vel & atten of comp waves—nearly sat soils

ELSEVIER

Soil Dynamics and Earthquake Engineering 12 (1993) 391-401 0 1994 Else&r Science Limited

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Velocity and attenuation of compressional waves in nearly saturated soils

J.P. Bardet & H. Sayed

Civil Engineering Department, Kaprielian Hall 210, University of Southern California, Los Angeles, California, 90089-2531, USA

Communicated by M.D. Trifunac

(Received 16 August 1990; revised version received 2 August 1993; accepted 18 August 1993)

Based on the two-phase theory of Biot, we present exact and approximate expressions for the velocity and attenuation of compressional waves within nearly saturated poroelastic media. We use the approximate solutions to model the low- frequency compressional waves within nearly saturated soils. The model accounts for the effective stress, degree of saturation, and void ratio, and is capable of describing experimental results on Ottawa sand. The three-phase theory of Vardoulakis and Beskos and the two-phase theory of Biot similarly describe the velocity and attenuation of compressional waves in most soils. However, the former theory breaks down for nearly saturated gravels and dense sands.

1 INTRODUCTION

It has long been recognized that compressional waves, in contrast to shear waves, propagate in saturated soils with a velocity that is strongly affected by the water filling the interstices of soil grains. Hardin and Richart’ applied the two-phase theory of Biot,’ and studied the influence of the confining pressure on compressional waves in saturated sands. At first, Hardin and Richart’ measured similar velocities for the compressional waves within dry and saturated sands subjected to resonant column tests. Later, Allen et d3 obtained quite different speeds in dry and saturated sands by fully saturating sands in pulse chamber experiments. They established the effects of void ratio and degree of saturation on the velocity of compressional waves within nearly saturated sands. Ishihara4,5 also applied the Biot theory, and showed the effect of the compressibility of the solid-fluid system on compressional waves.

However, another physical aspect of compressional waves in nearly saturated soils, namely their attenuation, has been largely underestimated in soil dynamics, which is surprising in view of the extensive geophysical studies surveyed by 0gushwitz.6 Past works in soil dynamics have unanimously neglected the attenuation of compressional waves which arise from the interaction between soil grains and interstitial water. No theoretical model has yet been proposed to account for the combined effects of degree of saturation, void ratio, and confining pressure on both the velocity and attenuation of compressional waves in nearly saturated soils.

The existing models for the velocity of compressional waves 1,3,4 are based on the Biot theory2 which was adapted by Bear and Corapcioglu7 Bowen,* Ghaboussi and Wilson,’ Green and Naghdi,” Mei and Foda,” and Zienkiewicz et a1.‘2 Departing from the two-phase theory of Biot, Vardoulakis and Beskost3 also proposed a theory for three-phase materials applicable to nearly saturated materials. They compared the aforementioned adaptations, showed that they coincide with the Biot theory in the case of fully saturated poroelastic materials, but did not examine the effect of their theory on compressional waves. Hereafter, the theories of Biot (1956) and Vardoulakis and Beskos (1986) are referred to as the B and VB theories, respectively.

Following the introduction, the second section reviews the one-dimensional governing equations of the B and VB theories. The third section provides exact and approximate solutions for the velocity and attenuation of compressional waves. The last sections compare the results of the B and VB theories, and present a model for compressional waves in nearly saturated soils.

2 REVIEW OF TWO-PHASE FORMULATIONS

2.1 Definitions

In general, soils are three-phase media that consist of solid, fluid and gas constituents. The relative

proportions of constituent volumes are characterized

391

392 J.P. Bardet, H. Sayed

by the porosity n and the degree of saturation, S:

where Vt is the total volume, V,, is the pore volume, and Vf is the fluid volume. In the case of nearly saturated soils (0.95 5 S 5 l), there is only one homogeneous fluid phase because the gas is dissolved in the fluid. The specific gravity of the solid constituent is

G, = fi Pf

where ps is the solid unit mass, and pr is the fluid unit mass. In one dimension, the solid displacement is denoted U, while the fluid displacement is 21. Provided that the individual solid particles are much more incompressible than the assemblage of grains, the effective stress c’ of soil mechanics is14

c/=0--p (3)

where cr is the total stress tensor in the soil-water mixture, and p is the fluid pressure. In linear isotropic poroelastic solids, the change of effective stress AU’ is

A&Dd” ax

where the modulus D depends on Young’s modulus, E,

and Poisson’s ratio, u:

E(l - u)

D = (1 - 2v)(l + v) (5)

The pressure change Ap in the air-fluid mixture results from volumetric strain:

Ap=Dfg (6)

where Df is the bulk modulus of the air-fluid mixture.

