Mathematical Models and Methods in Applied Sciences c © World Scientific Publishing Company MEMORY EFFECT IN HOMOGENIZATION OF VISCOELASTIC KELVIN-VOIGT MODEL WITH TIME DEPENDENT COEFFICIENTS

Communicated by (xxxxxxxxxx) The paper is devoted to the modelling of the procedure of formation of a composite material constituted of solid fibers and of a solidifying matrix. The solidification process for the matrix depends on the temperature and on the reticulation rate which are used for the modelling of the mechanical properties of the matrix. The mechanical proper-ties are described by the viscoelastic media equation with rapidly oscillating periodic coefficients depending on the temperature and the reticulation rate. That is modelled as an initial boundary value problem with time-dependent elasticity and viscosity tensors to account for the solidification, and the termal ”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 coefficients including explicit memory terms, and prove that it admits a unique solution. Finally, we prove error bounds for the asymptotic solution, and discuss the related regularity properties of the homogenized solution