Rock and soil rheology: proceedings of the Euromech Colloquium 196, September 10-13, 1985, Bucharest, Romania

Lecture Notes in Earth Sciences Edited by Somdev Bhattacharji, Gerald M. Friedman, Horst J. Neugebauer and Adolf Seilacher

14 N. Cristescu H.I. Ene (Eds.)

Rock and Soil Rheology Proceedings of the Euromech Colloquium 196 September 10-13, 1985 Bucharest, Romania

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editors Prof. Dr. Nicolae Cristescu University of Bucharest, Faculty of Mathematics Str. Academlel 14, Bucharest 1 C o d 70109, Romania Dr. Horia I. Ene INCREST, Department of Mathematics Bd. Pacii 220, 7 9 6 2 2 Bucharest, Romania

ISBN 3 - 5 4 0 - 1 8 8 4 1 - X Spnnger-Verlag Berlin Heidelberg N e w York ISBN 0 - 3 8 7 - 1 8 8 4 1 - X Springer-Verlag N e w York Berlin Heidelberg

Thrs work ~s subject to copynght All rights are reserved, whether the whole or part of the material ~s concerned, specifically the nghts of translation, repnnt~ng, re-use of dlustratlons, recitation, broadcasting, reproductron on mlcrofdms or in other ways, and storage in data banks Duplication of th~s pubhcat~onor parts thereof ~sonly permitted under the provisions of the German Copyright Law of September 9, 1965, ~nits version of June 24, 1985, and a copyright fee must always be pawd Violations fall under the prosecution act of the German Copyright Law © Sprmger-Verlag Berhn Heidelberg 1988 Printed In Germany Printing and blndtng Druckhaus Beltz, Hemsbach/Bergstr 2132/3140-543210

CONTENTS

I. T H E O R E T I C A L

F. G i l b e r t

APPROACH .................................

1

C h a n g e of S c a l e M e t h o d s A p p l i e d in M e c h a n i c s of S a t u r a t e d S o i l s ............

3

The Use of the H o m o g e n i z a t i o n M e t h o d to D e s c r i b e the V i s c o e l a s t i c B e h a v i o u r of a P o r o u s S a t u r a t e d M e d i u m ............

33

A Statical Micromechanical Description of Y i e l d i n g in C o h e s i o n l e s s Soil ........

43

A M a t h e m a t i c a l M o d e l for the L i q u e f a c t i o n of S o i l s . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

The K i n e m a t i c s of S e l f - S i m i l a r P l a n e Penetration Problems in Mohr-Coulomb Granular Materials ......................

83

P. H a b i b

Slip

93

II.

AND APPLICATIONS ........................

117

U n d r a i n e d C r e e p D e f o r m a t i o n of a S t r i p L o a d on C l a y . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119

A C l o s e d - F o r m S o l u t i o n for the P r o b l e m of a V i s c o p l a s t i c H o l l o w Sphere. A p p l i c a t i o n to U n d e r g r o u n d C a v i t i e s in R o c k Salt ....................................

151

Horia

I. E n e

B. C a m b o u

L. D r ~ g u ~ i n

R. B u t t e r f i e l d

EXPERIMENTS

A. F. L. Hyde, J. J. B u r k e

P. B e r e s t

M.

P. L u o n g

R. R i b a c c h i

N. Cristescu, D. Fot~, E. Medve~

Surfaces

in S o i l

M e c h a n i c s .........

Characteristic State and Infrared V i b r o t h e r m o g r a p h y of S a n d . . . . . . . . . . . . . . . 173 N o n L i n e a r B e h a v i o u r of A n i s o t r o p i c Rocks ...................................

199

R o c k - S u p p o r t I n t e r a c t i o n in L i n e d Tunnels .................................

245

Y. A r k i n

D e f o r m a t i o n of L a m i n a t e d L a c u s t r i n e S e d i m e n t s of the D e a d Sea . . . . . . . . . . . . . . . 273

R. T r a c z y k

On the C o n s t r u c t i o n of a C o n s t i t u t i v e E q u a t i o n of S o i l s by M a k i n g Use of the DLS M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . .

279

LIST OF CONTRIBUTORS

¥. A r k i n Ministry of Energy and Infrastructure Geological Survey 30, Malkhe Israel Str. Jerusalem 95501 Israel. P. B e r e s t Laboratoire de M~canique des Solides Ecole Polytechnique,

F 91128 Palaiseau - C@dex

France. R. B u t t e r f i e l d Dept.

of Civil Engineering

The University Highfield Southampton,

S09 5NH

United Kingdom. B. Cambou Laboratoire de Mecanique de Solides Ecole Centrale de Lyon 36 Av. Guy de Collongue B.P.

163, 6 9 1 3 1 E c u l l y

- C@dex

France. N. C r i s t e s c u Dept.

of Mathematics

University of Bucharest Str. Academiei

14, Bucharest 70109

Romania. Lucia Dr~gusin Dept. of Mathematics Polytechnical

Institute

Spl. Independentei Romania.

313, Bucharest

Vi Horia

I. Ene

Dept.

of Mathematics,

Bd. P~cii

INCREST

220, 79622 Bucharest

Romania. F. J. G i l b e r t

Laboratoire

de M g c a n i q u e

Ecole Polytechnique,

des Solides

F 91128 Palaiseau

- C@dex

France. P. H a b i b

Laboratoire

de M e c a n i q u e

Ecole Po!ytechnique,

des Solides

F 91128 Palaiseau

- Cedex

France. A. F. L. Hyde

Dept.

of Civil E n g i n e e r i n g

University Bradford,

of Bradford BD7

IDP

United Kingdom. M. P. L u o n g

Laboratoire

de M ~ c a n i q u e

Ecole Polytechnique,

des Solides

F 91128 Palaiseau

France. R. Ribacchi

Inst.

di Scienza

Universit~

delle Construzioni

di Roma

Via E n d o s s i a n a

18, Roma,

Italy. R. T r a c z y k

Institute

of G e o t e c h n i q u e

Technical

University

Plac Grunwaldzki Poland.

of W r o c ~ a w

9, 50-370 Wroc~aw,

- C~dex

INTRODUCTION While studied

the complex m e c h a n i c a l

for

quite

a while,

established m a t h e m a t i c a l rimental tance

data.

creep,

reported have

were

were

either

the

loading number

are

not

behaviour

materials

as

for

soil

rheology

that

the

solving

modern

models,

technology

thus

the

was

study

that

is

or

models

for

the

specific

to

the

plastics. It

of rock

possible

solving

of

mechanical dilatancy these

test

very

only

involving

and

general

presence

of a great

a n d soils,

the mecha-

from

That must

and

by of

data

developed

instance

distinct

development

expe-

as for ins-

specific

in roks

aspects.

not

sound

experimental

problems

due

quite

that

Generally,

a

are

on accurate

the

for

etc.

for

existing

problems

and

as

solving

metals

the

major

damage

specific

of various

but

necessary the

made

for

pores

instance some

time

soils

decades

based

consequence,

term were

geomaterials

has

last

and

of geomaterials

materials

us remind

and/or

of

a of

these

models

Let

cracks

long

some

appropriate

histories. of

a as

generality

or by

empirical

in the

properties

for and,

c o m p r e s s i b i l i t y , long

therefore

by

studied

possessed

particular

nical

rheological

problems

properties

is only

of rocks

models were developed

incomplete

missed

engineering

and/or

Some

it

properties

that

is

of

why

also

be

other

rock

soil m e c h a n i c s

using

and

mentioned posed

time-independent

rehological

models

become

a b s o l u t e l y necessary. In

the

last

became

available

niques

and

of

scientific

the

growing

community.

development models,

decade

of

able

to

compressibility

or

as a result

interest

These

genuine

such

creep,

after various time intervals, Today rocks

can

effects cription in

are

is

clear

not

the

concepts

of

accurate

this

field

in

turn,

have

geomaterials, term

as

experimantal

of

for

properties

no the

Another

of

very

development

of

research

made

and

techin the

possible

mainly

creep,

damage

data

experimental

the

rheological

dilatancy failure

and/or

occurring

slip surface formation etc.

unless

included.

for long

that

formulated

of the concepts

someway

notion.

it

be

data,

models

describe during

so,

of the

damage

accurate

constitutive

dilatancy idea and

irreversible

is

equation

phenomena

the

failure

need

of

and a

of rocks,

dilatancy

or

for

the

time

better

des-

again

another

using

related

VIII

In soil r h e o l o g y into

consideration

routine

tests.

in

Also

it is clear that the scale effect may be taken order

to

in writing

obtain

to take

because

is a great variety

soils,

Rheology some

into account

granular or cohesionless The

of

and

therefore

research

and

problems

which

scientific

also are

approaches

That

is

interest

geology, sports, (cold,

waste

culture, hot and

reservoirs

under

true

for

seismology,

industrial

of

the

obtained

are

for

present in

volume

last

of

the

helpful

the it

phenomena,

or n o n s a t u r a t e d

to Rock and Soil

too,

years

the

major

not

Exchange

mainly

in

rheology

and

of

petroleum civil

geothermal

of

storages),

engineering, storage, goods

underground

etc. Some of the last obtained

results

review

field

of

solved

opinions

those

areas

and

where

is quite fast.

engineers,

of

to

yet

geomaterials,

energy

storage

is

in this

and the progress made

telecommunications,

refrigerated

the

some

geophlsics,

storage,

for soils

or local

196 devoted

consideration.

quite

mining

from

soil etc.

formulate

now

equations

of types of saturated

are c o n t r o v e r s i a l

especially

great

to

discussions

some

that

results

information

the m i c r o s c o p i c

aim of the E u r o m e c h C o l l o q u i u m

the main

corect

the c o n s t i t u t i v e

is n e c c e s s a r y there

a

domain

of

engineering nuclear

and

caverns

for

and

foodstuffs

oil and natural are m e n t i o n e d

gas

in the

present volume.

IrOn N~methi dealt with the difficult task of typing a large part of the m a n u s c r i p t using a Rank Xerox 860 word processor; we thank her for the excellent

job she did. Nicolae C r i s t e s c u Horia I. Ene

I. T H E O R E T I C A L

APPROACH

CHANGE OF SCALE METHODS APPLIED IN MECHANICS OF SATURATED SOILS F. GILBERT

RESUME

Deux a s p e ~ compl~me~aires de la description m~canique des sols sat~s, ut~ant d ~ m~thodes de changement d'~chelle, sont pr~sent~. Les m~thodes d'homog~n~isation pour un milieu polyphasique u t i l i sant un changement d' ~che~e par convolution s p a t h e sont tout d'abord ~tendues sous forme lagrangienne. Les ~quations de bilan au niveau macroscopique sont ainsi ~tablies ~ p a r t ~ du niveau local ( ~chelle des grains). Pour l ~ sols satur~s la vitesse de f i ~ r a t i o n ~ l e tenseur de c o n t r a i ~ e e f f e ~ v e sont int~odui~. No~ calculons explicitement l e tenseur de viscosit~ apparente et la force de flottabi~it~. Les p r o p h e t , s de p~m~abilit~ en r~gime harmonique de certains empilement~ bi-dimensionne~s de grains, qui rendent compte sch~matiquement par leur caract~re auto-similaire de la forte h~t~rog~n~it~ locale des sols, sont ensuite ~tudi~es. L'int~r~t de combiner ces deux types d'approches est soulign~.

ABSTRACT Present work deals with two compl~mentary ~ p e c ~ of mechanical d ~ c r i p t i o n of saturated s o i l s , using change of scale methods. Homogenization methods for multiphase media using change of scale by spatial convolution are f ~ t extended in Lagrangian forum. Balance equations at macroscopic scale are thus ~ t a b l i s h e d starting from corresponding equations valid at the local l e v e l (grains scale). For saturated soils seepage velocity and e f f e c t i v e stress tensor are i ~ o d u c e d . Apparent viscosity s t r ~ s tensor and buoyancy force are e x p l i c i t l y calc~ated. Permeability properties under h~monic conditions are then analysed for particular ~wo-dimensional g r a i ~ packings, whose s e l f - s i m i l a r i t y accounts schematically for strong local heterogeneity of s o i l s . Interest of combining these two kinds of approaches is emph~ized.

INTRODUCTION Predicting the macroscopic behaviour of saturated soils involves a lot of difficulties, part of which owing to the multiphase character of such a medium where solid and liquid parts are intimately mixed and interact in an intricate manner under unsteady or cycling loading conditions. generally be considered as quasi-homogeneous

They can

at macroscopic scale, in the

sense that they repeat themselves more or less in the space in a statistical manner, but appear always strongly heterogeneous at small length scale owing to the great complexity of their internal geometry. General mixture theories

(Truesdell and Toupin (1960), MOller (1975),

Bowen (1976)) have been used for soils by various authors. But apart from their perhaps too wide generality their usefulness lack of precise geometrical

is restricted by the

and physical interpretation of the various

terms. These theories must be supplemented

in any case by numerous pheno-

menological

(1980).

assumptions as made by Prevost

Furthermore the essential immiscibility character results in kinematical constraints upon the motions of the species. To account for this phenomenon theories with microstructural

content have been developed for po-

rous and granular materials and used in particular by Ahmadi Ahmadi and Shahinpoor

(1980) and

(1983). Review of theories of immiscible and struc-

tured mixtures may be found in Bedford and Drumheller

(1983). These theo-

ries need additional variables and equations which are hoped to account in a global manner for the geometrical arrangement of the constituents, its influence on the mechanical behaviour and its evolution. Another macroscopic approach for porous media, due to Biot (1961, 1962 a, 1962 b, 1977), uses a postulated

lagrangian formulation following

motion of the solid part. A discussion of theoretical and experimental results may be found in Coussy and Bourbie

(1984). However the underlying

homogenization process involved in such formulation is not very clear except for simple cases. Direct statistical assumptions

(Matheron (1965, 1967), Batchelor (1974))

or the hypothesis of fine periodic structure of the medium (Sanchez-Palencia (1974), Ene and Sanchez-Palencia

(1975), Auriault and Sanchez-Palencia

(1977), Bensoussan, Lions and Papanicolaou

(1978), Sanchez-Palencia

(1980),

Auriault

(1980), Avallet

(1981), Borne (1983))

have been used to study

various saturated porous materials. It is worth noting that part of the results so obtained are in fact much more general. Homogenization

processes using change of scale by

spatial convolution, as suggested in particular by Marle (1967, 1982), Ene and Melicescu-Receanu

(1984) and Gilbert

(1984, 1985), are hence

well suited for a comprehensive physical description of complex multiphase media as saturated soils. They allow to define in a natural way macroscopic quantities as semilocal ones linked to local quantities of each phase by accurate equations. This may be viewed as a generalization of previous works by Marle (1965), Slattery (1967, 1969, 1972), Whitaker Coudert (1973), Hassanizadeh

(1969), Gray and O'Neill

(1976),

(1979) and Hadj Hamou (1983) for instance.

This paper is organized as follows. Section 1 is devoted to definition of change of space by spatial convolution on a reference configuration and section 2 deals with basic geometrical and kinematic quantities at macroscopic scale in a multiphase medium. Balance equations at macroscopic scale are established

in sections 3 and 4, as well as associated

expression of the principle of virtual work. Application to saturated soils using actual configuration as a reference configuration is made in section 5 and stress tensors of interest are introduced. Apparent viscosity stress tensor is calculated in section 6. Explicit value of the buoyancy force is given and filtration processes are described in section 7. Unsteady permeability of particular self-similar structures is studied in the Last three sections. Hydraulic impedance of a two-dimensional narrow gap between two grains is calculated in section 8. Construction of a sort of compact grains packing is recalled in section 9 and method of solution for permeability properties is explained. Numerical results are presented in section 10 as well as possible application to periodic tices.

lat-

LAGRANGIAN

DESCRIPTION

OF

A MULTIPHASE

MEDIUM

1.- SPATIAL CONVOLUTION ON A REFERENCE CONFIGURATION FOR A MULTIPHASE MEDIUM

Let

E z and

Ex

ponding to local

be the initial and transformed physical spaces corres-

(grains scale) and semi-local descriptions

scale). Correspondence between them is made at fixed time of a positive even weight function support

D(O)

is equal to 1 ( F i g . l ) .

the position at time position at time

t

m(Z)

(macroscopic t

through use o whose integral over its bounded

It will be supposed in this part that

t of any particle is a continuous function of its : thus sliding is actually excluded.

0

/ I

/ 0

/ //

X1

Z1

D('O)

z2

t to

DIX

Figure I : Change of space by s p a t i a l convolution on a reference config~ion with a weight function m for a multicomponent medium. C o ~ t ~ t u e n t C i s found at time t i n part ~ a of ~ (X,t) whose image at time a t o i s D(X) , t h e t r a ~ l a t e d of D(O) by vector X. C o , a c t s u r f a c e of c o n s t ~ e ~ C with t h e other ones may be expressed as ~ = U P~(~) [ b ~ ) w h o s ~ images a t t are ~b (t). (For non reacting media ~ ' ~ ( ' ~ j = ~ ( ~ o J ). ~The same type of notation i s used for t h e i ~ e r n a l d ~ c o ~ t i n u i t y surfaces Z a ( t ) .

Note that except for particular purposes it is convenient to choose m of class C N on ~:~3 (with N being not too small) to ensure a sufficient regularity of macroscopic quantities. For the separate constituents considered one introduces for every constituent C a the function Ia(Z,t) defined as follows : Ia = 1 if the particle whose position is Z at t

at time t and I = O otherwise. a a Hence possible phase changes C a ~ C b along contact surface tab(t), such as o

belong to C

freezing of water in the pores (C a = water, C b = ice), may be considered. For a non reacting medium I a is clearly independent of

t.

To an additive quantity ~(a)(Z,t) relative to C a is associated an appa>a rent average < ~(a) at the macroscopic scale by the convolution product (in Lagrange variables)

a~Ia ~

=

The obtained quantity reflect small scale variations.

X,~

long trends of the medium regardless of

A sort of change of scale is thus achieved

(Fig.2)

for quantities of interest.

Q e-E z

~

Figure 2 : Macroscopic volume ~ vi~ed

"- Ex

i n E z and Ex

For instance if ~(a)- I one gets quantity

oa(%t)

:

which is to be interpreted as the volume fraction at t particles initially in D(X) which at time

t belong to

o

around X of the Ca

Derivatives of the semi-local quantities < O(a) >a are given by

(1.3)

(1.4)

:


a = (po(2)Ia([,t)) ~ ~.

•

Total mass at time t per unit volume at time t

e q u a l t o ~ - m a ( x , t ) . I t a p p e a r s as a q u a n t i t y a

o

around X is clearly o much more r e g u l a r t h a n l o c a l

mass perunit volume which is a discontinuous function in E

Z

at the grains

scale. It is convenient to define average velocity ~a(X,t) for constituent C a by reference to the apparent average

(1.1) of the momentum density a

8

where ~(a)(Z,t)

is the local velocity of C a . A mean position ~a+(X,t) for

Ca is obtained inserting in formula (2.2) positions instead of velocities. Motion of C a in E x is described by 1

(2.3a)

~a ( ~ , t )

= i~ ( x , t )

(2.3b)

~a ( X , ~ )

=

Xa ( X ) -

+

where Xa(X) is the value of x a at time to . For non reacting media the two functions ~ a ( X , t ) and ~ a + ( X , t ) a r e e q u a l a s s u g g e s t e d by i n t u i t i o n . This property is known to be false (except at particular time t ) for reacting o media. Average acceleration Ya(X,t)

for Ca in E x is taken as material deriva-

tive of Ua(X,t). Average displacement of C a is defined by

~2.~)

~

(~,t)

=

~ (~,~)

-

~(x)

and hence X is to be interpreted as a reference position. Gradient of de-

formation for C in E is a x

(2.s)

-o-a ( ~, ~)

= v~

( ~, ~)

.

Note that at time t o the gradient of deformation for C a is ~ a " fers s l i g h t l y

from t h e u n i t t e n s o r ~ . More p r e c i s e l y

if

which dif-

l o c a l mass p e r

unit volume at t o takes an uniform value for C a one gets for each constituent

(2.6)

V ~

Xa

=

I

-

I ya Sa .,.,

~

0

10 where the geometrical tensors ya(~) related to contact surfaces ra(t o) at time t

o

by

introduce a kind of departure of the medium from macro-homogeneity

at

scale of D(O).

3.- BALANCE EQUATIONS FOLLOWING MOTIONS OF THE SPECIES Balance of mass for C a is obtained through application of formula (1.4) to definition (2.1) as

Va

(3.1~

~ t) = ~'o a ,,,,

c o (x,~)

8 p . (t) '~

=

where the mass production rate ~a of C per unit volume of the reference o a configuration converts surface reactions in E z into volume reactions in E x. As a direct consequence of its definition one gets

(3.2~

v

7. c

a

( ~, t )

= o

which expresses conservation of mass for the whole medium following the motions of the various species. Balance equation of momentum for C

in E is obtained using the appaa x rent average (1.1) of the local balance equation. This procedure ensures automatically the compatibility between the two considered descriptions. One gets after some calculation material derivative of the momentum as

(3.3)

G oa ~ a

+ CV ao ~ a

=

moa F a

+

-Ro a

+

a]~Vo ,~, B a

where the body force per unit mass ~a" the interaction force per unit volume of the reference configuration ~a and the apparent Boussinesq stress o tensor Ba for Ca are defined in terms of corresponding local quantities by (3.4)


a

)

)

11 Observe that the interaction terms ~a appear quite naturally and that (ne0

glecting surface tension effects and adding formulas

7, Ro The p a r t i c u L a r L y

o

s i m p l e form o f e q u a t i o n ( 3 . 6 )

with corresponding expressions obtained for

Melicescu-Receanu 1984, G i L b e r t

of virtual

Evaluation in E

Z

and E

(Marle,

1982, Ene and

1984). V a n i s h i n g of the v e L o c i t y f L u c t u a t i o n

terms o b t a i n e d here i s cLearLy r e l a t e d the system t o be c o n s i d e r e d .

i s t o be noted and compared

Cauchy s t r e s s t e n s o r s at macros-

c o p i c s c a l e used i n f u L L y E u l e r i a n d e s c r i p t i o n s

o f the p r i n c i p l e

(3.5))

to the b e t t e r

To f o r m u l a ( 3 . 3 )

definition

used f o r

i s a s s o c i a t e d a s i m p l e form

work. X

at time t

0

and for a given macroscopic volume

of, say, the momentum of the solid part, yields slightly

different

re-

sults. Comparison has to be made between the two quantities ~(s) and U s

(3.8)

z

(3.9)

~

= ~

One can show ( G i l b e r t

~o~(~,~o)~(X,~)a~ x

1984) t h a t t h e d i f f e r e n c e

over the two small volumes C+ and C_ plying

to ~ the Serra t r a n s f o r m s

(Fig.3)

~s _ ~ ( s )

. involves integrals

obtained respectively

( M a t h e r o n , 1967, S e r r a ,

1982) t h r o u g h d i -

L a t i o n and e r o s i o n by the s y m m e t r i c a l volume D(O). Hence r e l a t i v e rence i s n e g l i g i b l e

if

diffe-

D(O) i s s m a l l enough w i t h r e s p e c t t o ,C~.

6N

8~Qo D(O)~

by ap-

--

C_ C+

Figure 3 : Volum~ contributing to t h e d i f f e r e n c e ~ - p i S l . Note t h a t for a symm~tric~ volume D(O) t he two Serra transforms ~ e given by the m e ~ o n e d Minkowski pseudo-addition and pseudos u b ~ a c t i o n.

12 Note that rigorous equations by equation in E

Z

in E

(3.9) and not by equation

x

as (3.3) involve momentum defined (3.8)

(which is the right definition

only).

4.- BALANCE EQUATIONS

FOLLOWING MOTION OF THE SOLID PART

One is generally essentially

interested

by motion relative to solid

part. We shall suppose the solid matrix to be chemically sake of simplicity

cartesian orthonormal

co-ordinates

inert and for

will be used in

the sequel. Geometrical

situation

is depicted

in figure 4 in E . X

Ex 0

Figure 4 : Companion i n E~ between the motion x~(X,t) of CA and i h e motion of a n o t h ~ c o ~ e n t Ca. Point~ of ~a belongin~ a t time t t o a c e r t a i n macroscopic volume i n E~, whose motion i s given by u (X~, t) , belong at fixed time t to ~he v ~ i a b l e volume

~ ( ~ t, Ca) given by ~ Let ~ be a macroscopic

={Y/Xa(

Y~-,t) = xs (Xs t ) j X e

,('l}

volume in E x given at time t o and moving by

assumption with velocity Us(X,t) of the solid part C s. Corresponding and momentum of C a at time part are (Fig.4) 4.1)

:

mass

t following motion x = x (X,t) of the solid S

13 With the dot

denoting derivative following in E

p a r t one g e t s f o r b a l a n c e of mass a f t e r (4.3)

• ~

transformation

X

motion of the solid

of f o r m u l a s ( 3 . 1 )

fi.,). •

(4.4a)

"¢'¢Ich ( X , ~ )

,

-F Ivl

CX;t~)

, for a # s

(4.4b)

a and M a are the masses of C a acquired by phase changes and convecwhere mch tion between t

0

and

t, per unit volume of the reference configuration ~.

Calculation yields

@

va

~

(4.~)

~ ° ~ (X,q = ~o C7,0-

(4.6)

Ma (X,t) = - divo N a ( X, t)

with

(X,~)

•

and thus classical Biot's structure is recovered. Quantity Ma(X,t)

for a

fluid C a is called fluid accumulation. Balance of momentum of Ca is expressed by

(4.+) P (o_>)

+

~(~,

~

•

o

where the surface i n t e g r a l corresponds to the open character of volume with respect to constituent Ca. Volume i n t e g r a l in formula (4.8) takes another form a f t e r transformation of formula (3.3) i n t o (note that s i m p l i f i c a t i o n f o r a = s is obvious)

m&

(4.9) # ~ + inch~

= ~ F. + ~t~'5-~ + di(o~.T~

where (4.10)

A"iR#t)

.~

is the i n t e r a c t i o n force f o r Ca at time t per unit volume of the actual configuration.

14

The apparent stress tensors ~a for the various C a are calculated as (note that for a = s, ~s = BS)

T a =

(4.11)

de~ ~s (X,~)

•

~(~ ~}.~(%~}.~ ~ (~,,~)

The principle of virtual work applied to ~, considering different virtual displacement fields ~x a for the various constituents,

a

states then

b~a

w i t h f o r any c o n s t i t u e n t Ca s u r f a c e f o r c e s a p p l i e d t o t h e t r a n s f o r m o f boundary 3~ only due to C a itself

(~.~,

~w0~f,,

Contact forces

-

f

due t o i n f l u e n c e

8~.~T ~. ~odAo of the other

.

constituents

Cb must no be

c o u n t e d t w i c e and h e n c e a r e t a k e n i n a c c o u n t by v o l u m e i n t e g r a l s

(4.13b)

W b

'

.0.

only

b~a

Virtual works of body forces, internal forces and inertia forces are expressed by

(4.13c)

~--,,ody ~J''

=

(4.13d)

8,~o,

:

d,F

~

S~)a

These e q u a t i o n s p r o v i d e a u s e f u l b a s i s f o r comparison w i t h B l o t ' s

theo-

ries. One can see in figure 4 a little difficulty unavoidable with this formulation : at time t = t

correspondence is made by x between the phases as O

= X

and thus ~ ~ X. It introduces a small distorsion in the evaluation

a s of mass and momentum of C a .