2.2 One-dimensional governing equations

Assuming Darcy’s law with permeability k, and incompressible particles, Vardoulakis and Beskos13 showed that the one-dimensional dynamic response of nearly saturated poroelastic materials in the low frequency range is described by the following equations:

1-n azu D+n

a2v Df @fDf@

>

= (1 - n)p, $ + npf $

a27J vfg &J au =“Pfs+k( )

---

at dt

(7)

(refer to Ref. 13 for derivations), g is the earth gravity. While Biot,2 Ghaboussi and Wilson,’ Van der Kogel,” Mei and Foda” and Zienkiewicz et al.‘* obtained eqn (7) exactly, instead of eqn (8), they found the following equation:

a*?J npfg dv 82.4 =Pf@+T( )

--- dt dt (9)

Equations (8) and (9) only differ in the inertial term of the fluid constituent; this term is npf in eqn (8), and pf in eqn (9). This difference comes from the constitutive assumptions made in the B and VB theories, which consider two-phase and three-phase materials, respectively. These different coefficients will later be shown to have more effect on the attenuation of compressional waves than on their velocity. Equations (7) and (8) become

i a2U a2w ;j@+s=“i (( > a224 2 t+x g-$+$ > (10)

d2U a2w g aw n@+s+kz=C:

whereas eqn (9) becomes

a214 1 a2w g dw

(

1 a224 1 d2W @+;z+kz=C; --+;s

n 8x2 1 (12)

where

w = n(v - u), x = g Df ’

c, is

and /3 = 1

n + (1 - n)GS

the wave velocity in the fluid:

I% C w= - V Pf

(13)

(14)

2.3 Particular case of zero permeability

The particular case of zero permeability is a useful point of reference. When k = 0, the interstitial fluid and the solid matrix move simultaneously without a relative displacement, i.e. u = v; eqns (10) and (12) become

a2U 2 a2u --C aX2- Odt2 (15)

where co is the wave velocity in the homogeneous mixture of stiffness D + ( l/n)Df and unit mass

(1 - n)pS + wf:

(8) co = c, Jc ) P x+; (16)

Compressional waves in nearly saturated soils 393

3 PROPAGATION OF COMPRESSIONAL WAVES

3.1 Exact expression

Compressional waves are plane harmonic waves with the following amplitude:

i

24(x, t) = A e -y ,iw(t-i)

(17) w(x, t) = Be

-y ,iw(f-:)

where i = a, c is the wave velocity (phase velocity), 6 its absorption coefficient, and w the circular frequency of the harmonic signal. It is appropriate to introduce I+!X

$=-2-i!!! C

(18)

so that

c=--&: and S = - gTtil (19)

where %‘{$} and JJ{$} are the real and imaginary parts of $J. By substituting eqn (17) into eqns (11) and (12), two linear equations with two unknown coefficients A and B are obtained for the VB theory:

+*(x+;)++)+B(l +$)=O (20)

,(,,$)+B(l+$-if-)=0

In order to have a non-trivial solution, the determinant of eqn (20) must be equal to zero, which gives the following characteristic equation:

;C+ag2+b=0

where the coefficients a and b are

1 0=x+--l-i&

np kw

Equation (21) always has four complex roots:

(21)

P-4

(23)

and $2,4 = * -an + Jn2a2 - 4bnx

2x

Therefore, there are two waves that have the velocity

of *cl and fc2, and the amplitude decay of f6i and *s,:

CU’ and 61,2 = f-

la{*,,21 I (24)

In the B theory, the coefficients a and b of eqn (21) are different from those in eqn (22):

and b=i_ 1 -i-&I nP kw P

(25)

3.2 Approximate expression

When (kw)/g < 1, eqn (24) is approximated as follows:

‘cl=co(l+O(~))

and Sl =bok(l+Oe))

’ cz=cd&l+G(!;))

and &=&~(l+o(!;))

(26)

where the velocities co and cd, and the amplitude decay & are identical for the B and VB theories:

co = c, , c,=&J;;~; (27)

and 6, = cd

However, the amplitude decays b. of the B and VB theories are different. For the B theory, So is

X- e(, - l))=

whereas for the VB theory, So is

nx*+

+(I -n)2G (G

.2 ss -

(28)

1) (29)

