a constitutive model for the stress-strain-time behavior of 'wet' clay
TRANSCRIPT
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BORJA, R. I. & KAVAZANIIAN, . (1985). Giotechnique 35 No. 3 283-298
A constitutive model for the stress-strain-time behaviour of
‘wet’ clays
R. I. BORJA* and E. KAVAZANJIAN, JR*
A constitutive model is developed to describe the
stress-train-time behaviour of ‘wet’ clays subjected to
three-dimensional states of stress and strain. The
model is based on Bjerrum’s concept of total strain
decomposition into an immediate (time-independent)
part and a delayed (time-dependent) part generalized
to three-dimensional situations. The classical theory of
plasticity is employed to characterize the time-
independent stress-train behaviour of cohesive soils
using the ellipsoidal yield surface of the modified Cam
Clay model presented by Roscoe and Burland. The
time-independent strain is divided into an elastic part
and a plastic part. The plastic part is evaluated using
the normality condition and the consistency require-
ment on the yield surface. The time-dependent (creep)
component of the total strain is evaluated by employ-
ing the normality rule on the same yield surface as in
the time-independent model and the consistency re-
quirement which requires that the creep strain rate
reduces to phenomenological creep rate expressions
for isotropic or undrained triaxial stress conditions.
The mathematical characterization of the constitutive
model is given by the constitutive equation expressed
in a form suitable for direct numerical implementation
(i.e. finite element formulation). The required soil
parameters are easily obtainable from conventional
laboratory tests. The constitutive equation is shown to
predict accurately the stress-train-time behaviour of
an undisturbed ‘wet’ clay in triaxial and plane strain
stress conditions.
Un modele constitutif a CtC dtveloppt pour dtcrire le
comportement contrainte-dtformation dans le temps
des argiles ‘humides’ soumises a des Ctats tridimen-
sionnels de contraintes et de deformations. Le modtle
est base sur une generalisation aux conditions
tridimensionnelles du concept de Bjerrum de la
decomposition de la deformation en une partie
immediate (indtpendante du temps) et une partie
retardee (dependante du temps). On utilise la thtorie
classique de la plasticite pour caracteriser le comporte-
ment contraintedtformation indtpendant du temps
des sols cohtrents, en se servant de la surface
d’ecoulement ellipsoidale du modele modifie de l’ar-
gile de Cam present& par Roscoe et Burland. La
deformation independante du temps est subdivisee en
Discussion on this Paper closes on 1 January 1986.
For further details see inside back cover.
* Department of Civil Engineering, Stanford Univer-
sity.
une partie tlastique et une partie plastique. On tvalue
la partie plastique d’apris la condition de normalitt et
la consistance exigee sur la surface d’tcoulement. La
partie dependant du temps (fluage) de la deformation
totale est &al&e d’aprts de la regle de normalite sur
la m&me surface d’ecoulement que dans le cas du
modble independant du temps, en employant en m&me
temps I’exigence de consistance qui veut que la vitesse
de la deformation de fluage se rtduise a des expres-
sions phtnomenologiques de vitesse de fluage pour des
conditions de contrainte triaxiales isotropiques ou
non-drainees. La caracterisation mathtmatique du
modele constitutif est donne par I’tquation constitu-
tive exprimee dans une forme qui est utilisable pour
l’application numtrique directe (c’est-a-dire pour la
formulation d’elements finis). Les paramttres du sol
necessaires s’obtiennent facilement a partir d’essais de
laboratoire conventionnels. On dtmontre que
l’equation constitutive predit de facon precise le com-
portement contraintedltformation dependant du
temps d’une argile non-remaniee dans des conditions
contraintedeformation triaxiales et planes.
KEYWORDS: clays; constitutive relations; deforma-
tion; finite elements; plasticity; time dependence.
INTRODUCTION
Consideration of the time dependence of the
stress-strain behaviour of cohesive soils is im-
portant in the evaluation of long-term per-
formance of geotechnical structures such as
embankments, tunnels and excavations. The
deformations and pressures that develop with
time are due to both hydrodynamic lag and
creep effects.
Numerous constitutive models have been
proposed (Bonaparte, 1981; Kavazanjian &
Mitchell, 1980; Pender, 1977; Roscoe, Schofield
& Thurairajah, 1963; Roscoe & Burland, 1968;
Tavenas & Leroueil, 1977) and numerically im-
plemented (Bonaparte, 198 1; Johnston, 198 1)
to characterize yielding of ‘wet’ clays using
effective stress quantities and the classical theory
of plasticity. The deformations predicted by
these models are analogous to those correspond-
ing to initial undrained loading and primary
consolidation or to those which would im-
mediately develop at the instant the imposed
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STRESS-STRAIN-TIME BEHAVIOUR OF WET CLAYS
285
model not only in axisymmetric (torsionless)
conditions but also in plane strain applications is
investigated using a finite element program.
