Analysis of fan waves in a laboratory model simulating the propagation of shear ruptures in rocks

The fan-shaped mechanism of rotational motion transmission in the system of elastically connected plates on a plane base is analyzed. This mechanism governs the propagation of shear ruptures in super brittle rocks of the Earth’s crust at seismogenic depths. A laboratory physical model was created which demonstrates the process of fan wave propagation. Equations of the dynamics of a fan system as a mechanical system with a finite number of degrees of freedom are obtained. A computational algorithm taking into account contact interaction between plates is worked out. Within the framework of a simplified continuous model, the approximate estimates of the length of a fan depending on the velocity of its propagation are obtained. It is shown that in the absence of friction a stationary fan can move with any velocity that does not exceed the critical value, which depends on the size, the moment of inertia of plates, the initial angle and the coefficient of elasticity of connection, and that the length of a fan decreases with increasing velocity. In the absence of distributed shear stress, when the system of plates is in a horizontal position, the fan stops due to the friction forces. The action of distributed shear stress leads to the incomplete disclosure of a fan, and besides the angle of opening decreases with increasing friction. In a system with friction the velocity of a traveling fan is uniquely determined by the opening angle, and in the case of neglecting friction it can take any value within an allowable range. On the basis of a discrete model, the computations demonstrating the incomplete disclosure of fans with different opening angles due to rapid or slow change in the velocity of rotation of the first plate are performed. Comparison of the results of computations of the length and velocity of the fan by means of a discrete model with computations based on analytical formulas and laboratory observations showed a good correspondence between the results.