Mathematical modelling of mountain shocks and earthquakes related to volcanism

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Introduction The destructive property of all earthly disasters is not their gigantic energy, destroying material and cultural values that take human lives, but their suddenness and the seeming unpredictability of earthquakes. Why earthquakes occur and what is their mechanism, and how to predict it? The occurring movements in the Earth’s core creates changes in the spatio-temporal geophysical fields themselves, due to the thermal and gravitational effect, both the temperature and pressure in the Earth’s core increase, and at depth catastrophic natural disasters occur subsequently on the Earth’s crust. The enormous pressure of the overlying thicknesses of plates and crust prevents the melting of rocks. Sometimes there are underground explosions or tremors associated with the movement of magma, which are considered as “volcanic” earthquakes. Near volcanoes, seismographs are installed, recording the first tremors before the eruption (sometimes eruptions begin before the tremors). With the growth of rock activity, closely spaced seismographs begin to record the fluctuations of “volcanic tremors”. It is a high-frequency vibration with an oscillation period generally less than 0.5 s. The first ‘tremor’ is expressed weakly and lasts no more than 2 hours, and increases as the activity of the volcano increases. The strength of the earthquake depends on the duration of the period during which the anomaly is observed. For detecting the shake of the Earth’s crust has different methods. Japanese seismologist Vadati invented one of the methods in which 12 or more seismographs are installed on one profile. At stations in New Zealand and the United States, another method was tested, based on the recording of P-wave intakes from individual seismic tremors within the estimated area of anomalous velocities occurring in the Earth’s core. All processes occurring in the core of the Earth are similar to those in the Sun, where there is a vortex twisting of magnetic drills, emissions of plasma flows, the release of elementary charged particles, etc., but without thermonuclear fusion. Shock waves of high-energy boiling magma generate high amplitude perturbations in the Earth’s surface, which contributes to strong earthquakes, which are accompanied by aftershocks. To determine these processes, it is possible to assume the Earth’s layers as an infinitely piecewise homogeneous two-layer plate, on which the normal gravitational load and internal pressure from the yard act, and to describe by mathematical modeling. In real conditions, the destruction of rocks is usually accompanied by shock loads, which creates P- and S-waves. P-waves determine the magnitude, and S- the intensity of the earthquake. There are some works of algorithms where the P-wave spectrum is used for at least four (Allen, 2003) and (Kanamori, 2005) stations. Let us consider the solution of oscillations of an infinite two-layer plate under the action of a normal load, the shock wave of the occurrence of volcanism, applied to the surface of this plate, as on the earth layers. The effect of normal load on an infinite piecewise homogeneous two-layer plate This work considers that mathematical modeling will help improve the model of events and unravel the relationship of seismic signatures of tremors. This method of modeling will create a three-dimensional image of the Earth’s crust and show the interactions between tectonic plates, as the forces that create and push the fault change over time. To do this, seismic terrain data must be entered so that the model matches observations of how the plate deforms before and during and after an earthquake. This will help draw conclusions about what forces act on the plate boundary, how the plate deforms, and transmitting oscillation information where at the point one plate dives into the Earth hot viscous mantle is. In this semi-molten layer, solid rocks ooze and behave in unexpected ways, so understanding the overall dynamics of the core state can help determine the relationship between stresses along the fault before and after an earthquake. The problem of the effect of moving loads on structures and their elements arises from the core of the earth by the shock force of boiling magma. Let a travelling wave spread along the X-axis with a constant speed along the surface of a piecewise homogeneous two-layer plate with a normal load of the form fz = F(x + V0t); fxy = fxy = 0. (1) In this case, the condition F(ζ) = 0 at ζ = 0. In this problem, the initial condition is absent, since this problem is for a flat plate. Due to the external influence of the