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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308

(Print), ISSN 0976 – 6316(Online), Volume 5, Issue 8, August (2014), pp. 20-31 © IAEME

20

RELATIVE INFLUENCES OF STRESS PATHS WITH DIFFERENT INITIAL

SOIL STATES ON STRESS –STRAIN RESPONSE USING MODEL TESTS

Sandhya Rani. R.1, Nagendra Prasad. K.

2, Vidhushimani. M.

3

1(Research Scholar, Dept. of Civil Engineering, SV University, Tirupati, India)

2(The Registrar, Vikrama Simhapuri University, Nellore, India)

3(Former Post Graduate Student, Dept. of Civil Engineering, SV University, Tirupati, India)

ABSTRACT

It is now widely established that the behavior of soils is described by elasto-plastic theories.

The original elasto-plastic model was called the Cam-clay model. Today there are numerous versions

of elasto-plastic models, which are basically modifications or improvements over the original model.

But the fundamental theory behind all these models is the same. Cam-clay (CC), Modified Cam-clay

(MCC) and Wheeler are considered in the present investigation. These models are essentially based

on critical state framework. The critical state framework unifies stress-strain characteristics so that

the behavior of soil under different loading conditions can be comprehensively understood. These

three models have been adopted for the purpose of making a comparative study of the stress-strain

characteristics for different stress paths which clearly brings out of the distinct differences in the

behavior.

The types of tests include a series of undrained test conducted on soils with different liquid

limits at the constant initial volume. Another series of undrained tests were performed by keeping the

initial mean principal stress constant. In another series of tests similar test conditions have been

applied by allowing the drainage. For bringing out further comparison, a series of constant p tests

have been conducted for similar initial test conditions as before. The model tests thus conducted

bring out salient features of stress-strain response with relative influences of stress paths.

Keywords: Cam Clay Models, Critical State, Elasto-Plastic, Model Tests, Stress Paths.

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING

AND TECHNOLOGY (IJCIET)

ISSN 0976 – 6308 (Print)

ISSN 0976 – 6316(Online)

Volume 5, Issue 8, August (2014), pp. 20-31

© IAEME: www.iaeme.com/ijciet.asp

Journal Impact Factor (2014): 7.9290 (Calculated by GISI)

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IJCIET

©IAEME



21

1. INTRODUCTION

Engineering is concerned with understanding, analyzing, and predicting the way in which real

devices, structures and pieces of equipment will behave in use. It is rarely possible to perform and

analyses in which full knowledge of the object being analyzed permits a complete and accurate

description of the object to be incorporated in the analyses. This is particularly true for geotechnical

Engineering (Wood, 1990).

Soils in situ usually possess natural structure which enables them to behave differently from

the same material in a reconstituted state e.g., Burland (1990), Leroueil and Vaughn (1990),

Cuccovillo and coop (1999). During isotropic compression, natural clays show a stiffer (lower

compressibility) response than their corresponding reconstituted sediments upto a higher pressure

(Yan & Li, 2011). Although soil structure may arise from many different causes ,their effects on

mechanical behavior have been shown to be similar (Liu and Carter, 1999).Various geological

processes as well as loading can cause a loss of soil structure either by induced yield or by removing

bonding agents. Indeed, significant difficulties have been encountered in cases where the structural

features of the soil dominate its engineering behavior.

The models considered here are conceptual models. Predictions can also be based on

physical models in which, for example, small prototype structures are placed on small blocks of soil.

Such physical models are also simplified versions of reality because it is not usually feasible to

reproduce at a small scale all the in situ variability of natural soils. The objective of using conceptual

models is to focus attention on the important features of the problem and to leave aside features

which are irrelevant. The choice of model depends on the application (Wood, 1990).

The scope of the present study includes understanding the model predictions from classical

constitutive models such as Cam clay, modified Cam clay and Wheeler. The parameters involved are

simple and easily determinable from the experimental results performed in routine soil

investigations. By making the comparative study, the salient features that can be captured are

understood for appropriate use in engineering applications. Accordingly a comparative study of the

models with different stress paths has been presented in the present analytical study.

