Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials.

It is shown that essentially nonlinear models for solids with complex internal structure may be studied using phenomenological and proper structural approaches. It is found that both approaches give rise to the same nonlinear equation for traveling longitudinal macrostrain waves. However, presence of the connection between macro- and microfields in the proper structural model prevents a realization of some important solutions. First, it is obtained that simultaneous existence of compression and tensile waves is impossible in contrast to the phenomenological approach. Then, it is found that correlation between macro- and internal strains gives rise to the crucial influence of the velocity of the macrostrain wave on the existence of either compression or tensile localized strain waves. Also, it is shown that similar profiles of the macrostrain solitary waves may be accompanied by distinct profiles of the microstrain waves. Finally, the dispersion curves for the waves belong to the different branches. This is important for internal structural deviations caused by the dynamical loading due to the localized macrostrain wave propagation and demonstrates a need in the development of the proper structural approaches relative to the phenomenological ones.