THEORETICAL BASES
Separation of ‘immediat e’ and ‘delay ed’ str ains
The concept of total strain decomposition into
an immediate (time-independent) part and a
delayed (time-dependent) part proposed by
Bjerrum (1967) for one-dimensional compres-
sion is illustrated schematically in Fig. 1. The
plots of void ratio e and effective vertical stress*
u, versus time t show an immediate component,
occurring at the same instant that the external
load is applied, and a delayed component that
persists indefinitely with time. This decomposi-
tion scheme does not consider the influence of
hydrodynamic lag.
Superimposed in Fig. 1 is Taylor’s (1948)
description of consolidation using a curve con-
sisting of two phases: a primary consolidation
phase for tst, in which excess pore pressures
dissipate and a secondary compression phase,
I-Consohdatmn-;P
t n
(b)
Fig. 1. Definitions of primary and secondary consoli-
dation and immediate and delayed compression
* All stress
quantities
used in this Paper are effective
stress quantities. To simplify notation, the prime sym-
bol commonly used to differentiate effective stresses
from total stresses will be omitted.
governed by the secondary compression coeffi-
cient C, (=$ In 10, where $ is the secondary
compression coefficient in the In t scale) for t 2
t,. The time t, corresponds to end of pore
pressure dissipation, or to 100% primary con-
solidation. During secondary compression, the
soil continues to deform at a constant effective
stress.
Bjerrum’s graphical representation of the
effect of delayed compression on the void ratio-
log vertical effective stress diagram for a one-
dimensional consolidation test is shown in Fig.
2. The diagram consists of contours of constant
time, each contour representing compression
after an equal duration of sustained loading. By
assuming a constant C,, the constant time lines
in Fig. 2 become equally spaced.
The effect of continued volumetric compres-
sion at a constant vertical effective stress for a
typical soil element portrayed in Fig. 2 is for the
soil to exhibit apparent stiffening and to develop
a quasi-preconsolidation pressure (T, during sub-
sequent loading.
M odif ied Cam Clay as a t ime-independent plas-
ticity model
The Cam Clay theory postulates the existence
of a unique state boundary surface, XXYY in
Fig. 3, representing the limit of all possible
states of a ‘wet’ clay in the void ratio e-
volumetric stress p-deviatoric stress 4 space, in
which the stress parameters p and q are defined
Instant
ompresslot
Fig. 2. Bjerrum’s model for one-dimensional com-
pression
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STRESS-STRAIN-TIME BEHAVIOUR OF WET CLAYS
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The deformations predicted by this model are
analogous to those corresponding to undrained
loading and primary consolidation, or to those
which would immediately develop at the instant
that the imposed load is applied, in the absence
of hydrodynamic lag.
Inclusion of creep deformations
Kavazanjian & Mitchell (1980) postulated
that time-dependent soil deformations can be
divided into distinct, but interdependent, volu-
metric and deviatoric contributions. They con-
sidered these creep deformations using the
following phenomenological volumetric and de-
viatoric expressions for creep.
Volumetric creep. Volumetric creep deforma-
tions were based on Taylor’s (1948) secondary
compression equation. The accumulated vol-
umetric creep E,’ in time period At in a typical
isotropic (or one-dimensional) consolidation test
is the integral (consult Fig. 2)
I
+At
t-
ICI
E” -
t
(l+e)t” dt
where
t,
is the volumetric age, relative to an
initial reference time (t.,)i, of the state point
associated-with the constant time contour on the
e-ln p plane similar to that shown in Fig. 2. If
there is no primary loading or unloading, the
soil ages linearly with natural time during this
period; hence At = At,.
Deviatoric creep. Deviatoric creep deforma-
tions were based on the Singh-Mitchell creep
function (1968). The accumulated axial creep
strain E,’
m time period At in a typical un-
drained triaxial creep test is the integral
E,‘=,.,‘+AtAe”G[~]m dt (9)
where A, di and m are the Singh-Mitchell creep
parameters, (f& is an initial reference time, td
is the deviatoric age ,relative to (f& and D =
(a1 - as)/(ol - a& is the deviator stress level. If
there is no deviatoric loading or unloading, the
soil ages linearly with natural time during this
period; hence At = At,.
In general, equ_ation (9) holds for stress levels
of about 0.2< D < 0.8, overestimates cat for
stress states near the isotropic condition (D + 0)
and underestimates_ F,’ for stress states near the
failure condition
(D +
1.0).
Kavazanjian & Mitchell (1980) further as-
sumed that the time-independent deviator
stress-axial strain diagram corresponding to the
reference time (t& in equation (9) is given by
Kondner’s (1963) hyperbola normalized with re-
spect to the confining stress (Ladd & Foott,
1974) (T, as follows
E,fl,
lS-(T3=
a+be,
(10)
where a and b are hyperbolic parameters ob-
tained from conventional triaxial shear tests. A
third parameter Rf, termed the failure ratio by
Duncan & Chang (1970), is introduced to force
equation (10) to pass through the failure point at
an actual finite strain.