15 APPLICATION

TO

SATURATED REFERENCE

SOILS

USING

REACTUALIZED

CONFIGURATION

5.- STRESS TENSORS Let us use the particular choice t o = t : fixed reference configuration coincides with actual configuration

(note that the description is slightly

different from a fully Eulerian one). It is then a simple matter to show that sliding between grains is now allowed (see also Gilbert 1984). Two constituents are to be distinguished for a saturated soil : the solid part C S and the pore fluid Cf. Actual pon is defined as ¢ ~(~,to ) and masses

rosity

(1 -n) Ps and npf

m °s and m of may be written as

respectivelyr with obvious notations for densities at

macroscopic scale. Balances of momentum

where unit

(3.3) read (no mass production term)

g i s the a c c e l e r a t i o n

of gravity

volume) e x e r t e d on the s o l i d

part

and

R the i n t e r a c t i o n

force

s

f

by the f l u i d

the s y m m e t r i c a l apparent averages of c o r r e s p o n d i n g tensors.

FormuLas ( 5 . 1 )

ture theories.

are s i m i l a r

part ; ~

and A

(per are

LocaL Cauchy s t r e s s

to f o r m u l a s p o s t u l a t e d

in general mix-

Note t h a t p h y s i c a l meaning of each term i s e x p L i c i t L y

known

here. It is however convenient to use in soil mechanics other stress tensors. One can define in an obvious way for the fluid part an average fluid pressure

p and an apparent viscosity stress tensor

(s z)

where

P

p(f)

=

Z f by

< Pcf)>f//

and ~(f) are the local fluid pressure and the local viscosity

stress tensor. Of interest for the solid part is the modified stress tensor

16

q

sd

insensitive to any uniform translation of local stresses along the

pressure axis (5.4)

:

,v ~$d

-v@'$ +

(~'q%)

~ -~-Z

,

Hence e f f e c t i v e stress tensor ~ r i s given by (5.5)

o"v

:

o"

+ roT.

:

o ''sd

+ ~f

where the stress tensor q for the whole medium is the sum of qs and q f . Dynamic equations

where vector (5.7)

(5.1) are now written in a more useful form

b , which w i l l appear useful,

is given by

D = R + p grad n

6.- EXPLICIT CALCULATION OF THE APPARENT VISCOSITY STRESS TENSOR The (relative) (6.1)

seepage velocity U(x,t)

is defined as

0 = n(uf - Us )

at a given point in E . For incompressible constituents of uniform densix ties Ps and pf formulas (1.3) yield for consolidation problems (6.2)

div U = - div

s

which expresses conservation of total volume of the medium. Let us now consider an incompressible newtonian fluid of uniform dynamic viscosity

q flowing with seepage velocity U(x,t) through a packing

of rigid grains having different velocities and spin vectors, lid part moves with average velocity Us(X,t). to

Using formulas

so that so(1.3) with

t one gets for the apparent viscosity stress tensor

where D denotes the symmetrical part of the gradient.

Note that formula

(6.3) is valid for any geometry of the porous medium and for any fields

17 and Us" When solid part is at rest or moves with uniform velocity it reduces to the result given by Gilbert (1984). The corresponding term in formula (5.6b) is (4 = Laplacian) :

and is thus found to be negligible (with respect to b) for practical applications. Although fluid movement is generally governed essentially by viscosity, corresponding macroscopic terms disappear : o v and qsd are almost equal and fluid stress is correctly represented in E

by a simple X

pressure

p(~f = - np~).

7.- FILTRATION PROCESSES The interaction term

R

is to be splitted in three parts : a static

one owing to the possible macroscopic inhomogeneity of the soil (called "buoyancy" force), a kinematic dissipative one (drag force) and a dynamic one corresponding to inertial coupling between fluid and solid parts (virtual mass effect) (7.1)

R

=

Rstatic

÷

Rkin.

÷

Rdyn.

In this section dynamic term Rdyn. will be disregarded. Neglecting variations of fluid density at scale of D(O) one gets (Gilbert 19C4) for the static part (7.2)

Rstatic = - p grad n + pfg. ~f

where influence of the geometrical tensor Yf given by (2.7) is very small, at least in mean value (see formula (2.6)). Equation (7.2) then reads (7.3)

Rstatic ~

- p grad n

Hence in that case b (formula (5.7)) and not

R (as postulated on in-

tuitive grounds in certain mixture theories) equals zero. Note the particularly simple expression of the buoyancy force (7.3) and its obvious geometrical interpretation. Estimates in (5.6b) of vector

b, or R - Rstatic"

yield Darcy's law under various forms. Note that an estimate of b is naturally not available for any porous medium under any flow condition.

18

Slow stationary a fixed stationary a symmetrical

filtration

random porous matrix yields

intrinsic permeability

tensor

newtonian

fluid through

(Marle 1967), as D(O) grows,

k given by

k-~ 1 ~Mih ~Mil >f ~h ® ~1 "" : ~. < ~zJ ? z J~'

(z.4~

as a function of the stationary function of

of an incompressible

random tensor M(z) which maps u(f)(z) as a

uf.

One can also treat by this method the corresponding periodic slow stationary

flow through a periodic

venient here to choose for

m the discontinuous

case of spatially

fixed matrix.

It is con-

function equal to 1/ID 1 in

the basic period (IDI being the volume of the basic period of the lattice) and to 0 elsewhere. (7.5)

Equation

(5.6b) then reads

~) = n(pfg - gra--d p)

However

~) and gra---d p are not constants whatever

x.

sary to use a double averaging process, which eliminates fluctuations,

the preceding

by introducing

(7.6)

B : b * m

(7.7)

P = p * m = ((p(f)If) * m) * m

Classical

variational

with a symmetrical (7.8)

It appears neces-

structure

is then recovered yielding

intrinsic permeability

tensor

Darcy's

law

k

~ = _ 1 k (grad P - pfg) r1-,-

Denoting by ~(z) a D-periodic

function and by

~ a constant vector one

has for the various pressures

(7.9b)

p

(7.9c)

P (~) : ~. ~ +

Observe

(~)

(Gilbert

:

~.XF(X

) l-cl~e~

1984) that fluctuations

of

p around

lated to geometry only, through the periodic abovementioned

P are thus retensor

value for a periodic medium is shown to be

(7.1o)

Y f : (~-~1) ~1 ®~I + (~-~z) ~-2® ~z + (~-~31~s®~

Yf whose

19 where

n is the (constant) volume porosity and n. the variable surface I i porosity of planes z i = x i ± ~ l~il (~g. 5). The difference between p and

P is small when the elementary period contains many grains since

surface porosities become progressively equal to volume porosity as geometrical disorder in the period grows.

pressure /

~

P(x 1) : slope

I I) : s l o p e n~ n l ( x I -+ -E 1 J~l j)

v

+ 11Zli

I

1 z1 X

D(x)

Figure 5 . Average p r e s s e s p and P i n a p a r t i c u l a r p e r i o d i c medium for ~ = ~I" The vectors -~, = l~il-e i ~ e basic v e c t o r s of the periodic lattice. Note that the pressure P may be identified with the first term Po of the asymptotic development of the pressure in successive powers of the small parameter c. which is postulated in the theory of homogenization of fine periodic structures.

20 HARMONIC

FLOW

A PARTICULAR

THROUGH

FIXED

STRUCTURE

8.- HYDRAULIC IMPEDANCE OF A TWO-DIMENSIONAL NARROW GAP To obtain more complete information about behaviourof

a saturated po-

rous medium one must use postulates or analyse by numerical methods particular structures of interest, which allow to go further. Such an example is presented here concerning harmonic flow of an incompressible newtonian fluid through narrow gaps between the various parts of a fixed solid matrix. For sake of simplicity problem is studied in two dimensions ; grains are roughly schematized as parallel cylinders. Let us consider a narrow gap of minimum width 2h.. between two locally i] regular motionless cylinders of parallel axes with radii of curvature a.] and

a. respectively in the vicinity of the narrowing ]

"L~.. A ,~ p C-o>)

Yi

t

~.,,S

+hii

- L ij .~

(Fig.6).

+ L ii .v

o t, -hij

~ ,
0

the second y i e l d i n g mechanism is actived.

and sign

is 7 - ( ~ )
0 , ~

>

-

7- ( ~ ) < 0 , ~

>

for

IT

For a loading without reorientation of principal axes, i t is easy to demonstrate that ( ~ ~= + dE=) takes the sign of (0-~ - (I':) ~, then volumet r i c strains are compressive. With an additional simplifying assumption i t can be demonstrated that fulf i l l i n g equations (20) and (21) leads to the stress-dilatancy relation defined by Rowe I l l } 4.3. Load!ng- s t r e s s path_wi}h_(3_z_C~_~~_constant The previous analysis points out s l i d i n g c r i t e r i o n s defined by CT~/(F=. In such a hypothesis, loadings with ~ / ( T = = c t , cannot provide irrecoverable s t r a i n s , which does not correspond to the observed experimental behav i o r o f granular materials. To have a b e t t e r adequacy of the model, i t is necessary to modify hypothesis (4) and (19) . In f a c t , the irrecoverable strains provide an evolution of the number of contact points, which is not proportionnal to the stresses applied. to

To take into account this phenomena t e r m ~ in relation (4) and (19).

~rU~,:) P°q

can be substituted

Term a is a parameter to be defined, Po equal to the unity of pressure makes i t possible to preserve the homogeneity of the formulae. The analyses presented in section 3 and 4 can be developped with the new assumption {5}. In this case, the yielding lines of the two mechanisms are s l i g h t l y curved.

57 5.

Elasto-plastic model.

The previous microstructural analysis leads, in the general framework, of elasto-plastic theory, to a model with two yielding mechanisms (Fig.9). . The f i r s t one defined in section 3 characterizes the evolution of mean contact forces. For a bidimensional material this mechanism depends on two scalar parameters : ~d which is a kinematic hardening parameter, is linked to the evolution of the deviatoric stresses. ~:which is an isotropic softening parameter, is linked to the reorientation of the principal axes. For a monotonical loading the evolution of this mechanism leads to extensive volumetric strains. Yielding surfaces are defined in section 3. The second one defined in section 4 characterizes the width of the distributions of contact forces, For a bidimensional material, this mechanism depends only on one scalar parameter which is an isotropic softening parameter. For a monotonical loading the evolution of this mechanism leads to compressive volumetric strains. Yielding surfaces are defined in section 4. For a f i r s t loading path, the l i m i t between the two domains of activation of the two mechanisms is given by (Fig.9) : A f t e r complex loading t h i s l i m i t can change• For real loading i t can be assumed t h a t the t r a n s i t i o n from one mechanism to the other occurs progress i v e l y around the defined bounds. 6.

Predictions o f the model and experimental r e s u l t s -

.

I s o t r o p i c loading :

Hypothesis taken i n t o account in section 4.3. allows to describe i r r e c o v e r a b l e s t r a i n s under an i s o t r o p i c loading. First monotically deviatoric loading: From experimental data Habib and Luong {9} have shown, t h a t in a stress space, a " chcutactP~tic" line can be defined. This line is the l i m i t between volumetric compressive strains and volumetric extensive strains, this l i m i t does not depend on the density of the granular material (Fig.lO). These experimental results are in agreement with the analysis proposed herein. The characteristic line is in our analysis the boundary of the two yielding mechanisms. I t is possible to determine experimentally the yielding surfaces of a granular material. I t can be shown on Fig.11 from Tatsuoka and al {12} that these surfaces are very similar to the yielding surfaces defined in this work (Fig.7). Fig.12 shows that at the beginning of a deviatoric stress path the hardening observed is isotropic and kinematic and for greater value of deviatoric stresses i t is only kinematic. This kind of behavior is in keeping with the previous analysis .

58

g mech.n°l

boundaries of the 2 mechanisms for a virgin material

\

kinematic #f"

i

isotropic

Mech.n°2

yielding mech.n°l

Fig. 9 :

q

LR// / /

~ f

~z

LC

Yielding mechanisms.

~Domaine surcaract~ristique

dilatant

-

_ ~

Domaine P subcaract~ristique > ~ontr~tant

~z~

3

Fig.lO : Definition of the "characteristic" line. From Luong {9}

59

Fig.11

:

/

Y i e l d i n g surfaces

Experimental r e s u l t s from Tatsuoka and I s h i h a r a

S

{12}

4

=r2

,m

0

* pwCO~Sl, -4t 'A ~ a ¢ O n s l ,

"so

~ effective

• *

~

[~, /

'~-:~

*$ "~

+

s

6

me'Qm prlmcipal sirius

P'(~,lcn't)

iso + kinem, hardening kinematic hardening ,,A,,

(~o+ k P~ Fig.12

3-

elastic

domain

: Compression -

Extension t r i a x i a l

tests

on sand

2-

~=

1.

/

15,4 kN/m3

60 -

cyclic loading :

For c y c l i c loading with large stress reversal, volume changes show a general tendency to compaction even for dense materials (Fig.13 from Thanopoulos {13}}. This can be explained in the framework of this study because, a f t e r large stress reversal, i t is easy to demonstrate that G and Ad have the same sign in equation (17), then the volumetric strains are necessarily compressive. 6.

Conclusion

The microstructural analysis proposed in this work has allowed us to define the y i e l d i n g mechanisms of an e l a s t o - p l a s t i c model. This model seems to be in agreement with phenomena observed in experiments on granular media. Only bidimensional media were considered here, the analysis of three dimensional material does not present any t h e o r i t i c a l d i f f i c u l t y but requires taking into account a greater number of y i e l d i n g parameters.

l q-o~ e~--o55

lO~P~

essa, 2

I 6 4

,3 - -2t

-3 -2 "3 "2 "t

Fig. 13

"t I

:

2 .....~'~4__~ G

7

E,%

Cyclic loading on dense sand.

(From Thanopoulos

{13} )

61

REFERENCES 1

CAMBOUB. (1982) " O r i e n t a t i o n a l d i s t r i b u t i o n s o f c o n t a c t forces as memory parameters i n a g r a n u l a r m a t e r i a l " IUTAM Symposium proc. "Deformation and f a i l u r e o f g r a n u l a r M a t e r i a l " Delft. CAMBOU B. (1984) " Microscopic aspects o f hardening i n g r a n u l a r material". I n t e r n a t i o n a l CHISA - Prague. CAMBOU B. - SIDOROFF F. (1983) " Failure criteria for granular m a t e r i a l based on s t a t i c a l M i c r o s t r u c t u r a l v a r i a b l e s " V i l l a r d de Lans - Juin 1983. CNRS. Symposium Proc. CAMBOU B. - SIDOROFF F. ( 1 9 8 4 ) " D i s t r i b u t i o n s o r i e n t ~ e s dans un milieu granulaire et leurs representations". "Proc. o f Journ~es de M~canique A l 6 a t o i r e appliqu~e ~ la c o n s t r u c tion" - Paris. CAMBOU B. " Les m~canismes de d~formations p l a s t i q u e s dans un sol granulaire" Revue FranGaise de G~otechnique n ° 31. CAMBOU B. - SIDOROFF F. ( 1 9 8 5 ) " D e s c r i p t i o n de l ' ~ t a t d'un mat~riau g r a n u l a i r e par v a r i a b l e s i n t e r n e s s t a t i q u e s ~ p a r t i r d'une approche d i s c r e t e " . Journal de M~canique Th~orique et Appliqu~e. V o l . 4 , N° 2 ,pp.223/242. CHRISTOFFERSEN J. MEHRABADIM o - NEMAT-NASSER S. - " a Micromechanical d e s c r i p t i o n o f g r a n u l a r m a t e r i a l b e h a v i o u r " . J. o f . Appl. Mech. Voi.48 , pp. 339.344. LECKIE F , A . - ONAT E.T. (1981) " Tensorial nature of damage measur i n g i n t e r n a l v a r i a b l e s i n physical non l i n e a r i t i e s i n s t r u c t u r a l analysis." Ed. J. Hult - J.Lemaitre - Springer B e r l i n , pp. 140/155. LUONG M.P. (1980) " Ph~nom~nes c y c l i q u e s dans les sols p u l v ~ r u l e n t s " Revue FranGaise de G~otechnique n ° I 0 . pp.39/53.

10

MEHRABADI M. - NEMAT-NASSER S. - ODA M. " On s t a t i s t i c a l o f stress and f a b r i c i n g r a n u l a r m a t e r i a l s " I n t . J. Num. Anal. Meth. Geom. 6. 1982, pp. 95/108.

description

11

ROWE P.W. (1969) "The r e l a t i o n between the shear s t r e n g h t o f sands i n t r i a x i a l compression, plane s t r a i n and d i r e c t shear". G~otechnique 19 - V o l . l . pp. 75/86.

12

TATSUOKA F . - ISHIHARA K. ( 1 9 7 4 ) " Y i e l d i n g o f sand i n t r i a x i a l compression " S o i l s and Foundations, 14, 2 pp. 63/76.

13

THANOPOULOS I . (1981) " C o n t r i b u t i o n ~ l ' ~ t u d e du comportement c y c l i q u e des m i l i e u x p u l v ~ r u l e n t s " . Th~se D . I . Grenoble.

62

14

THORNTON C. - BARNES D.J. " On the mechanics o f g r a n u l a r m a t e r i a l " C.R. IUTAM Symposium "Deformation and F a i l u r e o f g r a n u l a r m a t e r i a l " D e l f t , Balkema 1982 , pp. 69/77.

15

THORNTON C . - BARNES D.J. (1984) " The r e l a t i o n s h i p between stress and m i c r o s t r u c t u r e in p a r t i c u l a t e media". CoR. Du Congr~s I n t e r n a t i o n a l CHISA - Prague.

16

WEBER J. (1966) " Recherche concernant les c o n t r a i n t e s i n t e r g r a n u l a i r e s dans les m i l i e u x p u l v ~ r u l e n t s " . B u l l e t i n de L i a i s o n des Ponts e t Chauss~es n ° 20 , pp. 3 . 1 / 3 . 2 0 .

A MATHEMATICAL

MODEL

FOR THE L I Q U E F A C T I O N

Lucia

Polytechnical

OF S O I L S

Dr~gusin

Institute,

Bucharest,

Romania

Abstract. A unitary model for mechanical soil behaviour is set forth. By u s i n g the particular cases of the material constants occurring in the constitutive equation, the model is able to interpret the behaviour of both granular and cohesive soils, either normally consolidated or overconsolidated under monotonic loading. It accounts for the occurrence of dilatancy in cohesionless soils and of liquefaction in certain cohesionless soils undergoing quasistatic cyclic loading.

1.

INTRODUCTION

The behaviour describe from the

work

soils

the

behaviour

of

three

the

constitutive processes,

reloading

processes

a

saturated

loading

for

supplies

of

join

mathematical under

this

in

bi-phase

equations

of

the

The

stress

the

the

paths

the

order

model

an

processes

mechanical

In

the

rate-type,

stress

space,

for

loading.

material,

one for the unloading

processes.

up

model

monotonic

equation

initial

for

and another

described

by

stress

to

starts

one

these

state

of

a

process being equal to the final stress state of the previous process. Strain and stress will be taken positive The material i.e.

viscous

superposed only theory

as

and

rate

dot will

an

properties

will

effects

will

indicate rate,

ordering

parameter

in

of plasticity.

Monotonic

and

in compression.

be considered not

be

independent

covered

by

but in the latter, the

sense

cyclic

time

generally

loading

will

of time,

the

model.

is

intended

used be

A

in

the

considered

as quasistatic. The materials:

model

Relying model

may

describe

the

behaviour

cohesive soils and cohesionless on

highlights

constitutive the

way

in

equations which

the

of

two

large

classes

of

soils. of

the

hypoelastic

initial

state

of

type,

the

stress

and

strain has to be formulated. The dilatancy phenomenon of cohesionless existence

soils in evinced by the

of a value of maximum density P (for certain

loading paths).

64

The

occurrence

of

cohesionless

undrained

cyclic

loading

is

stability

in

constant

density

will

the

establish

where

the

entailed

effective

the pore pressure

The

liquefaction account

The

paths

of

the

condition

featured

undergoing lack

of

p : constant

by undrained

tests

is zero.

mathematical

MATERIALS

model

set

OF G R A D E T H R E E

forth

starts

from

the

following

:

i. soils initial

on

curves.

stress

2. A CLASS OF H Y P O E L A S T I C

hypotheses

soils

have

memory,

state and the stress iio for

a certain

their

behaviour

depending

on

the

history;

stress

history,

the

dilatancy

of

soils

may

occur; iiio

because

mechanical

energy

the

related

stress

history

for to

T = T(T),

normal

at

tensor

(L = -grad x

initial

stress

surface); is of

tensor

that

of

the

the work

depends

non-

done

on the

by

stress

~

= f ( (Ps)° tr(TD)dT]dVo, V o to Ps

(defined

the

spatial

the initial solid phase

(see Dr~gusin [I]). iv. for a closed (undrained)

the work

contribution

configuration

D = ½(L + L T)

being

particle); V o = V(t o) relative mass density volume)

the

the

we suppose

t = f (ftr(TD)dV]dT to V

T is the Cauchy body

soils

be ignored,

• ¢ [to,t ]

[w(T( - ))](t)

where

drained

can not

w is a conservative

function,

is

by means the

gradient

of the

rate of

of

the

internal

deformation

velocity

of

a

volume V;(Ps) o is the initial (mass of the solid phase/total incompressible namely

there

system

(p = po ),

is a function

@ so

that dw : tr(TD)dT

: SdT.

PROPOSITION 1.

The c o n s t i t u t i v e e q u a t i o n

~=(~1x2+6ox3+~--~7y+61xY -

Z)IDI+ (~2x2+62x3-~--y+63xY+64z)D +

6 -8 +( ~3x+ 65x2)IDT+( (~4x+66 x2 + ~ y ) t r ( T D ) I + ( ~ 5

2B9 2 67y)TD2D_____~T+ x - --~-x +

(I) e7 + (-T+

~

X)IDT2+(~6+68x)tr(TD)T+(-

+ (~7 + 6 9 x ) T 2 D 2 D T 2 -

~7 87- 89 ~-- + ~ x)tr(T2D)I +

B4tr(TD)T 2-67tr(T2D)T,

65 where I is

is

the u n i t

the Jaumann-Noll

I D = t r D i s the t r a c e of D; ~ : T + WT - TW

tensor; stress

r a t e , W being

the spin

tensor; T* = T

i s the d e v i a t o r i c s t r e s s tensor; x = t r T; y = t r ( T * ) 2 ;

2 81 a2 + a7 (9 a2+9 e5+4 a7 ) el = 27 (9a2+3e5+e7) '

-27 a2 ~5+a7 (9 a2 + 12e5+5 a7 ) 9 (9e2+3e5+a7)

a9 =

I 81 =- 1--~(384+2786+287+388-1169 ),

~ITI z = t r ( T * ) {, and

I 83 : ~(84+27B6+287+368-989 ),

85 = - ~(81 8o+2782+2786+287+368-589), i m p l i e s the e x i s t e n c e of

$(x,y,z)

two p o t e n t i a l s ~,~ so t h a t

x : ~ID,

~)(x,y,z)

: tr(TD).

They are

16a3x3-3ab2xy+2b 3z laI+ 16a3x~-3ab2xoYo+2b3Zo

~ ( x , y , z ) - ~ ( X o , Y o , Z o ) =--!-I i n b2d 2 2AXo-CYo

I

+ 2--~(

~ xo

2Ax2-CY),x3 (2)

i

~(x,y,z)-~(Xo,Yo,Zo )---!-l-b2 d In x 0

~

I

I [2AXo2-(c+D)Y0 2Ax2-(C+D)v

.j

-

where I

I

a= a2+~a5+~a7 ;

2

b= a5+~a7 ;

c: 3a4+a5+a6 ;

d:9 ai+3 e2+3 a3+3 e4+ a5+ a6--~a7 ; B:~(84+2786+487+388-889);

I C=~(84-986-388+289 );

D:-(188o+682+686+-~(87-89)); and

a,b,d,A,B,D,E, > O,

d-c > O,

I

A= 82-~89 ;

E:87-84, C+D > O,

b-3a > O,

b2-3ac > O.

PROOF. From the constitutive equation (I), because :

-

2X

X

2-

X

1 +

66 we obtain

½:[d~---qCx2+ ~ x 3 ] I D + ( C x - C x 2 ) t r ( T D )

_1 ~:[_ 3a9____.bbx33a(d-c-b)+b2 xy__~x 4 2

+

-

3a

I" ra 4

~z=[~x

+

-

9a

C2Dx2y]I D

+

+

[3a321Ox2

-

Y+ Ax 3 - ?xy]tr(TD)+(bx)tr(T2D) 6a-b 2 1--T'~- x

Y

+

~a(d-c)+b 2 x z 9a

+

~x 5

-

3A Bx3y 9

(3) -

6B+E ~xy

2

+

6B+9(C+D)-2Ex2 z E z 2a 3 ~xy - b2-3ac 2A 4 18 + 3Y ]ID + [ - T x + 3a z ---~x -

- T2B x 2 y- 6B+96C-4E.xz]tr(TD)+(ax2+Ax3+Bxy_Ez)tr(T2D) Then " 2 6a3x3+3a(2ac-b2)xy+2b(b2-3ac) z ]x-3a cx 2 - y+2abexz- + 3(2Ax2-3Cy) ID:3[ dx2(6a3x3_3ab2xy+2b3z) 2ADx 5

tr(TD)={6a3x3-3a[2a(d-c)+b2]xy+2b[b2+3(d-c)]z}~÷3a2(d-c)x2y dx(6a3x3-3ab2xy+2b3z)

_

2ab(d-c)xz + [2Ax2-3(C+D)~]i+(C+D)xy dx(6a3x3-3ab2xy+2b3z) 2ADx tr(T2D)=[ 2Ax4+ (6A-C-2D)x2y-9Cy2-9Dxz 6ADx 5

'

(4)

+ E 2Ax3y2-3Cxy3-18Dxz2-36C}/2Z]x + 12ADx5 (Ax3+Bxy-Ez) ]

I(C+2D)x2+3Cy CExy2-6CEyz+6BDx2zl z 4 Y+ ' [ 6ADx 4 + 12ADx (Ax3+Bxy-Ez)l 3(Ax3+Bxy-Ez) if x ~ 0, 6a3x3-3ab2xy+2b3z ~ O, The differential forms

x

: ~I D

and

Ax3+Bxy-Ez ~ O.

