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
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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)
:
<
!
where V denotes the gradient in the reference configuration and the prime the material derivative in E x or E z ; no(a) is the unit vector normal to F°(t) (corresponding to the unit vector normal to Fa(t) and directed outwards C
at t) and ~ the unit vector normal to Z°(t). The brackets [ a o a denote the corresponding discontinuity along Za(t) ; qO is the mass re(a) flux of Ca crossing Za(t) per unit time and per unit area of the reference configuration and ~o(a) the mass of Ca created by phase changes on Fa(t) per unit time and per unit area of the reference configuration. Local mass per unit volume in the reference configuration is denoted by po(Z). Formulas (1.3) and (1.4) use the surface distribution 6 S on ~
4 (S(t)
a surface varying with time t) defined by the following equality valid for any test function f(Z, t) with compact support in space-time (1.5)
(~$,1C)
f(2,t) aAt (2) dt
-~
~t)
where dAt(Z) is the area element of S(t). For further explanations the reader is advised to refer to the abovementioned references or to Estrada and Kanwal (1980);these references give similar formulas in Euler variables, instead of Lagrange variables used here.
2.- MASSES, POSITIONS AND VELOCITIES
The average mass of C a at time inE
IN E
X
t per unit volume at t o around X is
x
(2.I)
a
(~,t) : < ~o(~)> 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)
<
, (~,t)
(3.5)
(3.6)
B
~0 ~ , , a
_
:
Cx t) =
1~
. ~ Ca)
>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
~ ,<
tl x I, (+0,)
Figure 6 : Two-dimensional gap between l o c a l l y r e g u l ~ cylinders An equivalent diameter is introduced for the gap by d.. = 4(a] I + all) -I
(8.1) Quantity
1]
hij
gives
1
a natural
]
width
is given by (8.2)
L.. IJ
= (h...d..) IJ IJ
I/2
scale
;
an a p p r o p r i a t e
Length scale
21
and we investigate
asymptotic
hydrodynamic
behaviour of the gap when the
grains are very close one to the other, that is when parameter small with respect to one h.. (8.3) ~ _ i~ d.. ij Significative co-ordinates
.
scaling is obviously obtained by using dimensionless
x and
(8.4)
+0
~ is very
y
(inner variables)
x = Lijx
Inner asymptotic culated, matching
,
y = hijY
expansion of Navier-Stokes
equations
for ~+0 is cal-
it on outer conditions expressed at the infinity in in-
ner variables (8.5) where
P ÷ P(±~) Po
= PO ~ 2P cos ~t
is a constant pressure and
drop P cos~t between
,
as
X÷ ±~
P the amplitude of periodic pressure
left and right sides of the gap. Scalings are thus in-
troduced for pressure,
time and the two velocity components
principle of minimum degeneracy For sufficiently
is used.
small pressure drops stationary
be neglected and two-dimensional
flow rate q(t)
inertia forces are to 2 -I (in m .s ) past the gap,
which is a periodic function of time, is proportional Denote by
and classical
~ and
~ the complex quantities
to P.
associated
to periodic
flow
rate and pressure drop. An easy but somewhat tedious calculation yields
(8.6)
where function
~
:
Z
~]'r
1j . . . . . F
F is a complex valued integral of the ratio B of minimum
width 2h.. ij of the gap to "skin depth" ~ at circular frequency ~, given by (8.7)
~
=
( )I/~p~ci). ~
we call
z.. hydraulic impedance of the gap and F reduced impedance. 1j It is represented in figure 7 in log-log plot as a function of the dimensionless parameter B. Purely viscous solution of the problem is recovered for B ÷ 0 (all inertia terms dropped) fluid for B ÷ + ~
(asymptotic
cases B = 0 and B = + ~ ) .
as well as solution for an inviscid
expansions are easily calculated
for the
22
~m
F
100 (~e F _p2
10
0.1
T
10
I / / / /
/
001
Fiure
7 : Reduced impedance F as a f u n c t i o n of t h e parameter B. r e p r e s e n t approximations v a l i d f o r B ÷ 0 or B ÷ + ~ o b t a i n e d by e x p a ~ r o n of F. Note t h e i r accuracy : f o r B ~ 10-2 or B ~ I0 z r ~ a t ~ v e e r r o r i s l ~ s t h a n I%. Imaginary p a r t of F i s such t h a t , w h a t e v ~ B, 0.4444 B2 <JmF<0.5333 B2.
~'~hdd l i n e s
Behaviour of the pressure in the limit ~+0 was carefully investigated : it appears to be independent of
~ and tends very rapidly, as I~I grows,
towards upstream and downstream values given by formula (8.5). Thus an excellent overlap is obtained between inner expansion and outer conditions. It is worth noting link between such calculation and Biot's simple approximation of viscodynamic operator (8.8)
(Biot 1961) which predicts
~P-- c I + ic2~ q
where c I and c 2 are real positive constants related to dissipation rate and relative kinetic energy.
23 Extension to flow under unsteady conditions other than harmonic ones is straightforward using Carson-Laplace transforms of flow rate and pressure drop. Flow law with memory effects over pressure drop is obtained as expected.
9.- APPLICATION TO PERMEABILITY OF A LEIBNIZ PACKING OF CYLINDERS UNDER HARMONIC REGIME Preceding results are used to compute permeability of particular selfsimilar compact packings of two-dimensional grains, called Leibniz packings and obtained through a slight modification of Appollonius construction (Mandelbrot, 1982). Such packings, which were studied in particular by Adler (1985) and Gilbert and Adler (1985a,b), are build up as follows. Consider three cylinders with parallel axes, which are almost tangent and have arbitrary radii al, a2, a 3. The gap 2hij between any two of them is assumed to be a fixed small fraction c v of the radius of the smallest of the two adjacent cylinders (9.1)
2hij .. = ~r . min(ai, a.)j
and hence parameter ~(i,j)
lies between cv/8 and or/4. These three cylin-
ders constitute the generation number
n = O.
It is possible to insert between them a fourth cylinder of radius
a4
almost tangent to them, so that relation (9.1) still holds for gaps around it. This cylinder constitutes the generation number By the same process, in a second step
n = 1.
n = 2, three new cylinders num-
bered from 5 to 7 are inserted in the three interstices created at step n = I, relation obvious way
(9.1) still holding. This construction is continued in an
IFig.8)
with smaller and smaller cylinders and gaps ; total
number of cylinders N
n
after step
n rapidly increases as
n increases.
2,4
3
e~ / q23
' :31 C
2
A
q12 Figure 8 : L~bniz packing co~aining cylinders of generations 0 to 4. Harmonic flow rates ~ , qo~ and q~7 Cwith q12 + q23 + q~1 =0) are ~signed at the boundary o~ the pa~ing. Note the obvrous s e l f - s i m i l a r i t y of the packing. Suppose that harmonic flow rates q12" q23 and q31" whose sum is equal to zero, are assigned at the boundary of the packing. Hence pressure drops are observed between points A, B, C and complex pressure drops P12 and P23 are defined by
(9.2a~
~
~12 = ( PA- Pc)(t)
Clearly, for n = 0 (Fig.9} relationship between flow rates q12 and ~2~ and pressure drops Pl~ and P23 is that of a star with hydraulic impe.-(o) _(o) . _(6)" aances Z12 , L23 ana ,'31 of values Z12, Z23 and Z31 given by formula (8.6).
25
/I
c
t,,/
-V -a.x ]
A
(n=l)
A
Figure 9 : E q u i v a l e n t ~ e c t r i c a l d e s c r i p t i o n of t h e packing f o r n = 0 and n = I. Primal graph a s s o c i a t e d to t h e packing i s shown by dashed lines, For greater values of hydraulic
impedances.
the packing,
n the packing always appears as a network of
Its topology
is that of dual graph associated to
the primal graph being obtained as follows
: vertices corres-
pond to centres of the cylinders and an edge relates two vertices when corresponding
cylinders are close one to the other.
graph hence correspond to the gaps of the packing matter to state that, whatever packing,
so that for fixed
(9.3)
P12
P23
IFig.9).
It is a simple
n, there is a star equivalent
to the whole
n
= IZ12 + L31 L31 | (n) _ (n) _ (n)
\Z31
q12
L23 + L31
(n) _(n) _(n) where Z12 , L23 and L31 are the hydraulic star at step
Edges of the dual
\Cl23
impedances of the equivalent
n.
A program was written by us at the Laboratoire de M~canique des Solides to compute physical properties of such packings. grains
Problems
concerning
many
(for n = 10, N
= 29527 grains) are reduced in a recursive manner n to much simpler ones which are easily solved. For permeability the matrix procedure uses only the classical
triangle-star
transformation
already used by Adler (1985) for steady conditions.
for networks
A more intricate problem
was studied by Gilbert and Adler (1985a, b) using the abovementioned gram.
pro-
26 10.- NUMERICAL RESULTS A few numerical
results are presented
performed up to generation
impedances of the equivalent
; real and ima-
star at step
n
{Fig. t01. One gets with a good accuracy
7~)
(10.1)
,
/~2 a3/
-2 ~,-5/2 2"
= =
(10.2)
a,
.
.
c.
where c12 , c23 and c31 are dimensionless a3/a I and
were
n not too small) power laws as functions of the total number of
cylinders Nn, as expected
merical
CalcuLations
n = 10 for various configurations
ginary parts of the hydraulic obey (for
in this section.
c is a dimensionless
functions of the ratios a2/a I and
constant approximately
equal to 3.60. Nu-
evaluation of the two exponents ~ and ~ yields very different va-
lues (10.3)
~ = 2.01
,
~ ~ 0.469
Observe that, for
n not too small, {~e.Z~ ) is almost independent of i] (n) of viscosity ~. This is closely related to formula ~/~Zij
Pf and w and (8.6) and properties
of reduced impedance
F. Dependence
upon parameter cl
is explained in the same manner and is valid for sufficiently
low values
of c w . Closer examination
of the recursive procedure
reveals that behaviours
of
~eZ}~ ) " and ~ Z } ~ ) " as functions of N must be, in some sense, indeij ij n pendent one from the other as n grows. Explanation of this phenomenon is the relative smallness of the p~rameters at step ~(n) eLij
B corresponding
to gaps created
n. Note that initial configuration is always reminded for ~-~ (n) only ; ~.k~Zij appears to be isotropic and size invariant.
Scaling arguments may be used to obtain estimates of the exponents and ~, as they are independent generation
from initial
n whose three first cylinders
viewed as the sum of the three packings (a 1, a 4, a 3) of generation neglecting
differences
configuration.
have radii
The packing of
(al, a2, a 3) may be
(al, a 2, a4), (a 4, a 2, a 3) and
n - 1. Putting for instance a I = a 2 = a 3 and
between the various
c. 13
coefficients
one gets,
27
('n) ~mZij ( Pa x s /
m2)
10 20
(~e
Z
(n)
ij
1 0 l°
(n)
2., Zii
15 10
Figure 10 : Hydraulic impedances of the e q u i v ~ e ~ star ~ step n
-lo g
for
10. 10
a 1
=
3.10 -3
a;
=
2.10
m
-3 m
a3 =
1. 1 0 - 3 m
~:'
=
10 -3
w
=
10 3
rad/s
Pf
=
10 3
kg/m 3
q
=
10 -3
Pa
x s
..1,2") 1-80
I , , 347 01 2
I
I
I
I
I ,
1
16 3
43 4
124 5
367 6
1096 7
3283 8
I
1
9844 29527 9 10
Y Nn n
28 using formulas (10.1) and (10.2) for the four packings involved, the following estimates of the exponents (10.4)
~ = 2.11
,
~ = 0.465
which are in excellent agreement with numerical estimates given by (10.3), taking into account the simplifying assumption of isotropy for
~ e Z }ij )~. "
Preceding estimate of ~ is also obtained theoretically for a network of pure inductances having the same value for all gaps regardless of the radii of the cylinders involved. Note that it is more or less the case here : it explains the observed isotropic and size invariant character of ij N e g l e c t i n g conductance o f gaps which do not i n v o l v e one o f t h e t h r e e
first cylinders yields for ~ the value 2.16. Obtained results are interesting to compare with Biot's type approximation. For particular values of the ratios a2/a I and a3/a I one can draw periodic lattices of such packings, whose permeability properties are easily studied starting from preceding results.
CONCLUSION Empirical approaches use the notion of representative elementary volume (REV) and postulate average balance equations on intuitive grounds. Convolution methods give a rigorous form to these estimates and allow to introduce in a consistent and natural way macroscopic quantities of interest and balance equations for complex heterogeneous media. Explicit physical meaning of the various quantities used is known and thus they provide a useful framework for the discussion of constitutive relations. Obtained equations, as they are supplemented by constitutive relations, reflect the fact that soils generally exhibit statistical quasi-homogeneity at macroscopic scale.
2g
Relationships between this work and other theoretical approaches, as general mixture theories, Biot's theory or homogenization of fine periodic structures are also to be mentioned. The shortcoming of the formulation used is naturally connected with the difficulties encountered to express precise constitutive
relations.
These difficulties arise from the simplification of the geometrical description of the medium involved by this change of scale process, which obviously implies lack of information. To allow to go further, obtained results must be supplemented by direct macroscopic postulates, geometrical assumptions as local periodicity depending upon a small parameter or use of simple cell models representative of the medium. In this framework analysis of simple self-similar structures, which account for the strongly heterogeneous scale, is of particular interest.
character of soils at small length
Despite their apparent great complexity
these structures often lead to tractable calculations.
Another sort of
change of scale is thus achieved. It seems to us that combining these two methods of change of scale accounting for the two abovementioned essential geometrical aspects of soils will appear fruitful for the study of soil rheology.
30 REFERENCES
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COUDERT, J.F., (1973), Th6orie macroscopique des 6coulements multiphasiques en milieu poreux, Revue Inst. Fr. du P@trole, vol.28, n°2, marsavril 1973, pp. 171-183 and voi.28, n°3, mai-juin 1973, pp. 373-398.
31
COUSSY, O. and BOURBIE, T., (1984), Propagation des ondes acoustiques dans les milieux poreux satur~s, Revue Inst. Fr. du P~trole, vol.39, n°l, janv-f~v.1984, pp. 47-66. ENE, H.I. and SANCHEZ-PALENCIA, E., (1975), Equations et ph~nom¢nes de surface pour l'dcoulement dans un module de milieu poreux, Journal de M~canique, vol.14, n°l, pp. 73-108. ENE, H.I. and MELICESCU-RECEANU, M., (1984), On the viscoelastic behaviour of a porous saturated medium, Int. J. Engng. Sci., voi.22, n°3, pp. 243-246. ESTRADA, R. and KANWAL, R.P., (1980), Applications of distributional derivatives to wave propagation, J. Inst. Maths Applics, voi.26, pp. 39-63. GILBERT, F., (1984), Description des sols satur~s par une m~thode d'homog~n~isation, Ecole d'Hiver CNRS-IMG "Rh~ologie des G~omat~riaux", Aussois, France, 33 p. GILBERT, F., (1985), A mechanical description of saturated soils, International Symposium Physical Basis and Modelling of Finite Deformation of Aggregates, Paris, sept-oct 1985, 16 p. GILBERT, F. and ADLER, P.M., (1985a), Spin viscosity of heterogeneous suspensions, 5th International Conference on Surface an d Colloid science, Clarkson, Potsdam, N.Y., june 24-28, 1985. GILBERT, F. and ADLER, P.M., (1985b), Rotation tensor of fractal suspensions, to appear in J. Coll. Int. Sc. GRAY, W.G. and O'NEILL, K., (1976), On the general equations for ylow in porous media and their reduction to Darcy's law, Water Resources Research, vol.12, n°2, april 1976, pp. 148-154. HADJ-HAMOU, A., (1983), Contribution ~ l'~tude du comportement des sols pulv@rulents sous chargements cyclique et dynamique, Th~se DocteurIng~nieur, Paris, ENPC, d~c. 1983. HASSANIZADEH, M., (1979), Macroscopic description of multi-phase systems A thermodynamic theory of flow in porous media, Ph.D. Thesis, Princeton, sept. 1979. MANDELBROT, B.B., (1982), The fractal geometry of nature, Freeman, SanFrancisco. MARLE, C.M., (1965), Application des m@thodes de la thermodync~nique des processus irr~versibles a l'~coulement d'un fluide ~ travers un milieu poreux, Bulletin Rilem, nouvelle s~rie, vol.20, pp. 107-117. MARLE, C.M., (1967), Ecoulements monophasiques en milieux poreux, Revue Inst. Fr. du P~trole, voi.22, n°lO, oct. 1967, pp. 1471-1509. MARLE, C.M., (1982), On macroscopic equations governing ~ltiphase flow with diffusion and chemical reactions in porous media, Int. J. Engng. Sci., vol.20, n°5, pp. 643-662.
32
MATHERON, G., (1965), Les variables rdgionalis~es et leur estimation, Th~se Docteur-~s-Sciences Appliqu~es, Paris. MATHERON, G., (1967), Elements pour une th~orie des milieux poreux, Masson et Cie, Paris. MULLER, I., (1975), Thermodync~nics of mixtures of fluids, Journal de M&canique, vol.14, n°2, pp. 267-303. PREVOST, J.H., (1980), Mechanics of continuous porous media, Int. J. Engng. Sci., vol.18, pp. 787-800. SANCHEZ-PALENCIA, E., (1974), Comportement local et macroscopique d'un type de milieux physiques h~t~rog~nes, Int. J. Engng. Sci., vol.12, pp. 331-351. SANCHEZ-PALENCIA, E., (1980), Non-homogeneous media and vibration theory, Lectures notes in Physics, 127, Springer. SERRA, J., (1982), Image analysis and mathematical morphology, Academic Press. SLATTERY, J.C., (1967), Flow of viscoelastic fluids through porous media, A.I.Ch.E. Journal, vol.13, n°6, nov. 1967, pp. 1066-1071. SLATTERY, J.C., (1969), Single-phase flow through porous media, A.I.Ch.E. Journal, vol.15, n°6, nov. 1969, pp. 866-872. SLATTERY, J.C., (1972), Momentum, energy and mass transfer in continua, Mc Graw Hill. TRUESDELL, C. and TOUPIN, R.A., (1960), The classical field theory, Handbuch der Physik, III, I, S. FIOgge ed., Springer. WHITAKER, S., (1969), Advances in theory of fluid motion in porous media, Industrial and Engineering Chemistry, vol.61, n°12, dec. 1969, pp. 14-28.
THE USE OF THE H O M O G E N I Z A T I O N DESCRIBE
METHOD
THE V I S C O E L A S T I C
BEHAVIOUR
POROUS S A T U R A T E D
MEDIUM
Horia
TO
OF A
I. ENE
INCREST, D e p a r t m e n t of M a t h e m a t i c s Bd. Pacii 220, 79622 Bucharest, Romania
Abstract. Using the homogenization method, we obtain the constitutive equation for a mixture formed by a viscoelastic skeleton and a viscous incompressible fluid. The macroscopic constitutive equation give us the effective stress tensor as the difference between the mean value of the stress tensor in the skeleton and the pore pressure multiplied by the porosity. The motion of the fluid is described by a Darcy's law with memory depending on the pressure gradient and the inertia force. It is also deduced the form of the conservation of mass and momentum.
1. I N T R O D U C T I O N 1.1. G e n e r a l i t i e s In
the
consider
small
distribution The
is well
the
orders
the
topological
case
of
the
In well
law
it
case
fluid
of
a
method
mixture
incompressible
and
fluid
the
periods
a great
variety
the
coefficients
of
the
a
[I, 2] we
formed
fluid.
parts are
of
that
by
a
The
geo-
periodic,
with
associated
with
is
then
problems very
are
an
the
of are
mixture
mixture
appears
only
we
can
body
macroscopic
[2]. and
a
stress
[2, 3],
arise
different,
different
elastic
law with memory
consider
one,
viscous
and
fluid). in
the
orsmall
viscosity
and
smaller
magnitude
is of
consequence
of
of
but
or In
if if the
viscous
tensor
depending
is only
tensor.
our
holds
that of
by a v i s c o e l a s t i c
as the
slightly
viscous solid
properties
vibration
on the strain
the h o m o g e n i z a t i o n
motion
e. known
fluid,
also
a
the
of m a g n i t u d e
barotropic given
and
of
of
the
dimensions
parameter
It
of
skeleton
periods.
the small
framework
problem
viscoelastic metric
general
the
of the
fact
that
that the
In
fact
ease
of
possible order that
it
the
viscosity is well
large large
E z then the
solid
part
of the known
viscosity velocity. the
displacement
connected,
fluid
[2, 4] and
More
larger
is
is small that
small
in
the
(a
Darcy's
velocity,
precisely,
is of order
vector
as
if the ~o.
As a
solid
part
34
is
of
order
order
e °,
we
take
the
velocity
~a~. !3 3xj
~2u. i 8t 2
PS
(1.2)
aS S lj = aijkh ekh(U)
ekh(U)
of
the
same
the equations
fluid
are:
+ b~, 8u 13kh ekh ~-~
i ~Uk ~u____hh = 2 (~-~h + ~Xk)
f is the exterior
satisfy
body force ' and the coefficients of symmetry and positivity;
a~jkh,
b sijkh'
the usual properties
(1.3)
as s s s ijkh = ajikh = ajihk = akhij
(1.4)
as ijkh eij ekh Z ~eij eij;
and similar
relations
~2u. ~t 2i
~ > 0
for b s ijkh"
In the fluid part,
the equations
~
= fi
Of
(1.6)
af 2 ~jh ij = -p ~ij + e ~ ~ik ekh(V)
(1.7)
div v = O, Moreover,
~u v = ~-~. at
the
interface
must have the continuity [u] = O,
are:
~a{.
(1.5)
between
of displacement
[aijnj]
We must adjoin
(I.i0)
part
= fi
(I.i)
(1.9)
fluid
solid with a viscous
In the solid part of the mixture,
(1.8)
the
and the viscosity of the fluid of order e 2. 1.2. Mixture of a viscoelastic
where
in
the solid
and
the
fluid,
we
and stress:
= O.
initial and boundary
conditions:
u = 0 on 8~ 8u ~ = ~-~ = 0 for t = 0
where R is the domain
occupied
by the mixture,
and is formed
by Rs and
af. The v a r i a t i o n a l +(1.9)+(1.10) is: find that:
formulation of the problem (1.1)+(1.5)+(1.8)+ u, f u n c t i o n o f t w i t h v a l u e s in H ~ ( a ) , such
35 2 u.
~u
(i.ii) fp ~ ~t 2
w i dx + a(u,w)
w
+ b(~-~, w) - f p div wdx = f f w dx ~ f
H ~ (~) t)
(1.12) div v = 0 (1.13)
in
f
s = f aijkh ekh(U) eij(w) dx s
a(u, w)
(1.14) b(v, w) = f bijkh ekh(V ) eij(w ) dx C (1.151
bijkh =
ijkh
in a s
e2 ~/ ~ik ~jh 1.3.
We variables Ys'
Two-scale
consider Yi
asymptotic
a
by
a
.
process
parallelipipedic
(i = i, 2, 3)
separated
in ~f
formed
by
a
Y
part
boundary F. We look e in the variable y = x/e: (x) ~ p(x/e),
the
appropiate
that
of
flow
vector
asymptotic
viscoelastic through
porous
expansion
region.
On
natural
to
respect
to
in
the
media.
the
introduce
it
the
solid:
we
search
But
the
does does y,
solid
not
of one
coef-
aijkh (x/e)
÷ 0, we assume part
is
the
depend
on
y
fluid
displacement y,
that
analogous
analogous
of
t) = u°(x,
space
a solid
a~jkh(X ) ~
term
the
two-scale
the and
Y-periodic
fluid part
first
in
relative
a
the
in the
ur(x, for
in
for
process e
and
solid
contrary,
the
consequently
expasion
mixture,
of Yf
smooth
ficients and beijkh(X) H bijkh(X/e). In order to study the asymptotic
of
period fluid
to
to that
displacement in
the
solid
region.
It
is
then
of
fluid
with
the
t) - u°(x,
asymptotic
expansion
t),
and
suitable
in ~ef as well as in ~es: (1.16) ue(x, where
t) = u°(x,t)
y = x/e
takes values
and
all
in HI(y)
+ ur(x,y,t)
functions
+ e ul(x,y,t)
are
and is zero on Y
Y-periodic s
+ . in
y.
The
vector
ur
and on F.
For the pressure we have also: (1.17)
pe(x,t) Now,
= p°(x,t)
with standard
e, the p r o b l e m
(i.ii),
+ epl(x,y,t) notation
+ . . .
in h o m o g e n i z a t i o n
(1.12) may be c o n s i d e r e d for u
theory, for e and p .
fixed
36 2. M A C R O S C O P I C 2.1.
Balance
EQUATIONS of mass
If we replace
(1.16)
into
(1.12) we have
(2.1)
div
(2.2)
diYx(V O + v r) + divyv I = 0
in ~ef"
Note
in
Y
vr = 0
that
(2.2)
only
holds
fact that f divy v I dy = f v I n d s Y 8Y
the
fluid
part.
But
using
the
= 0
we take the mean value of (2.2) over Y and we have: (2.3)
n div x V O + divx ~r = (i/iyl) fy divy v I dy;
n =
IYfl
s
which is the balance of mass. After that, taking test functions depending on e in the form (2.4)
w(x)
at order
= w°(x)
e °,
from
+ wr(x,y)
+ ewl(x,y);
the equation
div
analogous
to
Y
wr = 0
(i. Ii) , using
(1.16)
and
(1.17) we obtain: 2 (2.5)
f pe
o r s ~u~ ~u~ ~w9 (ui 8--2 +t ui~twiO + wi) r dx + £esaijkh(~-~h + ~ - ~ ) ( + xi . 3
~w~ + ~--~) dx - f p°(divx wO + divx wr + divyw I) dx + ~ef
+ f
bs
aw °
I
ijkh 8t (~-~h + ~-~h) (~-~' + ~ ) ~es 3
dy +
~u{ ~w{
+f
~cf
r ~ idx = f fi(w~ + wi)dx~ - - ~t ~Yj~ oyj
2.2. Relative velocity
(2.5) odic
The relative motion of the fluid may be obtained if we take in divy w = 0, ~ Y-periw o = wl = 0, wr = @(x)w(x/e), 0 e ~(~), and
zero
corresponding
on
Ys"
pressure
To
this
end,
term in (2.5)
f (gradxP° + gradypl)wrdx.
it into
is
also
useful
to
modify
the
37 Then
(2.5) gives: 82u°
0f
+ u r)
( ist 2
~oi 8 dx + ~f (gradx pO + grady pl) m 8 dx + er
~ef
8 u r. 8to.
8
1 ----! 8 dx = f fi ~°i @ dx
~ef and for ~
÷ 0 we have the local problem for the relative velocity:
If we define the space Vy = {u; , e HI(yf), UlF = 0, divy u = 0, Y-periodic } and Hy, £he completion of Vy for the norm associated with the scalar product (u'w) Hy = ~yf u i w i dy we
obtain
the
evolution
problem:
find v r,
function
of
t, with
values
in Vy, such that: vo
,~vr " P~-'~°)H
~D° - p ~ ) = (fi - 8x i f
+ ~(vr'~)V
(2.7)
y
y
f ~0 dy i Yf
Vw
~Vy.
v r (0) = 0 If
we
introduce
the
vectors
0 i (i = i, 2, 3),
elements
of
H
Y
defined by (2.8)
and the selfadjoint
operator
theorem with the form
(2.9)
Vw
fyf~i dy = (@i'W)Hy
e H
Y
A 1 of Hy associated
by the representation
(v,w) v , (2.7) becomes: Y
~V r vr ~P ° _ p i P 8"t--+ ~ A1 = (fi - ~x. ~-t--)# l vr(0)
= 0.
The solution of (2.9), by standard semigroup theory,
(2.10) v~r(t) = p-i ~ e 0 Taking
the
relative velocity:
-P-I~AI (t-s) 0 i
mean
value
of
8 v° (fi - ~x-~ - p ~ ! ) ( s ) i (2.10)
we
have
is:
ds.
the
macroscopic
38 t
~r
(2.11)
vk(t)
=
(2.12) gki(~)
~
av °
gki(t
-
s)
~PO _ p ~ !
(fi
= P-I(e-P-lgAI~
- ~x i
) (s) ds
0i,0k)H Y
REMARK
2.1.
(2.10)
gives vr(t)
as a functional
of exterior
body
forces, gradient presure and inertia term. The mean value (2.11) contains a well-defined function of ~, gki(~) which decreases exponentially as ~ -+ ~, and gki = gik" The proof is similar as in the case of acoustics in porous media [2]. 2.3. Stress tensor
In (2.5)
order
to
w O = w r = 0,
same way,
study
the
local
w I = e(x)~ (x,y),
the asymptotic
s o
+ I bs Ys ijkh Note
in
the
8 • D(~),
w
solid,
we
Y-periodic.
take In
in the
process e -+ 0, gives us:
f s - - dy + ysaijkh ~ ~yj
f (ai]kh s_ bs + ijkh --~) ~ y (2.13)
state
~0
i
~
dy +
~.
~Uk dy + pO I 6i j _ _ dy = 0. ~t ~ ~Yj Ys ~YJ
that p°(x,t)
is defined
in ~
(does not depend
on y).
In fact we
continue pe in the solid part, with the periodicity condition, and we use that ~ d i v y m dy = 0. If we introduce the space V of functions y Y from HI(y s) with zero mean value and the scalar product
(2.14)
8u k ~v i - - dy (u,V)~y = fYS b jkh ~Yh ~Yj
and A 2 • L(Vy,Vy),
m kh e Vy,
n kh e Vy,
(2.15)
(A2uI,(~) = f a s ~Ukl ~°~i dy Ys ijkh Y~h ~Y--~
(2.16)
(mkh,t~ _ = f a s .....i.. dy )Vy YS ijkh 3yj
(2.17)
8w. (nkh,~ ~ = f b s ....1 dy )Vy Ys ljkh ~yj
(2.18)
(Y
dy '~ )Vy~ = fYS 6iJ 8 yj
the relation
(2.13)
is equivalent
to:
cV
Y
by:
39
(2.19) (ult+ A2ul + kh U +.kh t
huk° +
pOT, 0J)Vy : 0
V~
~ %y-
Thus the first factor in (2.19) must be zero.
