Z Y Yin

P Y Hicher

Y Riou\

H W Huang

An Elasto-viscoplastic Model for Soft Clay

Keywords: Elasto-viscoplastic, finite element, overstress, Cam-Clay, time-dependent ., '

The purpose of this paper is to present the development of an elasto-viscoplastic model for normally consolidated clay. A three dimensional constitutive equation was derived based on the framework of Perzyna's overstress theory and Modified Cam-Clay model. It was implemented in a finite element program, and applied to analyze time-dependent behavior of soft clay. The influence of two viscous parameters, which can be determined by undrained triaxial compression tests, was studied. The proposed model was verified using observed laboratory responses of several natural clays. The results showed that the model can satisfactorily describe the stress-strainrate behavior such as strain rate effect, creep and stress relaxation.

Introduction

Several models integrating the viscous behavior of fine soils have been proposed in the literature (for example, [START_REF] Adachi | Constitutive equations for normally consolidated clay based on elasto-viscoplasticity[END_REF][START_REF] Hinchberger | The behaviour of reinforced and unreinforced embankments on rate sensitive clayey foundations[END_REF][START_REF] Fodil | Viscoplastic behaviour of soft clay[END_REF][START_REF] Zhang | Analysis of time-dependent behavior of soft clay and its application on the prediction of long-term settlements of tunnels[END_REF], but not widely used in geotechnical engineering. The purpose of this study is to develop a viscoplastic model simple enough to be easily calibrated and used in geotechnical projects.

This present paper has three objectives in studying the viscoplastic behavior of clay : (i) to deduce constitutive equations and implement them in a finite element program; (ii) to study the influence of the viscous parameters; (iii) to explore the implications of mode ling the time-dependent behavior of natural clays.

1 lnstitut de Recherche en Genie Civil et Mecanique, GeM UMR CNRS 6183, Ecole Centrale de Nantes, BP 92101,44321 Nantes Cedex 3, France; Tel: 0033-240371666; Email: zhenyu.yin@ec-nantes.fr 2 Department of Geotechnical Engineering, Tongji University, 1239 Siping Road, Shanghai 200092, PR China;Tel : 0086-21-65982986 ;Email: huanghw@n1ail.tongji.edu.cn Elasto-viscoplastic Modified Cam-Clay model

The Modified Cam-Clay model was proposed by [START_REF] Roscoe | On the Generalized Stress-Strain Behavior of Wet Clay[END_REF] and has been widely used for estimating time-independent behavior of soft clay. Formulations of this model are suitable for use in finite element analysis. For these reasons, the Modified Cam-Clay model was adopted as a basis for the formulation of a viscoplastic model.

Constitutive equations of the viscoplastic model. According to the work of Adachi and Oka 1982, we define a static yield criterion "Is" which represents a reference yield surface for the material. Its initial shape depends on the consolidation pressure ''p/". The expansion of the static yield surface, which describes the hardening of the material, is expressed by the variation of the consolidation pressure due to the inelastic volumetric strain "c;/P":

a s -s 1 + eo a vp s - ( 1 + eo vp ) 'Pc -Pc . . &V or Pc -Pco. exp --& V A-K A-K (1)
A dynamic yield criterion ''fd" is defined to represent the current state of stress and is expressed as follows.

2

fd = ! 2 +p'•(p'-p:)=O (2)
Based on ''p/" and ''pcd", the scaling function "J.l<l>(F)", which controls the amplitude of the viscoplastic strain rate, is taken as the power form or exponential form as follows.

(3)

where, "f.!" and "N" are the viscous parameters of this model. The flow rule for the viscoplastic strain rate in a simple case of infinitesimal strain field follows the form proposed by Perzyna 1966:

(4) { 0• for •F ~ 0
where, the function of MacCauley is expressed as: (F)= . The F• for•F > 0 principle of the viscoplastic model is presented in Figure 1. Taking into account the elastic stress-strain relations, constitutive equations of this viscoplastic model for normally consolidated clays are derived as follows: Finite element analysis. The basic finite element scheme with an elasto-viscoplastic model is similar to those presented by Oka and[START_REF] Oka | Two-dimensional consolidation analysis using an elasto viscoplastic constitutive equation[END_REF][START_REF] Hinchberger | The behaviour of reinforced and unreinforced embankments on rate sensitive clayey foundations[END_REF]. At time tn, for the time steps dtn=tn-tn-I, the viscoplastic strain increment using Euler numerical constant "8= 1" can be written as follows:

