The propagation of longitudinal stress waves in an infinite rod with a viscoelastic region

The problem of propagation of longitudinal stress waves in an infinite piece-wise homogeneous rod with a viscoelastic region of finite size is solved. A mathematical formulation of the problem and its analytical solution in stresses are obtained. The solution uses the method of the Laplace integral transformation with respect to time. The expressions obtained allow one to determine the stresses in an arbitrary cross-section of the rod at any time. The structure of the solution reflects the process of transformation of the incoming wave with its multiple refraction and reflection at the boundaries of the sections. The solution obtained makes it possible to quantitatively investigate the damping effect of the viscoelastic region and can serve as the basis for selecting the parameters of the insert material when it is used to extinguish the dynamic effects