(2.20) ~
U1
+ A2U
1
_mkh 8u~ kh 8 = ~ - n ~t
o 8Uk ~x h
pO~
ul(o) = O. The solution of (2.20) is: (2.21) U 1 = -n kh ~u~ + ~o e-A2(t-s) (rkh ~U~Sxh _
p O ~ ) (s) ds
(2.22) r kh = A2nkh - m kh ~Uko kh a u k _ pO T + r 8t ~x h 8x h o t _A2(t_s) (rkh ~u k pO~) (s) ds. f A2e ~ xh o
8u I kh (2.23) ~ = -n
-
2.2. The right hand side of the equation defined as function of u O and pO.
(2.23)
is well
The macroscopic stress tensor is defined as the mean value of
(2.24) ooij
s
= (aijkh +
bsbijkh~ta
) (~-~h + 8-~h)"
If we introduce the coefficients and the functions: 8 (rkh)e]~ (2.25) o ijkh = [aijkh - aijep ~yp 8 (nkh)e + bije p ~YP 1 (2.26) eijkh = [bijkh - bijep ~yp (nkh) e]~ 2 8~k]~ (2.27) ~ij = [bijkh 8Yh
(2.28) gijkh(~)
*
= [aijep ~yp(e
-A2~
8
(2.29) gij(~ ) = [bijep 8yp(A 2 e
r
kh)
e
_
~
bijep 8yp(A 2 e
-A2~
rkh)e]~
-A2~
~)e - aikep ~ p ( e -A2~
we have (2.30) 5 O o el ~u° ~ 2 pO + lj = c~ijkh ekh (uO) + ijkh ekh(~t--) - 13
~)e ]~
40 t + fgijkh(t
t - s) ekh(U °) (s) ds + ~ g i j ( t - s) p°(s)
O
REMARK elastic
term
pressure
2.3.
The
s° • a
viscoelastic
term
~2,
and
strain
and
decays
exponentially
law
pressure.
(2.30)
Now
two
terms
Because for
Balance
it
is
of
contains
instantaneous
with
long
memory
is
~ -~ ~
a positive
(also
than
for
g,
memory
g*,
defined
functions
operator,
g*(~)) . The
strain
easy
to o b t a i n
the
balance
for s
of
momentum.
÷ 0 we have
For
(2.33)
~
we
V w O e HI(~) o
Dw?
DoT -'-]----+-Lw° dx ~ij)wio dx = - a Dxj i
- n p0 6..
z3
z3
~2U9
~V{
DO?.
+ pf ~,~- DxZ3= fi" Dt~
3
REMARK [5,6].
this
DXjl dx - ~I n pO div w o dx =
= 59.
z3
stress
:
fi w? dx
oT.
of
g(~)
(1.2).
:f
(2.32)
an
~i , a
momentum
D ~O : -Ia ~ j j (Oij - n p O
In
2.4.
the
same
appears
3.
oT is the z3 time (2.32)
effective proves
the p o r e p r e s s u r e
(or
that
multiplied
total)
in
the
stress
tensor
effective
stress
by the p o r o s i t y .
CONCLUSION
The
macroscopic
described
by
the
fluid
velocity
~r.
of mass,
(2.11),
These
stress
The in
particular
homogenized)
displacement
in
effective
(or
the
pressure
as
(2.30)
with
D 2 ( u ~ + u~) ~w9 ~o 1 l z w oi dx + f oij ~--~ dx - n I pO d i v w O dx = Dt2 ~ 3
i3
same
equation
term
I P
f 50
tensor
A2
(2.5) w = w 1 = 0. Then,
in
(2.31)
constitutive
is very d i f f e r e n t
2.4.
take
ds.
O
tensor
coefficients the
case
p°(x•t)
quantites
Darcy's
of
vector
law,
and
satisfy and
motion
of
the
solid
in the
mean
the
equations
the
value
mixture
u°(x,t), of (2.3)
(2.33),conservation
the
case of an e l a s t i c
skeleton
in our
viscoelasticity results
reduce
be
pore
relative
conservation
of m o m e n t u m •
b e i n g d e f i n e d by (2.32) and (2.30). o a1 U i j k h and ijkh and the f u n c t i o n g i j k h homogenization
may
the
[2].
the
are
the
In
the
to those
41
obtained was
in
proved
consequence
[2], but the c o n v e r s a t i o n of mass that of
a
Darcy's
law
of
the c o m p r e s s i b i l i t y
for incompressible
the of
the
is different.
from
(2.11)
fluid,
but
is
In fact not
appears
only
it a
as valid
fluids.
REFERENCES io
A. B e n s o u s s a n , J. L. Lions and G. P a p a n i c o l a o u , As>~ptotic Analysis for Periodic Structure, N o r t h - H o l l a n d , A m s t e r d a m (1978).
2.
E. Sanchez - Palencia, Topics Vibration Theory, Lecture Notes Berlin (1980).
3.
Th. Levy,
4.
H. I. Ene et E. Sanchez - Palancia, (1975).
5.
M. A. Blot,
Indiana Univ. Math.
6.
M. A. Biot,
J.Geoph. Res. 78(23), p.4924
Intern.
J. Enging.
in Non-Homogeneous Media and in Physics 127, S p r i n g e r - V e r l a g
Sci. 17, p . 1 0 0 5 - 1 0 1 4 Jour. Mecan.
J. 21(7),
p.597 (1973).
(1979). 14,
(1972).
p.73-108
A STATICAL MICROMECHANICAL DESCRIPTION OF YIELDING IN COHESIONLESS SOIL B, CAMBOU ~
1.
Introduction
At present, the description of the behavior of cohesionless soil is not s u f f i c i e n t l y r e a l i s t i c and accurate in complex loadings, especially with r e o r i e n t a t i o n of principal stress axes. One reason of this limited success seems to be the lack of micromechanical analyses. The purpose of this work is to propose a micro s t a t i c a l analysis of a simple granular material to provide a better understanding of the microphenomena and of the fundamental variables which lead to irrecoverable s t r a i n s . P a r t i c u l a r i n t e r e s t is taken in loadings with r e o r i e n t a t i o n of principal axes. This study is limited to a bidimensional granular material made of c y l i n d r i c a l particles in an i r r e g u l a r array. 2.
Microstructural
analysis
In a granular medium, the fundamental mechanism of irrecoverable strains is the s l i d i n g between p a r t i c l e s . The following condition can be w r i t t e n for each contact point ( F i g . I ) :
IT~I
(1)
..< p. U ~
is the f r i c t i o n
c o e f f i c i e n t of the material of p a r t i c l e s .
Relation ( i ) between the components of contact forces leads us to the conclusion that a s t a t i c a l analysis seems to be the more appropriate to describe irrecoverable s t r a i n s . The macro s t a t i c a l variable is the stress tensor ((Z~K~) and~the micros t a t i c a l variables are the components of contact forces ( F o< ~ ). The~variables are described by the d i s t r i b u t i o n s ble o r i e n t a t i o n ( ~ ) (Fig.2). P(~) range
defined for each possi-
is the p r o b a b i l i t y of occurence of a contact o r i e n t a t i o n in the ~ ~3 (Fig.l).
F~'(~) are the components of forces applied on contact points defined by t h e i r orientations n . In a volume V the number of these con~acts is : i t is assumed to be s u f f i c i e n t l y large.
J~
P(~
j r is the number of contact points per u n i t area.
Professor
Ecole Centrale de Lyon
F-69410 EculIy
FRANCE
~
44
N
Fig. 1 : D e f i n i t i o n
"'"
/"
of forces-~ k
at contact point k
I
:,t
'
Fig. 2 : D e f i n i t i o n
I,,I
2P(n)s~(~)
of the d i s t r i b u t i o n
'
of P(~)F~ for a given o r i e n t a t i o n
45 These d i s t r i b u t i o n s are defined by t h e i r s t a t i s t i c moments. The accuracy of the description increases with the number of moments taken into account. A simplified description based on the f i r s t moment (mean values) is f i r s t proposed. 3.
Statistical
f i r s t - o r d e r description.
(Yielding mechanism n°l) The microstatical variables are described by the mean values of the d i s t r i b u t i o n s (Fig.2) ~ ( ~ ) ~(mi. 3.1.
Relations between micro and macro statical variables.
Weber {16}
, Cristoffersen and al {7} have demonstrated that :
Summation or s u b s c r i p t (Fig.i).
~ is extended to all c o n t a c t s in V volume
For c y l i n d r i c a l p a r t i c l e s with an average diameter ~ , assuming t h a t /3 and ~ are independent v a r i a b l e s , r e l a t i o n (2) can be w r i t t e n as :
This relation points out the product P ( R ) as the microstatical variable.
~(~)
which has been chosen
In previous works { i } {2} {3} {4} {5}, i t has been shown that this variable can be expressed in principal axes by :
(4)
?(~')Fa(~) = J~5 CI-a ~a (<5
Capital greek subscript A means that this relation is only valid in the principal axes.( and without summation on subscript ~ ) , ~ _ A In expression (4) CT'A defines the actual loading and #fD~a the i n t e r nal state of the material. From relation (3) and (4) following conditions :
i t is easy to show that
#A must s a t i s f y the V
n
I t can be note~ that easily from ~a(~).
~,(~) U~
expressed in any axis system can be defined
The value of ~a( ~)^t^~ or an isotropic material subjected to an isotropic loading is noted B~). I t can be shown {1} that
:
~(~
~
,
nA
The internal state defined by ~ o ) is named "the v i r g i n state". V
46 3.2.
Discretization of the internal state.
~a(R~) lar material.
d~fined by (4) characterizes the internal state of the granu-
In agreement with Leckie and Onat {8} . we have proposed {3} to express this function by the following expansion : (6)
~(~)
=
~<~
n~ + ~ff~
~# n~ n&
This expansion must s a t i s f y relations (5). For a bidimensional material i t has been shown {4} {5} {6} expressed from only four independent variables, noted A<~ :
t h a t ' < c a n be U
(7) U Variables / ~ # characterize the internal state of the material. I t can be easily shown that for a loading without reorientation of principal axes : A,~ = A ~ = 0 , the internal state is then only defined by : ~ , , and
The global stress tensor applied is decomposed as follows : : part of(~'due to normal components of the average contact forces. (~- (t)
: part ofGTdue to tangential components of the average contact forces. To define CT~~) the average contact forces are expressed from relations (4) and (7) _ -
The normal component ( ~ ) allows to define ~ ) cipal axes system of
(8)
i s t h e n calculated. Finally relation (3) from N~(~) , which can be written in the prin~# as :
60%J ~ ) = c r = ( s
÷ X,~]
+ o-, ( I _ X.~I
For a loading without reorientation of principal axes ~ = ~4 so the principal axes of ~ j ~ ) and of ( ~ are i d e n t i c a l . In this case
:
with
X,,,, =
X,t
<
+ x,.,_")
.-- (,112.] ( X,~ - X,.,')
= 0 ,
47 For a bidimensional material the mobilized angle of internal f r i c t i o n can be measured by ~ : -
"-
_
4"
+
OT/:}-
o-: ÷ CF~ Then B¢~)can be written as
:
From (9) i t can be noted that
Thus /~d characterizes the better yielding mechanism for any value of ~ . So we assume that ~ does not change from i t s i n i t i a l value ( A , ~ = 0) and thus A a remains the only internal state variable (for irrotational loadings). Then relation (9) becomes (I0)
~)
:
"I ~ + .~_k =Z
This relation shows that the evolution of ~3 is related to chanqes in the structure in such a way that deviatoric stresses would be borne in a greater part by the normal components of contact forces which are stab~ilizing elements on every contact point. This conclusion is in agreement with the results proposed by Thornton and Barnes {15} (Fig.3). For loadings with reorientations of principal axes, parameters /~zand ~ are not equal to zero. Cl-(-) Relation (8) shows that -~z ~ 0 so axes I and 2 are not principal for tensor (IFCn) . The evolution of these parameters can be considered in two different cases : a) When a reorientation of principal axes occurs. b) When ~4zand XzI are different from zero for a loading increment without reorientation of principal axes. Case a) can be analyzed as a change of axis system for parameters A~"
o
In a previous work {6}, i t has been shown that the rule of axis change can be expressed by :
(11)
B X~II
l
=
C
-DAB
D -O -~5 /~
I t is easy to demonstrate that
~
X.,
: /~-I implies that /~'4~: ~kLi
This relation suitable for the "virgin s t ~ e " , w i l l remain true for any change of axis system related to reorientation of principal axes.
48 ~r s o
S
n
From Thornton and BarnesIl41
t
i s not s p e c i f i e d
0.4
0.2 Fig. 3 : Contribution o f normal and 0.2
tangential components o f
0.4
contact forces to the mobilised i n t e r n a l angle o f friction
.
-F Sn
t
o S
0.4
~
:
20-
Our a n a l y s i s with Nd= 0.3
0.2
/~.= 0
S
0.2
0.4
PS1 , PS2 : Principal directions of the granular structure
Kf
PLI , PL2 : Principal directions of the loading
Pole of the
~
L
Fig. 4 : Representation of the internal state (z~=0.15 ,~a=O.30 ) by a circle (isotropic IoadingO-= I )
49 In case b) (lZ)
(T _ ~-(m) can be expressed by =
-
[
+
: +
+
- x
O]
(3-~(~) is a measure of the difference of the principal o r i e n t a t i o n s of (n) (3-~. and £T~ . I t is assumed that the evolution of the internal state I~ . I~ occurs in order to minimise this difference. Relation (12) shows that ( / ~ + /~z~ ) is the most representative parameter of the evolution of CF4m f~l . Thus i t is assumed that ( A~m - A~I ) = O. So i t can be w r i t t e n :
So I t can be considered that for every case, parameter A,~characterizes the evolution of o r i e n t a t i o n of principal axes. Thus expansion (7) can be w r i t t e n as : (13) The angle o~ between the two principal can be calculated :
axis systems of
~
and
(IF (~)
A~ Taking into account the only two parameters z~4 and / ~ , the rule of changing axis system ( I I ) becomes simpler and can be w r i t t e n : (14)
~d
~2.~
-/a4~2~
I
I~4
~) : angle between the two axis systems. 3.4.
Representation of_ t h e material_ intern_al_state=
For a given loading, internal state variables >xdand >x~. define the d i s t r i b u t i o n of vectors P ( ~ ) Fw(~). A convenient representation of t h i s d i s t r i b u t i o n can be provided in axes ~n'~, "~ ( F i g . l ) Normal and tangential components can be defined by :
In these axes the evolution of the internal state can be defined by the curve described by the end of vectSrs P ( ~ ' ) ~ (~). For a v i r g i n s t a t e , i t is easy to demonstrate that this curve is a c i r c l e (Fig.5a). A f t e r y i e l d i n g without r e o r i e n t a t i o n of principal axes, t h i s curve is no longer a c i r c l e but remains symmetrical about axis ~ (Fig.5a).
50
K•
: 3.86
q
: 1
2
•
.........
'-:~...2
3 ...."
4
./'~
K~
/~= 0
/~+= 0
(circle)
/~a = 0.3
~-- 0
(curve symmetrical about
/~a :
/~+=
0
~a = 0.3
KN )
0.15
~+=.0.15
L1 is the representative point of the mean contact force in the
K~
major principal
direction of the loading
'/ / 0.5
+
I
~
' I
-C& -_~
~..... "~ °+
"
%
\':.._
+
l
:
:
+T"
~/
.%/"
Fig. 5 : Representation of the state of the material by the curve described by the end of vectors P ( ~ ) - ~ a) For the actual loading b) for an i s o t r o p i c
loading ((3"= 1 )
51 After yielding with reorientation of principal axes, this curve is no longer symmetrical about axis ~ (Fig.5a). In fact, this curve gives two informations one about internal state, the other about applied loading. To define a convenient representation of the only internal state, an isotropic loading ( 0 - = 1) is considered. In this case
:
(15)
I t can be easily demonstrated that the end of vectors -P"(#) F ( ~ ) a c i r c l e with a radius equal to V ~
÷~
describes
(Fig.4).
The rule of changing axes (14) allows us to make sure that this quantity is independent of the axis system that is used. To define the internal state completely, i t is necessary to know one point of the c i r c l e corresponding to one contact orientation. For this purpose, i t seems convenient to define pole P of this representation (Fig.4). The distributions of mean values of contact forces under isotropic loading is then perfectly defined by this representation (circle and pole P). The principal directions of the structure can be defined : Contact directions for which mean values of contact forces are normal to the contact plane, for an isotropic loading. Two principal orthogonal directions can Be defined. I t is easy to drawn them on the geometrical representation (Fig.4). In this representation, i t is possible to define the major principal direction from equation (15) by angle ~ (point L1 on Fig.4) : :
_
(
I t can be noted that i f L1 does not coincide with S1 (Fig.4) the principal directions of the structure d i f f e r from the principal directions of the loading. We can see the representations of various internal states on Fig.5b. At f i r s t glance, this representation seems very similar to the Mohr representation of a stress tensor. In fact, i t is quite d i f f e r e n t , i n particular the direction of evolutidn of the representative point on the c i r c l e is opposite. This representation cannot be expressed from a second rank tensor.
Then t h e i n t ~ s t a t e of a granular m a t e r i a l can be defined by t h e two parameters A d and ~ w i t h t h e r ~ e of changing axes ~14), t h e g e o m e t ~ c a l r e p r e s e n t a t i o n of t h i s s t a t e i s a ~b~cZe i n axes ~ , ~ . The p r i n c i p a l d ~ e c t i o ~ of t h e s t r u c ~ e can be defined. I t can be noted that for a loading without reorientations of principal axes, the principal axes of tensor QF and of the structure are identical, for a loading with reorientation of principal axes, i t is not the case. 3.5. S1i d~ng_c r i t e r~onc Relation (1) gives the sliding condition at contact point k. By su~ation on a l l contacts defined for a given orientation f3 , mean values of contact forces must satisfy a similar condition : (16)
I V- )I
N
52
c&
3
~.= 0
i o~4
4J )
_~
0;2 . /
0.'4
0.8
,Xd
0,4
J
~: 2 0
°
o&
-0"4
~ ~ ~o,~ J °:4 0.4
Fig.6
~.
values of /~
Representation of the sliding criterion for different values of >~ and >~m (yielding mechanism n° 1)
53 I t will be noticed that condition (16) is necessary to have an admissible state, but i t is not sufficient. In fact, i t is the only one which can be written because only mean values of contact forces are defined. To go further, i t would be necessary to consider a more accurate statistical description of the distributions.of contact forces. In the principal stress space, the sliding criterion (16) can be written :
I t will be noticed that this criterion depends on the frictiRn at contact point ( ~ ) but also on the internal state of the material (~{~ or A and AA ). For Ad = ~ = 0 criterion (17) is a Mohr-Coulomb criterion with an internal f r i c t i o n angle equal to 6 ( ~L : t a n ~ ) . Fig.6 shows criterion (17) depending on parameters Ad and ~
.
I t can be noticed that for A~= 0 (loading without reorientation of principal axes)the yielding mechanism is of a kinematic type. For A~ = 0 the yielding mechanism is of a softening isotropic type. In the general case, the yielding mechanism is quite complicated (kinematic + isotropic). In the principal stress space, criterion (17) is represented by straight lines (Fig.7) the orientations of which depend on parameters A . Within the framework of an elastoplastic analysis, these lines represent the plastic-potential functions depending on parameters A~ and A ~ . 3.6.
Flow rule .
To define the flow rule of this material, i t is expressed that the dissipation due to plastic strains is equal to the work of forces at sliding contact points.
"-*a~(~: sliding at contact ~ ( ~ & c]~Q~--v > 0 quantity
~ ~ (n)
)
is defined by :
From relation (15) expressed in the principal axis system, i t is easy to define d ~a : a~
= (-~/~')
a£,,
:
(18) From (16) i t has been shown in by :
(17)
- Cd~ z
with
~
-
2_
{5} that volumetric strain can be expressed
+ (J E_:~ =_ (Tr
2_ G )~d
54
J s
s
~
,,-values of z~ d
~ Fig.7
:
. OI~values
Representation of the s l i d i n g c r i t e r i o n
of / ~
(yielding mechanism n ° 1)
(~ = 20 ° t
2 Vo = 0.2455
2(l' VO~ V ) 0.8
0.6 !
0.4
Fig.8
:
Representation of the s l i d i n g c r i t e r i o n
(yielding mechanism n° 2)
55
is a quantity with a sign identical to that of
(CT~- (]'z)
For a monotically increasing loading path, A d has the same sign as (CT: - (3-=) , then the volumetric strains are necessarily extensive. With an additional simplifying assuption i t can be demonstrated that f u l f i l l i n g equations (17) and (18) leads to the stress-dilatancy relation defined by RBwe {11} . For loading with reorientation of principal axes the volumetric strains depend on another term including ~ :
In t h i s case, the s t r e s s - d i l a t a n c y 4.
relation
is no longer accurate.
Secondorder s t a t i s t i c a l description (Y!elding mechanism n° 2)
The distributions of random variables ( P (~')F'<~ ) are described more accurately in this paragraph by the mean values P ( ~ ' ) F~(~) and the standard deviations P ( ~ ) 6~(~" ). 4. i . Di_sc_retiz_atio_n_of_ t h e second o r d e r _moments _~d!~__~_). : The f o l l o w i n g r e l a t i o n ,
(19)
P
similar
=
to (4) is proposed
:
IfS)
without summation on s u b s c r i p t ~ . Function
~(~)
characterizes the second order material
I t would be possible to define an expansion o f ~ ' ~ s i m i l a r proposed for ~ .
state, to the one
To simplify the analysis we only consider the f i r s t order expansion :
At v i r g i n s t a t e , the material
is i s o t r o p i c
so •
Vo 4.2.
Deviatoric loadings.
Distributions of contact forces are assumed to be symmetrical and bounded by P ( ~ ) [ ~ _t 2_ S~] 4.2.1. Sliding criterion : The sliding criterion can be written
When a virgin granular material is subjected to an isotropic ~oading, i;crecoverable strains occur, so criterion (20) is satisfied with ~ =
v.=Vo
T hen Vo can be defined : 2_Vo = V-2 sin ( ~ I Z } .
56 A f t e r a numerical analysis of r e l a t i o n (20) i t has been concluded that the two parameters V:( have a very similar e f f e c t . So i t is assumed that :
V=V Then V is the only parameter of an i s o t r o p i c y i e l d i n g mechanism. C r i t e r i o n (20) can be w r i t t e n with V d = Vo ( F i g . 8 ) . 4.2.2. Flow rule
:
I t is assumed that only
(with sign
+ if
7-(~)
> 0
the second y i e l d i n g mechanism is actived.
and sign
is 7 - ( ~ ) <
0).
Similarly to 3.5. plastic strains can be defined, and particularly volumet r i c plastic strains : c
with sign
+ for
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.
)
\
.,~
/
t
!
~'I
/
(P = c o n s t . )
f o r c, C > 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 <
4(b-~a)3(4b+21a)+ga2[(5b-15.9a)2+80.19a 2] 162abA(14b-9a)
As
lim G2(q) > 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
<~(q + 3T~)
q e (0,q o]
we acquire the curves shown in fig. 6a, 6b.
4. M A T H E M A T I C A L
MODEL
FOR
THE
MECHANICAL
BEHAVIOUR
OF
SOILS
Let PI be the set of unloading processes PI : {(P,q)IP : p(T), where
q : q(T),
t iI
is the initial moment (for the first unloading
process
T e [t i,t~]], I
while t fI one shall
the
~ < 0},
final
consider
moment (p(t~),
of
the
q(t~))
=
= (ps,qs) viz. the stress state "in situ". Let P2 be the set of reloading processes PR={(P,q)
I P=P(T),
q=q(T), f i ¢ ~0, t1:t2}.
Tc[t2,t
], ¢(p,q)<@(p(t I
Let P3 be the set of loading processes P3={(p,q)
I p:p(T), •
q=q(T), i
Tg[t~,t
], ~(p,q)_>¢(p(t I
f}.
¢_>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/,,<~ L
i
Figure 4
I Oa
_I
I
Finite element meshes and boundary conditions used in the analysis of shallow and deep layer problems
140
Creep
displacement
Cn 0
0
mm
0 0
0 0
0
0
i-,. ~Q
b-]
\
fD
\
\
fD rlfo
\
i~. ~.
\
fO 13"
\
\
fo
!.
fD ~m 0
\
\
\
\
0 Cr~ fb m
rf H ~) u) m
\
\
m o ~. m
\
~o~
\
o l-h
m
\
m i~.
1
o p~
\
I
l
rf i~ .
\ I
I
\
m
l
I O~
,
Lj ~
L~
I
U m o'~
n, rf
o
\
o
o
~
141
Creep displacement 0
0
mm
0
0
\-\
~D O~
CD 0
O~
\
0
0
\
0 CD
0
\
o
m 0
\
~t ~0 0 0 ~0
,\
I rt
0
~3 0
0 D~ o
m O~
C
\
~
PJ ct ;w~ i~Z 0
'I
0
~Y
I
i
0 ~h
to
I O~
O~
'
o, t~ ~t ~D
0
142
q--, Pc
0.700 San Francisco
],
Bay Mud
|1.44
I I
0.405
~
I !
i .~5 I
I
!
K euper Marl
I ! !
I
!
I
I
! 0.324
1 0.485
p! 1
Figure
7
Wet side y i e l d locus ellipsi San F r a n c i s c o Bay Mud
for Keuper Marl and
--
Pc '
143
Creep displacement 0 0
0
0
0 0
0
0 0
0
Qo
c~
o o o
0
0 0 0 o o o 0 Z
0
0
P.~m
~
.
0 0
ii t
N
......
i
Pi
II
0~
N P~ 0 0
g~ cl f~ 0 0 0
f't N
o c~ o
cl N m m
I
L<
0
I d 0 0
........
L
i
144
Creep
0
0
displacement
am
0
0
0
0
p,. u~ I-~ 0 ~0
n,
O i=,. 0
m
0 II
m o
O~ m m M
e M
N II m N
U
'
0
0
D
5
k
I
n. m
0
II
II
145 x
Y
i
~--~.~/
Strip load
45 Elements 160 Nodes
rd o
//,<<',, L
lOa
!
Figure 10
Finite element mesh and boundary conditions for the analysis of a strip load on a semi-infinite layer
J l
146
centreline
0
0
displacement/a
0
0
ua M C~ I t.u
I~- (1) m 0 (~
0
0
0
I IxJ
0
0
1
m
0
0
p-
~,-
fO m
C/ 0
0 0
o
Q M
0
L
c~ r,
~
ct"
0
0
0 m
f0 c~
c~ o t~
m ct M
E
/ H-
~h m
0
0 M
/ / 0
0
147
excess
pore
foundation
pressure stress
u
f~
0 I
0 m c~ m
0
p~ 0 m
o
I
m
'
t-,.
1
~- 0 i'~
01 o~
o
~- 0
0
;g
g
o (3 o
0 I
o t-~ (~
fO N
N
Q 0
Q 0
m cf
m ~
0
0
I
o o
(9
o~ N N
~
//
ct
m
/
m 0 ff 0 Q
0
II
o
N~
II
148
Vertical
movement/a
o
0
0
L,
L,
L
! c)
0
,,i
*
oa
#
m
cn o m o m
i S
0
m m o
o
w-
q m q 0
o
o
o
I
m
m m
I
Cl N f~
o~o
o
~J
"3
m
m fD m 0 N.I
0
q 0
m 0
0
rt"
m ri'~
0
0
f
i' I
m N i~. re
o
fl
149
Horizontal movement/a inward -0.04
-0.02
outward 0
0.04
0.02
\
X
--
=
-
ol_~dati~ ~h
i
a
J
)
i c'ee~ End of ramp loadinc T = 0.00572 non Linear consolid~ tion T = 5.72 non Linear consolid~ tion wit~ creep T = 5.7"
/
...... . /
/
/
//!
10 Figure 14
Horizontal movement below the edge of a strip load for nonlinear consolidation and creep analyses
A CLOSED-FORM
SOLUTION
OF A V I S C O P L A S T I C APPLICATION
FOR THE P R O B L E M HOLLOW
TO U N D E R G R O U N D
SPHERE
CAVITIES
IN ROCK SALT
P. BEREST LABORATOIRE
DE M E C A N I Q U E
ECOLE 91128
DES SOLIDES
POLYTECHNIQUE
Palaiseau
Cedex
- France
Su,m,m,ary : This paper closed
is d i v i d e d
form s o l u t i o n
in two parts.
for the p r o b l e m
viscoplastic
medium,
submitted
second part,
some case
In the first p a r t we give a
of a
to a v a r y i n g
hollow
sphere
internal
in a elasto-
pressure.
stories
and actual datas
are d i s c u s s e d
allow to try to b a c k - c a l c u l a t e
the c o n s t i t u t i v e
equation
In the ; they
of the rock
mass.
Ac,k,,nowledgment
:
Some results of the authQr lides.
of the p r e s e n t
and N g u y e n
paper
Minh Duc,
are based upon
Laboratoire
a former
de M ~ c a n i q u e
study
des So-
152
F I R S T PART BEHAVIOUR OF VISCOPLASTIC .
.
.
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.
.
.
.
.
.
.
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.
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.
.
.
.
.
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.
.
.
THICK WALLED SPHERES
.
.
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.
.
.
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.
.
INTRODUCTION The quasi and c y l i n d e r s mechanics. spherical
static
Indeed, voids
Tijani
of the applied tion,
many cavities
has been
(1978),
pressure.
considers
this
presented
the s o l u t i o n
submitted
to a v a r i a b l e
sphere will
solution
be given
We c o n s i d e r its
mechanical
the e x t e r n a l Initially,
are equal
rock
considered
as
(1963),
Aufaure
case of a m o n o t o n o u s (1960)
viscoplastic
variation
established
behaviour
B~rest
for a s p h e r i c a l internal
; but O t t o s e n sphere,
and N g u y e n M.D.
cavity
in an i n f i n i t e
pressure. A s o l u t i o n
a solu-
(1983) medium,
for a t h i c k w a l l e d
in this paper. sphere,
properties. radius
homogeneous
The internal
and i s o t r o p i c
radius
as regards
of the sphere
is i,
is p > i.
the m e d i u m
is in a natural
state
(strains
and stresses
to zero).