2. RECENT PLASTICITY MODELS FOR SOILS

There are significant developments in understanding the mechanics of structured soil. At a

fundamental level ,there have been useful advances in formulating constitutive models incorporating

the influence of soil structure ,such as those proposed by Gens and Nova (1993) ,Whittle(1993),

Rouainia and Muir Wood(2000).

A perfectly plastic material deforms continuously at constant stress with no change in

volume, once the yield stress has been reached. Soil and other particulate materials are rarely

perfectly plastic but are either work hardening or work softening. They undergo plastic volume

changes and the material with the changed volume may behave all together as a different material

with its own yield properties. Most of the current day elasto-plastic theories are extensions of perfect

plasticity by incorporating the effects of work hardening.

2.1 Cam-clay Model

The derived yield curve was of the form:

1)/ln(/ ''' =+ xppMpq … (1)



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The final expression for the yield surface is of the form:

( )[ ] ( )kpvkMpq −−−−+Γ= λλλ /ln '' … (2)

2.2 Modified Cam-clay Model

The yield curve for the modified cam clay model is given by

q2 + (p-px) M

2 p = 0 … (3)

2.3 Wheeler Model The formation and development of soil structure often produces anisotropy in the mechanical

response of soil to changes in stress. Destructuring usually leads to the reduction of anisotropy

(M.D.Liu and J.P.Carter, 2002). Anisotropic fabric is macroscopically manifested by the value of the

rotational hardening variable, which measures the rotation of the yield surface and plastic potential

surface in stress space. Fabric anisotropy for clays is abundant in nature and is encountered in all

cases of natural deposition under K0 gravity consolidation (Dafalias Y.F. & Taiebat M., 2014).

To make the adoption of anisotropic models for geotechnical design more feasible, an

alternative elasto-plastic model for soft clayey soils was proposed by Wheeler (1997) and

subsequently slightly modified by Näätänen et al. (1999). The main objective in developing the

model was to provide a realistic representation of the influence of plastic anisotropy whilst still

keeping the model relatively simple, so that there would be a realistic chance of widespread

application in geotechnical design.

The model is an extension of the critical state models, with anisotropy of plastic behavior

represented through a rotational component of hardening. Rotational hardening is a constitutive

feature of anisotropic clay plasticity models that allows rotation of the yield and plastic potential

surfaces in stress space in order to simulate, more realistically than isotropic models, the material

response under various loading conditions (Dafalias Y.F. & Taiebat M., 2013). For the sake of

simplicity, the model is presented here for the simplified stress space of the triaxial test, although it

has already been extended to general three-dimensional stress space.

The yield curve is sheared ellipse, as proposed by Dafalias (1987) and Korhonen & Lojander

(1987), defined by

f = (q - αp’)2 – (M

2 - α2

)(p’m – p’)p’ = 0 … (4)

The model incorporates two hardening laws. The first one describes changes in size of the

yield curve and it is similar to that of modified Cam clay:

κλ

ε

−

′=′

p

vm

m

dpvpd … (5)

The second hardening rule predicts the change of inclination of the yield curve produced by

plastic straining, representing the development of anisotropy with plastic strains. The rotational

hardening law is

dα = µ [(χν (η) - α) dενp + β(χd (η) - α) dεd

p ] … (6)



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The model involves a few soil constants: 5 conventional parameters from modified Cam clay

(N, M, λ, κand Γ) and two additional parameters relate to the rotational hardening (β and µ). In

addition, the initial state of the soil is defined by the stress state and the initial values of the

parameters pm and α defining the initial size and inclination of the yield curve. The model parameter

β defines the relative effectiveness of plastic shear strains and plastic volumetric strains in rotating

the yield curve and µcontrols the rate, at which α tends towards its target value (Wheeler, 1997). If

the initial value of specific volume v is also defined, this replaces the requirement to define a value

for the parameter Γ (the intercept of the critical state line in the v: ln p’ plane).

