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

Kukarskikh, L. A.

Nitsak, D. A.

Polenov, V. S.

Ural mathematical journal

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 130–136DOI: 10.15826/umj.2020.1.010THE DYNAMIC DEFORMATIONOF THREE-COMPONENT POROUS MEDIAVictor S. Polenov1, Lyubov A. Kukarskikh2, Dmitry A. Nitsak3Air Force Academy named after Professor N.E. Zhukovsky and Y.A.Gagarin,54a Old Bolsheviks Str., Voronezh, 394064, Russia1polenov.vrn@mail.ru, 2kukarskih.liubov@yandex.ru, 3dima nitsak@mail.ruAbstract: A mathematical model of the dynamic deformation of three-component elastic media saturatedwith liquid and gas, given by elastic moduli and coefficients characterizing the porosity and compressibility ofthe liquid and gas, is considered. Formulas for determining the propagation velocity of monochromatic wavesin ternary porous media are obtained. The existence of three longitudinal waves depends on the discriminantof a cubic equation and the velocity ratio.Keywords: Elasticity, Medium, Fluid, Stress, Deformation, Displacement.1. IntroductionThere 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. Hecreated the theory of elasticity and consolidation of a porous medium. This theory studies settle-ment 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 problemsin 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 classicBiot–Phrenkel model.When solving a considerable number of applied problems arising in various areas of humanactivity (soil, porous sintered composition materials, building materials, etc.), one has to deal witha 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 acceptedmodels (elastic medium—liquid—gas) have not been fully developed. Therefore, a mathematicalthree-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 porousmedium to the propagation velocities of the wave surface of the porous medium in the longitudinaland transverse directions. The interpenetrating motion of the elastic component, liquid and gas isperceived as the motion of liquid, and gas in a deformable porous medium. It is supposed that thepore size is small compared to the distance at which the kinematic and dynamic characteristics ofthe motion change significantly. This allows us to assume that all three media are continuous andthat 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 velocitiesessentially depend on the direction of propagation of the wave surface. Graphs of the dependenceof the velocity ratio on the porosity of the medium are constructed.The Dynamic Deformation of Three-Component Porous Media 1312. Main resultsConsider a system of equations determining the dynamic behavior of a three-component mediumsaturated 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 poresTij = λu(1)k,kδij + µ(u(1)i,j + u(1)j,i ) +mR(2)0 u(2)k,kδij +mR(3)0 u(3)k,kδij ; (2.1)• forces acting on the liquid and gas per unit area of the cross section of the porous medium:N = mR(2)0 u(1)k,k +mR(2)0 u(2)k,k +mR(2)0 u(3)k,k,P = mR(3)0 u(1)k,k+mR(3)0 u(2)k,k+mR(3)0 u(3)k,k;(2.2)• equations of motion of the porous mediaρ11ü(1)i + ρ12ü(2)i + ρ13ü(3)i = Tij,j,ρ21ü(1)i + ρ22ü(2)i + ρ23ü(3)i = N,i,ρ31ü(1)i + ρ32ü(2)i + ρ33ü(3)i = P,i.