The m e d i u m plastic
spheres
including
for a v i s c o e l a s t i c - v i s c o p l a s t i c
as erroneous.
a hollow
and
by W i e r z b i c k i
Madejski
elastic
a solution
of t h i c k w a l l e d areas,
can be a p p r o x i m a t i v e l y
for the special
internal
who gives
in m a n y
medium.
considered
adopting a simplified
(1985),
behaviour
importance
in an i n f i n i t e
This p r o b l e m (1975),
time d e p e n d e n t
is of great
exhibits
behaviour,
with
an i n s t a n t a n e o u s a Tresca's
elastic
viscoplastic
behaviour criterion
and a v i s c o and a s s o c i a -
ted flow rule. The sphere (or)
a pressure
haviour ternal
is loaded applied
by a p r e s s u r e outside
of the sphere only depends pressure
and the e x t e r n a l
logy of the medium. that the e x t e r n a l
Only the q u a s i - s t a t i c considered.
inside
However
the c a v i t y
between
rheo-
loss of g e n e r a l i t y ,
is zero. problem
and small d i s p l a c e m e n t s
be-
the in-
due to the c o n s i d e r e d
without
and
the m e c h a n i c a l
u p o n the d i f f e r e n c e
pressure,
We can then assume,
pressure
applied
the cavity.
will be
153
Under
these
as f o l l o w s
hypothesis,
equations
can be written
:
E -~ =
With
the constitutive
(i + ~ )
~-~
F(o)
- ~i-~2-2C
21
1 ~
=
F(o)
the n o t a t i o n s r,8,~
:
tr
if
~ ]-~
F(o)
II+E~
~> 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 <
r < x
zone
shows
the
x <
~
0
p,
where
BsvP ~t
cases
(fig.
i).
r <
eight
~evP --~
, where
-
0
pA 4-
ar
÷
e ++//J ÷
l/lip, ~ T i h A ++
1
I.lu,
A l
rr
B
4-
RESIDUAL
C[D1H
B
'111//1
* ÷
F
] G
E
I
~ t
Figure
1
elastic
: Evolution
A.
Fully
B.
Viscoplastic
C.
Viscoplastic
zone
elastic
e ~
1 ~ r <
Residual
E.
Fully
F.
Viscoplastic
G.
Fully
H.
Viscoplastic
zone,
elastic
e ~
• General
a-
can
be
being
considered
3u ~ +
r <
x,
residual
zone zone
x <
r <
p
x ~
r <
e,
(x < 0) elastic
zone
e ~
r ~
0
r ~ x,
residual
zone
x ~
r ~
(~ < o)
p,
1 ~
r ~
p,
u 2 ?
of )
residual
zone
x <
r <
e,
(x > 0).
o
=
and
of
whole
evp
(i - 2 ~)
with
(i - 2 9)
the
r ~ x,
:
a variable for
elastic
e,
1 <
integrated
-
x,
sphere
elimination (
boarder.
sphere
relationships
the
viscoplastic
r <
p,
r <
zone
residual
E u A(t)
1 ~
viscoplastic
E which
zone
1 ~
D.
zone
the
sphere zone
zone
of
or
(
respect
3 = - ~
to
as
r
(2')
(3)
leads
to
can
be
:
A(t) r3
The for
(i')
~°r~ + r -~- /
3 Or
(i - ~)
integration. sphere,
between
u
same and
function or must
A(t) be
continuous.
156
b- T h e
elimination
between
(2')
and
(3)
leads
to
:
~a __~_ r 1 ( i - 2 ~) o r = i ~ r --~- + ~ E v p
E ~r -
c- T h e n
of o
: ~a
r A(t) r --~- + 3 ~ Note
that
(4)
is t r u e
E e vp = 0 + i _ v
everywhere
(4).
in the m e d i u m
; the
flow
law has
not
y e t b e e n used. d- T h e nated
flow
law can be w r i t t e n
as f o l l o w s ,
once
o
has
been
elimi-
: . In a v i s c o p l a s t i c ~e vp ~---~
zone
1 < r ~°r n 2 ~r
In an e l a s t i c
:
2 ~ C >
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~
1 x3
dx = A(t)
(P)
C L ° g P3 } ]tc(P )
SPHERE
i'--B -"1
Figure
4 ~(t)
+
(4) a nd
sphere
,t
(B) .
(5') y i e l d s
:
1) p3
Moreover, ~vp -~t
[ x(t),t
=
4 ~ C x3 3
t h e n A(t)
On the o t h e r h a n d
: ~
[ x(t),t
0 = e ~i + ~i + 4 e 3 C
This case ends,
.... or -~(~r ~ - [ x ( t ) ' t ]= 4..~Cx
] =0,
1-
1
]=
:%
x3
x3 + 6
4 ~ C
) A
then
1 +Log
:
3 x 3 - x_~
.
w h e n r = p or w h e n x = 0. N.B. -x "3 m e a n s d x 3. dt
V , VISCOPLASTIC/RESIDUAL
SPHERE,
WITH DECREASING
PLASTIC
RADIUS
(C or F)
+
Pl
/ / / I /Lf'./ ., tc(~)
Figure
6 : Viscoplastic/residual r a d i u s (C or F).
In the r e s i d u a l to time,
I
I bcH
,
td(~)
zone,
then integrated
r < x(t), with
sphere,
with decreasing
(4) can be d e r i v a t e d
respect
to r b e t w e e n
plastic
with respect
r = x and r = p :
160
-
,
~t
[ x(t),t
] = i(t)
{ x3
Then (7) can be rewritten in the following way : ~ ~i + ~i = ( i - ~
) A(t) -~ 4 ~ C xL°g 3x3 + ~
This latter relation can be derivated with respect to time, as for ~~ [ x(t) ,t ] is known (see above) and -~~ r [ x(t) ,t] - 4 x~ C " Then ~i + ~'i =
(
1-7
1 ) ~.(t) + ~ ~(t) ( 1 x3
1 ) p3
(8).
In this relation remain two unknown quantities, namely A(t) and x(t). In order to eliminate x(t), we integrate (6) as in III: td(~) I i(t) exp (et) dt = 0 tc (~) Or, by derivating with respect to ~3 : .~(t)
exp(c~t)
td
= 0
x3(t) tc(~) Then (8) can be multiplied by exp(et) and integrated with respect to time between tc(~) and td(~) : [
1 ]td(~) exp(~t) { ~i(t)- A(t) ( l-~)}Jtc(~ ) = td (~) = ~ ;
i(t) {---!---ie x3(t)p ( ~ t )ix}
dt
te(~)
Let x M be the maximum value of x between tc(~) and td(~). Then : td(~) [ A(t) { - - i - i } exp(et) dt = x 3 (t) Jt c (~) 3 XM t (X) ! / 1 )[A(t) exp(~t) ] i dx 3 = 0 = ~-i x3(t) t (x)
161
Then the set of equations : [
]td[ i(t) exp(~t) x3(t)
{ =0
exp(~t)
( ) $i(t)-
}Itd
1-7
t
~(t)
t
= 0
c c
allows for calculating A(t d) and x(td) , provided A(t c) and x(t c) have been previously calculated. VI - VISCOPLASTIC/RESIDUAL SPHERE, WITH INCREASING PLASTIC RADIUS (H)
+
~÷+ +,+-+:-yr/r/,
iii
s~+~:+++, ++,fill
ii
...... '~÷ :
tc%"~)
tdCl )
11[IIII17111117111,II
tll C~")
Figure 7 : Viscoplastic/residual plastic radius (H)
i lti,,, t¢Cl )
,i
sphere, with increasing
As for gvp(~,tc(~)) = gvp(~,td(~)) , (6) yields : A [ tc(~) ] + ~ [
tc(~) ] = A [ td(~) ] + ~ ~ w [
td(~) ]
or, by derivating this relation with respect to ~ : [ tc(~) ] x'3[tc (~) ]
A [ td(~) ] x'3[td (~) ]
-
4c{ 3
~ [ tc(~)
] -~
[ td(~)
]
}
"
Now, (8) can be integrated with respect to time between td(~) and t (~) : c tc t c p3'itd Jt d
x a(t)
Let x m be the minimum value of x between td(~) and tc(~). Then : x3 td (~) ---L--I- 1 ] dt = 1 _ 1 A(t) dx3= 3 (t) x 3 (t) tc (~) jt c (~) x ~3 td (~) _ 4C [ ~ ] { Log x 3 _ x 3 3 tc(~ )
x~ } ~3
162 Or : [
~i+~i-A
(i
~)
]tc--~ 4C td
~ t c (~)
{ Log x3 - - x- +3 p3
1 }
Then the set of equations [ _A(t)
4C ~(t) Itd- 0 [~
x 3 (t) allows
3
i
t c
1
p 3 /Jtj c
for calculating i(t c) and X(tc) , provided i(t d) and x(t d) have
been p r e v i o u s l y
calculated.
• Case o f a c o n s t a n t l o a d
We consider oi(t)
:
the particular = ~
case of a load of the following
form
:
F Y(t)
Where F is a constant,
larger than
(i - i/p 3) and Y(t)
is the Heaviside
function. has the same sign as F ; without assumed.
Then
loss of generality,
F > 0 can be
Then the radius of the v i s c o p l a s t i c
zone is given by :
p3
p3
dt
+ ~
i + Log x 3
: • At the initial instant, e 3 (t = 0) -
03
p3 _ i
x 3 "jumps" to the initial value
F
• As t tends to infinity,
x 3 tends a s y m p t o t i c a l l y
to the zero
value of : F = 1 + Log x 3 If x 3 happens Then
x3 p3
to be equal to p3, unrestricted
plastic
flow occurs.
: • If F > p3 _ i, u.p.f, • If F > Log p3 , u.p.f,
occurs at the initial instant• occurs after a finite interval of time.
• If F < Log p3 , there is no u.p.f. The case p = ~ will be discussed
in the second part of the paper.
163
SECOND PART UNDERGROUND .
.
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.
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.
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.
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.
CAVITIES
.
.
.
.
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.
IN ROCKSALT .
.
.
.
.
.
.
.
.
.
.
.
INTRODUCTION For centuries, conventional kable
properties
rage
: rocksalt
to be w a s h e d
rocksalt
mining
which
are r e q u i r e d
is i m p e r v i o u s
out.
has been e x p l o i t e d
or by leaching.
and
The e x i s t e n c e
layers which,
pollution
by the stored products.
sands beds
of u n d e r g r o u n d
in return,
storage
for large
protect These
cavities
by
remarsto-
are easy and cheap
is due to i m p e r v i o u s
aquifer
advantages in
exhibits
scale u n d e r g r o u n d
large c a v i t i e s
of salt beds
overlying
as a raw material,
But this m a t e r i a l
layers have
from any
led to thou-
the w o r l d e x c a v a t e d
in salt
and salt domes. Some
typical
underground
• Lo a . d i n ~ p a r a m e t e r s Many factors cal b e h a v i o u r
cavities
have been d r a w n
on figure
8.
:
are
likely
to have a m a j o r
of such cavities,
for i n s t a n c e
influence
on the m e c h a n i -
:
- Shape of the cavities. - Composition - Vicinity
of the o v e r l y i n g
of others
But we will
consider
cavities,
stratas. if any.
more p r e c i s e l y
the i n f l u e n c e
of more q u a n t i -
tative p a r a m e t e r s -
Mean v a l u e and v a r i a t i o n s
- Depth -
of the i n t e r n a l
of the cavity.
Mechanical
properties
of rocksalt.
pressure.
164
GROUND
LEVEL
~100m
500m
1000 m
1500 m
V
'.J
v
Figure i.
Tersanne
5.
Huntorf
8.
Melville
ii.
(F) (FRG)
Hauterives
-
2. -
6.
8
: Salt
Etrez
(F)
Epe
(FRG)
(Can.)
-
9.
(F)
-
12.
Regina Salies
-
Caverns
3. -
Atwick
7.
(Can.) de
: (GB)
Eminence -
B~arn
I0.
-
4.
(USA Manosque
(F) .
Kiel
(FRG)
(F)
-
165
•
Internal
pressure
The
internal
:
pressure
dent o n the n a t u r e
- for liquid of l i q u e f i e d brine up to the surface, by an e q u i v a l e n t remain
filled
close
to
movement
is s t r o n g l y
tubing
of p r o d u c t s
depen-
is filled w i t h must
so that the cavity
be the stored q u a n t i t y
of pressure,
in MPa,
For natural
be b a l a n c e d
and the tubings
of h y d r o c a r b o n s .
due to losses of load d u r i n g so that the internal
The
injection
pressure
or
is q u i t e
H in meters)
gas,
only r e s t r i c t e d ; these
cavities
H
(i).
the internal
about
1500 m e t e r s
The two cases
products
oil)
The
can vary
storage
lead to select
and m i n i m a l
pressure deep,
8 MPa < Pi < 0.0166
(respectively
pressure
by safety rules.
two r e s t r a i n t s
(for tightness)
must
in a large remain
a relevant
the s e le c t e d
extent,
tight and sta-
maximal
(for stability).
pressure
In France,
rules were
for
:
H
(2).
(2) and
(3) can be summed up b r i e f l y
storage
the v o l u m e
(respectively
: in a gas
pressure)
of the
is kept constant.
Depth of t h e c a v i t y
:
Due to v i s c o p l a s t i c
properties
to think that in m a n y cases
to the s o c a l l e d
"geostatic
.... !3
P
~.
Instead mechanical
of r e c e n t
behaviour
,
,
• Mechanical p r o p e r t i e s
effects
have b r o u g h t
inconsistent We will
with
Many
about
equal
H
advances
authors
(3).
in the k n o w l e d g e
inadequate
of the
of l a b o r a t o r y
have n e g l e c t e d
of rocksalt,
behaviour
in the following
the actual
state of stresses,
interpretation
to be of u t m o s t
the actual
propose
to back c a l c u l a t e
salt,
reasonnable
in a salt dome or
:
in the b e h a v i o u r
from in situ o b s e r v a t i o n s sometimes
P = 0.023
considerable
of rock
it seems
:
of Rock S a l t
to be c o n t r o v e r s i a l .
the d e l a y e d
of an i s o t r o p i c
stress"
13
of rocksalt,
the state of stresses
salt bed is not v e r y d i f f e r e n t
appears
the c e n t r a l
of brine,
can be neglected,
P. = 0.012 1
ble
storage
:
:
(Pressure
-
products,
for any m o v e m e n t
up, w h a t e v e r
small v a r i a t i o n s withdrawal,
in an u n d e r g r o u n d
of the stored p r o d u c t s
tests
in the past
w h i c h has a p p e a r e d
importance
; hasty
interpretations
conclusions
and p r e d i c t i o n s
of the cavities• to use
properties
in situ d a t a s
of rocksalt.
in o r d e r
166
•
In
than
situ
datas
Many
of the considered
twenty
years
Some useful i-
Liquid
:
their
conclusions or
ii- S e v e r a l
(less
gas
storage,
• Eminence
Salt Dome,
internal
pressure
each year. total
After loss
. Kiel, This
varied
so t h a t
the
after
the end of the
28 000 m 3 of s u m p ) .
8 MPa
and
period,
Legreneur,
i - The three
during
cavities
1725 m e t e r s
40%
(Baar,
was
located
the brine pressure loss
9 ) :
and
1965 m e t e r s
with up by
several
; the cycles
36 m e t e r s
and
1977).
between pumped
vanished
of v o l u m e volume
after,
1300 m e t e r s
out
till
to zero.
the
and tubing
During
this
was
7500 m 3,
was
68 000 m 3, i n c l u d i n g
the cavity
45 d a y s
exhibited
an addi-
1974).
:
situated
between
i0 y e a r s
; the
one or two
Two main
(or g r o u p s
internal
have
of c a v i t i e s )
1500 m e t e r s varied
At the end 25% or 30%
c a n be m a d e
been
and
pressure
cycles.
to b e s o m e
inferences
have
1400 m e t e r s
annual
appeared
of v o l u m e ,
ii- t h e d e f o r m a t i o n
was
(the i n i t i a l
of v o l u m e
losses
about
(R~hr,
France
1980)•
at a very
(figure
:
28 MPa,
Five months
1900 m 3
loss
worked
damage
raised
; the
test
22 M P a w i t h
the
larger
and
internal
Two cavities, been operated
precisely, storages
has
cavity
collapsed
• Tersanne,
by a small
volume)•
7 MPa
leaching,
the roof
loss of
controlled.
:
experimental After
are affected
the b o t t o m
was
for m o r e
:
initial
between
between
of v o l u m e
test,
tional
located
out
suffered
U.S.A.
two y e a r s ,
R.F.A.
1400 m e t e r s • emptied,
have
in operation
has been precisely
storage
5% of t h e
or, m o r e
pressure,
cavity was
evolution
products
than
low minimum
The
have been
c a n be p o i n t e d
liquefied
convergency
the
and
cavities
larger
have
between
of t h i s (Boucly,
:
than
expected
for t h e s e
;
been delayed
on
several
months
or y e a r s .
167
0 O
E
1
0 ~ . ~
0
- S~
~,.
0
LII
"
_ - ~
0
_
-
~
i." :':':.:..'~.~" .: .:': ...........
_
. ' - ....
;...,.,--
0
o:
%
I'M
o
/)cJ ~ . , I
0
o
I--LU
0
"-1
I
I
I
I
I
,I
I
In:l 0
0
1=
.', ". • ." .'... • • :.
.. :-......
. . . .
_j
_..
O ,e,.-
>>" f
C:) ¢',i C~
I C~
I
I
I
,o
O
•' : ' " ""' "'" "'."''S"" • .: . ' : ", • . ' ; . . ' - - - - ' s
:.;.'.-.
I-i ¢)
0
• . • o • . .~. • , ...°°°°% _ _ _
e,':, C:) C',,J
• ;."-... • .°° %. •...,, ;..'-...,,,,
I
I
I
I
I
I
0
_J
~t
E O C'4
LLJ r....)
O O-
Z
m
L.IJ
O
Z
O
LL.I r
i
I
I
i O C:~
I
I
I
t
l C~ C:~
I
I
l
I
I C:~
I~"
168
A TENTATIVE
EXPLANATION
A r o u g h but simple ted to,
is the d i f f e r e n c e
the i n t e r n a l The ties.
index
pressure
figure i0 shows
for by p o i n t s P - Pi =
Vertical
this
segments
stand
because
suggested
the value
• Elastoplasticity
influence.
stress than
P -Pi
But,
intensity,
20 MPa.
v
are
H
(4) because
their
internal
part of the
salt beds
and
low.
of v o l u m e
have been o b s e r v e d
20 MPa.
4C E
load i n t e n s i t y
of volume convergency
suggest
the p e r f e c t
appears
are not p r o p o r t i o n a l is o b s e r v e d
a plastic plastic
~ Poisson's
( P - P i ) is g i v e n by 1 AV =
than
cavi-
etc...)
:
in deeper
losses
or a p p l i e d
facts
Let us c o n s i d e r
3
leached
is more
losses
for m i n o r
These
and E Y o u n g modulus, tion of
line
is free and then o f t e n v e r y
index
(i) and
:
The d i f f e r e n c e major
a s tr a i g h t
LPG,
storages take up the u p p e r
by figure i0 large
of the
(formula
for d i f f e r e n t (oil,
H = 0.011
they are u s u a l l y
pressure
depth,
of liquids
for gas storages,
Gas
load
are submit-
(3)).
versus
along
(0.023 - 0.012)
diagram,
when
index,
located
is not constant.
As
the o v e r b u r d e n
(2) or
the s t o r a g e s
pressure
their m i n i m a l
between
(formula
In this diagram,
standed
of the load w h i c h the c a v i t i e s
to have a to this
when P -Pi
is less
behaviour.
Tresca model,
with
ratio.
The
loss of v o l u m e
3(1 -~) 2
exp
(Q- i)
cohesion
C
as a func-
:
( i - 2 ~) Q
(5) . Q = ~C
(p - Pi )
If likely v a l u e s chosen,
this
gure i0)
if the c o h e s i o n
So the f r a m e w o r k magnitude work,
w i t h actual
m a t i o n must be e x p l a i n e d city.
(~ = 0.25,
is t a k e n as r a t h e r allows
loss of volume.
loss of volume
and filled with
consistent
parameters
is in good a g r e e m e n t
of p l a s t i c i t y
of the g l o b a l
the total
created
of elastic
simple m o d e l
the
is r e a c h e d
by r e s o r t i n g
E = 2.104 MPa) in situ d a t a s
(less than
for e x p l a i n i n g
the order
as soon as the c a v i t y Such an a s s u m p t i o n
The actual
history
to the f r a m e w o r k
are (fi-
3 MPa).
But in this t h e o r e t i c a l
stored products.
observations.
low
with
of
frameis is not
of the defor-
of v i s c o p l a s t i -
169
"C: - i ~'a
C= 0 , , . ,
¢1:
b~
<
~
I
[
]
I
C)
-~-I
O -
¢N 11
.
/
/
/
,
d/
/
/
I~l
~/
/
"
.'--
~ 0 L'q
.~
2
.~
"~ ~
',, ,, >
I
b"
C~
, ®
0
O
r,.)
170
Vis c o , , ~ , l a s t i c i t y Plasticity period
and v i s c o p l a s t i c i t y
between
viscoplastic
:
initial
give similar
and final o b s e r v a t i o n s
time constant,
results
only
if the
is larger
than
some
so that m o s t of the c o n v e r g e n c y
has been
achieved. As it has b e e n r e p o r t e d this
time c o n s t a n t
form s o l u t i o n submitted
above,
is high ; then
for the p r o b l e m
to a v a r y i n g
in situ e x p e r i e n c e
it was u s e f u l
of the e v o l u t i o n
internal
pressure
suggests that
to d i s p o s e
of a closed
of a s p h e r i c a l
cavity
(B~rest and N g u y e n M i n h Duc,
(1981)). A more general
solution
part of this paper. meters
(cohesion
of the gas
(P = 34 MPa)
? We must
The
internal by Q =
:
1AV 3 V
-
Heaviside
function
of volume
4C { 3
in the first
of 3 m a i n p a r a -
to the condi-
is a b o u t
is about
triplet
after
(1-2
close
E).
1500 meters
8 MPa.
ten years u n d e r
equations
3
~) Q - ~
We will
of p a r a m e t e r s
opera-
(t is actual
(i-9)
(C,
time,
3}
x
(6). dQ dx3 3 ~-~ + Q - 1 = dT + Log x
pressure
(3/4)
pressure
the f o l l o w i n g
radius)
E t = 2(i -~)
but rather
: the depth
question : what
a 30% loss
then solve
and x v i s c o p l a s t i c
defined
internal
has been g i v e n
of the values
n, and Y o u n g m o d u l u s
example,
of T e r s a n n e
the f o l l o w i n g
~, E) can e x p l a i n
depends
constant
a theoretical storage
and the m i n i m u m
try to a n s w e r
tion
of this p r o b l e m
solution
C, v i s c o s i t y
Let us select tions
This
is c o n s t a n t
(P-Pi)/C
= (I +s)
; so the solution
in our example ; then . Y(t), takes
w h e r e Y(t)
the form
let s be
is the
:
x3 I
d s - Log
T =
(7).
s Then, given t = i0 years be o b t a i n e d
for C and
(E, C, n) space gure
and A V / V = 0.3,
when
of p o s s i b l e
E is fixed. values
a differential
relation
The r e p r e s e n t a t i v e
for the p a r a m e t e r s
can
surface
in
is shown on fi-
ii. Now,
several
(personnal
authors
communication),
(Clerc-Renaud, infer values
range E = i0 000 to E = 30 000 MPa.
Dubois
(1978),
for Y o u n g ' s
Boucly
modulus
(1980))
in the
171
\ \ \ \ \
C
MPa)
6
.\
~\
\\
# \\
5
~ " .\
\\O k
\
/ o~ Y[-// /~9. /]~.
"x
\'@
\\~\
,
/
\O
\ ", \ \
L
4
\~
\
[.//0o I
I
\
t_._
"~/. /
\
\
\
\
2.
\ \
\
\
\
, - . ( G Po ,an) Z'I
J
s
5(
E ,~ GPa~) Figure
ii
: Values loss
of C,
E,
of v o l u m e
~ accountinq after
\
for a 30%
i0 y e a r s .
172
Figure tance.
ii shows that inside these limits E is of limited impor-
So, the error will be n e g l i g i b l e is we choose the mean value
E = 20 000 MPa. Then,
the o u t l i n e d curve C versus n on figure ii shows that
- either the c o h e s i o n C is high city
:
: we are close to p e r f e c t plasti-
; the a d d i t i o n a l loss of volume,
after
i0 years,
can be predic-
ted to be small. - Or, the c o h e s i o n C is low, and the v i s c o p l a s t i c i t y c o n s t a n t is high but remains finite
: r o c k s a l t behaves
like a N e w t o n ' s fluid
;
c o n v e r g e n c y will go on d u r i n g centuries and up to total closure.
REFERENCES
:
Aufaure, M., 1975, "Etude ~ l a s t o v i s c o p l a s t i q u e d'un r ~ s e r v o i r sph~rique ~pais soumis ~ des charges lentement croissantes", Journal de M~canique, G a u t h i e r - V i l l a r s Ed., vol. 14, n ° 2, pp. 221-235. Baar, C.A., 1977, "Applied Salt Rock M e c h a n i c s - I", E l s e v i e r Scientific P u b l i c a t i o n Company. B~rest, P., Nguyen Minh, D., 1983, "Response of a s p h e r i c a l cavity in an elastic v i s c o p l a s t i c m e d i u m under a v a r i a b l e internal p r e s s u r e " Int. J. Solids and Structures, P e r g a m o n Press, vol. 19, pp. 10351048. B~rest, P., N g u y e n Minh, D., 1984, "Deep U n d e r g r o u n d Storage Cavities in Rock Salt : I n t e r p r e t a t i o n of In Situ Data from F r e n c h and F o r e i g n Sites", The M e c h a n i c a l B e h a v i o r of Salt, Trans. Tech. Publications, H.R. Hardy and M. Langer Ed., pp. 555-572. Boucly, Ph., Legreneur, J., 1980, "Hydrocarbon storage in c a v i t i e s leached out of salt formations", S u b s u r f a c e Space, R o c k s t o r e 80, M. B e r g m a n ed. P e r g a m o n Press, pp. 251-258. C l e r c - R e n a u d , A., Dubois, D., 1978, "Long term o p e r a t i o n of underground storage in salt", V S y m p o s i u m on Salt, Hambourg. Madejski, J., 1960, "Theory of non s t a t i o n a r y p l a s t i c i t y e x p l a i n e d on the example of t h i c k - w a l l e d r e s e r v o i r loaded w i t h internal pressure", Arch. Mech. Stos., n ° 5/6, vol. 12, pp. 775-788. Ottosen, J., 1985, "Behaviour of v i s c o e l a s t i c - v i s c o p l a s t i c spheres and c y l i n d e r s - partly plastic vessel walls", Int. J. Solids and Structures, P e r g a m o n Press, vol. 21, n ° 6, pp. 573-595. R~hr, H.U., 1974, "Mechanical b e h a v i o u r of a gas storage c a v e r n in e v a p o r i t i c rocks", IV S y m p o s i u m on Salt. Tijani, S.M., 1978, "R~solution n u m ~ r i q u e des p r o b l ~ m e s d ' ~ l a s t o v i s c o plasticitY. A p p l i c a t i o n aux cavit~s de s t o c k a g e en couches salines profondes", Th~se de D o c t e u r - I n g ~ n i e u r , U n i v e r s i t ~ Paris VI. Wierzbicki, T., 1963, "A t h i c k - w a l l e d e l a s t o v i s c o p l a s t i c spherical c o n t a i n e r under stress and d i s p l a c e m e n t b o u n d a r y value condition", Arch. Mech. Stos., n ° 2, vol. 15, pp. 297-308.
CHARACTERISTIC STATE VIBROTHERMOGRAPHY
AND OF
INFRARED SAND
M i n h Phong Luong Laboratoire de M4canique des Solides, CNRS UA 317 Ecole Polytechnique 91128 Palaiseau Cedex France
Sum~Z Rheological properties and t r a n s i e n t solid
of granular soils
l o a d i n g can be i n t e r p r e t e d
particles
(contractancy)
interact
at the g r a n u l a r
leading to a global
or d i s a g g r e g a t i o n
subjected to vibratory l e v e l where the
irreversible
(dilatancy).
A cohesionless granular
soil
i s c o n s i d e r e d as a g r a i n assembly.
Observed macroscopic d e f o r m a t i o n s are d e r i v e d e s s e n t i a l l y structural
modifications,
inducing irreversible -
of s o l i d
from t h e i r
rearrangement of the c o n s t i t u t i v e
contractive
or d i l a t i v e
grains
volume changes :
Compaction mechanism c o r r e s p o n d i n g to the mutual t i g h t e n i n g
particles
which induces a c o n t r a c t i v e
- Distortion to a c o n t r a c t i v e deviatoric
i.e.
aggregation
stress
behaviour,
mechanism due to g r a i n s l i d i n g s
b e h a v i o u r f o l l o w e d by a d i l a t i v e
leading initially
b e h a v i o u r when the
l e v e l q = q / p exceeds the g r a i n i n t e r l o c k i n g
called characteristic
state
(stationary
threshold
volume change or zero d i L a t a n c y
rate), -
Attrition
mechanism subsequent to a s p e r i t y
c r u s h i n g m o d i f y i n g the r e l a t i v e effect
is a contractive
d e n s i t y under high s t r e s s e s .
The r e s u L t a n t
behaviour.
The c h a r a c t e r i s t i c a dilatancy
breakage and g r a i n
threshold
i s r e v e a l e d by the appearance of
loop when the load c y c l e crosses the d e v i a t o r i c
qc " Such o b s e r v a t i o n s enable the d e t e r m i n a t i o n c a p a c i t y of g r a n u l a r s o i l .
stress
of the entanglement
Level
174
Below the characteristic
threshold,
the intergranular contacts
are stable. The limited slips tend to a maximal aggregation. subcharacteristic
domain or contractancy zone, a hysteresis
In this loop occurs
when reloading. The mechanical behaviour depends upon the load history. Above the characteristic unstable,
leading to significant
A reload shows a dilatancy
threshold, the grain contacts become
slidings due to interlocking breakdown.
loop with memory loss of load history and a
softening phenomenon occurs. The infrared vibrothermography demonstrates
used in our laboratory LMS
the thermal dissipation of sheared granular soil characte-
rizing the sliding mechanism of grains when the granular interlocking structure breaks down on exceeding the characteristic This non-destructive records and observations
and non-contact
threshold.
testing technique allows
in real time of heat patterns produced by the
dissipation of energy due to friction between grains. The infrared vibrothermographic thermal energy. Additionally
test couples mechanical and
it offers the potential of directly
monitoring the stress state of particle rearrangements or characteristic threshold and of predicting the degradation or damage of granular materials by active heating.
175
I -
INTRODUCTION
Research in granular material behaviour has been considerably developed using constitutive cyclic and transient
laws in the case of dynamic, vibratory,
loading. The applicability of the stress-strain
analysis requires realistic understanding and ready determination of the significant
factors of the mechanical performance of granular ma-
terial response.
