  A multifield asymptotic homogenization technique for periodic
thermo-diffusive elastic materials is provided in the present study. Field
equations for the first-order equivalent medium are derived and overall
constitutive tensors are obtained in closed form. These lasts depend upon the
micro constitutive properties of the different phases composing the composite
material and upon periodic perturbation functions, which allow taking into
account the effects of microstructural heterogeneities. Perturbation functions
are determined as solutions of recursive non homogeneous cell problems emanated
from the substitution of asymptotic expansions of the micro fields in powers of
the microstructural characteristic size into local balance equations. Average
field equations of infinite order are also provided, whose formal solution can
be obtained through asymptotic expansions of the macrofields. With the aim of
investigating dispersion properties of waves propagating inside the medium,
proper integral transforms are applied to governing field equations of the
homogenized medium. A quadratic generalized eigenvalue problem is thus
obtained, whose solution characterizes the complex valued frequency band
structure of the first-order equivalent material. The validity of the proposed
technique has been confirmed by the very good matching obtained between
dispersion curves of the homogenized medium and the lowest frequency ones
relative to the heterogeneous material. These lasts are computed from the
resolution of a quadratic generalized eigenvalue problem over the periodic cell
subjected to Floquet-Bloch boundary conditions. An illustrative benchmark is
conducted referring to a Solid Oxide Fuel Cell (SOFC)-like material, whose
microstructure can be modeled through the spatial tessellation of the domain
with a periodic cell subjected to thermo-diffusive phenomena.
