THE DYNAMIC DEFORMATION OF THREE-COMPONENT POROUS MEDIA

A mathematical model of the dynamic deformation of three-component elastic media saturated with liquid and gas, given by elastic moduli and coefficients characterizing the porosity and compressibility of the liquid and gas, is considered. Formulas for determining the propagation velocity of monochromatic waves in ternary porous media are obtained. The existence of three longitudinal waves depends on the discriminant of a cubic equation and the velocity ratio.


Introduction
There are a number of papers [1-3, 8, 10] devoted to the propagation of elastic waves in twocomponent porous media. Among these studies, papers of M. A. Biot [1][2][3] should be noted. He created the theory of elasticity and consolidation of a porous medium. This theory studies settlement under the influence of a load of a porous medium containing a viscous fluid.
Phase states, laws of thermodynamics of porous systems, and attempts to solve wave problems in porous materials and moist soils were considered by Ya. I. Phrenkel [6], J. V. Reznichenko [13], and Kh. A. Rakhmatulin [12]. The studies of these authors played a huge role in creating the classic Biot-Phrenkel model.
When solving a considerable number of applied problems arising in various areas of human activity (soil, porous sintered composition materials, building materials, etc.), one has to deal with a three-component media. The complexity of describing the effects of the interaction of components, heat transfer, and other related processes has led to the fact that until now the generally accepted models (elastic medium-liquid-gas) have not been fully developed. Therefore, a mathematical three-component model that takes into account the porosity of the medium is of apparent interest.
The paper considers the ratio of the velocities of acceleration waves in a three-component porous medium to the propagation velocities of the wave surface of the porous medium in the longitudinal and transverse directions. The interpenetrating motion of the elastic component, liquid and gas is perceived as the motion of liquid, and gas in a deformable porous medium. It is supposed that the pore size is small compared to the distance at which the kinematic and dynamic characteristics of the motion change significantly. This allows us to assume that all three media are continuous and that at each point in space there are three displacement vectors.
It is proved that, in such a medium, in the general case, three waves propagate, whose velocities essentially depend on the direction of propagation of the wave surface. Graphs of the dependence of the velocity ratio on the porosity of the medium are constructed.

Main results
Consider a system of equations determining the dynamic behavior of a three-component medium saturated with liquid and gas in the motion of the components [9]: • complete stress tensor in the skeleton in the presence of liquid and gas in pores (2.1) • forces acting on the liquid and gas per unit area of the cross section of the porous medium: • equations of motion of the porous media Here λ and µ are the Lamé coefficients; u (α) i are the component displacements, where α = 1, 3 stands for the medium: 1 for the rigid component, 2 for the liquid, and 3 for the gas; the dots above the letters indicate the time derivatives; indices after the comma below the letter stand for the derivatives of the corresponding coordinates; δ ij is the Kronecker symbol; ρ 11 , ρ 22 , and ρ 33 are effective densities of the rigid component, liquid, and gas, respectively; ρ 11 < 0, ρ 12 < 0, and ρ 13 < 0 are the coefficients of dynamic coupling of the skeleton, liquid, and gas, respectively; R (2) 0 and R (3) 0 are compressibility moduli of the components saturated with liquid and gas, respectively; 0 ≤ m ≤ 1 is the porosity of a medium, m = 1 − m; and i, j, k = 1, 3. Suppose that ρ ij = ρ ji .
Hereinafter, the repeated indices assume a summation of one to three. An acceleration wave in a three-component porous media saturated with a liquid and gas is an isolated surface on which the stress, the forces acting on the liquid and gas, and the propagation velocities of the components are continuous while some of their partial derivatives have discontinuities.
Differentiating relations (2.1) and (2.2) in t, we obtaiṅ Let us write equations (2.3) and relations (2.4) in discontinuities [5,7,11,14]: where [·] denotes the difference in the values of a function on different sides of the discontinuity surface.
We apply kinematic and geometric consistency conditions of first-order to relations (2.5) on the discontinuity surface: Here s ik , η, γ, and λ (α) i are values characterizing jumps of the first derivatives of stresses, forces acting on the liquid and gas, and the propagation velocities of the components; ν i are the components of the unit normal to the wave surface; and G is the propagation velocity of the wave surface of the porous medium.
Using conditions (2.6), we write formulas (2.5) in the form Excluding the values s ij , η, and γ from (2.7), we get a homogeneous system for λ (1) k , λ (2) k , and λ (2.8) Similar to [8], system (2.8) enables deriving formulas for determining the velocity of longitudinal and transverse waves in the three-component porous media.
We find propagation velocities of longitudinal waves assuming that λ (α) k ν k = 0 on the wave surface. Reducing (2.8) by ν i and summing over the repeated index i, we obtain the homogeneous system of three linear equations for ω α = λ (α) i ν i : where Λ = λ + 2µ. Define (2.10) Taking into account (2.10), we write system (2.9) in the dimensionless matrix form: where z 2 l = c 2 l /G 2 l , c 2 l = M/ρ; c l are the propagation velocities of the longitudinal waves in the porous media; G l is the longitudinal component of the propagation velocity the wave surface in the porous medium; and z l is the longitudinal velocity ratio. The condition for system (2.11), homogeneous with respect to ω 1 , ω 2 , and ω 3 , to have a nontrivial solution is that its third order determinant must be zero: It is shown in what follows that condition (2.12) also defines three propagation velocities of the wave surface in the three-component porous medium.
We find the solution of the cubic equation (2.13) by the Cardano formulas [4]. Divide (2.13) by k and introduce a new variable On rearrangement, we get y 3 + 3py + 2q = 0, (2.14) where Let us calculate the discriminant D = p 3 + q 2 . If D < 0, then (2.14) has three distinct real roots expressed in terms of complex values. If D > 0, then (2.14) has one real and two imaginary solutions. If D = 0, then there are three real solutions, two of which coincide.