Acta Mechanica

, Volume 230, Issue 5, pp 1685–1715 | Cite as

On a micropolar continuum approach to some problems of thermo- and electrodynamics

  • E. A. IvanovaEmail author
Original Paper


A new nonlinear model of a micropolar continuum is suggested. The peculiarity of the model is that the constitutive equations depend only on the strain measures associated with rotational degrees of freedom, and at the same time, the stress tensor turns out to be different from zero. This mathematical model has been created with the view of its use for modeling various processes, including processes at the micro-scale level. Following the terminology of nineteenth-century scientists, we call our model the ether model, though in its mathematical content, it differs from the nineteenth-century ether models very significantly. There may be different points of view concerning the physical meaning of our model. On the one hand, one can suppose the same meaning that nineteenth-century scientists implied in their ether models. On the other hand, one can imagine a continuum consisting of quasi- or virtual particles. The choice of one of the aforementioned physical interpretations is not important for constructing the mathematical model. Our method of modeling thermo- and electrodynamic processes is as follows. 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. We show that under certain simplifying assumptions the equations of our model coincide with well-known equations, in particular, with Maxwell’s equations. We explore the properties of our mathematical model in its most general form, investigate what processes can be described in the framework of our model, and suggest a possible interpretation of these processes.


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  1. 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. 2.
    Dixon, R.C., Eringen, A.C.: A dynamical theory of polar elastic dielectrics—II. Int. J. Eng. Sci. 3, 379–398 (1965)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Treugolov, I.G.: Moment theory of electromagnetic effects in anisotropic solids. Appl. Math. Mech. 53(6), 992–997 (1989)Google Scholar
  4. 4.
    Grekova, E., Zhilin, P.: Basic equations of Kelvin’s medium and analogy with ferromagnets. J. Elast. 64, 29–70 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grekova, E.F.: Ferromagnets and Kelvin’s medium: basic equations and wave processes. J. Comput. Acoust. 9(2), 427–446 (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Zhilin, P.A.: Advanced Problems in Mechanics, vol. 1. Institute for Problems in Mechanical Engineering, St. Petersburg (2006). (In Russian)Google Scholar
  7. 7.
    Zhilin, P.A.: Advanced Problems in Mechanics, vol. 2. Institute for Problems in Mechanical Engineering, St. Petersburg (2006)Google Scholar
  8. 8.
    Ivanova, E.A., Kolpakov, Y.E.: Piezoeffect in polar materials using moment theory. J. Appl. Mech. Tech. Phys. 54(6), 989–1002 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bardeen, J., Cooper, L.N., Schrieffer, J.R.: Micromorphic theory of superconductivity. Phys. Rev. 106(1), 162–164 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Eringen, A.C.: Continuum theory of micromorphic electromagnetic thermoelastic solids. Int. J. Eng. Sci. 41, 653–665 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Galeş, C., Ghiba, I.D., Ignătescu, I.: Asymptotic partition of energy in micromorphic thermopiezoelectricity. J. Therm. Stress. 34, 1241–1249 (2011)CrossRefGoogle Scholar
  13. 13.
    Tiersten, H.F.: Coupled magnetomechanical equations for magnetically saturated insulators. J. Math. Phys. 5(9), 1298–1318 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. Elsevier Science Publishers, Oxford (1988)zbMATHGoogle Scholar
  15. 15.
    Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua. Springer, New York (1990)CrossRefGoogle Scholar
  16. 16.
    Fomethe, A., Maugin, G.A.: Material forces in thermoelastic ferromagnets. Contin. Mech. Thermodyn. 8, 275–292 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Zhilin, P.A.: Rational Continuum Mechanics. Polytechnic University Publishing House, St. Petersburg (2012). (In Russian)Google Scholar
  19. 19.
    Ivanova, E.A.: A new model of a micropolar continuum and some electromagnetic analogies. Acta Mech. 226, 697–721 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mech. 215, 261–286 (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component Cosserat continuum. Tech. Mech. 32, 273–286 (2012)Google Scholar
  23. 23.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ivanova, E.A.: Description of nonlinear thermal effects by means of a two-component Cosserat continuum. Acta Mech. 228, 2299–2346 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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
  26. 26.
    Kiral, E., Eringen, A.C.: Constitutive Equations of Nonlinear Electromagnetic-Elastic Crystals. Springer, New York (1990)CrossRefGoogle Scholar
  27. 27.
    Whittaker, E.: A History of the Theories of Aether and Electricity. The Classical Theories. Thomas Nelson and Sons Ltd, London (1910)zbMATHGoogle Scholar
  28. 28.
    Cosserat, E., Cosserat, F.: Theorie des Corps Deformables. Hermann, Paris (1909)zbMATHGoogle Scholar
  29. 29.
    Mandelstam, L.I.: Lectures on Optics, Theory of Relativity and Quantum Mechanics. Nauka, Moscow (1972). (In Russian)Google Scholar
  30. 30.
    Hassanizadeh, M., Gray, W.: General conservation equations for multi-phase systems: 3. Constitutive theory for porous media flow. Adv. Water Resour. 3, 25 (1980)CrossRefGoogle Scholar
  31. 31.
