A unified elasto-viscoplasticity model for clays, part I: Theory

Computers and Geotechnics 13 (1992) 71-87

A UNIFIED ELASTO-VISCOPLASTICITY P A R T I: T H E O R Y

M O D E L F O R CLAYS,

Robert Y. Liang and Fenggang Ma Department of Civil Engineering, University of Akron, Akron, OH 44325-3905

ABSTRACT

A unified elasto-viscoplasticity model is formulated for describing the stress-strainpore pressure-time behavior of saturated cohesive soils under general three-dimensional stress conditions. The limit surface and the conjugated static yield surface (elastic domain) form the basic framework from which the time- (aging and creep) and rate- effects can be conveniently accounted for by using a single internal state variable - preconsolidation pressure. The attendant evolution law for the internal state variable is formulated on the basis of phenomenological characteristics presented by Leroueil, et al, (1985) [1]. A detailed review of macroscopic characteristics of clays and mathematic derivations of the proposed viscoplasticity theory are presented in this paper.

INTRODUCTION The effects of time on the strength and deformation characteristics of saturated cohesive soils assume an important role in a variety of geotechnlcal engineering problems where either very rapid, or very long term response is of concern. Examples would include rapid drawdown in cohesive dam cores, excavations and underground tunneling in cohesive soils. Although the numerical methods (e.g., finite element method) are frequently employed to solve those complicated boundary value problems, the success of the numerical programs largely depends on the constitutive models incorporated in the computer codes. In this paper, a unified rate- and time- dependent elasto-viscoplasticity model is developed for predicting the stress-strain-pore pressure-time behavior of saturated cohesive soils in a general three-dimensional stress state. 71

Computers and Geotechnics 0266-352X/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain

72

P H E N O M E N O L O G I C ~ k L (MACI~OSCOPIC) C H & R ~ k C T E R I S T I C S Traditionally, study of time- and rate-dependent behavior of saturated clays has been carried out in the laboratory environment under either completely undrained or drained condition. This can be attributed to technical difficulties in maintaining controlled partially drained condition in a laboratory experiment as well as lack of adequate data interpretation procedure. As a result of this polarized approach, there has been abundant literature on either undrained or drained behavior of clays. A coherent framework of unifying the time- and rate-dependent behavior under both undrained and drained conditions has been rather scarce. Recently, Tavenas and Lerouell (1977) [2] presented a more unified framework for the time- and rate- dependent behavior of clays.

I. U n d r a i n e d B e h a v i o r Although somewhat artificial, manifestations of time- and rate- effects in undrained condition can be divided into three categories: undrained creep, stress relaxation, and strain rate effect. Undrained creep can be defined as continuous increase in pore water pressure and distortional deformation with passage of time under apparently constant deviatoric stress. Numerous researchers (e.g., Walker, 1969 [3]; Arulanandan, et al., 1971 [4]; Shen, et al., 1973 [5]; Holzer, et al., 1973 [6]; Febres-Cordero and Mesri, 1974 [7]; Tavenas, et al, 1978 [8]; and Bonaparte, 1981 [9]) have generally confirmed the validity of Singh and Mitchell (1968) [10] phenomenologically-based creep equation ~.

= A

exp(ctD)(t,/~)'~

where ~o is the tudal strain rate of creep; D is the stress intensity defined a s

(1) (0" 1 - - 0 " s ) / ( 0 " 1

--

0"~),~o®; tt = a reference time which is usually taken as unity; and A, a, and rn are the so-called Singh-Mitchell creep coefficients. The bnild-up of pore water pressure during undrained creep has been attributed to two mechanisms (Holzer, et al., 1973 [6]): (i) the applied principal stress difference, and (ii) arresting of secondary consolidation. In general, the "wet" clay tends t~ develop positive pore pressure, while the "dry" clay tends to develop negative pore pressure, as the pore pressure response in the undrained condition is intimately related to the volume change tendency (contraction or dilation) under the drained condition. In normally consolidated clays, as a result of positive pore pressure rise and the attendant decrease in effective stress, creep rupture may occur.

