Variational-asymptotic homogenization of thermoelastic periodic materials with thermal relaxation

A multi-scale variational-asymptotic homogenization method for periodic microstructured materials in presence of thermoelasticity with periodic spatially dependent one relaxation time is introduced. The asymptotic expan-sions 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 prob-lems 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. The variational-asymptotic scheme to establish an equivalence between the equations at macro-scale and micro-scale is developed. 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 established. 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.

Automated summary

A multi-scale variational-asymptotic homogenization method is developed for periodic microstructured materials in presence of thermoelasticity with periodic spatially dependent one relaxation time. The method establishes equivalence between macro-scale and micro-scale equations, leading to efficient computation of the microscopic fields in terms of the macroscopic ones. Results show good agreement with heterogeneous continuum theory.