Skip to main content
SpringerLink
Log in

Towards Micropolar Continuum Theory Describing Some Problems of Thermo- and Electrodynamics

  • Chapter
  • First Online:
Contributions to Advanced Dynamics and Continuum Mechanics

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 114))

Abstract

A new model of a micropolar continuum is considered. This mathematical model has been created in order to simulate thermo- and electromagnetic processes. Our method is based on the following reasoning. In the framework of our model, we introduce mechanical analogies of physical quantities such as temperature, entropy, the electric field vector, the magnetic induction vector etc. Then, we show that under certain assumptions the equations of our model coincide with well-known equations, in particular, with Maxwell’s equations. Next, we explore the properties of our mathematical model in its general form. Following the terminology of 19th-century scientists, we call our model the ether model, though in its mathematical content, it differs from the 19th-century ether models very significantly.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
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. Dixon, R.C., Eringen, A.C.: A dynamical theory of polar elastic dielectrics–I. Int. J. Eng. Sci. 2, 359–377 (1964)

    Google Scholar 

  2. Dixon, R.C., Eringen, A.C.: A dynamical theory of polar elastic dielectrics–II. Int. J. Eng. Sci. 3, 379–398 (1965)

    Article  MathSciNet  Google Scholar 

  3. Treugolov, I.G.: Moment theory of electromagnetic effects in anisotropic solids. Appl. Math. Mech. 53(6), 992–997 (1989)

    Google Scholar 

  4. Grekova, E., Zhilin, P.: Basic equations of Kelvin’s medium and analogy with ferromagnets. J. Elast. 64, 29–70 (2001)

    Article  MathSciNet  Google Scholar 

  5. Grekova, E.F.: Ferromagnets and Kelvin’s medium: basic equations and wave processes. J. Comput. Acoust. 9(2), 427–446 (2001)

    Article  Google Scholar 

  6. Zhilin, P.A.: Advanced Problems in Mechanics, vol. 1. Institute for Problems in Mechanical Engineering, Saint Petersburg (2006). (In Russian)

    Google Scholar 

  7. Zhilin, P.A.: Advanced Problems in Mechanics, vol. 2. Institute for Problems in Mechanical Engineering, Saint Petersburg (2006)

    Google Scholar 

  8. Zhilin, P.A.: Rational Continuum Mechanics. Polytechnic University Publishing House, Saint Petersburg (2012). (in Russian)

    Google Scholar 

  9. Ivanova, E.A., Kolpakov, Y.E.: Piezoeffect in polar materials using moment theory. J. Appl. Mech. Tech. Phys. 54(6), 989–1002 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. Ivanova, E.A., Kolpakov, Y.E.: A description of piezoelectric effect in non-polar materials taking into account the quadrupole moments. Z. Angew. Math. Mech. 96(9), 1033–1048 (2016)

    Article  MathSciNet  Google Scholar 

  11. Ivanova, E.A.: On micropolar continuum approach to some problems of thermo- and electrodynamics. Acta Mech. 230, 1685–1715 (2019)

    Article  MathSciNet  Google Scholar 

  12. Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Micromorphic theory of superconductivity. Phys. Rev. 106(1), 162–164 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  13. Eringen, A.C.: Continuum theory of micromorphic electromagnetic thermoelastic solids. Int. J. Eng. Sci. 41, 653–665 (2003)

    Article  MathSciNet  Google Scholar 

  14. Galeş, C., Ghiba, I.D., Ignătescu, I.: Asymptotic partition of energy in micromorphic thermopiezoelectricity. J. Therm. Stress. 34, 1241–1249 (2011)

    Article  Google Scholar 

  15. Tiersten, H.F.: Coupled magnetomechanical equations for magnetically saturated insulators. J. Math. Phys. 5(9), 1298–1318 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  16. Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. Elsevier Science Publishers, Oxford (1988)

    MATH  Google Scholar 

  17. Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua. Springer, New York (1990)

    Book  Google Scholar 

  18. Fomethe, A., Maugin, G.A.: Material forces in thermoelastic ferromagnets. Contin. Mech. Thermodyn. 8, 275–292 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  19. 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)

    Article  ADS  Google Scholar 

  20. Ivanova, E.A.: A new model of a micropolar continuum and some electromagnetic analogies. Acta Mech. 226, 697–721 (2015)

    Article  MathSciNet  Google Scholar 

  21. Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mech. 215, 261–286 (2010)

    Article  Google Scholar 

  22. 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, pp. 151–174. Springer, Berlin (2011)

    Chapter  Google Scholar 

  23. Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component Cosserat continuum. Tech. Mech. 32, 273–286 (2012)

    Google Scholar 

  24. 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)

    Article  MathSciNet  Google Scholar 

  25. Ivanova, E.A.: Description of nonlinear thermal effects by means of a two-component Cosserat continuum. Acta Mech. 228, 2299–2346 (2017)

    Article  MathSciNet  Google Scholar 

  26. Ivanova, E.A.: Thermal effects by means of two-component Cosserat continuum. In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of Continuum Mechanics, pp. 1–12. Springer, Berlin (2018)

    Google Scholar 

  27. Whittaker, E.: A History of the Theories of Aether and Electricity. The Classical Theories. Thomas Nelson and Sons Ltd., London etc (1910)

    MATH  Google Scholar 

  28. Cataneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus. 247, 431–433 (1958)

    Google Scholar 

  29. Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)

    Article  ADS  Google Scholar 

  30. Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (2001)

    Book  Google Scholar 

  31. Babenkov, M.B., Ivanova, E.A.: Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type. Contin. Mech. Thermodyn. 26(4), 483–502 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  32. Babenkov, M.B., Vitokhin, E.Y.: Thermoelastic waves in a medium with heat-flux relaxation. In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of Continuum Mechanics. Springer, Berlin (2018)

    Google Scholar 

  33. Sommerfeld, A.: Mechanics of Deformable Bodies. Lectures on Theoretical Physics, vol. II. Academic Press INC., New York (1950)

    Google Scholar 

  34. Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann, Paris (1909)

    MATH  Google Scholar 

  35. Whatmore, R.W.: Piezoelectric and pyroelectric materials and their applications. In: Miller, L.S., Mullin, J.B. (eds.) Electronic Materials. Springer, Boston, MA (1991)

    Google Scholar 

  36. Novik, V.K., Gavrilova, N.D.: Low-temperature pyroelectricity. Phys. Solid State. 42, 991–1008 (2000)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Mechanics, Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya 29, 195251, Saint Petersburg, Russia

    Elena A. Ivanova

  2. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, Bolshoy pr. V.O. 61, 199178, Saint Petersburg, Russia

    Elena A. Ivanova

Authors
  1. Elena A. Ivanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elena A. Ivanova .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Holm Altenbach

  2. Institut für Technische Mechanik, Johannes Kepler Universität Linz, Linz, Austria

    Hans Irschik

  3. Institute of Continuous Media Mechanics, Russian Academy of Sciences, Perm, Russia

    Valery P. Matveenko

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Ivanova, E.A. (2019). Towards Micropolar Continuum Theory Describing Some Problems of Thermo- and Electrodynamics. In: Altenbach, H., Irschik, H., Matveenko, V. (eds) Contributions to Advanced Dynamics and Continuum Mechanics. Advanced Structured Materials, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-030-21251-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation