Continuum Mechanics and Thermodynamics

, Volume 30, Issue 1, pp 175–194 | Cite as

A geometric rationale for invariance, covariance and constitutive relations

  • Giovanni RomanoEmail author
  • Raffaele Barretta
  • Marina Diaco
Original Article


There are, in each branch of science, statements which, expressed in ambiguous or even incorrect but seemingly friendly manner, were repeated for a long time and eventually became diffusely accepted. Objectivity of physical fields and of their time rates and frame indifference of constitutive relations are among such notions. A geometric reflection on the description of frame changes as spacetime automorphisms, on induced push–pull transformations and on proper physico–mathematical definitions of material, spatial and spacetime tensor fields and of their time-derivatives along the motion, is here carried out with the aim of pointing out essential notions and of unveiling false claims. Theoretical and computational aspects of nonlinear continuum mechanics, and especially those pertaining to constitutive relations, involving material fields and their time rates, gain decisive conceptual and operative improvement from a proper geometric treatment. Outcomes of the geometric analysis are frame covariance of spacetime velocity, material stretching and material spin. A univocal and frame-covariant tool for evaluation of time rates of material fields is provided by the Lie derivative along the motion. The postulate of frame covariance of material fields is assessed to be a natural physical requirement which cannot interfere with the formulation of constitutive laws, with claims of the contrary stemming from an improper imposition of equality in place of equivalence.


Constitutive relations Spatial and material fields in spacetime Stretching Stress rate Lie derivatives Spacetime transformations Invariance Covariance Form invariance Material frame indifference 