(2.3)Here λ and µ are the Lamé coefficients; u(α)i are the component displacements, where α = 1, 3stands for the medium: 1 for the rigid component, 2 for the liquid, and 3 for the gas; the dotsabove the letters indicate the time derivatives; indices after the comma below the letter stand forthe derivatives of the corresponding coordinates; δij is the Kronecker symbol; ρ11, ρ22, and ρ33are effective densities of the rigid component, liquid, and gas, respectively; ρ11 < 0, ρ12 < 0, andρ13 < 0 are the coefficients of dynamic coupling of the skeleton, liquid, and gas, respectively; R(2)0and 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 ρij = ρji.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 isan isolated surface on which the stress, the forces acting on the liquid and gas, and the prop-agation velocities of the components are continuous while some of their partial derivatives havediscontinuities.Differentiating relations (2.1) and (2.2) in t, we obtainṪij = λv(1)k,kδij + µ(v(1)i,j + v(1)j,i ) +mR(2)0 v(2)k,kδij +mR(3)0 v(3)k,kδij ,Ṅ = mR(2)0 v(1)k,k+mR(2)0 v(2)k,k+mR(2)0 v(3)k,k,Ṗ = mR(3)0 v(1)k,k +mR(3)0 v(2)k,k +mR(3)0 v(3)k,k.(2.4)Let us write equations (2.3) and relations (2.4) in discontinuities [5, 7, 11, 14]:λ[v(1)k,k]δij + µ([v(1)i,j ] + [v(1)j,i ]) +mR(2)0 [v(2)k,k]δij +mR(3)0 [v(3)k,k]δij = [Ṫij ],mR(2)0 [v(1)k,k] +mR(2)0 [v(2)k,k] +mR(2)0 [v(3)k,k] = [Ṅ ],mR(3)0 [v(1)k,k] +mR(3)0 [v(2)k,k] +mR(3)0 [v(3)k,k] = [Ṗ ],ρ11[v̇(1)i ] + ρ12[v̇(2)i ] + ρ13[v̇(3)i ] = [Tij,j],ρ21[v̇(1)i ] + ρ22[v̇(2)i ] + ρ23[v̇(3)i ] = [N,i],ρ31[v̇(1)i ] + ρ32[v̇(2)i ] + ρ33[v̇(3)i ] = [P,i],(2.5)132 Victor S. Polenov, Lyubov A. Kukarskikh, Dmitry A. Nitsakwhere [·] denotes the difference in the values of a function on different sides of the discontinuitysurface.We apply kinematic and geometric consistency conditions of first-order to relations (2.5) on thediscontinuity surface:[Tik,k] = sikνk, [Ṫik] = −sikG, [N,k] = ηνk, [Ṅ ] = −ηG,[P,i] = γνi, [Ṗ ] = −γG, [v(α)i,k] = λ(α)i νk, [v̇(α)i ] = −λ(α)i G.(2.6)Here sik, 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; νi are thecomponents of the unit normal to the wave surface; and G is the propagation velocity of the wavesurface of the porous medium.Using conditions (2.6), we write formulas (2.5) in the formλλ(1)k νkδij + µ(λ(1)i νj + λ(1)j νi) +mR(2)0 λ(2)k νkδij +mR(3)0 λ(3)k νkδij = −sijG,mR(2)0 λ(1)k νk +mR(2)0 λ(2)k νk +mR(2)0 λ(3)k νk = −ηG,mR(3)0 λ(1)k νk +mR(3)0 λ(2)k νk +mR(3)0 λ(3)k νk = −γG,ρ11λ(1)i G+ ρ12λ(2)i G+ ρ13λ(3)i G = −sijνj,ρ12λ(1)i G+ ρ22λ(2)i G+ ρ23λ(3)i G = −ηνi,ρ13λ(1)i G+ ρ23λ(2)i G+ ρ33λ(3)i G = −γνi.(2.7)Excluding the values sij, η, and γ from (2.7), we get a homogeneous system for λ(1)k , λ(2)k ,and λ(3)k:λλ(1)k νkνi + µ(λ(1)i + λ(1)j νiνj) +mR(2)0 λ(2)k νkνi +mR(3)0 λ(3)k νkνi == ρ11G2λ(1)i + ρ12G2λ(2)i + ρ13G2λ(3)i ,mR(2)0 λ(1)k νkνi +mR(2)0 λ(2)k νkνi +mR(2)0 λ(3)k νkνi == ρ12G2λ(1)i + ρ22G2λ(2)i + ρ23G2λ(3)i ,mR(3)0 λ(1)kνkνi +mR(3)0 λ(2)kνkνi +mR(3)0 λ(3)kνkνi == ρ13G2λ(1)i + ρ23G2λ(2)i + ρ33G2λ(3)i .(2.8)Similar to [8], system (2.8) enables deriving formulas for determining the velocity of longitudinaland transverse waves in the three-component porous media.We find propagation velocities of longitudinal waves assuming that λ(α)k νk 6= 0 on the wavesurface. Reducing (2.8) by νi and summing over the repeated index i, we obtain the homogeneoussystem of three linear equations for ωα = λ(α)i νi:(Λ− ρ11G2l )ω1 + (mR(2)0 − ρ12G2l )ω2 + (mR(3)0 − ρ13G2l )ω3 = 0,(mR(2)0 − ρ12G2l )ω1 + (mR(2)0 − ρ22G2l )ω2 + (mR(2)0 − ρ23G2l )ω3 = 0,(mR(3)0 − ρ13G2l )ω1 + (mR(3)0 − ρ23G2l )ω2 + (mR(3)0 − ρ33G2l )ω3 = 0,(2.9)where Λ = λ+ 2µ.The Dynamic Deformation of Three-Component Porous Media 133Defineσ11 =ΛM, σ22 =mR(2)0M, σ33 =mR(3)0M,σ12 = σ21 = σ23 =mR(2)0M, σ13 = σ31 = σ32 =mR(3)0M,M = Λ+mR(2)0 +mR(3)0 + 3mR(2)0 + 3mR(3)0 ;γij =ρijρ, ρ =∑iρii + 2∑i,j;i 6=j;i<jρij.