~ = tr(TD)

are exact. By integrating these relations we get the expressions

(2).

REMARK. In the axial-symmetrical case, the stress tensor T and the rate of deformation tensor D have the form

I! I T =

0 TI

0 0

0

T3

D =

lilop10D3:I

67 Then x = 3P,

Y : ~q2 ,

2 3 z = ~q

and from (2), (4) we have

$(p,q)= [(9ap+ebq)(9ap-bq)+6acqa]P-6acpqq dp(9ap+2bq)(9ap-bq)

+ ~(9Ap2-Cq2)p+eCpqq 81ADp4

~(p,q): [(9ap+2bq)(9ap-bq)-6a(d-c)q2]p+6a(d-c)Pqq dp(9ap+2bq)(9ap-bq) +

3[ 9Ap2- (C+D) q2 ]P+2 (C+D) Pqq 81ADp 4

~(P'q)-

•

(5)

+

~(P°'q°)=

llnIl~Pb2dol b2-3ac

I i27Ap2-Cq2 o o I93a3p3_27ab2pq2+2b3q3 Ia I 3a3 3 2-ab 2 _2+2b3^3 + I~-I-A-D p~ 9 Po- Y Polo ~o

27Ap2-Cq 2 ) p3 (6)

~(p,q)

- ~ ( p o , q o ) = ~--~dln

0

Po- • Poqo qo a(d-c) I " 193aSp3_27ab2pqe+2b3q~ +~

3 Po

-

p3

"

In this case, the system (3) may be inverted if p ~ 0,

9ap+2bq ~ 0,

9ap-bq ~ 0,

27Ap3+2Bpq2-~Eq 3 ~ 0.

We consider the set D = {(p,q) IP > 0,

9ap + 2bq > 0,

3. THE S T A B I L I T Y

9ap - bq > 0,

WITH R E S P E C T

~3p ---Ep9B 2_

TO THE INITIAL

243A2E > 0}.

STRESS S T A T E

DEFINITION I. Let D c R 2 × R be the set in which the equation F(p,q,k) = 0 implicitly defines the function p = f(q,k) and let q -+ fk(q) be the partial function. We say that q* is a critical point df k for

fk

if

condition

(d-~] q=q,-- O. I t

d2f k Id~)q=q.

i s a nondegenerate c r i t i c a l point i f

~ 0 is also satisfied;

degenerate c r i t i c a l point.

if

d2f k Ij)q=q,

the

= 0 it is a

68

DEFINITION ~f k* ) = O. ~q(q, with respect and

a

2.

We

say

that

- k I < 6(¢),

and

V .(E) for

q

qe c V .(¢), q

d2fk

d--~--(q*) Cristescu,

let

= 0. The

nondegenerate We

q*

the

function

f .

any E > 0 t h e r e

that

equation

for

any

is

q

is

an ~(~)

k,

for

which

~f ~-~(q,k) = 0

has

a~

> 0

Ik* -

least

a

< ¢ .

Dr~gu~in and

[2]

the

following

q* a critical with

proposition

point

respect

has

for which ~q(q*

to the parameter

been

, k* )

=

k in its

criticqal points.

shall

further

~(p,q)

: qb(po,q o) state

and

study

~(p,q)

from

this

point

: ~(po,qo ) with

of

view

respect

the

to

the

curves initial

(po,qo).

From of the continuity

equation -~ = I D one has -~ = 15 P •

This

relation

represents

stable

I

f , is stable

stress

means the

that

stress

the paths

for

characterizing

constant

thus

tests with pore pressure

PROPOSITION 2 .

function

p :

function

p

p(q,k)

pertaining

: 8cAD'

If

the

equation

then

the

to

the

¢(p,q)

which

undrained

total

= k,

where

mass

behaviour

of

k = ~(po,qo)

density a

P

remains

material

under

equal to zero. ~(p,q)

= k

critical

implicitly

degenerated

domain D are

(Pl

defines

points

- adC 3cAD'

of = 0)

ql

the the and

q2 : 32bcAD )"

PROOF. satisfy

for

so

the

(qg)

k* be fixed

function

if

q*

of the equation

we have

dq2

In

point

k,

of

which

d2fk ,

proved:

in the

to the parameter

neighborhood

solution

Let k* be fixed and q* a solution

The

critical

degenerated

points

of

function

p = p(q,k)

the relations dp 0, dq =

d2p 0 dq2 = "

We

rearch

shall

for

the solutions

to the domain D. From the relation

[243acADp3-dC(9ap+2bq)(9ap-bq)]

Ii

of this

(51) , we obtain

is equivalent

to

for

: 0

43acADp3-dC[(9ap+2bq)(9ap-bq)+bq(9ap-4bq)]

This system

system

= 0.

which

$(p,q)

belongs

= 0

69

Ii °

Ii Ea laIasCAD

:

:

~

-

-T~p~]

and

(7)

adC 3cAD yielding

9a I~P

the solutions

(pl,ql)

and

(p2,q2).

[d2p) 2 -adC+3cADp ,dq 2 q=0 - 27aAp 3Dp+d

As

dq 2 1q

'

cAD 3a-8--~p )] '"

:

8b2d2C2 [ a+/3a ( ~a~- 8cAD~-~-p) ] /3ai ~ a- ~-~-p )~cAD :+ 81acADp2116b2A(3Dp+d)+5dC(a

if

c,C > 0

for

: ~b [a + /3a(3a

are

maximum

points

maximum

the

on

points

on

the

the on

ado

Po < ~

± /3ai3a-8~cDp)] From (pl,ql) relation

the

points

points

on

= ~b [ a

q

c,C < 0, q = O;

then

and minimum

on the

axis

curve

q =

q = ~b [a -

adC q = 0; for Po < 3cAD ± o-,3a

~~ -c~A-~P .) ]

-

for

p ] and are maximum points

and

stress

¢(pl,ql ) = ¢(po,qo ) viz.

there

adC 3cAD < Po < ~8cAD

the

the minimum

maximum

points

the

points

on

on the curves

are

maximum points the

on

axis

q =

[a ±

instability

point

on the axis q = O.

(6) we find

for the initial

adC Po > ~

for

b [a+ 3F~a(~5a-o~--P)], ~cAD q : 49_~

there

along

curve

on the axis

curves

points

= O.

hand axis

minimum

are minimum

points

a-8

the relation

occurs

q

other

q = ~b[a-

for

on

axis

the

exist

maximum

points

are on the curve

curve

q = 0;

. - 8cAD ~-~-p)],

minimum

If,

points

there

_ gcAD ~-~--P)]" and minimum

/~3a

there

then

adC 3adC for ~c-AD < Po < ~r6-~ there

q = 0;

_

adC Po > -~'AD

-+ 3~a( ~a-o~-~--p)]~cAD (

that state

the

first

(po,qo)

which

verifies

the

70 9a3po3

2 2 27APo-Cqo adC In po+ab-~In 9a3p 3 _27ab2p^q 2 +2b3q 3 + d 81ADp3 : in -3cAD u

v

U

v

while the second instability point state which verifies the relation

U

(p2,q2),

for

the

initial

2~A 2 - 2 f Po-Uqo ac : in 3adC --:In 9a3p3 27ab2p q2+2b3q3 + d 81ADp~ 9a3p 3

in

po+a~-~in

0

+

C

0

stress

d32

+

0

15a2C-8b2A 18ab2C

PROPOSITION 3-

by the

0

aC cA

equation

If

¢(p,q)

the function q : q ( p , k ) then

= k,

its

is

i m p l i c i t l y defined

c r i t i c a l degenerated

points

from

domain D are the solutions of the system -

(81~ 2AE2o

E2Fo )2

+

81a 2A (EoF I - EIFo)(E2F I - ~ E o E

I) = 0 (8)

q

:

EoF

i_p

EIF o

(E2F o

81a2A E 2) - ~ ,

where

Eo=-b[540AD2(b2-3ac)p2+24dD(27a2C+11b2A-18acA)p+ +d2(171a2C+32b2A)], E1=9a[108AD2(3b2+ac)p2+9dD(18b2A+2acA-9a2C)p÷2d2(10b2A-9a2C)]

,

E2=243a2bA(153D2p2+82dDp+11d2), Fo=648A2D2(b2-3ac)2p2+36dAD(45a2b2C+8b4A-117a3cC-24ab2cC)

p +

+ d2(729a4C2+360a2b2AC+32b4A2), F1:-27abA[144AD2(b2-3ac)p2+dD(68b2A2+81a2C-108acA)p+ +2d2(9a2C+4b2A)]. PROOF. The critical verify the system dq O, dp =

degenerated

points of function

d2q 0 dp2 =

viz. [9Ap2(3Dp+d)-dCq2](9ap+2bq)(9ap-bq)+162acADp3q2

=

0

q : q(p,k)

71

A p ( 9 D p + 2 d ) (9ap+2bq) ( 9 a p - b q ) + a [ 9 A p 2 ( 3 D p + d ) - d C q 2] (18ap+bq)

+

+ 5 4 a c A D p 2 q 2 = 0. By obtain

the r e l a t i o n s

p. H e n c e tical

successively

the

(8).

functions

degenerated

eliminating Relation

q = q(p,k)

q

from

these

two

(8) I is an e q u a t i o n

may

h a ve

relations of d e g r e e

we 8 in

8 or 6 or 4 or 2 or no cri-

point.

PROPOSITION 4.

Equation $ ( p , q ) : 0 has no singular

points

in D

i f b2A - 3a2C > 0. PROOF. ly s a t i s f i e d .

In

its

singular

points,

equation

$(p,q) : 0 is i d e n t i c a l -

Then

243acADp3-dC(9ap+2bq)(9ap-bq)

= 0

9Ap2(3Dp+d)-dCq2](9ap+2bq)(9ap-bq)+162acADp3q2 This

system

is e q u i v a l e n t

: O.

to

gA~2~2(2b2+3ac)2p2 + 23~D[(2b2+3ac)(8b2A-15a2C)+9a3cC]p d

The

P l ,2=d

2 + (b A - 3 a 2 C ) ( 4 b 2 A - 3 a 2 C )

first

equation

+

= 0

has the r o o t s

-~[{2b2+3ac)(8b2A-15a2C)+9a3cC]± ~a2b2C2+4aC(2b2+3ac)(aC+cA ..... 6 A D ( 2 b 2 + 3ac) 2

F r o m the b2A - 3a2C > 0. The

condition

curves

¢(p,q)

that

the

= ¢(po,qo )

two

roots

are

are

negative,

graphically

we

represented

)

find

for

c, C > 0 in fig. 1 and for c, C < 0 in fig.2. The dotted q -- ~ b [ a

The

slope

line r e p r e s e n t s

the f u n c t i o n s

+ ~/3a(3a - 8~-~-p) cAD "~],

of the s t r a i g h t

6a2B + b2A 6a 3

]. - ~/3a ( 3a - o~-~--p)~CAD,

line q = mp is the s o l u t i o n

0 m 3 - ~Bm2Q - 243A 2E = " If E < b

q = ~b[a

9a then o < m < --6"

of the e q u a t i o n

72 For

a stress

(p,q)

belonging

to

the straight

line

q = mp,

the

soil sample becomes unstable.

O.

t

~//// ~ . "/

,; ,Y//" /

/ "k

1 2:

Fig.

g. U n d r a i n e d

(P = const.)

P

~2', \ ~ The curves @(p,q)

= ~(po,qo ) have the equation

ll~ p+a(d-c) b2 in

p3 27Ap 2- (C+D)q2 - d 93a3p3-27ab2pq2+2b3q 3 81ADp 3

a(d-C)l n

in p o ÷ ÷

Po3

^,3a3_3 2"ab 2 Po- f

27Ap2o-(C+D)q2

2+2b3_ 3 - d

Poqo

qo

tests

for c, C < 0.

81ADp 3

73 Their only points of maximum value are on the axis q = O. Fig. 3 provides a graphical representation of the curves ~(p,q) : = @(po,qo ) •

Fig.

3.

Curves

~(p,q)

= const.

f

\

X

We shall further consider the response of a material having the constitutive equation (I) in the case of some particular loading. PROPOSITION 5 .

stress

path T I :

-P=

Let

i" lq°+3Ti 13(2b-3a)

Po

L

be

(I)

T cI = c o n s t a n t .

the

Then,

constitutive

((b-3a)q-9aT~

lq+3T I

equation

and

the

in the domain D, we have

4(b-3a)

b-3a)qo-9aT 1

. t((3a+2b)q+9aTi t(3a+2b)~ 6b'~:Ti 3a+2b)qo+9aT~

)

"

f" ~(b2-ac) ( I I ] 3A-C ~ I q+3Ti)2 • exp '~- b2 d q+3T1c - qo+3T cI + ~

I

]+

c2 (qo+3T1)

27C(TI)2 4CTI I~ + --~((q+3T~)3

o 6b 2T cI

(qo+3T1)31 c )_

~ + 3 T cI

+11((.~,a 1 q+3Ti) 2 where

e : D 3 - D I.

((q+3T1 ........Ic)4 - (qo+3Ti)41)~,

4AD

- _----~-E_ cI l(b_3a)q_9aT

t

I(3a+2b)qo+9aT I I

1(

T c

3 (qo+3T1) 2) - ~

......

1

(q+3Ti)3

1 (qo+3T1) 3)'

(9)

+

74

PROOF.

For an axial-symmetrical

2TI + T3 3 '

P c

Then

T 1 = p- ,

c q = T3-TI'

T 3 = p+

,

loading

we have J

I D = 2D1+D3,

DI =

ID-~ 3 '

e = D3-D I. ID+2C 3 '

D3-

$ : 2T D1 ÷

2 • + T3D 3 = pI D + ~qc. Introducing

these relations

into

(5) we get

[(3a+2b)q+9aT~][(3a-b )q+9aT1]-6acqT c cI P

(q+3Ti)2[(3a+2b)q+9aTi][(3a_b)q+9aT1

]

3A(q+3TI)2-Cq(q-6TI) 1 +

(10)

AD(q+3TI)5

27aTi

c=q

q+3Ti)[(3a+2b)q+9aT1][(3a-b)q+9aT By integrating the

relation

points

the

q : GI(q)

(10)

I]

them we find the expressions that

solutions

the of

function the

p = p(q)

equation

(9). has

of

the

One

as

finds

its

5th

degree

- dCq(q_6Ti)]+18acADTiq(q+3Ti) for

c, C > 0

llm GI(q) ~-~

interval

(O,ql) , whilst

in this

interval.

the

(11)

< 0,

lim GI(q) ~ q2

function

p = p(q)

For

c, C > 0 this

root,

(2)

if G1(q)

curve

has

= 0 has

(3)

if G1(q)

= 0 has one positive

val

(q2,ql),

(4)

roots

in interval

negative

roots

if

< 0,

q2 = one at

lim GI(q) ~ 0 9aT I 3a+2b real

least

If c, C < 0, the function p = p(q)

one critical point in the interval (q2,0). The curve p = p(q) has been graphically real

G1(q)

(q2,ql),

in interval

= 0 (5)

plotted

the

shape

(I)

three

Foots

in the

and two negative has

three

if G1(q)

(q2,ql).

2 -

3

9aT I lim GI(q) > 0, lim G1(q) > 0, where ql - b-3a' ql q+~ lows that the equation (11) will have at least point

in

= 0, where

G1(q) : [(3a+2b)q+9aT~][(b-3a)q-9aT1][3AD(q+3T1)3+3dA(q+3T~)

Since

from

critical

positive

< 0,

' it fol-

root one

in the critical

has at least

in fig.4a.

if G1(q)

= 0 has

interval

roots and

in the two

= 0 has one positive

a

(0,ql) , inter-

negative and

four

75 For

c, C < 0 the curves

drawn. The dilatancy. as

stresses

The relation critical points

with

(I'),

(2'),

GI(q) = 0

(3'),

are

(4'),

related

to

(5')

have been

the

material

(10) 2 makes obvious that the function c = e(q) has the solutions of the third degree equation in q

G2(q) = O, where G2(q) = 54aAT~(q+3T~)3+(q-6T~)[(3a+2b)q+9aT~][(b-3a)q-9aT~]. The equation G~(q) : 0 has the positive roots e

TI

[-162aAT~+b(4b-3a)

±

qi,2 = 54aAT~+ (3a+2b) (b-3a) ±A'162abAT~(14b-9a)+16b4-60ab3-99a2b2+405a3b+272a

4]

which occur if

C

TI
O, q+q2 lows that in the

lim G2(q) > O, l i m G 2 ( q ) > O, G~(q I) < 0, it folq+O q÷ql plot of the function a = E(q) (fig.4b), the curve

(I") will occur if G2(q ~) > 0, while

(2") when G2(q *) I < 0

,,q

~-~

4~

Fig.

4a.

Curves

p = p(q)

for

T 1 = const.

Fig.

4b.

Curves c = c(q) T 1 = const.

for

76 REMARK. The stress path T I : TcI = constant responds to the "triaxial al compresion" experiments. PROPOSITIOII 6. the

stress

path

T3 :

Let Tc 3 :

us

take

the

constant.

constitutive

Then,

in

the

for q > 0 corequation

(I)

domain D we

and

shall

have

p S]2qo-3T~ 13(3a+b)12 (b-3a)q+gaT ~ (b-3a) Po= \12q-3T~ ~'(b-3a)qo+9aT ~

)ac

(6a+b) qo-9aT ~

.exp~ (b2-3ac) ( I b2d 2q-3T e3 CT~ ('(2q_3T3) Ic 3 + A--~-

I ) 12A-C I 8--'8-A-D-( 2q_3T~)2 2qo-3T~ (

I T c 2)+ (2qo-3 3 )

27C(T~)2 ]) (2qo_3T~)I 3 )+ 16AD ('(2q_13T~)4 (2qoZ3T~)4 ..... (12)

%+ 3bT31"#12q 12(b-3a) qo+9aT3 ~ u-3T$ ~ I

I

(6a+b) qo-9aT~ ~ I " l(6a+b)q-9aT~ I(6a+b).j - ~A((2q_3T~)2

_~,u(

........ I (2q-3T~) 3

I

C

I

I (2qo_3T~)2")

).

(2qo-3T~)3

e the equation of the stress path is p = PROOF• If T 3 = T3, 3T~ - 2q • From the relations (5) we get 6-[ (6a+b)q-9aT~][2(b-3a)q+9aT~]+9acT~q I~ = ~i .(2q-3T~)2[ ... (6a+b)q-9aT~][2( b-3a)q+9aT~ ] +2

3A(2q-3T)2-Cq(q+3 )) AD(2q_3T~)5

~%(

27aT~ : q 2q-3T3) C [ (6a+b)q-9aT~ ][2(b-3a)q+9aT~ ] + A(2q_3T~)t ~ " By integration we obtain (12).

(13)

77

The

functions

p = p(q)

5a and fig. 5b. One o b t a i n s and

(I'),

(2'),

(3'),

and

¢ = E(q)

the c u r v e s

(4'),

(5')

(I),

have

(2),

been

(3),

plotted

(4),

in

(5) for

fig.

c, C > 0

for c, C < 0.

,~t-~ Fig.

5a.

Curves

REMARK ponds

for

T 3 = ct.

Fig.

The

stress

path

T 3 = T c3 = const,

1.

to " t r i a x i a l REMARK

curves

shown

5b

q < 0,

for

p = p(q)

2.

extension" For

in fig. the

joint

4b

¢ = ~(q)

for

q < 0

will

look

for

corres-

experiments.

a cyclic 4a,

5b. Curves T 3 = ct.

loading,

for

these

q > 0 and

occurring

in point

curves

like

the

like

those shown in fig. 5a, e e (p = T I = T 3, q = 0). F r o m

Y ~t{b-3d Fig. 6a. Curves p = p(q) for cyclic loading q C [-qo,qo ] .

Fig. 6b. Curves E = ¢(q) cyclic loading q C [-qo,qo

for ] •

78

the relations (10) and (13) we notice that this joint is reached with a continuous derivative for the curves e : e(q) and a discontinuous one for the curves p = p(q). For a cyclic loading

0, t3=t I Assuming

that

constitutive :

the

mechanical

behaviour

of

soils

is

process

Pn

described

by the

equation

} #(n)X(P n ) n:1

where X(P n ) :

is

the

~0

characteristic

'

(P'q)

¢ Pn

,

(p,q)

¢

function

of

the

expression resulting from the equation ~i(I)' 8i(I) for the unloading~ processes, for reloading processes and ~ 3 ) •

while

~

is

the

(10) with the coefficients the coefficients ~k2)i" "' ~i

~(3) for loading processes

' Pi

The initial stress state for a process final stress state of the previous process.

will

be

equal

to

the

79 For processes

@(p,q) we shall

: ~(po,qo ) assume

(~)(2)

REMARK. the

tensor

)(3)

In drained

of effective

pressure).

loading, stress

If the undrained

maintains

constant

its

shape

in

various

: (_d.~)(2) : (--..-:-) d-c

the

while

stress

tensor

in undrained

will

loading

(the tensor of effective loading

total

mass

(3) (3)

(c~___~D)(~)(.q~_:)(2)(c~___~D).

C+D

to the tensor of total stress which

preserve

d-eb : (_~)(1)

a a (I) s:(~) :I~J~(2) :I~) ( 3 ) ;

d d (~)

to

that

be equal

to

it is equal

stress minus

pore

to the stress

path

confines

itself

density,

then the pore

pressure

is

zero and the tensor of effective stress equals the tensor of total stress. DEFINITION. under

undrained

to zero,

We

the whole

The

shall

cyclic

loading

undrained

unloading,..,

plotted

say

loading

that

a

sand

if the tensor

sample

to

liquefy

stress

tends

being taken over by pore pressure.

cyclic

test

of

loading,

unloading,

in fig. 7 will have the equations

¢(3)(p,q)

: ¢(3)(Po,0) '

q g [0,qo]

¢(1)(p,q)

: ¢(1)(pl,qo),

q e [O,qo] ,

q < 0

¢(2)(p,q)

= ¢(2)(p2,_qo) ' q ¢ [_qo,0],

q < 0

¢(1)(p,q)

= ¢(1)(p3,qo),

q ¢ [-qo,0],

q > 0

@(2)(p,q)

= ¢(2)(p4,_qo) ' q e [0,qo] ,

q > 0

q %

Fig. 7. Stress path for a cyclic undrained test

tends

of effective

m

reloading,

80

Owing

to

the

shape

change

of

curves

~(p,q)

= censt.,

one noti-

ces a steady decrease of effective pressure p.

5. C O M P A R I S O N

WITH E X P E R I M E N T A L

DATA

I) Constant density curves plotted in fig. I. (g for c, C > O, adC Po < 3-c-~ ) agree with the experimental results for undrained tests on Fuji

River

sand

fig.3,5,6),

(Nova,

loose sand

3,

fig.

13.6),

(Thurairajah,

dense

2) Constant density curves plotted with

the

experimental (Nova,

data

ated

kaolin

fig.

13.12) and Weald Clay

3,

3) The "Yield imental

results

for

fig.13.11),

a

in fig.2

for undrained

test

plotted

normally

(Thurairajah,

4,

(for c, C < O) agree

on a n o r m a l l y

overconsolidated

(Mroz, Norris,

function"

sand

4, fig.4.7).

kaolin

5, fig. 8.

(Nova,

3,

10).

in fig. 3 agrees with

consolidated

consolid-

kaolin

the exper-

(Nova,

3,

fig.

13.3). 4)

c T I = TI,

For

(c, C < 0)

agree

with

the

curves

the

(I'),

5) For T I = T~,

sand (Nova,

results

the curves

kaolin (I'')

cell

for

in a

(Nova,

fig.

4a,

constant

3, fig.

13.4).

(c, C > O) agree with

pressure

4b

cell

test on a Fuji

the

River

3, fig.13.7).

6) For T I = T cI < T3 ' curves

(2),

for a constant

plotted

results

pressure test on a normally consolidated

experimental

(I'')

experimental

(11) , (I~)

the experimental on dense sand

T3 = T c 3 < T I in figs.

for loading data

and

for drained

(Thurairajah,

(12) , (2~) triaxial

6a,

6b

(c, C > 0, the

for u n l o a d i n g

agree

with

compression-extension

tests

of c o h e s i o n l e s s

if c,

4, fig.1.2).

CONCLUSION

I) Our model

descibes

C > 0 and of cohesive dilatancy

soils

the behaviour if c, C < 0.

It

involves

The

emphasis

falls

on

the

importance

the initial stress and strain state (Po' qo' 3) The occurrence the

the occurrence

of

in c o h e s i o n l e s s soils.

2)

in

soils

shape

of

of instability

unloading

and

of

stress

history

and

Po' ~o respectively).

points accounts

reloading

curves

the

of

with

for

the change

respect

to

the

initial stress state. 4) curves rence

The

for of

materials.

sharp

the

the

small

change

in

values

of mean

liquefaction

phenomenon

shape stress

unloading

p accounts

in the case

and for

of some

reloading the occur-

cohesionless

81

REFERENCES

I.

L. Dr~gu~in, A 511, 1981.

hypoelastic

2.

N. Cristeseu, L. Dr~gu~in, On the stability with respect constitutive parameters, Rev.Roumaine Math. Pures Appl., 833,1984.

3.

R. Hova, A constitutive model for soil under monotonic and cyclic loading, Soil Mechanics-Transient and Cyclic Loads, 343, 1982.

4.

A. T h u r a i r a j a h ,

5.

Proc. 8th Int. Conf. Soil.Mech. Found. Enging. 1.2,439,1973. Z. Mroz, V.A.Norris, Elastoplastic and Viscoplastic constitutive models for soils with application to cyclic loading, Soil Mechanics-Transient and Cyclic Loads, 173, 1982.

Shear

model

behaviour

for

of

soils,

sand

Int. J.Enging

under

stress

Sci.19,

to the 29,10,

reversal,

THE

KINEMATICS

OF

SELF-SIMILAR

IN M O H R - C O U L O M B

PLANE

PENETRATION

GRANULAR

PROBLEMS

MATERIALS

R. B u t t e r f i e l d

U n i v e r s i t y of Southampton,

i.

INTRODUCTION

Throughout penetrated

to

angle

(#),

shear

and

and

the

this

be

a

such

point

lines are

(1931)

the

can

develop by

on

the

defined

material

continuum, planes

equation are

and Harkness'

on

being

friction which

the

velocity

'slip-line'

(1971)

relative

to

in velocity the

slip

which

line

and

generalisation

: "As successive m a t e r i a l points

any change

the

T = C + ~tan#

also

independently on either

statement

next,

assume

Mohr-Coulomb

so

is B u t t e r f i e l d

the

shall

related

Slip can occur

are considered and

we

characteristics

the kinematic model

a slip-line

slip

stresses

stress

of Geiringers'

paper

rigid-plastic,

that

direct

characteristics.

one

U.K.

occurs

field,

along

between

is

in

the

direction of the c o n j u g a t e slip-line". When extensive

a plane, bed

of

wedge-shaped

material

there

body is displaced are

three

vertically

distinct

into an

categories

p e n e t r a t i o n problem of interest.

