Continuum Mechanics and Thermodynamics

, Volume 26, Issue 3, pp 387–401 | Cite as

On the proper formulation of Maxwellian electrodynamics for continuum mechanics

  • Daniel S. Weile
  • David A. Hopkins
  • George A. Gazonas
  • Brian M. Powers
Original Article

Abstract

Despite the importance of electromagnetomechanical physics to processes ranging from piezoelectricity to the dynamics of electron beams, confusion abounds in the continuum mechanics literature as to how Maxwell’s equations of electrodynamics should be formulated in the material frame of continuum mechanics. Current formulations in the literature conflict as to the manner in which the authors define fields, derive constitutive relations, and interpret contradictory formulations. The difficulties persist even when the phenomena described are electrostatic. This paper will demonstrate that the perplexity arises from two sources: a misunderstanding of the limitations of material frame descriptions, and the failure to appreciate the centrality of relativity theory to the formulation of electrodynamic equations in the vicinity of mechanical motion. Two new formulations of Maxwell’s equations are provided that avoid the paradoxes of earlier formulations and thus describe the physics clearly and without self-contradiction.

Keywords

Convective coordinates Continuum mechanics Electrodynamics Relativity theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aris R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover Publications, Inc., New York (1962)MATHGoogle Scholar
  2. 2.
    Arthur J.W.: Understanding Geometric Algebra for Electromagnetic Theory. Wiley-IEEE Press, New York (2011)MATHCrossRefGoogle Scholar
  3. 3.
    Balanis C.A.: Advanced Engineering Electromagnetics. John Wiley and Sons, Inc., New York (2012)Google Scholar
  4. 4.
    Borisenko A.I., Tarapov I.E.: Vector and Tensor Analysis with Applications. Dover Publications, Inc., New York (1979)Google Scholar
  5. 5.
    Chadwick P.: Continuum Mechanics: Concise Theory and Problems. Dover Publications, Inc., New York (1999)Google Scholar
  6. 6.
    Clayton J.: A non-linear model for elastic dielectric crystals with mobile vacancies. Int. J. Non Linear Mech. 44(6), 675–688 (2009)CrossRefADSGoogle Scholar
  7. 7.
    Dorfmann A.R., Ogden R.W.: Nonlinear electroelasticity. Acta Mech. 174(3–4), 167–183 (2000)Google Scholar
  8. 8.
    Einstein, A.: On the electrodynamics of moving bodies. In: The Principle of Relativity. Methuen and Company, Ltd., London (1923). Translated from the original “Zur Elektrodynamik bewegter Körper”. In: Annalen der Physik 17, 891 (1905)Google Scholar
  9. 9.
    Eringen A.C., Maugin G.A.: Electrodynamics of Continua I. Springer, New York (1990)CrossRefGoogle Scholar
  10. 10.
    Jackson J.D.: Classical Electrodynamics. 3rd edn. John Wiley and Sons, Inc., New York (1999)MATHGoogle Scholar
  11. 11.
    Kelly, P.A.: Lectures in Solid Mechanics Part iii: Foundations of Continuum Mechanics. http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III (2012)
  12. 12.
    Lax M., Nelson D.: Maxwell equations in material form. Phys. Rev. B 13(4), 1777–1784 (1976)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Malvern L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Inc., Englewood Cliffs, NJ (1969)Google Scholar
  14. 14.
    McConnell A.J.: Applications of Tensor Analysis. Dover Publications, Inc., New York (2011)Google Scholar
  15. 15.
    Minkowski, H.: Die grundgleichungen für die elektromagnetischen vorgänge in bewegten körpern. Nachrichten von der Gesellschaft der Wissenschaften zu Göttigen, Mathematisch-Physikalische Klasse pp. 53–111 (1908)Google Scholar
  16. 16.
    Møller C.: The Theory of Relativity. Oxford University Press, London (1969)Google Scholar
  17. 17.
    Truesdell C.: The Elements of Continuum Mechanics. Springer, New York (1965)Google Scholar
  18. 18.
    VanBladel J.: Relativity and Engineering. Springer, Berlin (1984)CrossRefGoogle Scholar
  19. 19.
    Yang J.S., Batra R.C.: A theory of electroded thin thermopiezoelectric plates subject to large driving voltages. J. Appl. Phys. 76(9), 5411–5417 (1994)CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Daniel S. Weile
    • 1
  • David A. Hopkins
    • 2
  • George A. Gazonas
    • 2
  • Brian M. Powers
    • 2
  1. 1.Department of Electrical and Computer EngineeringUniversity of DelawareNewarkUSA
  2. 2.RDRL-WMM-B Army Research Laboratory, APGAberdeenUSA

Personalised recommendations