Effective Medium Theory for Elastic Metamaterials in Thin Elastic Plates
  An effective medium theory for resonant and non-resonant metamaterials for flexural waves in thin plates is presented. The theory provides closed-form expressions for the effective parameters of arrangement of inclusions or resonators in thin plates as a function of the filling fraction of the inclusions, their physical properties and the frequency. It is shown that positive or negative effective elastic parameters are possible depending on the symmetry of the resonance but, unlike it happens for bulk elastic waves, the responsible for the negative mass density behaviour is the monopolar term, while the negative Young's modulus and Poisson's ratio is due to the combination of monopolar and quadrupolar resonances, showing also that, at least for the first order in the scattering coefficients, the dipolar resonance plays no role in the description of the effective medium. Several examples are given for both non-resonant and resonant effective parameters and the results are verified by multiple scattering theory. 
Daniel Torrent, Yan Pennec and Bahram Djafari-Rouhani
cond-mat.mtrl-sci