Two scale homogenization of a row of locally resonant inclusions - the case of anti-plane shear waves

We present a homogenization model for a single row of locally resonant inclusions. The resonances, of the Mie type, result from a high contrast in the shear modulus between the inclusions and the elastic matrix. The presented homogenization model is based on a matched asymptotic expansion technique; it slightly differs from the classical homogenization which applies for thick arrays with many rows of inclusions (and thick means large compared to the wavelength in the matrix). Instead of the effective bulk parameters found in the classical homogenization, we end up with interface parameters entering in jump conditions for the displacement and for the normal stress; among these parameters, one is frequency dependent and encapsulates the resonant behavior of the inclusions. Our homogenized model is validated by comparison with results of full wave calculations. It is shown to be efficient in the low frequency domain and accurately describes the effects of the losses in the soft inclusions.