Skip to main content

Multiscale Modeling of Wave Propagation in Viscoelastic Phononic Crystals and Acoustic Metamaterials

Research Sponsor:

US National Science Foundation


Ruize Hu and Caglar Oskay


  • Phononic crystals and acoustic metamaterials exhibit extraordinary capability in controlling

    acoustic and elastic waves by manipulating band gaps that forbid waves to propagate within

    targeted frequency ranges.

  • Modeling of transient wave propagation in these composites using direct numerical simulations is computationally prohibitive for structural design and analysis.


  • Develop a multiscale computational framework for efficient modeling of transient wave propagation in periodic viscoelastic composites, capturing wave dispersion and bandgap phenomenon.
  • Investigate the effects of microstructural morphology, elastic and viscoelastic constituent material properties on the overall wave attenuation behavior.


Spatial-temporal nonlocal homogenization model (STNHM)

High order asymptotic expansion

  • Critical in extending the classical homogenization theory to short wavelength regime.
  • Introduce spatial nonlocal terms in momentum balance equations at successive orders.

Construction of gradient-type nonlocal governing equation

  • Derive temporal nonlocal terms in high symmetry directions of the first Brillouin zone.
  • Nonlocal model parameters are uniquely determined by minimization of asymptotic residual.

Nonlocal effective medium model (NEM)

  • Governing equation in the second order form with nonlocal features retained by the frequency-dependent nonlocal effective moduli tensor.
  • Simulation of transient wave propagation without need of high order boundary conditions.

Dispersion relation

  • Captures the acoustic branch, the stop band and the first optical branch in high symmetry directions.
  • Evanescent wavenumber is predicted within the stop band.

Anti-plane shear wave 


In-plane wave


Transient wave propagation

In-plane elastic waveguide modeling

  • Wave dispersion occurs at relatively low frequency.
  • Highly confined wave propagation when wave frequency is within the stop band.



Anti-plane shear wave propagation in viscoelastic phononic crystal

  • Nearly complete attenuation due to band gap formation and viscoelastic dissipation.
  • Stronger attenuation compared to elastic phononic crystal.


  • Compared to having viscoelastic dissipation only (LHM), band gap formation significantly improves wave attenuation.
  • Layered microstructure has wider stop band and stronger wave attenuation within the stop band.