2. T H E

INCREMENTAL

Fig.l(a) plest for

case

a

a

a

where lifted

the

slip-line

the

surface.

ground

locally smooth

field

This

_

I

A_

is

at an angle

WEDGE

i ~

B

~0o ~

(a)

is

to cases

surface

and

body

surchar-

in Figs.l(b,c)

OF A BURIED

sim-

wedge-shaped

horizontal,

'ground'

extended

for

of

smooth,

beneath ged,

depicts

DISPLACEMENT

up-

A

~i $'f

~

(p),

rough-faced

wedges respectively.

which

The

general

the

wedge-face

case,

in

friction

Figure

1.

(a),

(b).

B

of

84

angle

(6)

is

in Fig.l(d) define

B

0 < 6 < ~,

which

symbols

is

serves

used

shown

also

to

elsewhere

in

the paper. For angle

a

(8),

given ~,

statically field such

can that

ontal Figure

s £ = cos8

1.

(c),

(d).

that,

theorem

quoted

above

be

B

on

lies

when 6

BB',

the

~

= @

(i)

us to c o n s t r u c t

a con-

velocity for these

as

sketched

Fig.2(a).

fol-

lows

that

from this power

has

been

and

ensured the

incremen-

v

tal

volume

erial

ly

matby

vertical-

downwards

equal

to

is

that

up-

lifted

by

form

displacement

the

uni-

of

AB

Such

consistent

dographs

(Fig.2b).

(for

tical

be

ho-

iden-

slip-line

fields) Z.

of

displaced

AD moving

Figure

of

dissipation

that

~, t---t1

It

immediately

positive

~

in

- p

in

A

is

ensuring

= 0

cases

HOOOGRAPH

horiz-

which to

sistent

(o.)

a

slip-line

hodograph

B

semi-

values

constructed

equivalent

allows

p

admissable

_ c o s ( 4 5 ° - ~/2) -sir&0 -exp(~2 -tar~ ) ; cos(45 ° + ~/2)

The kinematic

wedge

and

always

surface

effect

6

can

drawn

for

ating

materials

which

the

cy angle V

also dilin

dilatan> 0 by

85

extending

the

increments

kinematic

should

(Butterfield

be

the m a t e r i a l

and the h o d o g r a p h more

Fig.3 full

Thus

illustrates solution

= 30 ° ,

with

the

for e =

# = 42 ° ,

= 12.5 °

and

slip-line

in

Fig.3(a),

hodograph Fig.3(b)

dilating

zone

become

curved

ing

an

simple

and

ABC)

which,

in

and

this

CONTINUOUS

this

tail,

see

~ values

wedge is

will

extended

DA

and

example,

be

OF

as p o i n t s

applicable

A FINITE

(possibly

deeply

is to

axis AB

value of the top a p e x - a n g l e For

cannot

AB map

displacement,

along

AB

becomes

to

a

the a

further

WEDGE

incorporating

submerged

and

not be a k i n e m a t i c a l l y

question

on

the

PENETRATION

the

'penetrometer'

surface

after

solution

below)

model,

be

free

3.

of the wedge.

DEEP

case

obvious can

the

Therefore,

displacement

material

the

Figure

along

it.

surface

solutions along

ki

CB

the

of p a r t i c l e s A'B'

An

A~

the velo-

incremental

shaped

B

lead-

in

for example,

In

field

'expanded'

hodograph

3.

the slip-line

(outside

the Rankine

curved

(Y > 0) both

and

All

slip-lines

curve

Fig.2(c).

(v = 21 ° )

Fig.3(c).

cities

characteristics,

material

hodograph

to

velocity

A

(v =

in

the

all

,,

erial

a

has weight

HOOOOPJ~PH

mat-

in

to

that

1971).

~ =

the n o n - d i l a t i n g

= 0)

v

require

p = 3°

the

field

at

to

0~-

and

1972).

above

be-

complex

(Butterfield Andrawes,

inclined

and Harkness,

When

come

statement

then

whether

encompass (Fig.4). (v has

to be

(8) between

when 6 = 0, via

this This

the

not

problem, turn,

zero!),

AB and

suitably

depth,

relevant

or

in

a

the with

for

our

previous B

located

specified

establishes

the p e n e t r o m e t e r

the relationship,

for

parameter.

e,

a unique axis.

86

sinS-exp(~.tan¢)

Fig.4(a)

shows

= s i n % - S i n ( 4 5 ~ - ~/2} ; sin(45 ° + ¢/2) the

slip-line

field

for

YI

and

Last,

1983)

together

hodograph

which

sistent the of

volume AD

sing AB,

i Figure

in i.e.

so

at

be

a

ditions

which

AB

is

%~\\

cros-

remains

straXY

to

AB,

penetrometer AB

surface these

preserve

the

the

which

can

(in

this

latter

con-

geometry

by the solid t

•

de-

crucially,

parallel

solid

the void o c c u p i e d

void

so with uni-

along

is

that ahead

displacement

does

the

It

con-

material

and

is sliding

a

confirms

each

that

to

smooth).

(Butterfield with

displaced

the

hodograph

relative

therefore

~.

AB that

Additionally,

the

(2)

equal-volume

as it moves,

material

case

the

and

velocity

ight.

a case

material

behind

increment

form

such

of

fills

veloped

g

= 90°+B+0

of

body and |

t

t__

X

• x',',,xx\,,, |xXxX'X~kX\

, ~ X\

,

,

,

{

. . . . . .

t

t

t

tt It ~titt

I d

t

!

t|

"]

i-

,,

,

\~.~tlY//

/ /, ,

I

/ I

i

i

t

E

f ,t,t

t t t t JJ_L,t (el

~ t / t~, t, ,

~ t ,t ,t,

OlSPLACEHENT SCAL£ O ~0 20 38

Figure 5. (a) Displacement resultant calculated from the hodograph presented by Butterfield and Last (1983) (~ = 4 2 ° and PEN = 6 . 0 n ~ ) . (b) Displacement resultant vectors (D/B = Z0.0, e = . 5 Z a n d PEN = 6 . 0 n ~ . (c) Displacement resultant vectors. Stationary probe (D/B=Z0.0, e=0.62 a n d P E N =6 . 0 ram).

87

thereby

validate It

by

is also

subtracting

the

the s t e a d y - s t a t e

velocity

stream

of

6 > 0

can

of

interest

its

velocity

origin

to

frictional be

examples.

The

Fig.5(a)

shows

X

flowing

streamlines

that,

from

for

the

this

velocity

fix

the p e n e t r o m e t e r

hodograph

the

past

into

if we

the

Fig.4b)

incorporated

the

to note (u)

in

fluid

solution.

solution a

smooth

are

becomes

as

in

the

displayed

vectors

derived

those

measured

transfers that

for

diamond-shaped

solution

case

(which

from

a

pier.

previous

in

Fig.5(c).

the

Fig.4(b)

O

hodograph model

(% = 42 )

penetrometer

and

test

Fig.5(b) in a dense

bed of sand

4. T H E S E L F - S I M I L A R P E N E T R A T I O N O F A N

The of

a

third

set

wedge-shaped

will

clearly

have

that

p

will

not

surface

will

have

of

body

solutions

concerns

from

surface

something now

be

the

in common

arbitrary

to satisfy,

at least,

(Mahmoud,

on

a

1985).

'INFINITE' W E D G E

the of

with

since

experimentally

continuous

the

those

the

bed. of

penetration

Such

solutions

Section

geometry

of

the c o n d i t i o n

2,

the

that

except

uplifted

the TOTAL

Figure 6. volume

of

equals

the

triangle

material TOTAL

of

by

that

necessitates cos(0 The condition

et

al.

the

(1947)

above

such

the

(i.e.

onset that

of the

motion, area

of

Fig.6a). the

# = 0,

with

B

smooth

on

the

wedge

material

case

and

surface,

that

= cos~;

smooth

from

material

ABF,

studied

+ sin~)

equivalent,

wedge,

requirement,

a unique p value + p)(I

the

uplifted

DEF = that of triangle

Hill showed

displaced

volume

wedge,

p = 8 - ~ non-dilating

•

(3) frictional

material

is that,

cos(0+p) (l+f-sin~)

= cos~

- (l-f2)-sin2O; 2f

with f = cos(45 ° - ~/2) -exp (~ tan% ) cos (45 ° + #/2)

p = 0-~2

(4)

88

This Shield

solution

(1953)

should

which

be

related

distinguished

to

a

smooth,

from

that

dilating,

published

by

associated-flow

model of M o h r - C o u l o m b material. However, have

the

that,

kinematic

of

the slip-line

at

all

exactly

scaled

problem

(with

by

the self-similar

additional

geometry 3)

for

Hill

et

stages

of

and

If we

consider

is s, a typical point velocity

v

(Hill

al.,

is

et

now

then

located

define

in which

penetration

following case

the

stage

scaled

everywhere

the

s

dd~tt =

invariant

fields

shall

Fig.6a) is

was

an

be

Such

first

a

solved

extension

of

his

with

the wedge

process typical

a suitably

by

a

point

the

ratio

wedge,

and

'unit

l:s,

velocity

which

must

a

diagram'

in this diagram

determined

in such a unit diagram, in

penetration

is located by r and moves with

whole

invariant during

on

the

self-similar).

in which

s = i, our

by p and moves

that its g e o m e t r y remains For

the

slip-line

(i.e.

material

in the m a t e r i a l

of all points

to

material weight and dilatancy.

the general

can

1947)

The v e l o c i t i e s any

we

(analogous

the

other

weightless

(1947)

al.

each

solution to include friction,

p e n e t r a t i o n problem we

in the s t e a d y - s t a t e p r o b l e m of Section

penetration, of

# = 0 and

requirement

field

replicas

continuous

u.

represents

therefore

ensure

the motion. in

the

supporting

medium,

we

have, r=s-p and hence dr = W = d-~

ds + p" ~-~

ds

p-

~-~

+

S-

u

in our unit diagram s = 1 and therefore,

u

ds

= v-

We

p-~-~

can

also,

without

loss

of

generality,

specify

the

wedge

ds velocity, d-t ' to be unity whence, u

=

The display v

v

-

p;

'unit for

any

point

superimpose same

in

it

'unit' follows

vectors v and p, Fig.7b. preserved

ds

~-~

=

1

for

within

(5)

the it.

wedge-medium

Fig.6c

shows

system

such

will

then

a hodograph

for

field of Fig.6b.

We now to the

=

hodograph'

all points

the slip-line

drawn

s

then

not

only

the unit hodograph on the unit diagram, scale, from

Fig.7a,

equation

If the g e o m e t r y will

points

on (5) of

within

which as

the

the

value

difference

the unit d i a g r a m it

corresponding

of

both u

for

between is to be to v = 0

89

move

radially

points

towards

along

the

local tangent,

the c o o r d i n a t e origin

free

surface

AB

must

(with u = -I)) but also all

move

in the

direction

of

their

Fig.7c. X

::~A

UHIT DIAORAM

O

T I

~Q

(b)

-_

×

A

~(P)

--I

(c)

PART OF UNIT DIAORAH

Figure Since same on

in

velocity

AB

produced.

the g e o m e t r y be

the

of Fig.7a

correct problem

and

solution

It We

is

important

this c o n d i t i o n (4)

fields

the

(6),

u(AB)

requires

and

in Figs.

continuous

AB

that,

unit

6(a,b)

particles

with

and

in

the

to lie

again 8, #

the

penetration,

of

move

and X has

= OX - p (AB).

therefore

shown

trajectories

along boundary

From and

diagram,

7(a)

are

a

self-similarity the

unit

diagram

to note

how such

trajectories

'focus'

on

their

images in the unit hodograph.

now

is no

to

points

equation

equation

Typical

all

in Fig.7(d).

corresponding wedge

by

case

to be a straight

from

slip-line

posed.

are sketched

has

Whence,

interrelated

hodograph

present

(OX), AB

7.

extend

longer

this

smooth

solution (~ > 0)

to

and

include

the

the m a t e r i a l

case has

in which

weight

the

(y > 0).

90 The

~ > 0

changes

requirement

the

(Fig.6b).

angle

The

does

not

at which

inclusion

the

of

introduce

any

new

slip-lines

meet

the

material

weight

does,

ideas, face

it

of

the wedge

however,

radically

affect

the

similar

self-

solution.

Whereas

it will resemble

that

shown

in Fig.3

self-

be

can

such

an

if

free

1.

longer

plane.

precise

form

is,

surface

of

is

of AB) the

only when

image

of

process.

The

produce

the tangent

P,

self-similarity

well

at

be

P'

at any point

in

preserved.

salient

a solution

features

the

is such

any

unit

clearly a

and

the

met-

gener-

assumed

(P) along AB passes

superimposed

This of

for

shape through

hodograph,

a

trial

solution

are

AB not

initially

(although

no The

of

hods used to equally

the

course,

known

will

with

'expanded'

1

ate Fig.3

only

preserved

hodograph

8.

cle-

arly

similarity

Figure

merely

will

and

error

sketched

in

Fig.8. Fig.9

shows

accurately 90 60

-~. A ~ ....-

surface obtained

!

B

a

an

recorded profile driving

by

steel

wedge

(0 =

O

-30 -60

=30 ) into a bed of dilatant, brass rods i

(~=25 °) (1970).

D

It

Andrawes is en-

tirely plausible

2~0

5

corresponding -5

that

tangents to this curve will pass through in

a

unit

points

hodograph

of the form described above and a tentative

Figure surface

9. H o d o g r a p h profile

and

sketch measured vertical displacement

contours.

solution

is sketched

91

in the figure. Measured about are

7%

of

seen

to

displacement

the

contours

penetration

agree

(for a d i s p l a c e m e n t

depth)

are

plotted

in

the

with

the

predictions

of

reasonably

increment of figure.

even

These

the

very

crude hodograph. It

is

incorporated example

solely

explained

the

around

admissable

the

steady-state

However,

the

because

flow

bounding

systems

3,

a

of

u

single

the hodograph

intimately

coupled

consistent

slip-line

velocity

from with

in self-similar

the

can

system

discontinuity

be (for

BCD)

as

the

and

field

can

Harkness,

is only

be

field

(Fig.7)

such valid

1971)

meaningful

slip-line

problems

specification

'free-surface',

slip-line

(Fig.4) and

angles

of

the kinematic a

(Butterfield

solution

dilatancy

regions

(1973).

it follows

in

and

values

different

different

and Harkness

that

in Figs.l

all

that in

fundamentally,

material

depicted

noting easily

in B u t t e r f i e l d

More of

worth quite

whereas

for u = 0.

geometries

there

as for

are

is a unique

field for each value of ~.

5. C O N C L U S I O N S

The

paper

self-similar, penetrating

distinguishes

planar

problems

rigid-plastic,

ticularly

the

kinematic

establish

the

validity

either of

steady-state

self-similar

fully

consistent

between of

Mohr-Coulomb boundary

of

any

incremental,

arbitrarily

materials.

conditions,

solutions

to

flow or self-similarity.

continuous solutions

penetration for

steady-state

rough,

and such

It

rigid

emphasises

hodographs, problems

In particular

and

wedges parwhich

involving

a new

study

is p r e s e n t e d which can generate

dilating,

Mohr-Coulomb

materials

with

weight.

6. R E F E R E N C E S

Andrawes, K.Z. (1970) A contribution to P l a n e Strain Model Testing of Granular Materials, Ph. D. T h e s i s , U n i v e r s i t y of Southampton, U.K. Butterfield, R. and Andrawes, K.Z. (1972) A C o n s i s t e n t A n a l y s i s of a Soil Cutting Problem, 4th. Intl. Conf. for Terrain Vehicle Systems, Stockholm. Butterfield, R. and Harkness, R.M. (1971) The Kinematics of M o h r - C o u lomb Materials, in Stress Strain Behaviour of Soils, Pub. G.T. Foulis. Butterfield, R. and Harkness, R.M. (1973) Idealised Granular Materials, Symp. P l a s t i c i t y and Soil Mechanics, Cambridge, U.K.

92

Butterfield, R. and Last, N.C. (1983) Continuous Penetration Testing in Granular Materials, a New Analytical Solution, Intl. Symp. on i n - s i t u Testing, Paris.

Mahmoud, A.M. (1985) Continuously penetrating bodies in granular dia, P h . D . Thesis, University of Southampton, U.K. Geiringer, H. (1931) Beitrag zum Vollstandigen abenen Plastizitas blem, Proc. 3rd Intl. Cong. App. Mech. Vol. 2, Stockholm. Hill, R., Lee E.H. and Tupper S.J. (1947) The tation of Ductile Materials, Proc. Roy. Shield, R.T. (1953) Mixed Boundary Quart. Appl. Math. V o l . I I .

Value

mePro-

Theory of Wedge IndenSoc. Series A, Vol.188.

Problems

in Soil

Mechanics,

SLIP S U R F A C E S

IN S O I L M E C H A N I C S

P. H a b i b L a b o r a t o i r e de M e c h a n i q u e des Solides (Joint L a b o r a t o r y E.P.-E.N.S.M.P.-E.N.P.C., A s s o c i a t e d with CNRS), Ecole P o l y t e c h n i q u e - P a l a i s e a u , France

Abstract. Isolated slip surfaces occur in softening soils, that is to say mainly stiff clays and dense sands. T h e y do n o t o c c u r for all plastic deformations. A mechanism of progressive development of discrete slip surfaces in a heterogeneous stress field is first described, in good agreement with actual physical observations. This mechanism makes it possible to analyse mechanical strength tests and to show that the validity of the shear test is especially assured for residual strength. For the triaxial test, the peak value is seen to have physical significance, but the softening slope is shown to be function of sample size, also residual strength is not correct when a localization of strains occurs. A correction is proposed. The occurence of slip surfaces questions the validity of classical formulae. Approximate practical corrections are proposed for bearing capacity of shallow foundations on sand and for active and passive earth pressures.

I. I N T R O D U C T I O N

The slip

surfaces

(Fig. l). theory del,

being

of

since

quasi-static

common

has

not

the

of

in

widely

soil

reported

heterogeneous

remains

observations

of

phenomenon been

disregards it

deformation

an

the

localization

important

physical are

hard

on

one

or

rock

mechanics

the

classical

and

several

deformation subject

but

phenomenon to

or

carry

of

out,

(Mana

dif-

the

ac-

soil

not

transparent. fact,

we

only

Practioners

surface

in

disappointing

is only

know

depth,

to

very

that

recently

the

tes of p r o g r e s s i o n

research

since

an the

of deep

a few data

slip

by

slip

drilling is

a

interfacing

perfectly.

of slip

of

accident,

observations

that

development

knowledge

after

rebond

themselves,

of the

have

operation,

tendency

surfaces

tics

very it

Plasticity

occurrence

emerge.

table

a

of

Nevertheless,

one,

In

slip

is

However,

1966).

ficult tual

localization

As

for

partial

have

surfaces

and

the

been

I to

a

have

a regret-

collected

of around

on

sometimes

isolated,

(1983)

they

and

progression and

Suemine

when

trenching

costly

surfaces

are

surfaces.

surfaces

of

slip

and

it

on the kineindicated

100m/h.

ra-

94

Figure

1. Slip lines in granite. surfaces in contact were opened by weathering,

The

Even tion,

as

for

orientation tion

of

are

lines

materialize of

about

deformed. initial

an

contradictory.

only

on the for

We

longer

no

soils,

configuration

know

is a certain

is p a r t i c u l a r l y

field

there was

rejects

that

whether

the

are

now

obviously

obvious two

test,

to the

reports

the

identification

One

reason

is

of

axes

are

slip

surface

scatter

after

the they

a strain

themselves to

the

one. in the

on Figure

later

the

this

that

reference

to relate

the

direcon

piece

conjugate

active

situa-

of a test

or to the s u b s e q u e n t

there

which

so

compression

Available

Indeed,

surface

accessible

in relation

difficulties.

lateral

and

triaxial

surfaces,

imprecise.

serious

geneous

caused

remains

elementary

or

slip

presents

one

of

uniaxial

sometimes

3-10%

This

a

quite

or several

stresses,

Moreover, lines.

apparently

example

of one

main

subject slip

for

Figure Z . Two f a m i l i e s of slip lines with reject on t h e f i r s t when the second one moved. Note the uncertain parallelism of the second family lines orientation.

2.

orientation

In an

families than

the

of slip

initially

of

slip

other.

homo-

surfaces, This

has

to form.

In addition

certain

surfaces

of

the

second

family

show

doubtful

95

parallelism,

somewhere

the two conjugate

around

families

15 °.

raises

On

Figure

complex

Figure 3. Scatter in orientations of slip surfaces in a granite rockmass. This Coulomb

dispersion

defines

the

tion of the

strict

function

its

bouring

up

If there

path

will

is

well

modest slip

the

It

almost

form.

elementary

is

quite

as

Landslide (B.R.G.M.

easy plane

strength

is

In

4.

failure

of

understand. the

facettes

a

calcula-

along

this

plane

obvious

that

the

as

the

of the material,

fact,

at photo).

to by

unfavourable

heterogeneity

scale,

but on in

problems

for

on active

surface

are

is a slight

small

the

of a certain incline.

planes

of

known,

remarkable,

of

orientation of

identification

failure

whose

critical

a slip

surface

mean

as a

neigh-

is

direction

surmade gives

orientation.

However,

situated

minimum

hesitatingly

of a series

the general

the

the

problems.

Figure Villerville

orientation

angle

inclined

plane. face

of

in

3

a as

raised

example, faults,

just

the

in other

as

This the

surfaces

epicentres

words,

as catastrophic,

building. much

that

by slip

on slip

Figure

spectacular

slip

are

of

earthquakes

surfaces.

4 shows accident

displacement

important.

is

the is about

It are

At a more

effect

of a

particularly one

metre,

96

whereas

the

slip

emergence

(Figure

5)

occurred

at

a distance

Figure

of about

5. Landslide

at

Villerville. Slip surface emergence (B.R.G.M. photo).

250

meters.

lized,

If

there

the

Finally, leul

1983,

with

the

or

with

zero

the

Desrues

occurs

the

others occur this

with

the

certain

case

correspond But faces for toe

for

example of

a

mean

occur,

All

debatable

loca-

(Duthil-

the

slip

lines

over

many

of d e f o r m a t i o n

Although

the

merge years,

(lines

study

interesting

stress are

or on this

or

and

the

loose

of

of

this

it is and

for

where

contrary, how

rocks

high

failure do not

difficult

state

the when

especially

1983).

where

occur,

they

rocks,

surfaces

the

are

stresses;

and

(Goguel,

materials

conti-

but not with

intermediate

overthrusting

a

ca-

surfaces

consider

for

soil

Sometimes,

under

If we an

with

slip

sands).

cases, in other

sands),

disappear

through

but

occurs

or dense

identified

cases

certain

example,

sands.

fields,

shows

For

clays

dense

well

in

surface,

resistance

muds

passes

are

there

landslines,

being

belived

surprise:

stresses

and g e o l o g i c a l

certain

pile.

(stiff

materials,

They

is

see how

of a slip

threshold

soils

transition form.

of

localization.

of m o d e r a t e l y

to faults

usually

the no

of

to the house.

it generates.

formation

moderate

surfaces

suggested?

the of

do

as was

we shall

with

brittle-ductile slip

(1970)

instead

surfaces

of the rates

difficult,

and

spread

no damage

problems,

lines,

source

(compressible

is

plane

first

exceeding

under

In stress

a

deformation

been

of slip

characteristic

very

is

had

practically

nature

consequences

There

observed

very

as Roscoe

seems

the important

ses,

been

1984).

lines

extension)

phenomenon

nuous

have

characteristic the

failure

deformation

would

slip

as under

problem

sur-

appear,

as the

becomes,

97

since at

the

one

the

criterion

and

mean

the

for slip

same

stress,

t~me,

the

surface

the

type

formation

constitutive

of

stress

must

take

equation

field,

into

of

and

the

account, material,

probably

boundary

conditions. We field

shall

before

first

studying

II. S L I P

SURFACES

Experience menon

in

a

material

the

constitutive

terial

is

to

stress

field:

hardening,

strictly

if

the

homogeneous

the

softening,

and

represented

indicate

the

stress

the

following

consitutive

deformation

homogeneous

6, w h i c h

a

FIELD

interpretation

characteristics

in Figure

in

i de n t i f y i n g

indicates

classical

never

happens

case.

contibuted

equation

The

the m e c h a n i c a l curves

has

indicates

surface.

what

IN A H O M O G E N E O U S

homogeneous

the

slip

examine

the general

is

there

then

by

of

homogeneous;

if

is formation

follows:

there

is

the

behaviour

pheno-

equation

the

slight

set

of

of a

real

ma-

scatter

of

stress-strain

of a series

o

of poten-

8 b

(c)

f

//

g

Figure tial

slip planes

tain

slip.

Let

through in

(c) An

an

increase

than

develop

before

the

there

plane.

stress

which

the test

contrary, exists

Under

which,

neighbouring

plane

a

hardening.

if

stress

around in

turn,

planes.

Even

failed

first,

~ it u n d e r g o e s

this

plane

can

generates if

it

the

seen

to

occur

increase

deformation

is

a cer-

only

is

less

extend

and

piece. the

plane

behaviour

where

the

curves greatest

show

a maximum

strength

is

(Figureached

the others. If

this

the

throughout On

re 7),

in

of

in d e f o r m a t i o n

in

near

Case

sample.

be the weakest increase

deformation

there

in a test

6.

the

plane.

Young's

deformation

This

modulus

brings of

a

soil

increases, about

a

is

much

the

stress

relaxation higher

at

in

begins all

to d e c r e a s e

the

unloading

test

than

in

pieces.

the

first

98

(,}

/

I

5 Figure loading now

modulus.

behave

there

like

nificance must

test

after be

of

blocks

curve

the maximum

replaced sand,

deformation

volumes

of

softening.

not

affected

which

slide

by

along

large

the

by

a

the

in F i g u r e

because

the

7 has no

slip

deformation

stress-displacement

diagram (Figure

longer

of the volumetric 8)

loses

its

maximum

replaced

by

the

of

perceptible

the

5

lume 8. Volume variation of a sand during failure (Triaxial test).

in the m o n o l i t h i c

blocks

the

must

curve

be

of AV

is

on

curve

on

not

after and

a slip

the as

localized

on

volume at

a

once

is

curve.

in the slip plane,

as soon as r e l a x a t i o n

it appears

other

be detected

is

stress-

physically

variation the

the

surface

the

maximum

the

a

~.

important

(since

develops

theless, are

of

strength

eded), can

as a func-

although

formation

and

sig-

triaxial

volume

remark

AV

AV

a

of d i s p l a c e m e n t

5

-strain

ceased

so

continuous.

significance

This

-

real

strain ~

because, ,

any

For

strength

function

J I

dense

planes,

is not

diagram.

variation

Figure

deformations

of deformation.

stress-strain

on dense

tion

the

rigid

is l o c a l i z a t i o n The

It

Then

7. C a s e

exce-

hand,

it

the

vo-

Never-

variations given

time

whereas

they

started.