(2.10)Taking into account (2.10), we write system (2.9) in the dimensionless matrix form:(Σz2l − Γ)−→ω = 0,Σ = {σij} ,Γ = {γij} ,−→ω = {ωi} , (2.11)where z2l = c2l /G2l , c2l = M/ρ; cl are the propagation velocities of the longitudinal waves in theporous media; Gl is the longitudinal component of the propagation velocity the wave surface in theporous medium; and zl is the longitudinal velocity ratio.The condition for system (2.11), homogeneous with respect to ω1, ω2, and ω3, to have a non-trivial solution is that its third order determinant must be zero:∣∣Σz2l − Γ∣∣ = 0. (2.12)It is shown in what follows that condition (2.12) also defines three propagation velocities of thewave surface in the three-component porous medium.Expanding the determinant (2.12), we obtain a cubic equation for z2l :kz6l + bz4l + dz2l + f = 0, (2.13)wherek = σ11(σ22σ33 − σ12σ13) + σ212(σ13 − σ33) + σ213(σ12 − σ22),b = γ11(σ12σ13 − σ22σ33) + γ22(σ213 − σ11σ33) + γ33(σ212 − σ11σ22)−−γ12(σ213 + σ12σ13 − 2σ12σ33)− γ13(σ212 + σ12σ13 − 2σ13σ22) + γ23 [σ11(σ12 + σ13)− 2σ12σ13] ,d = −σ11(γ223 − γ22γ33)− σ22(γ213 − γ11γ33)− σ33(γ212 − γ11γ22)−−2σ12(γ12γ33 − γ13γ23)− 2σ13(γ13γ22 − γ12γ23) + (σ12 + σ13)(γ12γ13 − γ11γ23),f = −(γ11γ22γ33 + 2γ12γ13γ23 − γ11γ223 − γ22γ213 − γ33γ212).We find the solution of the cubic equation (2.13) by the Cardano formulas [4]. Divide (2.13) byk and introduce a new variabley = z2l +b3k.On rearrangement, we gety3 + 3py + 2q = 0, (2.14)where3p =dk−13(bk)2, 2q = 2(b3k)3−bd3k2+fk.Let us calculate the discriminant D = p3 + q2. If D < 0, then (2.14) has three distinct realroots expressed in terms of complex values. If D > 0, then (2.14) has one real and two imaginarysolutions. If D = 0, then there are three real solutions, two of which coincide.134 Victor S. Polenov, Lyubov A. Kukarskikh, Dmitry A. NitsakThus, in the considered three-component porous medium, there are three types of longitudinalwaves can propagate depending on the discriminant of the cubic equation (2.14) and the velocityratios z(α)l.Knowing the propagation velocities cl of the longitudinal waves and the velocity ratios zl, wecan calculate the propagation velocity of the longitudinal wave surface in the three-componentporous media by the formula G(α)l = cl/z(α)l .In the absence of coupling between the liquid—gas and elasticity—gas components, i.e., ifγ13 = 0, γ23 = 0, σ13 = 0, and σ32 = 0, then equation (2.13) takes the form of biquadratic equationwith respect to z2l :k1z4l + b1z2l + d1 = 0, (2.15)wherek1 = σ11σ22 − σ212, b1 = 2σ12γ12 − σ11γ22 − σ22γ11, d1 = γ11γ22 − γ212.Equation (2.15) coincides with the equation from [8].Assume that λ(α)i νi = 0 in (2.8). Under the condition G = Gt, we obtain in the dimensionlessform(σ′11z2t − γ11)ω1 − γ12ω2 − γ13ω3 = 0,γ12ω1 + γ22ω2 + γ23ω3 = 0,γ13ω1 + γ23ω2 + γ33ω3 = 0;σ′11 = µ/M′, M′ = µ+mR(2)0 +mR(3)0 + 3mR(2)0 + 3mR(3)0 ,z2t = c2t /G2t , c2t = M′/ρ.