^ anc FILE cie DNA-TR-87-42

UNDRAINED COMPRESSIBILITY OF SATURATED SOIL

o Q S. E. Blouln «^j- J. K. Kwang

Applied Research Associates, Inc. 00 New England Division 09 Box 120A r" Waterman Road <J South Royalton, VT 05068

I Q ^ 13 February 1984

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UNDRAINED COMPRESSIBILITY OF SATURATED SOIL

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18 SUBJECT TERMS (Continue on reverse if necessary, and identify by block number) • - ''- "" - "' Saturated ^oils-. •'Liquefaction . Soil .Mechanics; Mixture Tfieory;

(U-r-, pr^s/i/-:; pCOT1*' h s 19. ABSTRACT (Continue on reverse if necessary and identify by block number)

sk series of equations for the bulk and constrained compressibility of undrained saturated granular soils are developed. Four sets of equations are developed, each set using a more complicated and realistic set of assumptions describing the soil behavior. A computational parametric analysis describes the applicability of the various assumptions as a functicn of soil parameters. The parametric analysis can be used to select the simplest suitable soil model applicable to a given problem.

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TABLE OF CONTENTS

SECTION

1

2

3

4

PAGE

List of Illustrations iv

Introduction 1

Material Models 3

Material Model and Parametric Analysis 26

List of References 59

Accession For

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By Distribution/

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Avail and/or Special

m

a. A«* A ^ mvim msm m*«. «"«:*•-..■*

LIST OF ILLUSTRATIONS

FIGURE PAGE

1 Saturated soil/water mixture under hydrostatic compression, schematic section view . 23

2 Schematic section view of the decoupled model under hydro- static compression 24

3 Schematic section view of the decoupled model under con- strained (uniaxial strain) loading 25

4 Constrained modulus vs. initial porosity for uncemented sand. . 34

5 Constrained modulus vs. in-tial porosity for coral 35

6a Undrained bulk modulus as a function of porosity, uncemented sand 36

6b Undrained bulk modulus for uncemented sand, deviation from fully coupled model 37

7a Undrained bulk modulus as a function of porosity, coral .... 38

7b Undrained bulk modulus for coral, deviation from fully coupled model 39

8a Undrained constrained modulus as a function of porosity, un- cemented sand 40

8b Undrained constrained modulus for uncemented sand, deviation from fully coupled model 41

9a Undrained constrained modulus as a function of porosity, coral. 42

9b Undrained constrained modulus for coral, deviation from fully coupled model 43

10a Undrained bulk modulus as a function of grain bulk modulus, uncemented sand at 35% porosity 44

ICb Undrained bulk modulus for uncemented sand at 35% porosity, deviation from fully coupled model 45

11a Undrained bulk modulus as a function of grain bulk modulus, coral at 35% porosity 46

TV

i^*gK«K5xa<SJi5lJklMJ?s3MJtSGgaaMkAk<M^

LIST OF ILLUSTRATIONS (concluded)

FIGURE PAGE

lib Undrained bulk modulus for coral at 35% porosity, deviation from fully coupled model 47

12a Undrained constrained modulus as a function of grain bulk modulus, uncemented sand at 35% porosity 48

12b Undrained constrained modulus for uncemented sand at 35% porosity, deviation from fully coupled model 49

13a Undrained constrained modulus as a function of grain bulk mod- ults, coral at 35% porosity 50

13b Undrained constrained modulus for coral at 35% porosity, deviation from fully coupled model 51

14a Undrained bulk modulus as a function of grain bulk modulus, uncemented sand at 50% porosity 52

14b Undrained bulk modulus for uncemented sand at 50% porosity, deviation from fully coupled model 53

15a Undrained bulk modulus as a function of grain bulk modulus, coral at 50% porosity 54

15b Undrained bulk modulus for coral at 50% porosity, deviation from fully coupled model 55

16 Undrained constrained modulus as a function of grain bulk mod- ulus, uncemented sand at 50% porosity 56

17a Undrained constrained modulus as a function of grain bulk mod- ulus, coral at 50% porosity 57

17b Undrained constrained modulus for coral at 50% nTosity, deviation from fully coupled model 58

vmttAA aun Am«Lii feA mj\ tw\9u\ Af • iwui i iiM\£m-irmvm.M^xnjenMmjä.MjaammtmAj^n&i^

VI

SECTION 1

INTRODUCTION

Analysis of high intensity dynamic loadings of saturated in situ soil

deposits requires mere sophisticated effective stress models than are used in

conventional soil mechanics. The low intensity loadings and slow rates of

application of concern in most conventional werk permit the use of simplifying

assumptions. For instance, the compressibility of the solid grains is usually

ignored and even the compressibility of the pore water can often be neglected.

Such simplifying assumptions, however, can lead to large errors when used to

describe soil response to high intensity, rapidly applied loads such as those

from explosive detonations.

Detailed modeling and analysis of the response of saturated soil

deposits to explosive loadings require development of finite difference and/or

finite element computer codes which delineate the stresses and strains in both

the pore water and the soil skeleton. Irreversible volume reductions in the

soil skeleton can result in liquefaction and drastically altered behavior of

the soil mass. Use of effective stress models which describe the action of

the soil skeleton within the pore water are required to properly model and

analyze this type of behavior.

While the simplifying assumptions used in conventional soil mechanics

are no*, relevant to the high intensity dynamic loadings, other simplifying

assumprions can sometimes be employed without significantly degrading the

accuracy of the dynamic models. These might include ignoring volume strain in

the soil particles resulting from effective stress and ignoring skeletal

strain resulting from compression of the individual soil particles by pressure

in the pcre water. Use of such assumptions can significantly simplify effec-

tive stress models and great'y reduce computational time and cost.