This paper describes an experimental approach in order to : 1.- Grasp the basic aspects of stress-strain
response of granular materials
under various loadings ; 2.- Consolidate the experimental data for a comprehensive definition of a stress domain in which the resultant effect of repeated loading is contractancy
(contractive behaviour) or dilatancy
(dilative behaviour)
3.- Describe the deformation process which influences the overall macroscopic behaviour of granular materials ; 4.- Visualize the distortion mechanism occurring in the grain structure by infrared thermographic analysis giving a physical meaning to parameters introduced in the constitutive
law ;
5.- Suggest the development of relevant parameters for use in analytical and numerical models ; 6.- Guide and interpret the main features of the cyclic behaviour of granular materials.
2,
DEFORMATION
MECHANISMS
In order to analyze and predict the macroscopic behaviour of granular material, it is necessary to understand how the individual microscopic constituent elements interact at the grain level.
;
176
A cohesionless granular material can be considered as a grain assembly where the discrete and solid granules are in contact and free to move with respect to their neighbours.
It is often assumed that the
constituent granules are in direct, elastic contact with one another. The inherent non linearity of Hertz relationships between two elastic bodies indicates great difficulties
in the application of contact theory
to the study of granular media.
Nevertheless, observed macroscopic deformations of the material are derived essentially from their structural modifications,
i.e. rear-
rangements of the constitutive grains inducing irreversible contractive or dilative volume changes : 1.- Compaction mechanism corresponding to the mutual tightening of solid particles inducing a contractive behaviour ; 2.- Distortion mechanism due to irreversible grain slidings leading initially to a contractive behaviour, then interlocking disrupture where the individual particles are plucked from their interlocking seats and made to slide over the adjacent particles with large distortion of the grain arrangement - inducing significant dilative volume changes ; 3.- Attrition mechanism subsequent to asperity breakage and grain crushing which modifies the relative density under high stresses. The resulting effect is a contractive behaviour.
Granular material deformation under load is due in part to elastic deformation of the solid particles. This elastic deformation often constitutes only a small fraction of the total deformation and is often obscured by deformation resulting from slippage, rearrangement and crushing of particles.
Classical elastic plastic theory assumes that the elastic and plastic components of strain can be isolated by loading and subsequent unloading. The recoverable strain is elastic. The total strain
c, is the sum of the
177
elastic strain c e and the plastic strain c p. However, in granular materials it is not possible to separate the elastic strains simply by unloading.
Even when recovery of strain in granular materials is a result of stored elastic energy, the strains recovered are not always purely elastic• Slippage at particle contacts may accompany strain recovery. Sometimes elastic and plastic deformations occur in parallel and cannot be isolated from each other experimentally•
3.
MECHANICAL
CHARACTERIZATION
Extensive laboratory tests are presented to assess various rheological responses of cohesionless granular materials subjected to vibratory, cyclic and transient loading under drained or undrained conditions•
When tests are performed at varying confining pressures or changing pore water pressures, the penetration of the protective membrane into the solid skeleton may cause significant errors in the measurements of volume changes and pore pressure developments. To overcome this difficulty, it was necessary to set up the arrangement shown in Fig. 1, the purpose of which is to check the calibration and the accuracy of the volumetric strains or pore pressures obtained usually with conventional test procedure• Thanks to the double membrane device, the thin annular part allows a perfect control of effective confining pressure and avoids the penetration of the inner protective membrane by maintaining the pore pressure u' of the outer part always equal to the pore pressure u of the inner part. For this test, membrane penetrations are located only in the annular part of the sample•
The loading parameters of axisymmetric triaxial tests are : • mean stress
p = (q
. deviatoric stress
q = q
. deviatoric level
n = q/P
a
a
+ 2 q )/3 r - q
r
178
The corresponding deformation parameters are defined by : • volumetric strain
c
• deviatoric strain
~
. dilatancy rate
6
v q
= E
a
= 2(c
+ 2 E
a
r
- c )/3 r
= ~P/~P v q
Tests are conducted either with a constant stress rate ~ or a constant strain rate ~ on saturated materials under drained or undrained conditions, c, c e and c p denote respectively total, elastic and plastic strain. For example, contraction or dilatation c
may be the resulting value of v elastic volume increase or decrease E e and plastic volume change c p due to v dilatancy and/or contractancy.
3.1.- CHARACTERISTIC THRESHOLD
Extensive laboratory tests on several sands show that the lowest point on the volume change axial strain curves (Fig. 2), - that is the point of minimum volume of the sample -, corresponds to a constant stress ratio nc [I] .
The stress peak or maximum of shear resistance occurring at maximum dilatancy rate has been analyzed and interpreted by the stress dilatancy theory [2] .
The asymptotic part of the stress strain curves determining the ultimate strength has suggested the well-known critical state concept [3].
For our concern, the transient and cyclic loading cases require the analysis of the prepeak part where the stress ratio nc at zero dilatancy rate (6 = 0) defines evidently the characteristic state of the granular material associated with an angle of aggregate friction ~c Eli
Under undrained conditions
onasaturatedsandapore
pressure increase
characterizes the first stage of any u n d r a i n e d t r i a x i a l t e s t irrespective of the initial sand density. As the deviatoric stress q = q
a
-q
r
increases, the pore
t79
pressure generation rate u decreases, passing through zero to become negative, followed by large irreversible axial strains. This pore pressure generation behaviour parallels phases of contractancy and dilatancy in a drained test [4] .
Thus the deviatoric stress level corresponding to either the inversion of pore pressure generation rate in an undrained test, or the zero of the volume change rate in a drained one, determines the characteristic threshold CT unambiguously
: the granular material is in a "charac-
teristic state" having the following properties
:
i. the volume change rate is zero ; ii. the stress level reached by the material is an intrinsic parameter which defines a characteristic friction angle ~c = sin-l[3qc/(qc + 6)] determining the interlocking
capacity of the grain assembly.
The value of ~c is independant of the initial sand density. At any point on the characteristic
line CL (Fig. 4) herely determined, the rate
of the irreversible volume change is stricly zero.
The position of the effective determines
its mechanical behaviour.
loading point relative to the CL line In particular, the material undergoes
large deformations after crossing the characteristic threshold CT into the surcharacteristic
domain in which the characteristic value qc is exceeded.
The characteristic
line CL divides the allowable stress-space into
two regions :(1) subcharacteristic
region corresponding to an interlocking
of grain structure or contractancy,
(2) surcharacteristic
region where di-
saggregation of granular material or dilatancy occurs. Thus every closed stress path in the subcharacteristic
domain exhibits a contracting soil be-
haviour illustrated by an irreversible
compaction
(or an irreversible
in-
crease of pore water pressure under undrained condition) whereas a closed load cycle in the surcharacteristic expansion
domain leads to an irreversible volume
(or an irreversible decrease of pore water pressure).
180 A very accurate experimental procedure of determination of the characteristic threshold qc follows which is readily available under either drained or undrained conditions : ~c is revealed by the appearance of a dilatancy loop with volume change or pore water pressure during a load cycle crossing the characteristic line CL (Fig. 5).
3.2.- CHARACTERISTIC PROPERTIES OF GRANULAR SOILS
Extensive laboratory tests using the axisymmetric triaxial apparatus on various sands : Fontainebleau sand, Loire sand, carbonate Channel sand [4], carbonate marine sediments [5] and Hostun sand [6] substantiate the following rheological properties : • Under Drained Conditions
I.- Adaptation or elastic response may be considered as obtained after a finite number of cyclic isotropic Loadings : 2•- Accommodation with stable hysteresis loop appears under radial loading or conventional loading at a stress level ~ = q/p smaller than the characteristic threshold qc" Stress-volume change curves of sandy soils exhibit a clockwise hysteresis loop after unloading and reloading. This hysteresis susceptibility becomes negligible when the number of cycles increases. 3.- For q greater than qc" the hysteresis loop moves toward large deformations and cyclic loadings cause ratcheting behaviour with increasing cumulative strains. The soil volume dilates and reflects the phenomenon of dilatancy of the grain structure. After unloading, a dilatancy loop is seen in an anticlockwise direction on any diagram where volume change is plotted. The dilatancy loop is a very practical and useful criterion for the detection of the characteristic threshold• 4•- Densification of dense sands may be obtained easily by cyclic loading at large amplitude exceeding both triaxial compression and extension characteristic thresholds. The high amplitude loading benefits in partial loss of strain-hardening during the dilating phase in the surcharacteris-
181
tic domain leading to a breakdown of the granular interlocking assembly. On each reload, the tightening mechanism induces new irreversible volumetric strains and recurs each time with a renewed denser material.
•
Under Undrained Conditions .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5.- Sand liquefaction occurs only when load is cycled alternately on both sides of zero deviatoric stress and has reached the characteristic
levels.
6.- Cyclic non-alternated deviatoric stress tests show a progressive tendency of the effective stress state to move toward the characteristic
level
and to stabilize there, i.e. cyclic softening is occurring. 7.- Cyclic hardening of sandy soils may be observed when undrained cycled in the surcharacteristic and the characteristic
loads are
region bounded by the failure line FL
line CL. It leads to a stabilization of the granu-
lar material on the characteristic thresthold as shown in Fig. 6. Irreversible strains accumulated during undrained
loadings depend on the
stress amplitude of cycles.
4.
THERMAL
DISSIPATION
Infrared thermography has been successfully employed as an experimental method for detection of plastic deformation during crack propagation under monotonic
loading of a steel plate or as a laboratory technique for investi-
gating damage, fatigue and creep mechanisms
This experimental
[7].
tool is used here to illustrate the grain sliding
mechanism of a granular material subjected to a shear loading which exceeds the characteristic threshold. The heat dissipation evidenced here is associated with a plastic work of distortion.
182
4.1.- HEAT PRODUCING MECHANISM
A consideration of the forces and deformations at each contact surface [8] may serve as one starting point in interpreting the thermomechanical
cou-
pling of sand behaviour under vibratory shearing.
For the simplest case of two like spheres, each of radius R compressed statically by a force N which is directed along their line of centers, normal to their initial common tangent plane, the contact theory due to Hertz predict~ a plane, circular contact of radius a = [3(1 - ~2)NR/4E] 1/3 where ~, and E denote respectively Poisson's ratio and Young's modulus of the sphere.
The normal pressure on the contact area is given by :
= 3N(a 2 - pz)1/212~a3
where p represents the radial distance from the center of the contact circle.
An additional force C ~ i s
assumed to act in the plane of contact and
its magnitude rises monotonically from zero to a given value. Because of symmetry the distribution of normal pressure remains unchanged.
If there is
no slip or relative displacement of contiguous points on a portion of the contact surface, the displacement of the contact surface in its plane is constant. The Mindlin's solution of the appropriate boundary-value problem shows that the tangential traction is parallel to the displacement the applied f o r c e ~ )
(and to
axially symmetric in magnitude, and increases without
limit on the bounding curve of the contact area. Slip is assumed to be initiated at the edge of the contact and to progress radially inward, covering an annular area. In accordance with Coulomb's law of sliding friction T = fq where f is a constant coefficient of static friction. The radius c of the adhered portion or the inner radius of the annulus of slip is given by c = a(1 - T/fN) 1/3 (Fig. I).
183
When the tangential f o r c e ~ d e c r e a s e s 0 <~*
from a peak value~*,
< fN, slip, once again, occurs, but its direction will be opposite
to that of the initial slip. An annulus of counter-slip is formed and spreads radially inward as the tangential force is gradually decreased. Its inner radius is b = all - ~
-~'~)/2fN]I/3. The inelastic character
of the unloading process appears evident since the annulus of counter-slip does not vanish when the tangential force is completely removed.
Under oscillating tangential force, the load-displacement curve forms a closed loop traversed during subsequent oscillation o f ~ limits ~ *
between the
providing that N is maintained constant. The area enclosed in
the loop represents the frictional energy dissipated in each cycle of loading. For small tangential forces, it has been suggested by Johnson [9] that the tangential displacement necessary to relieve the singularity in traction takes the form of an elastic deformation of the asperities. An increase in applied tangential force causes the asperities at the edge of the contact surface to deform plastically through relatively large strains, a process which leads to a marked increase in energy dissipation and to severe damage to the surfaces.
Thus at small amplitudes of the tangential force, energy is dissipated as a result of plastic deformation of a small portion of the contact surface, whereas, at large amplitudes, the Coulomb-sliding effect predominates.
In the conventional triaxial test, if the load is cycled within the subcharacteristic domain below the characteristic threshold qc the intergranular contacts are stable. Small slips lead to a maximum entanglement due to the relative approach of constituent granules. The dissipated work given by the hysteresis loop (a) in the (q,Ca) diagram is relatively small. The corresponding heat production is relatively low and negligible. On the contrary when the shear load is cycled above the characteristic trheshold, the intergranular contacts become unstable, leading to significant slidings due to
184 interlocking breakdown. A large frictional energy (B) is dissipated as shown in the Figure 8 and is transformed almost entirely into heat owing to the thermomechanical
conversion.
4.2.- VIBROTHERMOGRAPHY
Infrared thermography utilizes a photon-effect detector in a sophisticated electronics system in order to detect radiated energy and to convert it into a detailed real-time thermal picture on a video system. Temperature differences in heat patterns as fine as 0,1°C are discernible instantly and represented by several distinct hues.
This technique is sensitive, non-destructive ideally suited for records and observations
and non-contact, thus
in real time of heat patterns
produced by the heat transformation of energy due to friction between grains of sheared sand. No interaction at all (with the specimen)
is required to
monitor the thermal gradient.
The quantity of energy W emitted by infrared radiation is a function of the temperature and the emissivity of the specimen. The higher the temperature, the more important
is the emitted energy. Differences of radiated
energy correspond to differences of temperature, since W = h T 4 ~
where h
denotes a constant, T the absolute temperature and ~ the emissivity.
Soils present a low thermomechanical
conversion under monotonic
loa-
ding. However plastic deformation - whereby sliding between grains occurs creating permanent changes globally or locally - is one of the most efficient heat production mechanisms.
Most of the energy which is required to cause
such plastic deformation is dissipated as heat. Such heat development is more easily observed when it is produced in a fixed location by reversed or alternating slidings due to vibratory reversed applied loads. These considerations define the use of vibrothermography granular material damage.
as a non-destructive
method for observing
185
4.3.- EXPERIMENTAL SET-UP AND RESULTS
The thermomechanical
behaviour of a Stampian sand (Fontainebleau
sand) is studied when subjected to two types of vibratory loading. a. Conventional triaxial
loading : indirect shearing
A cylindrical dry sand sample (dry unit weight Ed = 15.7 kN/m3 ; void ratio e = 0.72 ; relative density I D = 0.62) confined under a constant isotropic pressure of 100 kPa is subjected to a vibratory force generated by a steel mass located at the top of the specimen excited by an electrodynamic vibrator. When the frequency reaches 87 Hz with a controlled displacement of 1 mm at the base, the specimen (70 mm diameter - 150 mm high) is subjected to stationary stress waves and presents a striction zone where the deviatoric stress level q exceeds the characteristic threshold qc of interlocking breakdown of the granular structure b. Cylindrical
(Fig. 9a).
loading : direct shearing
A tubular sand sample at the same initial density confined under a pressure of 50 kPa is directly sheared by a concentric steel cylinder exciter in an axial vibratory motion by the electrodynamic generator. In this case of loading, the principal stress axes rotate during
loading.
At the frequency of 80 Hz and with a controlled displacement of 1 mm, the characteristic
threshold is exceeded and hot colours due to heat production
by friction appear as shown in the Figure 9b. The temperature increase is about 6 ° Celsius for a test duration of 20 secondes.
3.4.- THERMOMECHANICAL COUPLING
Thermal effects occurring in these tests may be interpreted on the basis of the theory of internal variables. Assuming the case of small disturbances, the two laws of thermodynamics
and Fourier's
law of heat conduc-
tion can lead to the following equation of thermomechanical with tensorial quantities k,~ and o [10] :
evolution written
186
C~ = div(k grad T)- t c~ ~ + D + ¢
C
denotes the specific heat per unit volume
T
the absolute temperature of reference
k
the thermal conductivity tensor the coefficient of thermal expansion
a
the stress tensor
D
the intrinsic energy dissipation the density of heat sources.
In the second member of the transient heat flow equation, the first term governs the thermal diffusion. The second term represents the thermoelastic effect which may be significant
in the cases of isentropic
loading.
The third term concerns the thermal dissipation due to viscosity or irreversibility phenomena. The last term denotes the presence of thermal sources.
Only friction energy dissipation has been shown in the presented tests. It may be interesting to study further how the other parameters k and ~ can be affected in the course of loading.
5.
CONCLUDING
REMARKS
The characteristic
state concept, defined easily using conventional
tests, allows the prediction of drained or undrained properties of granular materials. a. The essential parameter for studying the rheological behaviour of granular materials subjected to cyclic and transient
loading is the develop-
ment of the volumetric strain during loading stages. b. The friction angle ~c is an intrinsic factor characterizing
the inter-
locking capacity of grain structure under drained conditions and the average mobilized friction angle under undrained loading.
187
c. The characteristic threshold is defined by a stress threshold qc at which level a dilatancy loop appears after unloading. d. This criterion becomes all important when considering questions of cyclic loading, facilitating the definition of a region of contractive behaviour for the granular material. Beyond that region of contractancy, in the surcharacteristic domain and up to the failure limit the granular material behaviour during load cycles is dilative. e. Infrared vibrothermography demonstrates the thermal dissipation of sheared granular material characterizing the sliding mechanism of grains when the granular interlocking structure breaks down on exceeding the characteristic threshold. This non-destructive testing technique allows records and observations in real time of heat patterns produced by the dissipation of energy due to friction between grains. The infrared vibrothermographic test couples mechanical and thermal energies. Additionally it offers the potential of directly monitoring the stress state of particle rearrangements or characteristic threshold and of predicting the degradation or damage of granular materials by active heating.
REFERENCES
[I]
KIRKPATRICK, W.M., (1961), "D~c~sion on Soil Properties and their Me~urement", Proc. 5th Conf. Soil Mech. and Found. Eng., III, pp. 131133, Paris.
[2]
ROWE, P.W., (1971), "Theoretical Meaning and Observed Values of Deformation for Soils. Stress Strain Behavio~r of Soils", Cambridge, March 1971, pp. 143-194, ed. R.G.H. Parry.
[3]
SCHOFIELD, A.N. & WROTH, C.P., (1968), "Critic~ State Soil Mechanies", Mc Graw Hill, London, G.B.
[4]
LUONG M.P., (1980), "Stress-Strain Aspects of Cohesionless Soils under C y ~ c and Transient Loading", Proc. Int. Symp. on Soil under Cyclic and Transient Loading, Swansea, Jan. 1980, U.K., PP. 315-324.
188
[5] NAUROY, J.F. & LE TIRANT, P., (1981), "Comporteme~ des S ~ d i m e ~ Ca~bonat~ sous l'Ac~ion de Chargement~ Cycliqu~", Proc. Xth ICSMFE, June 1981, Stockh61m, Sweden. [6]
THANOPOULOS, I., (1981), "Contribution ~ l'Etude du Comportement Cyclique des M ~ e u x P~v~rulents", Th@se de Docteur-Ing6nieur, I.M.G. Grenoble.
[7]
REIFSNIDER, K.L., HENNEKE, E.G. & SINTCHCOM, W.W., (1980), "The mechani~ of Vibrothermography" in Mechanics of Non destructive Testing, ed. by W.W. Stinchcom, pp. 249-276.
[8]
MINDLIN, R.D. & DERISIEWICZ, H., (1953), "Elastic Spheres in Contact unde~ Varying Oblique Forces", J. Appl. Mech. 20, pp. 327-344.
[9]
JOHNSON, K.L., (1955), "Surface I ~ c ~ a ~ o n between Elasti~ly Loaded Bodies under Tangential Forces", Proc. Roy. Soc., London, ser. 1, 230, pp. 531-548.
[10] NGUYEN, Q.S., (1984), "Thermodynamique des Milieux Cont~n~", Cours DEAENPC (1984).
189
C.L.
!
t inner
.r.....
cr
membrane
.~.~"11..i"."". •
C U
i!~ ~ i
'I i
U
outer
membrane
[
F~, ! - Double membrane device for tests under v a r y i n g confinement condition
190 CP~.RACTE RI S TI C STATE q = n
CRITICAL STATE
STRESS DILATANCY THEORY R=KD
C
iql = M p
D=
eP=o
d___~)
v
dE a
v
= F - kLnp
R = oi/o3 Stress level at
Stress level at zero dilatancy rate
ultimate
Grain interlocking
Pure distortion under shearing
capacity dv = 0
loading
dv = 0
at
large deformations
low deformations (before failure)
(after failure)
~a
Easily determined
Not readily determined by usual tests
by usual tests TRANSIENT BEHAVIOUR
ASY~IPTOTIC BEHAVIOUR
FAILURE BEHAVIOUR
Fig. 2. Location of basic concepts.
t
O,4
qMPa
t 0,2
Fontainebleau sand Yd =t6,3 kN/m 3 O'c= O'r = 0 16 MPa
/
CT
0
5
~
•
v
0
0,5
qc
/P)c I
1,0
q MPa
, ~ ~
Characteristic
~
Fig.
at
I
~
Ea°~
~'threshold
t
-10
tic
"
0,2
I
0
20
u kPa
3. Definition of the Characteristic Threshold under undrained condition.
191
FL/LCL (n")
/
ncy domain DD (surcharacteristic)
0 % ~ / ' ~ ' ~ _ C o n t r act a ncy domain CD (subcharacter istic)
Fig. 4. Characteristic Criterion.
192
q (7a Ic
~ch a r a c t e"r i s t i c /threshold
'q y/
t °r-,.- ~
0Lo(~ESP
>HLo /f /
£8
/
l\
,/"
] /
Lo
q = aa-cr r
/
\./
p = (%÷ 2 a r )/3 ,= p'÷u
q =q/p
~CL
Oilatancy a v = aa,,- 2 E r
fE v
p,p'
t
HL ° Hysteresis
Loop
(5a) Drained test Fig. 5. Dilatancy exceeding
CL ° Contractancy
loop
DL o Dilatancy
(5b) Undrained
test.
loop observed after an effective load cycle the characteristic threshold.
193 Under undrained condition
Under drained condition
q Cyclic ~/ CL hardening ~ _ , / / ~ u = 0 ,
Constant confinement stress path CCSP
///----j--~ Liquefaction
*Ada2tation Z,ccommodation
,Cyclic softening
P'=-
of
~'~f~J~ satu rated sands p -p' ~0
6c0
Fig. 6, Diverse cyclic behaviours of cohesionless soils readily obtained from the conventional axisvmmetric triaxial apparatus.
194
d 5/z \
~
.t ~r-~
il
5/2
~
~N
.t
•
a
.t 5
//
/ /
Fig. 7. Stress and strain on contact surface of two like spheres subjected to normal force N followed by a tangential force ~ (after Mindlin 1953).
Q
slip
@
sliding
@
195
q MPa
q MPa
0.6
0.6
qc O.Z,
0.2
-
//
/
///
/1
~f
Ea ~
o.~
0.2
!
t
q ~
-0.2-
Fontainebleau sand
Fig.
-0.2
10 = 62 ~/o
8. Triaxial test under small amplitude amplitude (B) of deviatoric stress.
u;
Orr = o.23 MPa
(a) and large
-~-
196
Accelerometer 0
Fig. 9a
M - :.-. ; ..... .
",
.°
•
Heat patterns on dry sand sample under indirect shearing
.*
. ° , . .
70 mm
E °°°
0
•
°°
~.~,"
0
°°
It__. ~
Vacuum
Load cell
15o cycles
Vibrator
4oo cycles
3 ooo cycles
5 °C
+1
Room temperature
1 500
cycles
1100
cycles
197
Fig. 9b
Accelerometer
D
M
I
Heat patterns on dry sand sample under direct shearing
m
-;--
i:.'
•:"'F":"
"~-. -:~
i:t" ,[__
m.,
•..
*!..
::'"
r'i::l
°. ,~
', ' ~
48mm
Vacuum
50 cycles Roo m temperature +l
2
3
~ ! ~ z, 5 °C
Vibrator
2 ooo cycles
25o cycles 1 50o cycles
1 ooo cycles
NON LINEAR BEHAVIOUR OF ANISOTROPIC ROCKS
R. R i b a c c h i
U n i v e r s i t y of Rome
I. I N T R O D U C T I O N
Large plants, of
engineering
have
the
been built
Alps
structures, in recent
(Martinetti,
the design stage During of
these
have
a complex
years
particular in
Dolcetta,
the
caverns
for
metamorphic
1982)
and
power
formations
others
are
now
in
(Piedilago cavern).
the
tion
1977;
in
investigations
sites
the
for
the
constituent
behaviour,
and
geomechanical
anisotropic
sometimes
characteriza-
rocks
surprising
were
found
discrepancies
to
bet-
ween the results of in situ and l a b o r a t o r y tests were observed. In this paper, some
interesting
after a review of simplified
experimental
results
into account
that such e x p e r i m e n t a l
ber of years
and
that
research-oriented exhaustive ations.
and
In
-designed
plan
approach.
are
some of
the most
of
This
insufficient
cases, tests
with could
had
explains
a technical
why
the
interpreting
knowledge
clarify
reliable models of the behaviour
It is to be taken
studies have d e v e l o p e d them
for
the
theoretical models,
are presented.
many
data some
available problems
of these anisotropic
over
a num-
rather are
than
often
a
not
ambiguous
situ-
today,
well-
and
a
produce
more
rocks.
2. E N G I N E E R I N G M O D E L S O F A N I S O T R O P I C R O C K S
In
an
anisotropic
rock
the
engineering
strain
and
stress
vectors are c o n n e c t e d by a symmetric c o m p l i a n c e m a t r i x M e =M~ where
C =
el e2 e3 Y23 3,31 3'32
O =
Sl (~2 a3 T23 T31 T32
(i)
200
In tropy,
Rock
that
isotropy)
Mechanics
is the o r t h o t r o p y
are
usually
scopic textural
as
the
in
simplified
and the polar
utilized.
a
The
reference
compliance
terms
of
"i/E 1
system
matrix
M
and ~
elastic
(Poisson's
to
the
aniso-
the
and metamorphic an
of
(or transversal
corresponds
1,2,3 of
models
anisotropy
latter
the engineering
li), G (shear moduli)
M
only
symmetry of many stratified
Assuming directions, written
problems
principal
orthotropic parameters
macro-
rocks. orthotropy
rock E
can
be
(Young modu-
coefficients).
-~21/E2
-~31/E3
0
0
0
I/E 2
-932/E3
0
0
0
I/E 3
0
0
0
I/G23 0
0
=
(2)
I/G13 0 I/G12 For circular
a transversally
symmetriy,
racterized
axes
isotropic
1 and
by 5 independent
rock,
if
axis
2 are equivalent,
elastic
parameters
3
and
is
the
axis
the rock
of
is cha-
(2 Young moduli,
E 1 and
E3, 1 shear modulus GI3, 2 Poisson's coefficients ~31 and 921 ). In the following, i, 2, 3 will be a reference system corresponding
to
the principal
lar anisotropy, corresponding respectively to it
to
the
minor
in the plane
(3) will
directions
3 is the polar of
axis,
the polar
that
principal transversal
also be indicated
the angle between
of elasticity.
In the case
is in practice
modulus.
The
isotropy
by // and
axis and a generic
directions
(i or
I. For
2)
and
transversal
direction
3' 2 I
2' Ij
Fig. I - Orientation of the reference axes in a transversely isotropic rock
lying normal
isotropy
will be
cated by 8 (Fig.l). 3
of po-
the direction
indi-
201
A generic reference system will be indicated by i', 2', 3' and M' will be the corresponding compliance matrix. The transformation formulas containing the direction cosines of i', 2', 3' with respect to !, 2, 3 are given by Lekhnintski, (1963). Even in the simplest anisotropic models (transversal isotropy), the determination of three main elastic parameters (Poisson's coefficients often have a small influence and are estimated a priori) poses a difficult problem. In some cases, often without any experimental evidence, the assumption is made that the shear modulus GI3 is not an independent parameter, following relationship I/G13 For
an
to
the
rock
other
rock
from
expressed
the
by
the
(3)
three
elastic
(3); the independent The
constant
= 1/E 1 + I/E 3 + 2~31/E 3.
orthotropic
related
but is related to the other (Saint Venant relationship)
principal
properties
elastic constants
deviation
of
the value
R s = GI3/GI3
=
by
moduli
relations
similar
would be reduced
the shear modulus
predicted
by the ratio
by
shear
to
be rel.
to 6.
of a transversally
the Saint Venant
can
isotropic
relationship
can
be
: M44 M11 + M33 _ 2M13
(4)
^
where 3).
GI3
is
pressed _
!
i
931/E3
containing of
the
tensor,
value
the
R G = I, are
the Saint Venant
characteristics
the shear modulus
the polar
for
axis;
orientation
in
of
the
relationship
anisotropy
can
(rel. be
ex-
ratios R E = EI/E 3 and R G-
invariant
every
and the lateral
couple
the normal
the
same
of
axes
compliance
way
as
the
compliance lying
varies
normal
in
as
a
MI3 plane
a function
component
of
a
that is 1
cos28
M~3 = ~'
E3
sin28
+
E----~
(Sl
In the general case,
instead,
1 cos2% sin28 E' = E------~+ E1
+ ~GI3 - ×GI3
If value
satisfying
main
by the anisotropy When
=
the
Therefore,
of
GI3 E'(%)
intermediate
becomes occurs
1
small not
1
enough on
the normal compliance
the
is given by:
sin28
4
(6)
(GI3 < E3/2(I + 931), polar
axis
(8 = 0)
the but
minimum for
some
angle 8.
In a similar
way,
the lateral compliance
931 931 ( 1 M{3 = E3 - E3 + G13
1 sin28 ^GI3)~
is given by:
(7)
202
The
anisotropy
orientation
of
its
anisotropy,
Various
averaging
of
utilized the
to
Km
evaluated
by
rock
and
its
estimate
and
being
minerals
which
M m.
