Self-organized criticality in a hierarchical model of defects development

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Abstract
We suggest a hierarchical model of defects development demonstrating critical behavior in a wide range of parameters that is naturally called the self-organized criticality. The kinetic equation is used for description of temporal evolution of the system. Conditions of appearance and healing of defects wholly govern the system behavior. Properties of the stationary solution are investigated. The model is found to show two kinds of behavior: stability and the self-organized criticality. The first one corresponds to small deformations in destruction experiments, when samples contain only the cracks of a few small ranges and there are no large cracks. The second one represents scaling properties of the world seismicity. The slope of magnitude-frequency dependence in the region of self-organized criticality is equal to unity for arbitrary parameters of the model. It is similar to the slope of the Gutenberg-Richter law defined for the world seismicity.