3. STRESS PATHS

The path of applied stress is an important parameter that influences the material behavior

(Desai and Siriwardane, 1984).

One-dimensional compression of soil: Some soils have been deposited rather uniformly over an

area of large lateral extent, for example, in marine or lacustrine conditions. For such soils, symmetry

dictates that soil particles can only have moved downwards during the process of deposition; lateral

movements would violate the symmetry. The deformation of such soils during deposition is entirely

one-dimensional, and the effective stress state can be reproduced in a conventional triaxial apparatus.

One-dimensional unloading of soil: One-dimensional unloading of soil produces a more rapid drop

of vertical effective stress. If it is supposed that soil behaves isotropically and elastically

immediately on unloading (which implies that there are no plastic deformations).

Elements on centerline beneath circular load: Elements of soil within a soil deposit of large lateral

extent and on the centerline beneath a circular load will be subjected to axially symmetric changes of

stresses. This is the only engineering situation for which the stress path can be followed precisely in

the conventional triaxial apparatus.

Element in long slope: If the slope is also of large lateral extent (in its direction of strike), then

deformations can be assumed to occur in plane strain.

Elements adjacent to long excavation: The total and effective stress paths for element adjoining

excavation are essentially the same as those for an element behind a retaining wall which is moving

forward, so the soil deforms actively with the vertical stress driving the deformation.

It should be clear that the range of geotechnical situations for which stress paths can be

qualitatively assessed with any confidence is rather limited. As the geotechnical structure becomes

more complex, the stress paths also become more complex and more uncertain, and the possibility of

reproducing them in a laboratory testing apparatus becomes more remote. That is not intended as a

cry of despair, however, because the whole object of developing numerical models for soil behavior

is precisely to provide a rational basis for the extrapolation from the known region of laboratory test

data towards the unknown region of actual field response. Numerical analyses will then give an

indication of the stress paths expected by the computation and reveal the extent of the necessary

extrapolation (Wood, 1990).



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Fig.1: Cam clay models and Wheeler model yield curves

Fig.1 shows the yield curves of various models and the critical state line. It may be found out

that initial yield values are higher for Wheeler and modified Cam clay models as compared to Cam

clay model.

4. MODEL TESTS

When the effective stress exceeds the remoulded yield stress, the compression behaviour of

reconstituted clays is controlled solely by the water content at the remoulded yield stress and the

liquid limit (Hong, 2012).

As reported earlier, three series of model tests have been performed to analyze the test results

obtained by conventional Cam clay models and to bring out the comparison of stress strain behavior

under different loading conditions. Soils with different liquid limits are considered. The range of

liquid limit considered is 40-80. The computed model parameters for different liquid limit values are

given in Table.1. The values of M are so chosen that the value decreases with increase in liquid limit

and some standard default value is selected for the parameter µ. The parameter α will be calculated

using equati 3

322

Moo

o

KK

K

−+=

ηηα … (7)

The value of ηko can therefore be estimated using Jaky’s simplified formula as

ηko = 3(1-Ko)/(1-2Ko) … (8)

Where, Ko=1-sinϕ and sinϕ = 3M/(6+M)

and β is given by )2(2

)4/3(3

00

00

22

22

KK

KK

M

M

ηη

ηηβ

+−

−−= … (9)



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The compression paths for different liquid limit values are shown in Fig.2. Normal pressures

for different soils for the same initial volume and the specific volumes for the same mean stress are

shown in figure. These paths are drawn by choosing different values for p and calculate the

corresponding values of specific volume using the equation, v = N-λ lnp.

Fig.2: Compression paths in v-lnp plot

4.1 Undrained Tests An undrained test usually assumed to be constant volume test (the term isochoric is

sometimes used), but it is more strictly a constant mass test (isomassic) because of closure of

drainage tap merely prevents any material leaving the sample (Wood, 1990). It is commonly adopted

in limit equilibrium analyses where the rate of loading is very much greater than the rate at which

pore water pressures that are generated due to the action of shearing the soil may dissipate.

