<p>We study a nonlinear model that describes the longitudinal deformation of an elastic rod with a power-law nonlinearity in the presence of nonstationary extreme external action. All invariant submodels given by invariant solutions of the equation of this model are obtained. For the submodels given by the explicitly found solutions, we found out whether or not the fracture of the rod over time. For specific parameter values included in these solutions the distribution longitudinal displacement in the rod are shown in the graphs. For other submodels, we study physically meaningful boundary value problems that describe non-linear longitudinal deformations of an elastic rod, for which at the initial moment of time at a fixed point either a longitudinal displacement and its gradient, or a longitudinal displacement and a rate of its change are given. These boundary value problems for some specific values of the parameters included in them are solved numerically. The results of solving these boundary value problems are shown in the graphs. The conducted research is especially relevant in rocket engineering, aircraft engineering, shipbuilding, construction of buildings and structures, in medical research related to traumatology and limb prosthetics.</p>