73 The term "stress relaxation" refers to the time-dependent stress and pore pressure responses of a specimen whose dimensions are held constant. A bulk of experimental studies have been reported (Murayama and Shibata, 1961 [11]; Vialov and Sklbltsky, 1961 [12]; Saada, 1962 [13]; Wu, et el, 1962 [14]; Lacerda and Houston, 1973 [15]; and Lacerda, 1976 [16]). Experimental findings in this aspect appear to be quite consistent. Mainly, the decay of stress is essentially linear with the logarithm of time, following the application of a constant strain. Further, the variation of pore pressure during undrained stress relaxation is very small. Regarding the apparent strain rate effect on the undrained strength, there have been reports (Crooks and Graham, 1976 [17]; Graham, 1979, 1983 [18] [19]; Vald and Campanella, 1977 [20]; Vald, et al., 1979 [21]; and Lefebvre and LeBoeuf, 1987 [22]) that an increase of 10% to 20% in the undrained shear strength could be resulted from a tenfold increase in strain rate. Part of the reason for this observed rate effect has been attributed to viscosity-induced lower pore pressure generation under faster strain rate (Casagrande and Wilson, 1951 [23]; Bjerrum, etal., 1958 [24]; Crawford, 1959 [25]; O'Neil, 1962 [26]; and Lefebvre and LoBoeuf, 1987 [23]).

II. D r a i n e d B e h a v i o r

Leroueil, e t a l .

(1985) [1], in their extensive experimental program involving the

use of four types of oedometer tests (constant rate of strain tests, controlled gradient tests, multiple-stage loading tests, and creep tests) on structural-sensitive natural clays, have confirmed the theological model originally proposed by Suklje (1957) [27]. In fact, Leroueil, etal. (1985) [1] suggested that the theological behavior of natural clays can be completely described by two equations ~ = f(~) fly --

~

(2)

i =

g(~o)

(3)

where Eq.(2) represents the dependency of the preconsolidation pressure on the volumetric strain rate, and Eq.(3) indicates a unique compressibility of clays when the effective stress is normalized with the strain-rate dependent preconsolidation pressure. These two equations are schematically shown in Figure 1. From this observation, it seems that one of the more convenient approaches to formulate a generalized viscoplasticity theory is to make the preconsohdation pressure to be rate-dependent. Experimental results

74

~.

~2

~

f

',

~1

I

~'~

00

~

1

~',',~.'.'/..1 2

•~; % ~0l ~i

~ ' ~

t

i, it

~,

Figure 1: The rheological model proposed by Tavanas and Leroueil by Leroueil, et al. (1985) [1] suggest that the functional relationship in Eq.(2) can be represented by a simple linear relationship.

~ t = (~)° + b ( ~ , ~ ,

- ~=(~)°)

(4)

where superscript a indicates any selected reference state, and b is the slope in the ~r~-lrti~ plot. In fact, parameter b possesses clear physical meaning, i.e., the capability of the clay skeleton to creep and the sensitivity to rate of straining.

III. U n i f i e d V i e w p o i n t A detailed laboratory investigation (Tavenas and Leroueil, 1977 [2]) involving both undrained and drained tests on natural clay specimens (Champlain clays) has led to a better understanding of the effects of time and rate of loading on the yielding behavior. Figure 2 depicts the essential features of the unified behavior model. Point B, with the corresponding void ratio eB and the effective overburden pressure pB, represents the state of end of soil deposition process. Curve Y0 represents the corresponding "young" limit state surface. Note that the limit state surface is defined by Tavenas and Leroueil as a

75 combination of yield surface for those stresses below the Mohr-Coulomb failure line and the peak strength envelope. The aging effect is shown by the path BB' in Figure 2(b), in which the void ratio decreses to eA at a constant overburden pressure PB.

