UNIVERSITY OF SOUTHERN CALIFORNIA Department of Civil Engineering

Transcription

1 UNIVERSITY OF SOUTHERN CALIFORNIA Department of Civil Engineering ON THE REFLECTION OF ELASTIC WAVES IN A POROELASTIC HALF-SPACE SATURATED WITH NON-VISCOUS FLUID b Chi-Hsin Lin, Vincent W. Lee and Mihailo D. Trifunac Report No. CE -4 September Los Angeles, California

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3 Abstract The effects of the stiffness and Poisson s ratio of the solid-seleton and boundar drainage are investigated for in-plane elastic wave incidences in a poroelastic half-space saturated with non-viscous fluid. Biot s theor of elastic wave propagation in a fluidsaturated porous medium is briefl summarized and applied to the models. Determination of the required poroelastic material constants is proposed. Both drained and undrained boundaries of the half-space are considered and formulated with appropriate boundar conditions. The amplitude coefficients, surface displacements, surface strains, rotations, and stresses are calculated and discussed. It is found that the stiffness, Poisson s ratio of the solid-seleton and the boundar drainage affect the reflection response of the halfspace significantl. i

4 Table of Content Abstract Table of Content i ii Introduction. Literature review. Obective and organization of this report Review of the theor of wave propagation in a fluid-saturated porous medium 3. Biot s theor of wave propagation in a fluid-saturated porous medium (low frequencies 3. Validit of Biot s theor 7.3 Material constants for fluid-saturated porous media.4 Wave velocities in fluid-saturated porous media 3.5 Boundar conditions for porous media 4 3 Response of a fluid-saturated porous half-space to an incident P-wave 6 3. The model 6 3. An eample of surface response analsis Results and discussion for open-boundar case 3.4 Results and discussion for sealed-boundar case Comparison of open-boundar and sealed-boundar cases Conclusions 53 4 Response of a fluid-saturated porous half-space to an incident SV-wave The model An eample of surface response analsis Results and discussion for open-boundar case Results and discussion for sealed-boundar case Comparison of open-boundar and sealed-boundar cases Conclusions 98 ii

5 5 Case stud and general conclusions 5. Case stud: strong motion records at Port Island during the Kobe Earthquae on Januar 7, Conclusions 5 Acnowledgements 6 References 7 Appendi A Potential-displacement-stress relations Appendi B Summar of notations 3 iii

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7 Introduction. Literature review The dnamic response of a fluid-saturated porous medium is of interest to applications in geophsics, petroleum engineering, geotechnical engineering, and earthquae engineering. Biot (956a formulated the wave equations for a fluid-saturated porous medium assuming an elastic solid-seleton and compressible fluid in the pores. He showed that two P-waves and one S-wave co-eist in this model. The dnamic behavior of a poroelastic medium depends on the material constants, including the elastic moduli and the dnamic mass coefficients. The determination of the poroelastic material constants has been discussed b Biot and Willis (957, Berrman (98 and Bourbié et al. (987. The elastic moduli have been measured eperimentall b Fatt (959 and Yew and Jogi (976. Deresiewicz and coworers published a series of papers discussing the effects of free plane boundaries on wave propagation in a poroelastic medium (Deresiewicz, 96, 96, Deresiewicz and Rice, 96. The studied amplitude ratios, phase velocities, and attenuation of the reflected waves. In geotechnical earthquae engineering, the drainage of a poroelastic medium boundar is an important factor for the initiation of liquefaction. Studies of boundar conditions for phsical boundaries of poroelastic media have been carried out b Deresiewicz and Sala (963, Lovera (987 and de la Cruz and Spanos (989. These authors emploed different approaches, but conservation of mass and continuit of momentum were the common principles.. Obective and organization of this report The obective of this report is to stud ( the effects of the stiffness and Poisson ratio of the solid-seleton, and ( the effects of drained and undrained boundaries on the reflection of waves from a poroelastic half-space saturated with non-viscous fluid (nondissipative case. In studies of wave propagation, the elastic bod waves are usuall decomposed into SH, P, and SV-waves. Because the SH-wave equation of poroelastic medium for - -

8 non-dissipative case is identical to the one for an elastic medium (Deresiewicz, 96, a poroelastic half-space has the same response as an elastic medium. Therefore, the investigation of incident SH-wave is not included in this report. In Chapter, Biot s theor of wave propagation in fluid-saturated porous medium is briefl reviewed. The relations among the material constants are discussed and suggested for numerical analsis. The boundar conditions are adopted from the description of open-boundar and sealed-boundar b Deresiewicz and Sala (963. In Chapter 3 and Chapter 4, the cases of incident plane P-waves and SV-waves are investigated for the amplitude coefficients, surface displacements, surface strains, rotations and stresses. The effects of the stiffness, Poisson s ratio of the solid-seleton and the effects of boundar drainage are discussed in detail. In Chapter 5, a case stud of strong motion recorded at Port Island during the 995 Kobe Earthquae is briefl eamined. We applied the model analzed in Chapter 3 and Chapter 4 to come up with a rough preliminar interpretation on this unique record of amplified vertical motions, but reduced horizontal motions. eneral applications of the results of this stud and brief conclusions are also stated. It is hoped that the present stud will provide useful theoretical estimates of some fundamental properties of porous media for engineering applications. - -

9 Review of the theor of wave propagation in a fluid-saturated porous medium. Biot s theor of wave propagation in a fluid-saturated porous medium (low frequencies.. Stress-strain relations for fluid-saturated porous medium Biot (956a considered an element of a solid-fluid sstem represented b a cube of unit size in which the stress tensors are separated into two parts solid and fluid. The solid-seleton is assumed to be isotropic and elastic, and the fluid is allowed onl dilatational deformation. This particular solid-fluid sstem has the following stress-strain relations. ( The stress tensor on the solid part of the cube is τ τ τ z τ τ τ z τ z τ z τ zz ( The stress tensor on the fluid part of the cube is σ σ σ where σ is proportional to the hdraulic pressure on the unit cube according to (- (- σ nˆp (nˆ porosit (-3 (3 The stress-strain relations are established as τ P τ λ τ zz λ σ Q τ τ z τ z where λ λ Q P λ Q λ P Q Q Q R µ µ zz z z γ γ γ zz ε γ γ z µ γ z τ, τ, τ, τ, τ, τ stresses of the solid-seleton (-4-3 -