In
random
Gm
(bulk modulus the
is
result the
preferred
when
"solid by
M m.
here,
have
the o r i e n t a t i o n
modulus)
and
or
indicated
isotropic)
shear
Voigt
the
discussed
following,
and
so-called
be
be
(statistically
of
"intrinsic"
will
cannot
the
is
averaging
the
compliance
techniques,
minerals
parameters
a
constituent
matrix"
been
of
the will
Reuss
elastic be
simply
estimates
of
K
and G. In many types of schistous m e t a m o r p h i c rocks the m i n e r a l s are not uniformly d i s t r i b u t e d in the mass, but the rock is composed of levels mainly formed by mica plates, and of levels of quartz and/or feldspaths. Whilst the micas always show a preferred planar orientation, the other minerals are often not m a r k e d l y oriented. To of of
such
obtain
an
schistous
alternating
randomly
of
quartz
technique
the
crystals,
content
by
oriented
were
Salomon
of
tipical
intrinsic
anisotropy
of R E and R G of a rock consisting
perfectly
proposed
The mica
about
the values
layers
oriented
averaging
indication
rocks,
in Fig.
2.
schists
it may be as much as 50% or more.
mica
determined (1968).
The
typical gneisses
plates by
and
of
of
the
means
results
is 10-20%,
are
shown
while
for
30
E (6Pa)
/
20
I: o
7
/ /
~RE
/ t0
2 / f
0
0
50
Z BIOTITE
:'./
lO0
50
"/. MUSCOVITE
100
Fig. 2 - Theoretical values of elastic parameters for a biotitic and a muscovitic schist This classical rocks,
kind
test
on
of
calculation
Rock
Mechanics
appears (Jaeger
to and
support Cook,
a
comment
1969),
the a n i s o t r o p y ratio R E "is rarely as high as 2".
that,
of
a
for
203
However, different
experimental
picture.
For
data
instance,
2
on m e t a m o r p h i c Fig.
rocks
3 summarizes
furnish
quite
a
the a n i s o t r o p y
t o PIEDIL,,~GO G N E I S S
GNEISSES
RV
A ENTRACQUE
2 o
• •
1
0
I 2
BACENO
o oe
,2
1
0
1,,o
~: -i.-~.
Z.
SCHt5T
.. o
• ........
3
GNEISS
5
3
t
t
0
r
SCHISTS
•
n. 0
1
z
2
8
6
9
10
11
t2
Fig. 3 - Anisotropy ratios for Alpine gneisses and schists determined on the basis of the squared ratios of principal velocities. Axis 3 is perpendicular to the m~in planar structure, axis 1 is aligned with the lineation ratios
obtained
schists.
They
R~ =
rock, RV
and
upon
Wenk
seismic
(1974) velocity
for
Alpine
gneisses
and
measurements:
(VI/V3) 2
axis and
Johnson
are based
(Vl/V2) 2
=
where
by
3
is
axis
probably
perpendicular
1 is directed
underestimate
to
along
the
the
main
planar
a p os s i b l e
corresponding
structure
lineation• static
The
of
the
values
anisotropy
of
ratios
RE • The
figure
even
in g n e i s s e s
low.
In some
isotropy
cases
very
the
high
anisotropy
intrinsic
are c o m p a t i b l e
whereas
some
ratios
anisotropy
other
with
is
a model
rocks
are
possible
certainly of
require
very
transversal
at
least
an
model•
These anisotropic
that
which
the data
(R~ = 1),
orthotropic
obsevations properties
microfractures following
shows for
(or
paragraphs.
suggest of
rocks
cracks);
that is
their
the
the
main
factor
presence
effect
will
of be
influencing oriented
sets
discussed
in
the of the
204
3.
ANALYTICAL
3.1.
Open
The
there
is
simplest
no
1969).
open
cracks
of
theoretical
interaction Cracks) For
Consistent) effect
of
p r o b a b l y because of their
(Dilute
and
CRACK-INDUCEDANISOTROPY
has
been
importance
more
frequently
for the g e o p h y s i c a l
related to seismic wave propagation.
The
D.C.
OF
cracks
influence
investigated, problems
MODELS
Model the
between
Model
large
various
utilized
(O'Connel
between
on
assumption
in
the
instance the
and
cracks
the
cracks
for
concentrations
applied
interaction
is based
the
was
crack
is
model
by
rock.
(1965
S.C.
(Self
so-called
1974);
approximately
matrix
having
the
average
elastic
properties
of
the
accounted
for by assuming that each crack behaves as if it were embedded mogeneous
This
Walsh
Budianskii,
is
that
in a ho-
the cracked
body. The adding
global
compliance
M
to the c o m p l i a n c e M m of
of
the
cracked
the matrix
obtained
by
the c o n t r i b u t i o n M Oc of
rock
is
the
open cracks M
= M m + M Oc
Let
us
negligible,
parameter,
assume
and
penny-shaped
(8)
that
that
the
cracks
of
the
anisotropy
rock
contains
variable
of a
radius,
the set
solid
of
r;
matrix
is
non-interacting
the
crack
density
e, of the set is expressed by:
1 3 = N
e = ~[r where
N is the number 3 average value of r • In
the
D.C.
(9) of
open
Model
the
cracks
only
in
unit
non-zero
volume
terms
of
and
the M °c
the
matrix
are: oc _ 16 1 - ~2m M33 3 E eoc m
(I0)
oc oc 16 1 - ~2m M44 = M55 = -~- E m ( l ~ ~m/2) where
Em
and ~m
coefficient. Fig.
respectively
characteristics
the of
intrinsic the
modulus
anisotropy
are
and
Poisson's
evidenced
in
4 and 5 (left side). Rel.
of
The
are
(ii) eoc
(i0)
parallel
because
it
and
cracks is
(ii) is
show that the a n i s o t r o p y a
characterized
particular by
type
3 elastic
of
induced by a system
transversal
parameters
only,
isotropy, instead
of
205
the
5
of
the
slightly
general
greater
transversal
than
1.0
isotropy.
k
PLANAR SET D.i. MODEL ....
0,8
\\,
0,5
The
value
of
RG
is
only
i.
P~LANARSET
.......
O.Z,
613/Gm
0,2
t 0
0
0.2
0"
05
08
12
0
02
0/
05
0,8
e
1.0
Fig. 4 - Young and shear moduli predicted by the D.C. and S.C. models for a rock containing a planar sey of cracks with a crack density e
I
PLANARSET D.C. MODEL 1
1
/
i 1I
/
/
Z 0
PLANAR SET S.C MODEL
R6
0.2
t0.6
0"
L
I "" J
~8
1.0
0
02
~
RG~
0z
.
.
0.6
.
k. 0.@ e [0
Fig. 5 - R
and R ratios predicted by the D.C. and S.C. models E for a ro~k containing a planar set of cracks
An neglected
important with
OC
simplification
respect OC
OC
M44 = M55 = M33
to
unity;
is in
this
obtained case
rel.
if
the
(ii)
term
9o/2
is
becomes:
(12)
206
Now,
the
shear
(RG = i)
modulus
and,
lateral
exactly
whatever
strain
the
satisfies
orientation !
compliance
the
terms MI2,
of
Saint
the
condition
reference
!
Mi3
Venant
and M23
are
axes,
the same
the
and are
equal to the uncracked matrix. The mation
above
even
relationships
for cracks
whose
can
be
shape,
circular,
provided
that an equivalent
the
and
perimeter
area
Budianskii,
rel.
of
with
the
radius
of
a
good
approxi-
the crack,
is not
(which is the function of
crack)
is
adopted
cracks
which
(O'Connel
and
1974).
Even Walsh
the
utilized
in the plane
for
the
(1965),
only
(i0) and
(ii)
two-dimensional a
modification
of
the
were
numerical
considered
by
coefficients
in
is required:
oc 4n I - 92m M33 = E eoc OC
OC
(13)
OC
M44 = M55 = M33.
In this last case the Saint Venant r e l a t i o n s h i p When matrix a
sum
various
M °c of
can
be
crack
sets
immediately
the M °c
of
the
are
present,
determined
single
sets,
is exactly verified. the
global
(in the D.C.
after
they
compliance
formulation)
have
referred
as
to
a
common reference system. The symmetry
elastic
of
instance,
the
symmetry,
fabric
a common
microfissure
deriving
situation
sets
in general,
having
from
the various
in schistous
different
from
each
cracks.
Both
the fabric and the elastic
easily
orthotropic
sets
superimposed
the g e o m e t r i c
of
cracks.
is the p r e s e n c e
onto
and
at
randomly
For
of two oblique
oriented
symmetry of this rock would
be
type.
However, be
other p o s s i b l y
rocks
to
characteristics,
angles
of m o n o c l i n i c
corresponds
if
shown type.
the
that
approximate the
Besides,
elastic
relation
(12)
is
accepted,
it can
symmetry
will
be
at
of
it will be c h a r a c t e r i z e d
least
by R G = i,
the
and by
a constant lateral strain coefficient: ~.. 13 = _mm E. E 1 m and
therefore
et al.,
4 elastic
parameters
only
(Berry
considerations
have
been
et
al.,
1974;
Crea
1976). The
formal
by
(14)
above
simple
way by K a c h a n o v
(1980), Oda
ter introduces a "fabric tensor",
(1982),
Oda
F, defined by
et
expressed al.
(1984).
in
a more
The lat-
207
Fij = 1~12~r3ninj
(15)
where n is the unit vector ing
the
approximation
correspond matrix.
to
the
normal
(12),
principal
In this reference
system,
8(i
-
of
of each crack.
directions the
of
the compliance
matrix
Accept-
this
orthotropic
tensor
compliance
is given by:
0
0
0
0
0
0
0
0
F33
0
0
0
0
0
F33+FII
0
F22 =
principal
direction
0
MOC
to the plane
the
92m)
F22+F33
37 E m
(16)
FIl+F22 The related
crack
A
particularly
statistically
E
isotropic
~
tensor
a
tensor F. For
= ~/~r3nin
eij
_
density
to the fabric
introduced instance,
is
closely cracks: (17)
important
crack
distribution
crack orientation
D.C. MODEL
\
0.6
Kachanov
j.
....
corresponds
("random cracks").
RANDOM CRACKS
Ere ~ 0.8
by
for penny-shaped
to
a
By means
RANDOM CRACKS 5.C. MODEL
\
O,Z,i
0.2
F\
0 0
0.2
Fig.
O.Z
0.5
6 - Young modulus D.C.
0.8
f.O
0
of a r a n d o m l y
a n d S.C. m o d e l s
0.2
-
E1
techniques
we obtain
0.6
08
cracked body predicted
as a f u n c t i o n
p a r a m e t e r e. D a s h e d line i n d i c a t e s proposed by BRUNER
of averaging
0.4
v
by
of the crack d e n s i t y the S.C. m o d e l
the relationships
16 1 - 9 In 2 1 - 0.3v m 1 - 0,59 m eoc E1 m + -9- E m
f.O
(Fig. 6, left):
(18)
208
1 G If
the
1 16 1 - u 1 - 0.2U m Gm + 9 Gm 1 - 0.5~ m eoc
assumption
tained,
obtai n e d cracks
by
6,
The
becomes
(Fig.
of
in the S.C. crack
between
much more
cracks
cannot
be
main-
complex.
formulations
Budianskii
(1974)
and
by
(1979)
Hoening
of
the
compliance
in
the
case
in
the
case
were
of of
random
a
set
of
4, right).
following
effect
(19)
and
right)
cracks
the
interaction
self-consistent
O'Connel
(Fig.
paral l e l
of
no
the situation Explicit
-
of
"
indications
cracks
models
density
on
can be drawn:
the
than
in
which
modulus
the
is of
reduction
D.C.
models,
practical
is
at
by
far
least
in
stronger the
interest
for most
induced
anisotropy
field
types
of
rock; -
for
a
single
racterized totally
planar
by values
different
However, models crack
are
it
adopted,
cracks,
that
found
the has
with
of
the
of R G m a r k e d l y
from
was
distribution
orthotropic,
set
by
principal
i, a trend
(1980)
compliance
first
than
by the D.C.
Kachanov
elastic (in a
lower
indicated
induced
directions
even
by a
which
is
formulations.
that
approximation)
is cha-
any
when
anisotropic
symmetry
corresponding
to
S.C.
that
which of
is
the F
of ~ tensors. The C.
models
cation
big cast
of
Hashin
of
The
on
doubts
theories.
who
showed
than
D.C.
those
model
always
prediction
centration, by Bruner
that
appears (1976),
introduction tions h i p
come
of
the
predictions of
the
quantitative
the
S.C.
that
predictions
their
composites
by other
undoubtedly
provides
but to
the
moduli
in which
an
cracks,
moduli
and
D.C.
of
furnished
rock,
the
validity
of
closer
of
application
properties
physically
the
between
the
The
elastic
of the cracked
dictions S.C.
some
the
(1970),
accurate
pliance
between
either
evaluation
more
differences
are
averaging a
lower
it cannot
be
the
results.
true
become
zero
unreasonable. incremental
seemingly crack
said
at
the
discussed
by
not
necessarily
procedures.
that
to
finite
pre-
con-
proposed for
satisfactory
(Fig.6,
the
crack
is adopted
a more
com-
instance,
modification
concentration
the
the S.C.
For
procedure
gives
for
limit
some
The
S.
appli-
models
was
and
left,
the
reladashed
curve). 3.2. C l o s e d C r a c k s The behaviour
presence which,
of
a
system
however,
compliance,
and
which
for loading
the rock.
closed
cannot
is strongly
be
cracks
induces
represented
dependent
on
the
an by
stress
anisotropic an
elastic
path
adopted
209
Only crack
in
the
surfaces
matrix
M cc
(non
is
for
the
realistic)
case
behaviour
a material
of
zero
elastic;
containing
a
the
friction
non-zero
single
set
of
between terms
the
of
parallel
the
cracks
are only cc oc = M55 = M44
cc
M44 In
these
conditions
.
(19)
the
anisotropy
would
of R E = 1 and a value of R G m a r k e d l y When only
a
friction
fraction
the crack;
where
of
this
~act an
between the
is
the
the
with
stress
active
normal
models
it
is
stress
path
(Brady,
1969).
For
rock
containing
triaxial
a planar
conditions,
increasing
axial
the axial c o m p l i a n c e
to
therefore
a given
an
value
is
taken
for
the
into
account,
deformation
of
the
plane
of
the
crack
angle.
for
so-called
a
by:
orientation
ple of
surfaces is
by
(20)
compressive
D.C.
deformability crack
stress
is given
characterized than I.
= T - a n tg#
and ~ is the friction In
the crack
shear
fraction
be
greater
set
of
aa,
to
calculate
distribution
when
closed
the
to the lateral
1 - 92 'cc 16 m M33 = 3 Em(l - 0.5~m)
a given
instance
at a constant
stress
and
and
possible
of
a cylindrical
cracks
is
confining
contribution compliance
the the sam-
loaded
in the
pressure
a c and
of
the
are given
cracks
to
by:
K(8)
(21) ecc
'cc _M~CC MI3 =
(22)
If (Oa - ~c ) (sin8 cos% K(@)
is g i v e n K(@)
Otherwise
= sin8 cos8 (sin8 cos@
K(e)
The
stress-strain
of
Only
in
Walsh
(1965)
conditions
closed
cracks;
cracks
are
rock.
stress
less A
path
Nemat-Nasser
will
(23)
- tg# cos2@)
will
a
loading,
same
effective
complete
(24)
an
than
ac/a a = const,
the open
the
was
of
slope
a certain
value
curve
compliance a
random
crack
ones
plain
change
above
stress-strain
a pp a r e n t
of
for
abrupt
only
containing
value
analysis
(1982 and 1983).
the
the
material
the
show
be m o b i l i z e d
evaluated
of
for
curve
cracks
uniaxial
stress
the
> 0
= 0.
the closed
linear.
- a c tg#
by
because a a.
- tg~ cos2~)
is
in
recti-
uniaxial
distribution
density,
in reducing
the
of
closed
the moduli
strain
conditions
presented
by
and
Horii
of
for
a
and
210
The
aforementioned
no residual the rock
normal
results
stress
exists
is m a c r o s c o p i c a l l y
stresses
at
stress which
the
of
the surfaces However,
the grains
microscopic
level,
by X-ray d i f f r a c t i o n
should
between
unstressed.
the elastic c h a r a c t e r i s t i c s
measured
are all based on the hypothesis
were produced
probably
by
in
the high
variable
some
(Friedman,
therefore be present between
of a crack
cases
1972);
stress
at
of
residual could
be
compressive
the faces of closed shear
when
the d i s h o m o g e n e i t y
originates
which
techniques
that
the
cracks, boundary
between d i f f e r e n t mineral grains. If
this
residual
still hold, but rel.
stress
(23)
is
indicated
is m o d i f i e d
(a a - ~C) (sin8 cos0 - tg~ cos2@) Now,
even
in
uniaxial
stress-strain
curve
loading
will
be
by Pc'
rel.
(21)
and
(,24)
in the following way: - (Pc + ~c ) tg~ > 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
"<<. 0/.
0
I
I
~2
O~
_
o.6
0.8
t,o
o
PLANAR SET 5. C. MODEL S A TURAFED
I
1
0.2
OZ
06
08
e
aspect S.C.
all m o d i -
06
PLANAR SET DC, MODEL SATURATED
matrix
and
O.8
0
~.
1977).
t.O
0.2
for also
(32)
the
typical
an o r d e r
are
regions
that
0.0____~3 a
Kw
a
conditions,
depends
of
the p o r e w a t e r For
ratio
bulk
cracks
. .1 .K w. Km where
isobaric
10
Fig. 7 - Young and shear moduli predicted by the D.C. and S.C. models for a saturated rock containing a planar set of cracks with a stiffness ratio ~ = 10
213
fied
(in
the
model).
As
reduced
in
greater
D.C.
model)
or
a consequence, undrained
than
1
is
the
only
slightly
anisotropy
loading
modified
ratio
conditions,
R E of
whereas
(in
the
RG
S.C.
is
much
rock
becomes
markedly
(Fig.8).
5 PLANAR SET D.C. MODEL
Z
the
PLANAR SET 5. C, MODEL
/
/
/
/
3
/
/
' ,/WRG
2
l
,
J
SATURATED
{
0
0.2
O
OZ.
06
SATURATED .........!
0.8
1.0
0.2
o.z
o6
o.o
1.o
o
Fig. 8 - R E and RG ratios in undrained conditions predicted by the D.C. and S.C. models for a saturated rock containing a planar set of cracks
The
bulk
undrained modified; by
the
modulus
loading as
a consequence
loading
conditions
_f%---t
I "",J
Em
08
0.2
of
a
randomly
conditions,
cracked
whereas,
the Y o u n g (Fig.9),
modulus
whereas
-
rises
markedly
shear
modulus
is o n l y
slightly
Poisson's
RA,,Do. CRAC.SF----
pc MOD L
rock
the
is
in not
affected
coefficient
beco-
RANDOM CRACK5 S.C. MODEL
!
i
t
f
I SA TURATED 0.2
5A TURATED 0,~
0.6
0.8
1.0
0
0,2
OL
a6
~8
LO
Fig. 9 - Young modulus of a randomly cracked saturated body in undrained conditions predicted by D.C. and S.C. models
m e s g r e a t e r t h a n t h a t of increasing crack density.
the
solid
matrix
and
tends
towards
0.5
at
214
3.5. Seismic V e l o c i t i e s The
measurement
of
or
in situ
is c o m m o n l y
used
of
the
rocks.
estimating the
For
the
"dynamic"
termined.
"static" elastic
of
seismic
for
instance,
However,
dispersion
the
many
the
sults o b t a i n e d
in many
results
100
j
of
the
which
have very
Italian
in
be
often high.
sites
been
been
proposed
on
the
far more
basis
easily
inconclusive
A summary
. Piani di Ruschio-iimestone o Timpagr=nde- Granite
samples
characteristics
mass
by
is p r e s e n t e d
Taloro -Granite En#racque - Anatexit¢ Entracque - Pa/eomy(on#e o Piedilego-Gneiss + Edolo-Schist
the
have
rock
can
laboratory
of
correlations
deformability parameters,
the c o r r e l a t i o n s
velocity
the e x p l o r a t i o n
of
the
for of de-
and
the
in situ
re-
in Fig.10.
-~
i
80
.~ + ....j
~
/
ZO-
o
/.
'°
"
.~
"t ;oe
."". .::'. ... -::L;: "" .o, "
$1A 5
I0
15 20 25 V 2 (kin/s)2
30
35
Fig. 10 - Correlation between seismic velocity in situ and Young modulus of the rock mass determined in various Italian sites (LEMBO-FAZIO and RIBACCHI, 1984) The compliances In
fact,
with the
the
respect crack
remain
velocities
stress to
level
the
in
surfaces
to
the
corresponding
situ
and,
are
solid
stress
or
therefore,
essentially
matrix
and
to
to the wave to
most
the of
related the
propagation
residual the
open
stress
closed
to
the
cracks. is low between
microfissures
inactive. The
pulse
seismic
corresponding
variation
orientation
elastic
compliance
anisotropy
is
must
of
the
present
deriving
negligible);
seismic
velocity
a s y m m e tr y from
the
therefore,
open in
as
a
function
corresponding cracks
dry
rocks
(if we
to that the
of
the
of
the
instrinsic
should
observe
2t5
an
orthotropic
anisotropy
distribution. simple
principal
set
single
the
more
of
cracks
determination
powerful
of
technique
directions
of
the
general is
situation,
superimposed
the seismic
to
fabric
velocity
determine tensor
of
the
or
onto
a a
is therefore
symmetry
the open
polar random
and
a
the
cracks
(Anderson
waves)
is
1974). A
strong
expected
difference
between
velocities
for
be to
in
seismic
saturated
the
rapid
velocities
conditions;
calculated
undrained
except
the
and
to
owing
compliance symmetry
dry
are
compliance,
-
in
a
The
and
et a l . ,
symmetry
when
on
the
loading
loading
(P
in
the
basis
conditions.
conditions
will
latter
of
the In
not
to
case
the
undrained
general,
show
be
the
orthotropic
when:
geometrical
symmetry
of
the
crack
systems
is
itself
ortho-
tropic; -
the
stiffness
aspect
ratio
When table,
the
ratio
in
situ
velocity
conditions.
This
the
why
difference
as the fissuring For values
a
of
high
measurements
are
be
strongly
can
and
the
between
(~ > I00),
that
is,
the
crack
partly
out
static
by
explain
"dynamic"
the
carried
influenced
moduli and
above
why are
the
varying the
always
the dynamic
water
saturation
relationships dispersed
moduli
and
increases
increases.
rock
the
very
will
"static"
is
low.
factor
between the
w
is very
containing
P-waves
are
a
planar
shown
in
set
of
Fig.ll
cracks, for
dry
the
principal
and
saturated
conditions. t.0
I/sat
V
~
Vf';3"
~ ~ ~ ~ ",........._ ..,....
0,8
I/sat
ll dry ,.,i _L saT"
0,6
oz
0.2-
o
o
PLANAR SET D.C MODEL
1
0.2
PLANAR SET 5.C t40DEL
I
o_z
06
0.8
It
1.0
0.2
OZ
0,6
0.8
1.0
Fig. ii - Seismic velocity (P-waves) in dry and saturated conditions for a material containing a planar set of crack. The results of the D.C. and S.C. models are compared We is
only
notice
slightly
that
the
velocity
influenced
by
the
along
the
saturation
plane and
of is
the
cracks,
almost
equal
Vl, to
216
that
of
the
intact
matrix.
conditions
is i n s t e a d
very
The
variation
of
orientation
is
shown
The
difference
strong
for
the
the m i n o r
seismic
in F i g . 1 2 .
We
between
dry
principal
velocity
notice
and
that
the
saturated
velocity,
with trend
of
V 3.
the
pulse
the
curves
;.0
t ,.,_ - F 5
.... Vm
sat
0.8
....1-
/
/ C ,
I 0.2 _PLANAR SET D.C. MODEL 0
PLANAR SET e=02 5.C. MODEL
e=0.6
30
60
90
O
I 30
0
] ........ 60 e
90
Fig. 12 - Variation of seismic velocity as a function of the orientation of pulse for a material containing a planar set of cracks. Crack density parameters were chosen in order to obtain approximately the same value.of V in 3 dry conditions for the D.C. and S.C. models
corresponding dry of
cracks
Nur
V 2 with
the
The
S.C.
(1971)
8 should
of a t e n s o r ;
shown
to
this
and
proved
be
D.C. that
similar
is c o n f i r m e d
seismic
in F i g . 1 3 .
It
to
models with
the
that
valid
by the
velocities
shape
(P waves)
is a p p a r e n t
is
that
somewhat
D.C.
for
in this
the
the
variation component
cracked
case,
in the
RANDOM
Vm
"-,,I~.._.....~
o z
I
~
o 13 -
rock
CRACKS
S.C. MODEL
l
I
0.2
oz
Seismic
06 velocity
I~ o8
t.o
(P-waves)
o in
o,2 dry
and
o.~
t 0,6
saturated
o8
are
preceding
'
02 -- RANDOMCRACKS D,C. MODEL 0
in F i g . 1 2 .
randomly as
For
normal
the c u r v e
__V__
Fig.
model,
for
of
different.
e
conditions
for a material containing random cracks. The results of D.C. and S.C. models are compared
1.o
217
one,
the
saturated
indicator
seismic
velocity
of the m i c r o f i s s u r i n g
is
not
conditions.
in
itself
The v e l o c i t y
a
sensitive
variation
~V
06
~V
_C. MODEL
Vm
O.k
0.2 D.C. MODE~"'~L~ ~ 0
I
0
0,2
0.4
0.6
0.0
1.0
v~y/vm Fig. 14 - Variation of seismic velocity between dry and saturated conditions predicted by D.C. and S.C. models for a material with random cracks
between value
dry
of
and
dry
saturated
velocity
for the a n a l y s i s 4.
i.
Rocks
The
rocks
tested
formations
Piedilago
gneiss. from
(01)
10%;
3.
the P i e d i l a g o
4.
stones,
poorly Ol
5.
near is in
gneiss Q
related
the
model
to
the
adopted
planar
is
cavern, and
was
fine-grained spite
of
("anatexite"). texture.
are
Mineral
400 m.
similar
B = 16%;
from
with
low
mica
is
explora-
a
to
that
0 = 26%;
quarry
for
i.
Ol = 28%; It
is:
Mineral
an
of about
taken
texture
Oligoclase
from
very
rock,
the
0 = 34%;
10%;
taken
composition
rock
the
enveloping
0.9%.
(O)
at a depth
texture
the site of rock
a
about
were
to
oriented
(biotite)
Orthoclase
samples
belonging
Alps.
mica
Porosity 20%;
The
The
is: B = 12%;
oriented
= 29%;
Edolo
It
texture
Entracque
of
Mineral
= 18%.
and W e s t e r n roughly
plant
gneiss.
ornamental
composition
of
schists
a
(B)
25%.
and
layers
Appearance
Quartz
oriented
shows
for a power
gneiss.
gneiss.
gneisses
feldspath.
(Q)
Ol = 38%;
Beola
independent
of the C e n t r a l
Biotite
Quartz
"Serizzo" of
were
It
of
is:
tory drift
almost
undulating
porphyroblasts composition
is
linearly
RESULTS
4.1. T e s t e d
deriving
2.
and
is almost
(Fig.14).
EXPERIMENTAL
metamorphic
conditions
a
a
markedly
content.
planar Mineral
Q = 26%.
fine-grained
composition
is:
rock
B = 8%;
with
a
0 = 30%;
= 28%.
schist.
Markedly
fissile
rock
with
an
oriented
but
218
undulating quartz 6.
San
texture.
(25%)
Fiorano
the E d o l o 7.
Baceno
and
feldspar
It
(35%),
summary
is
a
similar
in
fine-grained
texture.
4.2. U n i a x i a l A
is h i g h
(50%).
Other
minerals
composition
and
texture
oligoclase
Compression
of
samples
Mica
the
of
fissile
is a b o u t
50%.
rock
of
Alpine
uniaxial
gneisses
are
compression presented
tests
GNEIS'.
Em .0
6
f
f
J
.2
o
10
20
30 40 (MPa)
50
05
G'=O (~4Pa) 20 °
I 60 °
90" #
"SERIZZO- GNEISS
1
t
°
4
J
.4
,_.9_._-......11--------== f
.2
J iC=O(MPaj 0
10
20
30 40 (HPa)
an
minerals
in
E_
.4
with
Other
Tests on G n e i s s e s
PIEDILAGO
0
to
(15%).
results
various
markedly
content
I
0
are
(25%).
Quite
schist.
planar
are q u a r t z
laboratory
content
schist.
schist.
oriented
Mica
50
0°
30"
60"
90"
#
Fig. 15 - Results of uniaxial compression tests on various Alpine gneisses. The moduli are scaled with respect to the intrinsic values, calculated on the assumption that the mineral orientation is random
on
Fig.15;
219
ENTRACOUE
GNE/S,5
!
_E_ Ern .8
.6 f G" = 10 (M~a) .2
0
tO
20
20 ZO '5" (HPa .;
50
0°
"'BEOLA'"
!
30"
60"
30"
60 °
SOo
GNE/55
E,.~
.5
.2
0
0
I0
20
30 z,o (MPa)
50
0°
90 ° 4'
Fig. 15 - continued all
the
modulus
curves
have
been
Em
the
intact
comprised
of
between
Although symmetry in
was
Fig.15
passing
were
-
trends
The
taken
with
found with
the polar
to
texture be
the the
is
estimated various
planar,
orthotropic;
their
symmetry
the
ratio
but d e c r e a s e s
the
to
for
elastic rocks
is
axes
lying
the in
the
elastic
samples
analysed
one
of
axis of the m a c r o s c o p i c
peculiarities
of
the
various
the
planes
fabric.
rocks,
certain
are apparent.
anisotropy
levels,
respect
which,
macroscopic
Notwithstanding general
with
matrix,
70 and 80 GPa. the
generally
through
scaled
types
of
RE
is
when
rock;
often the
at
very
stress
20
MPa
high
(up
increases, RE
is
to
6)
at
low
at a rate
already
lower
increase
with
load
varying than
1.4
(Fig. 16). -
At
low
stress
levels,
the moduli
gradually
the
angle
220
5 ;) 2J 3) ~J
RE S
PIEDILAGO GNEISS SERIZZO GNEISS ENTRACQUE GNEISS BEOLA GNEISS
4'
Fig. 16 - Anisotropy ratios R in , E uniaxial compresslon tests for the rocks of fig. 15
3
10
e;
-
at
high
stress
levels,
50
instead,
to an o r i e n t a t i o n
of the sample
Above
certain
value
of
sample,
a
entation rate,
with
are equal -
20 ~o G a rMPaJ
20
At
low
the the
stress the
the
modulus
samples viour It
is
of
varia b l e
apparent
- the D.C.
determined
the and
for
a uniaxial
of
at
the
ori-
a
decreasing
50 MPa,
the moduli
axis
decrease
samples @ angles.
stress or
increasing along
The m i n i m u m
level
at low
at
loaded
of
5-10
e angles,
the value
MPa.
this
For
beha-
important.
that
the
behaviour
of
these
rocks
is
mainly
of the sets of cracks. of
both
analyzed,
available the
correspondence information particular
the
various
open
and/or
utilizing
with
could
the
be
simplest
from of
a
data,
theoretical real
one)
model
planar
the
factors, closed
- for
behaviour by
sake
for loading
the o r i e n t a t i o n
of of
cracks
is
theoretical cracks
of
with
simplicity
obviously
can only of
utilizing
(Fig.17A),
set
problem
simulations
obtained
the isotropic
The
dispersion
were
45-50 °.
