Sometimes dynamical strain processes turn out essentially nonlinear due to an influence of an internal structure of material. Then truncated power series in strains cannot be used in this case contrary to the weakly nonlinear case. Nevertheless they are formally used, in particular, for description of strains in seismic media and in paramagnetic crystals. The sources of abnormal nonlinearity are the presence of components with contrasting properties in the former case and an influence of magnetic field in the latter one. Some experimental data is presented to justify formal use of the truncated power series ex-pansions for description of essentially nonlinear processes. It is shown that model equation for essentially nonlinear longitudinal strain waves is of the same form for different media while transition from weakly nonlinear statement of the problem to the essentially nonlinear one gives rise to only one additional nonlinear term in the model equation. Its exact solutions allow us to define conditions of the strain localization using known values of the parameters of the models. An important feature of the solution of the model equation is simultaneous existence of compression and tensile strain waves. This is impossible in the weakly nonlinear case.