(6)

The stress increment vector can be written as follows:

(7)

We have the equations of nodal displacements and nodal loads:

[~t:L = [BL •[~dL, [ML = f[B]~ •[Lla'L •dV V
The incremental form of the finite element equation is written as:

[~L = §B]~ -an].([BL •[ML -Nfl •[&vpL))•dV=[KL •[ML- §B]~ •[lY]•Nfl •[&vpL •dV (8) V V
Once the increments of stresses and strains are calculated, the results can be obtained at time tn:

For a coupled consolidation analysis based on Biot theory, the relationship of the load increment is given by applying the principle of virtual work to the equilibrium equation as shown by [START_REF] Oka | Two-dimensional consolidation analysis using an elasto viscoplastic constitutive equation[END_REF]:

$&-t •[&Ji•dV= $&r •[M]•dV+ §&r •[~J•dV= §Mr •[~]•dV+ ~Mr •[&i•d5 (10) V V V V

S

The detailed information of coupled consolidation analysis can be found 1n Zienkiewicz and Taylor 1989.

Scaling function study

In viscoplastic models for soils, two types of scaling function are usually proposed: a power form and an exponential form (Eq.3). The test results show that the stress evolution for a given deformation is rather proportional to the logarithmic strain rate.

The model responses for these two types of scaling functions were carried out on undrained triaxial compression tests at different strain rates. The Modified Cam-Clay model parameters proposed by V aid and Campanella 1977 were taken (Table 1 a), and the values of "N" varied from 1 to 20. For each value of "N", we determined "J.t", in order to obtain the same maximum stress value at a strain rate of 1 0%/min. The results are presented in Figure 2. One can see that the exponential type scaling function allows to limit the increase of q with the strain rate and therefore enlarges the domain of application of the viscoplastic model, as pointed out by [START_REF] Fodil | Viscoplastic behaviour of soft clay[END_REF]. Thus, the exponential type relation was considered in the following study. The elasto-viscoplastic model comprises seven parameters which include: mechanical parameters which can be determined by triaxial compression test, i.e., Young's modulus "E", Poisson's ratio "u", slope of the critical state line "M"• ' mechanical parameters which can be determined by oedometer test and density test, i.e., preconsolidation pressure "Pco", compressibility index "(A.-K)/(1 +e 0 )", where A is the compression index, K is the swelling index, e 0 is void ratio; viscous parameters which can be determined by specific tests such as creep test, relaxation test and strain rate test, i.e., "J.t" and "N". In this study, the particular attention was paid on the influence of the viscous parameters. Undrained triaxial compression constant strain rate tests were carried out for studying the influence of the viscous parameters: the values of the Cam-Clay parameters proposed by [START_REF] Rangeard | Identification des caracteristiques hydro-mecaniques d'une argile par analyse inverse des essais pressiometriques[END_REF] for Saint-Herblain Clay were considered in this example (Table 1 b). The numerical simulations are presented in Figure 3. The main conclusions are the following:

The maximum deviatoric stress decreases when the value of "N" is increased. The influence on q max is greater for small "N" values and becomes less pronounced when N> 30. A smaller "N" represents a larger viscosity. The maximum deviatoric stress increases by increasing the "~" value. For small values of"~" (}1<10-5 ), qmax reached a constant minimal value. While high values of"~", an almost linear relationship between qmax and log(~) was obtained. A smaller"~" represents a larger viscosity. The relationship between qmax and the logarithm of the strain rate is almost linear. It is influenced in a different way by "N" and "~": the slope depends on "N", but not on "~". This will allow us to determine firstly the "N" value according to the slope of the "deviatoric stress -strain rate" curve, and then to determine "~" according to the value of the deviatoric stress. were reported by V aid and Campanella 1977. The samples were initially consolidated under an effective isotropic confining pressure of 515 kPa for 36 hours and then were allowed to stand for 12 hours under undrained conditions before starting the shearing test. The material parameters are summarized in Table la. Numerical simulations show a good agreement between measured and predicted results (Fig. 4. ).

