Mykulyak S. The regularities of the dynamics of structured geomedia: theory, model, experiment

The thesis is devoted to studying the properties of dynamics of geomedia taking into account their discrete and hierarchical structure. The geomedia considered in the thesis cover a broad class of natural discrete rocks massifs: from granular rock massifs e.g. sand to substantially heterogeneous and fragmented areas such as seismically active zones. The research is carried out within the framework of the approach considering the geomedium as a thermodynamically open complex system consisted of hierarchically embedded discrete elements with nonlinear and dissipative interactions. Because of continuum models are not able to reproduce the entire variety of dynamic behavior of such media the discrete models were applied for studying the structured media dynamics. The computer simulation of wave processes is performed using the discrete element method. The studies testify that in discrete media these processes fundamentally differ from similar ones in homogeneous solids and the wave fields depend on the type of structured elements packing. The simulation of the wave propagation in a layer of the granular medium with spherical grains discovered a rotating wave structure formation. It is shown that the deformation properties of granular media with elastic, elastic-viscous and elastic-plastic types of interaction between particles depend on the rate of deformation, on the package density and on the of the grain sizes. To explore the granular medium microcharacteristics, namely forces acting on individual structural elements at the bottom of the granular sample under dynamic load, the novel experimental technique is developed. As a result of experiments, the distributions exibiting the exponentially decaying maximum values of the forces with which the spherical grains act on the cylinder bottom in the range of large forces are obtained. Moreover, the computer simulation demonstrates that the exponential force distribution takes place throughout the sample and thus confirms the presence of correlations between intergranular forces in the process of dynamic loading of granular specimen. The time dependencies of the coordination number, the orientation order parameter, the correlation radius and the distribution of forces obtained in the calculations clearly indicate the non-equilibrium nature of the deformation process in the granular medium under impulse loading. To study the dynamics of hierarchically organized discrete media, the model of the system of embedded anharmonic oscillators is elaborated. Using the Poincaré section technique, it is stated that the system of equations for the model having three hierarchical levels with oscillators identical on each level and under the condition that they move simultaneously possesses the localized quasiperiodic and chaotic trajectories. The study of vibrating processes in the multilevel system is also carried out. It is shown that there exists a critical value of the structural parameter corresponding to the formation of comparative oscillations in the first and the last hierarchical levels. The effect of dissipation on the formation of oscillation processes in the hierarchical system is studied within the framework of the three-level model, when the highest structural level is subjected to the action of harmonic force. The analysis of obtained amplitude-frequency curves show that the hierarchical structure behaves as an amplifier of the signal applied to the highest level of the hierarchy. The earthquake model is developed based on two fundamental principles: the hierarchical structure of seismically active regions and the concept of self-organized criticality. The model reproduces the basic empirical properties of seismic processes: the frequency-energy scaling relation (the Gutenberg-Richter law), the generalized Omori law for temporal decay of aftershocks, the aftershock productivity law, Båth’s law for the mean value of the relative difference in magnitude between the major earthquake and its largest aftershock, the fractal distributions of hypocenters (epicenters) with power-law dependencies of the number of events on distances between hypocenters (epicenters), and, finally, the γ distribution for waiting times. In the model, the threshold energies depend on the block sizes and are distributed according to the Gauss law. Experiments and simulation of shear deformation of the granular medium formed by the cubic elements demonstrate a statistical similarity to the processes occurring in seismically active regions. These results open the prospect of the better understanding of natural seismic processes and possibilities for forecasting and controlled influence on them.