10 .. The wave equations γ, γ, γ, γ, γ, γ strains of the solid-seleton zz σ stress of the pore fluid z ε dilatational strain of the pore fluid z λ, µ, P, Q, R elastic moduli for the solid-fluid sstem (P λ+µ. To develop the wave equations for the low frequenc range, three assumptions are made (Biot, 956a: ( the relative motion of the fluid in the pores is laminar flow which follows Darc s law (Renolds number < ; ( the elastic wavelength is much larger than the unit solid-fluid element; (3 the size of the unit element is large compared to the pores. Based on the above assumptions and stress-strain relations, the wave equations of fluid-saturated porous medium are ~ ~ ~ ˆ ~ µ u + grad[( λ + µ e + Qε] ( ρ ( ~ u + ρu + b u U (-5a t t ~ ~ ˆ grad[ Qe + Rε ] ( ρ ( ~ ~ u + ρ U b u U (-5b t t where u ~ displacement vector for the solid-seleton U ~ displacement vector for the pore fluid ~ ~ e div(u, ε div(u ρ, ρ ρ dnamic mass coefficients, bˆ dissipative coefficient ˆ µ nˆ ˆ ( nˆ porosit, ˆ permeabilit, µˆ absolute viscosit. Net, we appl Helmholtz decomposition to the displacement vector u ~ grad( φ + curl( ψ ~ (-6a ~ U grad( Φ + curl( Ψ ~ (-6b where φ and Φ are P-wave potentials and ψ and Ψ are S-wave potentials for solid and fluid, respectivel. Substituting Eq.(-6 into Eq.(-5, and rearranging the terms, the following two sets of equations with respect to P-wave and S-wave potentials are obtained - 4 -

11 - 5 - for P-wave potentials: Φ Φ + Φ + Φ + Φ + Φ + ( ˆ ( ( ˆ ( φ ρ φ ρ φ φ ρ φ ρ φ t b t R Q t b t Q P (-7 and for S-wave potentials: Ψ Ψ + Ψ + Ψ + ~ ~ ( ˆ ~ ~ ( ~ ~ ( ˆ ~ ~ ( ~ ψ ρ ψ ρ ψ ρ ψ ρ ψ µ t b t t b t (-8..3 Solutions for P-waves In the present stud, we consider a non-dissipative case in which the porous medium is saturated with non-viscous fluid ( µˆ. For such a case, the last terms in Eq.(-7 and Eq.(-8 drop out. If the wave potentials have harmonic time variations, the potentials can be epressed as t i e z ω φ φ,, (, t i e z ω Φ Φ,, ( (-9a t i e z ω ψ ψ,, (, t i e z ω Ψ Ψ,, ( (-9b Substituting Eq.(-9a into Eq.(-7, and eliminating Φ, the P-wave equations for the solid-seleton become φ ω φ ω φ C B A (- where Q PR A Q P R B ρ ρ ρ + ρ ρ ρ C Eq.(- can be decomposed into (, + φ, (, (- where V,, ω (, (-a are the P wavenumbers, and

12 A (, (-b / B # ( B 4AC V, are the P-wave velocities From Eq.(-, it is seen that two P-waves eist in the medium. The general solution for the solid-seleton is φ φ + φ (-3 To determine the wave potential for the fluid component, substituting Eq.(-9a into Eq.(-7, the following is obtained where + Φ f φ + f φ Φ Φ (-4 f A/ V, ρr + ρq (, (-5 ρ R ρ Q..4 Solutions for S-waves Substituting Eq.(-9b into Eq.(-8 and eliminating Ψ, the S-wave equation for the solid-seleton becomes ( + ψ (-6 where ω (-7a V is the S wavenumber, and µρ V C (-7b is the S-wave velocit. The S-wave potential for fluid the component can be also obtained Ψ f 3 ψ (-8 where ρ f 3 (-9 ρ - 6 -

13 . Validit of Biot s theor.. Theoretical considerations Biot s theor is based on the principles of continuum mechanics, and assumes linear behavior of the materials. However, porous medium involves discontinuities between the solid and pores, for which the macroscopic laws of mechanics are not alwas applicable. For eample, if the incident wavelengths are short enough to feel the pores, then the diffraction of waves occurs and Biot s theor is no longer valid. Bourbié et al. (987 suggested that the minimum homogenization volume for porous roc is about three times the pore diameter. The range of pore sizes in porous roc is from -3 mm to mm, and for the stud of earthquae strong motion, the continuum mechanics representation is applicable, because the short wavelengths of earthquae shaing, e.g. m (assuming frequenc 3 Hz, V 3 m/s are still much longer relative to the minimum homogenization size of roc. Darc s law for pore fluid is another integral part of Biot s theor. If the frequenc of the incident waves eceeds a certain value, the laminar flow in the pores breas down. Biot (956a showed that this frequenc ma be related to the fluid viscosit and the size of pores as f t πνˆ (- 4d where d is pore diameter and νˆ is inematic viscosit. In the case of water, νˆ.3 [ -6 m /s, and we find that the maimum f t 4 Hz for d - mm, and f t 5 Hz for d [ - mm. To appl Biot s theor to geotechnical engineering, it is important to determine the range of its validit with respect to the grain size of the soil. In general, the soil consists of a collection of solid particles with various sizes. To determine the pore size of certain soil, the distribution of grain sizes needs to be analzed. The effective pore diameter for soil is assumed to be one-fifth of D in permeabilit studies (Cedergran, 989, where the D is the grain size that corresponds to % of the sample passing b weight. Permeabilit can be used to determine the mobilit of pore fluid in the soil. For - 7 -

14 poor drainage or impervious soils (cla or silt, relative fluid flow does not occur. In that case, Biot s theor is not applicable. Fig.. shows the effective pore size, permeabilit and frequenc f t with respect to soil grain size. It is seen that the effective pore size (D /5 range in which Biot s theor is applicable is from. to. mm. Therefore, soils consisting of sands and gravels, with grain size D.5 to mm are eligible to appl Biot s theor. rain size (mm ASTM soil classification Effective pore size d D /5 (mm Permeabilit (mm/s... ma. frequenc f t (Hz Cla Silt Fine Drainage Impervious Poor ood Range of effective pore size where Biot's theor can be applied Range of soils ma have grain size D. where Biot's theor can be applied. -3 Medium Sand Coarse - 5 ravel Cobbles Boulders Fig.. Soil grain size with corresponding (a soil classification, (b the effective pore size, (c permeabilit and drainage properties (Terzaghi and Pec, 948, (d frequenc f t, (e the range of effective pore size where Biot s theor can be applied, and (f the main range of soils ma have grain size D. to mm where Biot s theor can be applied... An eample of the geographical distribution of sands The preceding discussion suggests that the candidate sites for use of Biot s theor in geotechnical engineering must meet two criteria: ( the soils mainl to consist of sand or gravel deposits, and ( the soil stratum to be fluid-saturated. enerall, the areas with sand deposits near rivers, laes, reservoirs, or seashores are the most applicable sites for - 8 -

