We consider a one-dimensional layered structure that consists of an elastic material and a Maxwell fluid. It is assumed that the width of the layers is small compared with the size of the domain, where the problem is being solved. By employing the Nguetseng two-scale convergence technique, we have homogenized the equations that describe the mechanical system. For the homogenized equation, the limit cases of the Maxwell fluid with the small viscosity and with the small relaxation time are considered. The question of existence of the travelling wave solution for the homogenized equation is investigated.