The aim of this work is a developing of the physical and mechanical model of the cyclic hydroextrusion process, which is absent today, based on such modern concepts of mechanics and metal physics as the principles and patterns of growth and relaxation of stresses at the apices of concentrators not only always present inside real metallic materials, but also external ones presented in the case of hydroextrusion forming matrix. The experimental part of the work is carried out on Fe–Ni–C invar alloys containing 0.03% and 1.23% of carbon under hydrostatic conditions up to 1.6 GPa. Methods of metallography, electron microscopy, and durometry are used to study the features of surface structure formation and mechanical properties of sample tail parts in cases of the completed and incomplete last cycle after two hydroextrusion passes. The obtained results demonstrate the role of the external stress concentrator in the formation and interaction of plastic and elastic relaxation zones, as well as the hydrodynamic state of the material along the surfaces and inside the samples, depending on the pressure in the extruder container. The appearance of the hydrodynamic state reduces the friction forces along the surface of the matrix. It facilitates (makes possible) the process of hydroextrusion on the one hand, and on the other—provides the release (ejection) along the cylindrical surface of the sample material in the direction of its tail part which is in the matrix at the final stage of the last cycle of passage. In the case of an incomplete passage, this process is realized under conditions of considerably less weakening of the tail part. It is accompanied by the swelling of its surface under the influence of the opposite hydrodynamic pressure inside the sample and the formation of a significantly strengthened ‘knob’ that stops extrusion. The suggested model is in good agreement with the experimental and theoretical results and allows us to predict the processes taking place.