Skip to main content
SpringerLink
Log in

Homogenization of Harmonic Maxwell Equations with Allowance for Interfacial Surface Currents: Layered Structure

  • Published:
Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

The Maxwell equations for a composite two-component laminated material with a periodic structure in the field of a time-harmonic source acting along the layers are considered. Two-scale homogenization of the equations is performed with allowance for complex conductivity of interfacial layers and their thickness. The boundary-value problem for systems of differential equations with boundary conditions is reduced to a problem in a weakly variational formulation. Unique solvability of the problem is established. The case of low frequencies of interfacial currents of different intensities with allowance for the frequency-dependent wave length and skin layer length is analyzed. Macro-equations are derived, and effective material constants are determined, such as the dielectric permittivity, magnetic permeability, and electrical conductivities. Conditions at which the effective parameters depend on interfacial currents are described. It is found that the effective dielectric permittivity can be negative at specially chosen parameters of interfacial layers if it is determined on the basis of the effective wave number.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Y. Amirat and V. V. Shelukhin, “Homogenization of Time Harmonic Maxwell Equations: The Effect of Interfacial Currents,” Math. Methods Appl. Sci. 40 (8), 3140–3162 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. C. Tournassat, Y. Chapron, P. Leroy, et al., “Comparison of Molecular Dynamics Simulations with Triple Layer and Modified Gouy-Chapman Models in a 0.1 M NaCl-Montmorillonite System,” J. Colloid Interface Sci. 339, 533–541 (2009).

    Article  ADS  Google Scholar 

  3. Yu. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A Primer on Surface Plasmon-Polaritons in Grafene,” Int. J. Modern Phys. B 27 (10), 1341001 (2013).

    Article  ADS  MATH  Google Scholar 

  4. G. Kristensson, “Homogenization of Corrugated Interfaces in Electromagnetics,” Progr. Electromag. Res. 55, 1–31 (2005).

    Article  Google Scholar 

  5. V. V. Shelukhin and S. A. Terentev, “Frequency Dispersion of Dielectric Permittivity and Electric Conductivity of Rocks via Two-Scale Homogenization of the Maxwell Equations,” Progr. Electromag. Res. B 14, 175–202 (2009).

    Article  Google Scholar 

  6. V. V. Shelukhin and S. A. Terent’ev, “Homogenization of Maxwell Equations and Maxwell-Wagner Dispersion,” Dokl. Akad. Nauk, 424 (3), 402–406 (2009).

    MathSciNet  MATH  Google Scholar 

  7. N. Wellander, “Homogenization of the Maxwell Equations: Case 2. Nonlinear Conductivity,” Appl. Math. 47 (3), 255–283 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. N. Lagarkov, V. N. Kisel, and V. N. Semenenko, “Wide-Angle Absorption by the use of a Metamaterial Plate,” Progr. Electromag. Res. Lett. 1, 35–44 (2008).

    Article  Google Scholar 

  9. Y. Amirat and V. V. Shelukhin, “Homogenization of Time Harmonic Maxwell Equations and the Frequency Dispersion Effect,” J. Math. Pures Appl. 95, 420–443 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  10. L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Fizmatlit, Moscow, 2005; Pergamon, New York, 1984).

    Google Scholar 

  11. I. C. Bourg and G. Sposito, “Molecular Dynamics Simulations of the Electrical Double Layer on Smectite Surfaces Contacting Concentrated Mixed Electrolyte (NaCl-CaCl2) Solutions,” J. Colloid Interface Sci. 360, 701–715 (2011).

    Article  ADS  Google Scholar 

  12. G. Nguetseng, “A General Convergence Result for a Functional Related to the Theory of Homogenization,” SIAM J. Math. Anal. 20 (3), 608–623 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  13. K. Sarabandi and F. T. Ulaby, “Techniques for Measuring the Dielectric Constant of Thin Materials,” IEEE Trans. Instrum. Measur. 4, 631–636 (1988).

    Article  Google Scholar 

  14. A. H. Boughriet, C. Legrand, and A. Chapoton, “Non-Iterative Stable Transmission/Reflection Method for Low-Loss Material Complex Permittivity Determination,” IEEE Trans. Microwave Theory Methods 45 (1), 52–57 (1997).

    Article  ADS  Google Scholar 

  15. W. Brown, Jr., Dielectrics (West Berlin, 1956).

  16. D. R. Smith, W. J. Padilla, D. C. Vier, et al., “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 4 (18), 4184–4187 (2000).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Auvergne, Clermont-Ferrand, France

    Y. Amirat

  2. Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. V. Shelukhin

  3. Novosibirsk State University, Novosibirsk, 630090, Russia

    V. V. Shelukhin

Authors
  1. Y. Amirat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Shelukhin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Shelukhin.

Additional information

Original Russian Text © Y. Amirat, V.V. Shelukhin.

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 60, No. 4, pp. 3–20, July–August, 2019.

This work was supported by the Government of the Russian Federation (Grant No. 14.W03.31.0002).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amirat, Y., Shelukhin, V.V. Homogenization of Harmonic Maxwell Equations with Allowance for Interfacial Surface Currents: Layered Structure. J Appl Mech Tech Phy 60, 593–607 (2019). https://doi.org/10.1134/S0021894419040011

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0021894419040011

Keywords

Search

Navigation