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Acta Mechanica

, Volume 226, Issue 3, pp 697–721 | Cite as

A new model of a micropolar continuum and some electromagnetic analogies

  • E. A. IvanovaEmail author
Article

Abstract

A new model of micropolar continuum composed of two-spin particles is considered. In fact, this continuum represents a two-component continuum. The first component possesses the translational and rotational degrees of freedom, whereas the second component has only the rotational degrees of freedom. The main characteristic feature of the suggested model is that both the components are not infinitesimal rigid bodies. They are the body-points of a general type, which differ from infinitesimal rigid bodies by additional inertia parameters. A continuum, composed of such particles, has some additional properties compared with a conventional material. We suggest to use the continuum of two-spin particles as a mechanical model (or, in other words, a mechanical analogy) of the electromagnetic field in matter. This model does not pretend to be an explanation of the physical nature of electromagnetic phenomena. The interpretations of the electric charge, the electric field vector, the magnetic induction vector, and other physical quantities, which are given in accordance with the suggested model, are no more than the mechanical analogies. We show that the mathematical description of our model contains two special cases. Under one simplifying assumption, the suggested equations are reduced to the equations similar to Maxwell’s equations. Under another simplifying assumption, an analogue of the Lorentz force is obtained. We believe that in some cases the exact equations describing our mechanical model can be of interest for applications in electrodynamics.

Keywords

Lorentz Force Rotational Degree Inertia Tensor Material Tensor Rotation Tensor 
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.
    Whittaker, E.: A History of the Theories of Aether and Electricity. The Classical Theories. Thomas Nelson and Sons Ltd, London etc. (1910)Google Scholar
  2. 2.
    Treugolov I.G.: Moment theory of electromagnetic effects in anisotropic solids. Appl. Math. Mech. 53, 992–997 (1989)Google Scholar
  3. 3.
    Zhilin, P.A.: Advanced Problems in Mechanics, vol. 1, 2. Institute for Problems in Mechanical Engineering, St. Petersburg (2006)Google Scholar
  4. 4.
    Bardeen J., Cooper L.N., Schrieffer J.R.: Micromorphic theory of superconductivity. Phys. Rev. 106, 162–164 (1957)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Eringen A.C.: Continuum theory of micromorphic electromagnetic thermoelastic solids. Int. J. Eng. Sci. 41, 653–665 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Galeş C., Ghiba I.D., Ignătescu I.: Asymptotic partition of energy in micromorphic thermopiezoelectricity. J. Therm. Stress. 34, 1241–1249 (2011)CrossRefGoogle Scholar
  7. 7.
    Tiersten H.F.: Coupled magnetomechanical equations for magnetically saturated insulators. J. Math. Phys. 5, 1298–1318 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Maugin G.A.: Continuum Mechanics of Electromagnetic Solids. Elsevier Science Publishers, Oxford (1988)zbMATHGoogle Scholar
  9. 9.
    Eringen A.C., Maugin G.A.: Electrodynamics of Continua. Springer, New York (1990)CrossRefGoogle Scholar
  10. 10.
    Fomethe, A., Maugin, G.A.: Material forces in thermoelastic ferromagnets. Continuum Mech. Thermodyn. Issue 8, pp. 275–292 (1996)Google Scholar
  11. 11.
    Zhilin, P.A.: Rational Continuum Mechanics. Polytechnic University Publishing House, St. Petersburg (2012) (in Russian)Google Scholar
  12. 12.
    Shliomis M.I., Stepanov V.I.: Rotational viscosity of magnetic fluids: contribution of the Brownian and Neel relaxational processes. J. Magn. Magn. Mater. 122, 196–199 (1993)CrossRefGoogle Scholar
  13. 13.
    Ivanova E.A.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mech. 215, 261–286 (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ivanova, E.A.: On one model of generalized continuum and its thermodynamical interpretation. In: Altenbach, H., Maugin, G.A., Erofeev, V. (eds.) Mechanics of Generalized Continua, Springer, Berlin, pp. 151–174 (2011)Google Scholar
  15. 15.
    Ivanova E.A.: Derivation of theory of thermoviscoelasticity by means of two-component Cosserat continuum. Tech. Mech. 32, 273–286 (2012)MathSciNetGoogle Scholar
  16. 16.
    Ivanova E.A.: Description of mechanism of thermal conduction and internal damping by means of two component Cosserat continuum. Acta Mech. 225, 757–795 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Rosenberger, F.: Die Geschichte der Physik. Dritter Teil. Geschichte der Physik in den letzten hundert Jahren. Fr. Vieweg und Sohn, Braunschweig (1887)Google Scholar
  18. 18.
    Gliozzi M.: Storia della Fisica Storia delle Scienze, vol. 2. UTET, Torino (1965)Google Scholar
  19. 19.
    Cosserat E., End F.: Theorie des Corps Deformables. Hermann, Paris (1909)Google Scholar
  20. 20.
    Truesdell C.: The Elements of Continuum Mechanics. Springer, New York (1965)Google Scholar
  21. 21.
    Ericksen J.L.: Introduction to the Thermodynamics of Solids. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  22. 22.
    Lippmann H.: Eine Cosserat Theorie des plastischen Fliessens. Acta Mech. 8, 255–284 (1969)CrossRefzbMATHGoogle Scholar
  23. 23.
    Vardoulakis I., Aifantis E.C.: A gradient flow theory of plasticity for granular materials. Acta Mech. 87, 197–217 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Forest S., Sievert R.: Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160, 71–111 (2003)CrossRefzbMATHGoogle Scholar
  25. 25.
    Erofeev, V.I.: The Cosserat brothers and generalized continuum mechanics. Comput. Continuum Mech. 2, 5–10 (2009) (In Russian)Google Scholar
  26. 26.
    Truesdell C.: A First Course in Rational Continuum Mechanics. The John Hopkins University, Maryland (1972)Google Scholar
  27. 27.
    Zhilin, P.A.: Theoretical Mechanics. Fundamental Laws of Mechanics. Polytechnic University Publishing House, St. Petersburg (2003) (in Russian)Google Scholar
  28. 28.
    Eringen A.C.: Mechanics of Continua. Huntington, New York (1980)Google Scholar
  29. 29.
    Nowacki W.: Theory of Asymmetric Elasticity. Pergamon-Press, Oxford (1986)zbMATHGoogle Scholar
  30. 30.
    Rubin M.B.: Cosserat Theories: Shells, Rods and Points. Kluwer, Dordrecht (2000)CrossRefGoogle Scholar
  31. 31.
    Miller W. Jr: Symmetry Groups and Their Applications. Academic Press, New York (1972)zbMATHGoogle Scholar
  32. 32.
    Nye J.F.: Physical Properties of Crystals. Their Representation by Tensors and Matrices. Clarendon Press, Oxford (1979)Google Scholar
  33. 33.
    O’Keeffe, M., Hyde, B.G.: Crystal Structures; I. Patterns and Symmetry. Mineralogical Society of America, Monograph Series. Washington, DC (1996)Google Scholar

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Department of Theoretical MechanicsSaint-Petersburg State Polytechnical UniversitySaint-PetersburgRussia
  2. 2.Institute for Problems in Mechanical Engineering of Russian Academy of SciencesSaint-PetersburgRussia

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