Applicability of such assumptions is a function of the in situ soil properties

and the loading function, and should be cetermined on a case by case basis.

The following sections develop a series of equations for the bulk and

constrained compressibility of undrained saturated granular soils in terms of

1

11Q4»to1ii1lliliaiAltt^^ iyW^M&miMMMIIfliliäQi^^JSIXMnü&UaU&UDüttäklK^

parameters describing the pore water, the soil grains and the soil skeleton.

Four sets of equations are developed, each set being dependent on a more

complicated and realistic set of assumptions describing the soil behavior.

Finally, a series of parameter studies is presented which graphically descri-

bes the applicability of the various assumptions as a function of soil parame-

ters. These parametric studies can be used to aid in selection of the

simplest soil model applicable to a given problem.

■iievsmmttiMOTcnfmMVMMfevuMteiifemiMBMMa^^

SECTION 2

MATERIAL MODELS

2.1 MIXTURE MODEL.

2.1.1 Bulk Modulus.

The simplest applicable compressibility model for high stress, rapid

loadings of fully saturated soils is the mixture model in which only the

compressibility of the pore water and the compressibility of the solid grains

by the pore fluid pressure are considered. Strictly speaking, this model is

only applicable to suspensions, where intergranular (effective) stress is

zero. As will be demonstrated in Section 3, however, it is a reasonably

accurate representation of compressibility for uncemented low density

saturated soils.

The bulk modulus of the soil-water mixture. Km, is derived from the

relationships illustrated in Figure 1. A total pressure P acts on an ele-

ment of saturated so.l made up entirely of solid grains and water. The total

volume of the soil element is given by V^ and the total volumes of the pore

water and sol id grains within the element are giver, by Vw and Vg, respec-

tively.

Thus,

V » V + v t w g

(1)

and the porosity n is given by

V n = Jl

V (2)

t»Mi^MM.AmiuKmmmmmMt^mjmM&M.tLM.MM»M%jmmmmRMmjmMKMmjinMmM • inurx MW um wn titttrwrwrnwrt M* ■n-iminini wmm MiifunrMTMMWMP"*"—''''

Also,

V

t (3)

The total strain in the soil element, et» is given by

et -r m (4)

where Km is the bulk modulus of the soil-water mixture. The volume change in

the soil element, AV, equals the sum of the volume change in the pore water

AVW and the total volume change in the solid grains AVg' according to

AV = AV + AV w g (5)

The volume change can also be expressed in terms of total volume as

AV = et Vt (6)

Assuming that there is no flow of pore water into or out of the soil

element during loading, the volume change in the pore water is given by

AV = e V w w w (7)

where £w is volume strain in the pore water. In turn, pore water strain can

be expressed by

e P w = r

w (8)

where Kw is the bulk modulus of the pore water. Substituting expressions for

pore water strain from Equation 8 and the volume of water from Equation 2

into 7 gives

AV = -L nv w Kw nvt (9)

The total volume change in the soil grains is given by

AVg = cgVg (10)

where eg is the volume strain in the solid grains. Assuming that the volume

strain in the grains results solely from the action of the pore water pressure

P on each individual grain, the strain is gfven by

r . p IT B ——i

g Kg (ii)

where Kg is the bulk modulus of the solid grains. Expressions for strain from

Equation 11 and for granular volume from 3 are substituted into Equation 10

giving

AVg " IT (1 " n)Vt 9 (12)

It should be noted that the above assumptions ignore any contribution

of effective stress in the soil skeleton to the compressibility of the soil

element. Thus, only the properties of the soil-water mixture contribute to

the overall compressibility.

Substitution of Equations 6, 9 and 12 into Equation 5 gives

an expression for total strain of

£iiiMiMffiiMUMMwmi9<\nry.^luirMifu wu mXn&smmmam'mtMitxvjtuMyui** -AM mmi^s üxmüiüt üfnuuumfuiAAnjuumaiaBMl

et = p K + n(K - K ) w g w

K K g w

(13)

Further substitution of Equation 13 into 4 gives the bulk modulus of the

soil-water mixture as

K K Q W

Km " K + n(K - K ) w g w (14)

Equation 14 is a form of the Wood equation discussed by Richart et al.,

1970.

2.1.2 Constrained Modulus.

Since the assumptions used to derive the bulk modulus of the mixture

ignore any contribution of intergranular stress, the soil element must behave

as a dense fluid. Under uniaxial strain loading, the lateral pressure will

equal the applied stress, and the constrained modulus must equal the bulk

modulus. Thus, the constrained modulus of the mixture, Mm' is given by

M = K = m m

K K _3_w_ K + n(K - K ) w g w (15)

2.2 DECOUPLED MODEL.

2.2.1 Bulk Modulus.

While the compressibility of the soil-water mixture is a satisfactory

approximation of material compressibility for some loose uncemented soils, the

contribution of the soil skeleton is often of significance. The simplest

method of accounting for the influence of skeleton compressibility is to

assume that the skeleton acts independently of the soil-water mixture. This

model, called the decoupled model, is pictured schematically in Figure 2.

