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Elastic wave propagation in functionally graded periodically layered and damaged composites is investigated using the extended transfer matrix method and the boundary integral equation method.  Two different models are developed to approximate the damaged layer, namely, a periodic array of cracks and continuously distributed springs in the layer. The focus of this analysis is on the wave transmission and reflection, band gaps, localization and resonance phenomena due to the influences of functionally graded properties and crack-like damages.

metamaterials; phononic crystals; damages; band gaps; localization; resonance;