(2.16)For system (2.16) to have a nontrivial solution, its determinant must be zero.Expanding the determinant∣∣∣∣∣∣σ′11z2t − γ11 −γ12 −γ13γ12 γ22 γ23γ13 γ23 γ33∣∣∣∣∣∣,we obtain an expression for determining the ratio of the propagation velocities of the transversewaves in the three-component media:zt =√γ11γ22γ33 + 2γ12γ13γ23 − γ11γ223 − γ22γ213 − γ33γ212σ′11(γ22γ33 − γ223). (2.17)In the absence of coupling between the liquid—gas and elastic—gas components, i.e., if γ23 = 0and γ13 = 0, then (2.17) yieldszt =√γ11γ22 − γ212σ′11γ22. (2.18)Formula (2.18) coincides with the formula obtained in [8].3. Calculation resultsThe figure, using the data in the table, shows the dependencies of the ratio of the propagationvelocity of longitudinal waves in the three-component medium to the propagation velocity of thewave surface in the longitudinal direction on the medium porosity.The Dynamic Deformation of Three-Component Porous Media 135Table 1. Input data for calculating z(α)lm σ11 σ22 σ33 σ12 σ13 {γij}0.20.60.2 0.15 0.08 0.009γ11 = 0.7; γ22 = 0.32;γ33 = 0.1; γ12 = γ13 == γ23 = −0.020.4 0.15 0.1 0.025 0.0250.7 0.15 0.19 0.01 0.010.9 0.1 0.15 0.025 0.025Figure 1. Velocity ratios in the three-component porous mediaIt is seen from the figure that the ratios z(1)l and z(2)l change from 1.4 to 1.9 and from 0.7 to 0.9,respectively. The ratio z(3)l 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 valuesboth more and less than one.4. Conclusion1. In the three-component porous media, three longitudinal and one transverse waves propagatewhose velocities are defined by formulas (2.8) with λ(α)k νk 6= 0 or λ(α)i νi = 0.2. In general, ratios of the longitudinal velocity components in the three-component porousmedium depend on the coefficients and discriminant of a cubic equation.REFERENCES1. Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-FrequencyRange. J. Acoust. Soc. America, 1956. Vol. 28, No. 2. P. 168–178. DOI: 10.1121/1.19082392. Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher FrequencyRange. J. Acoust. Soc. America, 1956. Vol. 28. No. 2. P. 179–191. DOI: 10.1121/1.19082413. Biot M.A. Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys., 1962.Vol. 33. No. 4. P. 1483–1498.4. Bronstein K.A., Semendyaev I. N. Spravochnik po matematike dlya inzhenerov i uchashchihsya vtuzov[Maths Reference for Engineers and University Students]. Moscow, 1964. 608p. (in Russian)136 Victor S. Polenov, Lyubov A. Kukarskikh, Dmitry A. Nitsak5. BykovtsevG. I., VerveikoN.D. O rasprostranenii voln v uprugo-vyazko-plasticheskoi crede [On the prop-agation of waves in an elastic-viscous-plastic medium]. Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1966.No. 4. P. 111–123. (In Russian)6. Frenkel Ya. I. K teorii seysmichescikh i seysmoelectrichescikh yavleny vo vlazhnoi pochve [Toward atheory of seismic and seismoelectric phenomena in the moist soil]. Izv. ANSSR. Ser. Gegr. Geophis[Proceedings of the SSR Academy of Sciences. The Series of Geography and Geophysics], 1944. Vol. 8.No. 4. P. 133–150. (in Russian)7. Kukarskikh L.A., Polenov V. S. Advance of waves in fluid saturated elastic–viscoplastic porous medium.Bull. Voronezh St. Tekh. Univ., 2012. Vol. 9, No. 8. P. 57–60. (in Russian)8. Maslikova T. I., Polenov V. S. The propagation of unsteady elastic waves in homogeneous porous media.Mech. Solids., 2005. Vol. 40, No. 1. P. 86–89.9. Polenov V. S. Osnovnye uravneniya dinamichesckogo demphirovaniya trekhkomponentnykh sred [Thebasic equations of the dynamic deformation of three-component media]. Sovremennye tendentsii razvi-tiya estesvoznaniya i tekhnicheskikh nayk: Sbornik nauchnykh trudov po materialam MNPK. Belgorod[Modern trends in the development of natural