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Journal of Elasticity

, Volume 135, Issue 1–2, pp 261–293 | Cite as

A Causality Setting for Elasticity Theory

  • Roger FosdickEmail author
Article
  • 94 Downloads

Abstract

We give a causality approach to nonlinear elasticity theory within a pure mechanics setting. Based on the notion of a fixed frame in Open image in new window and Euclidean invariance considerations, we show how a broadly general and novel evolutionary hypothesis concerning ‘cause’ and ‘effect’ must reduce to the classical statement of the balance of energy, and we obtain all of the classical balance laws of continuum mechanics. The concept of mass and its balance is derived within this theory. The mass density naturally emerges from the theory without preconception as an inertial scalar field for the body which is associated with the speed of its material particles and a measure of their kinetic behavior. Aside from the causality hypothesis and its invariance, the fundamental notions of body, motion, force and internal power are primitive.

Keywords

Causality Continuum mechanics Elasticity Invariance Objectivity 

Mathematics Subject Classification (2010)

74A99 

Notes

References

  1. 1.
    Bartle, R.G.: The Elements of Integration Theory. Wiley, New York (1966) zbMATHGoogle Scholar
  2. 2.
    Bunge, M.: Causality, The Place of the Causal Principle in Modern Science. Harvard University Press, Cambridge (1959) Google Scholar
  3. 3.
    Favata, A., Podio-Guidugli, P., Tomassetti, G.: Energy splitting theorems for materials with memory. J. Elast. 101, 59–67 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fosdick, R.: A causality approach to particle dynamics for systems. Arch. Ration. Mech. Anal. 207(1), 247–302 (2013).  https://doi.org/10.1007/s00205-012-0567-7 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fosdick, R., Serrin, J.: The splitting of intrinsic energy and the origin of mass density in continuum mechanics. Contin. Mech. Thermodyn. 26, 287–302 (2014).  https://doi.org/10.1007/s00161-013-0301-1 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gibbs, J.W.: On the equilibrium of heterogeneous substances. In: Trans. Conn. Acad. 1875–1878. The Scientific Papers, Vol. I, pp. 55–353. Yale University Press, New Haven (1907) Google Scholar
  7. 7.
    Green, A.E., Rivlin, R.S.: On Cauchy’s equations of motion. Z. Angew. Math. Phys. 15, 290–292 (1964) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gurtin, M.E.: Thermodynamics and the possibility of spatial interaction in rigid heat conductors. Arch. Ration. Mech. Anal. 18, 335–342 (1965) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, San Diego (1981) zbMATHGoogle Scholar
  11. 11.
    Gurtin, M.E., Williams, W.O.: An axiomatic foundation for continuum thermodynamics. Arch. Ration. Mech. Anal. 26, 83–117 (1967) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jammer, M.: Concepts of Mass in Classical and Modern Physics. Harvard University Press, Cambridge (1961) Google Scholar
  13. 13.
    Noll, W.: On the continuity of the solid and fluid states. J. Ration. Mech. Anal. 4, 3–81 (1955) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Noll, W.: La mécanique classique, basée sur un axiome d’objectivité. La méthode axiomatique dans les mécaniques classiques et nouvelles, pp. 47–63, Paris (1963) Google Scholar
  15. 15.
    Podio-Guidugli, P.: Inertia and invariance. Ann. Mat. Pura Appl. (4) CLXXII, 103–124 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Serrin, J.: The equations of continuum mechanics as a consequence of group invariance. In: Ferrarese, G., Editrice, P. (eds.) Advances in Modern Continuum Mechanics, Bologna, pp. 217–225 (1992) Google Scholar
  17. 17.
    Šilhavý, M.: Mass, internal energy, and Cauchy’s equations in frame-indifferent thermodynamics. Arch. Ration. Mech. Anal. 107, 1–22 (1989) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Aerospace Engineering and MechanicsUniversity of MinnesotaMinneapolisUSA

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