The interaction of a constant cross-section rod with an elastic environment under dynamic loading is considered. The directly proportional to the displacement resistance forces impacted on the rod. Elastic and viscoelastic one-dimensional models describing longitudinal vibrations occurring in a rod located in the environment are developed. Numerical analysis of deformations and stresses in the rod in an elastic environment is performed, and deformation profiles in specific sections of the rod are calculated. The wave propagation and reflection from the free boundary of a finite-length rod are considered. The obtained results are compared with the experimental data and classical models of longitudinal vibrations of rods. A good correspondence of the calculation results with experimental data was revealed. Calculations describing the fracture of the rod are performed, and the possibility of spall fracture is shown. Fracture phenomenon is predicted by the incubation time criterion. In several special cases, the dependence of the threshold value of the load amplitude on the duration of the load is found.