15 strong motion response analsis b Biot s theor. Of civil engineering interest, geotechnical structures such as earth dams or man-made landfills at ports (for eample, Port Island in Kobe, and Terminal Island in Los Angeles are areas to stud the dnamics of fluid-saturated media. To illustrate the geographical etent of the areas where phsical conditions ma eist for use of Biot s theor in interpretation of incident seismic waves, we cite eamples from Trifunac and Todorovsa (998. Fig.. shows the geographical distribution of surficial sand deposits in Los Angeles-Santa Monica region of Southern California (after Tinsle and Fumal, 985. It is seen from Fig.. that large areas of metropolitan Los Angeles are covered b sand deposits. In those areas, when the water table is essentiall at ground surface (Tinsle et al., 985, the phsical conditions will eist for use of Biot s theor in interpretation of observed strong motion amplitudes. Fig.. eographical distribution of surficial sand deposits in Los Angeles Santa Monica region (shaded areas, after Tinsle and Fumal,

16 .3 Material constants for fluid-saturated porous media In Eq.(-5, two sets of material constants, elastic moduli and dnamic mass coefficients, are required in the model. Determination of the material constants for porous media is discussed in the following section..3. Elastic moduli Biot and Willis (957 proposed a method to calculate the elastic moduli of porous media from eperimental measurements. For an isotropic sstem, the four elastic moduli can be determined from the shear modulus, compressibilit of the solid-seleton (aceted compressibilit, compressibilit of the solid-fluid sstem (unaceted compressibilit, and compressibilit of the pore fluid (fluid infiltrating coefficient as follows µ µ s (- Q λ λs + R γ / κ ˆ s + n + ( nˆ( δ / κ s µ γ + δ δ / κ s 3 (- nˆ(nˆδ / κ s Q γ + δ δ / κ (-3 s s nˆ R (-4 γ + δ δ / κ where nˆ porosit ν s λs µ s Lamé constant for the solid-seleton ν s µ s shear modulus for the solid-seleton ν s Poisson s ratio for the solid-seleton κ s e p K e δ p s compressibilit of the solid-seleton compressibilit of the solid-fluid sstem - -

17 nˆ( ε e γ p nˆ( K f + e p n ˆ( κ f δ compressibilit of the pore fluid ( + ν K s µ s bul modulus of solid-seleton 3( ν K f bul modulus of fluid κ f fluid compressibilit.3. Dnamic mass coefficients In Biot s theor (956a, the framewor of the mass components in a unit cubic solid-fluid sstem is given as ρ ρ ρ nˆ + ( ρ s (-5a ρ ρ ρ ˆ + nρ f (-5b ρ ρ ρ ( nˆ ρ nˆ ρ (-5c + s + f where ρ unit mass of the solid-fluid aggregate ρ solid mass in the unit cube (unit mass of solid-seleton ρ fluid mass in the unit cube ρ s densit of the solid material ρ densit of the fluid f To determine Biot s dnamic mass coefficients, Berrman (98 proposed the following relations. ρ ( nˆ( ρ + τ ρ (-6a s r f ρ nˆ τ ρ f (-6b where τ r ρ f the induced mass due to oscillation of the solid particle in the fluid τ toruosit parameter The value of τ r has to be calculated from a microscopic model of the solid-frame moving in the fluid. For spherical solid particles, τ r.5 (Berrman, 98. Substituting Eq.(-6 into Eq.(-5, the following is obtained - -

18 ρ ( nˆ ρ nˆ( τ ρ (-7a s + f ρ nˆ( τ ρ f (-7b ρ nˆ τ ρ f (-7c and nˆ τ + τ r ( (-8 nˆ Eq.(-8 implies that τ as n ˆ, for pure fluid medium, and τ as n ˆ, for solid medium..3.3 Limits of the material constants and simplified formulae If the medium is pure fluid, then n ˆ, µ λ Q, R K f, and ρ ρ, ρ. In this case, the Eq.(-5a disappears, and Eq.(-5b gives the dnamic ρ f equation of motion of the fluid under the assumption of small displacements as grad U ~ ~ R [ div( ] ρ f U (-9 t If the medium is pure solid, then n ˆ, Q R, and ρ ρ, ρ ρ s. In this case, the Eq.(-5b disappears, and Eq.(-5a gives the Helmholtz wave equation for elastic solid as grad ( u ~ ( [ ] u~ λ + µ div + µ ρ u~ (-3 s t For dnamic analses in soil mechanics, it is often assumed that the compressibilit of the solid-fluid sstem can be neglected ( δ, relative to the compressibilit of the solid-seleton and the fluid. Therefore, Eq. (- through Eq.(-4 can be simplified as Q ν ( nˆ λ + K f (-3 nˆ s λs + µ s R ν s nˆ(nˆ Q (nˆ γ K f (-3 nˆ R γ nk ˆ f (

19 If we assume the solid-fluid sstem is formed b spherical solid particles (τ r.5, the material constants can be simplified as in table.. Table. The simplified formulas of material constants for porous medium. µ µ s Elastic moduli ν s ( nˆ λ µ s + ν s nˆ nˆ( nˆ Q ( nˆ K f γ ˆ n R γ nk ˆ f K f Dnamic coefficients of mass ρ ( nˆ ρ + nˆ( τ ρ s ˆ( τ ρ n ρ f ρ nˆ τ ρ f τ ( + nˆ f.4 Wave velocities in fluid-saturated porous media The wave velocit is one of the most important parameters in the stud of wave propagation. The relations among different wave velocities, depending on the combinations of material constants will influence the nature of the reflections at the boundaries. For porous medium with 3% porosit, for eample, and consisting of spherical solid particles (ρ s /ρ f.7, the wave velocities for various µ /K f ratios and Poisson s ratios of the solid-seleton can be computed using material constants shown in Table.. The µ /K f ratio represents the stiffness of the solid material with respect to the fluid bul modulus. The Poisson s ratio ma represent the consolidation status of the solid-seleton. In general, consolidated materials have small Poisson s ratios, and unconsolidated materials have large Poisson s ratios. The wave velocities for different combinations of material constants are shown in Table.. From Table., it is seen that the fast P-wave velocities are alwas larger than the S-wave velocities in the porous media, same as in elastic media. The slow P-wave velocities are larger than the S-wave velocities onl for soft, unconsolidated solids (e.g. µ /K f DQν s.4. If the porous medium is water-saturated (K f Mpa, the ρ s /ρ f ratio equals the specific gravit of the solid (the ratio between the unit masses of solid and water

