The Nature of Instabilities in Blocked Media and Seismological Law of Gutenberg-Richter

摘要

This paper studies properties of a continuum with structure. The characteristic size of the structure governs the fact that difference relations do not automatically transform into differential ones. It is impossible to consider an infinitesimal volume of a body, to which we could apply the major conservation laws, because the minimal representative volume of the body must contain at least a few elementary microstructures. The corresponding equations of motions are the equations of infinite order, solutions of which include, along with sound waves, the unusual waves propagating with abnormal low velocities, not bounded below. It is shown that in such media weak perturbations can increase or decrease outside the limits. The variance of structure sizes plays a double role. The intensity of instabilities decreases due to dispersion, thereby stabilizing the media, while the frequency range of unstable solutions expands, and disasters can occur at very low frequencies. The equation of equilibrium is not satisfied at any point in the medium. It is true only at an average. Hence there is a possibility to have a lot of micro-dynamic acts, in spite of static macroscopic state in average. This paper describes some of the conditions under which the possible occurrence of usual wave motion in media in the presence of certain dynamic phenomena. The number of complex roots of the corresponding dispersion equation, which can be interpreted as the number of unstable solutions, depends on the specific surface cracks and is an almost linear dependence on a logarithmic scale, as in the seismological law of Gutenberg-Richter.