Equation (26) defines the velocity and attenuation in terms of the physical parameters n, ,0, x, k and w. The coefficients co, cd, so and 6, depend on n, ,& and x, but not on k and w. Figures 1 and 2 show the variation of

cl/co, cz/cd, 6i/bo, and 62/& as a function of (kw)/g for three extreme values of x: 0.001, 1 and 1000. cl, c2, &, and S2 are calculated exactly by the VB theory (eqn (22)) while co, cd, So, and & are determined by using eqns (27) and (28). In Figs 1 and 2, the lines are straight, and coincide for (kw)/g < 0.1. For (kw)/g > 0.1, the lines become curved and depend on x, implying that the

J.P. Bardet, H. Sayed

.014 ““’ ““‘8 ““9 ...A .OOl .Ol .l 1 10

k w Ig

Fig. 1. Dimensionless wave velocities cl/c0 and c2/cd versus (kw)/g for various compressibility ratios x.

approximation of eqn (26) holds only for (kw)/g < 0.1. Similar results pertain to the B theory.

The first wave, with velocity cl and attenuation 6’, is not dispersive because co does not depend on w. Its attenuation 6, is inversely proportional to k and w, which implies that this wave is hardly attenuated when kwlg < 1. However, the second wave is dispersive because c2 and S2 are proportional to Jc;. Due to the term (kw)/g, the second wave is slower and more attenuated than the first wave. Hereafter the first and second waves are referred to as the fast and slow wave, respectively.

.OlJ “““’ “““8 “““‘I “....A .OOl .Ol .l 1 10

k w Ig

10000

T 1000

s s

3

5 100’

h C x

Ti P+

10

1 .00001 .OOOl .OOl .Ol .l 1

Air concentration (1-S)

Fig. 2. Dimensionless wave dispersions 6’ /So and &/64 versus Fig. 3. Velocity of compressional wave in water versus degree (kw)/g for various compressibility ratios x. of saturation, S, for various absolute hydrostatic pressures, p.

4 ATTENUATION OF FAST COMPRESSIONAL WAVES

When the fast wave propagates, its amplitude increases for So < 0, decreases for So > 0, and remains constant for ISol -+ co. The wave attenuation may be examined for the range of material parameters representative of nearly saturated soils. The attenuation of eqns (26), (28) and (29) depend on soil properties k, n, /3 and D, and the water properties c, and Df.

4.1 Water properties

The interstices of nearly saturated soils are assumed filled with a homogeneous fluid made of water and gas. The unit mass of the interstitial fluid, pf, is practically independent of S, and equal to the unit mass of water: pW = 1 g/cm3. However, the compressibility of the

interstitial fluid, /3; (Pf = l/Of) strongly depends on S, and is related to pf, the compressibility of the fluid alone:14

p; =pf+y

where p is the absolute pressure in the fluid. Allen et a1.3

used a different expression for p;, which however gives similar results when S is between 0.95 and 1. Figure 3 shows the wave velocity c, in the air-fluid mixture which is calculated by eqns (14) and (30) as a function of the air fraction 1 - S for various absolute fluid pressures. The theoretical velocity c, is in agreement with the experimental measurement of sound waves in air-fluid mixtures.‘6”7 The wave velocity is maximum in pure water, about 1500m/s. Strongly influenced by the air content 1 - S, it drops to 300m/s for S = 99.9%,

Theory 1

- 1OOOkPa

‘-‘- 100 kPa _._ ._._..... 10kPa

------ 1 kPa

Experiments

A \ 109 kPa (Campbell and Pitcher,l955) 0 109kPa (Silberman,1957)

Compressional waves in

---‘-- 100 kPa . .._....._ 1o kPa

------ 1kPa

109 kPa (Silberman, 1957)

109 kPa (Campbell and Pitcher, 1955)

.00001 .OOOl .OOl .Ol .l 1

Air concentration (1-S)

Fig. 4. Bulk modulus of water versus degree of saturation, S, for various absolute hydrostatic pressure, p.

lOOm/s for S = 99% and 30m/s for S = 90%. As shown in Fig. 4, the air-fluid modulus Df is 2200MPa for pure water, 100 MPa for S = 99.9%, 1OMPa for S = 99% and 1 MPa for S = 90%. The effect of S on Df

is reduced, but is not eliminated, by increasing the absolute pressure p. A small decrease of S from 1 to 0.95 multiplies x by a factor of 103. Therefore, the degree of saturation S strongly influences compressional waves within nearly saturated materials.

