Variational-asymptotic homogenization of thermoelastic periodic materials with thermal relaxation

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.