Object for study is multicomponent media with the internal exchanged processes. Intrinsic inhomogeneities give rise to the remarkable/unusual nonlinear behaviors demonstrated by these media at quasistatic and dynamic mechanical loads. The purpose of this book is to describe the dynamic behaviors of there media in terms of physically motivated models. New method of structure diagnostics by means of the nonlinear waves is suggested. The research of nonlinear evolution equations is carried out. The special attention is focused on development of models for describing both stress-strain properties of sandstone under quasistatic loading and vibration resonance in sedimentary rocks. The important result is theoretical prediction of a dynamic effect analogous to the known quasistatic effect of hysteresis with discrete memory. The models describe the experimental results and can be applied for future research. The reader may find some suggestions for future research herein, and open up questions that may be especially useful for young scientists. I hope that the reader will receive a more or less objective picture of the current state of the art of the wave dynamic of the structured media.