An unconventional model for the propagation of plane elastic-plastic waves taking into account unloading associated with plastic deformations is developed, and numerical calculations of the propagation of an elastic-plastic mechanical stress pulse are performed. The model is constructed based on analysis of the impact of a medium-power pulsed electron beam on an aluminum plate made of A995-grade material, which is close to a single crystal with a low yield strength, and the excitation of a shock-wave stress pulse within it, as well as consideration of such pulse formation in the nonequilibrium surface loading region. It is shown that rapid attenuation of the elastic-plastic wave amplitude occurs in this case, particularly during the propagation of short stress pulses. The adequacy of the proposed model for the process under study is confirmed by experimental data.