THE DYNAMIC DEFORMATION OF THREE-COMPONENT POROUS MEDIA

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 130-136

DOI: 10.15826/umj.2020.1.010

THE DYNAMIC DEFORMATION OF THREE-COMPONENT POROUS MEDIA

Victor S. Polenov1, Lyubov A. Kukarskikh2, Dmitry A. Nitsak3

Air Force Academy named after Professor N. E. Zhukovsky and Y. A. Gagarin, 54a Old Bolsheviks Str., Voronezh, 394064, Russia

polenov. vrn@mail.ru, 2kukarskih. liubov@yandex.ru, 3dima_nitsak@mail.ru

Abstract: 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.

There are a number of papers [1-3, 8, 10] devoted to the propagation of elastic waves in two-component porous media. Among these studies, papers of M. A. Biot [1-3] 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.

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

Tij = + + U<S) (2.1)

forces acting on the liquid and gas per unit area of the cross section of the porous medium:

N = wit,; </,&, + mil,: „,", + mlC "u, P = uilC + IRlC + &"/C equations of motion of the porous media

P11U(1) + P12U(2) + P13U(3) = Tijj,

P2iu(1} + P22u(2) + P23u(3) = Ni, (2.3)

••(1) , --(2) . -(3) D

P31Ui + P32Ui + P33Ui = P,i.

Here A and fi are the Lamé coefficients; u^ are the component displacements, where a = 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; ôij is the Kronecker symbol; P11, p22, and p33

are effective densities of the rigid component, liquid, and gas, respectively; p11 < 0, p12 < 0, and

p13 < 0 are the coefficients of dynamic coupling of the skeleton, liquid, and gas, respectively; R0

and R0 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 pij = pji. 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 obtain

Tij = Xv^Sij + rtv™

N = mR^vill + mlh2)v™ + m^v™, (2.4)

P = mR^v^l + mR^v^l + mi^g.

Let us write equations (2.3) and relations (2.4) in discontinuities [5, 7, 11, 14]:

+ [4ÏD = [TiA,

mR{o][vPk] + mi^g] +mR^[vPk] = [TV], mR^ [t;$] + mR^ [t;$] + m.R^ [t$] = [P], (2

P11[vv(1) ]+ P12 [vV(2) ] + P13[W(3) ] = [Tij-j ], P21[Wi(1)]+ P22 [vv(2) ] + P23[v)i(3)] = [N,i], P31[vv(1) ] + P32[Wi(2)]+ P33[v)((3) ] = [P,i ],

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:

[Tik,k] = SikVk, [Tik] = -SikG, [N,k] = nvk, [N] = -nG, ( )

[P,i] = YVi, [P] = -YG, [vg] = A(a)Vk, [vf)] = -A^G. (2.6)

Here sik, n, Y, and A(a) are values characterizing jumps of the first derivatives of stresses, forces acting on the liquid and gas, and the propagation velocities of the components; vi 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

A A;.& + niX^Vj + A^z/j) + mR{02) X^ vk6ij + mR{03) A[3) vk6ij = -SijG, m42)X[}]uk + mil,: X; i>h. ■ mil,: A,:&& //,. = -r]G, mR^ X[}] vk + mR^ A^ vk + rnR^ xf vk = -7G, PnA(1) G + P12A(2)G + P13A(3)G = -Sij v3 , P12A(1)G + P22A(2)G + P23A(3)G = -nVi,

P13A(1)G + P23A(2)G + P33A(3)G = -YVi.

(1) \\(2)

Excluding the values sij, n, and y from (2.7), we get a homogeneous system for Ak), A^

XX,1 I&I,. I&: + p(A^1} + A {pViVj) + mR^2) xf] vkUi + mR{03) X^ vkvi = = P11 G2A( ) + P12G2A( ) + P13G2A( ), mR{02) a[1} vkvi + mR^2) X{2) vkUi + mR{02) xf] vkvi =

= Pi2G2Af)+p22G2Af+p23G2Af), mR{03) vkvi + mRf] X[2) vkvi + mR^ xf] vkVi =

= Pi3G2Af)+p23G2Af)+p33G2Af).

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 Aka)vk = 0 on the wave surface. Reducing (2.8) by vi and summing over the repeated index i, we obtain the homogeneous system of three linear equations for ua = A(a)vi:

(A - piiG2)wi + (mR{2) - pi2G2)w2 + (mR^ - pnGj)^ = 0, (;mR{2) - piaG^wi + (■mR{2) - p22Gf)w2 + (mR^ ~ i^.Gj)^. = 0, (2.9)

tmRf - priGi)uJi + tniRf - p23G&f)uj2 + (mR^ - = 0,

where A = A + 2p.