4.2 Soil properties

Table 1 summarizes the typical range of variation for soil parameters. Typically, the porosity n varies between 0.12 and 0.55 for sands, gravels and silts and between 0.3 and 0.75 for clays.” The specific gravity of soil

grains G, has 2.2 and 3.2 for extreme values. The coefficient of permeability k has the largest range of variation of all the physical parameters. It varies from 0.01 m/s for gravel to lo-” m/s for clay. According to Scott,‘9 the bulk modulus D of soil skeletons varies typically between 0.05 and 200MPa. D depends on the effective pressure for all soils and on the overconsoli- dation ratio for clays. According to the ranges of variation for D in Table 1, and Df in Fig. 4, x may vary

nearly saturated soils 395

from 10e5 for saturated clays to lo2 for nearly saturated gravels. The ratio (kw)/g is calculated assuming that the circular frequency w varies in the range 1- 100 cycles per second, which is the range of vibration of interest in earthquake engineering.

4.3 Attenuation of compressional waves

As shown in eqn (28) the B theory finds that So is always positive. The fast wave is slightly attenuated in most cases, and becomes undamped in particular circum- stances. The attenuation vanishes independently of k

and w, when the denominator F of eqn (28) which is a perfect square, is equal to zero. As shown in Fig. 5 and Table 1, the values of n, x and G, corresponding to F = 0 are typical of dense sands and gravels.

For the VB theory, So is not always positive. The denominator F of eqn (29) is a function of n, x and G,:

x

+ (1 - n12 TWs - 1) (31)

1

1 !, ‘, '!i : \I\

0.6 - ! ‘,!.‘, \ \ ! 1 ; \‘(\ : \ ?’

n : \ 3.1 : ,\I : \!’ : \ ?I : \ ?\ 0.4 : ,I( : \ ?’ : \‘.I 1 ,..,...., ,.ll......_, ,.,,. 1

: \?I : !!\ : \?’ : \ ?I : \ ?I : \\\ \ ?’ 0.2

: : I\.\ : \ S.( : \ \\ I, : t.1, ‘L.\.:\\

'..>.Y,,, "..:L_\

.-....L. 0.0

10-4 10" 10'2 10-1 100 10' 101 10'

x Fig. 5. Values of x, n and G, generating undamped com-

. pressional waves for the Biot theory.

Table 1. Approximate range of variation for physical parameters of soils”

Soil type Porosity, G, Permeability Modulus, D kw n coefficient, k NW (S X1.0) (S 20.9)

(m/s) -T

Gravel 0.12-0.46 2.662.7 1o-2 200 10-l lo2 10~‘~10-3 Sand

dense 0.17-0.55 2.6-2.7 1o-4 50 lo-* 10’ 10~3-10-5

loose 1om6 5 1o-2 1 10-5-10-7

Silt 0.29-0.52 2.5-2.7 lo-* 2 1o-3 1 10-7-10~9 Clay 0.30-0.75 2.223.2 lo-‘O 0.05 1o-5 1o-2 1om9_10-12

'pr = 1.0g/cm3; l//?r = 2140MPa; c, = 1460m/s.


1.0

- Gs=l.Ol . . . . . . . . . Gaz2,

____---- Gs;~.J . .._..._. G1=.&

n

0.4 -

0.2 -

10-4 10.3 10.2 10" 100 10' 102 103 104

x

Fig. 6. Values of x, n and G, generating undamped compressional waves for the Vardoulakis and Beskos theory.

but is not a perfect square, and may change sign. F = 0,

F < 0, and F > 0 correspond to constant amplitude, amplification and attenuation, respectively. Figure 6 shows the transition F = 0 as a function of n and x for various values of G,. For a given G,, the waves are undamped on two curves, amplified in the region inside these two curves and attenuated outside. Waves are always attenuated in fully saturated soils since x is less than 0.1 in Table 1. However, they may become amplified in nearly saturated dense sands and gravels.

Since this latter case of unstable waves is improbable, one may conclude that the VB theory does not apply to all poroelastic materials. The finding in Fig. 6 does not abolish the VB theory, but completes it by specifying the range of its applicability in terms of admissible material constants n, x and G,.

