Numerous experimental studies on shock wave loading of metals have shown by electron microscopy that the crystal structure of the material can undergo transformation in a certain range of striker velocities. At the macroscale, these changes are observed as energy losses associated with the formation of a new structure. The losses are manifested on the temporal velocity profile of the back side of the sample surface which contains key information about the material properties. In this paper, a two-component model of a material with a nonlinear internal interaction force is proposed for the description of structural transformations, taking into account the periodic structure of the material. Dynamic equations are written with respect to the motion of the center of mass of the components acting as a measured macroparameter, as well as with respect to their relative displacement which is the internal degree of freedom responsible for structural transformations. The proposed model is applied to solve a quasi-static problem of the kinematic extension of a two-component rod in order to determine the parameters of a nonmonotonic stress-strain relationship that is often used in describing materials subjected to phase transformations. By solving a dynamic problem of nonstationary influence on the material by a short rectangular pulse, the effect of nonstationary wave damping is demonstrated which is associated with the wave energy dissipation in structural changes of the material. An analytical expression is obtained on the basis of a continuous-discrete analogy for estimating the duration of structural transformations and the parameter characterizing the force of internal interaction between the components. The conclusions are confirmed by a numerical solution of the nonlinear Cauchy problem within the finite difference framework.