-
model.
With
deriv i n g
at
the polar
containing
density
often
sharp
visualization
bodies
corresponds
whatever
increases,
a stress
(8 = 90 °) or at high
along
a better
for
modulus
8 of about
level,
modulus
by the c h a r a c t e r i s t i c s
For models
stress
moduli
very
occurs
loaded
the
is
is absent or less
controlled
the
stress;
levels,
plane
the
lowest
an angle
those of the solid matrix.
fall
schistosity of
applied
to 40-50%
stress;
of
the
with
the
have
rock;
various
fully
more
complete
stress-paths
(in
the samples.
shows
an
anisotropic
only;
to
take
the mica
not
a qualitative
plates
into
(to which
behaviour account
the
most
the
of
221
cracks
are
probably
connected)
a
uniform
distribution
of
1.0
orienta-
I
.....
E~
the
90"
/
0.8
0.6 J
~ f
r
60"
7
3 0 ~ ~
A
J 0.2 ~=0" ~aaar set ¢o~= 0.6
0
1 0
1
j.o~..____
O"
50
30 /.0 C5" (MPa)
20
10
30 °
60"
0=0
0°
I
L
90"
,)
2Q °
Emo.8 # = 60; 0.6
D
0.1, 0.2 pl~nari set e===O.61
00 Fig.
I0
20
~l"O
30 LO (MPa)
50
O"
30 °
60"
90"
O
17 - S i m u l a t i o n s of a n i s o t r o p i c b e h a v i o u r in uniaxial c o m p r e s s i o n t e s t s i n d u c e d by o r i e n t e d t e s t s o f m i c r o f i s s u r e s a i t h e r o p e n (with crack d e n s i t y e
) a n d / o r closed
(with a c r a c k d e n s i t y
oc e
tion
cc
)
within
an
angle
variability
of
the
nega t i v e
with
a
in
rel.(33)
of
average the
value
cracks
was
was
assumed.
represented
The by
fall
assumed of
residual
stress the
can be assumed
(33)
to be equal
the
curves
of
a
distribution
the
distribution
ratio
the
= eoc - exp(-p/pA)
p ~ was
The obser v e d
aspect
exponential
+ eoc(p) in which
±15 ° around
modulus of
in
Fig.15,
between crack
to 20 MPa.
the
density
the
initial
indicates crack
phase
that
surfaces
parameter
of
the
closed must
similar
loading
microcracks
be to
present. that
A of
222
e~c(p) The
= ecc - exp(-p/pB )
position
must
be
of
the
markedly
reasonable
choice.
or
open
of
both
"°
minimum
of
the
PA;
a
lower
than
The
effects
and
closed
of
E - ~a value
the
cracks
are
0.8
,
curves
of
2
presence shown
indicates
MPa of
that
appears
only
to
closed
in Figs. 17B
and
PB
be
a
cracks
17C;
the
,0 /
f
o. 1
E~
(34)
S
2.2L_.__~
....
C
"O Z.
0.2
planar Set ,©c=Cc¢ =,5 I I 0
Em
10
20
30
,~0 (MPa)
5D
°i
O•
50 ~
30 °
90"
#
l
0.8
05
D
°~6co.!
t
!
I
]
]p,.o~r ,~, ,o,:~,=x o
0
10
20
Z.O (MPa)
30
G
50
30 °
O"
60"
90"
Fig. 17 - continued
latter
already
However,
the
moduli
with
random
set
that
their
(33)
and
open
aspects (34) is
experimental
marked
many
to and
curves
of
RE
cracks
was for
of
be
for
indication
the of
increasing
and
by
"Beola"
the
major
the The
are
curves.
principal
therefore,
present.
set.
curves
experimental
of
modulus, must
planar the
the
values
represented the
Fig. 17D;
for
low
intrinsic
obtained
interesting
decrease
the
ratio
in
present
closed
adopted shown
characteristics
rocks
respect of
simulation
An
shows tested
same
a
assumed
distributions
results
rather
also
It was
of
similar
this to
the
gneiss.
these
simulations
stresses
observed
is in
that
the
the
samples
223
does
not
necessarily
imply
that
the
cracks
of
the planar
set
have
an
aspect ratio lower than that of the random cracks. Another curves
in
observation
the
requires
the
closure
samples presence
pressure
of
unreasonably
probable
that the effect
For aimed
at
that
low
the
trend
normally
large
number
of
Application
aspect
is due
to
of
ratios
of
the
experimental
the
planar
cracks
with
rel. (26)
for
some
to the presence
structure, a
or
very
(27)
cracks,
but
of tapered-end
low
would it
is
cracks,
in 3.3. the
the
conditions
a
(2-5 MPa).
indicate
as d i s c u s s e d
is
compressed
"Serizzo"
evaluation
of uniaxial
gneiss, of
the
a
more
complete
loading);
complete
investigation
compliance
matrix
was
(always
the tests were carried out
in
in a block
taken from the quarry. The initial reference system X Y Z connected to the block had the Z axis p e r p e n d i c u l a r to the planar structure of the rock and the X and Y axes in an arbitrary position. From direction, strain were
and
the
blocks,
were 2
extracted
transverse
determined.
terms
of
the
Each
cylindrical
and
tested;
strains
of
compliance
(together with analysis.
27
these
For
instance
at
in each
(according
strains
matrix,
their variance
samples
is
which
stress
45.7
-5.5
(2.2)
-6.6
the
linearly
and covariance) a
sample to
could
level
terms and their standard d e v i a t i o n s are
oriented the
different
longitudinal
scheme
in
Fig.l)
related
to
the
therefore
be
21
estimated
by means of a r e g r e s s i o n of
i0
MPa
the
compliance
(10 -3 GPa -I)
0.0
-6.0
-0.6
(1.2)
(I.I)
(2.9)
(3.8)
(3.2)
50.6
-6.6
-3.4
-1.3
-3.1
(1.7)
(i.i)
(2.7)
(3.6)
(3.2)
57.9
-6.6
-9.1
-0.9
(1.5)
(2.5)
(3.1)
(2.5)
M'. = 13 (s.d.)
in
137.2 5.1 (4.7)
4.9
(7.8)
(1.0)
121.8 (5.8)
0.9 (10.7) 107.7 (5.4)
By type Z
of
axis
test,
means
of
anisotropy is
a
of
principal
by comparing
(21 coefficient) regressors
statistical the rock. axis
of
tests For
it
is
possible
instance,
anisotropy
can
to
the hypothesis be
tested
the residual mean square of the c o m p l e t e
with
that of a 1 3 - c o e f f i c i e n t
analyse
regression,
that
with
the the
the
F
regression in which
(M~4 , Mi5, M~4 , M~5 , M~4 , M~5 , M~6, M~6 ) are eliminated.
8
224
To lated
determine
to
that
identify
the
direction
of
of
whether
an
the
behaviour
orthotropic
material,
principal the
strain
linear
directions
compressibility
of
the
it
is
1 2 3, tensor
rock
can
firstly that
which
is
be
assimi-
necessary the
to
principal
is d e f i n e d
by
the
relation
s~ Sij
s;/2 s;/2
=
Sl
S~/2
symm
(35)
S~
where 3
' = j[
=1 M'3i
Si
The
compliance
coordinate
system
elimination terms
in
variance Venant
of
1
13
matrix 2
of
the
Fig.18
does
shows
the rock
the
the
best
rock
is
conditions
corresponding
not
observations. for
of
Orthotropic
regressors,
rel. (2),
conditions
3.
M
a similar
(R G = i)
referred
to
the
are
satisfied
if
the
to
significantly In
then
the
zero
increase
way,
the
compliance
the
validity
residual of
Saint
is t e s t e d .
estimates
of
the
principal
strain
direc-
X
Y
Fig. 18 - Principal "strain directions" for a block of "Ser~zzo" gneiss as a function of the stress level in uniaxial compression tests tions that
as up
a
function
to a s t r e s s
considered
of
of level
the
the
stress
level.
of
20 M P a
the
orthotropic
Statistical
anisotropy
type
and
of
tests the
that
indicate
rock the
may
be
Saint
225
Venant
condition
transversally cipal
axis
(less
than
is
valid.
isotropic
of
the
However,
and,
the
besides,
compliance
rock
axis
matrix
cannot
Z
only
be
considered
corresponds
at
very
low
to
a
as
prin-
stress
levels
2.5 MPa).
The
values
directions
of
1 2 3 are
the
moduli
given
relative
in Fig.19.
The
to
the
trend
principal
of
30 E
strain
the curves
and
the
/"
6 ( MPa ]
1 1
~
f
I
I
E2~_-- "~
20
f
612
~.
70 623
0 1o
0
30
20 ~
ZO
rMP~ I
Fig. 19 - Values of the Young and shear moduli for the principal strain directions obtained by the statistical analysis of the uniaxial compression tests
observation and
3
that,
varies
structure
of
probably
than
20
the
well
MPa;
pulse
that
the
the
variation
saturated Pros
and
many
researchers 1974). Dry
sample
introduced
was
80°C first
in the
was kept
an
of
the
models data
seismic in
(Borelli,
et
conditions of
the
(1968),
(Bur
at
compliance
measured
conditions
Bur,
temperature
of
Babuska
orientation the
angle
of
rock
when
are
with
can
the
of
about no
stress
forced
axes
1
macroscopic
into
45 ° ,
longer
be
in greater an
elastic
are obtained.
The
in
the
agreement
values
of elastic
Seismic Velocity
was
in
experimental
results
orientation
sample
towards
4.3.
by
the
rock
stresses,
roughly
by means
if
abnormal
increasing
values
indicates
represented
model,
at
from
and
were
for
a
obtained few
subjected
vessel;
underwater.
1983). it
1973;
al.,
the
ve l o c i t y
spherical
days.
has
as
This
by To
to vacuum measurements
et
both
utilized al.,
storing obtain and
function
technique
been
Thill
a
samples,
the
was since
the and
proposed then
samples
deaerated
carried
of dry
Friedman
saturation
then
were
1973;
in
out
by and
at
a
conditions water
was
while
the
226
The further
results
obtained
for
the
Piedilago
gneiss
are
presented
down.
PIEDILAGO
GNEI55
DR)"
SATURATED
Fig. 20 - Isolines of seismic velocity (km/s) in dry and saturated spheres of the piedilago gneiss. The orientation of the measurements is shown on the left In directions
each
sample
(Fig.20).
the
The
velocity
shape
of
(P
the
waves)
isolines
was
measured
correspond
6
to
in
an
metry:
(km/s) 5
u
. - ...
..
,,
.
•
:::~
SAT
sym-
the
velocities
in
conditions
dry
equal
1.75,
km/s,
tions, 5.15
y
km/s. sis
to
4.70,
and
5.25
On of
30
60
~
9o
Fig. 21 - Seismic velocity for the Piedilago gneiss as a function of the pulse orientation will respect to the (approximate) polar axis. Lines A and B are the trends given by the relations V 2 = V 2 COS 2 8 + V 2 sin 2 @ 3 I
B) I/V 2 = cos28/V~ + sin2e/v~
the the
sition, o
mic the
in
condi-
ralogical
0
and and
saturated
DRY
to
3.15
3.45
I
prin-
cipal
were
A)
almost
orthotropic
V
55
compothe
seis-
velocity solid
of
matrix
(neglecting intrinsic
its aniso-
tropy)
was
mated
equal
5.9 km/s.
ba-
mine-
estito
227
The
deviations
this
rock
and
Therefore, of
the
that
can
the
1 V2 which
=
is
theoretical model;
the type
the
to
the
conditions
function
of
(Fig.22).
the
a
first
of
the
D.C.
it
approximation
is represented
curves
model
appears
relatively
is
(line
to
be
A)
and
for
analysis.
as a function
different
better
weak
from
even
more
represented
that from by
a
(line B) (36)
relation
valid
for
the
Young
modulus
of
a
the Saint V e n a n t relation.
variation
apparent
in
are
sin2e 2 V1
+
m a t e r i a l satisfying
saturated
shape
the
similar
If
neglected
The
cos2e 2 V3
conditions
the seismic velocity
S.C.
relationship'of - -
be
e.
by
of
axysymmetric
in Fig.21
angle
predicted
from
for
of
the
samples
corresponding
seismic
velocity
between
of all o r i e n t a t i o n s
dry
This behaviour
velocity,
is not
a
dry
are plotted
linear
trend
in agreement with
and as a
becomes
the theore-
L.
i AV (krn/s)
PIEDILAGO GNEISS I.......
3 • o ,. ,~'° o
"'i.... V~
i 2
3
6
Vdr)~ (km/s )
Fig. 22 - Variation of seismic velocity between dry and saturated conditions as a function of the pulse orientation (Piedilago gneiss) tical
prediction
probably
derives
random cracks, On analysis, in
the
for from
containing
a
planar
set
of
factors
cracks,
and
(influence of
slight d i s h o m o g e n e i t i e s of the rock). basis
in which
a matrix
rocks
the c o m b i n a t i o n of various
whose
the random cracks,
of
dry
the planar increased leads
velocity
values,
set of cracks compliance
to estimate
an
approximate
is assumed
derives
from
a crack density
S.C.
to be embedded the
presence
parameter
of
respec-
tively equal to 0.3 for the random set and to 0.2 for the planar set.
228
If
instead
for the random The less
a D.C.
set and
saturation
is adopted,
to 1.25
indications
reliable
model
given
because
even
can
lead
the
"dynamic"
estimates
for the planar by
the
a slight
to serious
equal
to 0.85
set.
saturated intrinsic
errors
are
seismic
velocities
anisotropy
in e s t i m a t i n g
or
are
incomplete
the m i c r o f i s s u r i n g
conditions. If velocities, GPa,
are
which
are
compared
triaxial
principal
equal
with
compression
601I
"static"
moduli
(Fig.23),
the
PIEDILAGO" GNEISS UNIAX1AL TESTS
[ ~ ----
TR,'AX,~L
" / /"// %11
C
from
GPa
and
found
in
latter
prove
the to
seismic about
25
uniaxial
to
be
or
markedly
] 7 I
TESTS
~
~
i,
'
J
i
R
8.5
! !
!I I
,V / / :/,"
derived to
the
tests
moduli
respectively
t
i
~
40
80
|
I
r20 ~" (MPa;
Fig. 23 - Static principal moduli of the Piedilago gneiss (as a function of the stress level. The curves for uniaxial conditions correspond to those shown in Scaled form in fig, 15 lower;
the
discrepancy
modulus
E 3 and no c o n v i n c i n g 4.4.
In
It
is
anisotropic denced
by
in situ
-
the
the
the
pressure
measurement
for
the
minor
principal
can be advanced.
to
compare
scale
of
tests
1984).
For
the
deformation
laboratory
samples
(Martinetti in situ
and
tests,
behaviour with
that
Ribacchi,
of evi-
1983);
careful
techniques
(having
a diameter
included:
application
depth)
the
high
Tests
plate-loading
to 0.5 m)
uniform -
at
Ribacchi,
which
explanation
Plate-loading
flattening
equal -
and
adopted
surprisingly
interesting
rocks
Lembo-Fazio were
Situ
is
and
smoothing
by means of
the
of
the
of a flat load
by
loaded
faced
means
area
rotary
of
a
bit;
flat-jack
to
ensure
a
distribution; of
in a c e n t r a l
the
displacement
borehole
below
at
various
the loaded
points
area.
(up
to
3 m
229
The For
each
load
is
stage
1
MPa/min;
order
to e v i d e n c e
(almost)
load
brought
it
tests
is
is
any
zero.
isotropic,
in
the
about
to
applied
possible
This are
\
\
\,\
cycle
,_
\
\
to
is
(usually
its
for
loading
twice. the
8 and
level i0
behaviour,
repeated
by
4,
maximum
constant
rheologic
performed
\\
stages
maintained
\ \\\\', \
three
or
at
When
rock
a
20
then
it
the
firstly
12
MPa).
rate
of
minutes
in
is
reduced
rock
is
an-
in a d i r e c -
\
\
\ \
/#,
,
\
:~\ -\
, -'\J~/,.
,_ , \ , \/S ", \
~.
"~ ~
\ \
/
~\ ~
i
'
i\'\,',.
\o. ~
\ ~ "\
\ }~
~
inside a drift in
~
an anisotropic rock
,,<\\N
tion
perpendicular
subsequently
to
the
in a p a r a l l e l
A typical
result
planar
direction
of a test
structure
(A
in
Fig. 24)
and
(B in F i g . 2 4 ) .
is s h o w n
in Fig.25.
12
PIE~QtL~GO GNEI55~
P (MPa)
///# t////
Fig. 25 - Example of the results obtained in a plate-loading test in the Piedilago gneiss (load applied,.~ to the planar structure, displacements measured at the surface below the centre of the plate)
/ "
'
l/z~
0
o;
0
In
Rock
"deformation
Mechanics
modulus"
the b e g i n n i n g the
of
the
immediately
phase.
The
02 03 DISPLACEMEN[ (.ram)
moduli
Ed test,
it on
and
recoverable are
is
the an
customary basis
of
"elastic"
displacements
calculated
on
the
practice
the
total
modulus measured basis
of
to
evaluate
displacements E e on in the
the an
a
from
basis
of
unloading
relationships
230
which
are
valid
for
a homogeneous
and
isotropic
medium
(Steinbrenner
approximation) .
1,,(.
2a
,,~
E_5"
h/a ~ ~
E 7.2
~
"~"7. h/a
o
1,0 ...... i
Fig.
26 - C o r r e c t i v e
factors
the s u p e r f i c i a l
.2
.Z
.6
.8
1.0
for a b e t t e r ~s
estimate
o f the true moduli
The been
Italy.
presence
and E
calculated
With
superficial estimated
E (6Pa; ;O-
the layer
in
all
layout
the
in
with
adopted
and the modulus
in
our
of
the S t a £ n b r e n n e r
respectively
superficial
situ
.~
layer below
investigations tests,
of the
the
of the underlying
the plate
carried
modulus
out
Es
of
Ed Ee
D O L OMI TE
~
} "~
~o
UNDERLYING ROCK MASS
e...~/ISUPERFICIAL ip------l-- - - ' ~ J LAYER
k
E
40-
8
12
p
{MPa)
TALORO GRANITE
Fig.
27 - Moduli
of the s u p e r f i c i a l
layer a n d o f the u n d e r -
(GPal
3020-
lying r o c k
~
' UNDER
100
~ SUPERFICIAL ---..o -----..o J LAYER 1 t i "~ 4 8 12 p (MPaJ
mass obtained
in twQ types of s t r o n g l y fractured
in the
rock mass E can be
separately.
. . . .
J
of a low modulus
evidenced
.6
rock E, o n the basis
a p p r o x i m a t i o n from the relative d i s p l a c e m e n t s points A and B and of the p o i n t s B and C
has
.~
ES/E
layer E s a n d of the in depth
of the a p p a r e n t m o d u l i
. '~
0
ES/E
rock masses.
231
Numerical analyse
the
and
errors
experimental
which
can
investigations
be
introduced
by
were
carried
simplified
out
to
interpreta-
tion procedures. The
influence
evaluation the
in
apparent
ficial wer
(Fig.27);
the
increase
(as
this
in depth,
always
but
on
load
in
are
approximation medium
is
the true moduli
the
maximum
error
in Fig.24, that
pond
to the minor
figuration
the
or
of
on
the
shown
both
in
moduli Fig.26;
for the super-
in most
the
B
cases
is
lo-
quantify;
but
out
in
to
The
it
also
on
the
always layer
is
rock
the
at
sometimes
errors to
mass
decrease they
the plate
transversely
interpretation E,
obtained
modulus the
corrective
of
introduced
be
taken
depends
into
not
only
initial
state
of
isotropic
rocks,
as
drift.
moduli
principal
decrease.
underneath
test
simplified
correspond the
to
carried
apparent
rock
superficial
sometimes
the
behaviour
in-depth the
stress
by
are a
tion
non-linear
in
the e x p l o r a t i o n
tests
evaluation,
a
difficult
state
applied
around
of
whereas
Fig.27)
the
When
show
moduli
factor that
shown
two-layer
E underestimate
loads,
account
stress
and
tests
increasing
the
Steinbrenner
20-25%.
The
by
the
dishomogeneous
moduli
layer
than
a
of
E 3 and
major
that
modulus
coefficients
is
based
on
the
assump-
A,
corres-
obtained
in con-
in c o n f i g u r a t i o n
of
those
E 1 . For Fig.28
a more can
be
accurate utilized
I AI
El
BI
E3
1.5
RG=2 B)
I
1.0 A)
i 2
.5
3
Fig. 28 - Relationship between the true moduli E l and E 3 and the apparent moduli El and E3 for plate-loading tests in trasversely isotropic rocks
4
El~E3
(Lembo F a z i o
and Ribacchi,
However,
the
two
1984).
types
of
test
shown
in
Fig.24
are
not
suffi-
232
cient that is
for
determining
is E 1 , E 3 and
usually
tively
GI3
assumed.
greater
therefore,
the
(or
When
and
the
all
RG)
and
than
anisotropy
elastic
parameters
therefore
R G = 1 the
smaller
true
relevant
true
the
an
RE
is
priori
E 1 and
apparent
ratio
a
moduli
of
values
greater
the
rock,
value
E 3 are El
and
than
of
E3'
the
RG
respecand
apparent
value. The
results
Piedilago
gneiss
ration
will
B)
plete
data
wide)
at
of
the
in situ
(6 tests
in
be d i s c u s s e d
are
of
configuration
in detail,
available.
a depth
plate-loading
The
380 m
site
below
and
because
of
the
ground
6
for
test
surface.
PIEDILAGO UNDERLYING ROCK MASS (• 25" 3.0 m )
A
tests
performed tests
this was
in
in the
configu-
rock more a
drift
Average
com-
(2.5
fracture
m
spa-
GNEISS
SUPERFICIAL LAYER fO.07- O.25 m )
LABORATORY SAMPLES E(secant)
50
~----
zo
Ed
3O L o..
0
" ~ / / j
uJ 20
10
± o-..___..~_.__.--~ io---o-
o
~
8
12
z
0
i
---o
o
o
~
,)
~
p (MPa)
Fig. 29 - Young moduli (secant values) for the Piedilago gneiss obtained in situ and for laboratory samples
cing the
of
the
water
excavation
about
the
pore
the
log
in
test
the
km/s
found
level
displacements from
any attempt
plates
the
the
of
the
obtained the
in
are the
Steinbrenner
at i n t r o d u c i n g
The
drift
was
modification
saturation
velocity
which
tests
moduli
m.
situated induced
conditions
of
below by
the
the rock
are not known.
borehole.
obtained
the
0.45
p r e s s u re
seismic
axial
were
and
in the l a b o r a t o r y
results
load
deriv i n g
drifts
the loading each
The each
was
but
For
V 3 = 4.65 values
mass
of
mass b e l o w
sonic
rock table,
was
Average are
determined
values
practically
tests on s a t u r a t e d are
summarized
evaluated 6
tests;
on
the
approximation
the c o r r e c t i v e
a
and
coincident
with
the
29 and
30.
For
of
average
samples.
basis
the
conventional were
factors
of
km/s
in Figs. the
by means
V 1 = 5.25
utilized,
of Figs.
relations without
26 and 28.
233
In
the
represented a value This
the
choice
be
-
factors
the
were
that
the
of
load
the
is
mainly
-
both
10-13
due
in
loading
29 and
they
that the
to
rock
mass
and
the
plate.
variable
the
following
and
if
(RE = 1.2)
is
anisotropy
reduction
of
much
lower
with
the
of the rock; than
greater
minor
cor-
independent
agreement
higher
times the
the
of the moduli)
composition
2
indications
modified
in good
about
strong
the
is
are
marked
at
by
is p r a c t i c a l l y
much
for
compression,
it
are
higher
is
plate.
for the c a l c u l a t i o n
samples;
samples
applied
owing
30,
mass
the
load
substantially
the plate
rock
laboratory
in u n i a x i a l
the
arbitrary,
be
the
times
to
to the
the
on
the
therefore,
respect
of
to
the
laboratory
samples;
and
not
of
moduli
from the m i n e r a l o g i c a l
of
laboratory modulus said,
below
of Figs.
ratio
applied
moduli
secant
introduced
the
value derived -
behaviour
extent
would
anisotropy
than of
the
the analysis (they
the
corresponding
state of stress
drawn
rection
of
stress
is to a large
Form can
figures
by means
of
uncertain
same
those for
modulus;
of the
the it
laboratory minor
of
the
major can
be
samples
modulus
with
in situ conditions;
situ
major
and
6
minor
moduli
decrease
at
increasing
_PIEDILAGO GNEISS
£1
5
h
~...._ LABORATORY SUPERFICIAL LAYER
~--E---d~---- _~UNDERL~NG I
0
"~"--r~--'~7~ L~ ~ 4 8
"-~ROCK MASS 12 p(MPa)
Fig. 30 -'~Anisotropy ratios fot the Piedilago gneiss obtained in situ and in laboratory samples
234
plate
pressure
in agreement with the general behaviour
all the types of rock masses -
the
behaviour
of
the
observed
in
(see Fig.27) ;
superficial
layer
is
intermediate
between
that of the rock mass at depth and that of the laboratory samples. The
results o b t a i n e d
are shown
i~ Fig.31;
80
in other
types of m a r k e d l y
the in situ behaviour
of the rock mass
BACENO SCHIST
EDOLO SCHIST
schistous
rocks
is similar
5. FIORANO SCHIST
f a b o r o t o r y test Ed Er
//
60
..... ~e
o,.,,.
//o %, g.
40
\
"%
m e
30 ~
".......... •~
t.--//
//,,.o
,-e
20
" °''°~-o_...o
fO -.t-,~
_L ~-.O-" 0
..i
i ~
0
i
r
D
8
p fMPa)
-.O-'O
L
J
Z
8
12
p(HPa)
0
_L
~
z
i
~ 8 p (MPa)
12
Fig. 31 - Youn~ moduli for various schistous rocks of the Alps obtained in situ and for laboratory samples. Only the values of the rock mass below the superficial layer are indicated for the in situ tests in many
respects
ratio varies
to that shown by the P i e d i l a g o gneiss.
greatly
from one
rock
type
to another;
the mica content is a p p r o x i m a t e l y the same In
the
perpendicular observed. is
to
the
A typical
represented
cycle,
plate-loading
in
delayed
layer and at depth,
it
a
after
are
is apparent
theologic
be
observed,
in
value),
were
in Fig.33
as
a
was
that
after
often gneiss
the
first
fully reversible. both
in
scaled
upon
function
that the delayed d e f o r m a t i o n s
direction
Piedilago
the
the load had been m a i n t a i n e d
arbitrary
a
behaviour
in the
noticed
recovered d i s p l a c e m e n t s
represented
out
from a test
may
displacements
(a c o n v e n i e n t
to the immediately values
Fig.32;
carried
fabric, taken
in spite of this,
(about 50%).
the d e f o r m a t i o n of the rock is almost The
minutes
planar
example
test
The anisotropy
down with
unloading. of
superficial
constant
the
are e s p e c i a l l y
for 10 respect
The average
load
level.
important
It for
235
the
superficial
layer
when
05
the
load
is
applied
normally
to
the
planar
..........
EO~Z, . . . . . . . . . .
Lt__. ~ r~ ~0,2
...........
:~
i . . . . . .
l .....
z
I
i
I
',
~ MPa
II
........
t
:/i
~ | .....
w....
|
:r ,
l 1 t ....
o
!
3
x
5
hours
Fig. 32 - Rheologic behaviour shown in situ by the Piedilago gneiss when loaded I-~ to the planar structure. The surface displacements at the center of the loaded area are indicated structure. behaviour
In is
the
underlying
less
important
rock and
is
mass
the
not
much
(short
term)
influenced
rheologic
by
the
load
direction. 30 PIEDILA GO GNEISS % J.
20
t "6 ~u 10
0
Fig. 33
To
explain
discrepancies of
the
p (MPa)
1)
Delayed deformations (after 10 minutes) scaled down with respect to the immediate recovered deformations, for the superficial layer and the underlying rock mass.
the
between
following
;
0
behaviour the
factors
of
in s i t u c a n be
the
and
rock
masses
laboratory
invoked:
and
values,
especially one
the
(or m o r e )
236
i)
an
ii)
an
influence
of
the
in situ
state
of
stress
on
the c o m p l i a n c e
of
in
the
in
the rock mass; influence
situ
(the
of
the
saturation
laboratory
conditions
samples
were
always
rock
tested
mass in
"dry"
conditions); iii)
an
irreversible
from iv)
a
the d i s t u r b a n c e
similar
storage As state
of
initial
to
fabric
after the
stress
on
Fig.34
walls
microcrack
a
rough
of
fabric
deriving
operation;
but
occurring
of the samples
point,
stress
behaviour
the
of the coring
the coring
the
of
modification,
first
isotropic
isotropic
modification
("delayed
evaluation
the drifts
(corresponding
gradually
can
of
be
during
rebound"). the
made
pre-existing
by
to the o v e r b u r d e n
assuming load)
an
and
an
a distance
of
of the rock mass.
shows
that
in
the
zone
~ \
~
.
.
.
comprised
between
111
INITIAL STRESS
.
STRESS APPLIED BY THE PLATE •
Fig. 34
0.25
and
1.0
Comparison between the axial preexisting stress on the walls of the drifts and the stress applied by the plate at the maximum load. m below
to the e s t i m a t i o n comparable some would
of
and
the
even
the
plate
(which
the moduli), lower
in the
the
than
which in situ
the
are
gives axial
the
greatest
stress
applied
pre-existing
stress.
oriented
conditions
normally and do not
to
contribution by
tests
is
Therefore,
the
plate
contribute
axis
to the
of the rock.