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  1. 1.
    Zaremba, S.: Le principe des mouvements relatifs et les équations de la mécanique physique. Bull. Int. Acad. Sci. Cracovie 614, 621 (1903)zbMATHGoogle Scholar
  2. 2.
    Jaumann, G.: Elektromagnetische Vorgänge in bewegten Medien. Sitzungsber. Akad. Wiss. Wien (IIa) 15, 337 (1906)zbMATHGoogle Scholar
  3. 3.
    Jaumann, G.: Geschlossenes System physikalischer und chemischer Differentialgesetze. Sitzungsber. Akad. Wiss. Wien (IIa) 120, 385–530 (1911)Google Scholar
  4. 4.
    Oldroyd, J.G.: On the formulation of rheological equations of state. Proc. R. Soc. Lond. A 200, 523–541 (1950)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sedov, L.I.: Different definitions of the rate of change of a tensor. J. Appl. Math. Mech. 24(3), 579–586 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Prager W.: Introduction to Mechanics of Continua. Ginn & Company, Boston, MA. (German Einführung in die Kontinuumsmechanik, Trans.). Birkhauser, Basel (1961)Google Scholar
  7. 7.
    Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics. Handbuch der Physik, Band III3. Springer, Berlin (1965), 2nd edn. (1992), 3rd edn. (2004)Google Scholar
  8. 8.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Prentice-Hall, Redwood City, CA (1983)zbMATHGoogle Scholar
  9. 9.
    Ryskin, G.: Misconception which led to the “material frame-indifference” controversy. Phys. Rev. A 32, 1239 (1985)ADSCrossRefGoogle Scholar
  10. 10.
    Liu, I.-S.: On Euclidean objectivity and the principle of material frame-indifference. Contin. Mech. Thermodyn. 16, 177–183 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, I.-S.: Further remarks on Euclidean objectivity and the principle of material frame-indifference. Contin. Mech. Thermodyn. 17, 125–133 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Murdoch, A.I.: On criticism of the nature of objectivity in classical continuum physics. Contin. Mech. Thermodyn. 17, 135–148 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Frewer, M.: More clarity on the concept of material frame-indifference in classical continuum mechanics. Acta Mech. 202, 213–246 (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, I.-S., Sampaio, R.: Remarks on material frame-indifference controversy. Acta Mech. 225(2), 331–348 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Frewer, M.: Covariance and objectivity in mechanics and turbulence. A revisiting of definitions and applications. arXiv:1611.07002 (2016)
  17. 17.
    Liu, I.-S., Lee, J.D.: On material objectivity of intermolecular force in molecular dynamics. Acta Mech. 228(2), 731–738 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Romano, G., Barretta, R.: Geometric constitutive theory and frame invariance. Int. J. Non-Linear Mech. 51, 75–86 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    Romano, G.: Geometry & Continuum Mechanics. Short Course in Innsbruck, 24–25 Nov 2014. ISBN-10: 1503172198.
  20. 20.
    Lee, E.H.: Elastic-plastic deformations at finite strains. ASME J. Appl. Mech. 36(1), 1–6 (1969)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Simò, J.C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: continuum formulation, part I. Comput. Methods Appl. Mech. Eng. 66, 199–219 (1988)ADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Lubarda, V.A.: Elastoplasticity Theory. CRC Press, Boca Raton (2002)zbMATHGoogle Scholar
  23. 23.
    van Dantzig, D.: Zur allgemeinen projektiven Differentialgeometrie I, II. In: Proc. Kon. Akad. Amsterdam 35, 524–534, 535–542 (1932).
  24. 24.
    Romano, G., Barretta, R., Diaco, M.: The geometry of nonlinear elasticity. Acta Mech. 225(11), 3199–3235 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Svendsen, B., Bertram, A.: On frame-indifference and form-invariance in constitutive theory. Acta Mech. 132, 195–207 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bertram, A., Svendsen, B.: On material objectivity and reduced constitutive equations. Arch. Mech. 53(6), 653–675 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Romano, G.: On Time and Length in Special Relativity. Rend. Acc. Naz. Sc. Let. Arti, in Napoli, May (2014).
  28. 28.
    Lie, M.S., Engel, F.: Theorie der Transformationsgruppen. In: Teubner, B.G., Leipzig, vol. 1–3, 2nd edn (1888-1890-1893) . Chelsea, New YorkGoogle Scholar
  29. 29.
    Dieudonné, J.: Treatise on Analysis, vol I–IV. Academic Press, New York (1969–1974)Google Scholar
  30. 30.
    Rodrigues, W.A., de Souza, Q.A.G., Bozhkov, Y.: The mathematical structure of Newtonian spacetime: classical dynamics and gravitation. Found. Phys. 25(6), 871–924 (1995)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Panicaud, B., Rouhaud, E., Altmeyer, G., Wang, M., Kerner, R., Roos, A., Ameline, O.: Consistent hypo-elastic behavior using the four-dimensional formalism of differential geometry. Acta Mech. 227(3), 651–675 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wang, M., Panicaud, B., Rouhaud, E., Kerner, R., Roos, A.: Incremental constitutive models for elastoplastic materials undergoing finite deformations by using a four-dimensional formalism. Int. J. Eng. Sci. 106, 199–219 (2016)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Samelson, H.: Differential forms, the early days; or the stories of Deahna’s Theorem and of Volterra’s theorem. Am. Math. Mon. Math. Assoc. Am. 108(6), 522–530 (2001).
  34. 34.
    Petersen, P.: Riemannian Geometry. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  35. 35.
    Romano, G., Barretta, R., Diaco, M.: Geometric continuum mechanics. Meccanica 49(1), 111–133 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Thompson, R.L.: A note on some insights from decoupling the time derivative of an objective tensor. Int. J. Eng. Sci. 82, 22–27 (2014)CrossRefGoogle Scholar
  37. 37.
    Bampi, F., Morro, A.: Objectivity and objective time derivatives in continuum physics. Found. Phys. 10(11–12), 905–920 (1980)Google Scholar
  38. 38.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, San Diego (1981)zbMATHGoogle Scholar
  39. 39.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  40. 40.
    Truesdell, C.: A First Course in Rational Continuum Mechanics. Academic Press, New York (1977) (2nd edn., 1991)Google Scholar
  41. 41.
    Romano, G., Barretta, R.: Nonlocal elasticity in nanobeams: the stress-driven integral model. Int. J. Eng. Sci. 115, 14–27 (2017)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Rivlin, R.S.: Material symmetry revisited. GAMM-Mitt. 1(2), 109–126 (2002)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Bertram, A., Svendsen, B.: Reply to Rivlin’s material symmetry revisited or much ado about nothing. GAMM-Mitt. 27(1), 88–93 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
  45. 45.
    Truesdell C., Toupin R.: The classical field theories. In: Handbuck der Physik, Band III/1, pp. 226-793 Springer, Berlin (1960)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Giovanni Romano
    • 1
    Email author
  • Raffaele Barretta
    • 1
  • Marina Diaco
    • 1
  1. 1.Department of Structures for Engineering and ArchitectureUniversity of Naples Federico IINaplesItaly

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