sciences and technical sciences: A collection of scientificpapers based on materials of the IPPC. Belgorod], 2018. P. 31–34. (in Russian)10. Polenov V. S., Chigarev A.V. Mathematical modeling of shock waves in inhomogeneous viscoelastic two–component media. Appl. Math. Phys., 2018. Vol. 6, No. 5. P. 997–1005. DOI: 10.4236/jamp.2018.6508611. Polenov V. S. Nitsak D.A. Metematicheskoe modelirovanie akusticheskoi emissii v nasyschennych zhid-kostew dvukhkomponentnykh sredach. In: Proc. XI Int. Sci. Pract. Conf. “Science of Russia: goalsand objectives”. Part 2. Oct. 10, 2018, Ekaterinburg, Russia. Ekaterinburg: NIC “L-ZHurnal”, 2018.P. 52–58. DOI: 10.18411/sr-10-10-2018-33 (in Russian)12. Rakhmatulin Kh.A. Gazovaya i volnovaya dinamika [Gas and Wave Dynamics]. M: MSU, 1983. 200 p.(in Russian)13. Reznichenko Yu.V. O rasprostranenii seysmichescikh voln v discretnykh i heterogennykh sredakh [Onthe propagation of seismic waves in discrete and heterogeneous media]. Izv. Akad. Nauk SSSR. Ser.Geogr. Geophys., 1949. Vol. 13. P. 115–128. (in Russian)14. Thomas T.Y. Plastic Flow and Fracture in Solids. New York: Acad. Press, 1961. 267 p.

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

Polenov, Victor S.

Kukarskikh, Lyubov A.

Nitsak, Dmitry A.

Ural Mathematical Journal (UMJ)

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 130–136DOI: 10.15826/umj.2020.1.010THE DYNAMIC DEFORMATIONOF THREE-COMPONENT POROUS MEDIAVictor S. Polenov1, Lyubov A. Kukarskikh2, Dmitry A. Nitsak3Air Force Academy named after Professor N.E. Zhukovsky and Y.A.Gagarin,54a Old Bolsheviks Str., Voronezh, 394064, Russia1polenov.vrn@mail.ru, 2kukarskih.liubov@yandex.ru, 3dima nitsak@mail.ruAbstract: A mathematical model of the dynamic deformation of three-component elastic media saturatedwith liquid and gas, given by elastic moduli and coefficients characterizing the porosity and compressibility ofthe liquid and gas, is considered. Formulas for determining the propagation velocity of monochromatic wavesin ternary porous media are obtained. The existence of three longitudinal waves depends on the discriminantof a cubic equation and the velocity ratio.Keywords: Elasticity, Medium, Fluid, Stress, Deformation, Displacement.1. IntroductionThere 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. Hecreated the theory of elasticity and consolidation of a porous medium. This theory studies settle-ment 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 problemsin 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 classicBiot–Phrenkel model.When solving a considerable number of applied problems arising in various areas of humanactivity (soil, porous sintered composition materials, building materials, etc.), one has to deal witha 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 acceptedmodels (elastic medium—liquid—gas) have not been fully developed. Therefore, a mathematicalthree-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 porousmedium to the propagation velocities of the wave surface of the porous medium in the longitudinaland transverse directions. The interpenetrating motion of the elastic component, liquid and gas isperceived as the motion of liquid, and gas in a deformable porous medium. It is supposed that thepore size is small compared to the distance at which the kinematic and dynamic characteristics ofthe motion change significantly. This allows us to assume that all three