However, while
of
microcracks
be closed
compliance
ANCHORS FOR DISPLACEMENT MEASUREMENT
an
undoubtedly
examination effective,
of
Fig.23
cannot
by
shows itself
that
such
explain
the
a
factor, high
E3
237
value o b t a i n e d As
to
exclu d e d
in situ. the
that
partly
influence
water
of
the
saturation
overpressure
"undrained"
loading)
in
are
the
conditions,
microcraks
induced
during
it cannot
be
(conditions
the
of
plate-loading
tests. In a rock results
of
undrained same
in
the
two
for
Let with
RE = 4
and
indicate
3 times
greater
rises
by
indicate
moduli In
than
2.
The
effect
in the
S.C.
behaviour
In
or
practically
set of cracks,
as
is
the quite
is shown
by
model. contains
shows
that a
E3
a
and
in drained
E 1 = 0.77 the
minor
conditions
condition
set
of
open
conditions
plate-loading
conditions
in drained
planar
test El,
(see
plate-loading
would
with
modulus
an
becomes
Fig.7), tests
Eu 3 = 2.4 E 3 and E 1 u = 0 83 E 1 with
and
would
an apparent
= 1.38.
is t h e r e f o r e
models.
could
situation
The
the
drained
8).
and
Fig.28,
this
in
6
= 1.25
value
moduli
ratio RE
E3
similar is
Fig.4 to
orientation,
modulus
rock
undrained
the
about
apparent
anisotropy
e = 0.6;
according
= 2.50.
be
a planar
the
a random
Young
on the D.C.
that
a density R G = i;
RE
(Figs.
based
with
would
the
containing
suppose
apparent
apparent
ter
conditions
example
us
tests
because
a rock
the following
cracks
microcracks
plate-loading
conditions,
different
RG
containing
also
It be
is
quite
important
to
noted
caused
be by
the
and would
that
an
be even
apparent
dissipation
of
grea-
rheologic
water
over-
pressure. However, her
the
partially mass
with
available
premeability
of
undrained
below
Regarding
the
that
high
core
when
coring
stub,
stress
evidenced axis
are
that
during
possibility
is
(Durelli
the
was
low
test,
of
effected
et
a
at
fabric
concetrations
al.,
strains
which
it is not p o s s i b l e
materials
to decide enough
least
to in
whetinduce
the
rock
layer.
stress
tensile
induced,
data rock
conditions
the s u p e r f i c i a l
well-known
initial
the
in
a
1968). in
sometimes
a
modification,
occur rock
In
mass
the
the
it
bottom
subjected
particular,
direction
cause
at
core
to
to
Stacey
parallel
to
break
is
of
a
high
(1982)
the
core
down
into
thin disks. It these
is
quite
stress
aligned
possible
concentrations
along
the
fissility
therefore cause
the
plane
of
that,
in
an
anisotropic
formation
or
the
thereby
rock,
extension
of
rock, cracks
increasing
its
anisotropy. Also coring
bit
gested
by
mechanical can p r o d u c e Tullis
stresses some
(1977),
in
and
effects order
on to
the
vibrations
the crack explain
induced
geometry,
the
strains
by
as was
the sug-
measured
by
238
various
researchers
stress-free) The also
a
after
during
its
extraction
Because
of
differential
stress
residual
up
this
after
and
a
but rock
suggested
anisotropy
at
MPa
of
of
has been
have
X-ray
of
the
the m a c r o s c o p i c
(Nur and Simons,
30-40
coring,
is not always clear-cut.
stresses
means
the
conditions
environment
sample
to
by
with
For
time
1970).
been
level Values
measured
diffraction
microfissures
some
of
instance,
in
techniques
a
samples Buen
The
during
(1979)
rate
likely a
to
slight
storage
observed
also their
velocity
decreasing
are
stresses;
and sometimes
storage.
at
residual
the
samples,
of
continued
of
of
variation
of gneiss
the
it
that
the
strength,
variation
was
throughout
may
the
about obser-
(3 months).
A good
indication
of
strain
unloaded
observations
instrumentation temperature
extension release
process.
velocity
and
and
gradual
increasing
in some
in are
drift,
and
(Teufel, 1982) which
high
humidity
vation period
Such
just
non-homogeneity
granites
the
decreased
of
occur
microfissuring
its o r i g i n a l
the
widening
facilitate
10-20%
(macroscopically
1972).
or
seismic
the
in a s t r e s s - f r e e
and
accompany thermal
of
from
minerals,
be present
(Friedman,
isolated
although direct evidence
of
can
quartzites
phenomena
modification
by many researchers, constituent
of
blocks of rock.
above-mentioned
slow
A
the overcoring
samples
difficult and
humidity.
are
the
have
easily a
rock
(delayed
of
the
perturbed recent
occurrence
reached
of
because
However,
confirmed cases
fabric variations would be the o b s e r v a t i o n
of
by
of
variations
of
accurate
delayed
a magnitude
rebound).
influence
investigation
rebound
of about
50%
strains,
the
instant
strains. In explain
the
case
of
to cause
such
a large
is not felt at all by for
the
how an extension
the
scarce
saturated
Piedilago
site,
however
and widening
of
the cracks,
modification the
seismic
sensitivity
conditions;
a
of
of
velocity,
even
is difficult important
the c o m p l i a n c e
this parameter
possible
it
characteristics,
if allowance
to cracking
explanation
is
to
enough
that
is made
intensity the
in
in
situ
c o n d i t i o n s were not fully saturated.
5. C O N C L U S I O N S
A marked boratory mechanism fied.
samples
difference has
controlling
been such
in c o m p l i a n c e evidenced behaviour
by
between the
could
in situ
rock
tests,
but
not
precisely
be
the
and
la-
effective identi-
239
On
the
regarding
other
the
real
hand,
in
similar
situ
difficulties
microfissuring
of sampling p r o c e d u r e s were met by other
in a paper by Wang and Simmons
Practical
implications
mechanisms discussed In fact, the
coring
of
parameters
effects
as is shown by a
by Tullis
(1977)
and by the
(1978).
greatly
if the m i c r o f i s s u r i n g or
depending
varlations
by delayed
rebound
of the rock m a t e r i a l s
large
should
the
on
which
of
the
in the p r e c e d i n g p a r a g r a p h applies.
disturbance
characteristics behaviour
vary
uncertainties
and
researchers~
review of many e x p e r i m e n t a l data presented discussion
and
conditions
engineering be
based
in
mainly
are not important
structures,
on
are
situ
caused
in unstressed
and
the
cores,
at all for the
choice
measurements
by the
of
only;
design
non-linear
effects will be less important. On
the
pressure situ
contrary,
effects
tests
in
would
if
the
cracks,
be
much
sity
behaviour
of
the
medium,
with
a
intact
rock
one of
the caverns
conditions
having
rock
than
those
corresponds
of
spaced
(Edolo plant)
in the fracture
showed
network
account
size
isolating
Piezometric
of
poro-
blocks
of in
water
pressure the exca-
not
known;
lengths depend on the size of
the
the slow rate of excavation,
defor-
measurements
"drained"
the
in
In fact,
a double
after
that the natural d r a i n a g e the
of
are
given
on
that
pore
the
the
blocks
however,
not
for
reached
variations
to
by
within
pressure
and
relevant
to
that
due
are q u i c k l y
Pore
blocks
is
obtained
of the caverns.
fractures
low p e r m e a b i l i t y .
vations.
the
moduli
the excavation
mass
network
very
behaviour
undrained
greater
m a t i o n of the rock mass during the
observed
the
excavations,
and
taking
into
fully drained c o n d i t i o n s p r o b a b l y
occur also w i t h i n the blocks. It cannot deformations
be
excluded
observed
in
however,
some
that
caverns
the not n e g l i g i b l e
may
be
partly
delayed
due
to
the
transition between p a r t i a l l y undrained and fully drained conditions. The an
complex
important
mechanical
influence
determinations
behaviour
also
in
the
of
the
anisotropic
interpretation
of
in rock masses which are based on the m e a s u r e m e n t
of the
In Italy, by
the state of
means
gauges are glued
of
the
to the
released by d e e p e n d i n g The requires the
scale
ficients,
of
at various
flattened
method
"overcoring".
sites of the Alps was (Leeman, 1965):
end of a borehole
and
strain
the stress
is
the hole with a coring crown.
knowledge about
stress
"doorstopper"
interpretation
the
have
stress
strain due to a c o m p l e t e stress release, o b t a i n e d with measured
rocks
initial
i0
both mm)
is of
based
on
the
elastic
and
of
the
linear
elastic
constants stress
of
models the
and
rock
concentration
(at
coef-
relating the initial stress to the induced stress at the bot-
240
tom of the hole. Until
recently,
coefficients methods and
were
of
Ribacchi,
1980).
solutions
can
1984;
Rahn,
found
because
this
it is not
of
taking
found
in
anisotropic
tions
in
into
easy
concentration and
(Ribacchi,
the
approximate
1977;
development
Martinetti
of
efficient
theoretical
anisotropic
rocks,
fabric
found
ticular
the
"secant"
moduli final
stress
path
and
taking
into
account
the
the
and
caused
moduli
for
the
sufficient
to decide
account
the
materials
elastic
(Borsetto
et
which
even
overcoring. cores
(starting stress
best
possible It
in the
the
technique
seemes
the
in
situ
tests
modifica-
reasonable
laboratory,
in
ca-
behaviour
crack
from u n s t r e s s e d
level of
the difficul-
stress-strain
worse,
by
all
is the
non-linear
the
values
to solve
and
to
in par-
conditions); should
stress,
be
but
the
chosen
they
are
arbitrary.
This
procedure
delayed
rebound
testing
of
determined site
stress media,
techniques,
for
is not
utilize
somewhat
with
element also
the
1984).
pable
the
of
isotropic
utilized
Recently,
Unfortunately ties,
for
were
finite
be
values
only
interpretation
three-dimensional
al.,
the
known
will
occurs
the
sample;
as soon
certainly
in the in
storage
this
as p o s s i b l e
give time
incorrect
between
situation
after
the
results
the
coring
compliance
the coring,
or even
if
and
should
better
a the be
at the
itself. It
dealing more
is
with
severe
true
that
brittle
similar
isotropic
in a n i s o t r o p i c
microfissure
sets
and
of
difficulties rock
rocks
types,
because
the m a r k e d l y
can
but
of
be
they
found appear
the greater
non-linear
also to
density
behaviour
they
when
be of
much the
induce
in the rock.
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Walsh, J.B. (1965 b) The effect of cracks on the uniaxial elastic p r e s s i o n of rocks. J. Geophys. Res. 70, 399-411. Walsh, J.B. (1965 c) The effect of cracks tio. J. Geophys. Res. 70, 5249-5257.
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in
on
Poisson's
partially
of
comra-
malted
ROCK-SUPPORT
University
IN L I N E D TUNNELS
INTERACTION
N. C r i s t e s c u D e p a r t m e n t of Mathematics,
of Bucharest,
D. F o t ~ and E. M e d v e ~ G e o m e c h a n i c s Laboratory, Bucharest,
I. I N T R O D U C T I O N After generally a
the
is
may
occur
by
has
occur
the
to
the
of
the
time
the
of
hand,
procedure
and
ances
its u l t i m a t e
and
port
have
the
linear
assumption used
a
various Minh
to
the
be
based
(wall
lining
of
the
time
and
sequence
to
the
failure
may
a
long
analysis
the
and
the
excavation
factors
support
rheological of
the
Since
after
any
the
influence the
on
tunnel
sequence.
sometimes
on
structural
exerted
support,
the
essential
deformation
properties
that
support
often
a
and in
or
also
therefore
are
have
of the
volume
rheological Nguyen
models
who
model
uniaxial
(1983),
plastic
of
of the
shape;
slowly
mechanical known
considered
viscoelastic
behavour
standard
design
in the
very
pressure
layout
If
of
pro-
support.
excavation
tunnel
perform-
layout
and
be
sup-
considered
in
analysis.
authors
linear
used
cribe
is
well
sequence
support
Most
is
application
application
the tunnel
place
the
of
is
tunnel
opening.
lining,
excessive
excavation
(or lining)
of the
the
the
excessive
destroying
on
of
presence
excavation
after
and it
of
section
of
of
the
appropriate
interaction rock
of the
result
takes
soon
possibly
the
other
an best
Romania
a support
security
absence
in
result
tunnel
quite
rock-support
perties On
of
the
the
even
For
the
tunnel,
closure
final
a
rock.
either
period
in
a
as
choose
conve r g e n c e occur
as
the
ensure
Sometimes,
may
lining one
to
competent,
convergence). failure
and
Romania
OUTLINE
of a deep
to limit
rock
not
failure
excavation
applied
squeezing
rock
AND H I S T O R I C A L
Bucharest,
and
constitutive
rock. a
For
linear
standard
models Minh
to d e s c r i b e
used et
al.
by
tunnel
Maxwell
several
(1983)
Baoshen model
Panet
et
have
al.
used
to des-
has
used
with
(1979) has
Berest
interaction.
analysis
order
(1979)
authors;
(1984)
the r o c k - s u p p o r t
in
(1979)
associated
Popcvic
model.
support
equations
instance
incompressibility;
linear
the
a
the have
revised
and
Nguyen
linear
visco-
246
A -support
simplified
approach
interaction
is due to A r d a s h e v
ment method ation. Booker
element
Rowe
et
others
method
stitutive
is
in used
in
equation of joints.
-support
interaction
to
tunnel
be e l a s t i c - p l a s t i c
considering bolts. were al.
uniform
Model
tests
simulated
internal
thin
with
rel-
by
al.
(1985),
(1985),
were
and
Sharma
Swoboda
having
in Hoek
to
a
in which
carried
Brown
(1980). the
rock
condition)
lining lining
in dry
con-
of rock-
assuming
yield
The
orthogonal
theory
and
(1985)
due
et
(1985)
interaction.
rocks
a Mohr-Coulomb
pressure
Moore
elastic/viscoplastic
for
found
excavation
papers,
stress-strain
also
time-independent
be
rockele-
et
an
of
finite
al.
by M G h l h a u s
(satisfying
of tunnel
by
can
view) the
rock-support
(1985)
complete
studied
et
the
Lee
analysis was
of
(1985);
used
Yufin
Kimura
study
and
A quite
stability
is
conjunction
by Sun
families
The
to
point
et al.
with a n o n l i n e a r
(1982),
(1985),
order
physical
method
al.
Rodriguez-Roa
many
same
in c o n j u n c t i o n
finite
(1982),
a1.(1985), and
is used
The
(from
or and
sand
to
and rock
rock
bolt
by Adachi
et
(1982). In
the
another
approach
relationship
tunnel
support
between
properties
cing
concept
the
(Kaiser
(1981))
(Detournay
and
Detournay
and
(Cividini
et
plastic
displacement
without
making
stiffness
for
an
this
and
the the
or
time
a
curve
introdu-
elastic
yield
is
rock model
Detournay
Mohr-Colomb
this
by
dilatant
and
i.e.
specific
This was done either
Vardoulakis
(1985)),
(1985)),
ground
reaction curve,
cohesive-frictional
(1982),
Vardoulakis al.
ultimate
equivalent
a
of ground
introduced
of the support.
Fairhurst
teristic curve".
(1982),
criterion
called
"charac-
In some of the examples given by these authors the as-
sumption
concerning
ever,
it will
as
was
of
or
concept
the
pressure,
mechanical
the
volume
be
shown
incompressibility below,
when
an
was
also
accepted.
elastic/plastic
or
How-
elastic/
/ v i s c o p l a s t i c model
is
not unique s i n c e i t
depends on t h e l o a d i n g h i s t o r y ( l a y o u t sequence of
e x c a v a t i o n and
used f o r t h e rock the ground r e a c t i o n curve
support
a p p l i c a t i o n and
certainly
on
the
is
mechanical
properties of the support). What the lining linear
concerns
relationship
linear r e l a t i o n s h i p nonlinear supports, by
the
subjected
only.
yielding
for
the
response
have
both
equation
of the support.
equations
either
by
wooden
consitutive
are
sliding strips.
equation
For
parts
the
a
linear and non-
encoutered
between
of
assumed
consider
supports see for instance W o o d r u f f The
used
all authors
we will
some
Below
model
pressure,
for the c o n s t i t u t i v e
constitutive
compressing
mathematical
to the rock
with of
the
The
yieldable support
description
of
or
such
(1966) Vol.2.
which
will
be
used
for
the
rock
is
247
e i t h e r of linear v i s c o e l a s t i c or of e l a s t i c / v i s c o p l a s t i c type s i n c e as a l r e a d y mentioned, time i s the major parameter in the r o c k - s u p p o r t i n teraction equation
analysis. allows
dilatancy more
a
to
(only
dilatancy
cedure
failure
constitutive
for the
and
as
the
tunnel drill meters
surface and
the
involved
cannot
excavation
application,
constitutive
describe
compressibility)
dilatancy.
constitutive
this
With
one
can
the
and
into
further-
failure
is
layout and optimal
taking
rock
elastic/visco-
describe
equation
the
due
instrumental sequence
account
the
pro-
specific
the depth etc.
sequence
operation
stress
machine in
to
such
becomes
blast,
volumetric however,
lining
during
viscoelastic it
equation,
properties of the rock, If
linear
analysis,
due
finding of optimal in
the
simpler
viscoelastic
possible
plastic
While
a much
the
free
used,
at a certain
(due to the excavation
excavation
etc.)
rock-support
then
one
interaction
of
moment
the
method
used:
the main
para-
analysis
the "time
is
of application of the support a f t e r e xcav at i on . I f t h i s timing i s too s h o r t , then t h e support may f a i l due to o v e r l o a d i n g ; i f t h i s timing i s too
long,
then
deformation.
a
failure
Generally
of
the
the
rock
is
possible
loading history,
whole
due
i.e.
to
excessive
the history
the variation of the pressure exerted
by the lining on the tunnel
face
the
is
of
great
significance
on
overall
evaluation
of
sur-
of
the
behaviour of the rock. In
the
presented The
case
following
following of
Cristescu
the
mainly
viscoelastic
et
and nonlinear
al.
(1988),
rock-support an
interaction
unpublished
rock
report
and nonlinear
while
the
support by C r i s t e s c u
case
of
by
support
analysis Cristescu was
will
be
(1985).
considered
elastic/viscoplastic
by
rock
(1988b).
2. RHEOLOGICAL MODELS FOR ROCK Two
reological
models
will
be used
for rocks:
an e l a s t i c / v i s c o -
plastic c o n s t i t u t i v e equation and a linear v i s c o e l a s t i c one. According
to
elastic/viscaplastie
the
the c o n s t i t u t i v e equation
3K where
the
)$I + ~ - ~
bulk modulus
model
(Cristescu
(1987))
is written as
K(e)
+ k may
vary
gently with
(2.1) the mean
stress e, G
is the shear modulus, H(o , o) : WI(t)
(2.2)
248
is the s t a b i l i z a t i o n boundary with WI(T) : ~ a(t)-~I(t)dt 0 the irreversible
stress work
~2 : a~ + ~ is
the
which
may
depend
stress
and
the
stress
on
by (Cristescu d(t)
Finally
(prime
stands
for "deviator"),
+ c~ - o1~ 2 - a2~ 3 - a3~ 1
equivalent
defined
= ~ a ( t ) ~ ( t ) d t + ~ -'(t)'~'I(t)dt=W~(T)+W~(T)(2.3) 0 0
k(q,~,d)
is
invariants
(2.4) the
viscosity
and
on
a
coefficient
damage
parameter
(1986)):
I : Wv(max)
- W~(t) .
(2.5)
= A + = ½(A + IAI). According
the volumetric
to
~V : k Let
us
the
rock
denote and
(2.1)
strain
the
variation
of
the
irreversible
part
of
existing
in
is
-
~-~.
by oP = a(t o) by a(t)
the
the
stress
initail field
"primary"
after
stress
excavation.
In the
points
of the rock where H(o(t)) an
elastic
state case
< H(o P)
(2.7)
unloading takes
place.
If
H(a(t))
= H(a P)
the
new
is on the same stabilization boundary and ~I = O. The case H(O(t)) > H(a P) will be called loading since a variation
according
of eI is possible.
3H to the sign of -~.
H(o(t)) ~H B--~ > 0
We
can distinguish
stress in
three
this
subcases
Thus
> H(a P) compressibility viscoplastic
~H ~--~ < 0
dilatancy
~H ~--6 = 0
compressibility/
(2.8) deformation
dilatancy The
second
boundary.
model
viscoelastic one of the form
which
will
(Massier
be
(1984)):
considered
is
a
linear
249
a :-kv(e
- -~Ko)
+ TK 6
(2.9) ~' where
are
: -k(e'
the
moduli,
a' I ~, - 2--G-~o) + ~-~
material
constants
K o and G o are
the
several
the
viscosity
moduli
positive) in the
coefficients.
inequalities K > Ko,
(all
G > Go,
relaxed
These
(see C r i s t e s c u
are:
K
and
G
state,
the
while
coefficients
dynamic k v and
must
k
satisfy
(1988a))
2G < 3K,
2G o < 3K o
(2.10) G-G o
kv This
K-K o
k,
model
0
can
describe
a
compressibility
of
the
rock
but
not
a
dilatancy. All or
(2.9)
the
constitutive
are d e t e r m i n e d
constants
or
functions
in the laboratory
involved
(Cristescu
in
(2.1)
in a rock
which
(1988a)).
3. F O R M U L A T I O N OF THE P R O B L E M Let is
us consider
squeezing
installed presence
in of
ly
lining
by
excavation
influenced The stress
aspects
displacement
cement
and
in
a
shape
certain
port
which
appropriate
would for
can
inner
At
time
We
to
blast, is
by
the the
creep
assume
from
time
of
creep
the
lining
kinds
of
time
of
the
becomes
rock
comes
initiation of
the
the
stopped
surface
the
to
is great-
Various
tunnel
etc.),
is
that
c.
properly
due
failure
elapsed
the
is
reduced
rock
radius
when
a
rock
in
of
the
is
not
for t ~ t o it is. problem
support,
at this
to
the
in
what
of rock
what
solved
interface, interface
kind
of the
and
a
economical of support
considered,
the
for
t > to
the
pressure
ultimate
therefore
of the
conditions
is the most
failure,
be
evolution
(if a s t a b i l i z a t i o n
which
be
the
lining.
an
to
(lining)
convergence
0 < t< t o
interval),
kind
rate
time
the
pressure
avoid the
For
excavated
times
the r o c k / l i n i n g
take place and when not,
a
to
that
while of
lining
time
the
and
the
in the at
ultimate of the
of
so
interaction.
of the
mate
is
lining
evolution
the
other
instance,
by the lining, main
support
analysed.
t = 0
the
while
b < a
be
for
a,
unless
convergence
presence
radius
due,
rock-support
the
the
of radius
Sometimes
lining,
(here
with
The
lining.
further
stress-free contact
time.
external will
a tunnel
convergence)
by the
delayed an
due the
altogether
has
(wall
are: and
displa-
the
ulti-
deformation
occurs
stabilization
will
design
sup-
would
optimal
of the
be the most
timing
t o and
250
generally
the
cation,
the
lining,
etc. It
material start that
best amount
is
the
of
the
certainly
from
by
procedure
which
assuming far
sequence
rock
which
assumed
the
that
support
axial
field
and
the
in
stresses
deteriorate
properties
are
For
known.
first
place
appli-
behind
mechanical
the
are
excavation-lining
may
is made,
symmetry;
in situ
of
of
that
(i.e. meters. and
it
We
assume
an
the
equilibrium for
at
of t u n n e l s
hydrostatic
component
any
a since fast
the
simplifying bcreholes
(-~
constitutive
equation
is
given
h
assumption
the
(3.1)
be c o n s i d e r e d
depth when at
equation
- ~ of
the
stability
used,
(3.2)
at r = a s a t i s f i e s
(3.3)
- -2---~)5 + "~O ~e
u : cer.
For
the
hydrostatic
("instantaneous")
then
reads
3Je"
rock
in
§6.
c o n s t i t u t i v e equation i s
constitutive
response
the
(3.1)
will
+ ~--CSe + k
"instantaneous"
for In
and
~e this
= Thus
state
instance).
elastic/viscoplastic
an
we also
(3.1)
(2.2)
renounce
failure If
for
is
satisfies
the
simplicity we
ah -~ (~v = yh and
the
excavation
far
the
field
stresses
stresses in the
(3.1),
rock
after
a
are
2 (~r : - ((~v - P ) % r
+ gv a
z
= a
(3.4)
v
2
ae ; and
(~v - P ) % r
therefore
at the
~r : p' with (3.5) iation
p
the in
G a
pressure we
on
get
pressure
r = a:
the
the
ez = Cv surface
r = a
relationship
to the
(3.5) of
the
rock.
relating
the
instantaneous
displacement
of the
Introducing
H(a
, ~
while
the
var-
interface
p__ - 2G"
According
Thus
interface
o@ : 2o v - p,
(3.3)
of the
+ av
(3.6) to
(3.1)
any
) : WI(t)It ~ instantaneous
ultimate
equilibrum
state
must
satisfy
~"
(3.7) response
relationship
is
unique,
but
251
certainly
depending
on
not
it d e p e n d s
on the
since Let
also
us
discuss
Cristescu
initial
the
data,
loading
the
history
formulation
(1988a)),
when
locus
the
of
of
ultimate
states
is
as well.
initial
the
conditions
elastic/viscoplastic
(see
constitutive
e q u a t i o n is used. T w o c a s e s can be d i s t i n g u i s h e d d e p e n d i n g on w h e t h e r P = ov < oo or o v > o o (in the m o d e l (2.1) eo is the p r e s s u r e w h i c h
o
closes each
all of
the
microcracks
these
stresses
two
at the d e p t h
or the s t a b i l i t y H(oP, O)
fig.3.1 in
in
all
a-f.
fig.3.1
ground
the
considered
satisfy
rock are
- see
fig.3.1
possible:
the s t a b i l i t y
the
-).
The b,
For
primary
equality (3.8)
< wIP
(3.9) -p ~
= O.
primary
while
reaction
The
stress
the
four
(3.7)
response
(3.6)
possible
satisfying
initial
curve
a as a full
as at
(3.8)
shape
of
a
border
the
cases
moment
the
is
are
shown
shown
as
a point
corresponding line
of
and
P
ultimate
that
excavation
in
of
is s h o w n
the in
line.
I
~v<~O
I
in
subcases
inequality
cases
"instantaneous" fig.3.1
two
= wIP
H(o",O)
since
existing
cases,
~v~o
1
"%
[
0
ol
o
o
0
o
P~-o b)
o
0
~-o ~ ~
o
c}
o
e)
q %
5
f)
d)
Fig.3.1. Formulation of the initial conditions for elastic/viscoplastic constitutive equations, depending on v a r i o u s possible primary stresses
In
a similar
way
the
case
when
inequality
(3.9)
is s a t i s f i e d
is
252
shown
in
(point
fig.3.1
as
dotted
excavation) which
is
ground
other
at
secondary
ground
stress
(3.8)
after
excavation.
Since
state
the
from
smooth,
initial
slope
(at point
instantaneous For
tion
of
the
the
the
initial
in
Q,
by
in
to
a
point
the
rock
and
an
curve
state
ulti-
curves
situated
this
the
the
under
remain
In
in
but all
however,
will
to
Q1
state
unique,
the
elastic
case
the
state,
but
state.
elastic/visco-
reaction
curves
have
an
with
the
slope
of
equation
the
formula-
c).
constitutive
is much
stress
If,
coinciding
(fig.3.1.
conditions
two
an e q u i l i b r i u m
ground
u = 0)
the
elastic/viscoplastic
hydrostatic ultimate
these
d.
after
point
instantaneous
not
e.
> o v o boundary is
then
the
fig.3.1
> o o)
viscoelastic
e;
to
fig.3.1
then
shown
p = o v and
response
linear
fig.3.1,
o
(stress
the
is c e r t a i n l y
q
leads
elastic
and
and
the
respectively)
in
represented
~
above
corresponding
(since qP
of
states
line)
curve
is
stress situated
Q2
shown
by point
is elastic
very
is
(or
point
case
increase
points
point
is
passes
passing
plastic
the
a
intermediate
smallest
< o o (point PI ) v primary s t a b i l i z a t i o n
(border
are
through
which
o
secondary
by
reaction
state
curve
An
the
boundary
each
pass
cases
curve line)
curves
primary
the
(full
ultimate
these
on
reaction
curve
intersect The
If
represented
located
respo n s e
both
corresponding
line.
are
mate
the
for
P2 ). A possible
shown
Q.
f
simpler.
First,
from
(2.9)
we
have
~e
08 ) + ~-~ 1 $8
1 - 76--
: -k(~e
(3.10)
o and
with
(3.4) 2 we
interface
u(t)
+~--~ a p(t)
This
equation
this
surface
exerted
: -k[u(t)
describes acts
by
the
+ 2~-(p(t) o
convergence
a pressure
excavation
instantly
from
variation
of the
for
t -+ ~ fied
differential equation for the rock~support
the
p(t)
- Ov)].
of the
which
may
(3.11)
tunnel
be
surface
variable
(the
when
on
pressure
by the lining). If
if
get
as
by
various and the
ov
to
(a
zero,
blast, then
displacement values
u -+ O,
of
from
ultimate
say)
according
is governed
p : const, (3.11)
values
of
the
the
to
(3.11)
by
after
follows
pressure
the
(3.6). a
long
at
the
time
p~
and
drops
instantaneous
O ~ the
relationship
pressure
r = a
other
hand,
interval to
be
when satis-
displacement
u~
253 2G P~ : dv ..... a ° u~.
(3.12)
This i s j u s t t h e e q u a t i o n of t h e u l t i m a t e ground r e a c t i o n curve which i s now unique;
it
is
shown in f i g . 3 . 2 as a border l i n e .
The two s t r a i g h t
fly Failure by overloading Rigid support Hard elastic support
"~ / j ~
P~o
l--L/ \ "~" ,,_
oI~
\~"
\
-,<
, I ! I
11
l
/ I
l
.
"~. ..- ~. f ~
I
Uo
Failure by excessive deformation
~'"
1
I I
a~v
I! I
Soft elast,c support
°~
a~v
2G Fig.3.g. on t h e
lines
(3.12)
uniquely initial
u
2Go
and
determined
Possible pressure-displacement tunnel surface and rock-lining (Cristescu et al. (t987}).
(3.6) if
have
certainly
the c o n s t i t u t i v e
relationship interface
distinct
equation
slopes;
both
is formulated
are
and
the
data are prescribed.
