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High-frequency homogenization in periodic media with imperfect interfaces.

Raphaël C Assier1, Marie Touboul2, Bruno Lombard2

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High-frequency homogenization is extended to 1D periodic media with imperfect interfaces, enabling analysis of elastic wave propagation with discontinuities. A uniform approximation is developed for both Dirac and non-Dirac points, validated numerically.

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Area of Science:

  • Solid Mechanics
  • Wave Propagation
  • Materials Science

Background:

  • High-frequency homogenization is a technique used to analyze wave propagation in periodic structures.
  • Periodic media with imperfect interfaces present challenges due to discontinuities in displacement and stress.
  • Understanding wave behavior in such media is crucial for designing advanced materials and devices.

Purpose of the Study:

  • To extend high-frequency homogenization to 1D periodic media with imperfect interfaces.
  • To develop asymptotic approximations for dispersion diagrams and wave fields.
  • To provide a uniform approximation applicable to both Dirac and non-Dirac points.

Main Methods:

  • Asymptotic analysis of periodic and antiperiodic solutions at the Brillouin zone edges.
  • Development of second-order approximations for higher dispersion branches and leading-order for wave fields.
  • Analysis of Dirac points and derivation of first-order dispersion and zeroth-order wave field approximations.
  • Numerical validation using the Bloch-Floquet approach for monolayered and bilayered materials.

Main Results:

  • Asymptotic approximations for dispersion diagrams and wave fields were derived.
  • A novel uniform approximation was developed, valid for both Dirac and non-Dirac points.
  • Numerical comparisons confirmed the accuracy of the uniform approximation, even away from Brillouin zone edges.
  • Convergence measurements validated the proposed homogenization approach.

Conclusions:

  • The extended high-frequency homogenization method accurately models elastic wave propagation in 1D periodic media with imperfect interfaces.
  • The developed uniform approximation offers a robust tool for analyzing wave phenomena in these complex structures.
  • The findings have implications for the design and analysis of acoustic and mechanical metamaterials.