The total pressure, P, applied to the soil element is resisted partially by

pressure in the pore water, u, and partially by effective stress in the soil

skeleton, P. The total pressure is given by

>feAA*BAja-j>%«.1fcKj)tt*,.n...a9LanjB ftKAjrAJtrflJi WJI R_M nj* ^JüSJ \Mr--.MVkMMJi ffLM fUt tUL-.P.Jtm ir^ + MnMKMILMnMK* Rh'RMMJlRJiRMHÜlUfRJtfmjfRMmÄmj^I

u + P (16)

which is the effective stress relationship. Assuming no flow during loading,

the strain in the soil skeleton, es' must equal the strain in the soil-water

mixture, em. Each in turn must equal the total strain; thus,

t m s (17)

The total strain can be expressed as

et = if (18)

where K^ is the decoupled bulk modulus of the soil element. The strain in the

soil-water mixture is given by

em " K u

m (19)

where Km is the bulk modulus of the soil-water mixture derived in the previous

subsection. The strain in the skeleton is given by

S = K (20)

where Ks is the bulk modulus of the skeleton. Substituting expressions for P,

u, and P from Equations 18, 19 and 20 into Equation 16 gives

e.K. = E K + E K t d mm s s (21)

mmmmmmm

Because all strains are equal, as shown in Equation 17, Equation 21 redu-

ces to

K , = K + K a m s (22)

Thus, the bulk modulus for the decoupled model is simply the sum of the bulk

modulus of the mixture and the bulk modulus of the soil skeleton.

2.2.2 Constrained Modulus.

The constrained modulus M^' for the decoupled model is obtained in a

manner similar to that used for the bulk modulus. Figure 3 schematically

depicts the uniaxial loading. The applied total stress, av' is resisted by

compression of the soil-water mixture due to the pore water pressure, u, and

by compression of the soil skeleton by the effective stress öv. Thus, total

stress is given by

% = u + 0v (23)

Though strain is constrained to the direction of the applied stress,

Equation 17 still holds, with

et ^ M, (24)

and

cs ^ M, (25)

where Ms is the constrained modulus of the soil skeleton. Since the soil'

water mixture behaves as a fluid, the strain in the mixture is given by

.mm*mm.*m*rmm.^]imr1t]VT-nTrirvmiint!rniftiimiFairfir ftn-mnwwiryiivifWify rw 'if i'Vfif •tifw'rf-ff> *m% i xvmur luru^MA AX mA MIMü MA «.I JU1

Equation 19. Combining Equations 19, 24, and 25 in the same way as

was done for the bulk modulus, gives a similar equation for the decoupled

constrained modulus.

M, = K + M d m s (26)

Thus, the decoupled constrained modulus is simply the sum of the bulk modulus

of the soil-water mixture and the constrained modulus of the soil skeleton.

2.3 PARTIALLY COUPLED MODEL.

2.3.1 Bulk Modulus.

The decoupled model adequately predicts the compressibility of

saturated soils over a range of porosities. At higher densities, however, it

may prove inadequate. A more realistic model can be derived by realizing that

the pore water pressure also acts to compress the soil skeleton. The pore

water pressure acting on each soil particle results in a volumetric strain,

€gU' of

u egu K

(27)

The strain in the soil skeleton resulting from the summation of all strains in

the individual particles, will also be egU. In other words, the soil skeleton

undergoes a uniform volume reduction proportional to egU caused by the reduc-

tion in volume of the individual particles. Thus, if a volume of saturated

soil, initially submerged at a shallow depth, is lowered to a deeper depth,

the soil skeleton will undergo a volume reduction with no increase in effec-

tive stress.

The compatibility equations for the partially coupled total strain

are:

P Et = IT (28)

P

SKm^WNM^m^jSS^^^

where Kp is the effective bulk modulus for the partially coupled model;

et ~ em K u

(29) m

where €m is the strain in the soil-water mixture due to compression by the

pv/r-e water pressure; and

^ JP_ . _u H = S = Ks Kg (30)

where the total strain in the soil skeleton equals the strain resulting from

the intergranular stress plus the strain resulting from skeleton compression

due to the pore water pressure given in Equation 27.

Substituting the value of P from Equation 16 into Equation 30

and setting it equal to 29 gives the pore pressure as

u =

IK K I \ m g /

(31)

Equating 28 and 29 gives

PK m (32)

Substituting Equation 31 into 32 gives

K = K + K - p m s

K K m s

(33)

or, the partially coupled bulk modulus equals the decoupled modulus modified

iy subtracting the last term in Equation 33;

10

•»fw^jri&.i tmim:WiJtr*Mt*m.njmr&mtmmmmMmmm-mm'mkmm..mmxmM-WLmm*ip» urn »« kf» kniu?m. ICH VS«I^ u£m.\xm\m

p d

K K m s K

(34)

2.3.2 Constrained Modulus.

In order to obtain the solution for the partially coupled constrained

loading, the influence of the pore pressure on the total strain in the skele-

ton must first be determined. Because lateral strains are zero by definition

in the constrained case, this influence is not intuitively obvious. For con-

venience, assume that the applied stress is vertical and that the major prin-

cipal effective stress in the skeleton is äv and the resultant radial stress

is är. Application of the vertical effective stress alone would result in ver-

tical and radial strains in the unrestrained skeleton of

ev E (35)

and

s (36)

where n is Poisson's ratio and Es is Young's modulus of the soil skeleton.

Assuming that radial effective stress is uniform, the minor and intermediate

principal stresses can both be given by är. Application of only the inter-

mediate and minor principal effective stresses result in a vertical strain o,

ev= -2lJ r (37)

and a radial strain of

e„ = o o

E ME s s (38)

11

««1U.IIM-!rSl«'M«M"JIM«M.«MÄÜ «MÄ«J!lW«»JrKJim««JIMJ% mit ^"MJlMJimÄl^MÄlLni^JU* 11^1^

Equation 38 takes into account that the intermediate and minor principal

stress are orthogonal to one another. Finally, the components of strain

resulting from compression of the individual soil particles by the pore water

pressure are given by

v 3K

and

u er 3K

In Equations 39 and 40, the skeletal strain is assumed uniform (i.e.

isotropic soil skeleton) and second order terms are ignored.