20 The shear wave velocities, V, and the overall nature of the material for different µ /K f ratios are illustrated in Table.3. Table. Wave velocit ratios for different Poisson s ratios of solid-seleton and µ /K f ratios in porous medium (assuming porosit.3, ρ s /ρ f.7. µ /K f µ /K f µ /K f. µ /K f. ν s V /V V /V V /V V /V V /V V /V V /V V /V Table.3 Shear wave velocities, V, for different µ /K f ratios in water-saturated porous medium (porosit.3, solid specific gravit.7. µ /K f Shear wave velocit V Field materials 3 m/sec Porous roc m/sec Porous roc. 3 m/sec Stiff soil. m/sec Soft soil.5 Boundar conditions for porous media In a wave propagation problem, emploing appropriate boundar conditions leads to specific solution. To establish boundar conditions for porous media, the solid-fluid interaction of the aggregate must be considered in addition to the elastic behavior of the solid material. Deresiewicz and Sala (963, Lovera (987, and de la Cruz and Spanos (989 proposed boundar conditions for two different fluid-saturated porous media in contact. These conditions were derived starting with two principles the conservation of mass and the continuit of momentum

21 The effects of a free plane boundar on wave propagation in a poroelastic halfspace have been investigated b Deresiewicz (96, where the boundar conditions for the free plane boundar include: ( zero stresses of the solid-seleton in both normal and tangential directions of the plane; and ( zero pore fluid pressure on the plane. In this case, the pores are open to the air and the pore fluid can be drained from the solid-fluid aggregate. The case of sealed-boundar in which pore fluid is trapped inside the aggregate is also important to stud for geotechnical engineering applications, because the pore fluid pressure ma build up and induce liquefaction. According to the wor of Deresiewicz and Sala (963, the boundaries for a free surface can be illustrated b the simplified diagrams in Fig..3. Fig..3(a represents the open-boundar case in which the pore fluid is not restricted. In Fig..3(b, the boundar is sealed with a thin membrane so that the pore fluid is restricted inside the aggregate. To stud the effects of sealed-boundar, the boundar conditions given b Deresiewicz and Sala (963 were adopted and compared with the ones for open-boundar case in Table.4. Air Air (a open-boundar (b sealed-boundar Fig..3 Simplified diagrams of the porous-air boundaries: (a open-boundar; (b sealedboundar. In the porous medium, the gra area represents the solid-seleton and the white parts are the pores. Table.4 Boundar conditions for open-boundar and sealed-boundar. Open-boudar Sealed-boundar τ τ nn nt σ τ τ u nn nt n + σ U n Note: subscript n: normal, t: tangential vector on the boundar surface - 5 -

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23 3 Response of a fluid-saturated porous half-space to an incident plane P-wave 3. The model Consider a fluid-saturated porous half-space subected to a plane P-wave with incidence angle θ as shown in Fig. 3.. We assume the wave has harmonic time dependence and, for brevit, we omit the i t e ω terms. Air Porous medium θ θ i φ θ r φ θ r φ r ψ Fig. 3. The fluid-saturated porous half-space subected to an incident P-wave. The incident P-wave potential is φ i i ( sin θ + cos θ ae (3-a and the reflected wave potentials are φ r i ( sinθ cosθ ae (3-b φ ψ r r i ( sin θ cosθ ae (3-c i ( sin θ cosθ b e (3-d The boundar conditions will be used to determine the unnown amplitude coefficients. To investigate the ground responses for drained and undrained boundaries, both open and sealed cases will be considered in the following

24 3.. Open-boundar case This case has been studied b Deresiewicz (96. Appling the conditions for open-boundar in Table.4, we can state the boundar conditions in Cartesian coordinates as τ τ σ (3- Substituting Eq.(3- into Eq.(3-, the equations can be simplified as in Eq.(3-3, where the notation used is summarized in Appendi A.,, 6,,, 6, a a a b From Eq.(3-3, the three real amplitude coefficients are obtained.,, 6, ( Sealed-boundar case Similar to the solution of the open-boundar case, the following boundar conditions are obtained from Table.4. τ + σ τ u U (3-4 After substituting Eq.(3- into Eq.(3-4, the unnown coefficients can be determined from, + 6,, + 6, a, +,, a a, ( f 4, ( f 4, ( f3 4 b ( f 6, 4, ( Surface response The following responses of the solid-seleton at the half-space surface are calculated

25 - 8 - Displacements The displacement responses of the fluid-saturated porous medium at the ground surface are obtained as i e b a a a u u 4 4, 4, 4, 3 3, 3, 3, (3-6 where θ θ θ sin sin sin is the apparent wavenumber on the halfspace surface. Surface strains Trifunac (979 and Lee (99 calculated the surface strains from the derivatives of displacements / / u u γ γ (3-7 Substituting Eq. (3-6 into Eq.(3-7, we obtain i e b a a a 8 8, 8, 8, 7 7, 7, 7, γ γ (3-8 Surface rotation Trifunac (98 calculated the rotation (rocing on the ground surface from i e b u u ψ ψ ψ (3-9 It is clear that ψ is associated onl with SV-wave motion. Therefore, the normalized rotation can be defined as (Trifunac, 98; Lee and Trifunac, 987 / ( / ( π λ π ψ ξ i i e b e i (3-

