Constitutive Model for Considering Time Effects in Clay Soils

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CONSTITUTIVE MODEL FOR CONSIDERING TIME EFFECTS IN CLAY SOILS

Sandeep Kumar Chouksey Research Scholar, Department of Civil Engineering, Indian Institute of Science Bangalore–560 012, India. E-mail: choukseysandeep@gmail.com G.L. Sivakumar Babu Associate Professor, Department of Civil Engineering, Indian Institute of Science Bangalore–560 012, India. E-mail: gls@civil.iisc.ernet.in

ABSTRACT: The paper presents an approach for understanding the stress-strain-time response of clay soils using the framework of modified cam clay model. The formulation is general and helps to predict the void ratio-compression path under various times, which lead to different pre-consolidation pressures and corresponding stress-strain response can also be obtained. The use of the approach is illustrated with a simple example, based on the established concept of saturated un-cemented soil behavior. It is as shown that, the complete stress-strain response is as a function of time and an important aspect in soft clays in designing and construction involving soft soils. 1. INTRODUCTION

Schmertmann (1991), in his Karl Terzaghi Lecture stated that everything on this earth has at least one thing in common—everything changes with time. All soils age and change. The stress–strain behavior of all soils exhibit also time dependent behavior. Although time dependency is small or negligible in sands, it is generally too significant to be ignored in clays (Bjerrum 1967 & Graham et al. 1983). Modeling time-dependent stress–strain behavior of soils has been intensively active research field for several researchers in soil mechanics. Several authors presented models based on the Elastic Viscoplastic (EVP) modeling framework of Perzyna (1963, 1966). In these cases, total strain rates were divided into elastic strain rate and viscoplastic strain rate components (for example, Desai & Zhang 1987, and Kutter Sathialingam 1992). Some researchers, for example Borja Kavazanjian (1985), divided total strain rates into three parts: elastic, plastic, and vis- coplastic strain rates. The approaches used to derive the scaling function S (describing the magnitude of the visco-plastic strain rates) in the models by Borja & Kavazanjian (1985) and Kutter & Sathialingam (1992) are similar. They both used the timeline concept proposed by Bjerrum (1967) and the framework of Modified Cam-Clay (Roscoe & Burland 1968). It is well known that the key approach is presented by the earlier work is based on either numerical analyses or using constitutive model for time dependent stress-strain behavior. Major of the work to understand the time dependent phenomena has been done on one-dimensional consolidation oedometer tests (for example, Bjerrum 1967, Leroueil et al. 1985 & Yin Graham 1989a). This paper focuses on time dependent stress–strain behavior and yield surface of soils under triaxial stress states and generalized constitutive

model. The model discussed in this paper is based on critical state soil mechanics framework.

This paper provides a description of the development of a generalized constitutive model, based on critical state concepts and accounts for mechanical compression and time dependent mechanical creep to calculate stress-strain response and yield surface for typical clay, with different variation of time in days.

1.1 Proposed Model

The following assumptions have been made in the develop- ment of the proposed constitutive model for soft clay: (a) The mechanical behavior follows elasto-plastic behavior

in the framework of critical state soil model with associated flow rule;

(b) The time dependent mechanical creep phenomena is governed by exponential function similar to the assumption of Gibson & Lo’s (1961) model, which is given by,

( )1Cctb p e−ε = ∆ −′ (1)

where b is the coefficient of mechanical creep; ∆p' is the change in mean effective stress, c is the rate constant for mechanical creep; and t is the time since application of the stress increment.

Let us consider the isotropic loading on typical clay as shown in Figure 1 on the e-ln(p’) plot. If the material is normally consolidated at A, the isotropic loading will follow the path ‘AB’. Let us now unload the sample to the mean effective pressure Ap′ . Because of the elasto-plastic nature, the unloading path will not follow loading path ‘AB’. Instead, the material

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Constitutive Model for Considering Time Effects in Clay Soils

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will follow the path ‘BD’ upon unloading. When the material is reloaded from pressure Ap′ to Bp′ , it will usually follow the same path that indicates the elastic behavior. The slope of the loading path is denoted by λ, and the slope of unloading-reloading path is denoted by κ. The vertical distance ‘AD’ shows the plastic component in the change in volume, and ‘DE’ shows the elastic component of the change in volume. Now we can write the total change in void ratio (e) during the loading-unloading cycle as shown below:

From Figure 1 total change in void ratio during loading path AB,

( )ln ln lnBA B B A

A

pe e e p p

p ′

= − = λ = λ −′ ′ ′ (2)

Change in void ratio in path BD.

