  A method for the homogenization of 3-D periodic elastic composites is
presented. It allows for the evaluation of the averaged overall frequency
dependent dynamic material constitutive tensors relating the averaged dynamic
field variable tensors of velocity, strain, stress, and linear momentum. The
formulation is based on micromechanical modeling of a representative unit cell
of a composite proposed by Nemat-Nasser & Hori (1993), Nemat-Nasser et. al.
(1982) and Mura (1987) and is the 3-D generalization of the 1-D elastodynamic
homogenization scheme presented by Nemat-Nasser & Srivastava (2011). We show
that for 3-D periodic composites the overall compliance (stiffness) tensor is
hermitian, irrespective of whether the corresponding unit cell is geometrically
or materially symmetric.Overall mass density is shown to be a tensor and, like
the overall compliance tensor, always hermitian. The average strain and linear
momentum tensors are, however, coupled and the coupling tensors are shown to be
each others' hermitian transpose. Finally we present a numerical example of a
3-D periodic composite composed of elastic cubes periodically distributed in an
elastic matrix. The presented results corroborate the predictions of the
theoretical treatment.