4. 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 V I S C O E L A S T I C ROCK For linear
reasons
let
us
rock and of various
4.1.
elastic
-shaped material the
methodological
viscoelastic
Linear
lining
of
(concrete
rock
lining/rock
on
the
support.
constant
thickness
or shotcrete, lining
interface,
and
then
say). u
the
is
consider
kinds
Let
us
b - c,
consider made
If p is the the
radial
constitutive
first
the
case
of
of linings.
of
a
thick
linear
pressure
exerted
displacement
equation
for
tubeelastic
the
of
by the
linear
254
elastic
support P =
E ( b 2 _ c 2) (I + ~ ) [ ( I - 2 ~ ) b 2 +
where
E
q
r~gidity
its
it
is
and
comes
~
are
in
correspond
the
and
elastic
uo
the
contact
constants
the
- q
of
displacement
with
hard
to
u - u° b
c 2]
of
lining.
supports
while
u - u° b
the
(4
material
the
rock
at
Relatively smaller
of
the
the
high
values
lining,
moment
when
values
of
q
on
the
I)
of
q
soft
to
supports. If
a
sudden
r = a
drops
r = a
becomes
the
walls
u =
is
u(t)
with
excavation
suddenly
to
(a~v)/2G
governed =
by
[1 + 2G °
a possible
zero
is
done,
and
the
(see
(3.11)
the
fig.3.2). with
Further
p = 0,
I I (2-G - 2--~o) e x p ( - k t ) ] a a
ultimate
pressure
displacement
of the
the
surface interface
convergence
of
i.e.
(4.2)
v
displacement
a~ u it÷ ~
u ~o
=
- 2G
p=0 If contact the
at
a
conveniently
the
rock,
displacement of
an
u(t)
(4.3)
o
with
presence
v
of
the
elastic
= Q
+
chosen
then
from
time
(4.1)
to
and
rock/support
the
(3.11)
interface
support we
comes
get
for
influenced
t ~ by
in to the
support
(u O - Q ) e x p [ P ( t
O -
(4.4)
t)]
with a
P
= k
I + 2G
q
b
o
,
Q
I + a__ 2G qb if
are
the
conditions
t = to:
u = Uo,
used.
When bov
u~
and
ultimate If
+
= a 2Gob
the
I + 2G b
p = 0
t -+ ®
the
ultimate
displacement
is
qu o (4.6)
+ aq
pressure
u = u o = const.,
(4.5)
a__q
2u~ o
initial
also
(~v + ~bUo
= ka
yields
support then
is the
from an
(3.12).
ideal
increase
of
rigid the
one,
pressure
i.e.
for
which
acts
t ~
to,
on
the
255
support
is g o v e r n e d
p(t)
If
to
:
is
under
is true The
determined
knowing
the
by
are
obtained to
very
high
collapse.
The
are rigid.
plane
in
(4.7)
excepted
marking
fig.3.2
by
an
in the l i n i n g
appropriate
the
failure
by
line)
can
condition
and
interrupted failure
~)
a
2~
(4.8)
'b2~' b 2 c 2"
z
for
a concrete
one can a c c e p t
(a r -
which
u-p
be
introducing
c2 b2p (I --~) b2-c 2 + r '
instance
(4.7)
may
distribution
ae =
failure
from
schematically
the s t r e s s
:
then support
supports
in
b2p (I a r = b2_c 2
For
the
for e l a s t i c
(shown
Gk - exp[~-- (t o - t ) ] } . o
small,
which
boundary
overloading be
2Go ~ Uo){1
(O v -
relatively
pressures, same
by
ae)2+
or
a Nadai
shotcrete (1950)
support,
in order
type of failure
to p r e d i c t
condition
(4.9)
(a e _ az ) 2 + (az - a r ) 2 : 2 [ a o ( a r + ae + az) + al ]2
where ac - ct
2aca t
ao
'
al =
ac + at with the
a c the u n i a x i a l uniaxial
condition the
influence
(4.8)
follows
surpass
compressive
tensile
seems
strength
appropriate of
the
that
the limit
(4.10) ac + at strength
of
for
the
concrete
hydrostatic
for
such
a
and a t the a b s o l u t e
concrete. since
pressure support
it
as
the
Such
takes
well. rock
kind
value
of
of
failure
into
account
From
pressure
(4.9)
and
must
not
value a 1 ( b 2 - c 2)
p <
(4.11) 2b2[ao(1
assuming by some
that other
+ ~) - (I - ~ + ~ 2 ) ½ ]
v > 0.1
say.
arguments,
Conversely,
then
one
can
if
the
determine
pressure the
is
minimal
prescribed thickness
of the wall of the l i n i n g 2p{(I c 2 < b211
- ~ + ~2)½
+
_ ao(1 + ~)} ].
aI
(4.12)
256
The
above
formula
can
be
used
for
instance
to
of
show
the
must
how
wall
be
lO
p).
:E
4.1
is
of
the
b r~
depth
For
with
the for
in
fig.
variation
maximum
pressure
of
the
tun-
three
kinds
of
(curves
a,b,c)
and
two
of
shotcrete
e).
The
are
in-
the
and ants
with
(increasing
radius
concrete
E
lining
instance
shown
nel
° c
thickness
the
increased
creasing
¢1 {:L
the of
(curves
material
given
d
const-
in Table
4.1
E X
and
g
clusion
c c = 10e t.
con-
the
Poisson
coefficient
of
the
lining
a
very
ence:
Fig.4.1. Variation of maximum pressure with tunnel radius for various linings
first
that
has tunnel radius (meters)
A
is
important
smaller
increase
the
sure.
increase
An
influ-
values
of v
maximum
pres-
of c c in-
creases the m a x i m u m pressure.
Table 4.1 Constants
Lining Material
v
used
a
in fig.4.1
(MPa)
Wall thickness
c
Curve in fig.4.1
(cm) Conc r e t e
0.25
35
30
a
Concrete
0.20
35
30
b
Conc r e t e
0.25
49
30
c
Shotcrete
0.25
35
5
d
Shotcrete
0.25
14
5
e
Further with
the
in
fig.4.2
tunnel
radius
interface.
The
shown
fig.4.1.
in
prescribed, lining
example
one
thickness
can
is
shown
for
various
shown
Thus,
how
in this
the
lining
pressures
figure
the
rock/lining
determine
for
each
tunnel
material
from
which
the
at
the
corresponds
if
once
thickness
radius the
rock/lining
to the
interface the
lining
varies
curve
pressure
a is
appropriate is made,
was
chosen. The excessive
boundary deformation
of
the cannot
domain
where
rock
failure
be obtained
with
the
may
present
occur
due
to
constitutive
257
equation.
This subject will be discussed
Knowing properties (i.e.
knowing
t o must way
of
elastic
the
support
q)
the
be chosen
that
(4. I )
the
the of
not
the
two
line cross
~
boun-
E°
daries mentioned above. In
order
example
of
formulae can us consider tunnel
of
to
how be a
a = 200
in
a
(coal)
the
depth
at
soft
constitutive
for
this
G
closure
of
~ i!
~
'E
m.
The
4 tu n n e l
G =
= 20 MPa,
any
_~
cm
280
the walls
of
30
rock
are
1.182.10-6s absence
9.°
constants
rock
= 300.8 MPa,
an
40
above
used, let circular
excavated
The
~
give
the
radius
50
in such a
straight
should
neither
timing
in the next section.
k =
in
the is
~1~ o.1
I0
F i g . 4 . Z . L i n i n g t h i c k n e s s as f u n c t i o n of tunnel r a d i u s for v a r i o u s p r e s s u r e s p.
slow
support
6 8 r a d i u s (meters)
/ 0
~---
ii //"
.~ o.osk
/
..".
..................
O0
1
2
0.8
.i.-:
3 time (days)
o._
22
4 '
. . . . . . . . . . . . .~. .. .. .. .. . . . .
..........
~ l b ............3'0
0.2
5'0
0
20 40 time (days)
F i g . 4 . 3 . a) V a r i a t i o n in time of r a d i a l d i s p l a c e m e n t in the absence of the support ( f u l l l i n e ) , of the r a d i a l d i s p l a c e m e n t in the p r e s e n c e of the support ( d o t t e d l i n e ) and of the p r e s s u r e on the support ( i n t e r r u p t e d l i n e ) for v a r i o u s moments of a p p l i c a t i o n of the s u p p o r t , b) U l t i m a t e p r e s s u r e as f u n c t i o n of the time of a p p l i c a t i o n of the s u p p o r t .
258
shown
in
than
two months.
slowed a
fig.4.3.a
down
and
d
shown
shown
in
leading
fig.4.3.a
by
the
the
lining
the
the
to
is
shotcrete
is
that
after
a
decrease in
In
these u o from
the
lasts
the
examples
is In
of
the
pressure
value
of
b is
t o = I day.
ultimate
ultimate This
is
curves
softer,
for
the
to
displacements.
the
(4.2)
more
closure
being
higher
time,
fig.4.3.b.
day
variation
of
of
one
creep
lining
the
longer
by
corresponding
but
shown
linings.
The
after
lines,
pressures
lines
shown
closure
dotted
a
smaller
installed
smaller.
This
is a p p l i e d
two
b = a - u o and
is
timing
line.
fig.4.1;
to
considered
from
certainly
by
interrupted
two
obtained
full
If a s u p p o r t
as
certainly
for
by
pressure
pressure
If
pressure P~/ev
is
with
decreases
quite
fast. 4.2. supports; is
Yieldable the
supports.
relationship
nonlinear.
The
between
support
support if it s a t i s f i e s
There
will
the
p be
t ~ tI :
p(t)
= q
several
and
u
for
kinds
such
of
kind
yieldable
of
supports
elastic~constant
called
following u(t)
to ~
are
pressure
conditions - u
o
b (4.13)
t ~ For
P = Pl
the
convergence
takes
convergence
is o b t a i n e d
t < to
to ~ the
tI ~
t ~
t I the
convergence
if
at
t = t I we u~
place
= const. place
according
according from
to
(4.4).
(4.2)
Finally
while for
for
t > 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 0
a2 ~** + a I Co+C I with
a o = 7 . 6 5 . 1 0 -4 MPa,
c o = 4 . 9 5 7 - I 0 -4 MPa, G o = 1.0996 MPa, and
d
Other
o, = I MPa,
history
ultimate
in WI(t)). obtain
~
K = 769.6 MPa,
variation
reaction
reduce
the
values
history
The
of
ground
If we
ultimate
previous
the
a 2 = 8 . 1 5 9 9 - 1 0 -3 ,
c I : 4 . 8 9 5 5 - 1 0 -4 MPa,
b o = 0.001MPa,
= 171.927 ° ,
¢ = -99.0680 ,
G = 300.8 MPa,
k = 6"10 -6 s -I
= 4 . 4 8 . 1 0 3 J m -3.
cr
other
a I = 0.55,
if~o~q
u~
threshold
of
which
will
the
certainly
loading
bigger
lead
history
p in s e v e r a l
are
is
successive
than
that
to
some
involved stages
obtained
in
we the
of p.
equation
failure
a
p
(since
pressure
of v a r i a t i o n
constitutive
of
(2.1)
process
can
due
be
to
used
to
estimate
when
excessive deformation
by
dilatancy may be r e a c h e d . For t h i s p u r p o s e we must c a l c u l a t e t h e value of
W~(t)
(see
(2.3))
for
When
this quantity = W Iv( t c r ) - W IP v = -dcr tcr
on.
This
timing
the
considered
reaches
a
history
certain
of
variation
critical
value
of
p.
W~(cr) V
then
failure
is o b t a i n e d
starts
taking
place
from
this
time
from
I~H tcr
H
= tc - ~
ln{1
-
~-~'a -- W--~, 3H
k~a where
again
failure such is
tc
a
failure
decreased
which
the
if
this
may
be
ment
is
in
a of
decrease avoided
u
. Also
to zero,
on
stars
to
the
is
from
has
the
expense of
in
(in of
a
of
the
failure, in ov
taken
progressive
to
p
from the point
when
steps, greater
in w h i c h of view
be
then
the
line
of
incipient
above
and
for
the
decrease
failure of
p
in
that
failure
displace-
is n e v e r
of a v o i d i n g
p
(during
shown
radial p
If
pressure
value
say),
The
erosion.
the
it can
fig.5.1
given
started.
of
ultimate
then
slightly
has
kind
case
its
several
of
rock
some
the
place)
variation
favourable
shown
creep
obtained
step
rock
histories
criterion
when
is
single the is
are more
By
moment
is a p r o g r e s s i v e
threshold
creep
(5.5)
(I - --~--) ~-6~
the
by d i l a t a c y
Wi(cr) } v
reduced failure.
according a
single
step. The reduced creeps also
to and
case
when
zero
for
by a
afterwards
considered.
First,
at
sudden
certain time
excavation time
to a yieldable
by c o m p u t i n g
(blast)
interval
support
W I and W I from v
the
during
pressure
which is
the
installed,
is rock was
264
~H TH---dP D~ 6
(5.6) DH dW I : (b__ qa + 2~ ) ~-~" ~--~----d a p
along
the straight
line
(4.13) I we can determine
val t o ~ t ~ t I a s t a b i l i z a t i o n take place.
the ultimate
ground
one. For instance
interrupted
ground
line
loading
line marked If
support
after
two
pressure-displacement as
lower
a
higher
ultimate The
threshold
reaction of
reaction
is certain-
in fig.5.1
is shown by
obtained
failure
with
is quite
the
close
pre-
to
the
by stars. a
installed
the
program.
inter-
of the creep or a failure threshold will
In the first case
ly different from the previous
sent
if in the time
dotted
line
tightening
made
of
hours
circular
and
variation
at
in fig.5.1. torque
pmf/~v = 0.0232. The horizontal
of
steel-yieldable
Pmi/Ov = 0.0161, the
The
the
interrupted
rock/lining
upper
nuts
line
dotted
for
which
corresponds
shaft
rings
pmf/Ov = 0.0143, interface line
is
shown
corresponds
pmi/Ov = 0.025 to the
is the to and
installation
of the support after one day. The
solution given above
obtained with the assumption since
the
compatibility
onents was disregarded. ical
procedure.
One
is but an a p p r o x i m a t e
of stress
equation
to
one since
constancy during
be
satisfied
by
creep
the
it was
and
strain
also comp-
A rigorous solution can be obtained by a numer-
must
integrate
the
system
of partial
differential
equations Do r
o r - a0 +
-
Dr
-
= 0
r
De 0 e0 - e r Dr + r
= 0
~K I ~r = ( - ~)~
=
0=(
I + k ~ + 2-G~r
-2--G)~ + 2 - G ~ e
I 3K
2G )~ +
+
z + k
k
_ HWI>DH 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~<.-p=O
(MPa)
/ 15
in
• 1
20"__
/
° 10"
the
neighwe
while the
the
inter-
crown
sidewalls
in
boun-
by
In the
distribution
stress
P and
= H(g P)
stress
=
the
shown
stabilization
H(o)
~h
obtain
primary
by
this
"
(5.4))
(y :
we
sin 2e
For
~v = yh
distribution
4]
)r-- + r 3
represent
h = 350 m,
and
stress
the
to
aOq.
for coal
the
dary
-
invariants
instance at
2V a2
2
10
get
along
stresses
are far in the d i l a t a n t domain. Thus
two
kinds
possible At
the
is
elastic
crown
surpasses
strength
the
stress
of
negative
the
tensile
the
rock
stress
plastic dilatancy an
state
and
takes
evolutive
possible. in
a
fig.6.2.a
for
90"q
.
.
.
.
.
#50"
\70( .'" ..
visco-
0
significant place;
is
"
walls
@IMPa)
thus
failure
This
090"
(here
the is
l
5
state
maximum
is
a t = 0.5 M P a ) . A l o n g the
are
(1986)).
the s t r e s s
while
principal and
of f a i l u r e
(Cristescu
is shown
p = O.
The
Fig.6.1. Stress distribution along the circumference of a tunnel
domain
at
the
crown
where
failure
by
268 ~-V I
Elastic
t
nin tension~
"-
\
O L.
~\ tO
/
O\\ Tunnel
>
Viscoplastic dilatant
) \\\
\
"- ~h \\ k
h=350m
x v ~ £ ~Od
Fig.6.Z a) F a i l u r e in tension and by dilatancy along the circumference of a borehole.
"-
P= 0.146 ffv G'h=0.2 ffv Failure ' after 0 37d -,~ 0
2 98 d Id
relolive radius
r/o
%
Elastic
Failure in tension
gl D
"",.
Viscoplastic dilatant "\'~d \ \
Fig.6.Z. b) Failure in tension and by dilatancy along the circumference of a lined tunnel
\
~J >
Boreho!.e p=O h=350m
0
0
Foilu offer 0.28d "-,,~ relative radius
rio
~re~~ I _ II 5.17d ld
269
tensile
stresses
ded
by
and
e = 0 ° after
by
thick
the
lines.
value
slowly
The
in
evolutive
0.28 days
if
i.e.
lines.
but
after
of
equal
lines
If
the
rock
weaknesses,
then
the
damage
the
shown
contour
same c o n v e n t i o n In
the
pressure
tensile where
decrease
es
wards
slowly
slightly
decreases
slowly
6.3.
If
is
shown
p = 0 of
from
increase
the
u/a
while
for
u/a
decreases
18%
from
increase We
conclusion in
danger
of
is
E
a constant
pressure
crown
where
(fig.6.2.b).
while
failure The
shrinking
the
and
by
domains
in size.
vertical
excavation
support
diameter
continues
to
p:
0.14s Cv
fol-
with
>~
maximum
.go
increase
an
failure
h : 350m
the
and
the damage
If however,
damage
by
exact
problem.
i
the
close
t (days)
to the
dilatancy determination
determination of the
damage
Therefore
Variation in t i m e o f displacement of the wall.
Fig.6.3.
decreas-
on the a c c u r a t e
estimation
an open
The
5%
increases
daager
the
failure.
of
p=O
initial
that
this
exact
portion
in
at
unity,
Also
thin
follows.
is
The
The
as
structure
0.02
3
which
Oh/a v
es.
at
some
by any
are however
suddenly
fig.6.2
larger.
at the
that
at
in fig.6.1.
in size
mention
5.17 days
p = 0.146Ov,
ratio
depends
domain
in
progresses
e = 33 ° . These
has
affected
with
is possible
arrived
by dilatancy.
be much
r = a
in
only
the
depth
the
fig.
in
initially
which
tunnel,
after
or
at
after-
the
decre a s e
can
starts
damage
shown
surroun-
is c h a r a c t e r i s e d
increas-
but
This
are
is not
diam-
excavation
time.
lows,
rock
r = a and
nonhomogeneous
increases
to
decreases
horizontal
first
at
damage
interesting
damage,
at
is used also
say,
is possible
further
the
eter
p = 0.146~v
opening
time,
the
of damage
2.46 days
lines
of a lined
a progressive
the
of
beginning
domain
by dotted
case
is
is
for the lines
stresses
It of
damage
the
is shown
d
r = 1.22 a,
boundaries,
of
"instantaneously"
= 4.48"103 J m -3. This evolutive cr time and reaches the u l t i m at e position
e = 0 ° and
full
is to be expected
of
the
of dcr , which reference
parameter
the above
at
examples
the are
is not an easy task.
configuration primary but
state
for
the
is still
illustrative.
270
7. CONCLUSION The analysis general
main
conclusion
rheological
for the support can
which
of the rock-support
of
for
the
rock
(possibly a yieldable
be considered
sequence
model
yields
interaction
in a similar
several
kinds
the
and a non
one).
way and
of
from
above
can be done
supports
linear
Any other
possibly
is
that
assuming
relationship
kind
of support
also a more
succesively
an
a quite
involved
applied
to
the
tunnel. After rock-support procedure
the
formulation
interaction,
can
be
of
obtained
by
order
to
avoid
design of tunnel out
and
pressure
failure
support
sequence all
the tunnel.
by
In other words,
steadily
decreasing pressure
step
p~,
the
model
support
methods.
a general
excessive
never
the entire
ultimate
of
for
If
the
rock
conclusion would
deformation
the
installation
the
reduce
the final by
possibly
simultaneously
circumference step by a
of
pressure
the
far
decrease
the
to
be that
zero
lay the
surface
of
p~ is to be reached
by
field done
future
is
appropriate
set up is the one in which the excavation
would
along
mathematical
mathematical
quite soft and squeezing very much, in
a
an optimization
stresses
to
succeslvely
the along
the contour.
REFERENCES Adachi,T., Tamura,T. Shinkawa,M. 1982, Analytical and Experimental Study on Tunnel Support System. Fourth I n t e r n a t . Conf.o~ Numerical Methods in Go omechanics, Edmonton, Balkema, Rotterdam, 513-522. Ardashev,K.A., Amusin,B.Z., Koshelev,V.F. 1985, Numerical methods of analysis of underground constructions. Fifth Internat. Conf. on Numerical Methods in Geomechanics, Nagoya, Balkema, Rotterdam-
-Boston, 1077-1084. Baoshen,L. 1979, A study of the mechanism of support. Fourth I n t e r n a t . Congr. on Rock Mechanics, Montreux, 1 9 7 9 , Balkema Books, I , 157-160. Berest,P., Nguyen,M.D. 1983, Time-dependent behaviour of lined tunnels in soft rocks. Eurotunnel'83 Conference, Basel, Switzerland, 57-62. Cividini,A., Gioda,G°, Barla,G. 1985, Calibration of a rheological model on the basis of field measurements. Fifth Internat. Conf. on Numerical Methods in Geomechanics, Nagoya, Balkema, Rotterdam-B~ston, 1621-1628. Cristescu,N. 1985, Mathematical modelling of the mining support and of the rock-support interaction. Report, University of Bucharest. Cristeseu,N. 1986, Damage and Failure of Viscoplastic Rock-Like Materials. I n t e r n a t . J. P l a s t i c i t y , 2, 2, 189-204.
Cristeseu,N.
1987,
Elastic-Viscoplastic
Constitutive
Equations
for
Rock. I n t e r n a t . J. Rock Mech. Min. S c i . & Geomech. A b s t r . 24, 5, 271-282.
271
Cristescu,N. 1988a, Rock press). Cristeseu,N.
Martinus
Rheology,
Nijhoff,
The
Hague
(in
1988b, Viscoplastic creep of rocks around a lined tunnel. J. Plasticity (submitted for publication).
Internat.
Cristeseu,~. Fotg,D., Medves,E. 1987 Tunnel Suport Analysis incorporating Rock Creep. Internat. J. Rock Mech. Min. Sci. & Geomech. A b s t r . , 24, 6. Detournay,E., F a i r h u r s t , C . 1982, G e n e r a l i z a t i o n of the ground r e a c t i o n curve concept. 2 3 r d U.S. Symp. on Rock Mechanics, Berkeley, 926-936. Detournay,E., Vardoulakis,I. 1985, Determination of the Ground Reaction Curve Using the Hodograph Method. Internat. J. Rock Mech. Min. Sci & Geomech. A b s t r . , 22, 3, 173-176. Hoek,E., Brown,E.T. 1980, Underground Excavation in Rock. Institution of Mining and Metallurgy, London. Kaiser,P.K. 1981, A New Concept to Evaluate Tunnel Performance-Influence of Excavation Procedure. 22nd U.S. Rock Mech. Symp., Boston, 264-271. Kimura,F., Okabayashi,N., Ono,K., Kawamoto,T. 1985, Rock-mechanical discussion for the mechanism of supporting system in severe swelling rock tunnel. F i f t h I n t e r n a t . Conf. on Numerical Methods in Geomechanics, Nagoya, Balkema, Rotterdam-Boston, 1265-1271. Massier,D.
1984, private communcation.
Moore,I.D., Booker,J.R. 1982, A Circular Boundary Element for the Analysis of Deep Underground Openings. Fourth Internat. Conf. on Numerical Methods in Ceomechanics, Edmonton, Balkema, Rotterdam, 53-60. M~hlhaus,H.-B. 1985, Lower Bound Solutions for Circular Tunnels in Two and Three Dimensions. Rock Mechanics and Rock Engineering, 18, 37-52. Nadai,A. 1950, Theory of Flow and Fracture of Solids, McGraw Hill, New York, Toronto, London, VoI.1. Nguyen,M.D., Berest,P., Bergues,J. 1983, Analyse du comportement differe des ouvrages souterrains. 5th Internat. Congr. on Rock Mechanics, Melbourne, Preprints Section D, D233-D239. Nguyen,M.D., Habib,P., Guerpillon,Y. 1984, Time dependent behaviour a pilot tunnel drive in hard marls. Design and performance underground excavation. ISRM/BGS, Cambridge, 453-459. Panet,M.
1979,
Fourth
Time-Dependent
Internat.
Congr.
Deformations in Underground on Rock Mechanics, Montreux,
of of
Works. Balkema
Books, I, 279-289. Popovid,B., Markovid,O., Manojlovi6,M. 1979, Stresses and Strains at the Contact of Rigid Tunnel Lining and Soft Rock. Fourth I n t e r n a t . Congr. on Rock Mechanics, Montreux, BalkemaBooks, I , 525-531. Rodriguez-Roa,F. Fifth
1985,
Internat.
Lining
ground
Conf.
on
interaction
Numerical
in circular
Methods
in
tunnels.
Geomechanics,
Nagoya, Balkema, Rotterdam-Boston, 1257-1264. Rowe,R.K., Lo,K.Y., Tham,L.G. 1982, The Analysis of Tunnels and Shafts in Dense (Oil) Sand. Fourth I n t e r n a t . C o n f . on Numerical Methods in Geomechanics, Edmonton, Balkema, Rotterdam, 587-596.
272
Sharma,K.G., Varadarajan,A., Srivastava,R.K.,
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Sun,J., Lee,Y.S. 1985, A viscous elasto-plastic numerical analysis of the underground structure interacted with families of multilaminate rock mass using FFM. Fifth Internat. Conf. on Numerical Methods in Geomechanics, Nagoya, Rotterdam-Boston,
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1127-1134.
Swoboda,G. 1985, Interpretation of field measurements under consideration of the three-dimensional state of stress, the visco-elasticity of shotcrete, and the viscpoplastic behaviour of rock. F i f t h Internat. Conf. on Numerical Methods in Geomechanics, Nagoya, Balkema, Rotterdam-Boston, 1651-1658. Vardoulaskis,I., Detournay,E. 1982, Determination of the Ground Reaction Curve in Deep Tunnels Using Biot's Hodograph Method. Fourth Internat. Conf. on Numerical Methods in Geomechanics,
Edmonton, Balkema, Rotterdam, 619-624. Woodruff,S.D. 1966, Methods of Working Coal and Metal Mines, Press, Vol.2.
Pergamon
Yufin,S.A., Postolskaya,O.K., Shavarehko,I,R., Titkov,V.I. 1985, Some aspects of underground structure mechanism in the finite element method analysis. Fifth Internat. Conf. on Numerical Methods in Geomechanics, Nagoya, Balkema, Rotterdam-Boston, 1093-1100.
DEFORMATION
OF LAMINATED
The
Dead
Sea
at more
SEDIMENTS
Y. A r k i n Survey of Israel,
Geological
surface,
LACUSTRINE
(Figure
I)
is
than 400m below
the
Jerusalem
lowest
LOCAT~O" MAP
3s °
[
/
K
K
E
on
the
/
~
earth's
representing 1
3:6 °
I ¢~
U
point
the level of the oceans, A
T
O F THE D E A D SEA
Haifa
;
Y
Adana.
~
o( Galilee t
-4
SEA
t Sea~
,#
O~
•
(>/
....0".
E G Y P T
U~D I
i
ARABIA 100
•
2.7
I o,,0ti0 "
1
200kin
~ outcrops "*l-Di...... ire
02 ~,
Facies
3d
, ~Aragonite Facies Al-G~t~sum Facies after 8egin et
aL, (1980), 0 36 °
50kin
" 'sd~;
' '
FIGURE I
the
modern
the M i o c e n e Upper Lisan Sea
day
tectonic,
and
clastic
(Neev
extending
of
of
flood
Arava
Pleistocene
Formation
(Sea
phase
- Jordan
lacustrine,
and Emery,
northwards
Galilee).
- evaporitic
and
These sequence
evaporite
Rift V a l l e y laminated
1967),
within
activity
are
began
(Picard,
1942).
sediments,
known
found
surrounding
the
rift
valley
sediments
are
part
which
inpart
was
which
to of
formed
as
in
the
the Dead
Lake
Kinneret
a
fluvial
by
seasonal
274
winter wadis,
floods
discharging
and periodic
into
the
rift
high evaporation.
valley
Three main south
I,IS ~ \ I.'lJi{ Xl \rl'l()\ [.()lil;\l\]'~l{ ( ; E t ) I . ( ) t ; I C A L
by
facies,
as
Aragonite (Begin
(Figure
2)
sequence
this
is overlain with
and (Kao-
dolomite, order
(Arkin,
of
det-
palygors-
and
in
abundance
a
montmoril-
some
quartz
gonite
of
(65-80%)
illite,
sequence
in and
(20-35%)
and
generally
This
deposited
essentially,
minerals
kite),
studied
environment
carbonate
linite,
with around
(Figure i).
of,
lonite
Facies
found
was
laminated
to
rec-
1980)
Formation
consists
clay
Gypsum
two
brackish
rital
are
Diatomite,
al.,
Lisan
main
in a north
and
et
the Dead Sea
a
the
the
latter
The
of
distribution,
ognised
the
way
of
1980).
by a varved
gypsum
and
minor
silt
araand
clay. Deformation in
of
mainly
the
composition,
FIGURE 2
1982).
This
overburden
consolidation
pressure
that
is considered has
been
by
the
porosity
geochemistry as
sequence
governed
of
affected
by
pore the
and water
amount
of
post-depositional
consolida-
tion
Starinsky,
(Arkin
and
to be m i n i m a l
applied
is
the
since
load
the only
of
several
meters of varved g y p s u m and aragonite. The
Lisan
sediment
42%, with a porosity the
pore
water
is
Dead
Sea
(Arkin
and
the
above
may
ranging
40
g/l
as
Starinsky,
conditions
have
a
natural
from 33-47%. compared 1982).
changes
to The
according
water
content
The total dissolved 320
g/l
shear to
in
the
range
salts
present
strength
the
exceeding
imparted in
In
by
relative
content of clay - silt - sand and the salinity of the pore water. overall range
in day
The
is 90 to 190 kN/m2.
general,
the
shear
strength
is
less
in
the
plane
of
275
(--i
o
8~
q'~'~ua.nS~
>-~i<
uo!saqoo
\
E
~L
C 8
q~uaa~S
~qs
uo!saqoa
\ r~
b~
?
> ~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).