26 i where λ is the wavelength of SV-waves, and i e is the displacement induced b an incident P-wave. From Eq.(3-, it is seen that the rocing is phase-shifted relative to the incident motion b π/. Stresses The stresses on the ground surface can be obtained as τ τ τ σ µ,, 5, 6,,, 5, 6,,, 5, 6, a a e 5 a b i (3-3. An eample of surface response analsis A fluid-saturated porous half-space subected to an incident P-wave with unit amplitude is investigated for surface response. We assume porous medium with 3% porosit consisting of spherical solid particles (τ r.5, and ρ s /ρ f.7. The relations among material constants in Table. can be used to calculate the surface response for various Poisson s ratios of the solid-seleton and µ /K f ratios. The surface response for the open boundar and sealed boundar cases are discussed in sections 3.3 and 3.4, respectivel. Fig. 3. through Fig. 3.5 and Fig. 3.7 through Fig. 3. show the amplitude coefficients, displacements, surface strains, rotations and stresses versus incident angle for various solid materials with different Poisson s ratios for open-boundar cases and for sealed-boundar cases, respectivel. Fig. 3.6 and Fig. 3. show comparisons for varing solid stiffness with the same Poisson s ratio, ν s.5, for open-boundar case and sealed-boundar case, respectivel. Fig. 3. illustrates the effects of open-boundar and sealed-boundar for the porous material with µ /K f ratio. and ν s.5. It is noted that the displacements and rotations are normalized b a factor of which is the displacement intensit of incident P-wave. The surface strains are - 9 -

27 normalized b a factor of which is the strain intensit of incident P-wave. The stresses are normalized b a factor of which is the stress intensit induced b incident P-wave (Pao and Mao, 97. All the above comple amplitudes are plotted as absolute values. 3.3 Results and discussion for open-boundar case 3.3. Amplitude coefficients From parts (a, (b and (c in Fig. 3. through Fig. 3.5, it is seen that the amplitude coefficients of the slow P-waves are much smaller ( -, than those of the fast P-waves. The coefficient variations with respect to Poisson s ratio are significant for the soliddominated case (large µ /K f ratio, and diminish with decreasing solid stiffness. Fig. 3.6(a, (b and (c show that the absolute values of a and b for an elastic medium are alwas larger than the ones for the porous medium. The effects of the incident angle decrease with decreasing solid stiffness. 3.. Displacements From parts (d and (e in Fig. 3. through Fig. 3.5, it is seen that the effects of variations in Poisson s ratio are significant for the displacement amplitudes in the soliddominated case. These effects diminish with decreasing solid stiffness. The pea displacement u has maimum value.89 for µ /K f and ν., and decreases with decreasing µ /K f ratio. For µ /K f., the pea displacement u reduces to.. The displacement u, alwas has amplitudes equal to two for vertical incidence and zero for horizontal incidence, which is same as in the case of an elastic medium. Fig. 3.6(d and (e show how the amplitudes of u decrease with decreasing solid stiffness and are alwas smaller than the amplitudes for an elastic medium Surface strains From parts (f and (g in Fig. 3. through Fig. 3.5, it is seen that the surface strain variations for different Poisson s ratios are significant for the solid-dominated case, and reduce with decreasing solid stiffness. The pea surface strain γ has maimum value of.78 for µ /K f and ν s., and decreases with decreasing µ /K f ratio. For µ /K f - -

28 ., the pea γ reduces to.83. The pea surface strain γ has maimum value of.56 for µ /K f and ν s.4, and decreases with decreasing µ /K f ratio. For µ /K f., the pea of γ reduces to.55. Fig. 3.6(f and (g illustrate how the surface strains decrease with decreasing solid stiffness and are alwas smaller than the corresponding amplitudes for elastic medium Rotations Parts (h and (i in Fig. 3. through Fig. 3.5 show the normalized rotation with respect to horizontal displacement and vertical displacement versus incident angle, respectivel. It is seen that the dependence of rotation on Poisson s ratio is significant for solid-dominated case, and reduces with decreasing solid stiffness. The ξ /u ratio alwas equals for vertical incidence and decreases with increasing incident angle. When µ /K f and ν., ξ / u has a minimum equal to.6 for horizontal incidence. For µ /K f., ξ /u remains close to, which means that the incident angle does not affect the ratio of ξ /u. The pea rotation ξ /u has a maimum equal to.63 when ν s. and µ /K f, and decreases with decreasing µ /K f ratio. For µ /K f., the pea ξ /u reduces to.8. Fig. 3.6(h and (i show how the rotations ξ / u approach and ξ /u tend to with decreasing solid stiffness Stresses From parts ( in Fig. 3. through Fig. 3.5, it is seen that the stress dependence on Poisson s ratios is significant for a solid-dominated case, and reduces with decreasing solid stiffness. The pea stress τ has maimum value of.5 as µ /K f and ν s., and decreases with decreasing µ /K f ratio. For µ /K f., the pea τ reduces to Fig. 3.6( illustrates how the amplitudes of τ decrease with decreasing solid stiffness, and are alwas lower than the amplitude for the elastic medium. - -

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42 3.4 Results and discussion for sealed-boundar case 3.4. Amplitude coefficients From parts (a, (b and (c in Fig. 3.7 through Fig. 3., it is seen that the amplitude coefficients of the slow P-waves are much smaller ( - than those of the fast P-waves. The coefficient variations with respect to Poisson s ratio are significant for the soliddominated case, and reduce with decreasing solid stiffness. It is noted that the coefficient a does not equal to zero for vertical incidence. Fig. 3.(a, (b and (c show that the pea absolute values of a and b for elastic medium are alwas larger than the ones for the porous medium. The effects of incident angle decrease with decreasing solid stiffness Displacements From parts (d and (e in Fig. 3.7 through Fig. 3., it is seen that the displacement variations caused b Poisson s ratios are significant for the solid-dominated case, and reduce with decreasing solid stiffness. The pea displacement u has maimum value.95 for µ /K f and ν s., and decreases with decreasing µ /K f ratio. For µ /K f., the pea displacement u reduces to.4. The pea displacement u has maimum value equal to.36 for µ /K f. and ν s., and decreases with increasing of µ /K f ratio. The pea of u reduces to.96 for µ /K f and ν s.. Fig. 3.(d and (e show how the amplitudes of u, decrease with decreasing solid stiffness, and are alwas smaller than the amplitudes for elastic medium. In contrast, for u components, the pea amplitudes are larger than those for the elastic case, ecept for µ /K f Surface strains From parts (f and (g in Fig. 3.7 through Fig. 3., it is seen that the variations in surface strain with respect to Poisson s ratio are significant for γ components for soliddominated case, and reduce with decreasing solid stiffness. The surface strain variations are alwas significant in γ components. The pea surface strain γ has the maimum value equal to.85 for µ /K f and ν s., and decreases with decreasing µ /K f ratio. For µ /K f., the pea of γ reduces to.96. It is found that the surface strain γ is no longer equal to zero for verticall incident P-wave. The pea surface strain γ has the