Summing Equations 35, 37, and 39 gives the total vertical

strain as

(39)

(40)

v - E; (% - 2"°r) + w t =

(41)

Summing Equations 36, 38, and 40 gives the radial strain as

E = 0 = -i r E ' - u (o + o )

r M^ v *■ I u

3K (42)

Solving Equation 42 for the radial effective stress gives

uE

r 1 - u v 3(l-u)K

12

(43)

. ■-. -,^^,>., ~. ^.^-^^ ^-^ ^.: ^.-..^ p.^W.-p.^^^.^^.^p^.^,^,^^^^^^^^^^-^.

Since radial strains are zero, the total strain equals the vertical strain.

Substituting Equation 43 into 41 gives

et Ev av Es(l - y) + Kg 3(1 - y) (44)

Using the elastic relationships between bulk modulus, constrained modulus and

Young's modulus;

K = 3(1 - 2y)

(45)

M = E(l - y)

(1 - 2y) (1 + y) (46)

K 1 + y M " 3(1 - y) (47)

Equation 44 can be rewritten as

0V

K - - _v s t M

+ inru s g s

Comparing Equation 48 to Equation 25 shows that the total skeleton strain

is modified by the last term of Equation 48 as a result of the partial

coupling assumptions.

The remaining strain compatibility conditions for the partially

coupled constrained loading are given bv

(48)

v t M_

(49)

13

mmmaumMxmjmemmmimea&MmeVM-'jMiiiiTmnr. im nm m ■ mi > i im»ill II 11 ■ n »iMMiiiriTwmmmp»*"*"*«*«'««'^«^'M*«<^»»»i«qimanmm wii^

Where av is the total applied stress and Mp is the partially coupled

constrained modulus, and

u et = K m (50)

Substituting the value for effective stress

a = a - u v uv

(51)

into Equation 48, and equating 48 to 50 gives the pore pressure as

u a K K v m g

KM + K K - K K g s m g s m (52)

Equating 49 and 50 gives the partially coupled constrained modulus as

o K

P u (53)

Substitution of Equation 52 into 53 gives

K K M = K + M 1~ p m s K (54)

Thus, the partially coupled constrained modulus is simply the decoupled

constrained modulus from Equation 26 modified by the same term used to

obtain the partially coupled bulk modulus in Equation 34;

K K M - M m S MP - Md ■ ir- (55)

14

temTutmntmvt i i jnt^k m. n-mM -mJsmMmM vmmmmMmM.uJ£mMmM «Jt *MmMMMmMmMmMM£&

2.4 FULLY COUPLED MODEL.

2.4.1 Bulk Modulus.

In the case of the partially coupled model, the influence of the pore

water pressure on the volume change of the skeleton was incorporated into the

modulus equations. In the fully coupled model, the influence of effective

stress on volume strain in the soil-water mixture is added to the assumptions

included in the partially coupled model. Effective stress in the soil grains

results in compression of the individual grains and an additional volume

decrease in the soil-water mixture not accounted for in the previous models.

From Equation 6, the volume strain in the soil-water mixture is

defined as

e = AY (56)

In this case there arc two components contributing to the net volume change;

AV = AV + AV m g (57)

where AVm is the volume change in the soil-water mixture due to the pore water

pressure given by

AV = — v m K vt m (58)

and AVg is the volume change in the solid grains resulting from the effective

stress P. This intragranular stress, äg( is given by

o = g 1 - n (59)

15

k\m%\m%iÄVViMA\vwvuiajiAJin^^

where 1 - n represents the ratio of the volume of the solid grains to the

total volume from Equation 3. Because the grains take up only a portion of

the total volume, the actual stress in the grains is greater than the effec-

tive stress, P. Strain in the grains, eg, due to the effective stress, is

given by

a j r _ _2 „ F g " K„ " (1 - n)K^ (60)

and the volume change equals the strain in the grains multiplied by the volume

of the grains from Equation 3,

_ p AV = :f- V

5 Kq t (61)

Substitution of Equations 58 and 61 into Equation 57 gives the net

volume change as,

Av = — v + — V a K t K t m 9 (62)

Further substitution of Equation 62 into 56 gives the strain in the soil-

water mixture;

u P et " K + K" m g

(63)

The other strain compatibility equations are;

P

'' = ^ (64)

where Kf is the bulk modulus for the fully coupled model, and

16

i»*i^B5aarwaifi.srtt«M-jraM Pw«*«M«MJi«M«Äifc:Smj!r*JSM.ft «j^^niL^

Gt K K s g

165)

Equation 65 represents the volume strain in the soil skeleton resulting from

the effective stress plus the action of the pore water on the individual

grains, and is identical to the compatibility equation used in the partially

coupled model.

Setting Equation 63 equal to 65 and solving for P gives

P= ßku (66)

where

K (K - K ) a - s g m Pk ~ K (K - K )

m g s (67)

Substituting Equation 16 into 64, equating 64 and 63, and solving for

the bulk modulus, Kf, using Equation 66 gives

(B + DK K Af K + BKm g in

(68)

Using the value for ß^ from Equation 67 and a bit of manipulation gives the

bulk modulus as

K, = K + K f m s

K K m s K

+ K K m s

K K K + K - -£-5- - K m s K g

g m s

This is simply the bulk modulus of the partially coupled model from

Equation 33 modified by the last term in 69. Kf can alternatively be

expressed as

17

(69)

irA«uautrj(iuifuiiun«nj(Ä*!Uk%jcJiiuuouuuo^

K.. = K + K K p m s

K + K m s

K K m s K

- K

K K m s

(70)

Equations 69 and 70 agree with results derived through different

approaches by Gassmann (1951), and Merkte and Dass (1983), and also presented

by Hamilton (1971) and Clay and Medwin (1977).