As can be seen, a quasi-

preconsolidation pressure P, develops (Bjerrum, 1967 and 1973 [28] [29]; Burland, 1971 [30]), causing the expansion of the "young" limit surface Y0 to the aged timit surface Y1 (Figure 2(a)). The effect of strain rate during the application of shear stress is manifested by the displacement of the limit surface. With an increase in loading duration (i.e. a decrease in strain rate), the limit surface has a tendency to displace toward a reduced precosolidation pressure and a smaller shear strength, as shown by those dashed lines in Figure 2(a). To illustrate the concept of the soil behavioral model, consider an element of soil in a natural slope that was formed by the erosion process. The state of stress of this soil element is assumed at point A to indicate a state of overconsolidation due to stress relief. The behavior of this soil element when subjected to changes in effective stress and/or time will vary depending upon which zones of state of stress it has been brought to. In zone I, failure will occur immediately. In zone II and zone III, large consolidation and creep will develop, respectively. In zone IV, creep deformation will develop at first. With the passage of time, during which the limit state surface will move from Y1 to Y0, the soil will fail eventuraily in an apparent creep failure. In zone V, small deformation will occur with no opportunity for soil failure. One of the critical experimental observations made by Tavenas, et ai. (1978) [8] is a further validation of existence of contours of equal strain increments in a stress space as shown in Figure 3. Their experimental results showed that at a given time, the equal creep strain increment contours are more or less homothetic to the limit surface. The strain increment was found to be dependent upon the 1~uclidean distance between current stress and the image stress on the limit surface. The closer the current stress to the limit surface, the larger is the strain increment observed. Although existence of elastic nuclei was not clearly identified in Tavenas, et al. (1978) [8] data, the small strain increment contours (e.g., 5 * lO-~%/mirt) can be effectively considered as the boundary of the elastic domain. Again, the homothetic nature between the elastic nuclei, the equal strain increment contours, and the limit surface suggests that a single surface can be employed to define the relevant surfaces in the viscoplasticity formulation. This fact will be utilized in the derivation of the proposed viscoplasticity theory.

76

(b)

(a) GI--O~

~

2

~

. . . . . . .

Mohr-Coulomb Agingli

Fallur ~ L ~ -

'~, Zone I

~

~/~ e

~

.~o "~-

,o,J /

0.01~

\\, II

~

.......

\x\\\ "

P, ~

I

I ~ Virgin Compression ~ Line

+~ 2

"~

~

1

I\

I, Pe

~~ Pc

I

~~ Pc ~

~'i-~

t

}

log P

~

Figure 2: Strain rate contours as obtained from lab. test

25¸

Mahr-Coulamb Failure L l n ~

20

:

"~

~-,:~o-3 \ ~

lO

A" ,-

15

!

~",'-~--

L ~ . S,a,e

/

I

5~I~,//

,

I

I

20

25

30

35

40

~1"['oa' --

45

2

Figure 3: The proposed ellipsoidal limit surface and the associated elastic nucliei

77 R E V I E W OF E X I S T I N G E L & S T O - V I S C O P L ~ k S T I C I T Y M O D E L S

A v~riety of viscoplasticity formulations have been suggested in the literature. However, two of the more popular formulations are the endochronic theory proposed by Vaianis (1971) [31] and the unified elasto-viscoplasticity theory developed by Perzyna (1963, 1966) [32] [33]. The essence of endochronic theory lies in an incorporation of an intrinsic time scale related to material deformation to account for time-dependent plastic behavior. Perzyna's formulation, on the other hand, employs a time-rate flow rule on both a plasticity yield function and an overstress function. Other viscoplastic formulation, as reviewed by Katona (1984) [34], include those developed by Katona (1980) [35], Bodner and Partom (1975) [36], and Phillips and Wu (1973) [37]. In developing the unified elasto-viscoplasticlty model, Perzyna adopted the following assumptions: (i) the visco behavior becomes active only after entering the plastic state; (ii) the inelastic viscoplastic strain increment can be evaluated from the overstress function. Mathematically, Perzyna's viscoplastic flow rule can be expressed as .~ Of ~,j = ~ < ~ ( F ) > 0~-~

