We apply the early developed analytic approach which was used to model the macroplastic deformation of polycrystalline solids, for analysis of evolution of our visible Universe. Such evolution was considered as grand non-stationary stochastic process providing transferring the energy of the Big Bang in space. As consequences of the first and second laws of thermodynamics necessities were shown, respectively, to endless existence of life phenomena in the Universe as a form of the energy transferring together with — conservation of individual life for each living creature to provide spatial homogeneity conservation for the transferring. Proceeding from the spherical symmetry of the Big Bang products distribution and the principally limited time period of the Big Bang development, spherical ring structure of the whole Universe was proposed. Based on the structure, two possible variants of the material objects motion within the whole Universe was considered: 1) - practically free of interactions with the other products, motion in radial directions; 2) — strongly affected by the interaction the curved trajectories motion. Calculated percentages for the contributions in the energy flux for the various types of elementary particles as the energy carriers are in accordance with the known contents of the baryonic, Dark matters and Dark energy in the visible Universe. Early predicted countless number of the approach constitutive equation solutions were associated with the corresponding number of the energy carriers and the mono-universes. A mechanism for participating the mono-universes in the expansion of the whole, poly-universe was proposed. The observed non-monotonic time dependence for the speed of our visible mono-universe expansion is in accordance with the approach predictions. Time dependencies of the square density of the energy carriers and the stress caused the Big Bang energy gradient were calculated and their reciprocal mutual dependence was shown at finishing stage of the visible Universe evolution, in accordance with the known Chaplygin gas models.