43 maimum value equal to 8.8 for µ /K f. and ν s., and decreases with increasing µ /K f ratio. The pea γ reduces to.543 for µ /K f and ν s.3. Fig. 3.(f and (g show how the amplitudes of γ, decrease with decreasing solid stiffness, and are alwas lower than the amplitudes for elastic medium. For γ components, unlie for the elastic medium, the surface strains are not zero for vertical incidence. In the case of soft solidseleton, it is found that the pea surface strains γ eceed the pea surface strains for elastic medium Rotations From parts (h and (i in Fig. 3.7 through Fig. 3., it is seen that the variations in rotation in terms of Poisson s ratios are significant for the solid-dominated case, and reduce with decreasing solid stiffness. The ξ /u ratio equals.993 for vertical incidence and decreases to.75 when ν s. and µ /K f. For µ /K f., the ξ /u remains.3 which is larger than in the open-boundar and elastic cases. The pea rotation ξ /u has the maimum value equal to.63 when ν s. and µ /K f, and decreases with decreasing µ /K f ratio. For µ /K f., the pea of ξ /u reduces to.8. Fig. 3.(h and (i show how the pea values of ξ /u var for different solid stiffnesses and tend to be insensitive to the incident angle with decreasing solid stiffness Stresses From parts ( and ( in Fig. 3.7 through Fig. 3., it is seen that the pea stress τ has the maimum value equal to.576 for ν s. and µ /K f, and decreases with decreasing µ /K f ratio. For µ /K f., the pea of τ reduces to for ν s.. The pore pressure σ has the same amplitude as the stress τ, but with opposite direction, due to the nature of boundar conditions. The pea pore pressure has the maimum value when ν s. and µ /K f, and a minimum value equal to.74 - for µ /K f. and ν s.4. Fig. 3.( shows how the pea stresses τ decrease with decreasing solid stiffness and are alwas lower than the corresponding stresses for the elastic medium

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57 3.5 Comparison of open-boundar and sealed-boundar cases Fig. 3. shows the displacements, surface strains, rotations, and stresses for elastic medium, for open-boundar and sealed boundar cases in porous medium for ν s.5 and µ /K f. (simulating soils. The pea values of displacement u, surface strains γ, γ, rotation ξ /u and stress τ for elastic medium are larger than those in porous medium. enerall, the response for sealed-boundar case is larger than the response for open-boundar case. For sealed-boundar case, the surface strain γ and stress τ are no longer equal to zero for verticall incident P-wave. For soft solid cases, it is found that the pea amplitudes of γ and τ are induced b vertical incidence. The pea value of u for sealed-boundar case eceeds the pea value, equal to two, for the elastic medium. In this case, the amplification of sealed-boundar case is larger than for the elastic medium in the vertical displacement component. From Fig. 3.(f and the parts (i in Fig. 3. through Fig. 3., it is found that the ratio of rotation to vertical displacement described b Eq. (3- is consistent with the case for elastic medium (Trifunac, 98. ξ sin (3- θ u From Eq. (3-, it is seen that the ratio of rotation to vertical displacement is associated with the ratio of wave number to and the incident angle for P-wave

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60 3.6 Conclusions The wave propagation amplitudes in fluid-saturated porous medium are influenced significantl b the stiffness and Poisson s ratio of the solid-seleton, and b the boundar drainage. The solid stiffness dominates the amplitudes of elastic waves in porous medium. For solid-dominated case (large µ /K f ratio, the porous medium behaves lie elastic solid medium. The porous medium behaves lie a fluid medium for fluid-dominated case (small µ /K f ratio. enerall, the variations caused b the Poisson s ratio are significant for the solid-dominated case, but are less significant with decreasing solid stiffness. For open (drained boundar, the pea amplitudes of displacements, surface strains, rotations, and stresses are alwas smaller than the amplitudes for an elastic medium. In this case, the fluid plas a passive role in the poroelastic sstem and reduces the dnamic response of the solid-seleton. For sealed (undrained boundar, the first remarable effect is that the pea values of normal displacement u are no longer equal to two as in the elastic and the drained-boundar cases. The displacement responses for most µ /K f ratios are amplified more than in the case of the elastic medium. Secondl, the surface strains γ are no longer zero for vertical incidence, and the pea values of γ eceed the values for elastic medium when the solid-seleton is soft. The pore pressures are not zero, and have the same amplitudes as the normal stresses τ for the sealed boundar case. For soft and unconsolidated solids, it is found that the pore pressures eceed the stresses τ (e.g. µ /K f DQν s DOWKRXJK their amplitudes are small. In this case, if the solid-seleton is formed with cohesionless granular particles, the initiation of liquefaction is possible because the pore fluid pressure ma cause loss of effective stress between the particles near the ground surface

61 4 Response of a fluid-saturated porous half-space to an incident plane SV-wave 4. The model Consider a fluid-saturated porous half-space subected to a plane SV-wave with incidence angle θ as shown in Fig. 4.. We assume the wave has harmonic time dependence and, for brevit, we omit the i t e ω terms. Air Porous medium θ θ θ r φ i ψ θ r φ r ψ Fig. 4. Fluid-saturated porous half-space subected to an incident SV-wave. The incident SV-wave potential is i ψ i ( sin θ + cosθ b e (4-a and the reflected wave potentials are φ φ ψ r r r i ( sinθ cosθ ae (4-b i ( sin θ cosθ ae (4-c i ( sin θ cosθ b e (4-d The apparent velocit along the half-space surface is V V V V (4- sin θ sin θ sin θ

62 From Table., it is seen that the fast P-wave is alwas faster than S-wave. If the incident angle is beond the critical angle θ cr, the reflected angle of the fast P-waves becomes comple, because sinθ (V /V sinθ >. The critical angle for the fast P- wave (the first critical angle is determined from θ sin ( V / V (4-3 cr It is seen that the critical angle depends on the ratio of P-wave and S-wave velocities. A critical angle for the slow P-wave (the second critical angle does not usuall eist, because V /V > onl in a soft, unconsolidated solid-seleton (see Table., µ /K f DQν.4. We rewrite Eq.(4- as follows. i ψ φ φ ψ r r r i+ ν b e (4-4a iν ae (4-4b iν ae (4-4c b e iν, (4-4d where sinθ sinθ sinθ is the apparent wave number, and i cotθ i ν (4-5a ν ν i when V V ( θ θ cr i cotθ (4-5b when V < V ( θ cr < θ i when V V ( θ θ cr i cotθ (4-5c when V < V ( θ cr < θ The value of ν is imaginar for θ,. The value of ν is imaginar when θ θ cr, and the potential remains as a harmonic wave. The value of ν becomes a negative real number when θ cr θ, and the potential describes a surface wave decaing eponentiall with depth ( 6LPLODUFRQLWLRQVDOVRKROIRUν. The boundar conditions will be used to determine the unnown amplitude coefficients. To investigate the ground responses for drained and undrained boundaries, both open and sealed boundaries will be considered in the following

