Nature of instability of block media and distribution law of unstable states

The paper studies the properties of a structured continuum. The result of finite structure size is that difference relations fail to automatically pass into differential ones. Consideration of an infinitely small volume of the medium with laws of conservation is found impossible. The representative volume is only that volume of finite dimensions which contains a certain minimum set of elementary mesostructures. The impossibility to merely replace difference relations by differential relations lead to equilibrium equations and equations of motion of infinite order due to an infinite number of degrees of freedom in block media. golutions of these equations contain, in addition to ordinary elastic waves, a set of waves with widely different velocities, including extremely low velocities unbounded below. As shown earlier, small vibration in these media can be both decreasing and unlimitedly increasing. Hence, small vibrations are not always harmless. A dual role belongs to structure size dispersion. The latter weakens unstable vibrational phenomena, but extends the range of vibration frequencies involved in a catastrophic process such that catastrophic events may arise at quite low frequencies. The equilibrium equation can not hold in each infinitesimal volume of the medium because it is not representative for the medium as a whole. The equilibrium equation is valid only on average for sufficiently representative volumes. Hence, individual dynamic events are possible in the medium even if it is at equilibrium as a unit. This phenomenon is termed acoustic emission. The paper describes conditions under which acoustic emission initiates wave processes in their ordinary sense, i.e., initiation of waves under quasistatic stresses. The set of complex roots of the dispersion equation which are possible to interpret as the number of unstable solutions depends on the specific crack surface. At the logarithmic scale, this relation is almost linear and fits the Gutenberg-Richter earthquake repeatability law well-known in seismology.