2.4.2 Constrained Modulus.

The difference between the fully and partially coupled constrained

moduli is in the strain compatibility equation for the volume strain in the

so'l-water mixture. Volume strain in the fully coupled case results from

volumetric compression of the pore water and soil grains by the pore fluid

pressure, u. plus volumetric compression of the solid grains by the principal

effective stresses, assumed to be ay and är as in the partially coupled case.

The overall volume strain is given by

M. (71)

where Mf is the fully coupled constrained modulus. The volume strain in the

soil skeleton is identical to that for the partially coupled case given in

Equation 48 and repeated here.

K

M. K M g s

u (72)

The volume strain in the soil-water mixture is derived in the same manner as

that for the fully coupled bulk modulus except that the mean effective stress

in the skeleton, än, is used in (.lace of the effective pressure, P. Thus, the

compatibility Equation corresponding to 63 is

18

MMDwjaaaaaQflQaMüfl J.ilWXiPfct

o _ u . m

m g (73)

where

am = av + 2ör

(74)

Substitution of Equation 43 into 74 gives the mean effective

stress in terms of the vertical effective stress as

n - « 1 •>• V 2 '"s n % " öv 3(1 - y) " 9(1 - y) K^ U

(75)

Using Equations 45 and 47, Equation 75 can be expressed as

K K / K

% ■ 5v ir + ir hr - J <,3 s g \ s

(76)

Substituting Equation 76 into 73, and equating 73 to 72 gives the

effective vertical stress as

o = ß u v m (77)

where

t = 9 s m s "s m s g m s m K^„(K,, - K )

m g g s (78)

Equations 77 and 51 are combined to give

v in (7S)

19

MvmarviryarujCHM. r ■ r in -i n i nunirrrini 'innrnrimnTninf irnn" irr'rniri'r niiiciminrini~"rrif~Mn>nirrrnini

The constrained modulus from Equation 71 can be expressed as

(ß + 1)K M nf K + B K s m g

(80)

using Equation 72 for et, 79 for av, and 77 for äv. Substitution of

Equation 78 into 80 gives

M,. = MK2 + KK2+KK2-MKK - 2K K K s g m s m g s m s m g s

K - K K g m s (81)

Further manipulation gives the constrained modulus in the form

(82)

which is a modification of the partially coupled modulus, M«' from

Equation 54 given by

M, = M + K K f p m s

K + K m s

K K m s K - K

K K K m s

(83)

The modifier in Equation 83 is exactly the same as the modifier for the bulk

modulus in Equation 70. These results agree with those derived by Merkle

and Dass (1983) using a somewhat different approach.

2.5 SUMMARY.

Equations for the undrained bulk and constrained modulus of saturated

granular soil were developed based on four sets of assumptions governing the

20

material behavior. These assumptions and the resulting equations are sum-

marized in Table 1. The first and simplest set of assumptions is that the

soil behaves as a dense fluid. Applied stress is resisted solely by the

compression of the pore water and compression of the individual soil grains by

the pore water pressure. The resultant mixture model neglects any contribu-

tion to stiffness from the soil skeleton.

The second, or decoupled model, assumes that applied stress is

resisted by the stiffness of the soil skeleton acting in parallel with the

stiffness of the soil-water mixture from model one. The resultant moduli are

simply the sum of the mixture modulus and the bulk or constrained modulus of

the soil skeleton, as appropriate.

In the third model, designated the partially coupled model, the

assumptions for the decoupled model are modified by including compression of

the soil skeleton due to pore water pressure acting on the individual grains.

The added strain due to the skeleton compression causes the partially coupled

moduli to be somewhat smaller than the corresponding decoupled moduli.

The fourth and most complex model, designated the fully coupled

model, incorporates all the assumptions of the partially coupled model and

also includes the reduction in volume of the soil-weter mixture due to the

action of effective stress on the individual particles. This additional

reduction in volume strain results in values of fully coupled moduli somewhat

lower than those of the partially coupled model.

21

lntf^Rf»a(UMiaAaiiaiiaMAa«MAMfi '^«vöucnKMwaKiyauogoUf^^

3 ■o o E

x> ttJ c ia i- T3

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to

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19

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*^r«iH".ai;MM<MiJrt*drK »..ir«..^ <n.-M *-M »» cmitm--mLnm mmmm mm-m'mM-^mMmmatmmMmmmm

SECTION 3

MATERIAL MODEL AND PARAMETRIC ANALYSIS

3.1 MATERIAL MODEL.

A number of numerical models can be developed based on incremental

expressions of the various equations derived in Section 2. Obviously, the

complexity of these numerical models will depend on the complexity of the

assumptions used in deriving the basic equations. It is desirable to use as

simple a model as is adequate for any given problem, thus keeping calculation

times and costs to a minimum. In this section, the four models for fully

saturated undrained soil, developed in Section 2, are compared as a function

of variations in assumed material properties of the soil skeleton. From these

comparisons the adequacy of a particular model can be assessed for a par-

ticular set of material properties.

In order to study the influence of changes in various soil proper-

ties, simplified material models for the soil skeleton were assumed. Models

for both uncemented and cemented sands were developed which should bound the

behavior of most saturated granular materials. These models express the ske-

leton bulk and constrained moduli, Ks and Ms as functions of the porosity, n.

