  Homogenization of the equations of motion for a three dimensional periodic
elastic system is considered. Expressions are obtained for the fully dynamic
effective material parameters governing the spatially averaged fields by using
the plane wave expansion (PWE) method. The effective equations are of Willis
form (Willis 1997) with coupling between momentum and stress and tensorial
inertia. The formulation demonstrates that the Willis equations of
elastodynamics are closed under homogenization. The effective material
parameters are obtained for arbitrary frequency and wavenumber combinations,
including but not restricted to Bloch wave branches for wave propagation in the
periodic medium. Numerical examples for a 1D system illustrate the frequency
dependence of the parameters on Bloch wave branches and provide a comparison
with an alternative dynamic effective medium theory (Shuvalov 2011) which also
reduces to Willis form but with different effective moduli.