63 Open-boundar case This case has been studied b Deresiewicz (96. Appling the conditions for open-boundar in Table.4, we obtain the boundar conditions in Cartesian coordinates as σ τ τ (4-6 Substituting Eq.(4-4 into Eq.(4-6, the equations can be simplified as in Eq.(4-7, where the notation used is summarized in Appendi A. 6, 6,,,,, b b a a (4-7 The three amplitude coefficients are determined as 3 ] det[ / / / c c c b b b a b a (4-8 where ν ( 4 S i c (4-9a ν ( 4 S i c (4-9b ] ( ( [ ( S M S M S S S S c + + ν ν ν ν (4-9c ] ( ( [ ( 4 4 ] det[ S M S M S S S S + ν ν ν ν (4-9d with M, S (, defined in Appendi A. In Eq.(4-8 and Eq.(4-9, the amplitude coefficients are real for incident angle θ < θ cr. From Eq.(4-9a and Eq.(4-9b, it is seen that no P-waves are reflected when, in which case, the incident angle θ equals 45. This result is consistent with the elastic solid media (Achenbach, 973. From Eq.(4-9c and Eq.(4-9d, the ratio

64 of / b b eits in the case of incident SV-wave beond the critical angle of the slow P-wave (both ν and ν are real in a soft, unconsolidated solid-seleton (V /V >. 4.. Sealed-boundar case The following boundar conditions for sealed-boundar case are obtained from Table.4. + U u τ σ τ (4- Substituting Eq.(4-4 into Eq.(4-, the equations can be simplified to Eq.(4-, where the notation used is summarized in Appendi A , 4,,, 6,, 6,, ( ( ( ( f b b a a f f f (4- The three amplitude coefficients are determined from 3 ] det[ / / / c c c b b b a b a (4- where ν ν 3 ] ( [( 4 f f f i c (4-3a ν ν 3 ] ( [( 4 f f f i c (4-3b ν ν ν ν ν ( ] ( ][ ( [( ] ( ][ ( [( f f S M f f f S M f f f c (4-3c ν ν ν ν ν 3 3 4( ] ( ][ ( [( ] ( ][ ( [( ] det[ f f S M f f f S M f f f (4-3d In Eq.(4- and Eq.(4-3, the amplitude coefficients are real for incident angle θ < θ cr. From Eq.(4-3a and Eq.(4-3b, it is seen that no P-waves are reflected when

65 ( ( 3 f f f. From Eq.(4-3c and Eq.(4-3d, the ratio of / b b eits in the case of incident SV-wave beond the critical angle for the slow P-wave (both ν and ν are real numbers in a soft, unconsolidated solid-seleton (V /V > Surface Response The following responses of the solid-seleton at the half-space surface are calculated. Displacements Once the amplitude coefficients are determined, the displacement responses of the fluid-saturated porous medium at the ground surface can be obtained as i e b a a b u u 4 4, 4, 4 3 3, 3, 3. (4-4 Surface strains The surface strains are defined b the following derivatives of displacements / / u u γ γ (4-5 Substituting Eq. (4-4 into Eq.(4-5, we obtain i e b a a b 8 8, 8, 8 7 7, 7, 7 γ γ. (4-6 Surface rotation The surface rocing for an incident SV-wave is i e b b u u ( ψ + (4-7

66 and the normalized rotation is (Trifunac, 98; Lee and Trifunac, 987 ξ π ψ ( λ / π i ( b + b e (4-8 i i e where λ is the wavelength of SV-wave, and i i e is the displacement induced b incident SV-wave. It is seen that the rotation is phase-shifted relative to the incident motion b π/ for θ < θ cr. Stresses The stresses at the ground surface can be obtained as τ τ τ σ b a e a b,,,, µ 5 5, 5, 5 6, 6, i ( An eample of surface response analsis A fluid-saturated porous half-space subected to an incident SV-wave with unit amplitude is investigated for surface response. We assume porous medium with 3% porosit consisting of spherical solid particles (τ r.5, ρ s /ρ f.7. The relations among material constants in Table. are used to calculate the responses for various Poisson s ratios of the solid-seleton and µ /K f ratios. The surface response for the open boundar and sealed boundar cases are discussed in sections 4.3 and 4.4, respectivel. Fig. 4. through Fig. 4.5 and Fig. 4.7 through Fig. 4. show the amplitude coefficients, displacements, surface strains, rotations and stresses versus incident angle for various solid materials with different Poisson s ratios for the open-boundar cases and sealed-boundar cases, respectivel. Fig. 4.6 and Fig. 4. show the effects of variable solid stiffness, with the same Poisson s ratio, ν.5, for the open-boundar case and sealed-boundar case, respectivel. Fig

67 4. illustrates the effects of open-boundar and sealed-boundar for porous material with µ /K f ratio. and ν s.5. It is noted that the amplitude coefficients become comple numbers beond the critical angle. The displacements are normalized b a factor, which gives the displacement intensit of the incident SV-wave. The surface strains are normalized b a factor b a factor, which is the strain intensit of incident SV-wave. The stresses are normalized, which is the stress intensit induced b incident SV-wave (Pao and Mao, 97. All the above comple amplitudes are represented in the plots b their absolute values. 4.3 Results and discussion for open-boundar case 4.3. Amplitude coefficients From parts (a, (b and (c in Fig. 4. through Fig. 4.5, it is seen that no P-waves are reflected at incident angle of 45. The pea amplitude coefficients of the slow P- waves are much smaller ( - than those of the fast P-waves for µ /K f and increase to.49 for µ /K f.. The coefficient variations with respect to Poisson s ratio are significant for the solid-dominated case (large µ /K f ratio, and diminish with decreasing solid stiffness. The amplitude spies at the first critical angle also diminish with decreasing solid stiffness. Fig. 4.6(a, (b and (c show how the amplitude coefficients var with decreasing solid stiffness, and relative to the amplitudes for elastic medium Displacements From parts (d and (e in Fig. 4. through Fig. 4.5, it is seen that the effects of variations in Poisson s ratio are significant for displacement amplitudes in the soliddominated case. These effects diminish with decreasing solid stiffness. The displacement u is zero at incident angle of 45. The spies of the displacement u at the first critical angle have a maimum value 5.34 for µ /K f and ν s.. These spies diminish with decreasing of µ /K f ratio. The pea displacements u, are between.55 and.69. Fig. 4.6(d and (e show how the amplitudes of u decrease with decreasing solid stiffness