It will be shown subsequently that the ‘trace’
of the Cam Clay yield surface corresponding to
reference time (tJi on the deviator stress-axial
strain plane for soils in triaxial compression can
be described, approximately, by the hyperbolic
curve (equation (10)) when (t,&= (t,)i.
DEVELOPMENT OF THE CONSTITUTIVE
EQUATION
Definitions of stress and strain parameters
Representation of
the
general three-
dimensional state of stress requires appropriate
definitions of stress and strain parameters to
encompass all the components of the stress and
strain tensors gli and E,+
The volumetric effective stress is defined as
P = coct
= & = +I,
(11)
where I, is the first invariant of the stress tensor
c,i
The deviatoric stress q is defined as
4 = 5 T0,t
= [f(crd)ii(~d)i,]1’2= (311,)“’
(12)
where ILd is the second invariant of the de-
viatoric stress tensor (uJii. In the triaxial stress
condition, this definition for q reduces to ul-
(TV, equation (2). In the undrained plane strain
condition, where u2 = f(u, + u3), the definition
for q reduces to A((+,-u&2.
The volumetric strain is given by
E, = 3&,,, = Ekk = I,
(13)
where I, is the first invariant of the strain tensor
Eij.
The deviatoric strain y is defined as
r=&&..=
r%&(GJkJ” = (411EdY
(14)
where II,, is the second invariant of the de-
viatoric part of ekl. In the undrained triaxial
condition where the principal strains are
(E,, -;Ea, -I
, the definition for y reduces to
E,.
In the undrained plane strain condition
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BORJA AND KAVAZANJIAN
Fig. 4. Development of quasi-preconsolidatioion
where e3 = -cl and F~ = 0, the definition for y
reduces to 2&J&.
Growth of preconsolidation pressure
Assume that the size pF of the yield surface is
given by the function
Pc = Pc(. “> tv)
(15)
Equation (15) states that the preconsolidation
pressure pc does not only grow because of time-
independent strain hardening but also expands
with time, resulting in the development of a
state of quasi-preconsolidation.
Consider the consolidation curve of Fig. 4.
Along any line of constant t,, say at tV= (tJi, the
time-independent plastic volumetric strain incre-
ment is given by
in which equation (16b) is the Taylor series
expansion of equation (16a). Taking the limit of
Ap,/h~,”
as As.,‘--+ 0,
aPC
l+e
a&,”
A--K
P=
(17)
Again, consider the consolidation curve of Fig.
4. If a soil element at a state point at A, with
volumetric age t,,
volumetrically creeps in
time period At = At, at a constant effective stress,
the soil would shrink by an amount
which is easily verified from geometrical con-
struction in Fig. 4.
Solving for ApJp,
(19a)
f... (19b)
where I/J is the secondary compression coeffi-
cient, while equation (19b) is the binomial series
expansion of equation (19a). Taking the limit of
ApJAt, as At,-+ 0
aPc II, Pi
_--
et,- h-K t,
20)
Hence, the rate of growth of pc decreases as the
soil ages.
Equations (17) and (20) are respectively the
hardening rules that describe the time-
independent and the time-dependent compo-
nents of the rate of growth of the size p= of the
yield surface.
General
formulation
Let the strain rate tensor Ekl be decomposed,
i.e.
Et, = &I’ + &,” + &,
(21)
where the superscripts e and p denote the
time-independent elastic and plastic parts re-
spectively, while the superscript t denotes the
time-dependent (creep) plastic part. In principle,
the above decomposition employs Bjerrum’s
scheme of separating the total strains into im-
mediate and delayed components (Bjerrum,
1967).
Applying the associative flow rule on &,
dF
i,,P = l l
aukI
(22)
where 4 is a proportionality factor. Setting I = k
aF
FkkD
4 ~
aakk
GW
or
,,%,~
(23b)
Rewriting equation (21) explicitly
aF
& = (c;J1&,j i-4 -+ EL,l
afl,, (24)
or
cri, = CRk,
(25)
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STRESS-STRAIN-TIME BEHAVIOUR OF WET CLAYS
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where c$, is the fourth-rank elastic stress-strain
tensor.
Consider the plastic potential F = F(p, q, p=) =
F(u,,, p,) given by equation (6). The consistency
requirement on F demands that the time rate of
chance
(26)
where
(274
(27b)
in which the symbol lo implies differentiation
with the quantity inside the parentheses held
fixed, while the terms in equation (27b) are
obtained on substitution of equations (17) and
(20) in equation (27a).
Substituting equations (25) and (27b) in equa-
tion (26) and solving for 4
a
aFPc 4
-&,(&-&y+---
au
apt
t h - K
1
(28)
where
1
a
aF aFaFl e
-=-_~k,-____
x aa,,
&Tkt aPc aP A - K
PC
(29)
Substituting 4 in equation (25) and
simplifying
. t
(Tii = cy& - Ufj
(30)
where cT[ is the fourth-rank elasto-plastic
stress-strain tensor and I?,,’ is the stress relaxa-
tion rate given respectively by
aF aF
e
c~,=c~kl-x C:pq~gCrskl
(31)
p4 IS
and
f
(T,, = c%,&lf+x
@P, 9
--~
ap, t, A - K
Gkl g
(32)
It can be seen that c$ has the major symmetry
if and only if czkl has the major symmetry.