A mR02) mR03)

ni42) mR^

Cl2 — <721 — <723 — —7T—, <713 — <731 — <732 (2.10)

M = A + rnR^ + mR{03) + SmR^ + 3mR{03)-,

Hj = y, P = Pa + 2 Pvi i,j;i=j ; i<j

Taking into account (2.10), we write system (2.9) in the dimensionless matrix form:

(Szf - r)<2 = 0, S = {<j} , r = {Yij} , 1 = {wj , (2.11)

where zz2 = c^/G^2, c^ = M/p; cl are the propagation velocities of the longitudinal waves in the porous media; Gi is the longitudinal component of the propagation velocity the wave surface in the porous medium; and zl is the longitudinal velocity ratio.

The condition for system (2.11), homogeneous with respect to Wi, w2, and w3, to have a nontrivial solution is that its third order determinant must be zero:

|Sz2 - r| =0. (2.12)

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.

Expanding the determinant (2.12), we obtain a cubic equation for z2

kzf + bzf + dz2 + f = 0, (2.13)

k = ^11(^22^33 - 712<13) + ^22(^13 - O33) + ^23(^12 - <22), b = 711(^12^13 - <22<33) + Y22(^23 - ^11^33) + Y33(^22 - ^11^22)--712(^23 + 712<13 - 2^12^33) - 713(^22 + 712<13 - 2^13^22) + 723 [<7n(<712 + <13) - 2^ 12^ 13] ,

d = -<11(723 - 722733) - 722(713 - 711733) - 733(712 - 711722)--2712(712733 - 713723) - 2<13(713722 - 712723) + (<12 + <13)(712713 - 711723), f = -(711722733 + 2712713723 - 711723 - 722713 - 733712).

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

y = zt + —y3 + 3py + 2q = 0, (2.14)

d 1 fb\\2 ,, Y—V-— L

On rearrangement, we get where

Let us calculate the discriminant D = p3 + q2. 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.

Thus, in the considered three-component porous medium, there are three types of longitudinal waves can propagate depending on the discriminant of the cubic equation (2.14) and the velocity ratios z(a).

Knowing the propagation velocities cl of the longitudinal waves and the velocity ratios zi, we can calculate the propagation velocity of the longitudinal wave surface in the three-component porous media by the formula G(a) = ci/z(a).

In the absence of coupling between the liquid—gas and elasticity—gas components, i.e., if Y13 = 0, 723 = 0, 013 = 0, and 032 = 0, then equation (2.13) takes the form of biquadratic equation with respect to zp:

fcizf + bizp + di =0, (2.15)

kl = 011022 - 012, bi =2012712 - 011722 - 022711, di = 711722 - 7l2"

Equation (2.15) coincides with the equation from [8].

Assume that = 0 in (2.8). Under the condition G = Gt, we obtain in the dimensionless

(0&izt2 - Yii)wi - 712W2 - 713W3 = 0, 712 Wi + 722W2 + 723 W3 = 0,

a&n = n/M&, M& = ii + >mR(02) + mR(03) + ?ymR(Q) + ZmR^, z2t=cl/G2u c* = M&/p.

For system (2.16) to have a nontrivial solution, its determinant must be zero.

Expanding the determinant

& 2 0iizt2 - Yii -712 -Yi3

Yi3 723 733

we obtain an expression for determining the ratio of the propagation velocities of the transverse waves in the three-component media:

Zt= / 711722733 + 2712713723 ~ 711723 ~ 7227i3 ~ 7337i2 ^

V 0n(722733 -7I3)

In the absence of coupling between the liquid—gas and elastic—gas components, i.e., if y23 = 0 and 713 = 0, then (2.17) yields

* = (2.18) V 011722

Formula (2.18) coincides with the formula obtained in [8].

The figure, using the data in the table, shows the dependencies of the ratio of the propagation velocity of longitudinal waves in the three-component medium to the propagation velocity of the wave surface in the longitudinal direction on the medium porosity.

Table 1. Input data for calculating zl

Figure 1. Velocity ratios in the three-component porous media

It is seen from the figure that the ratios z(1) and z(2) change from 1.4 to 1.9 and from 0.7 to 0.9,

respectively. The ratio z^ demonstrates a weak dependence on the porosity and is close to 1.1. Thus, in the three-component porous media, the ratios of longitudinal velocities can take values both more and less than one.

REFERENCES