99

III. SLIP S U R F A C E S

We will taking When

the

of

loading A,

a

solid

is

then

first

FIELD

in a n o n - u n i f o r m

subjected

increases,

criterion

in a point in ~

now examine what happens

example

the

failure

IN A N O N - H O M O G E N E O D S

to

any

stress

loading

field,

(Figure

9).

the

/

reached

in a small doma-

s u r r o u n d i n g the point A.

If this domain is s u f f i c i e n t l y small, the

stress

uniform mation is

field

there;

tinuous

deformation

if

and

con-

it undergoes

We will consider the case

slight

sical,

be a for-

if the body

softening,

of Soil M e c h a n i c s tly

considered

surfaces

to

hardening.

be

there will

of slip

subject

can

with

softening

for

elastoplastic

considered

as

a

a sufficienthe

clas-

solution

good

to

Figure 9 . C r o s s i n g

be

the criterion.

plasticity

approximation

of the problem studied *). Let Figure the

It

and

examine

9 and

straight

deformed. B.

us

extending line

is m a x i m u m

(d)

happening

in

front

(Figure

perpendicular

for

occurs

and

displacement Figure

is

The d e f o r m a t i o n

B a slip

lines

(a)

What

is greater

the

line

between

(e).

~(s)

The

and

to

(c)

the

slip

the

In

a along

half

the

of

defficiency

of

At

a

crack

passes the

increases

slip

surface

certain is

through

not

approaching

B.

which

Between crosses

B

towards are

plotted

on

the explanation,

but

it would

be

shear

A.

A

the

t(s)

corresponding

from

of

distance,

practically

(b) when

crack

AB

The

for the case where ~ ~ 0).

classical

constant

10). the

which

10 (assuming ~ = 0 to simplify

easy to generalize

the

for the line

lips

~

on

AB;

domain

elastoplastic

there ~

would

. In

in the AB crack

be

the

solution

correpondlng

case

of

plane marked

the

shear

would

shear

force

F

softening by the

there

shaded

is

area

be

in each a

shear

on figure

10, whose total value: AF = f (tmx - t(s))d-s AB an

upper

strength, segment

bound

of

tr

the

AB.

is

which

is

residual

Equilibrium

can

(tmx - tr)~ , where

tmx

shear

~

only

be

*) This would not generally be true Of s t r e n g t h after the maximum is very

strenght

and

ensured

by a

is

the

the

greatest

length

of

the

displacement

of

the

in Rock Mechanics important.

where

collapse

100

crack

tip

towards

distu r b a n c e

of

approximations

the

the of

right,

elastic

Figure

in

fracture

field

10,

the

mechanics

beyond

shaded

this,

areas

be

so

mode that

equal.

II

with

with

The

the

displace-

$

ment AF

Figure

lO. D i s p l a c e m e n t

A~

the

of

by

crack

bringing

displacement

tip

and

allows

additional

A~

shear

stress

near

compensation

strength

A~-t r.

a slip

of

the

An

upper

surface.

shear

deficiency

bound

of

the

is therefore:

or again: A~

tmx

~- < t

-I r

If the

the

softening

lengthening

is

10%.

of

resisting

dotted

This

area In

reaches

of

crack

mechanism the

the

is

increase

of Figure

the

corresponds

the

possible elastic

case,

boundary

movem e n t

the

slip

surface

by

on another

slip

surface.

This

bourhood

of

sibility

of

only

resist

the

restricted,

first,

resisting t . The r

since

equal is

failure

remains

of

if

the

it

not

is

to

to the shaded total,

mx' equilibrium

the

is

to

the

slip

surface

if

plastic

deformation

difficult

if

area.

However,

in the

capable

say

mechanism

the be

of 0.9 t

"ligament" that

preceding

may is

the

occurs.

allowing

strength

reestablished

stress,

yielding

and

deformation

trate

ensures

10a is at least

contrary free

only of

to a residual

which

blocks to

concen-

immediate imagine

t there, whereas the first surface mx o c c u r e n c e of a m u l t i t u d e of d i s c r e t e slip

the

neigh-

the can

posnow

surfaces

101

side

by

side

expected.

They

difficult

to

use

are

they

inside mass, back

can

be

will

be

see

to

yielded

they

the

lead

us

concept

of

quasi-homogeneous formation. plies

I

situated

the but

I

beca-

I

,~

com-

A

practical

experience. Figure

11

represents

multitude

of

L6ders

lines

surface

of

steel

at

Figure

which

is

a

lines.

of

condensation,

such

aluminlum in

following

the

discrete even

are to

L~ders

or mild

stage.

multiplicity

surely

have

distinct

yielding slip

of

slip

lines

which

which

occur

appear

immediately

field

under

Their

occurrence

on Figure

localized

families.

around

lines,

p r o g r e s s i v e l y widen

homogeneous

plastic

of

not

lines,

steel,

They then

the

double

more

two

Contrary

exceeded.

picture

lines

process

component

good

Lfiders

elongation

limitis

]Oa. Ultimate strength the ligament.

the

tunnel.

as

edge

ligament

stamped

plate,

section

$

a

discrete

a

I

de-

This

with

tmx tr

the

11 shows

a

but

are in

caused

ceratin

after

effect

that

the

by

a

metals

the elastic

to completely in

circular extensive

of

invade

a

hardening

shape

of

a

slip lines would

in

Other

examples are seen later. We

must

insist

progressive slip

line

also even

(or

if

this by

The

blocks.

punching

"blocks"

unique

slip

lines)

observation, is

sometimes

ultimate slice

spirals

assumptions

the

Rendulic accept

the m o v e m e n t

between

In fact, foundations

shows can

design

method,

indeed

extensive

failures,

a few

that

a

circle,

logarithmic

very

fact

of

approach

Fellenius

of

the

with

certain

intuitions.

rigid

of

correlates

masked

as

on

elongation

be

the practice or that

natural that

the

subjected

of

slope

Figure 11. Multitude of discrete L~ders lines in mild stamped steel.

moving to

large

deformations.

For

smaller

102

structures and

such

passive

sion the

of

earth

the

of

lead

beads

slip

following

from

family emerges the

just

it

been

stopped surface

Figure surface remains

1Z.

glass,

Slip

surface.

free

understand

development envelopes

a

and

of

the

progres-

a

of

scientists

the of

developa

mesh of

surface

etc..).

mesh in

of

plane

Grenoble of a mass

Figure

12,

foundation.

shows

punch,

the

follow

method,

is clearly

when

the

active

development

by Darve

at

in

research

to

recently

(1961)

with the

radiography

traced

Habib

a large

to

during which

and

under

It

many

deformation

surface

and

under

possible

principal

by

and

densimeter

mass,

walls,

of force

shown

dilatancy,

of a mesh

descreases

is

been

Scheebelli-Dantu

by Chazy

on the

(y

its

of

the

surfaces

displacement

intuition have

work

of slip

has

earth

a pane

retaining

identified

by

observation

Figure drawn

been

the

or

progression

techniques

in

behind

of rollers

the

surface

placed

photographic

foundations

length

has

different

the

deformation by

surface

surface

very

ment

surface

pressure,

slip

failure

using

as

the

where

visible

that

longest

on this

surface

that

development

the

is

the

figure

slip

appeared

a

that

approached.

other

they

of

surface

By

surfaces after

the

them all.

13a

and

13b

illustrate

formation

can

be

observed

some

and

cases

others

where where

isolated the

slip

deformation

quasi-homogeneous. The

case

of

mass

(Figure

13b)

the

influence

of

stresses

in

the

the d e v e l o p m e n t

expansion

or near the

the

of

range,

up to the free

circular

surface

ligament

elastic

the

which

of can

that surface

is

it

(Figure

resist to

cavity

say

or

within

13a) not

which

of an isolated

an

earth

clearly

shows

the

increase

allows, sllp

or

surface.

of

not,

103

///~//f/" t

Shallow hunch

Deep munch

Pile

////////~ Cylindrical Landslide

expansion

k ~ , ~

near the surface

~ / ~ ~

Cylindrical

expansion

at depth from from the Passive earth Active earth pressure pressure

surface Cylindrical contraction / / / / / / ) / / / / / / j

a- SLIP SURFACE b-HOMOGENEOUS Figure

13.

Different forces,

cases of deformation (the dark the light arrows displacements).

DEFORMATION

arrows

indicate

IV. D I R E C T S H E A R T E S T

A the

classical

rectilinear

cularly rather explain

the

during

the

remain

fixed

stress

in

mastered the

Casagrande

box

test.

into

disuse,

In general, in

direct

the the

main is

of the main The

stresses

It

shear

seems there

shear

stresses

triaxial

shear

the

controlled

without

direct

and is not known, test.

measuring with

test,

smallest

tensile

of test

fallen this.

way

shear

test.

whereas stresses,

a soil and

this

test

method

any

good

reasons

change

in the triaxial or sometimes

whereas

intermediate

the

has to

for the fact that

direction,

the

is

parti-

that

during

and hence

of

being

is criticized Also

variable

strength

displacement

test.

It

they

principal cannot

be

test it is equal to to the greatest

the deformations

are perhaps

in a not

104

always

homogeneous

horizontal leads

to

shear, is

faces a

not

the

often

the

edge

the

the

first

the

of

not

the

the

triaxial

the

teeth

test

pore

Finally,

is

on

piece.

pressure

although

apparatus

motives

box

often

stresses

varies

shear

not

during

cannot

The

are

the This

during

this

reason

undoubtedly

very

corresponds

earthworks

stresses

uniform.

attach

ensuring

test.

three

shear

main

of

which

of

a

because

less

than a shear machine...

extensive

main

maybe

boxes

triaxial

costly

in

direction

is

half

less

of

intermediate field

boxes,

acknowledged,

and

situation the

the

to

However,

At

of

the

quasi-impossibility

contrary

cumbersome

in

work

mastered

test

real

(Figure

for

goes and

14)

since

conditions.

excavation,

as

be

to

convincing

example,

forward.

the

The

deformation

however,

raises

a

6

Figure

certain

number

displacement) surface

in

of

failure

stress

If

with

which

a

clay),

we

for

a

displacement

box.

The

sllp

the

understand

and

since

the

that a

the

initial all we

the cannot

know

in

complete of

the

impossible which

is

the

is

the

shear

of curve

interpret

thickness

of

a

slip

course in

then

of

place

a

Casagrande

starting

is in slip

tr

typically

different the

of

residual

(or tmx and

reached

classical

by

dilatancy

to

in

is found

sometimes shown

f o r c e - box

identify

for a sand

maximum

erratic, This

to

maximum,

be defined

is

to

(shearing

stopped

a

millimeters

slope more

is

strength

back.

obtaining

curve

of d e f o r m a t i o n

can

dilatancy

at

test.

difficult

presents

the

few

the is test

area

curve

and Cr

of

plane

before

Finally,

modulus

Cmx

sometimes

movements

the

find

it

if

a lenticular

plane.

for

front,

and

shear

When

hardening,

box,

experimentation, a

Direct

problems.

shows the

14.

slip

at

the

swaying plane.

difficult terms

of

to slip

"surface"

it

105

must

be related. Torsion

15

we

have

cylinder

the

are

¢ = 0

a

crack

were

to occur,

place

(and

in

The

not

the

as the

of torsion nder

solid

if

horizontal

on

a

helicoid

of

¢ ~ 0).

of

the

e

-

two (a)

shows

if we

had

(b)

added

elastic

Figure

curve

IS.

Torsion

15a)

and

(Figure

the c y l i n d r i c a l of

the

15b).

the

rigid-plastic

Moreover,

test

piece

notched

test piece.

test

on the solid cyli-

(Figure

thickness

Figure

a

c,

take

curves

behaviour

On

on

This

a

comparison

together

tests

with

would

case

failure

experiment.

torsion

deformations it

along

plane

of wax.

of

clearer

two

of

Consequently,

localization

much of

test

material

hardening.

a

solid

The

made

is

results

horizontal a

thickness.

is

of

a by

pieces

as

cylinder

a

with

formed

zero

a

and

cylinder notch

of

plotted

coiled

test

curve

during

the

around

In

the

case

of

the

crack

was

practically

imperceptible

piece,

There

was

no

between

of

the

a

the

test

piece

slip

of

zero

generatrices in spirals.

rejection

before

notched

crack

tests,

the

the

on

both

the w e a k e n e d

sides

section

yielded.

torsion was

test,

infinitely

mediate value

of

of

immediately

towards

upper

the

slope box

initiating run

to mode

However,

initial

between

possible

corre s p o n d s

grande

from

of

on

the

the

the

half-box

the

shear

very front

middle

is

and

correspond

test

simple.

of

back

the

the

and

and

stress-strain

edges

test

to unequal

curve

of

the

test

The

erratic

of

towards

test

is not

in

torsion

follows

using

progression

of

piece

mode

of

the two

the

Casacracks

occurs;

tiltings

progression

imyield

by the

interpretation

piece. rates

the

effects.

crack shear

the

plane

the

piece

shear

the

direct the

II

at

generated

test

the

of

whereas

slip

after

by kinematic

test

mode

the

during

the

only

curve

of

progression

A

crack

of

hardening

periphery

obvious

the

occurred

of

kinetic

torsion

of

thickness

non-uniform

propagation,

analogy

becomes

the

made

since III

shape to

this

lips

the

movements

The

corresponds

yielding

the

since

Relative

in a d e f o r m a t i o n

Comparison

II.

normal

failure.

probably

progr e s s i o n the centre

was

small.

approach

very

rotation

which

they

of

according

the to

106

the

quality

the

stress-strain

this

of

the

yielding

Figure

10

shear

curve

and

to

filling

of

corresponds

not

the

stress

initial

just

distribution

to

any

Casagrande

box.

the

the

we

The

initial

irreversible

elasticity

test,

in

the

slope

of

progression

of

phenomenon.

can

produce

horizontal

a

plane

By

applying

diagram

(Figure

of

the

16).

The

representativeness this

diagram

conditions the I I I I

i l I

~

I 1

I t I I

I

......

l!i,.m.ntl

stress

teeth lower

are

1

The

mean

measured, soil.

but

coming the

it

There

progr e s s i o n

value does

of

failure

giving

producing (Figure

a

in

correspond

a

a

the

of

curves, is

only

this

define

value.

If

sample the

the

the

magnitude

a mean

we

test

the The

force

assume

zero

most

of

test.

thin of

that,

other after

cement,

that pieces

thickness

representative

the

t . r and

When the

diagram

S

in

shear

to find curve

two humps

Figure the

completely

linear

shear to

observe

certain

merges

stregth. 17. E x p e r i m e n t i n g

the two

we

linear

residual

out,

adapted

idealdirect

hand, a

the

give

othed

an

of

17). On

Figure

the

strength

in

increasingly

obviously

maybe

of

edges

the

would

the

in way

However,

maximum

effect box

the

is important.

all,

the

residual

a series

path

to

large

stress-strain

after

scale

very to

with

stress-strain

is,

obviously

effect

attacking

stress

exists

closer

is u n d e r t a k e n

interlocked

the

not

certainly

increasingly

test

of

the

porous

uniform

possible.

boxes

of

deformation

most

the

test

transmitting

plane

the

16. D i s t r i b u t i o n of shear in the failure plane of a direct shear test.

Figure

The and

at

with

the

piece.

slip

'

of

upper

aimed

t moan

of practical

increases

thickness

stones _~

to

will

strength

value

cracks

joint

in

shear

16

value

test

displa-

shear

with

the are of t r.

smot

mean The

is

perfectly

measuring

residual

107

V. THE T R I A X I A L Now

we must

a triaxial If

the

test

stress

plane

is

plane,

see

will

if the d e t e r m i n a t i o n

draw

field

is

constant.

Figure

TEST

the

same

uniform,

the

Accepting

the

adapted

to this

10 is

of m a x i m u m

criticism shear

as the stress

existence case

of

on

a

strength

linear a

point

18 and

by

test.

potential

weak

in F i g u r e

tmx

shear

slip

in

this

shows

that

7

m

initiatian

Figure the

18.

Progression of failure in a triaxial progressive development of the failure

deficiency

Failure this

in

should

is

much

experimentation weak

the

the

cannot brittle

occurs

be

supported

type

rocks

or

maximum

value

shear

is

test.

maximum

or

as

of

and

soon

base

halos,

limited

by

as

the

onset

and ligament.

occurs,

strength

and

to

the tests

are

in

this

plane This

for of

way

a

compliance

slip

any

notch.

slip

materials

the

perfectly really

starting

This and

the

not

of

triaxiai

progressively

crticism

at

explains

deterioration,

In

develops same

with

development

planes

with

compression. which

prone

measured

of

number

brittle

better

singularity.

as

uniaxial

this

therefore

in

development the

on

under

Certainly

value

a

observed

concrete

compression

the

near

successive

one

as

logical

accept

by

of

cone

more

situated

formation

failure

the

to

point

proba b l y

in

of

piece

is not so. It

a

strength

be

test plane.

as

test under

of

the

but

does

the

to

the

strength

carried

out

repetitive,

correspond

that

a as

of the material? Triaxial sure. in

a

the of

Test

pieces

routine base

a

test

with

triaxial

hooping.

slip

on

plane

This

short

test

slenderness test

so

also

(Figure

I).

pieces ratio

that

left A

the

the

were

between central

2 and

was

of

free

ratio

lower

make

were

used

remote

from

2.5

part

possibility

slenderness

to

development than

1.5

is

108

acceptable plane

is

Paris

inclined

failure were

for

plane

prevent

the

least

the

test

test

Sannoisian

30 ° on with

the

the no

pieces.

where

clay test

on

researchers

for

for

consists

sand grease

soils,

concretes

between

19a);

have

of placing the

the

This

endeavoured and

rocks.

a thin

test

failure

pieces axis.

the

If

nothing

it

would

that

to h a r d e n i n g

stress-strain

is

and

not

to

The

sheet

piece

solution

this or at

curve

of

build

solution

of rubber the

friction-free we

have

resting

supporting

perfect,

but

chosen

on a film

steel

point

represents

a

good

•/

test piece rubber membrane ~ film e! grease / I sheath

ol .-: ..........-

i :) :i F •

....~..:-~,

!

::::- :) ../.,

.

.

(b)

-~ - / / : . / / / / / / / . / ' / ' / ~

0

test

demonstrate

susceptible

maximum

the

piece

friction,

We will

to m a t e r i a l s

before

pieces

prone to softening.

supports

(Figure

about

supports

be limited part

Many

of

impose

clay

45 °. For

at

use of shorter

must

materials

almost

inclined

to

statement to

at

is

possible

Sparnacian

le)

Figure

19.

Triaxial

test

on

a

short

test

piece.

a p p r o x i m a t i o n of a normal stress at the contact surface. The

results

short

sand

forms

round

the test

test the

(Figure

obtained

piece,

the

initial 19b),

were

as follows:

material

cylinder

when

is

which

during

pushed probably

the deformation

the crushing

sideways

and

disturbs

exceeds

a

the

of a bulge

end

15 or 20%.

In

of the

case of dense sand, a great number of slip lines appear on the bulge. The s t r e s s - s t r a i n softening

for

pieces

the

of

extensive same

accompanied

by

localization

of

these

tests

maximum

was

strength

curves of short deformation,

density

an

present

isolated

deformation. the

following

of long test

a

slip The

samples

whereas

curves

maximum

followed

plane,

in

particularly

quantitative pieces

of dense

the

was

sand

by

other

equal to

the

test

softening, words

interesting

result:

show no

for long

with

aspect

value

the stage

of

of

the

value

for

the short test piece (Figure 20). Crushing clay

of

uniaxial

low

tests

on short

plasticity

compression

this

taken

test from

material

pieces

were

sampling relevated

also

at

carried

great

localization

out on a

depth. of

Under

deforma-

109

/

0' 1 _ o 3

~mx: 42 o

~

short

sample

long sample

Figure

ZO.

deformation

Triaxial

on a failure

Figure Z1. Localization deformation in a stiff clay test piece,

tically

brittle

Considering presence pushed passing

of out

the a by

through

with

tests

on

short

and

plane

(Figure

21),

of

a

Figure

quasicomplete

stiffness grease the

film

pressure

the

test

of

the

in

piece

to

test

pieces.

was prac-

ZZ. Multitude of broken fragments in a short stiff clay test.

loss

of

with

generate

axis.

sand

but its behaviour

material,

contact

and

long

strength we

tried

the

clay,

tension

Another

means

after to

failure. avoid

likely on

the of

to

the be

planes avoiding

110

friction layers

was of

used

ensures

Figure the

The

normal

22

tests

is

with

onset

clearly

on

by

test

the

Figure

not

There

is

samples

presented

device. still

tensile lubricant

short

is

o~

molybdenum

The

of

It

anti-hooping

or

base.

Failure

fractures.

23.

consisted

influence

piece

confetti.

It

grease

does

of the

multiple

shows

indicated

samples.

lubricated

the

of

short

contact

is no hooping

the

which

of

confetti,

stress

with

heads

clay-confetti

so there

occurred

the

aluminium

bisulphide. strength,

on

The

on

result

a certain

of

softe-

MPa

~l" :35,75 m m H= 74,5 mm

l

long sample

/J~ =102mm 39.4mm

short sample ! \

/

AI mm

, 0 Figure ning

Z3. U n i a x i a l

after

practically

the

Van concrete. would

be

know

that

softening In

short

for

(1984)

These

in

the

on

sample,

long

short

but

test

and

the

pieces

long

test

value and

pieces

~f

of the maxima

the

short

one

is

with

obtained

results with

this

are

the

results

obviously

short

case,

similar

test

different

pieces

strength

with

and

is

triaxial

from

friction

much

the at

higher

tests one

which

supports.

(Hudson,

on

We

Brown,

1971). sand

with

sample.

material.

same

Mier

For obtained

of the

test clay.

8

protection.

obtained

Fairhurst

6

compression

failure

anti-friction

the

4

2

It

and

the

the

it seems

triaxial

therefore

However, curve

clays,

the

remains course

that

the

value

of m a x i m u m

apparatus

is

independent

corresponds

to

an

physical

of

intrinsic

significance

of

the

strength shape

property

the

slope

of of

of the the

uncertain. of

this

article

we

have

thrown,

then

lifted,

111

doubt

on

the

maximum

consequently value

the

value

slip

In a triaxial

surface

with

angle of internal We

Cmx

of Cr in the triaxial

correction. the

on

strength

can

also

that

6 is all

origin

calculation,

means

that

After

softening,

friction

the

Cr

critical of

a

¢c'

-strain

accept is

slip

smaller

a

but

the on

the

will and

now

show

generally

on sand,

test

and

that

the

calls

the angle

is

all

the

as

the

density

Without wishing

slip

this

test

Coulomb

line

equation

characteristic

is accomplished,

plane

maximum

smaller

for 6 of

as

of

the

sand

is

to take sides

as

corresponds,

direction plane

piece

is

with

to complete

6 = ~ - ¢ of

the

is still

therefore friction

(which

stresses).

the residual for

angle of

example,

to

6. The Mohr larger

angle

t

the

circle

than

Cr'

~

in

the

Figure

24

~-%

stress-

which

the

r

/,

a,m~-a 3 rain

defines

from

corresponding

axis

triaxial

different

the

Cmx;

the

dilatancy

slip

curve

Cmax - ~3' obtain

to

The

on

piece

increases.

line

the

the

magnitudes. strength

can

homogeneous

indicates

We

a

of slip lines and simply to be able

when

in

corresponding case

we slip

value

sand.

a

is incorrect

in compression

test

higher or as the d i l a t a n c y

the

of

in

friction ¢ is wide.

say

to the physical

measured

test

test

the

value

-~3

we

strength

to

the

test

stage defines el min - q3' giving an angle ¢' which is greater

than

the

resi-

dual strength Cr" A

simple,

gthy

but

rather

computation

lena3

0

yields

o' I

°'1 rain

mx

the following equation:

Figure

Z4.

Mohr circles of maximum residiual friction.

and

+ I . tan I~ + ~2) With

Figure

25 we

can calculate

to the ¢' m e a s u r e d value to determine This

correction

Soil Mechanics;

- tanCr

only

entails

it can become

wide m a x i m u m friction angles.

the correction

¢' - Cr to make

the value of Cr" a

few

degrees

quite c o n s i d e r a b l e

in

normal

cases

in Rock M e c h a n i c s

of for

112

VI. P R A C T I C A L A P P L I C A T I O N S We

have

just

stated

that

in

the Soil

Mechanics

correction axial

6

test

small,

~ ~ = 6 0

slip

laboratory

to make to

determine

around plane

to the

the tri-

Cr

is

I ° or 2 ° , when

has

settled

in

a

the

test piece. The

same

phenomenon

much

greater

the

slip

plane not

surfaces

and

the

uniform,

bearing m...O

OJ

10

20

30

40

SO

so

4;

path of

on

residual

Z5.

Correction ( , - ~r ) for friction (triaxial

test).

are

a

simply

say

that

numerical

tables show

functions

the

example give

order

will

make

Cmx the

with

zation,

friction

the critical qu'

(or sand

to Cmax" However,

progressively

capacity

of

this

of

softening.

range

discontinuous

this

NY(¢c = 32 °) = 31

of m a g n i t u d e

phenomenon

reduce

bearing

of of

the

localiap-

~, so we can

calculated

by

the

< qu < Y~Ny(¢mx)"

(1956)

the

limit

surface and

formula of the surface term y~Ny lies between two values:

Y~Ny(¢C) A

the

slip

failure

dilatancy

proaches classical

shallow

orientation

corresponding Figure

of

the

non-cohesive

Generalized

the

the

not

fields

instance

on

of

when

are

capacity

materials. occurs

be

stress

for

foundations

where

can

importance

by

nature

It

a

of

the

clear and

:

obviously

theoretical deformation.