When creep is ignored, &t= 0, Ic,= 0 and
hence, ai,’ = 0. If the response of the soil is
perfectly elastic
uij = cp&& - &’
(33)
where
c
a:, = Ciik,Eklf
(34)
Thus, whatever the behaviour of the material,
the creep-inclusive constitutive equation for
‘wet’ clay can always be written in the rate form
by defining the tensors ciikl and ai,’ approp-
riately.
Evaluation of derivatives
The following are the derivatives of F (refer
to eauation (6))
aF
-=2p-p,
ap
aF 2q
-=2
aq M
aF
Gc=-p
(36)
(37)
(38)
By definitions (11) and (12) for p and q re-
spectively
8P
_=1
aui,
&, (39)
a4 3
1
- = -
aa, 2q
(
ui, - - a &sii
3
)
(40)
where IS,~ s the Kronecker delta.
Thus, the normal at any point on F is given by
a aF ap aF aq
-=
-- --
auzi ap au,, aq au,,
(414
(41b)
Elastic
soi l
constants
The elastic stress-strain tensor c& requires
at least two independent elastic material proper-
ties for complete definition. Two properties are
adequate by assuming homogeneity, material
isotropy and major/minor symmetries in c&.
The two elastic constants chosen herein are the
bulk modulus K’ and the shear modulus pe.
The elastic bulk modulus is obtained from the
volumetric Cam Clay model by noting that along
the swelling-recompression line
or
(424
(42b)
Thus the bulk modulus increases linearly with
the volumetric stress p, necessitating that p be
always positive (i.e. always compressive).
Assuming that the ‘trace’ of the modified Cam
Clay yield surface on the q-y plane (or the
deviator stress-axial strain plane in triaxial stress
condition) is a hyperbola of the form (cf. equa-
tion (10))
YPC R
q=-
f
afby
(43)
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BOWA AND KAVAZANJ IAN
lsotrop~cconsolidatw
4 =
MP
(4
hyperbola
4
4
y from hyperbola
Ibl
Fig. 5. Evafuation of a) volumetric and b) deviatoric
ages
the elastic shear modulus is back calculated from
the initial tangent modulus of the hyperbolic
curve as follows
dq
G
P& _ 3 dr,,t
I =---
=33CLe
-v=”
a
2d
(44a)
Yoct
or
e _ P&
P -3a
(44b)
Thus, the shear modulus increases linearly with
the size pc of the yield surface, necessitating that
the soil be initially preconsolidated to develop a
non-zero elastic shear stiffness.
Creep strain rate
The quantities in equation (32) that remain to
be evaluated are the volumetric age t, and the
components of the creep strain rate tensor &lf.
The direction of &lf is obtained from the
normality rule applied to the equivalent yield
surface associated with the stress state (p, q) as
follows
~-
Eklf=qE
dukl
(45)
where cp is a proportionality factor and F is the
equivalent yield surface evaluated using equa-
tion (6), whose size p0 is the ‘equivalent precon-
solidation pressure’ given by
which equals pc for normally consolidated soils
(refer to Fig. 4).
The magnitude of Fklf is obtained by de-
generating this tensor either to an isotropic ten-
sor or a triaxial tensor and appropriately scaling
cp dF/duk,) sing either the C, creep law for the
isotropic condition, or the Singh-Mitchell creep
equation for the triaxial stress condition. These
two methods are herein called volumetric scaling
and deviatoric scaling respectively.
Volumetric scaling
The magnitude of the trace of Etlf along the
volumetric axis p is a measure of the volumetric
creep
rate for the soil. The volumetric part of
&,’ is
(E,f)k, = f&Q
1 aF
Id=-(P- d
3 aa, ,
(47)
Recalling that the rate of secondary compression
is governed by the index C, (+ in the natural
logarithm scale)
or
t
E”
E= JI
aa, ,
(1-t
e)t, 48)
(49)
On substitution of cp in equation (45), the creep
strain tensor is obtained as
ik'=
1
+
F
-
(l+e)(2p-p,)t, aok,
50)
This expression for ik,’ is singular when p =
p,/2 (i.e. when the point is on the critical state
line) because the normal to
F
at this point is
vertical and cannot be scaled in the (horizontal)
p direction. If 4 is assumed constant, equation
(50) will predict higher deviatoric creep strain
rates at higher deviatoric stress levels as p ap-
proaches p,/2.
Volumetric age. The volumetric age of the soil
is obtained by examining its location in the
e-p-q space relative to the position it would
occupy if it were normally consolidated. The
volumetric age is back calculated on the basis
of the secondary compression coefficient C,
and the void ratio distance of the state point
from the state boundary surface.
