Dynamic homogenization theory for nonlocal acoustic metamaterials
Résumé
We present a homogenization method for periodic acoustic composites based on the Plane Wave Expansion (PWE) method. We show that the description of periodic acoustic composites needs constitutive parameters which depend on frequency and wavenumber, meaning that the effective material is resonant and nonlocal. Also, an anisotropic mass density and an additional constitutive parameter, called the Willis term in analogy to its counterpart in elasticity, are found. Numerical calculations compare the present method with the traditional multiple scattering method, showing a good agreement between both theories. However, the method presented here overcomes the limitations of the multiple scattering method, where the incorporation of anisotropy and non-locality implies solving the scattering problem of anisotropic objects with additional boundary-conditions. A final example showing the importance of nonlocal effects is provided. This work shows that acoustic metamaterials are nonlocal materials in general, and provides a tool for the proper modeling of the nonlocal constitutive parameters.