Memory effects in homogenization of viscoelastic Kelvin-Voigt model with time dependent coefficients

International audienceThis paper is motivated by modeling the procedure of formation of a composite material constituted of solid ¯bers and of a solidifying matrix. The solidi¯cation process for the matrix depends on the temperature and on the reticulation rate which thereby in°uence the mechanical properties of the matrix. The mechanical properties are described by a viscoelastic medium equation of Kelvin Voigt type with rapidly oscillating periodic coe±cients depending on the temperature and the reticulation rate. That is modeled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidi¯cation, and the mechanical and/or thermal forcing. First we prove the existence and uniqueness of the solution for the problem and obtain a priori estimates. Then we derive the homogenized problem, characterize its coe±cients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and establish some related regularity properties of the homogenized solution