Figure 5 shows an overconsolidated soil ele-
ment A with co-ordinates (e,, p, q) beneath the
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BOFUA AND KAVAZANJIAN
creep parameters are usually obtained from un-
drained triaxial creep tests, equation (62) will be
used to define the ultimate strength.
To be compatible with equation (62), the
hyperbolic curve (43) should yield the same
failure strength at an infinite deviatoric strain,
i.e. qUlt= q/,-.
This requisite condition can be
used to back calculate the relationship between
the stressstrain parameters b and R, as follows
b 21-“‘”
_-
R,-
M
(63)
where K, A and M are the Cam Clay parame-
ters.
Deuiatoric age. Consider the same point A in
Fig. 5 which is now given by the co-ordinates
(q, yr) on the q-y plane. If the soil is normally
consolidated, the stress q would locate A on the
hyperbolic curve (43) at
w
-Y?_=-
PS- qb
(64
The deviatoric age td is computed from equa-
tion (9) based on y2 as
t =
(y,-Y&-m)
Ae”“(t&“’
1
(‘bmJ f
mf 1
(654
= (
td),
exp
if m=l
(65b)
Numerical implementation
The development of equation (35) allows the
resulting effective stress-based constitutive equ-
ation to be incorporated into Biot’s (1941) gen-
eral three-dimensional hydrodynamic theory of
consolidation. This consolidation theory uses
Darcy’s law to characterize the transient pore
pressure dissipation condition, i.e.
(66)
where ic; is the ith component of the velocity
vector ic, p is the pore pressure field, k, is the
i, j)
component of the permeability tensor k,
and y_, is the unit weight of water (the negative
sign implies that the flow is in the direction of
decreasing gradient).
It can be seen from equation (66) that the
displacement field u and the pore pressure field
p are not independent, but satisfy the relation-
ship given. A finite element solution can then be
formulated (Borja, 1984; Christian, 1977; John-
ston, 1981) in which the nodal unknowns inter-
polating u and p are coupled using a virtual
work or a variational formulation. Such a
‘mixed’ type of formulation has been numeri-
cally implemented by Borja (1984) for solution
of two-dimensional boundary value problems of
axisymmetric (torsionless) and plane strain con-
figurations using the above constitutive equa-
tion. The program, called SPIN 2D, incorpo-
rates creep contributions by explicitly evaluating
the stress relaxation term hilt in equation (35) at
the start of each time increment. These con-
tributions can be treated in a manner similar
to or along with temperature stresses in the
finite element matrix equations, giving rise to a
pseudo-force term F”“’ which can be explicitly
evaluated and added to the applied nodal force
arrays (Borja, 1984).
Remarks
This numerical approach of expressing creep
contributions as artificial forces allows stationary
creep problems (e.g. isotropic undrained stress
relaxation experiments and undrained triaxial
creep tests) to be numerically simulated. The
treatment of hydrodynamic lag also allows the
separation of time-dependent deformations into
components due to pore pressure dissipation
and due to creep.
Example
The following example is based on the result
of a drained triaxial compression test on Weald
Clay reported by Bishop & Henkel (1962). The
test consisted of consolidating a cylindrical soil
sample to an isotropic stress of p= = 207 kN/m*
and then shearing the sample very slowly while
maintaining the radial stress constant. The test
results, shown in Fig. 6, are duplicated by SPIN
2D by suppressing the effects of creep and using
the following material properties: K = 0 .031
A = 0.088, M = 0.882, a = 0.023,
Rf =
1.00 and
ea= 1.31.”
To illustrate the influence of creep, the vol-
umetric scaling option on creep strain was emp-
loyed (4 = 0.22 artificially selected) using a four-
node axisymmetric finite element. The bold lines
in Fig. 6 illustrate what the deviator stress-axial
strain and the volumetric strain-axial strain be-
haviour would have been if the soil specimen
had been allowed to undergo stress relaxation
increment bc and loaded to failure (cd). During
the stress relaxation increment bc, the yield
surface continually expands with time owing to
creep, resulting in the development of a quasi-
preconsolidation pressure p=. Concurrently, the
*The same test results
have been duplicated by the
program PEPCO developed by Johnston (1981) for
the case when creep is ignored.
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size p. of the ‘equivalent yield surface’ continu-
ally shrinks because of stress relaxation (see
equation (46)).
When shearing is resumed the expanded yield
surface is again engaged, showing initial stiffen-
ing of the stress-strain response expected due to
quasi-preconsolidation. Further shearing causes
the element to fail at d.
APPLICATION OF THE MODEL TO THE
BEHAVIOUR OF UNDISTURBED BAY MUD
In this section the constitutive model is
evaluated on the basis of its ability to predict
accurately the results of simple triaxial and
plane strain laboratory tests on undisturbed San
Francisco Bay Mud (UBM). Both drained (free-
flow) and undrained (no-flow) problems are
numerically analysed using single finite elements
whose convergence characteristics have been
previously established.*
The soil parameters used to model UBM are
summarized in Table 1. These soil properties
were taken from a comprehensive summary of
Bay Mud properties presented by Bonaparte &
Mitchell (1979). Except for e, (the void ratio at
p = 1 on the isotropic consolidation line) which
locates the ‘immediate consolidation line’ on the
e-ln p plane, all the soil properties were directly
obtained from the results of conventional
laboratory tests.