68 4.3.3 Surface strains From parts (f and (g in Fig. 4. through Fig. 4.5, it is seen that the surface strains are zero for incident angle of 45. The surface strain γ variations for different Poisson s ratios are significant for the solid-dominated case, and reduce with decreasing of solid stiffness. The variations of surface strain γ for different Poisson s ratios are significant for all conditions. The spies of surface strains γ at the first critical angle have a maimum value 3.37 for µ /K f and ν s.. These spies diminish with decreasing of µ /K f ratio. The pea surface strain γ has the maimum value of.84 for µ /K f and ν s.4, and decreases with decreasing µ /K f ratio. For µ /K f., the pea of γ reduces to.6. Fig. 4.6(f and (g show how the surface strains decrease with decreasing of solid stiffness and are compared to the amplitudes for the elastic medium Rotations The parts (h and (i in Fig. 4. through Fig. 4.5 show that the normalized rotation and its phase with respect to the angle of incident motion. It is seen that the dependence of rotation on Poisson s ratio is significant for the solid-dominated case, and reduces with decreasing solid stiffness. The pea normalized rotation ξ alwas equals at incident angle of 45, for all the cases. The phase equals to π/ when θ < θ cr. From parts ( and ( in Fig. 4. through Fig. 4.5, it is seen that the amplitudes ξ / u blow up at incident angle of 45, due to u, and the ratio ξ /u equals sinθ for all conditions. Fig. 4.6(h and (i show how the rotations var with variations in solid stiffness, and relative to the rotations in the elastic medium Stresses From parts (l in Fig. 4. through Fig. 4.5, it is seen that the dependence of stress on Poisson s ratios is alwas significant. The spies of stress τ at the first critical angle have maimum value of 7.5 as µ /K f and ν s., and diminish with decreasing µ /K f ratio. For µ /K f., the pea τ reduces to 3. Fig. 4.6( shows how the amplitudes of τ var with the solid stiffness, and relative to the amplitudes in an elastic medium

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84 4.4 Results and discussion for sealed-boundar case 4.4. Amplitude coefficients From parts (a, (b and (c in Fig. 4.7 through Fig. 4., it is seen that the coefficient variations with respect to Poisson s ratio are alwas significant for an set of conditions. The incident angles leading to no reflections of P-waves are no longer 45, because of the solid-fluid interaction. The amplitude spies at the first critical angle diminish with decreasing solid stiffness. The pea amplitude coefficients of the slow P- waves are much smaller ( - than those of the fast P-waves for µ /K f and increase with decreasing µ /K f ratio. The effects of the second critical angle become significant for soft, unconsolidated solid-seleton (µ /K f DQν s.4. The spie of amplitude a at the second critical angle equals.99 as µ /K f. and ν s.4. Fig. 4.(a, (b and (c show how the amplitude coefficients var with decreasing solid stiffness, and relative to the amplitudes for an elastic medium Displacements From parts (d and (e in Fig. 4.7 through Fig. 4., it is seen that the displacement variations caused b Poisson s ratios are significant for the solid-dominated case. These effects diminish with decreasing solid stiffness ecept for µ /K f DQν s.4. The zero-amplitude displacement u near incident angle of 45 can be found in the soliddominated case, but not in the fluid-dominated case. The amplitude spies at the first critical angle also diminish with decreasing solid stiffness. The effects of the second critical angle become significant for soft, unconsolidated solid-seleton (µ /K f DQ ν s.4. The spie of displacement u, at the second critical angle, approaches. as µ /K f. and ν s.4. The pea displacements u, are between.66 and.69. Fig. 4.(d and (e show how the amplitudes of u var with respect to the solid stiffness. It is found that the pea value of u for the sealed-boundar case is larger than in the case for an elastic medium

85 4.4.3 Surface strains From parts (f and (g in Fig. 4.7 through Fig. 4., it is seen that the variations in surface strain with respect to Poisson s ratio are significant for the solid-dominated case, and reduce with decreasing solid stiffness ecept for µ /K f DQν s.4. The zeroamplitude strain γ near incident angle of 45 can be found onl for µ /K f. The amplitude spies at the first critical angle also diminish with decreasing solid stiffness. The effects of the second critical angle become significant for soft, unconsolidated solidseleton (µ /K f DQ ν s.4. The spies at the second critical angle approach.86 for γ component, and.964 for γ component as µ /K f. and ν s.4. Fig. 4.(f and (g show how the amplitudes of surface strains var with the solid stiffness. It is found that the pea value of γ for the sealed-boundar case is larger than that for an elastic medium when the solid is soft Rotations The parts (h and (i in Fig. 4.7 through Fig. 4. show the normalized rotation and its phase versus incident angle. It is seen that the variations in rotation in terms of Poisson s ratios are significant for the solid-dominated case. Unlie the open-boundar case, the pea normalized rotation ξ no longer equals. The phase equals π/ when θ < θ cr. The effects of the second critical angle become significant for soft, unconsolidated solid-seleton (µ /K f DQν s.4. From parts ( and ( in Fig. 4.7 through Fig. 4., it is seen that the amplitudes ξ / u are large around 45 since u is small, and the ratio of ξ /u equals to sinθ for all conditions. Fig. 4.(h and (i show how the rotations var respect to solid stiffness and relative to the amplitudes for the elastic medium Stresses From parts (l and (m in Fig. 4.7 through Fig. 4., it is seen that the stress dependence on Poisson s ratios is alwas significant. The spies at the first critical angle have maimum value of.4 for stress τ and.3 for pore pressure σ when µ /K f and ν s.. These effects diminish with decreasing µ /K f ratio. The effects of the second critical angle become significant for soft, unconsolidated solid-seleton (µ /K f DQ