Characterization of wave propagation in periodic viscoelastic materials via asymptotic-variational homogenization

• Published in: International journal of solids and structures 2019-11, Vol.172-173, p.110-146
• Main Authors: Del Toro, Rosaria, Bacigalupo, Andrea, Paggi, Marco
• Format: Article
• Language: English
• Subjects: Asymptotic series; Differential equations; Dispersion curve analysis; Dynamic variational-asymptotic homogenization; Homogenization; Mathematical analysis; Mechanical properties; Microstructure; Nonlocal continuum; Periodic materials; Perturbation; Propagation; Reliability analysis; Scaling; Tensors; Unit cell; Viscoelastic materials; Viscoelasticity; Wave propagation
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2019.03.007

A non-local dynamic homogenization technique for the analysis of wave propagation in viscoelastic heterogeneous materials with 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-scaling relation into the micro-field equation. Based on a variational approach, the macroscopic field equation 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 is investigated in case of a bi-phase layered material with orthotropic phases and axis of orthotropy parallel to the direction of layers as a representative example. In such a case, the local and non-local overall constitutive and inertial tensors are determined analytically. Finally, in order to test the reliability of the proposed approach, the dispersion curves obtained from the non-local homogenized model are compared with the curves provided by the Floquet-Bloch theory.