The value of e, can be obtained indirectly by
extrapolating the primary consolidation line to
p = 1 and moving up along the void ratio axis, to
the immediate line, according to the secondary
compression coefficient 4 (see also Kavazanjian
& Mitchell, 1980). However, Borja (1984)
showed that the stress-strain and pore pressure-
strain curves do not show appreciable sensitivity
to the value of e,. Hence, the primary consolida-
tion curve may also be taken as the ‘immediate
line’ without introducing serious numerical inac-
curacies.
Drained ttiaxial tests
The response of the soil is said to be fully
drained when the rate of loading is much smaller
than the pore fluid diffusion rate. In this case,
the pore pressure degree of freedom can be
suppressed, allowing the problem to be solved
using a single-phase continuum formulation.
To demonstrate the predictive capability of
the constitutive model in drained triaxial situa-
tions, four drained triaxial compression tests on
*See Kavazanjian, Borja & Jong (1984) for an
analysis of a test embankment problem involving the
combined effects of consolidation and creep using a
mesh of multiple finite elements.
.
200 -
150.
“E
t
y ioo-
ti
Whop 8. Henkel (1962),
and SPIN 2D (no creep)
/
- SPIN 2D with volumetric
creep
8
w’
PEPCO.
9 =
MP
Yield Surtace
p’ kNlm”
Fig. 6. Drained test on Weald Clay
UBM performed by Lacerda (1976) were simu-
lated. In his tests, the specimens were initially
isotropically consolidated to confining pressures
(T, of 53.9 kN/m’, 102.9 kN/m*, 156.8 kN/m’
and 313.6 kN/m’ and sheared at a strain rate of
E, = 3.2
x
lo-‘% per minute.
Lacerda showed that the deviator stress versus
axial strain diagram can be normalized by divid-
ing the deviator stress by the initial consolida-
tion pressure. A plot of this normalized curve
and the volumetric strain-axial strain curve are
shown in Fig. 7.
Numerical tests were run with and without
creep effects using a single axisymmetric finite
element. This element represents the upper
quadrant of a triaxial specimen and approxi-
mates the displacement field using a bilinear
interpolation.
The results of numerical analyses with and
without creep effects are shown by the full and
open circles in Fig. 7, respectively. The creep-
inclusive analysis, performed using the de-
viatoric scaling option, significantly improves the
prediction for volumetric strain E, particularly at
larger values of E,.
Undrained triaxial compression tests
Lacerda (1976) also performed a series of
undrained triaxial compression tests at different
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294
BORJA AND KAVAZANJIAN
Table 1. Model parameters for undisturbed Bay Mud
Value
ymbol
A
C,=log,,h
c, = log:, K
ICI
c, = log,, IL
a
b
4
A
cu
m
h, k
4’
M
ea
O,)i
I i
Parameter
Virgin compression index* 0.37
0.85
Recompression index*
0.054
0.124
Secondary compression coefficient*
0.0065
0.0150
Hyperbolic stressstrain parameters?
0.0062
1.36 (=1.23 from equation (63))
0.95
Singh-Mitchell creep parametersl
3.5 X 10-s per min
4.45
0.75
Permeability*
Variable
Angle of internal friction? 34.5”
M from equation (5)
1.40
2.52oid ratio* at p,= 1 kPa
Instant volumetric time
1.00 min
1.00 minnstant deviatoric time
* From triaxial isotropically consolidated or conventional consolidation test.
t From isotrooicallv consolidated undrained test with pore pressure measurement.
$ From isotropically consolidated undrained creep test.
strain rates, ranging from 1.1%
per minute to
7.3~ 10m4 per minute. He observed that both
the initial tangent modulus Ei and the ultimate
strength quit of the deviator stress-axial strain
curves varied linearly with the logarithm of the
axial strain rate E,. Similar observations have
been reported by Kondner (1963). Hence, it
may be inferred that the hyperbolic curves given
by equation (lo), obtained from conventional
undrained triaxial tests, do not exactly represent
the time-independent behaviour because they
also contain creep components.
Interpolating E, and quit from Lacerda’s plots
for & values of 1.0 per minute, 0.1 per
minute, 0.01 per minute and 0.001 per
minute, the corresponding transformed hyper-
bolic curves are plotted in Fig. 8. Also plotted in
Fig. 8 is the hyperbolic curve obtained by
Bonaparte (1981) from conventional stress-
controlled undrained triaxial compression tests.
