Overall response of viscoelastic composites and polycrystals: exact asymptotic relations and approximate estimates

This paper is devoted to the effective behavior of linear viscoelastic heterogeneous materials with a particular emphasis on their transient response. First, two new asymptotic relations for the overall creep function are derived at short and large times. They are related to the retardation spectrum of the composite and involve second-order moments per phase of the stress field for the purely elastic and purely viscous problems. In the context of harmonic loadings, these relations provide exact frequential asymptotic conditions on the overall storage and loss moduli. Second, by making use of these asymptotic results, an approximate model is proposed. It consists in approximating the retardation spectrum of the composite by a single discrete Dirac mass. Its accuracy is assessed by comparison with exact analytical results, full-field simulations and collocation results for several classes of composites and polycrystals.