a constitutive model for the stress-strain-time behavior of 'wet' clay

8/9/2019 A constitutive model for the stress-strain-time behavior of 'wet' clay

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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

283


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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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287

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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289

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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293

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


12/16

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


13/16

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


14/16

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


15/16


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 &


16/16

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.

REFERENCES

Biot, M. A. (1941). General theory of three-

dimensional consolidation. J. Appl. Phys. 12, 155-

164.

Bishop, A. W. & Henkel, D. J. (1962). The menyure-

ment of soil properties

in t he

triaxial test, 2nd edn.

London: Arnold.

Bjerrum, L. (1967). Engineering geology of Norwegian

normally consolidated marine clays as related to

settlements of buildings. 7th Rankine Lecture.

Geotechnique 17, No. 2. 83-117.

Bonaparte, R. (1981). A time-dependent constitutive

model for cohesive soils. PhD thesis, University of

California, Berkeley.

Bonaparte, R. & Mitchell, J. K. (1979). The properties

of San Francisco Bay Mud at Hamilton Air Force

Base California. Geotechnical Engineering Re-

port, University of California, Berkeley.

Borja, R. I. (1984). Finite element analysis of the

time-dependent behavior of soft

clays.

PhD thesis,

Stanford University.

Christian, J. T. (1977). Two- and three-dimensional

consolidation. In Numerical methods in geotechni-

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Duncan, J. M. & Chang, C. Y. (1970). Nonlinear

analysis of stress and strain in soils. J. Soil Mech.

Fdns Div . Am. Sot. Civ. Engrs 96, SM5, 1629-

1653.

Johnston, P. R. (1981). Finite element consolidation

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Kavazanjian, Jr, E., Borja, R. I. & Jong, H.-L. (1985).

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St ress-relaxa t i on and creep

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PhD thesis, University

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a constitutive model for the stress-strain-time behavior of 'wet' clay

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