Undrained strengths are typically used in traditional plastic collapse analyses for geotechnical

structures which involve the rapid loading of clays. An example of this is rapid loading of sands

during an earthquake or the failure of a clay slope during heavy rain, and applies to most failures that

occur during construction. As an implication of undrained condition, no elastic volumetric strains

occur, and thus Poisson's ratio is assumed to remain same throughout shearing.

Table.1: Critical state constants for the range of liquid limit considered for the model tests LL

(%)

Cc

(=0.009(LL-10))

λ

(=Cc/2.303) κ (=λ/4) M

e

(=wG/Sr)

v

(=1+e)

N

(=v+λ ln1)

Γ

(=N-λ+κ) α β µ

40 0.27 0.117 0.0293 1.3 1.272 2.272 2.272 2.184 0.49 0.86 30

50 0.36 0.156 0.0390 1.2 1.590 2.590 2.590 2.473 0.45 0.75 30

60 0.45 0.195 0.0488 1.1 1.908 2.908 2.908 2.761 0.42 0.63 30

70 0.54 0.234 0.0586 0.9 2.226 3.226 3.226 3.050 0.35 0.37 30

80 0.63 0.273556 0.0683 0.85 2.544 3.544 3.544 3.339 0.33 0.30 30



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Table.2: Model conditions for the same initial volume

Volume,v0 2.0 2.0 2.0 2.0 2.0

Mean Principal Stress, p0 10 40 100 198 298

Liquid Limit,% 40 50 60 70 80

Table.3 Model conditions for the same initial mean principal stress

Volume,v0 1.73 1.87 2.01 2.15 2.28

Mean Principal Stress, p0 100 100 100 100 100

Liquid Limit,% 40 50 60 70 80

4.1.1 Undrained Tests for the samples with same initial volume First, a series of model undrained tests were conducted by keeping the volume constant for all

the soil samples. The values of normal stress for the same initial specific volume as obtained from

the Figure 1 are presented in Table 2.

It may be seen that as the liquid limit increases the mean principal stress required for holding

the sample at same volume also increases.

Fig.3: Predictions of various models for same initial volume under Undrained condition

The model test results for different soils representing maximum shear stress and

corresponding shear strains are shown in Fig.3. The Cam clay model predictions are relatively lower

compared to modified Cam clay and Wheeler. The Wheeler predictions are found to be higher as it

takes into account the natural structure of the soil which will be additional component of resistance.

The shear strain at maximum stress is higher for Cam clay model. This turns out that the Cam clay

predictions are conservative compared to modified Cam clay and the Wheeler model gives rise to

higher stresses and the corresponding strains are lower.

4.1.2 Undrained Tests for the samples with same initial mean principal stress A series of model tests were conducted to predict the undrained shear response for different

soils, when tested with same initial mean principal stress. This enables the comparative study to

understand the effect of normal stress on the stress strain response of different clays. The stress state

for this condition can be seen in the Table 3.

It may be seen from the table that for the same initial normal stress, the specific volume

increases with increase in liquid limit. It turns out that as the liquid limit increases, the soil is held at

higher volume, as water holding capacity is more for the same mean principal stress.



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Fig.4: Predictions of various models for same initial mean principal stress under Undrained

condition

It may be seen from Fig.4 that the maximum deviatoric stress increases with decrease in

liquid limit value unlike in case of constant initial volume tests. The Wheeler predictions give higher

values of deviatoric stress and lower values of shear strain at maximum stress as compared to Cam

clay and modified Cam clay models which are similar to constant initial volume tests.

(a) (b)

Fig.5: Stress-Strain response of various models under undrained condition at LL of 80% for

(a) same initial volume and (b) same initial mean principal stress

It may be seen from Fig.5 that the Wheeler predictions give greater values of shear strength

as compared to Cam clay and modified Cam clay models. Both Cam clay and modified Cam clay

predictions characterize the strain hardening behaviour, where as the Wheeler’s predictions

characterize strain softening behaviour.