Fig. 1: Consolidation Behavior in e-ln p’ Space

( )ln ln lne BD E B A

A

pe e e p p

p ′

= − = κ = κ −′ ′ ′ (3)

Increment in volumetric strain due to elastic and plastic are derived from Figure 1.

Increment in total volumetric strain is given by,

0 01 1vde dp

de e p

λ ′ε = − =+ + ′

(4)

Hence, the elastic volumetric strain is written as,

1 1

eev

de dpd

e e pκ ′ε = − =

+ + ′ (5)

And, increment in plastic volumetric strain is written as,

2 22

1pv

dp dd

e p M

λ − κ η η′ ε = + + ′ + η (6)

The formulations for increments in volumetric strain due to elastic and plastic strains are well established in critical state soil mechanics literature (Wood 1990). There is need to

extend the elasto-plasticity concepts considering time dependent mechanical creep in soft clays.

1.1.1 Extension of Existing Model

In addition to elastic and plastic behavior of soft clay, considering compression due to time dependent mechanical loading, the total volumetric strain is expressed as,

cpevd d d d

v v vε = ε + ε + ε (7)

where

,edv

ε pdv

ε and

cdv

ε are the increments of volumetric

strain due elastic, plastic and time dependent mechanical creep.

From, Eqn. (1) increment in volumetric strain due to creep is written as,

c ctd cb p e dtv

−ε = ∆ ′ (8)

Using Eqns. (5), (6) and (8) substituting in Eqn. (7) total increments in strain is given by,

2 22

1 1v

ct

dp dp dd

e p e p M

cb e dt−

κ λ − κ η η′ ′ ε = + + + +′ ′ + η

+ ∆σ

(9)

On simplification of Eqn. (9),

( ) ( )( )00

0

1

exp 1 11

cte epq Mp b p e e

p e

λ λ−κ−

− ′ = + ∆ + −′ ′ +′

(10) where p' is the mean effective stress and 0p′ is the pre-consolidation pressure. The Eqn. (10) is the proposed new model for time-dependent behavior of soft clay, which is an extended form of modified cam clay model that predicts the stress-strain behavior and yield surface under loading considering time effects. In addition to elastic and plastic strains, the total volumetric strain (εv) includes mechanical compression under loading and time dependent mechanical creep given by Eqn. (1). M is the frictional constant, e0 is the initial void ratio and e is the void ratio after load increment. In case of undrained test the volume of the specimen remains constant, therefore void ratio after load increment remains same as initial void ratio. Additional terms b coefficient of mechanical creep; and rate constant for creep (c) can be obtained from consolidation tests. Thus Eqn. (10) represents the deviatoric stress for soft clay under load time dependent mechanical creep.

1.2 Analysis and Discussion

A typical soft clay in the west coast of India with physical properties with properties such as initial water content = 100%, Liquid limit = 110% and Plasticity limit = 25%

Constitutive Model for Considering Time Effects in Clay Soils

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friction angle 24° is used for the analysis. The compression index of soil is given by cc = 0.2343 × eL, where eL is the void ratio at liquid limit.

The values of b and c are taken as 0.0018 m2/kN and 0.057 per day respectively. The objective of this paper is to demonstrate the importance of time dependent behavior on soft clay using critical state soil mechanics framework. The proposed model is used to calculate the overconsolidation ratio with respect to time in days. Figure 2 shows the variation time-OCR response of typical clay. It was observed that the increase in overconsolidation ratio is very low at initial stage and it increases linearly with time after 50 days to a value in the range of 2 for 400 days. This behavior of normally consolidated clays is attributed to the effect of constant effective stress for long periods of time; develop upon subsequent loading, an apparent pre-consolidation pressure which is significantly higher than the sustained effective stresses which acted on initially. The effect of overconsolidation ratio has been the subject of numerous experimental investigations. One of the significant roles is in prediction of stress-strain response and yield surface with variation of time has been discussed in the following sections.

