Mathematical modeling of shear seismic vibrations of an array consisting of recultivated material deposits of the mining industry (tailings)

Abstract

Relevance. When the tailings, where toxic waste from the processing of the mining and metallurgical industry was constantly dumped and accumulated, collapse, an environmental threatoccurs. Sometime, the waste array is located in a narrow floodplain of the river close to the river stream. In seismotectonic situations, a potential environmental hazard may be created for the region river system due to the possible collapse of toxic material composing the massif into the river stream. Aim. Formulation and solution of two initial boundary value problems of mathematical physics modeling shear seismic vibrations of an array consisting of waste material of a tailings dump, when the movement of the array is caused by the fall of a seismic wave on its underlying surface. Methods. One problem is solved within the framework of the hypothesis of E.S. Sorokin regarding the dissipation of energy due to internal resistance in the medium. Another problem is posed and solved within the framework of the Kelvin-Voigt hypothesis. One coefficient in the differential equations of shear oscillations is a complex quantity. The other coefficient represents a function depending on the spatial coordinate. These features in the presented boundary value problems create mathematical difficulties when trying to solve them by analytical methods of mathematical physics.Due to these features, with an arbitrary dependence of the coefficients on the spatial coordinate, the tasks set have not been solved by anyone all over the world. In this paper, we have found a way to solve these problems in the special case with an exponential dependence of the array width on the coordinate. In this particular case, the variable coefficients degenerate into constant values and thus the way to solve the tasks is significantly facilitated. Results. When solving the two tasks, a set of calculation formulas for calculating seismic stresses and seismic displacements in the body of the array was obtained for each. According to the obtained calculation formulas, computational experiments were performed, using computer technologies to determine seismic stresses and displacements on the underlying surface and on the ridge of the array, depending on the frequency and amplitude of the incident seismic wave. The results obtained make it possible to perform further estimated calculations on the violation of the integrity of the massif and its partial or complete collapse into the river stream.