(5)

where

~(F)

viscous flow function : { v(E) 0

if F > 0 if F < 0

(6)

and 7 is a material constant representing its viscous property; ~o(F) is called viscous flow function, and its argument F -- ~ - 1 is a measure of overstress, representing the difference between the rate-sensitive (dynamic) loading function and the static yield function. The current direction of an inelastic strain increment is given by the gradient of f. F -- 0 (i.e., f -- k) represents the current static yield locus, which is regular and convex, and may evolve according to some hardening rules with ]~ as a hardening parameter. When F > 0 occurs, inelastic strain rate can be obtained from Eq.(5). Adopting Perzyna's elasto-vlscoplasticity formulation, several unified vis¢oplasticity models have been forwarded for cohesive soils (Adaehi and Okano, 1974 [38]; Zienkiewicz, et ai, 1975

[39];

Oka, 1981 [40]; Adachi and Oka, 1982 [41]; and Katona, 1984 [34]).

Adachi and Okano (1974) [38], on the basis of Roscoe's (1963) [42] original energy theory, derived a static yield locus for a normally consolidated clay. The dynamic loading surface, considered to be homothetic with the static yietding surface, was derived by adding an isotropic hardening and a strain rate dependent increment Ak to the static yielding locus. However, the determinations of A/¢ remain problematic. Oka (1981) [40] adopted the yield locus of the Cambridge theory (Roscoe, 1963 [42]) as the funetionai form of

78 both static yielding surface and dynamic loading surface. A more definite derivation of the form of both static yielding and dynamic loading surfaces was given by Adachi and

Oka (1982) [38]. These foregoing three models adopt the same functional form of ~ ( F ) as follows.

~a(F) = cexPim(f,~ - f,)]

(7)

in which fd and f, are the dynamic loading surface and the static yield surface, respectively; c and m are curve fitting constants. Zienkiewicz, et al., (1975) [39] indicated that any rate-independent yield surface F can be used as the static yield surface in the Perzyna's visco-plasticity formulation. When F is larger than zero, the inelastic strain rate can be evaluated from Eq.(5) in conjunction with the following visco-plastic flow function. F ~ ( F ) = 7~00

(8)

in which F0 is a reference value of stress and 7 is a positive constant. Specifically, a static yield surface combining the Mohr-Coulumb, or the Drucker-Prager yield condition with the Cam-Clay plastic model as the cap has been used. Katona (1984) [34] adopted the following two popular forms for the viscoplastic flow function: T(F) -- ( ~ ) ~

(9)

and

~(F) = exP( ~o)N -- 1

(10)

in which N = an empirical exponent; F0 = a normalizing constant to make flow function dimensionless.

FORMULATION

OF VISCOPLASTICITY

MODEL

One of the key ingredients in the Perzyna's visco-plasticity formulation is the plasticity yield function, since such functions influence both the magnitude and direction of the inelastic strain rate. In cohesive soils, it is well known that yielding takes place during the early stage of loading, thus making it difficut to identify the initial yield surface from traditional triaxial tests. However, studies by Tavenas and Leroueil (1977) [2], and Tavenas, et al. (1978) [8] indicated that limit state surfaces can be defined more clearly from experimental results. In fact, the development of the limit state surface (Roscoe and Burland, 1968 [43]) within the classical rate-independent plasticity has been well

79 advanced. From this standpoint, the concept of limit surface seems to be more suitable for deriving the dasto-viscoplasticity formulation. In the present formulation, a rate-dependent limit surface is adopted as a reference surface from which the elastic domain (static y i d d surface) and then the dynamic loading surface are defined. The genesis of the Limit surface is to some extent analogous to the bounding surface in the bounding surface plasticity. The same type of application of the bounding surface concept in visco-plasticity has been reported by Moosbrugger (1988) [44] in his study of metal materials. The main functions of the limit state surface in the proposed viscoplasticity formulation are: (i) its outward normal direction defines the direction of the inelastic strain rate; (ii)_its evolution is postulated to control the evolution of the elastic domain (i.e., the static yield surface); and (iii) it defines the overstress function used in the flow function. The total strain-rate tensor is assumed to consist of the elastic and inelastic (viscoplastic) components.

