Sapehin V.N. Using the method of integral fourier transform to solve the problem of nonstationary deformation of an elastic medium

Details

Parent Category: Geo-Technical Mechanics, 2019

Category: Geo-Technical Mechanics, 2019, Issue 147

Geoteh. meh. 2019, 147, 111-120

https://doi.org/10.1051/e3sconf/201910900080

Using the method of integral Fourier transform to solve the problem of nonstationary deformation of an elastic medium

¹SapehinV.N.

¹Institute of Geotechnical Mechanics named by N. Poljakov of National Academy of Sciences of Ukraine

UDC 539.3         

Language: English

Abstract.

For the first time in the article, the method of integral Fourier transform is used to solve the wave equation. As a result of applying the direct and inverse Fourier transforms, analytical expressions are obtained for calculating displacements and stresses in wells under the action of variable internal pressure. Features of the second kind were excluded by the half division method. The calculation of the Fur'ye- Bessel integrals of rapidly oscillating functions was carried out according to the standard program using the Gauss-Kronrod quadrature formula taking into account the periodic system of infinite discontinuity. It is established that the nature of the change in stress over time, after the termination of the force, is oscillatory and decaying. It was found that the shorter the pressure relief time, the greater the tensile stresses on the inner contour of the well cavity, which vary in hyperbolic dependence. For given parameters of the elastic medium with a fall time of less than 0.01 s, a sharp increase in radial and tangential stresses is observed. It is established that the nature of the change in tangential stresses in time after the termination of the force is oscillatory in nature, with the same period of damped oscillations. The greater the value of radial compressive stresses at the time of complete pressure relief, the more accumulated elastic energy passes into tensile radial and tangential stresses. With an increase in the inner radius of the cylindrical cavity in the studied range from 0.05 m to 0.3 m, the radial stresses linearly increase after the release of the internal pressure. Radial tensile stresses are directly proportional to the amplitude of the internal load (pressure) before it is discharged from the well. An analytical solution of the plane problem for calculating displacements and stresses is obtained, which allows them to be calculated in semi-infinite and infinite elastic media. The solution is stable when the ratio of the external and internal radii of the elastic medium is greater than or equal to 1000. In this paper, we consider the initiation phase, which precedes the layered separation of the elastic medium. The reliability and practical value of the proposed calculation method is confirmed by data from mine and laboratory studies obtained by other authors.

Keywords: stress, pressure relief, Fourier integral transform method.

 

REFERENCES

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About the author

Sapehin Volodymyr Mykolaiovych,Candidate of Technical Sciences (Ph.D.), Senior Researcher in Department of Mineral Mining at Great Depths, Institute of Geotechnical Mechanics named by N. Poljakov of National Academy of Sciences of Ukraine(IGTM, NAS of Ukraine), Dnipro, Ukraine, vladimir This email address is being protected from spambots. You need JavaScript enabled to view it.

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