Dynamic homogenization of composite viscoelastic materials

A non-local dynamic homogenization technique for the analysisof a viscoelastic heterogeneous material which displaysa periodic microstructure is herein proposed. The asymptoticexpansion of the micro-displacement field in the transformedLaplace domain allows obtaining, from the expressionof the micro-scale field equations, a set of recursive differentialproblems defined over the periodic unit cell. Consequently,the cell problems are derived in terms of perturbationfunctions depending on the geometrical and physicalmechanicalproperties of the material and its microstructuralheterogeneities. A down-scaling relation is formulated in aconsistent form, which correlates the microscopic to the macroscopictransformed displacement field and its gradients throughthe perturbation functions. Average field equations of infiniteorder are determined by substituting the down-scale relationinto the micro-field equation. Based on a variationalapproach, the macroscopic field equations of a non-local continuumis delivered and the local and non-local overall constitutiveand inertial tensors of the homogenized continuumare determined. The problem of wave propagation in case ofa bi-phase layered material with orthotropic phases and axisof orthotropy parallel to the direction of layers is investigatedas an example. In such a case, the local and non-local overallconstitutive and inertial tensors are determined analyticallyand the dispersion curves obtained from the non-localhomogenized model are analysed.