Skip to main content

Hypocontinua

  • Chapter
  • First Online:
Continuous Media with Microstructure 2

Abstract

Earlier proposals for a theory of continuous bodies for which the tacit axiom of enduring permanence of material elements fails to apply (continua called appropriately ephemeral) are briefly reviewed and, in some detail, corrected. The consequent derivation of a simpler class echoing that of hypoelastic bodies introduced by Truesdell is preened and put on surer footing for possible progress.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Biró, T.S.: Is There a Temperature? Conceptual Challenges at High Energy, Acceleration and Complexity, Fundamental Theories of Physics 171. Springer, New York (2011)

    Google Scholar 

  2. Biscari, P., Cercignani, C.: Stress and heat flux in non-inertial reference frames. Contin. Mech. Thermodyn. 9, 1–11 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brocato, M., Capriz, G.: Clockwork, ephemeral and hybrid continua. Phys. Mesomech. 14, 124–144 (2011)

    Google Scholar 

  4. Capriz, G.: A quest for an ‘extended’ continuum mechanics. Note Matemat. 27, 27–41 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Capriz, G.: On ephemeral continua. Phys. Mesomech. 11, 285–298 (2008)

    Article  Google Scholar 

  6. Capriz, G.: Hypocontinua. In: Albers, B. (ed.) Continuous Media with Microstructure, pp. 61–70. Springer, Berlin (2010)

    Chapter  Google Scholar 

  7. Capriz, G., Fried, E.: The ephemeral nature of Navier-Stokes-\(\alpha \beta \) continua. Rendiconti Lincei 22, 73–87 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Capriz, G., Mariano, P.M.: Objective fluxes in a multi-scale continuum description of sparse medium dynamics. Physica A 415, 354–365 (2014)

    Article  MathSciNet  Google Scholar 

  9. Celik, I.B.: Introductory Turbulence Modeling. West Virginia University (1999)

    Google Scholar 

  10. Cohen, H., Muncaster, R.G.: The Theory of Pseudo-Rigid Bodies. Springer, New York (1988)

    Book  MATH  Google Scholar 

  11. Dunwoody, J.: Comment: The bogus axioms of continuum mechanics [Bull. Inst. Math. Appl. 17, 98–102 (1981)] by Woods, L. C, with a reply by Woods, L. C. Bull. Inst. Math. Appl. 17, 219–220 (1981)

    Google Scholar 

  12. Edelen, D.G.B., McLennan, J.A.: Material indifference: a principle or a convenience. Int. J. Eng. Sci. 11, 813–817 (1973)

    Article  MATH  Google Scholar 

  13. Eringen, A.C.: Micromorphic theory of turbulence. ZAMP 61, 119–132 (2009)

    Article  MathSciNet  Google Scholar 

  14. Green, A.E.: A note on axioms of continuum mechanics: The bogus axioms of continuum mechanics [Bull. Inst. Math. Appl. 17, 98–102 (1981)] by Woods, L. C. Bull. Inst. Math. Appl. 18, 7–9 (1982)

    Google Scholar 

  15. Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mariano, P.M., Galano, L.: Fundamentals of the Mechanics of Solids. Birkhauser, Boston (2015)

    Book  Google Scholar 

  17. Mueller, I.: Extended Thermodynamics. A multifield theory for excellence. In: Capriz, G., Mariano, P.M. (eds.) Advances in Multifield Theories for Continuua with Substructure, pp. 115–126. Birkhauser, Basel (2004)

    Chapter  Google Scholar 

  18. Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistichen Mechanik. J. Ration. Mech. Anal. 4, 627–646 (1955)

    MathSciNet  MATH  Google Scholar 

  19. Noll, W., Virga, E.G.: Fit regions and functions with bounded variation. Arch. Ration. Mech. Anal. 102, 1–21 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oldroyd, J.G.: Non-Newtonian flow of liquids and solids. Rheology, Theory and Applications 1, Ch. 16. Academic Press, New York (1956)

    Google Scholar 

  21. Truesdell, C.A.: Hypoelasticity. J. Ration. Mech. Anal. 4, 83–133 (1955)

    MathSciNet  MATH  Google Scholar 

  22. Truesdell, C.A., Noll, W.: The Non-Linear Field Theories of Mechanics. In: Flügge, S. (ed.) Encyclopedia of Physics III/3, pp. 1–602. Springer, Berlin (1965)

    Google Scholar 

  23. Truesdell, C.A., Toupin, R.A.: The Classical Field Theories. In: Flügge, S. (ed.) Encyclopedia of Physics III/1, pp. 226–858. Springer, Berlin (1960)

    Google Scholar 

  24. Woods, L.C.: The bogus axioms of continuum mechanics. Bull. Inst. Math. Appl. 17, 98–102 (1981)

    MathSciNet  MATH  Google Scholar 

  25. Woods, L.C.: More on the bogus axioms of continuum mechanics. Bull. Inst. Math. Appl. 18, 64–66 (1982)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This research is part of the activities of the Project PON 04a2-F: “BE&SAVE AQUASYSTEM SIGLOD-Programma operativo nazionale di Ricerca e competitività 2007–2013”. The support of GNFM-INDAM is also acknowledged.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Pasquale Giovine .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Capriz, G., Giovine, P. (2016). Hypocontinua. In: Albers, B., Kuczma, M. (eds) Continuous Media with Microstructure 2. Springer, Cham. https://doi.org/10.1007/978-3-319-28241-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-28241-1_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28239-8

  • Online ISBN: 978-3-319-28241-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics