Wave propagation in liquid-saturated porous solid with micropolar elastic skelton at boundary surface

Applied Mathematics and Mechanics (English Edition), 007, 8(3):337 39 c Editorial Committee of Appl. Math. Mech., ISSN 053-87 Wave propagation in liquid-saturated porous solid with micropolar elastic skelton at boundary surface Rajneesh Kumar, Mahabir Barak (Department of Mathematics, Kurukshetra University, Kurukshetra 369, Haryana, India) (Communicated by DAI Tian-min, Original Member of Editorial Commitee, AMM) Abstract The present study is concerned with the reflection and transmission of plane waves at an interface between homogenous invisicid liquid half space and a micropolar liquid-saturated porous solid half space. The reflection and transmission coefficients of various reflected and transmitted waves with the angle of incident have been obtained. Numerical calculation has been performed for amplitude ratios of various reflected and transmitted waves. Micropolarity and porosity effects on the reflection and transmission coefficients have been depicted graphically. Some particular cases have been deduced from the present formulation. Key words micropolar liquid-saturated porous solid, reflection, transmission, amplitude ratios, micropolarity and porosity effects Chinese Library Classification O37 000 Mathematics Subject Classification 7A, 7B, 7F, 7J Digital Object Identifier(DOI) 0.007/s 083-007-0307-z Nomenclature λ, μ, K, α, β, γ material constants for solidliquid aggregate ρ the density of the medium b a dissipation function j the micro inertia ρ,ρ,ρ dynamical coefficients u the displacement vector in the solid part with components u,u,u 3 U the displacement vector in the liquid part with components U,U,U 3 e =divu, ɛ=divu corresponding dilatations φ the microrotational vector Q measure of coupling between volume change of solid and that of the liquid R measure of pressure that must be exerted on the fluid to force a given volume of it into the aggregate while total volume remains constant Introduction Dynamic analysis of liquid-saturated porous media is a subject with applications in numerous branches of science and engineering, including geophysics, seismology, civil and mechanical engineering. Liquid-saturated porous materials are often presented on and below the surface of the earth in the form of sandstone, limestone and other sediments permeated by groundwater or oil. Biot [,] studied the propagation of plane harmonic seismic waves in liquid-saturated porous solids. Biot [3] presented a unified treatment of the mechanics of deformation and acoustic propagation in porous media, and treated the liquid-solid medium as a complex physico-chemical system with resultant relaxation and viscoelastic properties. Deresiewicz [] and Deresiewicz and Rice [5] studied the reflection at the plane traction-free surface of non-dissipative and dissipative liquid-saturated porous solids respectively. Received May 7, 005; Revised Nov.7, 006 Corresponding author Rajneesh Kumar, Professor, Doctor, E-mail: rajneesh kuk@rediffmail.com

338 Rajneesh Kumar and Mahabir Barak The discrepancy between the results of classical theory of elasticity and the experiments appears in all cases when the microstructure of the body is significant i.e., in the neighbourhood of cracks and notches the stress gradients are considerable. The discrepancies also appear in granular media and multimolecular bodies such as polymers. The influence of the microstructure is particularly evident in the case of elastic vibrations of high frequency and small wave length. Eringen and Suhubi [6] and Suhubi and Eringen [7] introduced theories of micropolar elastic solids in which the micromotions of the particle contained in a macrovolume element with respect to its centroid are considered. Materials which are affected by such micromotions and macrodeformations are known as micromorphic materials Eringen [8]. Eringen [9] developed a theory for a subclass of micromorphic materials which are called micropolar media and these materials show microrotation s effect and microrotational inertia. Gauthier [0] revealed that the micropolar waves can be excited and detected in typical solids. Yang and Lakes [] suggested with a reasonable degree of confidence that human bone can adequately be used as a model for theory of micropolar elasticity. Many researchers [ ] studied the problems of wave propagation in micropolar elastic solid. The behavior of plane harmonic waves propagating in a liquid-saturated viscoelastic porous media with micropolar skeleton was studied by Konczak [5]. Konczak [6] also worked on thermo-mechanical effects on liquid-saturated porous viscoelastic media with a micropolar skeleton. Kumar and Singh [7] investigated the reflection and refraction of micropolar elastic waves at an interface between a porous elastic half space and a micropolar liquid-saturated half space. Kumar and Deshwal [8,9] studied the wave propagation in micropolar liquid-saturated porous medium and surface wave propagation in micropolar liquid-saturated porous solid layer under a uniform layer of liquid. Reflection and transmission of elastic waves at viscous liquid/micropolar solid interface has been investigated by Kumar and Tomar [0]. In the present investigation, we study the reflection and transmission of plane waves at an interface of homogenous liquid half space and micropolar liquid-saturated porous solid half space. Basic equations Following Eringen [] and Konczak [5,6] the field equations for a micropolar liquid-saturated porous solid in the presence of dissipation are given by (λ +μ + K) ( u) (μ + K) ( u)+k( φ)+q ( u) = t (ρ u + ρ U)+b (u U), t () (Qe + Rɛ) = t (ρ u + ρ U) b (u U), t () (α + β + γ) ( φ) γ ( φ)+k( u) Kφ = ρj φ t. (3) List of symbols is given in the nomenclature. Assuming time harmonic variations (e it,w is the angular frequency) and considering the Helmholtz s resolution of displacement vectors as u = q + H, H =0 () and U = ψ + G, G =0 (5)

Wave propagation in liquid-saturated porous solid 339 and then eliminating ψ, ψ, φ and φ from the resulting expressions, we obtain the following equations: (A + B + C )q =0, (6) [A + (Rρ ρ Q)+ib(R + Q)]q Fψ =0, (7) ( + D + E )H =0, (8) ( + E + pr )H p( r 0 + r )φ =0, (9) H =( H) y, φ =( φ) y, A = PR Q, ( B = P ρ + i b ) ( + R ρ + i b ) Q ( ρ i b ( C = ρ + i b )( ρ + i b ) ( ρ i b F =(ρ Q ρ R)+ i b (R + Q), D = E + r + (pr r 0 ), ( E = E r r ) ( 0, E = ρ + ρ ρ +ib ρ +ib +ib ρ + ρ ρ +ib r 0 = K γ, r = ρj γ, r = K γ. We assume the solution of Eq.(6) as ), ), P = λ +μ + K, p = K μ + K, ), E = E μ + K, q = q + q, (0) q and q satisfy ( + δ)q =0, ( + δ)q =0, () δ, = λ,, λ, = B (B AC) /. A Hence in an unbounded medium, solutions of Eq.(0) correspond to two coupled longitudinal waves. The wave corresponding to q being the faster one, is called fast longitudinal displacement (or P f ) wave propagating with the phase velocity λ and that corresponding to q being the slower one, is called slow longitudinal displacement (or P s ) wave propagating with the phase velocity λ. With the help of Eqs.(0) and (), from Eq.(7) we have ψ = μ q + μ q, () μ i = F ( Aλ i +(Rρ Qρ )) + i b (R + Q)q i, i =,. Take the solution of Eq.(9) as H = H + H, (3) H and H satisfy ( + δ3 )H =0, ( + δ )H =0, ()

30 Rajneesh Kumar and Mahabir Barak δ3, = λ 3,, λ 3, = D (D E) /. Using Eqs.(3) and () in Eq.(8), we get φ = μ 3 H + μ H, (5) μ 3, = δ 3, (δ 3, E pr ) p( r 0 + r. ) In an unbounded medium, the solutions of Eq.() correspond to two coupled transverse and microrotational waves propagating with velocities λ 3 and λ. Following Eringen[9] and Konczak [5,6] the constitutive relations in micropolar liquid-saturated porous solid are given as t kl =(λu r,r + QU r,r )δ kl + μ(u k,l + u l,k )+K(u l,k ɛ klr φ r ), (6) m kl = αφ r,r δ kl + βφ k,l + γφ l,k, (7) σ = Qe + Rɛ. (8) Formulation of problem We consider a homogeneous inviscid liquid half space (medium M) and a micropolar liquidsaturated porous elastic half space (medium M ) in welded contact along a plane interface. Rectangular Cartesian coordinate system (x, y, z) is chosen with plane interface z =0andzaxis is pointing into medium M. The medium M through which incident takes place occupies the region z>0andtheregionz<0 is occupied by the medium M. The problem is two-dimensional in xz-plane, therefore we take u =(u, 0,u 3 ), (9) φ =(0,φ, 0). (0) For the liquid half space, the equation of motion in terms of displacement potential φ is given by φ y + φ z = φ α t, () α = λ /ρ is the velocity of the liquid. pressure p are given by The displacement components u, u 3 and u = φ x, u 3 = φ z, p = ρ φ z. () We consider micropolar elastic waves (coupled longitudinal (P f or P s ) and coupled transverse and microrotational waves), propagating through the micropolar liquid-saturated porous elastic half space and incident at a plane interface z = 0, with the direction of propagation making an angle θ 0 with the interface. Corresponding to the incident wave, we get reflected P f wave, P s wave and a set of two coupled transverse and microrotational waves in medium M and transmitted longitudinal wave in the medium M. The complete geometry of the problem is shown in Fig..

Wave propagation in liquid-saturated porous solid 3 z = 0 O ' A θ θ θ 0 θ θ3 θ ' x B A' B z B B 3 Fig. The complete geometry of the problem 3 Boundary conditions The boundary conditions are the continuity of the normal stress components, liquid pressure, component of velocity normal to the interface averaged over the bulk area, vanishing of the tangential force stress and couple stress at the interface between the inviscid liquid and the micropolar liquid-saturated porous half space, i.e., at z = 0. The appropriate boundary conditions are (i) (σ zz ) +(σ) =( p) II, (ii) (σ zx ) =0, (iii) (m zy ) =0, (3) (iv) β (σ) =( p ) II, (v) ( β )( u 3 ) + β ( U 3 ) =( u 3 ) II at z =0. The potential functions satisfying the boundary conditions (3) are as follows: (a) For micropolar liquid-saturated porous solid half space (medium M), q = A exp[iδ (x cos θ 0 z sin θ 0 ) i t]+a exp[iδ (x cos θ 0 z sin θ 0 ) i t] B exp[iδ (x cos θ + z sin θ ) i t]+b exp[iδ (x cos θ + z sin θ ) i t], ψ = μ A exp[iδ (x cos θ 0 z sin θ 0 ) i t]+μ B exp[iδ (x cos θ + z sin θ ) i t] + μ A exp[iδ (x cos θ 0 z sin θ 0 ) i t]+μ B exp[iδ (x cos θ + z sin θ ) i t], H = A exp[iδ (x cos θ 0 z sin θ 0 ) i t]+b exp[iδ (x cos θ + z sin θ ) i t] + B 3 exp[iδ 3 (x cos θ 3 + z sin θ 3 ) i 3 t], φ = μ A exp[iδ (x cos θ 0 z sin θ 0 ) i t]+μ B exp[iδ (x cos θ + z sin θ ) i t] + μ 3 B 3 exp[iδ 3 (x cos θ 3 + z sin θ 3 ) i 3 t]; () (b) For liquid half space (Medium M ), φ = A exp[iδ (x cos θ z sin θ ) i t], (5) A, A =0, for incident P f wave; A, A =0, for incident P s wave; A, A =0, for incident coupled transverse and microrotational wave. Snell s law is given as cos θ 0 V 0 = cos θ λ = cos θ λ = cos θ 3 λ 3 = cos θ λ = cos θ α, (6)

3 Rajneesh Kumar and Mahabir Barak δ (λ )=δ (λ )=δ 3(λ 3 )=δ (λ )= = kc, at z =0, (7) λ, for incident P f wave, V 0 = λ, for incident P s wave,, for incident coupled transverse and microrotational wave. λ Making use of potentials given by Eqs.() and (5) and the Snell s law given by Eq.(6), boundary conditions (3) reduce to a system of five non-homogeneous equations represented as 5 a ij z j = b i, i =,, 5, (8) j= a i =[λ + Q + μ i (Q + R)+sin θ i (μ + K)]δi, i =,, a j = (μ + K)δj cos θ j sin θ j, j =3,, a 5 = ρ, a i =(μ + K)δi cos θ i sin θ i, i =,, a j =(μ + K)δj cos θ j sin θ j (μ + K)δj cos θ j + Kμ j, j =3,, a 5 =0, a 3 = a 3 = a 3 = a =0, a 3j = μ j δ j sin θ j, j =3,, a 35 =0, a i = (Q + Rμ i) β δi, i =,, a 5 = ρ, a 5i = {( β )+β μ i }δ i sin θ i, i =,, a 5j = {( β ) β α 3 }δ j cos θ j, j =3,, a 55 = δ sin θ, α 3 = ρ +i b ρ i b and b i (i =,, 3,, 5) are given as follows: (i) for an incident P f wave, (ii) for an incident P s wave, b = a, b = a, b 3 = a 3, b = a, b 5 = a 5 ; b = a, b = a, b 3 = a 3, b = a, b 5 = a 5 ; (iii) for an incident coupled transverse and microrotational wave, b = a, b = a, b 3 = a 3, b = a, b 5 = a 5. The amplitude ratios of various reflected waves are z j = B j, j =,, 5, (9) A A, for incident P f wave, A = A, for incident P s wave, A, for incident coupled transverse and microrotational wave, and z j (j =,, 3, ) are the amplitude ratios for the reflected P f wave at an angle θ, reflected P s wave at an angle θ, the reflected coupled transverse and microrotational waves at angles θ 3 and θ, respectively, and z 5 is the amplitude of transmitted longitudinal wave (P wave ) at an angle θ.

Particular cases Wave propagation in liquid-saturated porous solid 33 Neglecting porous effect, our results reduce at liquid/micropolar elastic solid in- a ij z j = b i, i =,,, (30) Case I terface: j= a =[λ +(μ + K)sin θ ]δ, a i = (μ + K)cosθ i+ sin θ i+ δi+, i =, 3, a = ρ δ, a =(μ + K)cosθ sin θ δ, a i =(μ + K)δi+ sin θ i+ cos θ i+ (μ + K)δi+ cos θ i+ + Kμ i+, i =, 3, a =0=a 3 = a 3, a 3i = μ i+ δ i+ sin θ i+, i =, 3, a = δ sin θ, a i = δ i+ cos θ i+, i =, 3, a = δ sin θ, μ 3, = δ 3, (δ 3, E pr ) p( r 0 + r, δ3, ) = λ 3,, λ 3, =[D (D E) / ]/, D = C E = C, λ = + r + (pr r 0 ), E = ρ λ +μ + K, C [r r 0 ], and b i (i =,, 3, ) are given as follows: (i) for an incident longitudinal displacement wave, C = μ + K, ρ b = a, b = a, b 3 = a 3, b = a ; (ii) for an incident coupled transverse and microrotational wave, b = a 3, b = a 3, b 3 = a 33, b = a 3 ; The amplitude ratios of three reflected waves are z i = B i A 0, i =,, 3,, (3) { A, for incident longitudinal displacement wave, A 0 = A, for incident coupled transverse and microrotational wave, and z j (j =,, 3, ) are the amplitude ratios for the reflected longitudinal displacement wave at an angle θ, reflected coupled transverse and microrotational waves at angles θ 3 and θ and refracted longitudinal wave at an angle θ, respectively. Case II By neglecting micropolarity effect, we obtain a system of four non-homogeneous equations in liquid half space/liquid-saturated porous half space as a ij z j = b i, i =,, 3,, (3) j=

3 Rajneesh Kumar and Mahabir Barak a i =[(λ + Q)+μsin θ i +(Q + R)μ i ]δi, i =,, a 3 = μδ cos θ sin θ, a = ρ, a i =μδi cos θ i sin θ i, i =,, a 3 =(μcos θ sin θ μ cos θ )δ, a =0, a 3i = (Q + μ ir)δi β, i =,, a 33 =0, a 3 = ρ, a i = (( β )+β μ i )δ sin θ, i =,, a 3 = (( β )β α 3 )δ cos θ, a = δ sin θ, δ, = λ,, λ, = B (B AC) /, δ = λ A, λ C = μ(ρ + i b ( ), A = PR Q, P = λ +μ, B = P ρ + i b ) ( + R ρ + i b ) Q μ, = B [ Aλ, +(ρ R ρ Q)+ i b (R + Q)], and b i (i =,, 3, ) are given as follows: (i) for an incident P f wave, (ii) for an incident P s wave, (iii) for an incident Sv wave, b = a, b = a, b 3 = a 3, b = a ; b = a, b = a, b 3 = a 3, b = a ; b = a 3, b = a 3, b 3 = a 33, b = a 3. The amplitude ratios z j (j =,, 3, ) for various reflected waves are given as A, A = A, A, ( ρ i b ), z i = B i A, i =,, 3,, (33) for an incident P f wave, for an incident P s wave, for an incident Sv wave, and z i (i =,, 3, ) are the amplitude ratios of reflected longitudinal P f wave at an angle θ, reflected longitudinal P s wave at an angle θ and reflected Sv wave at an angle θ, reflected wave at an angle θ, respectively. 5 Numerical results and discussion Following Gauthier [0], we take the following values of the relevant micropolar constants as ρ =.9 g/cm 3, γ =.68 0 5 dyn, j = 0 6 cm, K =.9 0 N/cm, /0 =0.

Wave propagation in liquid-saturated porous solid 35 Following Yew and Jogi [] and Fatt [3], the following values of relevant parameters have been taken for kerosene-saturated sandstone: ρ =.37 g/cm 3, ρ =.96 37 g/cm 3, Q =7.635 0 N/cm 3, ρ = 0.00 37 g/cm 3, ρ =0.5 337 g/cm 3, R =3.6 0 N/cm 3, λ =.339 0 5 N/cm, μ =.765 0 5 N/cm, f = b =0.7 57. ρ For the liquid half space, ρ =g/cm 3, λ =.5 0 6 N/cm For the above values of relevant parameters, the system (8) for the amplitude ratios z i (i =,, 3,, 5) are solved by Gauss elimination method for different angles of emergence of the incident P f (fast P -wave), P s (slow P -wave) and coupled transverse and microrotational waves starting from grazing incidence (θ 0 =0 ) to the normal incidence (θ 0 =90 ). The variations of amplitude ratios for micropolar liquid-saturated porous (), micropolar elastic () and elastic with porous () have been shown by solid line, long dashed line and small dashed line, respectively. The variations of the value of amplitude ratios z i (i =,, 3,, 5) for and z i (i =,, 3, ) for and with the angle of emergence θ 0 of the incident P f wave (Figs. 6), incident P s wave (Figs.7 ) and incident coupled transverse and microrotational wave (Figs. 6) are shown graphically for Kerosene-saturated sandstone. Amplitude ratios of z.0.0.00 Amplitude ratios of z 3 0.99 90 0 90 (a) z for (b) z for, and Fig. Angle of emergence of z for incident P f wave Amplitude ratios of z 0.05 0.00 0.05 0.00 0.005 Amplitude ratios of z 3..0 0.8 0.6 0. 0. 0.000 90 0.0 90 Fig.3 Angle of emergence of z for incident P f wave (in deg) Fig. Angle of emergence of z 3 for incident P f wave (in deg)

36 Rajneesh Kumar and Mahabir Barak Amplitude ratios of z 0.3 0. 0. Amplitude ratios of z 5 3 Fig.5 0.0 90 Angle of emergence of z for incident P f wave (in deg) Fig.6 0 90 Angle of emergence of z 5 for incident P f wave (in deg) Amplitude ratios of z 0 8 6 Amplitude ratios of z.5.0.5.0 0.5 Fig.7 0 90 Angle of emergence of z for incident P s wave (in deg) 0.0 90 Fig.8 Angle of emergence of z for incident P s wave (in deg) Amplitude ratios of z 3 3 Amplitude ratios of z 7 6 5 3 Fig.9 0 90 Angle of emergence of z 3 for incident P s wave (in deg) 0 90 Fig.0 Angle of emergence of z for incident P s wave (in deg) Amplitude ratios of z 5 0 5 0 5 Fig. 0 90 Angle of emergence of z 5 for incident P s wave (in deg) Amplitude ratios of z 35 30 5 0 5 0 5 0 90 Fig. Angle of emergence of z for incident coupled transverse and microrotational wave (in deg)

Wave propagation in liquid-saturated porous solid 37 Amplitude ratios of z 0.7 0.6 0.5 0. 0.3 0. 0. Amplitude ratios of z 3 3.5 3.0.5.0.5.0 0.5 0.0 90 0.0 90 Fig.3 Angle of emergence of z for incident coupled transverse and microrotational wave (in deg) Fig. Angle of emergence of z 3 for incident coupled transverse and microrotational wave (in deg) Amplitude ratios of z 6 5 3 Amplitude ratios of z 5 5 0 5 0 5 0 90 0 90 Fig.5 Angle of emergence of z for incident coupled transverse and microrotational wave (in deg) Fig.6 Angle of emergence of z 5 for incident coupled transverse and microrotational wave (in deg) Case I Incident P f wave The amplitude ratio z for has larger values in comparison to z for in the range 0 θ 0 and 3 θ 0 90, less in the range 5 θ 0 33. The amplitude ratio z for has smaller values in comparison to z for in the range 0 θ 0 90 (Fig.(a)). The values of amplitude ratio z for are more in comparison to z for in the range 0 θ 0 56 and 67 θ 0 85 andlessintherange57 θ 0 66 and 86 θ 0 90, and these variations are shown in Fig.3. Figure shows the variations of the amplitude ratio z 3 of reflected coupled transverse and microrotational wave for which has larger values in comparison to z for in the range 0 θ 0 6 and 9 θ 0 6 and smaller in the range 7 θ 0 8 and 65 θ 0 86. The amplitude ratio z for has smaller values in comparison to z 3 for in the range 0 θ 0 90,and z 3 for in the range 0 θ 0 5 to 69 θ 0 90 and has larger values in the range 5 θ 0 68 as depicted in Fig.5. The values of amplitude ratio z 5 for are monotonically decreasing in the whole range and have greater values than amplitude ratios z for. The values of amplitude ratio z for are smaller in comparison to z 5 for for the range 0 θ 0 90,asshownin Fig.6. The original values of z 3 for are multiplied by 0. Case II Incident P s wave Figure 7 shows that variations of amplitude ratio z for, and. The amplitude ratio z for has smaller values in comparison to z for and in the range 0 θ 0 8 but greater than the values of amplitude ratios z for in the range 83 θ 0 90. The amplitude ratio z for has larger values in comparison to z for in the range

