Homogenizing the Viscoelasticity Problem with Long-Term Memory

The system of integro-differential equations describing the small oscillations of an $\varepsilon$-periodic viscoelastic material with long-term memory is considered. Using the two-scale convergence method, we construct the system of homogenized equations and prove the strong convergence as $\varepsilon \to 0$ of the solutions of prelimit problems to the solution of the homogenized problem in the norm of the space $L^2$.