Multifield constitutive identification of non-conventional thermo-viscoelastic periodic Cauchy materials

The present work proposes a variational-asymptotic homogenization technique for non-conventional thermoviscoelastic periodic microstructured materials. According to second sound theories, the heat flux vector linearly depends upon the history of temperature gradient and the heat conduction tensor represents the kernel of the relative hereditary integral, thus overcoming the paradox inferred from the usual Fourier’s law of thermal waves propagating at infinite speed. Down-scaling relations have been provided in the frequency domain, relating the transformed micro displacement and relative temperature fields to the corresponding macro variables and their gradients through frequency-dependent perturbation functions which convey the influence of the underlying microstructural heterogeneity. Average field equations of infinite order have been derived. Transformed field equations of the equivalent first-order medium have been obtained as Euler– Lagrange equations of a suitable functional whose first variation is properly truncated. They are characterized by frequency-dependent overall constitutive tensors, whose closed form has been provided. A benchmark test assesses the capabilities of the proposed homogenization method, where the solutions relative to a periodic, bi-phase, thermo-viscoelastic material and to the equivalent homogenized medium match under the hypothesis of periodic source terms.