38 Rajneesh Kumar and Mahabir Barak 0 θ 0 andlessintherange8 θ 0 90. The values of amplitude ratios z for are larger in comparison to z in the range 0 θ 0 5 and less in the range 55 θ 0 90 for as depicted in Fig.8. In Fig.9, the values of amplitude ratio z 3 for are greater than those of amplitude ratio z for in the range 0 θ 0 90 as shown in Fig.9. Figure 0 shows that the values of amplitude ratios z for are less than z 3 of in range 0 θ 0 90 but greater than z 3 for in the range 0 θ 0 90. Figure shows the variations of amplitude ratios z 5 for with amplitude ratios z for and in the range 0 θ 0 90. The original values of z and z 5 for are multiple by 0 and 0 respectively. Case III Incident coupled transverse and microrotational wave The values of amplitude ratios z for are larger than z for and in the range 0 θ 0 57 to 68 θ 0 90 and less than the values of amplitude ratios z for in the range 57 θ 0 67 and these variations are shown in Fig.. In Figs.(3) and () the values of amplitude ratios z and z 3 for are larger than those of and respectively. The amplitude ratio z for has smaller values in comparison to z for in the range 0 θ 0 83, z 3 for in the range 0 θ 0 85. The amplitude ratios z 3 for have smaller values in the range 0 θ 0 57 to 77 θ 0 90 and have larger values in the range 58 θ 0 76 as shown in Fig.5. The amplitude ratios z 5 for have larger values in comparison to z for, 0 θ 0 56 and 83 θ 0 90 and have smaller values in the range 57 θ 0 8. The values of amplitude ratios z for are larger than z for in the range 0 θ 0 3 and 8 θ 0 90, and less than in the range 30 θ 0 80, as depicted Fig.6. The original values of z for are multiple by 0. 6 Conclusion The theory of micropolar liquid-saturated porous solid developed by Eringen [9] and Konczak [5,6] has been used to study the problem. The analytical expression for reflection and refraction coefficients of various reflected waves has been derived and these variations have been depicted graphically for a particular model. Some particular cases have been deduced. From Figs.() (6) it may be concluded that micropolarity and porosity have significant effect on the amplitude ratios of various reflected and transmitted waves. This problem through theoretical may be of some use in engineering, seismology and geophysics etc. References [] Biot M A. General solutions of the equations of elasticity and consolidation for a porous material[j]. J Appl Mech, 956, 3:9 96. [] Biot M A. Theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low-frequency range,ii.higher frequency range[j]. JAcoustSocAm, 956, 8:68 9. [3] Biot M A. Mechanics of deformation and acoustic propagation in porous media[j]. J Appl Phys, 96, 33:8 98. [] Deresiewicz H. The effect of boundaries on wave propagation in a liquid-filled porous solid I: Reflection of plane waves at a free plane boundary (non-dissipative case)[j]. Bull Seism Soc Am, 960, 50:599 607. [5] Deresiewicz H, Rice J T. The effect of boundaries on wave propagation in a liquid-filled porous solid III. Reflection of plane waves at a free plane boundary (genaral case)[j]. Bull Seism Soc Am, 96, 5:595 65. [6] Eringen A C, Suhubi E S. Non-linear theory of simple microelastic solids I[J]. Int J Eng Sci, 96, :89 03. [7] Suhubi E S, Eringen A C. Non-linear theory of simple microelastic solids II[J]. Int J Eng Sci, 96, :389 0.

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