media are continuous andthat 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 velocitiesessentially depend on the direction of propagation of the wave surface. Graphs of the dependenceof the velocity ratio on the porosity of the medium are constructed.The Dynamic Deformation of Three-Component Porous Media 1312. Main resultsConsider a system of equations determining the dynamic behavior of a three-component mediumsaturated 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 poresTij = λu(1)k,kδij + µ(u(1)i,j + u(1)j,i ) +mR(2)0 u(2)k,kδij +mR(3)0 u(3)k,kδij ; (2.1)• forces acting on the liquid and gas per unit area of the cross section of the porous medium:N = mR(2)0 u(1)k,k +mR(2)0 u(2)k,k +mR(2)0 u(3)k,k,P = mR(3)0 u(1)k,k+mR(3)0 u(2)k,k+mR(3)0 u(3)k,k;(2.2)• equations of motion of the porous mediaρ11u¨(1)i + ρ12u¨(2)i + ρ13u¨(3)i = Tij,j,ρ21u¨(1)i + ρ22u¨(2)i + ρ23u¨(3)i = N,i,ρ31u¨(1)i + ρ32u¨(2)i + ρ33u¨(3)i = P,i.(2.3)Here λ and µ are the Lame´ coefficients; u(α)i are the component displacements, where α = 1, 3stands for the medium: 1 for the rigid component, 2 for the liquid, and 3 for the gas; the dotsabove the letters indicate the time derivatives; indices after the comma below the letter stand forthe derivatives of the corresponding coordinates; δij is the Kronecker symbol; ρ11, ρ22, and ρ33are effective densities of the rigid component, liquid, and gas, respectively; ρ11 < 0, ρ12 < 0, andρ13 < 0 are the coefficients of dynamic coupling of the skeleton, liquid, and gas, respectively; R(2)0and 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 ρij = ρji.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 isan isolated surface on which the stress, the forces acting on the liquid and gas, and the prop-agation velocities of the components are continuous while some of their partial derivatives havediscontinuities.Differentiating relations (2.1) and (2.2) in t, we obtainT˙ij = λv(1)k,kδij + µ(v(1)i,j + v(1)j,i ) +mR(2)0 v(2)k,kδij +mR(3)0 v(3)k,kδij ,N˙ = mR(2)0 v(1)k,k+mR(2)0 v(2)k,k+mR(2)0 v(3)k,k,P˙ = mR(3)0 v(1)k,k +mR(3)0 v(2)k,k +mR(3)0 v(3)k,k.(2.4)Let us write equations (2.3) and relations (2.4) in discontinuities [5, 7, 11, 14]:λ[v(1)k,k]δij + µ([v(1)i,j ] + [v(1)j,i ]) +mR(2)0 [v(2)k,k]δij +mR(3)0 [v(3)k,k]δij = [T˙ij ],mR(2)0 [v(1)k,k] +mR(2)0 [v(2)k,k] +mR(2)0 [v(3)k,k] = [N˙ ],mR(3)0 [v(1)k,k] +mR(3)0 [v(2)k,k] +mR(3)0 [v(3)k,k] = [P˙ ],ρ11[v˙(1)i ] + ρ12[v˙(2)i ] + ρ13[v˙(3)i ] = [Tij,j],ρ21[v˙(1)i ] + ρ22[v˙(2)i ] + ρ23[v˙(3)i ] = [N,i],ρ31[v˙(1)i ] + ρ32[v˙(2)i ] + ρ33[v˙(3)i ] = [P,i],(2.5)132 Victor S. Polenov, Lyubov A. Kukarskikh, Dmitry A. Nitsakwhere [·] denotes the difference in the values of a function on different sides of the discontinuitysurface.We apply kinematic and geometric consistency conditions of first-order to relations (2.5) on thediscontinuity surface:[Tik,k] = sikνk, [T˙ik] = −sikG, [N,k] = ηνk, [N˙ ] = −ηG,[P,i] = γνi, [P˙ ] = −γG, [v(α)i,k] = λ(α)i νk, [v˙(α)i ] = −λ(α)i G.(2.6)Here sik, 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; νi are thecomponents of the unit normal to the wave surface; and G is the propagation velocity of the wavesurface of the porous medium.Using conditions (2.6), we write formulas (2.5) in the formλλ(1)k νkδij + µ(λ(1)i νj + λ(1)j νi) +mR(2)0 λ(2)k νkδij +mR(3)0 λ(3)k νkδij = −sijG,mR(2)0 λ(1)k νk +mR(2)0 λ(2)k νk +mR(2)0 λ(3)k νk = −ηG,mR(3)0 λ(1)k νk +mR(3)0 λ(2)k νk +mR(3)0 λ(3)k νk = −γG,ρ11λ(1)i G+ ρ12λ(2)i G+ ρ13λ(3)i G = −sijνj,ρ12λ(1)i G+ ρ22λ(2)i G+ ρ23λ(3)i G = −ηνi,ρ13λ(1)i G+ ρ23λ(2)i G+ ρ33λ(3)i G = −γνi.