Dynamic asymptotic homogenization for periodic viscoelastic materials

A non-local dynamic homogenization technique for the analysis of a viscoelastic heterogeneous material which displays a periodic microstructure is herein proposed. The asymptotic expansion of the micro-displacement field in the transformed Laplace domain allows obtaining, from the expression of the micro-scale field equations, a set of recursive differential problems defined over the periodic unit cell. Consequently, the cell problems are derived in terms of perturbation functions depending on the geometrical and physical-mechanical properties of the material and its microstructural heterogeneities. A down-scaling relation is formulated in a consistent form, which correlates the microscopic to the macroscopic transformed displacement field and its gradients through the perturbation functions. Average field equations of infinite order are determined by substituting the down-scale relation into the micro-field equation. Based on a variational approach, the macroscopic field equations of a non-local continuum is delivered and the local and non-local overall constitutive and inertial tensors of the homogenized continuum are determined. The problem of wave propagation in case of a bi-phase layered material with orthotropic phases and axis of orthotropy parallel to the direction of layers is investigated as an example. In such a case, the local and non-local overall constitutive and inertial tensors are determined analytically and the dispersion curves obtained from the non-local homogenized model are analysed.