Abstract

Here processes of wave propagation in a two-component Biot’s medium are considered, which are generated by arbitrary forces actions. By using Fourier transformation of generalized functions, a fundamental solution, Green tensor, of motion equations of this medium has been constructed in a non-stationary case and in the case of stationary harmonic oscillation. These tensors describe the processes of wave propagation (in spaces of dimensions 1, 2, 3) under an action of power sources concentrated at coordinates origin, which are described by a singular delta-function. Based on them, generalized solutions of these equations are constructed under the action of various sources of periodic and non-stationary perturbations, which are described by both regular and singular generalized functions. For regular acting forces, integral representations of solutions are given that can be used to calculate the stress-strain state of a porous water-saturated medium.

Highlights

  • The processes of wave propagation are most studied in elastic media

  • Based on the Fourier transformation of generalized functions, we constructed fundamental solutions of oscillation equations of Biot’s medium. It is Green tensor, which describes the process of propagation of harmonic waves at a fixed frequency in the space–time of dimension N = 1, 2, 3, under the action concentrated at the coordinates origin

  • Bfj 1⁄4 ρ22υfj, bsj 1⁄4 ρ11υjs: This form is very convenient for constructing originals of Green tensor

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Summary

Introduction

Various mathematical models of deformable solid mechanics are used to study the seismic processes of earth’s crust. The class of solved tasks to them is very limited and mainly associated with the construction and study of particular solutions of these equations based on methods of full and partial separation of variables and theory of special functions in Mathematical Theorems - Boundary Value Problems and Approximations the works of Rakhmatullin, Saatov, Filippov, Artykov [6, 7], Erzhanov, Ataliev, Alexeyeva, Shershnev [8, 9], etc. Based on the Fourier transformation of generalized functions, we constructed fundamental solutions of oscillation equations of Biot’s medium It is Green tensor, which describes the process of propagation of harmonic waves at a fixed frequency in the space–time of dimension N = 1, 2, 3, under the action concentrated at the coordinates origin. They can be used to calculate the stress-strain state of a porous water-saturated medium by seismic wave propagation

The parameters and motion equations of a two-component Biot’s medium
Biot’s law for stresses σij 1⁄4 Àλ∂kusk þ Q∂kufkÁδij þ μÀ∂iusj þ ∂ jusiÁ (2) σ 1⁄4 Àmp 1⁄4 R∂kufk þ Q∂kusk
Problems of periodic oscillations of Biot’s medium
Green tensor of Biot’s equations by stationary oscillations
Fourier transform of fundamental solutions
Stationary Green tensor construction: radiation conditions
Generalized solutions by arbitrary periodic forces
Green tensor of Biot’s equations by non-stationary motion
Generalized solutions of Biot’s equations by non-stationary forces
10. Calculation of the stress state of Biot’s medium
11. Conclusion

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