Figure 4 shows constrained skeleton moduli from static loadings of

5 different sands as a function of initial porosity. The moduli are the

secant moduli measured at a strain of 1.5%. This strain represents the strain

at an approximate total stress loading of 1 kbar. A fit to these

data was made using the equation

M. *'{l-t) m

(84)

where Mg is the constrained modulus of the solid grains, u is the porosity,

n0 is a limiting porosity, and m is an exponent controlling the shape of the

26

MMtiyitiMbSMtitä

data fit. At n equal to zero the constrained modulus will equal the modulus

of the solid grains.

A constrained modulus of the grains of 9000 ksi was computed using

Equation 45, and an assumed bulk modulus, Kg' of 5000 ksi and an assumed

Poisson's ratio, ji, of 0.25. The value is representative of typical mineral

constituents of many sands. As shown in Figure 4, values of m = 8 and n0 of

0.8 gave a good fit to the uncemented sand data. The data fit is

thus expressed as

M (ksi) = 9000 s (-Ä)

8

(85)

From this equation, an equation for the bulk modulus of the skeleton can be

established,

K (ksi) = 5000 I1"-) 8

(86)

Equation 86 assumes that the Poisson's ratio of the skeleton is the same as

that of the solid grains (n = 0.25). In actuality, Poisson's ratio of the

skeleton is somewhat greater than Poisson's ratio of the solid grains.

Typical values for Poisson's ratio of the skeleton are 0.3, while values for

the solid grains range from 0.12 to 0.30. However, in order to simplify the

model, Poisson's ratio of the skeleton was assumed equal to that of the

grains.

Figure 5 shows values of constrained modulus as a function of poro-

sity for cemented Enewetak coral computed from data given by Pratt and Cooper,

1968. This coral data represents an upper bound for cemented sand, as uncon-

fined compressive strengths ranged from 2.7 to 5.7 ksi. Using an assumed Kg

of 5000 ksi, and a representative Poisson's ratio for this material of 0.2, a

fit to these data (shown in Figure 5) is given by

27

inrnvturnvKMurKM «u »^f^j»WKiftAAy»wm\f Ä-iäisasvyutfiftÄK/^^

M (ksi) = 10000 s (>-&) (87)

The corresponding equation for the bulk modulus of the skeleton is given by

K (ksi) = 5000 (-Ä) (88)

For purposes of this parameter study, Equations 85 through 88

express the moduli solely as functions of porosity. While this is a

simplifying assumption, these equations represent reasonable upper and lower

bounds to the behavior of actual materials.

3.2 INFLUENCE OF MATERIAL PARAMEÜERS.

3.2.1 Modulus As A Function of Porosity.

Using the equations for skeleton moduli derived in tha previous sub-

section, the undrained bulk and constrained moduli for the four models sum-

marized in Table 1 are computed as functions of porosity. These results for

bulk modulus are shown for the uncemented and cemented cases in Figures 6a

and b and 7a and b, respectively. Corresponding plots for the constrained

modulus are given in Figures 8a and b and 9a and b. A bulk modulus of water

of 300 ksi was assumed in all calculations.

Variation in undrained bulk modulus in the uncemented sand is plotted

as a function of porosity for the four models in Figure 6a. For porosities

greater than 30%, differences between the four models are insignificant. At

the higher porosities, the influence of the soil skeleton is very small, with

the overall behavior primarily governed by the soil-water mixture. At the

lower porosities, significant differences develop. The lower bound is repre-

sented by the mixture model which ignores the stiffness of the soil skeleton.

The upper bound is represented by the decoupled model. In the decoupled model

the skeleton modulus from Equation 86 is simply superimposed on the mixture

modulus. The influence of pore pressure on the skeleton behavior is ignored.

28

i <\£MwrkinijrHjnac«xiu{M)ixAwnraixi^^

Due to this simple summation, at the limit n = 0, the decoupled modulus

approaches 2Kg, In actuality, the modulus should approach the solid grain

modulus. Kg. Thus, the decoupled model introduces large errors at low porosi-

ties.

In the decoupled model, the pore pressure acts to reduce the volume

of the individual grains and in turn reduces the overall volume of the soil

skeleton. This influence of the pore pressure reduces the modulus from that

of the decoupled model. At the zero porosity limit, the partially coupled

model satisfies compatibility with the solid grain modulus. In the fully

coupled model, the modulus is further reduced by additional volume reduction

due to the effective stress acting on the solid grains.

Figure 6b compares the mixture, decoupled, and partially coupled

models to the fully coupled model. Percent deviation of each of the three

models from the fully coupled bulk modulus is shown as a function of porosity.

Above a porosity of 26%, deviation in bulk modulus for all the models is less

than 10%. This porosity range is representative of nearly all uncemented

sands. The decoupled model appears to be entirely adequate in this range of

porosities. At 30% porosity, the decoupled bulk modulus is within 4% of the

fully coupled modulus.

Bulk modulus as a function of porosity for the cemented material is

shown in Figure 7a. Because the skeleton in the cemented case is much

stiffen than that in the uncemented case, there is a significant influence of

skeleton stiffness over the entire range of porosities. The general trends

are the same as those observed in the uncemented material; i.e. the mixture

model gives a lower bound modulus, the decoupled model an upper bound modulus,

and the oartially coupled model a slightly higher modulus than the fully

coupled modulus. At the limiting zero porosity, all models except the

decoupled are compatible with the solid grain modulus.

Deviation of the mixture, decoupled, and partially coupled moduli

from the fully coupled modulus in the cemented material is shown in Figure 7b.