Caquot-Kerisel

NY(¢m x = 40 °) = 114,

theoretical

is

the

uncertainty extremely

approach However,

which

linked

to

difficult

to

because there

of

the

remains

the

possibility of proposing approximations. If we look angle (or

of in

the

sentative tation

slip

of

the

used

strength

first at the upper bound,

internal

of

in the

friction

¢c

surfaces) state

the

of

slip

failure

is rapidly

and rigid

that

it is quite clear that the reached

Cmx

blocks.

is By

method,

in

other

surface,

we

can

not

the

doubt

analogy

words

try

in

to

slip

surface

only

repre-

the

compu-

with

by

taking

the

shear

lower

the

upper

bound

tan¢ c

va'lue

by

the

coefficient

to tanCmx

bring

friction

back

to

its

true

1t3

value. The of

situation

friction

too

¢c corresponds

short,

function width

so

of

L of

the

remains

consider

slip

coefficient

We

must

proportional network

the

the

line

which

remember

to

of

the angle

width

is

is

a

that

the

B

the

of

characteristic

stress

~I e(~/2)tan¢

+

that

L

arc,

proposed

that

intervenes the

to

two

an

twice,

first

define

the

slip

function

of L to choose

the calculation to

increasing

define

stress

surfaces

are

is unboudtedly

for the increasing of slip

of

length

to

the

homothetic

closer

coefficient

the

normal

not

circles,

the

we of

slip

but

the

to

L 2 than

to

L,

lower

bound

can

be

as follows: /L(*mx)]

Cmx e~tanCmx

2

tan2 ( ~ + --~-)

two corrections

similar

results,

ensure

a

obtained.

proposed

which

must

for

theoretical

pretention

calculation,

expressed

is

(Nq not

satisfactory The problems

that

confidence Meyerhof

surface in

on Figure

good

26.

They

as a demonstration in

the

(1961)

term.

had

It

correlation

approximation given

was with

provide but they a

devoid the

semi-

of

slip

any

circle

as follows:

1) t a n 1 . 4 ¢

surprinsing

:

[tan2(~

that

+ ~)e ~tan¢

the

two

-

1] t a n 1.4¢.

corrections

proposed

show

agreement. same

to

of

the but

indicated

be decisive

degree

remember

formula

Ny :

are

cannot

certain

We

-empirical

it

with

secondly

correction

means

the function

by analogy

Naturally,

effective

So

is

classical

Whereas

strength,

increasing

extension.

discharge the

to define

resisting

do

surface

Again

the

surface.

I{

tan

can

The

slip

an

bound.

we can formulate:

coefficient.

which

consider

side

for the lower

to the residual

can

Using

L : There

we

the

foundation. lines,

is similar

approach

propose

prone

to

softening.

earth

pressure

of non-cohesive

normal

can

be

corrections We

can

take

considered to

the

to a vertical

material.

The active

for

classical example

wall

other of

and

retaining

and passive

Ka = tan2

Mechanics

for

materials

active

(4-~) "

and

passive

a horizontal

pressure

are: Ko : tan 2 ({ + ~ )

Soil

formulae

mass

coefficients

114

B lqu=¥.~Ny N~,(4p) tg4'c

~

t,4,

I. _

"~__.++Y...__~

j(dp)e"tg+ Ny(~c)

Ny

500' 4o( 30o

J(~c)e**t04'c

2o0 F i g u r e Z6. C o r r e c t i o n s proposed for bearing capacity of shallow foundation on n o n - c o h e s i v e material w i t h an internal angle of friction ¢.

lO0

50 40 3O

Ny(+c) . . . . . . . . . .

20 10

14,c 10

With

passive

surface

30

20

earth

to occur,

40

pressure

5(3 e

when

the

2~__qc) We

can

propose

before,

the

is dense

enough

for

a slip

¢mx

same

decreasing

coefficient

of

the

upper

bound

as

that is : Cmx

tan 2 (~ + -7-) and

sand

the following range can be formulated:

for

the lower

(~ + +)

(H

simply

take

stress

normal

bound a coefficient

being

a linear to the

of the slip plane

tan¢c

×tanCmx

the

height

function slip

is fixed

of of

surface by the

function

the

of the length ~ =

retaining

-~ since

there

wall). is no

For

to

foot

We can

of the wall).

this

reason

to be proportional

Y

H/cos

for

we the

(the depth except

an

115 increasing coefficient expressed as: Y(¢mx )

Y(~c ) and as a correction:

f~ *c) The two corrections proposed are indicated on Figure 27. They are

tg¢ c

Kp 10

5 i=tg2(-~+ 2 3 L ) ~

iI /

10

Figure

20

30

40

50

ZT. C o r r e c t i o n s proposed for normal passive earth pressure wall retaining an e a r t h m a s s o f s a n d w i t h i n t e r n a l angle of friction

still close information. The

enough

to

transposition

each to

other the

to

case

obviously immediate by changing (~ + ~ ) t o

VII.

provide of

sufficient

active I~-

earth

on a

practical

pressure

is

~2)"

CONCLUSIONS

We have shown that the phenomena of deformation localization

in

116

one

or

several

slip

surfaces

consequences

in the laboratory

is

for

proposed

the triaxial

the

test,

but this test

In practice, mass

causes

of

the

is particularly

the occurrence

for more

important

A correction

shear

strength

interesting

useful

and

of discontinuities

discrepancy

compared

in

for deter-

even

irreplace-

with

at failure

of an

classical

beha-

from the hypothesis

of defor-

The corrections proposed here for the surface punch

and for active earth pressure are obviously call

have

strength.

the values calculated

mation homogeneity.

soils

residual

is especially

of residual

extensive

viour and modifies

in

In the same way, the direct shear test, either

or in torsion,

able for the determination earth

failure

and also in site practice.

determination

mining maximum strength. rectilinear

during

precise

definition

rough approximations

in the cases

be extended to other routine applications

examined.

which

They can also

in foundation engineering.

REFERENCES

Caquot A.; Kerisel, J., (1956), Traite de Mecanique hier-Villars, Paris, Ch. XVI, p. 389.

des

Sols,

Gaut-

Chazy, C.; ~abib, P., (1961), Les Piles du Quai de Floride, 5eme Congres Int. de Mec. des Sols, Paris, juillet 1961, Com. 6/27, p. 669.

Darve, F . ; Desrues, J . ; Jaequet, H., (1980),

Los Surfaces de Rupture en Mechanique des Sols en tant qu'Irreversibilite de Deformation, Cahiers du G.F.R., V. 3, j a n v i e r 1980.

Desrues, J . ,

(1984),

Materiaux

La Localisation de la Deformation dans les Granulaires, These de D o c t o r a t - e s - S c l e n c e s , INPG

Grenoble, mai 1984. D u t h i l l e u l , B., (1983), Rupture Progressive:Simulation Physique et Numerique, These de Docteur-Ingenieur, INPG Grenoble. Goguel, J., (1983), Etude Mecanique des Deformations Geologiques, B.R.G.M. Orleans, Manuels et Methodes n°6, ch. 6:Rupture Discontinue, Rupture et Glissement, p. 85. Hudson, J.A.; Brown, E.T.; Fairhurst, C., (1971), Shape of the Complete Stress-Strain Curve for Rock, 13th S)~np. on Rock Mechanics, sept. 71, pp. 773-795. Mandel, J.. (1966), Mecanique des Milieux Continus, Gauthier-Villars, Paris, Deformation Plastique Heterogene, tome II, p. 708. Meyerhoff, G.G., (1961), Fondations Superficielles: Discussion, C.R. 5eme Cong. Int. Mecanique des Sols -(Paris), tome III, p. 193. Roscoe, K.H., (1970), The Influence of Strains in Soil Geotechnique, vol. XX, 2, june 1970, pp. 129-170.

Mechanics,

Suemine, A., (1983), Observation Study on Landslide Mechanism in the Area of Crystalline Schist (part I). An Example of Propagation of Rankine S t a t e . Bull. of the Disaster Prevention I n s t . , s e p t . 83, vol. 3, Part 3, PP. 105-127, Kyoto University, Japan. Van Mier, J.G.M., (1984), Complete Stress-Straln Behaviour and Damaging Status of Concrete under Multiaxial Conditions, Int. Conf. on Concrete under Multiaxial Conditions, U n i v e r s i t e Paul Sabatier, mai 1984, vol. I, P. 79, Toulouse (France).

II.

EXPERIMENTS

AND A P P L I C A T I O N S

UNDRAINED

CREEP

STRIP

DEFORMATION

LOAD

A . F . L . H y d e ~) m)Loughborough m*) IBM (United

OF A

ON CLAY

and

J,J,Burke ~)

University Kingdom)

of T e c h n o l o g y

Ltd

S~OPSIS

Time-dependent creep effects can play an important role in the stress distributions and deformations of foundations.

Using a

phenomenological model, the analysis of undrained creep behaviour has been introduced into an elasto-plastic finite element programme.

The

creep behaviour of a strip load on a finite layer of soil has been illustrated and a study has been made of the effects of small changes in the values of the creep parameters on the overall analysis of creep deformation.

Time dependent creep deformations of a strip load are also

compared with those occurring due to consolidation. The treatment of creep behaviour has been restricted to the modelling of deviatoric creep.

When comparing creep effects on different

clays, the shape of the yield surface is an important consideration. Sensitivity analyses on the creep parameters revealed a necessity for their accurate evaluation.

Small variations in these parameters caused

correspondingly large variations in predicted settlements.

The inclusion

of creep behaviour in a consolidation and creep analysis resulted in a marked increase in settlements, creep settlements causing heave at points distant from the loading.

Consolidation and creep settlements

have opposite effects on horizontal displacements below the edge of a strip load.

120

Notation semi-width of strip foundation coefficient

of consolidation

(two dimensional)

slope of the logarithm of strain rate versus logarithm of time p'J

effective mean normal

q'

invariant

Pc

preconsolidation

qf

invariant

t

time

t1

unit time

At

time step size

A

strain rate at time t I and D = O

D

deviator

stress

shear stress pressure

shear stress at the point of critical states

(projected value)

stress

ratio of deviator stress to deviator E

Young's Modulus

Ko

coefficient

M

slope of the projection

stress at failure

of earth pressure at rest of the critical

state line in

q', p' space N

specific volume on normal normal effective

Of,

consolidation

line for unit mean

stress

value of the slope of the linear portion of a plot of logarithm of strain rate versus deviator stress ~qf

6ij

Kronecker

£:

direct strain

sC

creep strain

13

delta

creep strain rate tensor

121

k

swelling index compression

index

"o

Poisson's

°x,y,z

direct stresses

oij

stress tensor

1"

specific volume on the critical

ratio

normal effective

stress

state line for unit mean

122

Introduction Time-dependent stress distributions Kaufman and Weaver I0-15

creep effects can play an important and deformations

of foundations

and embankments.

(.1967) studying deformations which occurred over

years of the Atchafalaya

levee on the Mississippi

field data with nonlinear elastic and elasto-plastic analyses.

role in the

River compared

finite element

The results of their comparison showed that creep effects

should be included in these kinds of analyses.

Lo et al.

(1974)

monitoring a test embankment near Ottawa attributed more than half of the settlements

to creep behaviour.

Using a phenomenological Singh and Mitchell been introduced this programme

(1968) the analysis of undrained

into an elasto-plastic

creep behaviour has

finite element programme.

Using

the creep behaviour of a strip load on a finite layer of

soil has been illustrated small changes

model for creep behaviour proposed by

and a study has been made of the effects of

in the values of the creep parameters

analysis of creep deformation. accompanied by consolidation creep deformations

Creep deformation

settlements

on the overall

is likely to be

and so the time dependent

of a strip load are also compared with those occurring

during the consolidation process.

Creep Model Researchers

studying creep behaviour of cohesive soils have tended

to adopt one of two methods of analysis. soil behaviour has been developed data to check the applicability or experimental

Either a rheological model of

followed by the analysis of empirical

of the model

(Murayama and Shibata,

data has been analysed on a phenomenological

give predictive equations (Singh and Mitchell,

connecting the various measured

1968).

1958),

basis to

parameters

Any model of creep behaviour which is

123

adopted must use easily determined parameters preferably obtained from standard soil tests, must be applicable to a reasonable range of creep stresses and must describe the behaviour of a range of soil types.

The

phenomenological approach, particularly if normalised soil parameters are used, meets these criteria and lends itself to easier use for the prediction of soil behaviour. Creep tests on many soils such as London Clay (Bishop, 1966), Osaka Alluvial Clay (Murayama and Shibata,

1958) and remoulded illite

(Campanella, 1965) show a linear relationship between logarithm of creep strain and logarithm of time (Figure l(a)) and also between logarithm of creep strain rate and applied deviator stress

(Figure l(b)).

Upon

analysing experimental data on a number of clays, Singh and Mitchell

(1968)

derived an equation which was held to be valid irrespective of whether clays are undisturbed,

remoulded, normally consolidated or overconsolidated

or tested drained or undrained.

The equation expresses the strain rate,

~c, as a function of time, t, and sustained deviator stress, D.

~c

where :

=

Ae ~D (tl) m t

. . . . . .

A

=

strain rate at time t I and D = O

(i

=

value of slope of the linear portion of a plot

(i)

(projected value);

of logarithm of strain rate versus deviator stress; tI =

unit time;

m

slope of logarithm of strain rate versus

=

and

logarithm of time. The authors have chosen to use this model as it needs few parameters

to

define it and these may be determined by carrying out a small number of creep tests at different stress levels on triaxial samples.

124

Equation

1 can be written in a more useful form as:

~c

=

Ae (~ D)

(tl)m t

...

(2)

where D is the ratio of deviator stress to deviator stress at undrained failure, =

q'/q~;

=q~;

and

A, tl, ~ and m are defined above. The use of parameters

~ and D instead of a and D is more convenient

because they are both dimensionless greatly with moisture content. wide range of conditions

and the value of ~ does not vary

Thus predictions

of behaviour over a

can be made from a limited number of creep

tests. To use Equation 2 a starting value of elapsed time must be specified.

Typical values

day and one month.

for practical problems may range between one

The creep strain rate predicted by equation 2 for a

given point in the material

under a time varying stress level is shown

schematically

Under a level of stress, DI, the creep strain

in Figure 2.

rate is initially represented by point 1 and this value gradually decreases until the rate is represented by point 2. increase in load produces

At this time an

a stress of D2 giving rise to a strain rate

represented by point 3 in the figure.

The predicted

strain rate

gradually decreases with time as shown as long as no further disturbance is introduced. The predicted creep rate, ~ c a triaxial sample.

Problems

essentially uni-dimensional more dimensions overcame

is the vertical

arise when attempting

to apply this

creep strain rate to situations

(such as plane strain conditions).

these difficulties

creep strain rate of

involving

Chang et al.

by making the following assumptions:

(1974)

125

(i)

no volume

(ii)

change occurs

the principal proportional

due to creep strains;

shear strain rates are directly to the corresponding

principal

shear

stresses; (iii)

the principal deformation;

(iv)

the strains

strain axes do not rotate under and

are small.

The flow rule for creep strain

rates resulting

from these assumptions

is

where

Proper of plane

.c eij

_

3~ c , 2q' (aij - P'6iJ )

i,j

=

~c

is given by Equation

6

is the Kronecker

p'

is the mean normal

q'

is the invariant

x,y,z and axx'

consideration

direction

ax,' etc.; (2);

delta; effective

stress;

and

shear stress.

must be given to the constraints

(z-direction).

For plane

strain

in the out of plane direction

strain

relationship

written

=

(3)

"'"

strain

is zero.

in the out

conditions

The non-plastic

for a material when creep is continuing

the total stress-

can be

as:

For plane

ex

=

i g{a x, _ ~ ( o ~

+

o~)}

+

e xc

(4(a))

Ey

=

I , ~{ay - v(a~

, + a x)}

+

eyc

(4(b))

ez

=

~{azl' - v(a~ + a~)} + ezC

(4(e))

strain

conditions

e z = O and so from Equation

a~ = v(a~ + a'y) - EE~

(4(c)): (5)

126

and on substituting Ex

=

=

Ey

this in Equations

{(I - v2) ~

{(I

- ~(

~

c

(6(a))

-EV2)o (I + v)o') c ~ec . Y - ~ T x~ + Ey +

(6(b))

the brackets

o~}

c

+ E x + ~e z

The quantities within

~

(4(a)) and (4(b)) one obtains:

of Equations

(6) are the elastic

strains and so to account for the out of plane creep strains equal to ~e~ is added to the strains

a strain

in the x- and y-directions.

!

Equation

(5) is used to calculate o z .

The three creep parameters necessary deviatoric

creep behaviour

cylindrical

can be obtained

triaxial samples,

initial stress conditions.

to define the model of from a minimum of two identical

at the same moisture

content and same

The samples must be subjected

tests under different deviator stresses

covering a range,

to creep say, of 30%

to 90% of the maximum deviator stress depending on the stress history. Under these sustained Mitchell

loads,

strain is observed with time.

Singh and

(1968) expand on the subject of parameter evaluation

Appendix I.

in their

It should be noted that the parameter, m, is not unique

for a given soil and may vary depending on whether

the soil sample is

on the 'wet' or 'dry' side of critical

states.

results for Keuper Marl which indicate

a value of m of 0.86 on the 'wet'

side and 1.00 on the 'dry'

side of critical

Hyde

(1974) has obtained

states.

The computer program used was developed by Burke program allows nonlinear creep analyses

analyses

and at any stage of a load deformation

of equivalent

forces.

analysis

time

the creep strains are

above and then these are converted The equivalent

of

To model creep displacements,

an increment of time is allowed to elapse, as described

This

to be carried out independently

dependent behaviour may be introduced.

calculated

(1983).

into a set

loads are added to the external

127

load vector and the solution for the end of a time step involves a re-solution.

Because creep response under working load situations is

generally a decay process, progressively larger time steps may be used. Equation 2 was modified to include a lower cut-off for creep strains whereby values of D lower than 0.3 did not cause creep flow.

A flow

chart summarising the basic solution algorithm as stated above is shown in Figure 3.

Creep Deformation of a Strip Load on a Finite Layer of Soil To show the kind of behaviour one is likely to expect from Singh and Mitchell's (1968) creep model, it has been applied to the analyses of a strip load of width 2a underlain by a clay layer of thickness 3a and 6a (Figure 4).

The two materials used in the study were San

Francisco Bay Mud and Keuper Marl (the material parameters of which are stated in Table i) and creep deformations under different load intensities were investigated. The start of the analysis was taken at 7 days after the application of the loads.

The time stepping sequence began with a time increment of

i day and subsequent time step sizes were ever increasing and had a value of 1.5 times the previous value. Figures 5 and 6 show the creep behaviour for both depths of layer for San Francisco Bay Mud and Keuper Marl, respectively, at various loading pressures.

At a loading pressure of 50 kN/m 2 the amount of

predicted centreline creep displacement for each material is similar. As the loading increases, however, the creep displacements of San Francisco Bay Mud increase more than those of Keuper Marl.

In the case

of Keuper Marl doubling the loading pressure from 50 to iO0 kN/m 2 has the effect of increasing the centreline creep displacements at 2960 days by a factor of approximately 1.3 and 1.5 for the deep and shallow layers, respectively.

For San Francisco Bay Mud the same loading

increase causes increases in the creep displacements at 2960 days by

128

a factor of 1.4 and 2.0 for the deep and shallow layers, In doubling the loading pressure

respectively.

from IO0 to 200 kN/m 2 these factors

become 2.3 and 2.4. The above analyses

show that San Francisco Bay Mud is more prone

to creep than Keuper Marl and this could have been deduced from an inspection of their creep parameters.

However,

at low stress intensities

the amount of creep for each material

is shown to be similar.

in part due to the fact that although

the loading is identical Keuper

Marl has a lower undrained (this is illustrated

failure stress for a given stress history

in Figure 7 where the wet side yield locus ellipsi

for the two materials therefore,

This is

are shown).

For a given stress intensity,

this would imply that the ratio, D (see Equation 2), would

be higher for Keuper Marl than it would be for San Francisco Bay Mud resulting in enhanced

Sensitivity Analyses

creep strain rates for Keuper Marl.

on the Creep Parameters

When using any model of soil behaviour it is worthwhile

considering

what effect small changes in the values of the material parameters will have on an analysis parameters

so that material testing yields values of material

to the desired accuracy.

Francisco Bay Mud such a sensitivity the creep parameters problem

Using the parameters

for San

analysis has been carried out on

,a, and ,m, for the strip load on the deeper layer

(depth = 6a) at a loading pressure of 200 kN/m 2.

Figure 8 shows the effect of varying the parameter

,~,.

Values

of ,~, have been taken at 10% and 20% above and below the actual value for the material. centreline

An increase of 10% and 20% causes an increase in

creep displacements

at 2960 days of 25% and 54%, respectively.

A decrease of 10% and 20% causes - decrease in centreline

creep displace-

ments at the same time of 21% and 39%, respectively.

Increasing

therefore has a greater effect on creep displacements

than decreasing

,= .

This may also be explained with reference

,6,

to the equation of creep

129

strain rate (Equation 2). exponent,

Because

,~, appears in the equation as an

increasing the value by any amount will have more effect than

decreasing

it.

This partly explains why San Francisco Bay Mud is more

prone to creep than Keuper Marl which have values of 5.40 and 1.13, respectively. Figure 8 shows the effect of varying the parameter

,m .

Again,

values of the parameter have been taken at 10% and 20% above and below the actual value for the material. a decrease in centreline

An increase of 10% and 20% causes

creep displacements

of 28% and 48%, respectively.

A decrease of 10% and 20% causes an increase

in centreline

displacements

Thus decreasing

of 38% and 90%, r~spectively.

of ,m, has a greater influence on creep displacements

creep the value

than a similar

increase and this may be explained by the fact that ,m, appears in the creep strain rate equation as an exponent

to the reciprocal

of time.

Figure 9 also shows that creep rupture is associated with low values of am

•

The above sensitivity

analyses show that the creep parameters

and ,m, must be carefully determined because

the prediction of creep

displacements

is sensitive to small changes in their values.

Consolidation

and Creep

To examine the effects of creep during the consolidation the authors have analysed the problem of a semi-infinite Francisco Bay Mud supporting a flexible, Consolidation

in Table I.

(1983).

boundary conditions

layer of San

(elasto-plastic)

The material parameters

The finite element mesh used to approximate

layer and boundary conditions

process

porous strip load of width 20m.

settlements were computed using a nonlinear

analysis developed by Burke

,~,

are shown

the semi-infinite

are shown in Figure i0 and the additional

are that free drainage was allowed only along the

130

upper surface boundary, impermeable. throughout

the vertical

and lower boundaries

The soil had an initial vertical

its depth,

of earth pressure

ratio had a value of 0.444.

was assumed to be lightly overconsolidated ratio of 1.2.

stress of -150 kN/m 2

the value of the coefficient

at rest, Ko, was 0.8 and Poisson's

being

The soil

with an overconsolidation

The initial values of bulk and shear moduli were 4717

and 455 kN/m 2, respectively,

and the horizontal

were assumed to be 1.15 x 10 -5 m/day.

and vertical permeabilities

The coefficient

of consolidation,

c, of 5.72 x 10 -3 m2/day was calculated using the following formula:

c

-

(K +

)

. . . . . .

(7)

~w where

k

is the permeability;

Yw

is the bulk density of water;

and

K,G are the bulk and shear moduli,

respectively,

of the soil skeleton. Two analyses were carried out assuming: (a)

consolidation

of a nonlinear

(elasto-plastic)

soil skeleton;

and

(b)

consolidation

of a nonlinear soil skeleton with the inclusion of

creep effects. A uniform ramp loading was applied such that the full loading pressure of 100 kN/m 2 was obtained after the first ten time steps.

The time

stepping scheme was as follows: i0 steps of At

=

i0 days;

9 steps of At

=

102 days;

9 steps of At

=

103 days;

9 steps of At

=

i0 ~ days; and

9 steps of At

=

105 days.

The initial time step size of I0 days violated the stability criterion of Vermeer and Verruijt

(1981) for the consolidation

analysis,

131

however no problems

such as oscillating excess pore pressures were

encountered with this value of time step size.

This may, in part, be

attributed to the fact that the criterion strictly only applies

to

regular finite element meshes

problems.

for one-dimensional

The example used herein is two-dimensional is graded.

Another important

consolidation

and the finite element mesh

feature of the analysis

is that the loading

was not applied suddenly but gradually over the first ten time steps, thus reducing any tendency towards oscillating excess pore pressures. Creep effects were considered loading was complete.

from the point in time at which the ramp

The time at the end of the analysis

corresponds

to a value of time factor, T, of 57.2 calculated using the following formula: T

where

-

ct a2

° . .

c is the coefficient

(8)

. . .

of consolidation;

t is the elapsed time; and a is a reference

length

(e.g.

the semi width

of a strip load). Figure II shows the development

of settlements

at the centre of the

strip load with time factor for the two analyses performed. porportion of the total settlement

occurred during the loading period and

may be attributed to the fact that dissipation pressures was allowed during the relatively at the end of loading the settlements settlement

of the excess pore

long loading period.

due to the consolidation

The inclusion of creep behaviour

shows a marked increase in settlement Figure 12 shows the dissipation

Thus

comprise both the 'immediate'

due to the load and a contribution

of the underlying soil.

A large

in the analysis

at all times. of excess pore pressure with time

factor at a point below the strip load (position A in Figure i0).

The

effect of creep behaviour is to increase the peak value of excess pore pressure and cause this peak to occur at a later time than those shown

132

by the nonlinear consolidation

analysis.

to delay still further the subsequent

Also, creep behaviour

dissipation

tends

of the excess pore

pressures. Figure 13 shows the surface settlement loading and at T = 5.72 for the nonlinear analysis

including creep.

profile at the end of

analysis

and the nonlinear

The figure shows that settlements

creep may be significant when compared to consolidation

due to

settlements

and

also that creep may cause heave along part of the surface distant from the strip load. Figure 14 shows profiles the strip load (~ = I).

of horizontal movements below the edge of

It can be seen that the consolidation

effects are in opposition,

consolidation

creep causing an outward movement. times and at other locations other or cause oscillations

and creep

causing an inward movement and

It is conceivable

that at certain

in the soil these effects may cancel each in horizontal

be needed when horizontal movements known that the underlying material

movements.

are monitored

Care may therefore

in the field if it is

is prone to creep.

Conclusions The treatment of creep behaviour has been restricted modelling of deviatoric analysis,

creep.

to the

Of the two basic approaches

to creep

the authors have used the approach utilizing a phenomenological

model and in particular

that of Singh and Mitchell

When comparing creep effects on different

(1968).

clays at low stress

levels the shape of the yield surface and its effect on the stress ratio is an important Sensitivity parameters

consideration. analyses were carried out on the Singh & Mitchell

,~, and ,m, (because they appear as exponents

in Equation

creep (2))

of San Francisco Bay Mud and showed that they must be carefully determined. An over- or under-estimate

of the value of the parameter

cause an increase or decrease,

respectively,

in predicted

,~, by 10% may creep settlements

133

of the order of 25%.

An over-estimate of the value of the parameter

,m, by 10% may cause a decrease in predicted creep settlements of the order of 30%; an under-estimate of the value by the same amount may cause an increase in predicted creep settlements of the order of 40%. This last finding is consistent with the fact that creep rupture is associated with low values of m. The effects of combining creep with a nonlinear (elasto-plastic) consolidation analysis have been studied.

The inclusion of creep

behaviour resulted in a marked increase in settlement.