The attenuation or amplification of a wave is also conveniently characterized by the distance c that it travels before dividing or multiplying its amplitude by two:

< = 1% Wgl~ol kw2

(32)

Figure 7 represents the variation of [ versus S for typical gravels (k = 0.01 m/s, G, = 2.66, D = 150 MPa, n = 0.33, p = 100 kPa and f = 1OHz). In the B theory, the minimum value of < for a 10Hz wave is 200m for S = 0.98, and < becomes infinite for S = 0.998. In the VB theory, the same wave is attenuated for S < 0.989 and S > 0.998, but is amplified for 0.989 I S 5 0.998. For instance, when S M 0.996, [ = 2 km, i.e. the wave amplitude is predicted to double every 2 km. Therefore, as shown in Fig. 7, the VB theory breaks down in the range 0.989 5 S < 0.998.

Both the V and VB theories fail to describe the attenuation of fast waves under particular conditions. It seems that dynamic transition from perfect saturation to near saturation cannot be achieved from general balance consideration. One may conjecture that the existing theories should be improved by adding constitutive assumptions about energy dissipation in compression waves.

0.98

----- attenuated (Vardoulakls 6 Beskos, 1666)

-_‘-‘-. amplified (Vardoulakis & Beskos, 1966)

_ 0.99

S degree of saturation

1 .QO

Fig. 7. Variation of C versus degree of saturation predicted by Biot and Vardoulakis and Beskos for nearly saturated gravels.

Cornpressional waves in nearly saturated soils

5 A MODEL FOR COMPRESSIONAL WAVES 5.2 Application to Ottawa sand

Based on the approximate solutions of Biot, a model is now proposed for the velocity and attenuation of fast and slow compressional waves in nearly saturated soils.

5.1 Model

The compressional wave velocity c of dry soils depends

on the effective stress a’ as follows:“20

Figure 8 shows the variation of wave velocity measured by Hardin and Richart’ for dry 20-30 Ottawa sand over a large range of confining pressure and initial void ratio. By using a Davidson-Fletcher and Powell nonlinear optimization procedure with23 m = 0.25, it is found that A = 168.28 m/s and e, = 1.6284 when c is expressed in m/s and cr’ in kPa. As shown in Fig. 8, eqns (33) and (34) reproduce well the experimental data.

!??I c = w#J (33)

where the exponent m is close to l/4 and the constant we depends on the initial void ratio eo:21,22

Figure 9 compares theoretical and experimental results for atmospheric water pressure (p = 100 kPa). The experimental points are compiled from wave velocity measurements3,‘* or calculated from stiffness

measurements.*’ The model predictions are calibrated

for a dry medium-dense Ottawa sand (ee = 0.55). As shown in Fig. 9, the model describes well the wave velocity cl for S = 1. It also predicts that S has a very important effect on cl. A decrease in S from 1 to 0.999 causes a large decrease in c,, especially at effective stresses smaller than 100 kPa. When S decreases further toward 0.95, cl approaches the wave velocity c of dry sands (S = 0). However, the curves corresponding to S = 0.98 and 0.95 intersects the S = 0 curve instead of merging asymptotically with it; they are obtained by using the saturated unit weight psat instead of the dry unit weight pd. For effective stress larger than 100 kPa, practically all the curves associated with S between 0.999 and 0.95 coincide with the S = 0 curve. This extreme sensitivity of the wave velocity cl to S explains why Hardin and Richart’ measured similar velocity

wo = A ec - e0

1 + ee

where A and e, are two material constants. Since the dry

unit mass pd is

The modulus of dry soils is

(36)

Equations (26)-(28) (30), (34) and (36) provide a model for the velocity and attenuation of the fast and slow compressional waves in nearly saturated soils. The model constants are A, e,, eo, m, G, and k and depend on the variables S, p, w and 0’. These constants will now

be calibrated for the Ottawa sand.

397

Q No.25 (e=0.55-0.56)

l No.23 (e=0.65-0.63)

. No.27 (ez0.57)

e No.22 (ez0.58)

. No.26 (e=0.54-0.56) q No.24 (e=0.56-0.57)

A No.21 (e=0.53-0.59) b No.lc (ezO.5)

. No.12 (ez0.52)

- Theory (ez0.50)

-- - Theory (ez0.64) =I-

1000

Effective confining pressure (kPa)

Fig. 8. Velocity of compressional wave in dry Ottawa sand versus effective confining pressure for various initial void ratios (after Ref. 1).


1000

100

J _____ s.o.95

0 Dry (Lambe 6 Whitman. 1982)

A Saturated (Allen et al., 1980)

A Dty (Ko 6 Scott, 1967)

0 Saturated (Lambe 6 Whhman, 1982)

lo- “““I . . .“‘,I’ “““I “.‘.“I . ‘.,.A .l 1 10 100 1000 10000

Effective stress (kPa)

Fig. 9. Velocity of fast wave in Ottawa sand versus effective confining pressure for various degree of saturation (p = 100 kPa).

for compressional waves within dry and saturated sands.

