An intrinsic Lagrangian statement of constitutive laws in large strain. Comput. Struct. 84, 1125–1133 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Simo J.C., Ortiz M.: A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput. Methods Appl. Mech. Eng. 49, 221–245 (1985)CrossRefzbMATHGoogle Scholar
  17. 17.
    Yavari, A., Marsden, J.E., Ortiz, M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 042903-1–042903-53 (2006)Google Scholar
  18. 18.
    Yavari A., Ozakin A.: Covariance in linearized elasticity. Z. Angew. Math. Phys. 59, 1081–1110 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yavari, A., Marsden, J.E.: Covariantization of nonlinear elasticity. Z. Angew. Math. Phys. 63, 921–927 (2012)Google Scholar
  20. 20.
    Xiao H., Bruhns O.T., Meyers A.: Existence and uniqueness of the integrable-exactly hypoelastic equation and its significance to finite inelasticity. Acta Mech. 138, 31–50 (1999)CrossRefzbMATHGoogle Scholar
  21. 21.
    Weinberg S.: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, New York (1972)Google Scholar
  22. 22.
    Landau L.D., Lifshitz E.M.: The Classical Theory of Fields, fourth edition. Elsevier, Amsterdam (1975)zbMATHGoogle Scholar
  23. 23.
    Panicaud B., Rouhaud E.: A frame-indifferent model for a thermo-elastic material beyond the three-dimensional Eulerian and Lagrangian descriptions. Contin. Mech. Thermodyn. 26, 79–93 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rouhaud E., Panicaud B., Kerner R.: Canonical frame-indifferent transport operators with the four-dimensional formalism of differential geometry. Comput. Mater. Sci. 77, 120–130 (2013)CrossRefGoogle Scholar
  25. 25.
    Murdoch A.I.: On objectivity and material symmetry for simple elastic solids. J. Elast. 60, 233–242 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liu I.S.: On euclidean objectivity and the principle of material frame-indifference. Contin. Mech. Thermodyn. 16, 177–183 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Murdoch A.I.: On criticism of the nature of objectivity in classical continuum physics. Contin. Mech. Thermodyn. 17, 135–148 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sidoroff, F.: Cours sur les grandes déformations, Ecole d’été Sophia-Antipolis, Rapport GRECO (1982)Google Scholar
  29. 29.
    Dogui A., Sidoroff F.: Kinematic hardening in large elastoplastic strain. Eng. Fract. Mech. 21, 685–695 (1985)CrossRefGoogle Scholar
  30. 30.
    Badreddine H., Saanouni K., Dogui A.: On non-associative anisotropic finite plasticity fully coupled with isotropic ductile damage for metal forming. Int. J. Plast. 26, 1541–1575 (2010)CrossRefzbMATHGoogle Scholar
  31. 31.
    Jaumann, G.: Geschlossenes System physikalischer und chemischer Differentialgesetze. Sitzber. Akad. Wiss. Wien, Abt. Iia. 120, 385–530 (1911)Google Scholar
  32. 32.
    Green A.E., Naghdi P.M.: A general theory of an elastic-plastic continuum. Arch. Ration. Mech. Anal. 18, 251–281 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Forest, S.: Lois de comportement en transformations finies, in Aussois (2008)Google Scholar
  34. 34.
    Levi-Civita T., Persico E., Long M.: The Absolute Differential Calculus. Dover Phoenix Editions, New York (2005)Google Scholar
  35. 35.
    Schouten J.A.: Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications. Springer, New York (1954)CrossRefzbMATHGoogle Scholar
  36. 36.
    Misner C.W., Thorne K.S., Wheeler J.A.: Gravitation. W. H. Freeman and Co Ltd., New York (1973)Google Scholar
  37. 37.
    Boratav M., Kerner R.: Relativité. Ellipses, Paris (1991)Google Scholar
  38. 38.
    Bressan A.: Relativistic Theories of Materials. Springer, Berlin (1978)CrossRefzbMATHGoogle Scholar
  39. 39.
    Capurro M.: A general field theory of cauchy continuum: Classical mechanics. Acta Mech. 49, 169–190 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ferrarese G., Bini D.: Relativistic kinematics for a three-dimensional continuum. Lect. Notes Phys. 727, 169–206 (2008)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Grot R.A., Eringen A.C.: Relativistic continuum mechanics part i mechanics and thermodynamics. Int. J. Eng. Sci. 4, 611–638 (1966)CrossRefGoogle Scholar
  42. 42.
    Kienzler R., Herrmann G.: On the four-dimensional formalism in continuum mechanics. Acta Mech. 161, 103–125 (2003)CrossRefzbMATHGoogle Scholar
  43. 43.
    Lamoureux-Brousse L.: Infinitesimal deformations of finite conjugacies in non-linear classical or general relativistic theory of elasticity. Phys. D 35, 203–219 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Maugin G.: Un modèle viscoélastique en relativité générale. Compt. Rendus Acad. Sci. 272A, 1482–1484 (1971)MathSciNetGoogle Scholar
  45. 45.
    Maugin G.: Sur une possible définition du principe d’indifférence matérielle en relativité. Compt. Rendus Acad. Sci. 275A, 319–322 (1972)MathSciNetGoogle Scholar
  46. 46.
    Müller I., Ruggeri T.: Rational Extended Thermodynamics. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  47. 47.
    Ottinger H.C.: Relativistic and non-relativistic description of fluids with anisotropic heat conduction. Phys. A 254, 433–450 (1998)CrossRefGoogle Scholar
  48. 48.
    Tsypkin A.G.: On complicated models of continuous media in the general theory of relativity. PMM USSR 51, 698–703 (1987)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Vallée C.: Relativistic thermodynamics of continua. Int. J. Eng. Sci. 19, 589–601 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Williams D.N.: The elastic energy momentum tensor in special relativity. Ann. Phys. 196, 345–380 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Arminjon M., Reifler F.: General reference frames and their associated space manifolds. Int. J. Geom. Methods Mod. Phys. 8, 155–165 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Brunet M.: Analyse non-linaire des matriaux et des structures. INSA de Lyon, Lyon (2009)Google Scholar
  53. 53.
    Sidoroff F., Dogui A.: Some issues about anisotropic elastic-plastic models at finite strain. Int. J. Solids Struct. 38, 9569–9578 (2001)CrossRefzbMATHGoogle Scholar
  54. 54.
    Labergère C., Saanouni K., Lestriez P.: Numerical design of extrusion process using finite thermo-elastoviscoplasticity with damage. Prediction of chevron shaped cracks. Key Eng. Mater. 424, 265–272 (2010)CrossRefGoogle Scholar
  55. 55.
    Müller I.: On the frame dependence of stress and heat flux. Arch. Ration. Mech. Anal. 45, 241–250 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Landau L.D., Lifshitz E.M.: Fluid Mechanics, second edition. Elsevier, Amsterdam (1987)Google Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  • B. Panicaud
    • 1
  • E. Rouhaud
    • 1
    • 2
  • G. Altmeyer
    • 1
  • M. Wang
    • 1
  • R. Kerner
    • 2
  • A. Roos
    • 1
  • O. Ameline
    • 1
  1. 1.Institut Charles Delaunay (ICD) - CNRS UMR 6281 Laboratoire des Systèmes Mécaniques et d’Ingénierie Simultanée (LASMIS)Université de Technologie de Troyes (UTT)Troyes cedexFrance
  2. 2.Laboratoire de Physique Théorique de la Matière Condensée (LPTMC)Université Pierre et Marie Curie (UPMC)ParisFrance

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