(2.7)Excluding the values sij, η, and γ from (2.7), we get a homogeneous system for λ(1)k , λ(2)k ,and λ(3)k:λλ(1)k νkνi + µ(λ(1)i + λ(1)j νiνj) +mR(2)0 λ(2)k νkνi +mR(3)0 λ(3)k νkνi == ρ11G2λ(1)i + ρ12G2λ(2)i + ρ13G2λ(3)i ,mR(2)0 λ(1)k νkνi +mR(2)0 λ(2)k νkνi +mR(2)0 λ(3)k νkνi == ρ12G2λ(1)i + ρ22G2λ(2)i + ρ23G2λ(3)i ,mR(3)0 λ(1)kνkνi +mR(3)0 λ(2)kνkνi +mR(3)0 λ(3)kνkνi == ρ13G2λ(1)i + ρ23G2λ(2)i + ρ33G2λ(3)i .(2.8)Similar to [8], system (2.8) enables deriving formulas for determining the velocity of longitudinaland transverse waves in the three-component porous media.We find propagation velocities of longitudinal waves assuming that λ(α)k νk 6= 0 on the wavesurface. Reducing (2.8) by νi and summing over the repeated index i, we obtain the homogeneoussystem of three linear equations for ωα = λ(α)i νi:(Λ− ρ11G2l )ω1 + (mR(2)0 − ρ12G2l )ω2 + (mR(3)0 − ρ13G2l )ω3 = 0,(mR(2)0 − ρ12G2l )ω1 + (mR(2)0 − ρ22G2l )ω2 + (mR(2)0 − ρ23G2l )ω3 = 0,(mR(3)0 − ρ13G2l )ω1 + (mR(3)0 − ρ23G2l )ω2 + (mR(3)0 − ρ33G2l )ω3 = 0,(2.9)where Λ = λ+ 2µ.The Dynamic Deformation of Three-Component Porous Media 133Defineσ11 =ΛM, σ22 =mR(2)0M, σ33 =mR(3)0M,σ12 = σ21 = σ23 =mR(2)0M, σ13 = σ31 = σ32 =mR(3)0M,M = Λ+mR(2)0 +mR(3)0 + 3mR(2)0 + 3mR(3)0 ;γij =ρijρ, ρ =∑iρii + 2∑i,j;i 6=j;i<jρij.(2.10)Taking into account (2.10), we write system (2.9) in the dimensionless matrix form:(Σz2l − Γ)−→ω = 0,Σ = {σij} ,Γ = {γij} ,−→ω = {ωi} , (2.11)where z2l = c2l /G2l , c2l = M/ρ; cl are the propagation velocities of the longitudinal waves in theporous media; Gl is the longitudinal component of the propagation velocity the wave surface in theporous medium; and zl is the longitudinal velocity ratio.The condition for system (2.11), homogeneous with respect to ω1, ω2, and ω3, to have a non-trivial solution is that its third order determinant must be zero:∣∣Σz2l − Γ∣∣ = 0. (2.12)It is shown in what follows that condition (2.12) also defines three propagation velocities of thewave surface in the three-component porous medium.Expanding the determinant (2.12), we obtain a cubic equation for z2l :kz6l + bz4l + dz2l + f = 0, (2.13)wherek = σ11(σ22σ33 − σ12σ13) + σ212(σ13 − σ33) + σ213(σ12 − σ22),b = γ11(σ12σ13 − σ22σ33) + γ22(σ213 − σ11σ33) + γ33(σ212 − σ11σ22)−−γ12(σ213 + σ12σ13 − 2σ12σ33)− γ13(σ212 + σ12σ13 − 2σ13σ22) + γ23 [σ11(σ12 + σ13)− 2σ12σ13] ,d = −σ11(γ223 − γ22γ33)− σ22(γ213 − γ11γ33)− σ33(γ212 − γ11γ22)−−2σ12(γ12γ33 − γ13γ23)− 2σ13(γ13γ22 − γ12γ23) + (σ12 + σ13)(γ12γ13 − γ11γ23),f = −(γ11γ22γ33 + 2γ12γ13γ23 − γ11γ223 − γ22γ213 − γ33γ212).We find the solution of the cubic equation (2.13) by the Cardano formulas [4]. Divide (2.13) byk and introduce a new variabley = z2l +b3k.On rearrangement, we gety3 + 3py + 2q = 0, (2.14)where3p =dk−13(bk)2, 2q = 2(b3k)3−bd3k2+fk.Let us calculate the discriminant D = p3 + q2. If D < 0, then (2.14) has three distinct realroots expressed in terms of complex values. If D > 0, then (2.14) has one real and two imaginarysolutions. If D = 0, then there are three real solutions, two of which coincide.134 Victor S. Polenov, Lyubov A. Kukarskikh, Dmitry A. NitsakThus, in the considered three-component porous medium, there are three types of longitudinalwaves can propagate depending on the discriminant of the cubic equation (2.14) and the