Advertisement

Journal of Mathematical Sciences

, Volume 238, Issue 2, pp 129–138 | Cite as

Equations of the Local Gradient Electromagnetothermomechanics of Polarizable Nonferromagnetic Bodies with Regard for Electric Quadrupole Moments

  • V. F. Kondrat
  • O. R. Hrytsyna
Article
  • 4 Downloads

We formulate a complete system of relations of the local gradient electromagnetothermomechanics of electrically conductive nonferromagnetic polarizable solid media. The nonlocal character of the constitutive relations of the proposed mathematical model is explained by the presence of electric quadrupole moments in the polarization current. As a result of taking into account these moments, the space of parameters of the thermodynamic state of the body is expanded by including a pair of additional conjugate parameters, namely, the quadrupole moment and the gradient of the vector of electric-field intensity. It is shown that the developed model takes into account the electromechanical interaction for materials with high level of symmetry (isotropic materials) and describes the flexoelectric and thermopolarization effects. We also present the key system of equations for a physically and geometrically linear medium.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. M. Bredov, V. V. Rumyantsev, and I. N. Toptygin, Classical Electrodynamics [in Russian], Nauka, Moscow (1985).Google Scholar
  2. 2.
    Ya. Burak, V. Kondrat, and O. Hrytsyna, Foundations of the Local Gradient Theory of Dielectrics [in Ukrainian], Lira, Uzhgorod (2011).Google Scholar
  3. 3.
    S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam (1962).zbMATHGoogle Scholar
  4. 4.
    V. Kondrat and O. Hrytsyna, “Linear theory of the electromagnetomechanics of dielectrics,” Fiz.-Mat. Model. Inform,. Tekhnol., Issue 9, 7–46 (2009).Google Scholar
  5. 5.
    G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Dover, New York (2000).zbMATHGoogle Scholar
  6. 6.
    L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Butterworth-Heinemann, Oxford (1984).Google Scholar
  7. 7.
    I. E. Tamm, Fundamentals of the Theory of Electricity, Mir, Moscow (1979).Google Scholar
  8. 8.
    A. M. Fedorchenko, Theoretical Physics. Classical Electrodynamics [in Russian], Vyshcha Shkola, Kiev (1988).Google Scholar
  9. 9.
    F. Bampi and A. Morro, “A variational approach to deformable electromagnetic solids,” Acta Phys. Polon., B17, No. 11, 937–949 (1986).MathSciNetGoogle Scholar
  10. 10.
    A. C. Eringen and G. A. Maugin, Electrodynamics of Continua. Vol. 2: Fluids and Complex Media, Springer, New York (1990).CrossRefGoogle Scholar
  11. 11.
    E. P. Hadjigeorgiou, V. K. Kalpakides, and C. V. Massalas, “A general theory for elastic dielectrics. II. The variational approach,” Int. J. Non-Linear Mech., 34, No. 5, 967–980 (1999).CrossRefzbMATHGoogle Scholar
  12. 12.
    C. B. Kafadar, “The theory of multipoles in classical electromagnetism,” Int. J. Eng. Sci., 9, No. 9, 831–853 (1971).CrossRefzbMATHGoogle Scholar
  13. 13.
    V. K. Kalpakides and E. K. Agiasofitou, “On material equations in second gradient electroelasticity,” J. Elasticity, 67, No. 3, 205–227 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    V. K. Kalpakidis, E. P. Hadjigeorgiou, and C. V. Massalas, “A variational principle for elastic dielectrics with quadrupole polarization,” Int. J. Eng. Sci., 33, No. 6, 793–801 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    C. V. Massalas, V. K. Kalpakidis, and G. Foutsitzi, “Some comments on the extended Tiersten’s theory of thermoelectroelasticity,” Mech. Res. Comm., 21, No. 4, 343–351 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    G. A. Maugin, “Nonlocal theories or gradient-type theories: A matter of convenience?,” Arch. Mech., 31, No. 1, 15–26 (1979).MathSciNetzbMATHGoogle Scholar
  17. 17.
    G. A. Maugin, “The method of virtual power in continuum mechanics: Application to coupled fields,” Acta Mech., 35, Nos. 1-2, 1–70 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. Prechtl, “Deformable bodies with electric and magnetic quadrupoles,” Int. J. Eng. Sci., 18, No. 5, 665–680 (1980).CrossRefzbMATHGoogle Scholar
  19. 19.
    J. S. Yang, S. X. Mao, K. Yan, and A.-K. Soh, “Size effect on the electromechanical coupling factor of a thin piezoelectric film due to a nonlocal polarization law,” Scripta Mater., 54, No. 7, 1281–1286 (2006).CrossRefGoogle Scholar
  20. 20.
    X. Wang, E. Pan, and W. J. Feng, “Anti-plane Green’s functions and cracks for piezoelectric material with couple stress and electric field gradient effects,” Eur. J. Mech. A /Solids, 27, No. 3, 478–486 (2008).CrossRefzbMATHGoogle Scholar
  21. 21.
    X. M. Yang, Y. T. Hu, and J. S. Yang, “Electric field gradient effects in anti-plane problems of polarized ceramics,” Int. J. Solids Struct., 41, Nos. 24-25, 6801–6811 (2004).CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. F. Kondrat
    • 1
  • O. R. Hrytsyna
    • 2
  1. 1.Sahaidachnyi Academy of Army Ground ForcesLvivUkraine
  2. 2.Center of Mathematical Modeling, Pidstryhach Institute for Applied Problems in Mechanics and MathematicsUkrainian National Academy of SciencesLvivUkraine

Personalised recommendations