  A non-local dynamic homogenization technique for the analysis of a
viscoelastic heterogeneous material which displays a periodic microstructure is
herein proposed. The asymptotic expansion of the micro-displacement field in
the transformed Laplace domain allows obtaining, from the expression of the
micro-scale field equations, a set of recursive differential problems defined
over the periodic unit cell. Consequently, the cell problems are derived in
terms of perturbation functions depending on the geometrical and
physical-mechanical properties of the material and its microstructural
heterogeneities. A down-scaling relation is formulated in a consistent form,
which correlates the microscopic to the macroscopic transformed displacement
field and its gradients through the perturbation functions. Average field
equations of infinite order are determined by substituting the down-scale
relation into the micro-field equation. Based on a variational approach, the
macroscopic field equations of a non-local continuum is delivered and the local
and non-local overall constitutive and inertial tensors of the homogenized
continuum are determined. The problem of wave propagation in case of a bi-phase
layered material with orthotropic phases and axis of orthotropy parallel to the
direction of layers is investigated as an example. In such a case, the local
and non-local overall constitutive and inertial tensors are determined
analytically and the dispersion curves obtained from the non-local homogenized
model are analysed.
