  A multiscale asymptotic homogenization method for periodic microstructured
materials in presence of thermoelasticity with periodic spatially dependent one
relaxation time is introduced. The asymptotic expansions of the
micro-displacement and the micro-temperature fields are rewritten on the
transformed Laplace space and expressed as power series of the microstructural
length scale, leading to a set of recursive differential problems over the
periodic unit cell. The solution of such cell problems leads to the
perturbation functions. Up-scaling and down-scaling relations are then defined,
and the latter allow expressing the microscopic fields in terms of the
macroscopic ones and their gradients. Average field equations of infinite order
are also derived. The efficiency of the proposed technique was tested in
relation to a bi dimensional orthotropic layered body with orthotropy axis
parallel to the direction of the layers, where the mechanical and temperature
constitutive properties were well stabilised. The dispersion curves of the
homogenized medium, truncated at the first order are compared with the
dispersion curves of the heterogeneous continuum obtained by the Floquet-Bloch
theory. The results obtained with the two different approaches show a very good
agreement.
