  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.