Bonaparte’s tests were completed in about
250-300 min to allow pore pressure equaliza-
tion, compressing the samples to about 10
axial strain during this period. This is roughly
equivalent to compressing the samples at a strain
2.57
2.0-
1.5-
bU
B
l.O-
SPIN 20
0
No creep
. Creep Included
r:;~
Fig. 7. Drained test on UBM
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STRESS STRAIN TIME BEHAVIOUR OF WET CLAYS
295
rate of about 0.01% per minute. This is con-
firmed by the observation that Bonaparte’s results
plot very closely to the hyperbola corresponding
to this strain rate.
Numerical analyses using the properties in
Table 1 were performed with a single finite
element that uses a biquadratic displacement
interpolation and a linear pore pressure interpo-
lation.* Excellent agreement was achieved be-
tween Lacerda’s results and the numerical ex-
periments, as shown in Fig. 8. It can be verified
from Fig. 8 that the ‘trace’ of the Cam Clay
yield surface on the deviator stress-axial strain
plane under undrained triaxial compression is,
approximately, a hyperbola whose strength and
stiffness are a function of the imposed strain
rate.
When creep is ignored, the data points define
a unique hyperbola regardless of whether the
numerical tests are stress controlled or strain
controlled. It can also be observed that the
hyperbola obtained from conventional stress-
controlled triaxial compression tests does not
generally represent the immediate soil be-
a-
6
- Lacerda (1976)
(strain controlled)
-
Bonaparte (1981)
controlled)
duratfon
f 240-300 ml”
Stress controlled, no creep
Straw controlled, no creep
n Strain rate = 0 1% per mn
A Stress controlled,
duratfon = 300 mn
2 4
Q6 a lo
Fig. 8. Undrained triaxial tests on UBM performed at
various strain rates
*It has been generallv observed (Christian, 1977;
Johnston, 19817 that to obtain compatible coupled
fields the displacement interpolation should be one
order higher than the pore pressure interpolation.
haviour,
since it may contain a significant
amount of creep deformation.
Ufldrained plane strain. test
Plane strain test results are not as commonly
reported in the literature as triaxial test results
although the former can be more useful in the
analysis of actual geotechnical structures such as
dams, embankments and long excavations.
An undrained plane strain test on UBM was
performed by Sinram (1985) to simulate a deep
pressuremeter test stress condition (Figs 9 and
10). In this case, the normal off-plane stress
changes as a result of Poisson effects. Sinram
measured these stress changes as well as the
in-plane strains and pore pressures in his tests.
To mimic the imposed loading history shown
by the points on the total stress path in Fig. 9, a
numerical test was performed using a single
bilinear displacement, constant pore pressure
element and the deviatoric creep option for
creep strains. Fig. 10 shows that the induced
pore pressures due to volumetric compression
and shearing are initially overpredicted. Conse-
quently, the deviator stresses during the initial
shearing stage (oa) in Fig. 9 are also over-
predicted.
Overall, however, the pore pressure-strain
and stress-strain curves manifest the stress his-
tory to which the soil specimen was subjected.
Further, the rupture strength ((r,-~~),,,~._ the
pore pressure at the near-failure condition and
the additional excess pore pressure during the
Fig. 9. Stress-strain diagram for a plane strain test on
URM
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296
BORTA AND KAVAZANJIAN
9
200-
; itjo-
z‘
Points on TSP
-
I
120
--~
inram
1985)
-SPIN 2D
k
I’
2
0
0
I
0
2 4 6 8 10 12
SKiIn e22 %
Fig. 10. Pore pressurestrain diagram for a plane
strain undrained test on UBM
isotropic loading stage BC were reasonably well
predicted.
The same overprediction of the deviator stres-
ses in the triaxial stress condition was also re-
ported by Roscoe & Burland (1968). They
suggested that the Cam Clay model be modified
to account for plastic shear deformations which
may also develop beneath the state boundary
surface.
Undrained creep tests
Several undrained triaxial creep tests have
been performed at different deviatoric stress
levels over periods of several logarithmic cycles
of time. Results of three tests performed by
Lacerda (1976) are shown in Fig. 11 for values
of fi of 0.8, 0.7 and 0.5, obtained either by
increasing the axial load or by decreasing the
lateral (confining) pressures from the initial iso-
tropic condition.
Superimposed in Fig. 11 are the numerical
tests obtained using a single triaxial element that
interpolates the displacement field biquadrati-
tally and the pore pressure field linearly, and the
deviatoric scaling option on creep strain is emp-
loyed. The excellent agreement between the ex-
perimental and the numerical test results
affirmed the validity of the Singh-Mitchell creep
function for undrained triaxial conditions for
values of fi ‘within the range of engineering
interest’. It should be noted that for the condi-
tion fi = 0.8, the experimental (full) curve starts
to deviate from the Singh-Mitchell creep func-
tion for t > 1000 min, an observation which can
be even more apparent in the failure and near-
failure conditions (D -+ 1.0).