As in Fig. 9, Fig. 10 shows the variation of cl with a large absolute pressure (p = 1000 kPa). cl is slightly less influenced by the degree of saturation in Fig. 10 than in Fig. 9. The S = 0.9999 and S = 1 curves almost

coincide. However, a variation of S from 1 to 0.999 is still capable of causing a large decrease of cl, which implies than an increase in pressure p does not eliminate the effect of degree of saturation on the velocity of compressional waves during laboratory experiments.

P = 1000 kPa

_____

0 A A 0

Dry (Lamba 6 Whitman, 1982)

Saturated (Allen et al., ISSG)

Dry (Ko 6 Scott, 1967)

Saturated (Lambe 6 Whltman, 1962)

,” , .._., . . . . . ., .., .l 1 10 100 1000 10000


Fig. 10. Velocity of fast wave in Ottawa sand versus effective confining pressure for various degree of saturation (p = 1000 kPa).


5e-8 f q 1 Hz C-.\

, / \ : /I

\ .‘I-“-- 6=0.9999 \

: I \ 1’ ‘\

‘\ __________ l&g.gg ‘\

‘\ _._._._._._., I&o.g*

z ‘\

‘\ \. to

.l 1 10 100 1000 100 00


Fig. 11. Attenuation of 1 Hz fast wave in Ottawa sand versus effective confining pressure for various degrees of saturation.

Figure 11 shows the variation of attenuation l/6’ 2350 km. From a practical point of view, it is concluded versus effective stress (T’ for a 1 Hz wave (p = 100 kPa). that the fast wave is not attenuated when it propagates The permeability k of Ottawa sand is fixed to within saturated Ottawa sand. 8 x lop5 m/s (Ref. 18) and its specific gravity G, is Figure 12 shows the velocity c2 as a function of 2.66. The maximum value of l/S1 is 4.7 x lo-* s/m and effective stress 0’. The slow wave is much slower than the corresponding minimum value of < is equal to the fast wave; it does not exceed 14m/s. For S = 0.99

16

14

f=l Hz

10 100 1000


Fig. 12. Velocity of 1 Hz slow wave in Ottawa sand versus effective confining pressure for various degrees of saturation.


2

z UJ

N (& =;

1

0

f = 1 Hz \ \ \ \

‘\ _____----- s=0*gg

‘\

.l 1 10 100 1000 10000


Fig. 13. Attenuation of 1 Hz slow wave in Ottawa sand versus effective confining pressure for various degrees of saturation.

the wave velocity is about 2m/s and is practically independent of 0’. As shown in Fig. 13, the attenuation l/S, increases as S decreases and becomes independent of 0’ for 0’ > 100 kPa. Since the average < is 0.1 m, the slow wave is much more attenuated than the fast wave. To our knowledge, there are no experimental data points on any soils to calibrate the proposed model for slow waves. The only experimental result on slow waves24 was obtained for fused glass beads, and at a very high frequency (500 kHz); it does not pertain to the frequencies encountered in soil dynamics and earthquake engineering. There is a definite need for experimental research to exhibit slow compressional waves in nearly saturated soils.

6 CONCLUSION

The two-phase theory of Biot2 was used to derive exact and approximate expressions for the velocity and attenuation of compressional waves within nearly saturated poroelastic media. Based on these approxima- tions, a model was constructed for compressional waves in nearly saturated soils. This model accounts for the effective stress, degree of saturation, and void ratio, and is capable of simulating the experimental results on Ottawa sand. The wave propagation was also analyzed by using the three-phase theory recently proposed by Vardoulakis and Beskos.13 Biot finds that compressional waves are slightly attenuated in most saturated soils,

and become undamped under particular conditions. Vardoulakis and Beskos’ theory predicts similar results in most soils, but breaks down in nearly saturated gravels and dense sands. The results aforementioned are useful for soil dynamics and earthquake engineering; they are especially relevant to the dynamic analysis of ground motion amplification in geotechnical engineering.

ACKNOWLEDGMENTS

The financial support of the National Science Foundation is acknowledged (grant CEES-8800735). The authors are thankful to the reviewers for their comments, and to M. Harris for proofreading the manuscript.

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vel & atten of comp waves—nearly sat soils

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