"" ÷ ~s ~ij : eli where

(11)

dlj denotes the infinitesimal strain rate, superscripts e and i represent the elastic

and inelastic parts, respectively. The total strain rate of Eq.(11) is evMuated by applying (i) the generalized, isotropic, nonlinear elastic theory to compute the elastic part, and (ii) Perzyna's viscoplasticity theory in conjunction with the limit surface concept to obtain the inelastic part. I. F l o w R u l e The flow rule for the inelastic strain rate has been proposed as follows. ~ _ ~,(f) OF K 0~ij

f ~,(f)>O iff>O ~ 7'(f)=0

if f < 0

(12)

where K is a constant characterizing the fluidity (viscosity) of a material under shear straining; ~ ( f ) is the flow function dictating the current magnitude of the viscoplastic strain rate. The argument of flow function, f , is a measure of overstress, representing the difference between the (rate-sensitive) dynanfic loading function and the static yield function. The cnrrent direction of the inelastic strain rate is given by the gradient vector of a limit surface F at the image stress state. (i.e., the mapping stress according to the adopted mapping rules). An elllpse, as shown in Figure 4, is used in the model to represent the limit surface, which is expressed as follows. 2 R-2 J2 F - (f)2 _ g I 0 f - ----ff--Io~ + (R - 1 ) ' ( W ) = 0 2v .1%

(13)

80

Limit

Surface(Ellipse)

•

~

.

r

Io/R

Io

J

Figure 4: Limit Surface and Elastic domain on I - J plane where R and N are model parameters. The projection of the limit surface on the ~r plane is shown in Fig.5. The parameter R controls the ratio of the two m a j o r axes of the ellipse; I0 is a measure of the pre-consolidation history and considered as an internal variable; () represents a quantity on the limit surface; the first and second stress invariants, I and J , and the Lode angle O are defined as

I = O'ij~ij J =

(~SijSij )1/2

1 o. - 1 ~ 3 v / ~ , 4 , ~ , O= ~,n [--~-tT/l

(14)

( 15) (16)

and

2cN¢

(17)

N ( O ) = 1 + c - (1 - c)Sin30 where Sij = qij - 1-I6ij 3

(18)

1 1/s G = (~S~jS~&~)

(19)

6ij is the kronecker delta; c = N,/N~; and N~ and N~ are the slopes of critic~ state lines in extension and compression, respectively. Repeat indices imply s u ~ a t i o n . The stress 8i) on the limit surface is related to the current stress ai~ via the r a d i ~ ,

mapping rule, in which the mapping center is presently fixed at the origin of the stress

81 0" 1

(~2

(~3

Figure 5: Limit Surface on ~r Plane invariant space. Thus, the radial mapping rule can be expressed as ~ = ~q

(20)

Substituting the above equation into Eqs.(14-16), the relationship between current stress invariants and the stress i n ~ i a n t s on the l i ~ t surface can be obt~ned as fo~ows. [ = ~I

(21)

] = ~J

(22)

~ = ~

(23)

Substituting Eqs.(21-23) into Eq.(13), the mapping parameter can be obt~ned as follows B1 fl=~÷

~ B1 ~ 1 2 (~) ÷~-~(1-~)I0 I B1 = ~

B2 = I S + (R - 1)~(NJ----)~

(24) (25) (26)

82 The elastic domain is assumed to be conjugated with the limit surface such that _r~ = S

(27)

rb

where r~ and rb are the Euchdean distance from the mapping center (in the present formulation, it is assumed to be the origin of the stress-invariant space) to the elastic domain and the limit surface, respectively; and S is assumed to be a constant, implying that the elastic domain evolves according to the same mechanisms as that of the limit surface. From Eq.(27), the function f can be derived as follows 2S

f : 12 - ~ - I 0 !

S2(R -

R

2)1°~ ÷ (R -

_~

1)2(_)2

(28)

when current stress gives f < 0, then it is within the elastic domain; whereas, when f > 0, current stress exists outside the elastic domain. The numerical value of f provides a measure of overstress: when current stress gives f > 0, viscoplastic flow will occur. The specific flow function is adopted as shown below.