0

0.5

1

1.5

2

2.5

0 200 400 600 800Time (days)

(OC

R)

Fig. 2: Variation of OCR with Time (days)

In this section stress-strain behavior of soft clay is discussed using proposed model. Figure 3 shows the corresponding variations of stress-strain response. In general it is observed that the undrained strength increases significantly with time. The maximum undrained strength found in 1 day is 51 kPa whereas it increases to 93.6 kPa after 400 days. This is a significantly change in undrained strength with increase in time. This increase is strength is attributed to time dependent mechanical creep effect and increase in overconsolidation ratio as shown in Figure 2. As the time passes the soil becomes more and stiffer due to high overburden pressure. This shows that the oveconsolidation ratio influence undrained strength considerably. These observations are in agreement with the reported observations from field.

Also, for better understanding of time effects on soft clay response, yield surface’s are plotted for different time periods. The yield surface is defined as the boundary of a region in stress space in which both unloading and reloading produce

elastic strains only. In the region of stress space occupied by the family of loading surfaces, reloading produces plastic strains, while unloading produces only elastic strains. The yield surface is an ellipse given by Eqn. (10) and its initial size or major axis is determined by the pre-consolidation stress, ( 0p′ ). The higher preconsolidation stress, the larger is the initial ellipse. Based on the proposed model knowing initial condition or physical properties of soil preconsolidation pressure ( 0p′ ) with respect to time is calculated, it is assumed that the soil is normally consolidated therefore, over-consolidation ratio with respect to stress invariants is less than 2. Figure 4 shows the e-ln p’ plot for different time periods, which includes loading as well as unloading path. The subscript denotes time period in days for change in void ratio and preconsolidation pressures.

0

25

50

75

100

0 3 6 9 12 15 18 21Strain (%)

Dev

iato

ric

stre

ss (k

Pa)

1 day 3 days 7 days

10 days 20 days 50 days

100 days 200 days 400 days

Fig. 3: Variations of Stress-Strain Response for Different

Time Periods (days)

Fig. 4: Variation of Void Ratio (e) and Pre-Consolidation

Pressure (p'0) in e-ln p Plot for Different Time Periods

Figure 5 shows the variation of yield surface for different time periods in (days). The proposed constitutive model enables the calculation of yield surfaces, which gives the variation of

Constitutive Model for Considering Time Effects in Clay Soils

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shear strength expressed in q–p’ form as shown in Figure 5. It can be noted that with in the yield surface, the proposed model predicts higher deviatoric stress (q) for the higher time period. The difference is attributable to the effect of time dependent mechanical creep which results in higher deviatoric stress.

0

50

100

150

200

250

300

0 50 100 150 200p' (kPa)

q (k

Pa)

1day

3 days

7 days

10 days

20 days

50 days

100 days

200 days

400 days

Fig. 5: Variations of Yield Surface

2. CONCLUSIONS

In the present study, a generalized constitutive model is proposed based on critical state soil mechanics framework. The model accounts mechanical compression and time dependent creep effects for typical soft clay. The following are the major conclusions that can be drawn from the present study: (a) The time dependent behavior is well presented in the

form of time-OCR plot. It is observed that the over- consolidation ratio increases gradually with increase in time.

(b) The proposed model can be used to predict the time dependent stress-strain behavior of soft normally con- solidated soil. It is observed that the undrained strength increases significantly with time.

(c) The proposed constitutive model enables the calculation of yield surfaces, which gives the variation of shear strength expressed in q-p’ form. It is observed that the size of yield surface is higher for higher time period.

REFERENCES

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Desai, C.S. and Zhang, D. (1987). “Viscoplastic Model for Geologic Materials with Generalised Flow Rule”, International Journal for Numerical and Analytical Methods in Geomechanics, 11(6): 603–620.

Gibson, R.E. and Lo, K.Y. (1961). “A Theory of Soils Exhibiting Secondary Compression”, Acta Polytech. Scand., C-10, 1–15.

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Perzyna, P. (1966). “Fundamental Problems in Visco-plasticity”, Advances in Applied Mechanics, 9: 244–368.

Roscoe, K.H. and Burland, J.B. (1968). “On the Generalised Stress-strain Behaviour of Wet Clay”, In Engineering Plasticity. Edited by J. Heyman and F.P. Leckie. Cambridge University Press, New York, pp. 535–609.

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Yin, J.-H. and Graham, J. (1989a). “Viscous–elastic–plastic Modeling of One-dimensional Time-dependent Behaviour of Clays”, Canadian Geotechnical Journal, 26: 199–209.