The paper presents the study of elastic composites acting as resonant metamaterials. The composites are made of long bers of elliptic cross section. The condition for obtaining resonant metamaterials is presented in the context of homogenization by using asymptotic expansion. It is shown that inner resonance occurs in the case of soft inclusions embedded within a stier matrix, both components having mass densities of the same order of magnitude. The localization problem coming from the asymptotic expansion is solved in the case of bers with elliptic cross sections, leading to the dynamic density of the material. The dynamic density is characterized by resonance frequencies and the related eigenfunctions of the inclusions, that are obtained by using an expansion along Mathieu functions. The method allows us to obtain the (orthotropic) dynamic densityrelated to various cases of elliptic cross section, recovering the extreme cases of circular cross-sections and nearly plate inclusions.