Results of stress-relaxation tests on the Flumet clay were provided by Fodil et al.1997. An axial strain rate of 0.15%/h was applied. The stress-relaxation tests were performed on the same sample for each test at about 1%, 3. 5% and 6. 7% of axial strain and lasted at least 24h. The parameters are summarized in Table 1(c). The viscoplastic model predicts reasonably well the results of the stress-relaxation tests, as shown in Figure 5.

Anisotropically consolidated undrained triaxial compression tests (CAU) on Sackville clay provided by Hinchberger 1996 were also simulated. The axial strain rates ranged from 0.009%/min to 1.14%/min. In addition, CAU triaxial creep tests were performed at deviatoric stresses from 35 kPa to 50 kPa after an initial anisotropic consolidation for 24 hours. The parameters used in the analysis are summarized in Table 1 (d). A good agreement between measured and calculated responses was obtained (Fig. 6).

-;;-""' 0,8 0,7 -Deviator stress-strain rate 0 """' """' ""j ........_-Cal~ula~ c 0,6- c::T' 0,5 0,4 .f-----'---'---'-.Uu.lif-~~LLllj-~'-'--LLLUf-----'---'--'-L.llllf--~L.LLLU.j 0,0001 0,001 0,01 0,1 dfldt (%/min) 10 Deviator stress-time were provided by Hinchberger 1996 on Sackville Clay. Drainage was allowed at top and bottom of a 2.0 cm thick specimen. A permeability value "k=1.5x10-8 m/min" was given by Hinchberger. Fig. 7 presents the comparison of measured and predicted results for a vertical loading increment from 50 kPa to 100 kPa. The material parameters are summarized in Table 1 e.

200 150 ---- ~ 100 - c::T' -o m=l% '0-o--~--~--------------~--------------~ 0
Fig. 8 presents results obtained on Gloucester Clay (Lo et a1.1976). The height of the specimen was 2.0 cm. One-way drainage was allowed during the consolidation and the permeability "k" was equal to 7 .2x 1 o-8 m/min. These results show that the viscoplastic model is capable of calculating with good accuracy the long term settlement of natural clay. 3.8 m 50-100 kPa 0.76 -3% : In summary, the constitutive equations based on Modified Cam-Clay model and overstress theory were evaluated in the present study and found to satisfactorily describe the time-dependent behavior of several natural clays, isotropically and anisotropically consolidated, under various stress-strain-time loadings.

:> ' w -4% ! -5% • [~;~;;;.,"1u • • •~""• • . u u -6% --- ------~---- - calculated -7%-------, - -I% -2% : -3% :> -4% w -5% : : -6% -• Set -11/o I -8% 2 -9%

Conclusions

An elasto-viscoplastic model developed to describe the time-dependent behavior of normally consolidated soft clay was presented in this paper. The constitutive equations were based on Modified Cam-Clay elastoplastic model and Perzyna's overstress viscoplastic theory. The exponential type scaling function was adopted to enlarge the domain of application of this model.

The model has seven parameters, 5 describing the elastoplastic behavior, which can be determined directly from laboratory tests such as triaxial tests and oedometer tests, 2 describing the viscous behavior, which can be determined by curve fitting.

The proposed model was employed to predict a number of undrained triaxial tests and long-term oedometer tests on several isotropically and anisotropically consolidated clays: Saint-Herblain clay, Haney clay, Flumet clay, Sackville clay and Gloucester clay. Good agreement between measured and predicted results was achieved for different loading conditions: constant strain rate tests, stress-relaxation tests and creep tests.

The predictive capability of this model needs further validation, such as the prediction of infrastructures built on soft clay.

Figure 1 .

 1 Figure 1. Schematic behavior of the elasto-viscoplastic Modified Cam-Clay model during CA U triaxial compression and triaxial creep tests.

  Figure 2. Comparison of exponential and power type scaling function

Figure 3 .

 3 Figure 3. Influence of viscous parameters "N" and "~" at strain rate test

Figure 4 .Figure 6 .

 46 Figure 4. Strain rate test on Haney clay Figure 5. Relaxation test on Flumet clay

Table 1 .

 1 Summary of constitutive parameters

	~~~i~~~~~ Depth Stress lncrem:nt ~"',~::~'6'-, 2.4 m 38.5 -80.8 kPa
	4.3 m	43.2-82.7 kPa	:	o

Acknowledgements

The research reported in this paper is funded by a grant from "Conseil Regional des pays de la Loire".