interface homogenization

We propose a homogenization method based on a matched asymptotic expansion technique to obtain the effective behavior of a periodic array of linear viscoelastic inclusions embedded in a linear viscoelastic matrix.  The problem is considered for shear waves and the wave equation in the harmonic regime is considered.  The obtained effective behavior is that of an equivalent interface associated to jump conditions, for the displacement and the normal stress at the interface.  The transmission coefficients and the displacement fields are obtained in closed forms and their validity is inspected