It also resulted

in increased values of peak excess pore pressures (at a later time) and these excess pore pressures took longer to dissipate.

When considering

vertical movements, creep settlements appear to cause heave along part of the surface distant from the loading.

Consolidation and creep have

opposite effects on horizontal displacements below the edge of a strip load, consolidation causing an inward movement and creep causing an outward movement.

Care may therefore be needed when analysing in-situ

horizontal displacement records if it is known that the underlying material is prone to creep.

REFERENCES BISHOP, A.W. (1966), "The strength of soils as engineering materials", Geotechnique, Vol. 16, pp. 91-128. BURKE, J.J. (1983), "A non-linear finite element analysis of soil deformation", PhD Thesis, Loughborough University of Technology. CAMPANELLA, R.G. (1965), "Effect of temperature and stress on timedeformation behaviour in saturation clay", PhD Thesis, University of California, Berkeley. CHANG, C.-Y., NAIR, K. and SINGH, R.D. (1974), "Finite element methods for the nonlinear and time-dependent analysis of geotechnical problems", Proceedings of the Converence on Analysis and Design in Geotechnical Engineering, Austin, Texas, Vol. ~ ,

pp. 269-302.

134

HYDE, A.F.L. (1974), "Repeated load triaxial testing of soils", PhD Thesis, University of Nottingham. KAUFMAN, R.I. and WEAVER, F.J. (1967), "Stability of Atchafalaya levees", Proceedings of the Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM4, pp. 157-176. LO, K.Y., BOZOZUK, M. and LAW, K.T. (1974), "Settlements resulting from secondary compression",

Report RR211, Research and Development Division,

Ministry of Transportation and Communications, ~RAYAMA,

Ontario, Canada.

S. and SHIBATA, T. (1958), "On the rheological characteristics

of clay, Part i", Bulletin No. 26, Disaster/Prevention

Research Institute,

Kyoto, Japan. SINGH, A. and MITCHELL, J.K. (1968), "General stress-strain time function for soils", Proceedings of the Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SMI, pp. 21-46. VERMEER, P.A. and VERRUIJT,A~I981),

"An accuracy condition for

consolidation by finite elements", International Journal for Analytical Methods in Geomechanics, Vol. 5, pp. 1-14.

135

,.1•

t~

0

ol

O

I~,~

N

~

M

T

~o u

0 0

01 N

o

r~

0

°' [I

~p a~a

0

I

:0

I,

~

I

:"I

•

I

~1

o

1

0 ~o

0

i 0

vl

",,.J

~1

o

,-~1 I~ ~ t~

~ •~

e

co

=11o, o

ul

'~'

I

M

~

Q

o

I ,

b~

o

~.~ ~

U

ol ~1 = ol

o

o

o

I

~

-I¢

136

Log ~c

q3 q2 ql

q3 > q2 > ql

Log t Creep strain rate versus time

(a)

t 1

LoZ £c

t3 > t2 > tl

~

- ~ ~ / ~ ' ~ /

t2 t3

A3

(b)

Creep strain rate versus deviator stress

Fig I. Typical creep strain relationships

q

137

02

log ~c

51

13

log t

Figure 2

Schematic representation of creep response to a varying stress level

138

(

START

D

Obtain initial conditions

i

I

Calculate stiffness matrix and reduce it I

I Specify a set of loads

I

1

I

Solve for displacements & stresses J I I Calculate residual] force vector I I Add residual loads to the previously applied loads

I

an increment I[Specify of time I Calculate creep strain increments and

corresponding residual loads -

I

Add residual loads to ] the previously applied loads

I y

I

Solve for displacements I and stresses 1

I

YES

l

iOutput results for the time i?crement ]

lOutput results for I Ithe load increment ]

YES NO

~

Y

E

S

C STOP •

Figure 3 Flowchart of the basic solution procedure for elastoplastic and creep analyses by finite elements

139 x

~S IIII

Y~

44 Elements 163 Nodes

strip load

IIIII

Jllll I

]11~ 10a

L F

66 Elements 233 Nodes

ql--

strip load

Y

9 1 q

"7

(0

,I"/ q,,,

,I/,, 0 ;

= 0

if

F(o)

~< 0

:

spherical

coordinates

g

strain

tensor

with

principal

components

:

stress

tensor

with

principal

components

: °l

E

Young's

C

cohesion

F

yield

n

viscosity

:

unit

> ~2

g3 > ~3

ratio of t h e m a t e r i a l

criterion

(Tresca)

constant

absolute value strain rate

lI

e2,

modulus

Poisson's

?t

sl,

of t h e e q u i v a l e n t

viscoplastic

tensor

• S t a t e m e n t o f t h e problem : Since

we have

ses ; m o r e o v e r therefore

~u ~r

Su Dr

where strain

Or,gS,o ~ are the principal is p u r e l y

radial,

u =u(r,t)

stres;

and

u r

g@ = c

"

The yield stresses

symmetry,

; the d i s p l a c e m e n t

: Sr

Let v -

spherical

o 8 =~

limit

is r e a c h e d

simultaneously

by two pairs

of p r i n c i p a l

: f~ ~ ~ ( O

- o r ) - 2C=0

f8

- Or)

~ w(O8

~ = ± 1 has rate

viscoplastic

is,

the s a m e in t h i s

strain

- 2C= sign

as o

particular

rates

:

~I _ i < f ~ > + < f8 ~t ~ 2

~ v p = co ~t ~o

0

~

>

2

- or = o 8 - o r . The case,

the

total

plastic

sum of two associated

154

Then

the

constitutive ~v _ ~r

E

Str

E vV =

( ir- ~ )

Sl i %-E = ~ < ~ Since with

- m E

qo --~ -

~ ~t

state

: (1)

(2)

> (I)

and

(2) c a n b e

t = 0 and any other ~I ~-~ (r,t)

~(r,t)

follows

St

is n a t u r a l ,

It =

as

I ~l + ~ ~ E S--~

- ~r ) - 2 C

to t i m e b e t w e e n

~VP(r,t)

can be written

2 v

(~

at t = 0, t h e respect

relationship

dt

instant

integrated t.

If w e

take

,

0

~;VP(r,t)

where

positive

is

or n e g a t i v e ,

E - ~~u -=

we

then

viscoplastic

obtain

strain,

whose s i g n

can be

•

crr - 2 "9 o0 - E v p

E Ur =

with

the equivalent

(i - v )

(i')

1

o0-~

o r + ~ E eVp

(2')

: ~evP 1 { ~ n

~t Moreover, ~r ~r

the

~

equation

2

- 2 ~C

r

}

if

of equilibrium

I o

-~

~

reduces

I > 2C

r to

:

r

(3) •

r

NOTATIONS Once

the viscoplastic

zone will ved

develop

later).

leaving types

Later

residual

of zones -

criterion

inside

on,

may

this

strains

must

Elastic

zone

- Zone with

loading

considered boarder

zone

regress

a viscoplastic

(this w i l l

and eventually

it h a s

reached.

Then,

be pro-

disappear, three

:

residual

:~evP/St strains

kinds of zones may

parameter

dividing

~ 0

: ~evP/~t

= 0 , ev p ~ 0

zone").

oi = ° i ( t ) "

; they are determined

x = x(t)

exceeded,

the medium

: ev p = 0

(or " r e s i d u a l

of the

region

in t h e v o l u m e s

be distinguished

- Viscoplastic

These different

has been

f r o m the c a v i t y

:

exist, More

by the

depending

precisely,

evolution

on the evolution 8 cases

must

be

of t h e v i s c o p l a s t i c

155

The

- the

viscoplastic

- the

non

following

zone

viscoplastic example

1


zone

(5)

:

cv p = 0

(5')

. In a r e s i d u a l ~evP ~t .

0

-

:

and

v i 6 c o p ~ r ~ o n s ~ p s

In a v i s c o p l a s t i c a- the

zone

elimination

e vp ~ 0

:

zone, of

(4) -and

~Or/~r

~t : ~ = E/(2~

b- T h e e l i m i n a t i o n < ~r ~r The b o a r d e r s gration

with

Setting

(4)

+ 4 ~ C

and

(5)

:

leads

to

= 0

(6)

(I - v ~ . vp

between

(4) and

(5) l e a d s

to

4 ~ C ) + ~2~r + 3 - - = 0 r ~ r4 of the v i s c o p l a s t i c

respect

°x + ~°x ~t

of

(5) can be c o m b i n e d

between

r~

Setting

(5")

to r b e t w e e n

r = 1 and

e 4 ~3 C L ° g x 3 - A ( t )

: ax = o r

[ x(t),t

] .

zone

are

r = 1 and r = x .

r =x

leads

to

< ~ 3 - 1 > = ~ oi(t)

Then

inte-

: + ~i (t)

(7).

157

SOLUTION In

the

governing the

following, the

evolution

quantities Let

of

can

~ be

between Let

the

then

any p).

tc(~)

an

instant

an

t =

I - FULLY

~

be

ELASTIC

when =

when

~

x(t)

SPHERE

a

first

boarder

differential

A(t)

(and, if

x(t)) . The

(~ c a n

becomes {

,

~

x(t)

,

be

1 or

other

p or

viscoplastic

undergoes =

order

quantity

equation necessary,

unknown

deduced.

radius

x(t)

instant

td(~)

find

unknown

viscoplastic

particular

1 and

td(~)

will of the

easily

t = tc(~) An

we

evolution

any

value

:

> 0 a viscoplastic

x(t)

unloading

:

< 0

(A)

~t

Figure

2

In

this

case,

(4)

holds

between

r =

1 and

r =

p yields

~i(t) This r

= A(t)

case

ends

~dr/~r

=

4 ~ C.

Such

a

zone

ai = -

4 w C/3

with

elastic

VD~. =

0

sphere

(5') . I n t e g r a t i o n

of

:

1 when

necessarily (I -

: Fully (A).

i/p3).

a viscoplastic

appears

for

zone

r =

appears,

I, w h e n

i.e.

(4)

158 II - F U L L Y

RESIDUAL

OR RESIDUAL/ELASTIC

,

Figure

In t h o s e

two

This

case

III-

FULLY

1,,). ÷ t~

triO) %0)

tdU) toO)

residual

cases,

ends

or

3gvP/~t

respect

to

r = p :

(

= ~(t)

ai(tc)

4.

"tt~* ÷ ~///ll

r = 1 and

~i(t)

(D o r G)

'1f~r'-÷÷÷Vltllttl

(4) w i t h

r between

j

3 : Fully

derivating

SPHERE

i - 7

when

+ -~

residual/elastic = 0

(5")

time,

1 - --

zone

SPHERE

+ ~.

Moreover,

derivating

exp

and

(at),

yields

can

= A

(

be

used

1 _ ~i

(6) w i t h

integrating

44"k"

viscoplastic with

)

c(p) or

:

exp(~t)

: w(td)

i

respect

dt = 0

x =

p , and

to time, to

time

.,t

td(p)

sphere

_ a 4 ~ C Log

respect

with

t d (P) i(t)

by to

4-4-........... 1

:

It

Then,

respect

1 -

to(P)

4 : Fully

(7)

G) •

(E)

,,t

td, p) td(P)

i.e.

+--

1

case,

with

appears,

I

In t h i s

sphere.

integrating

e ( t c) = a i ( t d )

o 3

Figure

(D o r

')

a viscoplastic

VISCOPLASTIC

sphere

in the whole

then

~t

(E) o

then

~x = 0 :

p3

then

multiplying

between

tc(p)

and

by td(p)

159 td 0 = [ exp(~t)

So the i n s t a n t

{ oi(t)

t d w h e n this

IV - V I S C O P L A S T I C / E L A S T I C

I

case e n ds

can e a s i l y be c a l c u l a t e d .

(B)

.....

• t

5 : Viscoplastic/elastic

In the e l a s t i c

zone x < r~ 0.

conditions,

observed,

a

with

(25)

non-linearity

a decrease

of

of

the

the modulus

at increasing stress. 3.3. E f f e c t s of C r a c k C l o s u r e Penny-shaped normal

cracks

stress

and

2D

reaches

elliptical

respectively

cracks the

close

when

critical

the

critical

values

given

by

(Berg, 1965): nE m Poc = 4(1 - 9 2

e

(26)

E 2(1 - 9~) e

(27)

m) Poc where e

is the aspect ratio, that is the ratio b e t w e e n the m a x i m u m thick-

ness and the diameter When this the

the

critical crack

At

normal

value,

is

constant.

(or length)

not the

stress

the

of the crack.

is g r a d u a l l y

aspect

modified critical

and

ratio

therefore,

stress

the

increased

decreases the

crack

but

but

stays

the

diameter

compliance

completely

M Oc

closes

below of

remains and

its

effects disappear. This the

aspect

tropic

loading

luations al.

simple ratio

were

(1975),

situation of

(for

the

which

carried

Feves

and

would

cracks,

out

the for

Simmons

permit

subjecting

closed

cracks

instance (1976).

The

main

drawback

in these

(Mavko and Nur,

by

The

they should be c o n s i d e r e d merely as being

of the cracks

to d e t e r m i n e

by

are

the spectra of

sample

to

inactive).

Morlier results

(1971), are

an

iso-

Such

eva-

Simmons

interesting

et but

indicative.

analyses

1978).

the

is the effect

It is true

of

the shape

that at a given stress

211

the effect their

of

aspect

gradually

thin cracks ratio;

close

depends

however,

at

only

upon their

non-elliptical

increasing

pressure,

length

cracks,

varying

and not

with

their

tapered length;

upon ends,

there-

fore, even a rock c o n t a i n i n g cracks of the same aspect ratio would show a

gradual

thus

increase

simulating

ratios,

cracked

rock

the

sure. at

crack The

the

=eoc.

the

and in p a r t i c u l a r In general,

of

of

will

pressure

define,

an

an

isotropic

permit

(isotropic)

spectrum

e as

a

representing

will

be

anisotropic

in a similar way,

the

of

load,

crack

test

aspect

in a r a n d o m l y

integral

function

the

of

distribution

the closure

pres-

open crack density surviving + + by eoc(P); o b v i o u s l y eoc(0)=

indicated

crack

loading

to e v a l u a t e

parameters

p,

increasing

the cracks having very low values of e.

simply

density

at

of a continuous

therefore,

distribution,

For

stiffness

the p r e s e n c e

distribution,

it

will

be

possible

the integral d i s t r i b u t i o n F +ij(P)

to

of the fab-

ric tensor. For

stress

is more complex.

paths

different

At a given

from the

stress

level,

isotropic one the situation the global c o m p l i a n c e

is the

sum total of the following compliances: -

the c o m p l i a n c e of the matrix;

-

the c o m p l i a n c e d e r i v i n g

-

the

compliance

from the surviving open cracks;

deriving

from

the

sliding

of

some

of

the

cracks

initially open which were closed by the stress; -

the

compliance

cracks

when

deriving

the

shear

of

the

a D.C.

formulation

deformability

deformability.

The

of

each

final

the

stress

friction between the crack In

from

sliding

induced

by

of

the

initially load

closed

overcomes

the

surfaces. it

is

easy

crack

(or

to evaluate set

of

the

cracks)

contribution

to

the

results will o b v i o u s l y not c o r r e s p o n d

global to that

valid for an elastic body. For of

cracks,

tribution,

instance,

let us suppose

normal stresses between fore

be

resents

that

a rock

contains

a planar

set

some of which are open, with a given crack d e n s i t y dis+ eoc(P ). In the closed cracks of the set, d i f f e r e n t residual

characterized the crack

the faces will be present; by

density

an

integral

the set will there+ d i s t r i b u t i o n ecc(P), which rep-

relative only

to the cracks with

a residual

normal stress greater than p. For

a uniaxial

compressive

loading

path,

the axial

and

lateral

c o m p l i a n c e are given by: 1 16 1 - ~2 _ 2m m + + E-- + -3 - eoc(PA) c°s28 + 16 3 1- {eoc - eoc(PA) m E E m m + % + ecc - ecc~PB~K~A~. ,. ,-,

M33

=

+ (28)

212

!

Vm

=

MI3

16 1 - 92 m 3 Em

Em

+ {eoc - e o c ( P A )

+ }K(8) + ecc - ecc(PB)

(29)

where K(e)

= s i n e c o s e (sine c o s e

K(e)

= 0

o>~

- tg~ cos28)

(3O)

P A = ~a c ° s 2 e

PB = aa

sin6 cos6

- t@# cos28 tg~

3.4.

Loading"

Up in

the

load

"Undrained

to n o w

case

is

in S a t u r a t e d

the d e f o r m a b i l i t y

that

so

(31)

the

slow

cracks

that

no

of

are

water

dry

Rocks

the

rock

or

"drained",

overpressures

has

been

evaluated

that

are

is

induced

only

when

the

within

the

cracks. In

undrained

is a l l o w e d a planar on

out

of

set of

the a s p e c t

and of

Km

and

ratio

models

of

are

of

the

not

the c r a c k s

rock,

only

through

on

is w h e n the

the

the

no

normal

crack

flow

of w a t e r

compliance

density,

stiffness

but

parameter

shown

of

bulk

moduli

(O'Connel value

e = 10,

magnitude

in Fig.

7.

The

respectively

and B u d i a n s k i i ,

of

1976

corresponding

10 -3 , the shear

results

modulus

GI3

the and

to of

a

crack

the

is

solid

D.C.

not

at

" tI

to

a a = -2-~--(pI - g v ) + [u I + 2-G--(Pl - ev )] e x p ( - k ( t I - t)) (4.14) o o

u(t)

cement

takes

= P(tl)

is

have

u(t I) = u I and

obtained

either

from

P(tl)

(4.14)

= PI"

for

The

t~

~

or

ultimate from

displa-

(3.12)

for

Pl = P~"

first

A

example of

circular

steel-yieldable

together

by

certain

of

magnitude,

pressure pressure

is

kept

reaches

Pmi and

means

stops

Pmf

U-bolts

the under

the

yieldable

shaft

the

clamps.

sliding

If

joints

control.

value

support

rings;

(Cristescu

the

are

The

when

the

pressure

has

yielding

et al.

dropped

2n~dM = L R g d (sin~ + ~deOSa)

ground

sliding

2n~sM : L R g d (sin~ + ~seOS~)

is

the

overlapped

one ends

pressure so

starts

that when

made

from

are

held

reaches

a

the

ground

the

ground

(1987)) (4.15)

to (4.16)

259 Here

n

torque the two

is the of

a

number

nut,

of U-bold

the

shaft

I P[ "'. "

llol

d

the

bolt diameter, a the

II

angle

t

between

the

c"

surfaces

ents

respectively. pressures

I/ I

.

.

I I ~. I

x~ ~

1/ AM,,

O0 ,af v U0 I 20 i~

are

illustrated in fig. 4.4. The pressure

%@,,:,,

-/ U ......

.

pmf V --~ -- - - t I l #

~s and ~d are the static and dynamic friction coefficiThese

I I "o q~ I /~/~ " ~ / / / ~ 7 ~ ~L"

"d'

I .

(typical for the kind of shaft ring) and

u,=:

i\< -

--~-[---.~• . . . . . . LI I

--T -- -- - - - ~ I I

" \

ujluflul •2

a

aft v 2 Oo

( 6-v_ pm i )

,4

Fig.4.4. Variation of pressure and radial displacement in a yieldable steel shaft ring.

varies between these values

~TL~ // •_ _ "~

and

two

tightening

ring, li'~." Jl '

friction

M the

"~ ?°

g a constant (ranging between O. 15 0.20),

in a joint,

L

spacing between rings, along

thetunnel'RtheCv~ radius of curvature of

clamps

until

stabilization is obtained. It is easy to find the number of slip-creep cycles, the time during which a cycle takes place, imate pressure etc.

the ult-

~

/

/

F

~ plank

siders a lining made of circular

segments

concrete

of

panels

reinforced with

wooden 0

strips inserted between the longitudinal joints of the panels

(see fig.4.5)

(1988 a)).

(Cristescu

It is assumed

that

the reinforced concrete panels are much

more

wood

for

and

rigid this

than

the

reason

the

entire possible deformation of the lining

is essentially

to the compressibility

due

of the

wood planks. First was studied

Fig.4.$. panels

Reinforced concrete lining with inserted wooden strips.

u

260

the

compressibility

to the fibres.

Three

of

the wood

planks,

cases were

considered:

and wet planks which were previously tionship

between

when

the compressive

compressed

perpendicular

dry fir planks,

wet

already once compressed.

stress

and the reduction

planks

The rela-

of thickness

w is found to be of the form = Aw 3 + Bw 2 + Cw with A, B and C material arized, planks

for the

vergence

constants.

convenience,

inserted

between

(4.17)

in

pressure

the at

in the lining

This relationship

rock/support

of this interface

analysis.

circumference,

the rock/lining

can be also line-

then

interface

If there

the

and

the

radial

linearized

versions

equation

of

analysis.

Several

sure

prescribed

was

the

'>

wet

and dry

also

which

examples

timing t o the lining 4.6;

are

support and

can

were

by

a

possible.

con-

be

back

were

is

the

constitutive

included

in

the

rock/support

In

analysis

considered

(4.18)

This

considered.

is to be installed.

planks

n

become

P = b ~ c [A(2~(u n- u°) )3 + B(2~(u n- u°) )2 + C 2~(u n- u°)] and

are

relationship

it

all was

The results

the

ultimate

computed

at

are shown

for two cases:

4 or

•

wet

A

o

wet

2 nd [oading

A

o

wef

nonlineor

linearized

in fig.

mode[

model

I

,~- - ~ ' relative

..o~7./ o +

.4:+

u

displacement

u°

F i g . 4 . 6 . Variation of pressure and radial displacement at the rock/lining interface for 4 and 10 wood planks inserted along circumference.

what

10 planks

]

i

pres-

261

inserted 4 cm.

in

The

the circumference.

geometry

is:

a = 170 cm,

the

case

of

of

the

for the timing

planks

the initial

number

radial

planks

of

the

)u~ - o

a

the

cases

formulae

convenient

rock

by

+ Pl

v

rock/support

pressure

(or thicker)

rigid i t y

due

of

to creep

2Go +

the

followed

above

version

depth

For

350 m

instance

(4.18)

we

in

obtain

(4.19)

the most

(loading

all

linearized

displacement

a complete that

under

of

the

R = 156 cm.

was

2Go

%

yield i n g

at

planks

o I

R[(~_~ u° = ql

is

in coal

and

of the

I 2G

-

ao v

Thus

a

thickness

excavated

c = 150 cm

and

I t o = ~ in Uo

clusion

initial

to: 1 2--G

when

tunnel

b = 166 cm,

wet

The

which

unloading)

support

(ql

considered.

in

All

are d e t e r m i n e d

can

lining,

from

that

be done.

point

of

obtained

with

a

are

material

already

wet.

(4.20))

in laboratory

A major

the

previously

which

formula

the

was

(4.20)

analysis

were

and

support

b - c --S1Wl]. R

-

is

without

In

is

the

constants

con-

view

of

greater

compressed

this

case

the

smallest

involved

from

in

the

tests.

5. R O C K - S U P P O R T I N T E R A C T I O N A N A L Y S I S FOR E L A S T I C / V I S C O P L A S T I C ROCK For

elastic/viscoplastic

consitutive

equation

(2.1),

differential equation describing the lining~rock interface a s

and

(3.2)

we g e t

= -

Thus while

the

+

the

k

instantaneous

the u l t i m a t e Two

solution stresses further

-

in

response

of

(sudde~

reaction the

the

rock

convergence

in

the

of constant

pressure

during

absence

of

were

the

creep.

a lining

is in place

blast)

satisfies

problem

surrounding

constant

(3.5)

(5.1)

simplified was obtained

called

remain

~e

ground

solutions

from

the motion of

with tunnel

(i.e.

(p = const.)

governed

by

(3.6)

(3.7).

given

With

is

(Cristescu the vary

this

(1988b)).

assumption at

that

excavation

assumption

p = 0)

or

is governed

when by

the a

A the but

wall

support

262

W IP u(t)

p - c

a

assuming of

WI

=

20

that

for

H(I v +

p remains

the

primary

8H

- --~--)~c e 8H-a

constant stress

8H'c't {I - e x p [ ~ T ~

during state

creep:

and

t

here is

_

k c

W IP

the

t)]}

is

(5.2)

the

moment

value

when

the

C

deformation

by

t ~+ ~ we o b t a i n

creep

begins.

the equation

If

in

(5.2)

for

p = p~ = const,

of the ultimate ground

we

make

reaction curve

W IP ,8 H

u, ~-

H(I - T

P" -- Ov =

2G

or the r o c k / s u p p o r t

+

.......

interface

)Ta-~oI

8H.o

T@

.........

(5.3) P = P~

stabilization

curve

corresponding

to

the

particular loading \ 0.2

tory

\\ \

ci~Ib>~

\ \

his-

consid-

\

ered, i .e. due to a fast

\

excavation sudden

dec-

rease u~

from

of o

0.1

p~

which

afrecons-

obvious the

a

value

terwards mains

p

to

v

certain em c O

a

that u~,-

p~

relationship 0

o

@ F-= ' 0.008 E v 0.01 o 0.015 20 a relotive displacement { u )

0.02

(5.3) a line

Fig.5.1. Variation of pressure and radial displacement of the rock/lining interface for yieldable steel shlaft ring and various loading histories, Crosses mark incipient failure, while the border line and the dotted l i n e a r e two p o s s i b l e ultimate ground reaction curves (Cristescu (1988b)).

coal

is s h o w n for w h i c h

by

border

line

the

ultimate

ground

reaction

not

straight and t h a t

this line passing

is by

the point u = 0, P : ~v" = AS an example

5.1

is

in

obtained

fig.

for

263

H(a,~)£

~a°

CoSin(~+¢)+c

~ )2+bo~ * +

(

I

if 0DH Do r

I -

DO 0

~-~z"

(5.7)

265

This system was integrated with a p p r o p r i a t e initial and b o u n d a r y conditions using finite d i f f e r e n c e method (Cristescu (1988a)). The initial c o n d i t i o n s are obtained from the elastic solution since it is assumed that due to a fast excavation, in a very short time interval the rock response is e s s e n t i a l l y elastic. The b ou n d a r y c o n d i t i o n s are formulated at infinity (at far distances stresses and strains are those from the primary state) and at the t u n n e l / s u p p o r t interface (the pressure varies a c c o r d i n g to the r e l a t i o n s h i p c h a r a c t e r i s t i c for the kind of l i n i n g considered). If the pressure at the tunnel surface d e c r e a s e s s u d d e n l y from to a certain value which is afterwards kept constant, then the V

u lti m a t e ground r e a c t i o n curve which is obtained for a tunnel excavated 350

m,

in coal at the depth

is

shown

in

by curve b (dash-circle).