~(f) = exp(nf/fo) - 1

(29)

where f0 is a reference value of stress; and n is a calibration constant. Substituting Eq.(29) into Eq.(12), the flow law can be obtained as follows. '~ ~ =

{ ~p(~/0)-~ 0__~_~ if f > 0 o,,~ 0 if f < 0

(30)

or

d~ = < Lt > L~

(31)

Lt = E x p ( n f / f o ) - I K

(32)

where

OF Lij = 0~-~

(33)

and < > represents the heavyside function. II. I n t e r n a l

State Variable

(Io)

The internal variable, the pre-consolidation pressure, is affected by the aging (time) and rate of loading (rate). The aging-induced void ratio change and the development of quasi-preconsolidation pressure are accounted for via the following expression. e , . __( e ~ )e. i

(Io)t = Ioexp( ~

) A

~

(34)

83 where subscript t represents an arbitrary reference time; and ~ and ~ are the slopes of the

e

-

[np ~ curve under virgin loading and reloading, respectively.

Adopting the linear relationship between the natural logarithm of strain rate and the pre-consolidation pressure, as suggested by Leroueil, et al. (1985) [1] and discussed previousely in Eq(4), the following form is obtained. (35)

Io = b(In~k - l n ( ~ ) ~) + I~

where b is a model par~meter controlling the degree of rate effect; and ( ~ ) ~ and ~ are reference str~n rate and preconsolidation pressure, respectively. Substituting Eq.(35) into Eq.(34), the following relation is obtMned.

(~0)~ = [ b ( ~ -

( i~ _t ~ . eei ~

~ ( ~'~) ~) + ~ ] ~ (

~

)

(3~)

T~king derivatives with respect to time on both sides of Eq.(36) and neglecting the higher order derivative terms, the r~te-dependent evolution law of the intern~ state v~ria~le 10 can be expressed ~s ~ de ~o = [ b ( l n ~ - I n ( ~ ) ~) + I ~ ] E z p ( ~ )

-~ $ _ ~

(37)

Because the rate of inelastic void ratio change is rdated to the rate of inelastic volumetric str~n via the following equation ~' = -(~ + ~ 0 ) ~

(3S)

the rate-dependent harde~ng rule becomes ~o = [ b ( l ~ i , - ~ ( ~ ) ' )

+ ~g]E~( -(1 ~ , _

)(1 +~_~0)~i,

(3~)

where ¢0 is the i ~ t i ~ void ratio of matefi~s. Note that when b = 0, Eq.(39) becomes =

+

Eq.(40) can Mso be obt~ned from the traditional isotropic consolidation results (i.e., e ~ Inp' curve).

III. C o n s t i t u t i v e E q u a t i o n Using the isotropic el~tic Hook's law and Eq.(31), the constitutive equation can be expressed as

~ = ~ + ~ = C~+

< L~ > L~j

(4~)

84 or

~

= Ei~,t[~,t- < Lt > L~]

(42)

where CiVet and Eight are the elastic compliance and the modulus matrix, respectively. IV. U n d r a i n e d C o n d i t i o n All the foregoing equations are defined in terms of the efffective stress. However, under undrained conditions, excess pore water pressure in clay will develop. Assuming that both water and clay particles are incompressible, then the undrained condition implies zero volumetric strain, i.e., gk~ = g ~ + g 2 + ~ 3 = 0

(43)

Substituting Eq.(43) into Eq.(42) and following some algebraic manipulations, we have /: = - 3 K b < Lt > Lk~

(44)

where /t'b is the bulk modulus of materials. According to the effective stress concept, effective stress alj is related to the total stress (rltj and the pore water pressure u as follows (r~. '3 ---- O'ij

Q- ~iJ u

(45)

Substituting Eq.(45) into Eq.(44), the generation of pore water pressure under undrained conditions can be expressed as 1.t it = K~ < Lt > L ~ + ~ a ~

(46)

The undrained problems now can be solved by combining Eq.(46) with the effective stress based constitutive equations. CONCLUDING

REMARKS

A unified elasto-viscoplasticity model is formulated for describing the stress-strainpore pressure-time relationship of saturated cohesive soils in generai three-dimensional stress conditions. The major contribution of the proposed formulation lies in the recognization of a unique nature of pre-consolidation pressure in deternfining the limit state surface as well as the time- and rate- dependency of the pre-consolidation pressure. As a result, a consistent, and yet physically sound, viscoplasticity formulation is derived from which both time- and rate- sensitivities of cohesive soils can be naturally accounted for.

