Continuum Mechanics and Thermodynamics

, Volume 28, Issue 6, pp 1759–1780 | Cite as

Micropolar continuum in spatial description

  • Elena A. Ivanova
  • Elena N. VilchevskayaEmail author
Original Article


Within the spatial description, it is customary to refer thermodynamic state quantities to an elementary volume fixed in space containing an ensemble of particles. During its evolution, the elementary volume is occupied by different particles, each having its own mass, tensor of inertia, angular and linear velocities. The aim of the present paper is to answer the question of how to determine the inertial and kinematic characteristics of the elementary volume. In order to model structural transformations due to the consolidation or defragmentation of particles or anisotropic changes, one should consider the fact that the tensor of inertia of the elementary volume may change. This means that an additional constitutive equation must be formulated. The paper suggests kinetic equations for the tensor of inertia of the elementary volume. It also discusses the specificity of the inelastic polar continuum description within the framework of the spatial description.


Multipolar medium Spatial description Tensor of inertia Structural transformations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Batchelor, G.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1970)zbMATHGoogle Scholar
  2. 2.
    Birkhoff, G.: Hydrodynamics. A Study in Logic, Fact and Similitude. Princeton University Press, Princeton (1960)zbMATHGoogle Scholar
  3. 3.
    Chen, K.: Microcontinuum balance equations revisited: the mesoscopic approach. J. Non-Equilib. Thermodyn. 32, 435–458 (2007)ADSzbMATHGoogle Scholar
  4. 4.
    Daily, J., Harleman, D.: Fluid Dynamics. Addison-Wesley, Massachusetts (1966)zbMATHGoogle Scholar
  5. 5.
    Dlużewski, P.: Finite deformations of polar elastic media. Int. J. Solids Struct. 30(16), 2277–2285 (1993)CrossRefzbMATHGoogle Scholar
  6. 6.
    Eremeyev, V., Lebedev, L., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Eringen, A.: Simple microfluids. Int. J. Eng. Sci. 2(2), 205–217 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eringen, A.: Continuum Physics, vol. IV. Academic Press, New York (1976)zbMATHGoogle Scholar
  9. 9.
    Eringen, A.: A unified continuum theory of electrodynamics of liquid crystals. Int. J. Eng. Sci. 35(12/13), 1137–1157 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eringen, A.: Microcontinuum Field Theory I. Foundations and Solids. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Eringen, A.: Microcontinuum Field Theory II. Fluent Media. Springer, New York (2001)zbMATHGoogle Scholar
  12. 12.
    Eringen, A., Kafadar, C.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. IV, pp. 33–63. Academic Press, New York (1976)Google Scholar
  13. 13.
    Eringen, C.: Mechanics of Continua. Robert E. Krieger Publishing Company, Huntington (1980)zbMATHGoogle Scholar
  14. 14.
    Ivanova, E.: Rigid body oscillator: a general model and some results. Acta Mech. 142, 149–193 (2000)CrossRefGoogle Scholar
  15. 15.
    Ivanova, E.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mech. 12, 261–286 (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Ivanova, E.: On one model of generalised continuum and its thermodynamical interpretation. In: Mechanics of generalized Continua, pp. 151–174. Springer, Berlin (2011)Google Scholar
  17. 17.
    Ivanova, E.: Description of mechanism of thermal conduction and internal damping by means of two component cosserat continuum. Acta Mech. 225, 757–795 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Kafadar, C., Eringen, A.: Micropolar media I. The classical theory. Int. J. Eng. Sci. 9, 271–305 (1971)CrossRefzbMATHGoogle Scholar
  20. 20.
    Loicyanskii, L.G.: Mekhanika Zhidkosti i Gaza (Mechanics of Fluids, in Russ.). Nauka, Moscow (1987)Google Scholar
  21. 21.
    Loret, B., Simões, F.: A framework for deformation, generalized diffusion, mass transfer and growth in multi-species multi-phase biological tissues. Eur. J. Mech. A/Solids 24, 757–781 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Malvern, E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall Inc, Englewood Cliffs (1969)zbMATHGoogle Scholar
  23. 23.
    Mindlin, R.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mindlin, R., Tiersten, H.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. Academic Press, Elsevier, Amsterdam (1993)zbMATHGoogle Scholar
  26. 26.
    Oeve, W., Schröter, J.: Balance equation for micromorphic materials. J. Stat. Phys. 25(4), 645–662 (1981)ADSCrossRefGoogle Scholar
  27. 27.
    Pietraszkiewicz, W., Eremeyev, V.: On natural strain measures of the nonlinear micropolar continuum. Int. J. Solids Struct. 46(3–4), 774–787 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Prandtl, L., Tietjens, O.: Hydro- und Aeromechanik. Springer, Berlin (1929)zbMATHGoogle Scholar
  29. 29.
    Toupin, R.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Truesdell, C.: A First Course in Rational Continuum Mechanics. The Johns Hopkins University, Baltimore (1972)Google Scholar
  31. 31.
    Truesdell, C., Noll, W.: The nonlinear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, pp. 1–602. Springer, Berlin (1965)Google Scholar
  32. 32.
    Truesdell, C., Toupin, R.: The classical field theories. In: Flügge, S. (ed.) Encyclopedia of Phycics, vol. III/1. Springer, Heidelberg (1960)Google Scholar
  33. 33.
    Wilmanski, K.: Thermomechanics of Continua. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  34. 34.
    Wilmanski, K.: Continuum Thermodynamics Part I. Foundations. World Scientific, Singapore (2008)CrossRefzbMATHGoogle Scholar
  35. 35.
    Zhilin, P.A.: Racional’naya mekhanika sploshnykh sred (Rational Continuum Mecanics, in Russ.). Politechnic University Publishing House, St. Petersburg (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Peter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia
  2. 2.Institute for Problems in Mechanical EngineeringRussian Academy of SciencesSt. PetersburgRussia

Personalised recommendations