Physical Mesomechanics

, Volume 20, Issue 3, pp 263–279 | Cite as

Micropolar theory from the viewpoint of mesoscopic and mixture theories

  • W. H. MüllerEmail author
  • E. N. Vilchevskaya


This paper takes a nontraditional look at micropolar media. It emphasizes the idea that it may become necessary to abandon the concept of material particles if one wishes to describe micropolar matter in which structural changes or chemical reactions occur. Based on recent results presented by Ivanova and Vilchevskaya (2016) we will proceed as follows. First we shall summarize the theory required for handling such situations in terms of a single macroscopic continuum. Mne of its main features are new balance equations for the local tensors of inertia containing production terms. The new balances and in particular the productions will then be interpreted mesoscopically by taking the inner structure of micropolar matter into account. As an alternative way of understanding the new relations we shall also attempt to use the concepts of the theory of mixtures. However, we shall see by example that this line of reasoning has its limitations: A binary mixture of electrically charged species subjected to gravity will segregate. Hence it is impossible to use a single continuum for modeling this kind of motion. However, in this context it will also become clear that the traditional Lagrangian way of describing motion of structurally transforming materials is no longer adequate and should be superseded by the Eulerian approach.


micropolar media spatial description rational mixture theory characteristics transport equations 


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  1. 1.
    Eringen, A.C. and Kafadar, C.B., Polar Field Theories, Continuum Physics IV, London: Academic Press, 1976.Google Scholar
  2. 2.
    Eringen, C., Nonlocal Continuum Field Theories, New York: Springer, 2002.zbMATHGoogle Scholar
  3. 3.
    Eremeyev, V.A., Lebedev, L.P., and Altenbach, H., Foundations of Micropolar Mechanics, Heidelberg: Springer, 2012.zbMATHGoogle Scholar
  4. 4.
    Ericksen, J.L., Conservation Laws for Liquid Crystals, Trans. Soc. Rheol., 1961, vol. 5, no. 1, pp. 23–34.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Leslie, F.M., Some Constitutive Equations for Liquid Crystals, Arch. Ration. Mech. An., 1968, vol. 28, no. 4, pp. 265–283.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Atkin, R. and Leslie, F., Couette Flow of Nematic Liquid Crystals, Quart. J. Mech. Appl. Math., 1970, vol. 23, no. 2, pp. 3–24.CrossRefzbMATHGoogle Scholar
  7. 7.
    Ivanova, E.A. and Vilchevskaya, E.N., Micropolar Continuum in Spatial Description, Continuum Mech. Therm., 2016, vol. 28, no. 6, pp. 1759–1780.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Müller, I., In Memoriam Frank Matthews Leslie, FRS, Continuum Mech. Therm., 2002, vol. 14, pp. 227–229.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lebedev, L., Cloud, M., and Eremeev, V., Tensor Analysis with Applications in Mechanics, New Jersey: World Scientific, 2010.CrossRefGoogle Scholar
  10. 10.
    Müller, I., Thermodynamics, Boston: Pitman, 1985.zbMATHGoogle Scholar
  11. 11.
    Eringen, A.C., A Unified Continuum Fheory for Electrodynamics of Polymeric Liquid Crystals, Int. J. Eng. Sci., 2000, vol. 38, pp. 959–987.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ivanova, E.A., Derivation of Theory of Thermoviscoelasticity by Means of Two-Component Medium, Acta Mech., 2010, vol. 61, no. 1, pp. 261–286.CrossRefzbMATHGoogle Scholar
  13. 13.
    Ivanova, E.A., AAAAA On One Model of Generalised Continuum and Its Thermodynamical Interpretation, Mechanics of Generalized Continua, Altenbach, H., Maugin, G.A., and Erofeev, V., Eds., Berlin: Springer, 2011, pp. 151–174.Google Scholar
  14. 14.
    Zhilin, P.A., Rational Continuum Mechanics, St. Petersburg: SPbPU, 2012.Google Scholar
  15. 15.
    Ivanova, E.A., Description of Mechanism of Thermal Conduction and Internal Damping by Means of Two Component Cosserat Continuum, Acta Mech., 2014, vol. 225, pp. 757–795.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Müller, W.H., Vilchevskaya, E.N., and Weiss, W., Micropolar Theory with Production of Rotational Inertia: A Farewell to Material Description, Phys. Mesomech., 2017, vol. 20, no. 3, pp. 250–262.CrossRefGoogle Scholar
  17. 17.
    Kafadar, C. and Eringen, A., Micropolar Media I. The Classical Theory, Int. J. Eng. Sci., 1971, vol. 9, pp. 271–305.CrossRefzbMATHGoogle Scholar
  18. 18.
    Truesdell, C. and Toupin, R.A., The Classical Field Theories, Heidelberg: Springer, 1960.CrossRefGoogle Scholar
  19. 19.
    Courant, R. and Hilbert, D., Methods of Mathematical Physics. VII: Partial Differential Equations, New York: Interscience, 1962.zbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institute of MechanicsTechnische Universität BerlinBerlinGermany
  2. 2.Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia
  3. 3.Peter the Great Saint-Petersburg Polytechnic UniversitySt. PetersburgRussia

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