Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the volume average of energy and net power

  • 101 Accesses

  • 2 Citations

Abstract

This paper presents an analysis of volume average relations in connection with internal energy, net power and the balance of energy in continuum mechanics. The role of the so-called Hill and Hill–Mandel conditions in connection with the averaged energy balance equation has been elucidated. The significance of volume averages of stress moments for the calculation of the average net power is demonstrated.

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

References

  1. 1

    Hill R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 12, 213–222 (1965)

  2. 2

    Mandel, J.: Plasticité classique et viscoplasticité. Springer-Verlag, Vienna (CISM Lecture Notes, Udine, Italy), (1971)

  3. 3

    Stoltz, C.: On micro-macro transition in non linear mechanics. In: Inan, E., Markov, K.Z. (eds.) Continuum Models and Discrete Systems (CDMS9), Proceedings of the 9th International Symposium in Istanbul, pp. 753–766. World Scientific Publishing Co, Singapore (1998)

  4. 4

    Stoltz C.: General relationships between micro and macro scales for the non linear behaviour of heterogeneous media. In: Zarka, J., Gittius, J. (eds) Modelling Small Deformations of Polycrystals, pp. 89–115. Elsevier, New York (1983)

  5. 5

    Hill R.: Elastic properties of reinforced solids; some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)

  6. 6

    Hill R.: On constitutive macro-variables for heterogeneous solids in finite strain. Proc. R. Soc. Lond. A. 326, 131–147 (1972)

  7. 7

    Havner K.S.: A discrete model for the prediction of subsequent yield surfaces in polycrystalline plasticity. Int J. Solids Struct. 7, 719–730 (1971)

  8. 8

    Suquet P.: Elements of homogenization for inelastic solid mechanics. In: Sanchez-Palencia, E., Zaoui, A. (eds) Homogenization Techniques for Composite Mechanics., pp. 193–278. Springer, Berlin (1987)

  9. 9

    Signorini A.: Sopra alcune questioni di statica dei sistemi continui. Ann. Scuola Norm. Pisa 2(2), 231–257 (1933)

  10. 10

    Signorini A: Sulle proprietà di media comuni a tutti I sistemi continui. Rend. Accad. Ital. 2, 728–734 (1941)

  11. 11

    Signorini A.: Un’espressiva applicatione delle proprietà di media dello stress conume a tutti sistema I sistemi continui Studies in Math And Mech Presented to R v Mises, pp. 274–277. Academic Press, New York (1954)

  12. 12

    Nečas, J.: Les Méthodes Directes en Théorie des Equations Elliptiques. Masson Cie, Editeurs, Paris, Academia Editeurs, Prague, (1967)

  13. 13

    Truesdell C., Toupin R.: The classical field theories, encyclopedia of physics, vol. III/1. Springer, Berlin (1960)

  14. 14

    Hill, R.: On macroscopic effects of heterogeneity in elastoplastic media at finite strain. Math. Proc. Camb. Phil. Soc. 95, (1984).

  15. 15

    Lidström, P.: On Volume Average Relations in Continuum Mechanics. Part I. Kinematics, mass, momentum and moment of momentum. Average consistence and Hill’s condition. Research report ISRN LUTFD/TFME-06/1001-SE(1-55). Division of Mechanics, Lund University, (2006)

  16. 16

    Kouznetsova V.G., Geers M.G.D., Breckelmans W.A.M.: Multi-scale constitutive modelling of heterogeneous materials with gradient-enhanced computational homogenization scheme. Int. J. Numer. Methods Eng. 54, 1235–1260 (2002)

  17. 17

    Kouznetsova, V.G.: Computational Homogenization for the Multi-scale Analysis of Multi-phase Materials Proefscrift. Technische Universiteit Eindhoven (2002)

  18. 18

    Bowen R.M., Wang C.-C.: Introduction to Vectors and Tensors, vol 1. Plenum Press, New York (1976)

Download references

Author information

Correspondence to P. Lidström.

Additional information

Communicated by Stefan Seelecke.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lidström, P. On the volume average of energy and net power. Continuum Mech. Thermodyn. 23, 275–304 (2011). https://doi.org/10.1007/s00161-010-0175-4

Download citation

Keywords

  • Continuum mechanics
  • Volume average relations
  • Average consistency