Stress Relaxation in a Cellular Model of Elements with Nonlinear Interaction

A model of stress relaxation in a system of discrete elements is analyzed. The model suggests analyzing a small-time scale of the process when external supply of additional stresses in the system are negligibly small. The nonlinear interaction of elements is similar to the interaction of elements in the open dissipative OFC model. Toppling conditions are determined by the static fatigue effect. It is shown that at a high level of element coupling, the model is characterized by a power-law decay of the drop frequency in time, similar to that observed in aftershock sequences of earthquakes. This regularity slightly depends on the initial distribution of stresses in the system, its exponent is p = 0.85–10 for the element coupling parameter α = 0–0.25. The analysis of the value of the time delay c for the formation of a power-law drop frequency decay shows that this parameter correlates with the duration of large-amplitude drops at the initial step of the relaxation process. The value c is defined in this case by the parameter α. Calculations also shows that relaxation of the mean stress σ(t) in the system of elements follows the relation \(t \propto {{e}^{{ - \gamma \sigma }}}\) with a time delay corresponding to the value for the drop frequency dependence. At the same time, there is no delay in the time series of the mean stress decrease during the drop of an individual element \(d\sigma \). The dependence \(d\sigma \left( t \right)\) is defined by the relation \(t \propto {{e}^{{ - \beta \Delta \sigma }}}\) in the entire temporal interval of the relaxation process. The value β linearly decreases with the increase of element coupling α in the model.