4.2 Drained Tests In drained tests, drainage can occur freely from the sample and the volume occupied by the

soil structure can change freely as it deforms. Drained tests are more appropriate for analysing the

long-term stability of the structures. The drained shear strength is the shear strength of the soil when



28

pore fluid pressures, generated during the course of shearing the soil, are able to dissipate during

shearing. A series of model drained tests were performed to bring out relative comparisons in

constitutive behavior of different models for the range of soil samples considered.

4.2.1 Drained Tests for the samples with same initial volume A series of drained tests were conducted by keeping the volume constant for all the soil

samples.

Fig.6: Predictions of various models for same initial volume under Drained condition

It may be seen from Fig.6 that similar to undrained tests, the stresses increase with increase in

liquid limit in drained tests as the mean principal stress increases with increase in liquid limit value

even in the drained tests. The maximum stresses predicted by Wheeler are higher but the strains

experienced by the samples are lower.

4.2.2 Drained Tests for the samples with same initial mean principal stress A series of model tests were conducted to predict the drained shear response for different

soils, when tested with same initial mean principal stress.

Fig.7: Predictions of various models for same initial mean principal stress under Drained

condition



29

It may be seen from Fig.7 that the stresses increase with decrease in liquid limit value unlike

in case of constant initial volume tests, as observed in the case of undrained tests. The Wheeler

predictions give greater values of stresses as compared to Cam clay models predictions. The shear

strains at the maximum stress are lower for Wheeler model compared to rest.

4.3 Constant ‘p’ Test This test also comes under drained test, as the initial volume of the samples keep on changing

with shearing. This test path is unique and special in the sense that, this test brings out the shear

strength characteristics in a unique manner as the mean principal stress always remains constant.

Accordingly, another series of model tests were conducted by keeping the mean principal stress

constant.

4.3.1 Constant’ p’ Tests for the samples with same initial volume

Fig.8: Predictions of various models for same initial volume under Constant P condition

It may be seen that the features of stress-strain response for constant ‘p’ test are similar to

conventional drained tests. However, the constant p test gives rise to lower strengths compared to

normal drained tests, as this test path reaches the critical state at a faster rate compared. As a

consequence, the stress values show lower values compared to drained tests.

4.3.2 Constant’ p’ Tests for the samples with same initial Mean principle stress

The features explained as in the case of drained tests hold good. The only distinctly different

feature is that the loading rate in the case of constant ‘p’ test is faster compared to conventional

drained test. The Wheeler predictions match quite well with the predictions given by Cam clay

models unlike in the case of drained tests.



30

Fig.9: Predictions of various models for same initial mean principal stress under Constant P

condition

5. CONCLUDING REMARKS

Based on the analysis of model tests conducted, using the models considered, the following

general concluding remarks may be made.

• Qualitative stress paths for a number of field situations would indicate that different

geotechnical constructions would load and deform soil elements in different ways.

• The results showed that even something as apparently straightforward as undrained strength

of soils is not independent of the stress path.

• The model tests conducted for different stress paths indicate that constitutive relation is

dependent on rate of loading and stress path followed.

• The liquid limit increases the mean principal stress required for holding the sample at same

volume increases.

• The Wheeler predictions give greater values of shear strength as compared to Cam clay and

modified Cam clay models.

• The stresses increase with decrease in liquid limit value in constant initial mean principal

stress tests unlike in case of constant initial volume tests.

• The Wheeler predictions give rise to lower strains at maximum stress compared to Cam clay

and modified Cam clay models.

• The stresses increase with increase in liquid limit as the initial mean principal stress increase

with increase in liquid limit value even in the drained tests.

• The features of stress-strain response for constant ‘p’ test are similar to conventional drained

tests. However, the constant p test give rise to lower strengths compared to normal drained

tests, as this test path reaches the critical state at a faster rate compared.

• The Wheeler model predictions match quite well with the predictions given by Cam clay and

modified Cam clay for constant p test.



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ISSN Online: 0976 – 6316.

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