    Altenbach, H., Naumenko, K., Zhilin, P.A.: A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Contin. Mech. Thermodyn. 15(6), 539–570 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Gadala, M.: Recent trends in ale formulation and its applications in solid mechanics. Comput. Methods Appl. Mech. Eng. 193, 4247–4275 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Dettmer, W., Peric, D.: A computational framework for free surface fluid flows accounting for surface tension. Comput. Methods Appl. Mech. Eng. 195, 3038–3071 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Filipovic, N., Mijailovic, A.S., Tsuda, A., Kojic, M.: An implicit algorithm within the arbitrary Lagrangian–Eulerian formulation for solving incompressible fluid flow with large boundary motions. Comput. Methods Appl. Mech. Eng. 195, 6347–6361 (2006)CrossRefzbMATHGoogle Scholar
  35. 35.
    Khoei, A., Anahid, M., Shahim, K.: An extended arbitrary Lagrangian–Eulerian finite element modeling (X-ALE-FEM) in powder forming processes. J. Mater. Process. Technol. 187–188, 397–401 (2007)CrossRefGoogle Scholar
  36. 36.
    Del Pin, F., Idelsohn, S., Onate, E.R.A.: The ALE/Lagrangian particle finite element method: a new approach to computation of free-surface flows and fluid object interactions. Comput. Fluids 36, 27–38 (2007)CrossRefzbMATHGoogle Scholar
  37. 37.
    Ivanova, E.A., Vilchevskaya, E.N.: Description of thermal and micro-structural processes in generalized continua: Zhilin’s method and its modifications. In: Altenbach, H., Forest, S., Krivtsov, A.M. (eds.) Generalized Continua as Models for Materials with Multi-scale Effects or Under Multi-field Actions, pp. 179–197. Springer, Berlin (2013)Google Scholar
  38. 38.
    Vuong, A.T., Yoshihara, L., Wall, W.: A general approach for modeling interacting flow through porous media under finite deformations. Comput. Methods Appl. Mech. Eng. 283, 1240–1259 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Brazgina, O.V., Ivanova, E.A., Vilchevskaya, E.N.: Saturated porous continua in the frame of hybrid description. Contin. Mech. Thermodyn. 28(5), 1553–1581 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ivanova, E.A., Vilchevskaya, E.N.: Micropolar continuum in spatial description. Contin. Mech. Thermodyn. 28(6), 1759–1780 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ivanova, E.A., Vilchevskaya, E.N., Müller, W.H.: Time derivatives in material and spatial description—what are the differences and why do they concern us? In: Naumenko, K., Aßmus, M. (eds.) Advanced Methods of Continuum Mechanics for Materials and Structures, pp. 3–28. Springer, Berlin (2016)Google Scholar
  42. 42.
    Ivanova, E.A., Vilchevskaya, E.N., Müller, W.H.: A study of objective time derivatives in material and spatial description. In: Altenbach, H., Goldstein, R., Murashkin, E. (eds.) Mechanics for Materials and Technologies. Advanced Structured Materials, vol. 46, pp. 195–229. Springer, Cham (2017)CrossRefGoogle Scholar
  43. 43.
    Müller, W.H., Vilchevskaya, E.N., Weiss, W.: Micropolar theory with production of rotational inertia: a farewell to material description. Phys. Mesomech. 20(3), 250–262 (2017)CrossRefGoogle Scholar
  44. 44.
    Müller, W.H., Vilchevskaya, E.N.: Micropolar theory from the viewpoint of mesoscopic and mixture theories. Phys. Mesomech. 20(3), 263–279 (2017)CrossRefGoogle Scholar
  45. 45.
    Einstein, A., Infeld, L.: The Evolution of Physics. Cambridge University Press, London (1938)zbMATHGoogle Scholar
  46. 46.
    Einstein, A.: The Collected Papers, vol. 6. Princeton University Press, Princeton (1997)zbMATHGoogle Scholar
  47. 47.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. Vol. 2. Mainly Electromagnetism and Matter. Addison Wesley Publishing Company, London (1964)zbMATHGoogle Scholar
  48. 48.
    Sommerfeld, A.: Electrodynamics. Lectures on Theoretical Physics, vol. 3. Academic, New York (1964)Google Scholar
  49. 49.
    Tonnelat, M.-A.: The Principles of Electromagnetic Theory and of Relativity. D. Reidel Publishing Company, Dordrecht-Holland (1966)CrossRefGoogle Scholar
  50. 50.
    Malvern, E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall Inc, Englewood Cliffs (1969)Google Scholar
  51. 51.
    Truesdell, C.: A First Course in Rational Continuum Mechanics. The John Hopkins University, Baltimore (1972)Google Scholar
  52. 52.
    Eringen, C.: Mechanics of Continua. Robert E. Krieger Publishing Company, Huntington (1980)zbMATHGoogle Scholar
  53. 53.
    Batchelor, G.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1970)Google Scholar
  54. 54.
    Loitsyansky, L.G.: Fluid Mechanics. Nauka, Moscow (1987). (In Russian)Google Scholar
  55. 55.
    Daily, J., Harleman, D.: Fluid Dynamics. Addison-Wesley, Boston (1966)zbMATHGoogle Scholar
  56. 56.
    Zhilin, P.A.: Applied Mechanics. Foundations of Shells Theory. Tutorial book. Politechnic University Publishing House, St. Petersburg (2006). (In Russian)Google Scholar
  57. 57.
    Cataneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. C. R. 247, 431–433 (1958)Google Scholar
  58. 58.
    Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)CrossRefGoogle Scholar
  59. 59.
    Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  60. 60.
    Purcell, E.M.: Berkeley Physics Course. Vol. 2. Electricity and Magnetism, vol. 2. McGraw-Hill, New York (1965)Google Scholar

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Authors and Affiliations

  1. 1.Department of Theoretical MechanicsPeter the Great St. Petersburg Polytechnic UniversitySaint-PetersburgRussia
  2. 2.Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSaint-PetersburgRussia

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