As noted above, the mixture modulus deviates significantly from the

29

»«»ufciiM«*_^M»i^v.ni»sKriiraM*MÄanM».«««*vjr*>rair\rii^rtf^^ TVUV

fully coupled modulus over the entire range of porosities. At a porosity of

45%, the decoupled modulus is 10% greater than the fully coupled modulus, with

deviation increasing rapidly at lower values of n. The partially coupled

modulus gives a good approximation of the fully coupled modulus with

deviations of less than 10% over the entire range of porosities.

The undrained constrained modulus for the four models in the unce-

mented material is shown as a function of porosity in Figure 8a. As was the

case with the uncemented bulk modulus, there is good agreement between the

four models at higher porosities. The mixture and decoupled moduli represent

the lower and upper bounds at lower porosities. At zero porosity, however,

the mixture modulus no longer converges to the same value as the fully and

partially coupled moduli. As shown in the equations of Table 1, the mixture

modulus converges to the bulk modulus of the solid grains while the other two

moduli converge to the constrained modulus of the solid grains. The decoupled

modulus converges to the summation of the bulk and constrained moduli of the

solid grains.

The deviation from the fully coupled constrained modulus in the unce-

mented soil is shown in Figure 8b. The mixture modulus exceeds the fully

coupled modulus by more than 10% at porosities of 34% and less. As was the

case with the bulk modulus, the decoupled model gives good agreement with the

fully coupled model over porosities normally exhibited by uncemented sands.

At 30% porosity, the decoupled modulus is only about 3.5% higher than the

fully coupled modulus.

In Figure 9a, the constrained moduli in the cemented material are

compared over a range of porosities. Due to the high stiffness of the

cemented skeleton, the mixture model significantly underestimates the overall

modulus at all porosities. Convergence at zero porosity is similar to that in

the uncemented material, with the mixture modulus converging to Kg, the par-

tially and fully coupled moduli converging to Mg' and the decoupled modulus

converging to Kg + Mg. The deviation from the fully coupled model is shown in

Figure 9b. Clearly, the mixture model is a poor representation. Deviation

of the decoupled model exceeds 10% below porosities of 34%. The partially

30

i^mssrj&M^Vü^^MMä^v^^^

coupled model, however, is in good agreement with the fully coupled model over

the entire range of porosities, with a maximum deviation from the fully

coupled model of less than 5.5%.

From the comparisons in Figures 6a and b through 9a and b it appears

that the mixture model is an adequate modulus approximation for many unce-

mented saturated sands. The decoupled model represents a good modulus

approximation for all fully saturated uncemented sands. Suitability of the

decoupled model for cemented sands will depend on the in situ porosities and

the degree of cementation. For the strongly cemented material presented in

this study, the partially coupled model would be a significantly better choice

than the decoupled model.

3.2.2 Modulus As A Function of Grain Properties.

Examination of mineral properties reveals a wide variation in moduli.

The bulk moduli of materials making up common rocks and sands vary from about

4000 to 15000 ksi, or from about 13 to 50 times the bulk modulus of water. In

this subsection, the influence of variation in bulk modulus of the solid

grains on the undrained bulk and constrained moduli for the four models is

examined. Again, both uncemented and cemented materials are represented,

using the skeleton models developed in the first subsection. Two porosities

are studied, n = 35%, representative of a dense sand, and n » 50%, represen-

tative of a loose sand.

Figure 10a shows bulk modulus as a function of normalized solid

grain bulk modulus for the four models at a porosity of 35%. The solid grain

modulus is normalized by dividing by the bulk modulus of water (300 ksi).

Thus, while normalized modulus varied from 1 to 50, the range from about 13 to

50 represents the range of moduli experienced in common materials. With the

exception of the mixture model, the models are in good agreement over the

entire range of normalized modulus. The stiffer the grain modulus, the more

inadequate is the mixture model. As was the case for modulus variation as a

function of porosity, the mixture model represents the lower modulus bound,

the decoupled model the upper bound, with the fully coupled model falling just

below the partially coupled model.

31

(fjtiuiiui^fA.^ii?ji;uinjuuuurj^jraji(J>^^

Percent deviation of the mixture, decoupled and partially coupled

models from the fully coupled model for bulk modulus in uncemented soil is

shown in Figure 10b. For normalized grain bulk moduli above 35 (10500 ksi),

the mixture modulus is more than 10% lower than the fully coupled modulus.

The corresponding bulk modulus plots for cemented sand at 35% poro-

sity are shown in Figures 11a and lib. Because of the greater stiffness of

the skeleton, the modulus variation at low values of normalized bulk modulus

is less pronounced than in the cemented soil. For the same reason, the mix-

ture model is much softer than the other models which take skeleton stiffness

into account. There is somewhat more difference between the coupled

models than was the case in the uncemented material. As shown in Figure lib,

only the partially coupled model is consistently close to the fully coupled

model. Deviation is 8% or less over the entire range of normalized bulk modu-

lus.

The constrained modulus and deviation plots for the uncemented sand

at 35% porosity are shown in Figures 12a and b. The trends are the same

as those of the bulk modulus plots of Figures 10a and b, with cfose agreement

between all the coupled models over the full range of normalized grain modu-

lus. Figures 13a and b are plots of constrained modulus and modulus deviation

for the cemented material at 35% porosity. The trends are similar to those

for bulk modulus in the cemented material. The decoupled modulus shows less

variation from the fully coupled modulus, and the partially coupled modulus is

within 6% of the fully coupled modulus over the entire range of grain moduli.

Plots of modulus variation in the uncemented and cemented materials

at an initial porosity of 50% are shown in Figures 14a and b through 17a and b.