form (1), the stress-strain states of the plate from the coordinate y does not depend. The problem is reduced to solving the approximate equation for lateral displacement W points of the contact plane of the two-layer plate obtained in the first chapter [1; 3] (2) where is the coefficient Qj are determined by formulas (2). Since there are no initial conditions in the problem posed, it is necessary to look for a general simple solution to equation (2), passing to the moving coordinates associated with the moving coordinate system by the well-known Galileo’s transformation ξ = x + V0t. Then equation (2) goes over to the ordinary differential equation (3) The general solution of equation (3) is sought in the form (4) where ξ - dimensionless frequency. We introduce dimensionless parameters: (5) where h0, h1 - the thicknesses of the upper and lower layers of the plate are respectively; V0, V1 - approximate Poisson’s ratios of the upper and lower layers of the plate; μ0, μ1 - approximate Lame coefficients; b1 - the speed of spreading cross waves. The characteristic equation differential equation (3) has the form A1ξ6 + A2ξ4 = 0, (6) where are the coefficients A1, A2 and Q′j equal to: (7) and Since the first four roots of the algebraic equation (6) ξ1 = ξ2 = ξ3 = ξ4 = 0, and the roots ξ5 and ξ6 imaginary. Therefore, the general solution of the homogeneous equation (3) is equal to (9) Similarly, the general solution of the inhomogeneous equation (3) is W0 = Whomo + Wpart, where Wpart - a particular solution of the inhomogeneous equation and we find depending on the type of function of the external influence. If the right side of equation (3) is equal to (10) then a particular solution to equation (3) is represented as (11) Then, the general solution of differential equation (3) will be (12) where To determine the constant coefficients, Cj we use the boundary conditions for the case V0 > a0, which have the form: (14) (15) Substituting a general solution of the inhomogeneous differential equation (9) into the boundary conditions (11), we obtain (16) From the condition of bounded oscillations at infinity, where the coefficients C2 = C3 = C4 = 0. Thus, the solution to the problem of oscillation of an infinitely two-layer plate when exposed to a moving load has the form: (17) The impacts emanating from the core of the Earth from occurring volcanism, creating travelling waves in the earth’s crust, are described by the final formula (17) given above. The mathematical concept of interpretation can be applied to the concept of occurring events in the core for determining the shock force P-waves, intensity S- and location when predicting natural disasters on Earth. For an elastic two-layer plate, the sixth-order operator splits into the product of the second-longitudinal and fourth-order transverse-wave operators if the thicknesses of the plate components satisfy the derived equation containing the parameters of these components. As well as the mathematical concept of interpretation, the concepts of occurring events in the nucleus - determining the shock force P-waves, intensity S-waves and the place in predicting natural disasters on Earth can describe the concepts. Thus, the above model can describe the propagating shock force of a wave occurring earthquake from volcanism. To understand the concept and prediction of natural disasters, it is necessary to consider the processes taking place in a single time, this mechanism of action allows you to approach the development of reliable principles for predicting the place and time of disasters in the earth’s crust. Conclusion As a result of overvoltage of layers and pressures in the earth’s core a system of cracks and voids appear in the layers, they expand, and various gases rush into them. Gases instantly heat up, heat and melt the earth’s rocks, merging them into the core before the eruption of the volcano. P-waves, S-waves, and travelling waves appear, with the simultaneous origin of radioactive degassing on the surface of the earth’s crust.

About the authors

Rustam Kh. Rakhimov

Institute of Materials Science, SPA “Physics-Sun”, Academy of Science of Uzbekistan

Email: rustam-shsul@yandex.com
Dr. Sci. (Eng.); Head at the Laboratory No. 1 Tashkent, Republic of Uzbekistan

Mmatmatisa L. Jalilov

Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-Khorazmiy

Email: mamatiso2015@yandex.ru
Jalilov, Cand. Sci. (Eng.); Head at the Department “Computer Systems” Fergana, Republic of Uzbekistan

Asatulla U. Makhsudov

Institute of Materials Science, SPA “Physics-Sun”, Academy of Science of Uzbekistan

Email: asaduz50@rambler.ru
senior researcher Tashkent, Republic of Uzbekistan

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