Combined
creep
and
str ess relaxat i on t ests
To illustrate the ability of the model to ac-
count for the time and stress history dependence
of the material response to applied loads, a
numerical test was performed to simulate un-
drained triaxial creep and stress relaxation tests
on UBM performed by Lacerda (1976). Biquad-
ratic displacement, linear pore pressure interpo-
lations were used on a single axisymmetric
quadrilateral element and the deviatoric scaling
option on creep strain was employed.
10
r
CR-l-2 D = 0.8
0, I”creaslng,o, Consranl
SR-1-2 D = 0 7
m, constanlp, decreasing
SR-l-3 u = 05
v, constant,o, decreasmg
- Lacerda (19761
Fig. 11. Undrained creep tests on UBM
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STRESS-STRAIN-TIME BEHAVIOUR OF WET CLAYS
297
The test consisted of initially consolidating the
element to an isotropic stress of 78.4 kN/m’
and shearing the element undrained at a con-
stant axial strain rate of F = 0.38% per minute.
At this point, the deviatoric stress q =
42.6 kN/m’ (point A in Fig. 12).
After about 3000 min of stress relaxation
(AB) during which ca= constant, shearing was
resumed at the same strain rate until the stress
level was close to failure (C). The strain was
then held at about 2.3% for 1320 minutes after
which the specimen was sheared again at a
reduced strain rate of e, = 1.6 x lo-‘% per mi-
nute.
Very good agreement can be observed during
stages OA, AB, BC and CD on the stress-strain
curve of Fig. 12. During reloading at a reduced
strain rate (DE), however, the creep strain rate
compute_d_ from the Singh-Mitchell function
&’ zz AeaD
exceeded the actual strain rate of
0.016% per minute for values of DaO.342.
Thus the numerical solution had to be termi-
nated at that point.
The predicted pore pressure curve of Fig. 12
shows an offset of about AB = 10 kN/m’ beyond
point B. The overestimation of pore pressures
during this stress relaxation stage is due to the
arresting of secondary consolidation in a near-
isotropic condition (B), a phenomenon which
can be expected to occur but was never ob-
served in this laboratory test. The offset AB may
be attributed to the fact that the deviator& scal-
ing option overpredicts F,t for values of D close
to zero, resulting in the overprediction of pore
pressures as well.
CONCLUDING REMARKS
A constitutive model for ‘wet’ clays capable of
accounting for time-dependent (creep) effects in
general three-dimensional stress conditions has
been developed. This model is characterized
mathematically by the following effective stress-
based rate constitutive equation
u;j = C$&&k, ai,t
and has been incorporated into Biot’s (1941)
three-dimensional theory of consolidation to ac-
count for the influence of hydrodynamic lag.
The constitutive model defines the tensor of
moduli ciikl
and the stress relaxation rate ail’,
before and during yielding, based on Bjer-
rum’s (1967) concept that the total deformation
can be decomposed into time-independent and
time-dependent parts. The time-independent
part of the total deformation is evaluated using
the classical theory of plasticity and the ellipsoi-
dal yield surface of the modified Cam Clay
model presented by Roscoe & Burland (1968).
60
“E
t
Y
6-40
20 ---
Lacerda (1976)
- SPIN 2D
0
1 2 3 4 5 6
- -- Lacerda (1976)
-SPIN 20
OL
1 2 3 4 5
Am strain ‘72’ %
Fig. 12. Combined creep and stress relaxation test on
UBM
The time-dependent part is evaluated on the
basis of the phenomenological creep rate ex-
pressions for isotropic or undrained triaxial
stress conditions (volumetric and deviatoric scal-
ing respectively).
If the secondary compression coefficient 4
is constant, the volumetric and the deviatoric
scaling procedures cannot give identical results
because rC, cannot be made to vary with the
deviator stress level. An approach presented by
Borja (1984) for the plane strain case forces the
tensor Eer’ to satisfy both the volumetric and the
deviatoric creep requirements without applying
the normality rule.
The above constitutive equation has been
numerically implemented into a consolidation-
creep finite element program capable of solving
two-dimensional boundary value problems in
plane strain and axisymmetric (torsionless) stress
conditions. The tensor of moduli c,~~,contributes
to the global stiffness matrix, while the stress
relaxation rate term &
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298
BOFIJA AND KAVAZANJIAN
effects, demonstrating that creep deformations
can become a major contributor to the total
deformations for ‘young’ clays or in conditions
under which the time of sustained loading is
comparable with the geologic age of the soil. By
imposing the condition of incompressibility, it
has likewise been shown that undrained creep
can be of major importance in the prediction of
excess pore pressures as a result of either the
arresting of secondary compression or shearing.
While the examples discussed in this Paper
showed simple boundary conditions simulating
triaxial or plane strain laboratory stress condi-
tions, the validity of the above model has also
been investigated and has been shown to work
in more complicated axisymmetric applications
(Borja, 1984) and in an actual plane strain field
situation (Kavazanjian et al., 1984).
Further research is currently under way as
part of a Stanford University-University of
California, Berkeley, collaborative research
project on the stress-strain-time behaviour of
cohesive soils, funded by the National Science
Foundation.
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