Effective Dynamic Properties of Microstructured Heterogeneous Materials

Central to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. While the theory of static effective property calculations goes back about 60 years, progress in the actual calculation of dynamic effective properties for periodic composites has been made only very recently. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.

Application of the method of asymptotic homogenization to an acoustic metafluid

The method of asymptotic homogenization is used to find the dynamic effective properties of a metamaterial consisting of two alternating layers of fluid, repeating periodically. As well as the effective wave equation, the method gives the effective equation of motion and constitutive relation in a natural way. When the material properties are such that resonant effects can be present in one of the layers, it is found that the metamaterial changes dynamically from a metafluid with anisotropic density and isotropic stiffness at low frequency to one with anisotropic stiffness when the frequency is near to one of the local resonances. In this region of frequency, the resulting metamaterial is not a pentamode material and thus does not belong to the class of metafluids that can be transformed to an isotropic fluid by a coordinate transformation.

Sub-wavelength plasmonic crystals: dispersion relations and effective properties

We obtain a convergent power series expansion for the first branch of the dispersion relation for sub-wavelength plasmonic crystals consisting of plasmonic rods with frequency-dependent dielectric permittivity embedded in a host medium with unit permittivity. The expansion parameter is η = kd =2 πd / λ , where k is the norm of a fixed wavevector, d is the period of the crystal and λ is the wavelength, and the plasma frequency scales inversely to d , making the dielectric permittivity in the rods large and negative. The expressions for the series coefficients (also called dynamic correctors) and the radius of convergence in η are explicitly related to the solutions of higher order cell problems and the geometry of the rods. Within the radius of convergence, we are able to compute the dispersion relation and the fields and define dynamic effective properties in a mathematically rigorous manner. Explicit error estimates show that a good approximation to the true dispersion relation is obtained using only a few terms of the expansion. The convergence proof requires the use of properties of the Catalan numbers to show that the series coefficients are exponentially bounded in the H 1 Sobolev norm.