Because the soil skeleton has very little influence on the composite behavior

of the uncemented soil, all four of the models for both the bulk and

constrained modulus, shown in Figures 14a and b and 16a and b are in very close

agreement. The moduli for the cemented material, shown in Figures 15a and b

and 17a and b, show the same trends as those at 35% porosity, but with closer

overall agreement Both the decoupled and partially coupled moduli are within

8% of the fully coupled modulus over the full range of variation in grain modu-

lus.

32

In summary, the conclusions drawn from the parametric analysis as a

function of porosity hold independent of variations in the modulus of the

solid grains. That is, the decoupled model is a satisfactory model for unce-

mented sands, over all variations in grain modulus. The partially coupled

model is a satisfactory model for the cemented material over all variations of

grain modulus; though in some instances, the simple decoupled model is also

satisfactory.

33

fj*)fwV&<VJflttiiiKiÄKMXjvWMArj%^^

<A

O 2

C

c o O

düü i i n

A Enewetak Sand, ARA, 1984

+ Ottawa Sand, Lambe & Whitman, 1969

• Minnesota Sand 1 * Pennsylvania Sand ■ Davissen, 1965 1

D Sanqamon Sand

150 l\ B J ■ |vis =9cx;o [[■ QQ )

\

\ *

100 •

\+

V

50

n

0 *\

\ A

D \*

1 1 1 30 50 40

Porosity,n (%)

Figure 4. Constrained modulus vs. initial porosity for uncemented sand.

34

t KlrnjlAXIUUILULM Rjf AJBRSMAM aMMLM-j^aifRMTuniiAjasjmumsomMaKiunuaQauaiufleiK

10000

8000

- 6000 V)

•o o

"g 4000

c o Ü

2000

! j

• Enewetak Coral (Pratt and Cooper, 1968)

— Ms=IOOOO{|-^g)3

20 40 60

Porosity, n (%)

80 100

Figure 5. Constrained modulus vs. initial porosity for coral

35

«MMM»aö4«Ä»IUt1IMILM»jniMXM»KTUIIlif»U«J^U

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36

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41

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LIST OF REFERENCES

Clay, C.S., and H. Medwin, Acoustical Oceanography; Principles and Applications, John Wiley & Sons, New York, 1977.

Gassmann, F., "Über die Elastizität Poröser Medien," VIERTELJAHRSSCHRIFT DER NATURFORSCHENOEN GESELLSCHAFT IN ZURICH, Jahrgang 96, Heft 1, March 1951, pp. 1 - 23.

Hamilton, E.L., "Elastic Properties of Marine Sediments," J. Geophys. Res., Vol. 76, No. 2, January, 1971, pp. 579 - 604.

Merkle, D.H., and W.C. Dass, "Fundamental Properties of Soils for Complex Dynamic Loadings: Annual Technical Report No. 3," Applied Research Associates, Inc., Albuquerque, NM, December 1983.

Richart, F.E., Woods, R.D., and J.R. Hall, Jr., Vibrations of Soils and Foundations. Prentice-Hall, Inc., Englewood Cliffs, NJ, 1970.

59

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MCDONNELL DOUGLAS CORP AHN: RHALPRIN

MITRE CORPORATION ATTN: J SULLIVAN

PACIFIC-SIERRA RESEARCH CORP ATTN: H BRODE, CHAIRMAN SAGE AHN: L SCHLESSINGER

PACIFICA TECHNOLOGY ATTN: RALLEN

PATEL ENTERPRISES, INC AHN: M PATEL

PHYSICS INTERNATIONAL CO ATTN: HWWAMPLER

R&DASSOCIATES AHN: C K B LEE ATTN: C MCDONALD ATTN: D OBERSTE-LEHN ATTN: D SIMONS ATTN: J LEWIS AHN: J RADLER AHN: TECHNICAL INFORMATION CENTER

R&DASSOCIATES ATTN: G GANONG

RAND CORP ATTN; P DAVIS

RAND CORP ATTN: B BENNETT

S-CUBED AHN: BPYATT ATTN: C DISMUKES ATTN: DGRINE ATTN: LIBRARY ATTN: R LAFRENZ AHN: T RINEY

S-CUBED ATTN: B L RISTVET

SCIENCE & ENGRG ASSOCIATES, INC ATTN: JSTOCKTON

SCIENCE APPLICATIONS INTL CORP AHN: H WILSON ATTN: M MCKAY ATTN: RSCHLAUG ATTN: TECHNICAL LIBRARY ATTN: W SCOTT ATTN: WWOÜLSON

SCIENCE APPLICATIONS INTL CORP ATTN: D MAXWELL

SCIENCE APPLICATIONS INTL CORP ATTN: JCOCKAYNE ATTN: WLAYSON

SOUTHWEST RESEARCH INSTITUTE AHN: A WENZEL

SRI INTERNATIONAL ATTN: DKEOUGH

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DNA-TR-87-42 (DL CONTINUED)

TELEDYNE BROWN ENGINEERING ATTN: DORMOND ATTN: F LEOPARD

TERRA TEK, INC ATTN: SGREEN

THE 8DM CORPORATION ATTN: J MERRITT ATTN: LIBRARY

TRW SPACE & DEFENSE 2 CYS ATTN: N LIPNER

ATTN: PBHUTA AHN: TECH INFO CTR, DOC ACQ

WASHINGTON, UNIVERSITY OF ATTN: J KATZ

WEIDLINGERASSOC ATTN: J ISENBERG

WEIDLINGER ASSOC, CONSULTING ENGRG ATTN: TDEEVY

WEIDLINGER ASSOC, CONSULTING ENGRG ATTN: ISANDLER ATTN: M BARON

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