The

ulti m a t e

ob-

ground

tained

with

solution

Thus

the

more

significant with

solution. lines

the

marking

tions same

lead

a

tion

how

volume be

to the

for

P/a v < 0.3

behind

the

can

ask

the

ques-

is

the

of the rock which

will

microcracking

The

damaged and

the a d d i t i o n a l

how

big

loading amount

by

of

this

failed

rock

is

For

relatively

small

of p the amount

of the

damaged cant

rock

mainly

shown

is

on the

5.2.b. values

may

~A

0 0

be

o.ol

u

o:02

relative displacement ~Fig.5.Z. a) Ultimate ground reaction curves obtained with simplified method (curve a) and with numerical method (curve b), onset of failure curves obtained with simplified method (curve c) and with numerical method (curve d), at the depth of 350

m.

is taking

important

ultimately

lining.

lining

Ih3 m t

÷A

the

solu-

practically

one

,~ 0.1

threshold two

of constant pressure place,

÷&

is

result.

failure

0.2

the one

concerns

the

Though

÷&

~

simplified

the

failure,

cb I! II

u') u)

a

ulti-

method

than

What

\[

obtained

numerical

obtained

line

the

displacement

with

of

as

':

O.3

simplified

shown

(dash-dot). mate

reaction

the

is

04

fig.5.2.a

in

fig.

signifi-

if there are also

. ~>0.3 Q.

u)

o.2

CL

>~ o.1 _o

0 1.1

relative radius rio Fig.

S.Z. b) The amount of damaged rock behind the lining.

112

266

some

weak

structural How L@I

~ 5

', I

',1

"1

other

weaknesses

etc.

fast

damage

•

planes,

propagates

in

the

in

fig.5.3

of

pressure.

of

the

the

for

damaged

= 0.78

01

fast

I

1.2 relotive

radius

two Fig.5.3. Propagation of the damage in the rock surrounding the tunnel surface.

levels absence

amount

rock

is

and

5 days,

mainly

days.

of

quite

lasts

for

though

the

deterioration

place

rio

shown

it starts at t =

days

than

the

the

the

is

three

In

lining

significant:

more

rock

in

The

takes

the

first

presence

of

a

constant pressure is certainly

reducing

the

the

damaged

amount

rock

of

behind

the

lining.

6. T U N N E L

OR B O R E H O L E

In

to

order

study

STABILITY

possible

the rocks s u r r o u n d i n g a tunnel hole,

failure

and

loosing

of stability

of

(in the presence of a lining) or a bore-

let us consider a case which is very unfavourable.

The most impor-

tant factor influencing possible ~ailure and s t a b i l i t y is the ratio between the horizontal a h and vertical ~v far field stresses.

Let us con-

sider a case when this ratio is quite distinct from unity, Sh = 0"2~v"

The

depth of the tunnel

encing s i g n i f i c a n t l y

is c e r t a i n l y another

failure and stability,

for instance factor

influ-

besides the m e c h a n i c a l

pro-

perties of the rock. In order to make an analysis sary first

to study moments

the

stress

following

ation the rock response

of the failure,

distribution excavation.

around

Assuming

is elastic,

the

it is first neces-

circumference

that

after

a fast

the stress distribution

in the excav-

is obtained

from the well known formulae a

arr

2

h +2 a v ( 1

= p--~ + r

-

-~- a ~2)

+ ~h

2- ~ v ( 1

-

4a22 + ~ 4 4 ) c o s 28

r

r

r

(6.1) a2

gee

:

-

P --2

r

oh + av +

2

(1

2

+ ~'~) r

-

~

h

- ~ 2

v (1

+

~a r

) cos

2£

267

ev

ah are

=

- 2a2

2

(- I

r2

3~ 4-) sin 20 + r

(6.1) = 0 h - W(O h - O v ) 2 ~ e o s r

OZZ and

the

displacements

I + ur

from

(see

2e

Cristescu

(1988a))

2 a4] - V ) ra - - r 3 Cos

a2 ah + av a 2 a h - av[ { - p r-- + + 4(1 2 r 2

V

~

2e} (6.2)

I + ~ ah - qV[2(1 ue =

E It

of

the

is u s e f u l

depth

: 0.2a v = 0.0196)

(see

fig.6.1.

The

is m a r k e d initial

point

rupted

line.

bourhood tensile

of

the

stresses,

plane

lO~-~i
~L

E

--

uo!s~oo

276

FIGURE 4 Outcrop of Lisan Fro.showing convolute lamina in a specific bundle of lamina, FoJdsare tilted downstream.

FIGURE 5 Enlarged view of a convoIute fold. Note thrusting at the forward end of fold.

FIGURE 6 Low angJefauR in a specific bundle of tamina. Note undeformed overlying and underlying tamina.

277

deposition on

the

of

than at an angle to it, and the form of d e f o r m a t i o n depends

direction

deposition

specific to

the

plane

bundle

lamina

of

Shear

stress

convolute

folding

and

thrust

(Figures

4,5,6).

Shear

deposition the

restriction

the

accompanying

was

the

result

remoulding shear

of

and

sensitive

clays

only

an

Formation,

since

achieve.

Thus

flow

dependent

on

a two-fold

extrusion

by

an

saline

later

to

at

type

plane in

an

angle

faults

which

the

is

dilution

process

1952).

of

be

specific

leading

to of

in the present

in

the

for

the

to

Lisan

difficult

seem

pore

of

present

the

is

deformation

case

causing loss

related

water

saline

lamina

load)

often

However, pore

of

deformation

consequent

determined

saline

of dilution

that

(external

characteristic can

bundles

suggest

floods

pore

and Northey,

load,

specific

and

conditions

external

the

features

underlying

water

sensitivity

complete

in

stress

normal

structures

freshwater

of

(Skempton

to and

deformation

flash,

apparent

rise

overlying

sedimentary

This

3).

in press).

of

dilution

strength.

(Figure

gives

laminae

(Arkin and Michaeli,

and

to

stress

of

into

The

case

the to

bundles

dissipate

of

leads

to

water

weight

of

be and the

body of water of the flash flood.

REFERENCES

Arkin, Y. 1980. Underconsolidated tion. Sedom, S o u t h e r n Dead Qiryat Anavim, Israel.

sensitive clay in the Lisan Sea Basin. 5th Conf. Min.

Arkin, Y., and M i c h a e l i , L. (in press). The significance strength in the d e f o r m a t i o n of laminated sediments. area. Spec. I s s u e , E n g . Geol. I s r . J. of Earth S c i .

FormaEng.

of shear Dead Sea

Arkin, Y., and Starinsky, A. 1982. Lisan sediment porosity and pore water as indicators of original Lake Lisan composition. Current Research 1981 G e o l o g i c a l Survey of Israel. Begin,

Neev,

Z.B., Nathan, Y., Ehrlich, A. 1980. S t r a t i g r a p h y and facies stribution in the Lisan Formation, new evidence from south of the Dead Sea, Israel. Isr. J. of Earth Sci. Vol. pp. 182-189.

dithe 29,

D., and Emery, O.K. 1967. The Dead Sea, depositional processes and environments of evaporites. Bull. No. 41. Isr. Geol. Surv.

Picard,

L. 1942. Structure and evolution of Palestine, with comparative notes on neighbouring countries. Geol. Dept. Hebrew Univ., Jerusalem.

Skempton, A.W., and Northey, R.D. technique Vol. 3. No.l.

1952.

The

sensitivity

of clays.

Ge0-

ON THE CONSTRUCTION EQUATION

INTRODUCTION

In have

the

been

great

of

in

be

to

describe

engineering

cannot

hand,

many

practice.

solutions

determined

in

S. Dmitruk,

model

The physi c a l

This

[i, 2, 3]. interest suggested

and

soils.

And

initiated

was

a

the

by

loading

himself,

as

well

A. K w a s n i k - P i a ~ c i k

soil

Dmitruk,

from

from

intensive

[4-8,

to name

of or

the not

use

soil

[i]

study a

of

the as

literature

[4-6].

the

and

form

Lysik

models hand,

of

in

mechanics

the one

a

1969 model

raised

as

conventional

reported

great

approach involving

and

rheology

studies

have

DLS

model.

It

to a soil

results

data,

here

was

equation

compression

experimental

nonlinear

of been

some of these].

equivalent

in a triaxial

soil other

ago,

mathematical

experimental

consitutive the

years a

so

cannot

disadvantages.

in

a

On

proposed which

few

these

to

approximate

been

A

a

wide

behaviour.

developed

of

the

classical

why

had

Suchnicka

constructing the

from

linear

model

formulation

differed

of

involved

a

authors

making

area

to

of

have

Despite

limitation

constants

eliminates

on

1977

exposed

by

the

a

models

received

major

apparatus.

distrust

is

elementary

of

and

whether

established

that

number

representing

method

objective

of

models

great

which

soil. not

Apart

linear

test

of

a

models.

H. S u c h n i c k a

in

is

of

(DLS)

definitions

before

There

a body

a

rheological

have

as

developed

hand

that

The question

of

resulted

one

by

and

models

linear

standard

of soil

Such

on

notions

a

concept

model

[i, 2].

the

with

B. Lysik

rheo l o g i c a l

for

different

properties

these

regarded

of

yielded

many the

investigations,

soil

other

tions

decades

uses - s p e c i f i c a l l y

approaches,

far,

three

suggested

acceptance

the

past

number

their

USE OF THE DLS MODEL

Roman Traczyk of Geotechnique, T e c h n i c a l U n i v e r s i t y of Wroc~aw, pl. G r u n w a l d z k i 9, 50-377 Wroclaw, Poland

Institute

i.

OF A CONSTITUTIVE

OF SOIL BY MAKING

of was

sample

apparatus. obtained

specifically

to

answer

soil

might

considered which The by

those

had

the be an been

considerathe

author

reported

by

280

2. C O N C E P T

While physical

many

and

It seems,

OF

THE

IDENTIFYING

reports

mathematical

therefore,

PROCEDURE

are

available

model,

these

advisable

on

are

to present

so that no c o n f u s i o n occurs as to what

the

assumptions

mostly

written

the major

to

in

the

Polish.

formulae

involved

is included.

The e q u a t i o n of state of the DLS model may be w r i t t e n as Ro

7 = A(g where g

Ro

gR~)

+

(i)

is density of external

R° g

stress, which takes the form

~ & ~ i ( ~ ) H ( t - ~i) t - ~. i=l 1

=

R° or

~ ~' (~'~(~ ~ ~)d~

g

0+e

and gR~ denotes d e n s i t y of internal changes, gR~ =

where

~ A e i ( { ) H ( t - ~i ) i=l t - ~i

t indicates

gR~

time of o b s e r v a t i o n of loading effects,

in the state of stress,

is

the

for

being reserved When

function

of

H is H e a v i s i d e

unbalanced

internal

(3)

u shows

function,

changes,

A

time

and e(~) and

e(~)

for the parameters of the model.

the loads

kmi n defined

which is defined as 2 e'(~)H(t - ~)d~ = 0+e t -

or

of variation used

(2)

=

in

the

will be zero. Thus,

acting on the soil are smaller physical

model,

the

density

than the value of

of

for o = const ~ kmi n e q u a t i o n

internal

changes

(i) becomes

% = A T The

(4)

investigations

that kmi n may be,

reported

in principal,

in

the

literature

[4]

have

The concept of the i d e n t i f i c a t i o n p r o c e d u r e was d e v e l o p e d Kwasnzk-P1asczk model

[4].

parameters,

evaluate tests

A,

the

wherein

constant rate

tended

e({)

function

A

e(~),

to zero.

out

by making

availed

herself were long

kinds

involved

of

and

of

determination experimental

the

small

The d e t e r m i n a t i o n

at various

the

use of

applied

time

results

their

of

loading

i.e.

assessment

test results,

derivation

To

creeping

tended

The

the

data.

stain

compression stresses.

of

by A. of

and

of A enabled

on the basis of triaxial

expressions are given

to

a

strain of

the

which for A in the

[4-6].

Laboratory carried

involved

a sufficiently

the formulae

literature

8(~),

loads

after

had been o b t a i n e d and

and

procedure

authoress the

value

The

shown

identified with l o n g - t e r m strength.

on

investigations two

types

of

reported

samples.

Some

by

Kwasnik-Piascik of

these

consisted

were of

281

Jaroszow

clay

of

bentonite.

While

measure

the

of

describing shown

a

making only

the

of

scattering

use

of

of

the

of

it

A

on

intensity

intensity

values.

of

prepared

were

and water

a

given

interval

type

for

the

hand,

the results

obtained

for prepared

samples,

that

anticipation

of

by and

if at

regarded

Taking

moisture

of

the

one

soil.

have

time

content,

(Aw = 2.5%)

obvious

on

of

a for

proved

of

that c o e f f i c i e n t A may be

for

as

used

tests

[4]

samples

becomes

was

independent

intensity

believes

variability

A was

from

taken

strain

Kwasnik-Piascik

that

stress

constant

were

The results of p l a s t o m e t r i c

analysis

depending

small

stress the

others

clay

scattering hand,

the

Kwa~nik-Pia{cik

Jaroszow

structure,

stress,

regression

characteristic

account

values

state

slightly

a

undisturbed

the state of strain.

all. Thus, as

an

into

content

and

the on

A being

of

greater the other

a constant

ought tO be reconsidered. According values

of

the

to

the

e(~)

assumptions

function

were

be

described

by

the

the

found

variation of the state of stresses. e(~)

of

model,

to

the

vary

and

form

and

the

depend

on

the

K w a ~ n i k - P i a s c i k has suggested that Ro of g which takes the form of G ~

integral

and may be written as

GT = Analysis plot,

n-i Ro ~ gi A~ i=0

of

regressions

t f 0+e

=

8(~)

= f(G T)

Ro g

d~.

(5)

revealed

that

they had a similar

irrespective of what kind of loading had been applied. Although

the

GT or

high

the

investigations

probability

equation

of

soil

accurate

formulation.

considerable 2-the the

undisturbed

becomes

of

of

What

small

here

seem

constructing

DLS

model,

they

accounts

for

this

results

for

prepared

of

clay

during

that

when

the

difference

Jarosz6w

changes

obvious

success

terms

scattering

unusually

volumetric

in

discussed

between

samples

exposure

similar

the

which

to

the do

constitutive not

shortcoming soil

physical made

isotropic

investigations

to c o r r o b o r a t e

were

them

enable is

samples,

and

parameters

of

resistant

to

stress. needed

an

l--the

Thus,

to

it

examine

a

wider v a r i a b i l i t y interval for the physical properties of soil. The necessity those

of

fact, of

that

volume

distinguishing

the deviator.

variations the

Hence,

effects

occurred, of

the parameter

the of

accounted

spherical the model

for

the

tensor

and

were defined

as

where

A = f(w,e,Oo(~) ,t)

(6)

8 = f(A,w,e,~(~),t)

(7)

w

indicates

water

content,

e

denotes

porosity

factor,

ao(~)

282

describes deviator

variation stress,

of

isotropic

~ stands

for

stress, T(~)

time of

is used

stress

for variation of

variation,

and

t is time

during which stress effects are observed. The

choice

equations

(6)

significant

of

and

the

(7)

effect

terms

results

on

the

from

and,

consequently,

predicting

the A and e values,

incorporated

assumption

that

of

A

To

to

and

8.

examine

a series

of

the

they

in

exert

determine

possibilities

triaxial

a

this

compression

of tests

were carried out.

3. L A B O R A T O R Y

The

parentheses

the

values

influence

(with outflow)

in

TESTS

available

the state of stress

/

1 = ~

data

sets

show

that

it is c o n v e n i e n t

in terms of stress intensity.

_ a2)2

(al

2

+ (~2 - a3)

+

to describe

Hence, we have

2

(a 3 - ~i)

(8)

which becomes =

(o I

-

03)

(8a)

for axially s y m m e t r i c a l stress. To

describe

the

state

of

strain

the

author

availed

himself

of

the term of strain intensity. Thus, /2/ = T

(¢I - ¢2)

2

+ (e 2

and for axially symmetrical

_

¢3

)2 +

(e 3

- el)

2

(9)

strain

= ~(¢ 1 - e3). Equations variations necessary

(7)

occur

and

in

to perform

(9a)

the

(8)

are

volume

measurements

valid of

at

the

of

the

assumption

sample.

Hence,

volumetric

changes

in

that

no

it

became

the

course

of the triaxial c o m p r e s s i o n tests. The prepared Poland.

laboratory from

The

clay

method

homogeneity,

thus

consolidation.

investigations collected

of

creeping

to

tests

involved

the

preparation

contributing

The

in

Edmund

applied the were

in

samples the

abatement carried

which

openpit study

of

had

enabled

of d e f o r m a t i o n out

been

Jarosz6w,

by making

high

during use

for

two loading schemes. Scheme 1 The

samples

were c o n s o l i d a t e d

at d i f f e r e n t

pressures.

Following

283

completion axial

of

the

consolidation

procedure

loading

was

applied.

results

relations Scheme

A = f(e)

The

tests

A

identical

the

Axial

for

The soil

which

the

in

closed

and

determining

pressure

the

began

determination

A = f(e)

samples

of

of

different

to

makes

24 h,

rise.

Water

it p o s s i b l e by

block

and

specimens procedure

was

0.5 h,

porosity

relationship

same

to obtain

apparatus

after

to loading.

the

the

consolidation

the

applied

of

from

in order

was cut off prior

compression

increased

of had to

72 h and

144 h

outflow

from

to define

e({)

conducting

triaxial

tests.

Failure shearing

at

was

nature

cut

After

pressure

loading

the

were

12 days

properties. the

moment

the sample

defining

Samples

at 0.20 MN/m 2 for

physical

0.35 MN/m 2.

at

~o(~).

completed,

for

used

was

and A = f(t).

aimed

and

consoli4ated

form

were

outflow

2

between

been

The

water

at

tests

a

were

constant

at a c o n t r o l l e d

carried

rate

rate of

of

loading

out

table

by

the

motion

following

(method

(method

II).

scheme

1 revealed

I),

two

methods:

and

shearing

4. R E S U L T S

4.1.

COEFFICIENT

Creeping of

A = f(e). A

tests

The

A

involving

results

are

plotted

in

the

Fig.l.

To

linear

behaviour

investigate

[~e/M~;

0D2

0020

0D15

0010 ~

,

,

,

i

~05

Figure

1.

Coefficient

,

,

I

730

i

~___J

i

,

i

L

~75

A as a function of Jarosz6w clay samples.

,

,

I

i

, __

~20

porosity

~

I

725

factor

for

the

284

behaviour

of

press u r e

A = f(t)

analysis.

observation,

t,

the h y p o t h e s i s The large

longer

study.

the values

than

of

Despite

these,

to use

effect

of A

the

in

Oedometric

coefficient

300 h.

of

Obtaining

convenient

of

were

this

factor function

c re e p i n g There

and

a

were

subject

processes

were

no

A = f[ao(~)]

number

[7] it

on

constant

with

reasons

to

time

for

of

takes

a

this

the

kind

of

of

variation

time

of

ignoring

isotropic

the

desired

and

the

stress

of

for

the

it

A is

to d e s c r i b e

relation may

of

that

density

stress

using

further

dependence show

a

initial

when

calls

earlier

Thus,

state

requires

identical

to d e s c r i b e

performed

stress.

of

of

results

long

is p o s s i b l e

integral

relationship

samples

investigations

the

and

the

experiments

described

on ~o(~).

porosity

of lack of correlation.

number

method

constant

Analyzed

investigation

properties.

the

a

in the apparatus,

regression

the

at

be

between

written

as

follows A[~o(~ ) ] = A O - f[G ~O] . For relation

the

was

C

is

= A o - CG

constant.

determined

beings.

do

no.14

an error

easy

apparatus. yield

to

INTERVAL

following

8(~)

and

plot

of

error

which

shows

values

of

lower

porosity

a

method of

method

the

at the

initial

eliminate

only

sufficiently

of

was

analyzed

be

of

e

for

attributed

the m o m e n t

by the plot

internal factor.

of

was

using I

is

primarily

at which

the

the e - f u n c t i o n

changes It

by

method

markedly for

this

lower reason

I.

which

Unfortunately,

GT

function

should

illustrated

for

CHANGES

determination

is best

application at

The

considerable

II replaced

moment

included

the

n o r m a l l y c o n s o l i d a t e d and Go the A - G relation is linear.

between

scale.

inaccurate

This

samples

The

not

The

necessarily

sample

2,

clays

relationship logarithmic

that m e t h o d

is

for

The

in Fig.2.

exact

Hence,

OF UNBALANCED

process

scheme

(ii)

FUNCTION

the

to

,

4.2.

given

for

0o

of stresses,

in

than

according

variation

plots

to

performed

established

A[So,t] where

test

(i0)

II

process

if

the

accurate

stage the

made

it

began, of

the

force

apparatus

possible but

is

measured

until

to

define

~ - ~

investigation.

available

indications

to

the

in

the

higher

the

relation The

error

side

the

author

did

stresses

were

285

z;C

3.0

2.0

1.0

,6/

08 O.e

02

c~N~ a05

Figure

Z.

That

appropriate.

is One

why of

two these

r e l a t i o n s h i p was linear In

the

other

020

Ct30

0~0

Function of unbalanced internal changes by triaxial compression tests involving (at constant rate of table motion).

determined

applied.

0.;0

method,

methods made

of

use

calculating

of

in the initial

the

first

the

~0 (~)

assumption

stage of the

two degrees

of

no

internal

changes

occur

for

there

loading

two

seemed

that

to

be

the ~ -

testing

as they showed values lower than kmi n ([i]). Hence, that

versus GT method I

procedure.

were

neglected

it has been assumed degrees

of

loading

(kmi n < 0.030 MN/m2). When behaves

the

as shown

~ - ~

relation

by Fig.3.

is

assumed

The d e p e n d e n c e

to

be

linear,

the

plot

of e on G T may be described

with great accuracy as e = CI(GT)C2 + C 3. While varied so

C1

increased

only slightly.

they

should

be

(12) with

the

increasing

porosity

The values of C 3 showed no relation

regarded

as

incidental,

taking

into

factor,

C2

to porosity, account

measuring problems dealt with at the initial stage of shear.

the

286

2.00

;.40 '

18

OflO O.6O

0/0

Q20

o.,o

/2

0,08 eo6

78,,."

0,02

0,0;

by

triaxial of

Fig.4.

0,02

0,05

030

0"20

0,30

Figure 3. Function 0(~)versus GT d e t e r m i n e d compression tests involving method II (at constant ~ncrease intensity) and making use of the assumption that the T y relation is linear at the beginning of the process.

Neglecting

the

These

be

parallel

0.03

may

first

two

interpreted

or convergent,

which

degrees as

leads

of

loading

a pencil

of

gave

the

straight

to expression

(13)

plots

lines

and

of

either

expression

(14), respectively, e = CI(GT) C2

(13) Co

e = CI(G T + C2 ) Using the

relation

increasing

slight

of

to separate

only.

(14) for

(14)

(13),

porosity

variations

by virtue

. the C 1 values

factor,

whereas

The d e t e r m i n a t i o n the plots

the q u a n t i t i e s

of Fig.4

occurring

were the

found

to

values

of

of the CI, failed,

as

in parentheses.

C 2,

increase C2

with

underwent

C 3 constants

it was It has

impossible only

been

287

220

1/,o 12o ~o O.6O

0.~0

020

0,to OZ~ 0.o6

024

~;

&02

0,03

~05

~

~

Figure 4. F u n c t i o n by method

shown

that,

II

at

if

the

C1

assumption beginning

and

C3

are

030

8(~]

that no internal of the process.

assumed

increase with a rise of the porosity

versus OT determined changes occur at the

constant,

the

values

of

C2

factor.

5. C O N C L U D I N G C O M M E N T S

Analysis in terms of Gkmin

of

results

(GT - Gkmin), ~ = i= 1

kmi n iA~ A~

enables

the

8({)

function

to

be

described

G kmin being given by Gkmi n or

=

t f 0+¢

k ~ min t .

Relation e = f(G T) is then likely to take the form

(15)

288

8 = C I ( G ~ - Gkmin) c2 H(G T - Gkmin), w h e r e H(G T - G kmin) Taking on our e

the

into

account

invariability

interpretation can

is H e a v i s i d e

be

ourselves

makes

represented

f u n c t i on. plot

C 2 for

use by

of the s t a t e m e n t

Fig.5 gives

the

of CI,

(16)

of the

the

of

relation

a wide

statement

quantity

_

_

(16),

that the c h a n g e

that

the we

may

conclude Thus,

if

variability

of

may

of p o r o s i t y

the p l o t s of 8 for a s s u m e d

we

of p o r o s i t y .

G kmin ,

200

~/*0 _

range

also

avail

is included.

kmi n, CI, C 2 at

jl

_

700 0£0

~60

0~0

Q2O

0 . 7 0 _ _

~ 1 ~

~06

4

2

O~ ...........

/

002 ~

i

I/

Cr[MNI~J 007

ao2

o~

o o~

070

020

0~

Figure 5. F u n c t i o n e (~) versus G T c a l c u l a t e d by v i r t u e of e q u a t i o n (16).

&~ = 0.015 M N / m 2 and &{ = 0.166 h.

6. SUMMARY The

considerations

presented

in

this

paper

indicate

that

the

289

results

of

despite

the

equation

of

the

investigations

fact soil

adequately.

But

parameters

of

that in it

the

terms seems

the

performed

problem of

the

to

be

equation

of

DLS of

may

so

far

establishing model

has

particular be

deserve

not

a

constitutive

yet

been

importance

determined,

attention

using

solved

that

the

standard

laboratory apparatus.

REFERENCES I.

S. Dmitruk, B. Lysik, H. Suehnicka, Fundamental soil strength, Archiwum Hydrotechniki, 1973, Vol. (in Polish).

2.

S. Dmitruk, B. Lysik, H. Suchnieka, Problems relations in soil mechanics, Studia Geotechnica, fasc. I.

3.

S. Dmitruk, Problems of Representing Geological and Engineering Processes in Openpit Mining. Warszawa, Wydawnictwa Geologiczne, 1984 (in Polish).

4.

A. Kwagnik-Piaseik, Some problems dealt with in the identification of the DLS model. Ph.D. thesis. Technical University of Wroclaw, Institute of Geotechnique, 1978. PWr I-I0/K-242/78 (in Polish).

5.

A. Kwagnik-Piadeik, On the identification of the DLS model for a granular medium, Archiwum Hydrotechniki, 1981, Vol. 28, fasc.

6.

A. Kwa~nik-Piagcik, On the identification of the DLS model for a granular medium, Archiwum Hydrotechniki, 1981, Vol. 28, fasc. 2 (in Polish).

7.

R. Traezyk, Analysis of the indentification procedure for the DLS model. Ph.D. Thesis, Technical University of Wroc~aw, Institute of Geotechnique (in Polish).

problems of 20, fasc. 4

of physical 1973, Vol. 4,

I (in P o l i s h ) .

8.

R. Traezyk,

On

the

identification

procedure

Archiwum Hydrotechniki, 1982, Vol. 29, fase.

for the DLS model, I-2 (in Polish).