In this paper, we present a theoretical formalism for the study of damped periodic materials. First we consider a lumped parameter model consisting of two masses in the unit cell whereby the masses are connected by springs and dashpot viscous dampers. We then extend our analysis to the study of a two-dimensional phononic crystal, modeled as an elastic continuum, and consisting of a periodic arrangement of square inclusions distributed in a matrix base material. For our damping model, we consider both proportional damping and general damping. Our results demonstrate the effects that damping have on the dispersion relation and damping ratio band structure.

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