velocityratios z(α)l.Knowing the propagation velocities cl of the longitudinal waves and the velocity ratios zl, wecan calculate the propagation velocity of the longitudinal wave surface in the three-componentporous media by the formula G(α)l = cl/z(α)l .In the absence of coupling between the liquid—gas and elasticity—gas components, i.e., ifγ13 = 0, γ23 = 0, σ13 = 0, and σ32 = 0, then equation (2.13) takes the form of biquadratic equationwith respect to z2l :k1z4l + b1z2l + d1 = 0, (2.15)wherek1 = σ11σ22 − σ212, b1 = 2σ12γ12 − σ11γ22 − σ22γ11, d1 = γ11γ22 − γ212.Equation (2.15) coincides with the equation from [8].Assume that λ(α)i νi = 0 in (2.8). Under the condition G = Gt, we obtain in the dimensionlessform(σ′11z2t − γ11)ω1 − γ12ω2 − γ13ω3 = 0,γ12ω1 + γ22ω2 + γ23ω3 = 0,γ13ω1 + γ23ω2 + γ33ω3 = 0;σ′11 = µ/M′, M′ = µ+mR(2)0 +mR(3)0 + 3mR(2)0 + 3mR(3)0 ,z2t = c2t /G2t , c2t = M′/ρ.(2.16)For system (2.16) to have a nontrivial solution, its determinant must be zero.Expanding the determinant ∣∣∣∣∣∣σ′11z2t − γ11 −γ12 −γ13γ12 γ22 γ23γ13 γ23 γ33∣∣∣∣∣∣ ,we obtain an expression for determining the ratio of the propagation velocities of the transversewaves in the three-component media:zt =√γ11γ22γ33 + 2γ12γ13γ23 − γ11γ223 − γ22γ213 − γ33γ212σ′11(γ22γ33 − γ223). (2.17)In the absence of coupling between the liquid—gas and elastic—gas components, i.e., if γ23 = 0and γ13 = 0, then (2.17) yieldszt =√γ11γ22 − γ212σ′11γ22. (2.18)Formula (2.18) coincides with the formula obtained in [8].3. Calculation resultsThe figure, using the data in the table, shows the dependencies of the ratio of the propagationvelocity of longitudinal waves in the three-component medium to the propagation velocity of thewave surface in the longitudinal direction on the medium porosity.The Dynamic Deformation of Three-Component Porous Media 135Table 1. Input data for calculating z(α)lm σ11 σ22 σ33 σ12 σ13 {γij}0.20.60.2 0.15 0.08 0.009γ11 = 0.7; γ22 = 0.32;γ33 = 0.1; γ12 = γ13 == γ23 = −0.020.4 0.15 0.1 0.025 0.0250.7 0.15 0.19 0.01 0.010.9 0.1 0.15 0.025 0.025Figure 1. Velocity ratios in the three-component porous mediaIt is seen from the figure that the ratios z(1)l and z(2)l change from 1.4 to 1.9 and from 0.7 to 0.9,respectively. The ratio z(3)l 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 valuesboth more and less than one.4. Conclusion1. In the three-component porous media, three longitudinal and one transverse waves propagatewhose velocities are defined by formulas (2.8) with λ(α)k νk 6= 0 or λ(α)i νi = 0.2. In general, ratios of the longitudinal velocity components in the three-component porousmedium depend on the coefficients and discriminant of a cubic equation.REFERENCES1. Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-FrequencyRange. J. Acoust. Soc. America, 1956. Vol. 28, No. 2. P. 168–178. DOI: 10.1121/1.19082392. Biot M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher FrequencyRange. J. Acoust. Soc. America, 1956. Vol. 28. No. 2. P. 179–191. DOI: 10.1121/1.19082413. Biot M.A. Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys., 1962.Vol. 33. No. 4. P. 1483–1498.4. Bronstein K.A., Semendyaev I. N. 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THE DYNAMIC DEFORMATION OF THREE-COMPONENT POROUS MEDIA

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The Dynamic Deformation of Three-Component Porous Media

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