Mathematical modeling of the weak low-frequency impact of seismic waves on the glacial massif

Abstract

Introduction. The Kolka glacial massif repeatedly broke off from the freezing surface and thereby formed powerful glacial mudflows that caused great material damage to the Republic of the Republic of Alania with numerous human casualties. In the work of Muzaev I. D. and Sozanov V. G., the first step was taken in mathematical modeling of seismic vibrations of the glacial massif and the statement was proved that weak earthquakes of 3-3.5 MSK points can provoke bending and shear stresses several times higher than the destructive stresses of ice on the underlying surface of the glacial massif. The considered mathematical model proposed in the article has been amended, significantly clarifying the scientific results. The purpose of the work is to formulate and solve an initial boundary value problem that resembles the problems of mathematical physics and simulates the fluctuations of a glacial massif. The oscillation of the array is caused by the freezing of a weak traveling harmonic seismic wave with an intensity of 3-3.5 MSK points falling on it and on its underlying surface. Materials and methods. Due to the fact that the longitudinal dimensions of glacial massifs are an order of magnitude larger than their transverse dimensions, the article assumes that the mechanics of such massifs are most adequately modeled if their geometric configurations are considered as girder bodies with their physical and mechanical characteristics. The losses of mechanical energy due to internal resistance in the array material are taken into account in accordance with the hypothesis of S. A. Sorokin. The elastic rebound force from a seismically oscillating base is taken into account in accordance with Winkler’s hypothesis. Due to the fact that the coefficient of the differential equation of vibrations of the beam array is complex, the equation splits into a system of two fourth-order differential equations with real coefficients, where the desired functions are the real and imaginary parts of the originally sought complex function. The initial boundary value problem is solved analytically. Results. On the example of the Kolka glacier (North Caucasus Highlands), which collapsed in 2002 and claimed 126 human lives, computer experiments were performed to calculate induced stresses. Discussion. Comparative computational experiments have proved the claims that the seismic stresses caused by the Winkler elastic rebuff of the seismically oscillating surface of the base: a) are significantly greater than the stresses from the bending component of the vibrations of the array; b) can significantly exceed all the values of the destructive stresses of the ice. Conclusion. It is proved that the superiority of the Winkler stress over the destructive stress of ice can also occur in weak earthquakes with an intensity of 3-3.5 MSK points. Such superiority can play the role of a trigger for the breakaway of the glacial massif from the freezing surface and thereby the formation of a powerful glacial mudflow, as happened on the Kolka glacier. Suggestions for practical application and directions for future research. The paper presents a method for directly determining the cause of the breakaway of the glacial massif from the freezing surface. In the future, it is desirable to check the effect of a sharp induced maximum stress for steels intended for mechanical engineering purposes. It is desirable to study in detail the time dependence of the effect of inducing maximum stresses in arrays using low-intensity waves.