Full-band exact homogenization of one-dimensional elastic metamaterials

Abstract

Metamaterials extend the realm of materials’ properties by carefully designed structural inclusions. By targeting the extraction of effective properties from composite materials, homogenization theory plays an important role for metamaterials in their design and characterization. However, conventional homogenization methods are limited to the long wavelength limit. Here, we introduce an exact homogenization scheme valid for one-dimensional metamaterials over the full frequency band. In this scheme, with the aid of eigenstates’ characterization, a set of explicit formulas for effective mass density and effective elastic modulus are obtained by matching the surface responses properties of a metamaterial’s single structural unit with a piece of effectively homogenized material. In the frequency regimes beyond the conventional homogenization theory, new features, such as the imaginary parts of the effective parameters, have been found. Applying this scheme on a layered structure, the predicted transport properties and displacement fields from the effective parameters show excellent agreement with numerical simulations.