85 The traditional Perzyna's unified viscoplasticity formulation is employed as the basis of the proposed model. Phonomenological (macroscopic) characteristics as reviewed in this paper provided the physical basis for defining the static and dynamic yield surface via. conjugation with the limit surfaces. Furthermore, the time- and rate- dependent evolution of the internal variable (preconsolidation pressure) is derived to control the expansion of the limit surface as well as the elastic domain. The mathematic derivations of the proposed model are given in detail in this paper. In the companion paper, the model's predictive capabilities are demonstrated by comparisons between the model predictions and the experimental results. It will be shown that both time- and rate- dependent behavior of cohesive soils such as creep, stress relaxation, and rate-effect can be accurately captured by the proposed model. REFERENCES [1]. Leroueil, S., Kabbaj, M., Tavenas, F., and Bouchard, R. (1985). "Stress-strain-strain rate relation for the compressibility of sensitive natural clays." G~otechnique 35, No. 2, 159-180. [2]. Tavenas, F. and Leroueil, S. (1977). "Effects of stress and time on yielding of clays." 9th ICSMFE, Tokyo, Japan, Vol. 1,319-326. [3]. Walker, L. K. (1969). "Undrained creep in a sensitive clay." G~otechnique 19, No. 4, 515-529. [4]. Arulanandan, K., Shen, C. K., and Young, R. B. (1971). "Undrained creep behavior of a coastal organic caly." G~otechnique 21, No. 4, 359-375. [5]Shen, C. K., Arulanandan, K., and Smith, W. S. (1973). "Secondary consolidation and strength of a clay." Proc. ASCE, Vol. 99, No. SM1, 95-110. [6]Holzer, T. L. (1973). "Excess pore pressure during undrained clay creep." Can. Geotech. J. 10, 12. [7]. Febres-Cordero, E., and Mesri, G. (1974). "Influence of testing conditions on creep behavior of clay." UILU-ENG-74-2031, Report No. FRA-ORD-75-29, Nov. [8]. Tavenas, F., Leroueil, S., Rochelle, P. La, and Roy, M. (1978). "Creep behavior of an undisturbed lightly overconsolidated clay." Can. Geotech. J., 5, 402-423. [9]. Bonaparte, R. (1981). "A time-dependent constitutive model for cohesive soils." dissertation presented to University of California, Berkeley, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. [10]. Singh, A., and Mitchell, J. K. (1968). " General stress-strain-time function for soils." Journal of the Soil Mechanics and Foundation Engineering Division, ASCE, Vol. 94, No. SM1, 21-46. [11]. Murayama, S. and Shibata, T. (1961). "Rheological properties of clays." Proc. 5th Int. Conf. Soil Mech. and Found Engrg. Paris, Vol. 1,269-273. [12]. Vialov, S. S. and Skibitsky, A. M. (1961). "Problems of the theology of soils." Proc. 5th Int. Conf. Soil Mech. and Found Engrg. Paris, Vol. 1,387-391. [13]. Saada, A. S. (1962). "A rheological analysis of shear and consolidation of saturated clays." Highway Research Board Bulletin No. 342, 52-75. [14]. Wu, T. H., Douglas, A. G., and Goughnour, R. D. (1962). "Friction and cohesion of saturated clays." Journal of the Soil Mechanics and Foundation Div., ASCE, Vol. 88, No. SM3, 1-32. [15]. Lacerda, W. A. and Houston, W. N. (1973). "Stress relaxation in soils." Proc. 8th Int. Conf. on Soil Mech. and Found. Engrg., Moscow, 1/34, 221-227.

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