Hyperbolic constitutive model for tropical residual soils

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

(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 3, May - June (2013), © IAEME

121

HYPERBOLIC CONSTITUTIVE MODEL FOR TROPICAL RESIDUAL

SOILS

Nagendra Prasad.K1, Sulochana.N

2

1Professor, Dept. of Civil Engineering, SV University, Tirupati, India, (corresponding author)

2Sulochana.N , Research Scholar, Dept. of Civil Engineering, SV University College of

Engineering, Tirupati & Lecturer in Civil Engineering, Govt. Polytechnic for Women,

Palamaner, Chittoor District, A.P.

ABSTRACT

The stress-strain response of natural soils depends on soil state, stress history and

drainage conditions. Many constitutive models are available for describing the stress-strain

relationship for different soil types. It is desirable to have a comprehensive model, based on

sound principles of continuum mechanics, capable of describing the soil behaviour under any

type of loading. The model parameters involved in such models most often, require elaborate

experimental procedures to evaluate them. There are many instances when a problem posed

to an engineer may not necessarily require such a complex material model. For example, a

simple undrained analysis may be sufficient for the immediate or end of construction (this

will be always critical condition) of structures on clayey soils. Depending on specific field

situation, it may be possible to analyze the problem with much simpler model. Therefore,

there is a need to develop a realistic and simple model whose parameters can be determined

easily with simple procedures. The cardinal aim of the present paper is to develop a simple

constitutive relationship using hyperbolic approach, based on analysis of test results on five

different types of soils. Combination of stress ratio and mean principal stress is identified to

capture the strain softening behaviour of residual soils. The model developed is applied to

predict the stress-strain response for other soils found in literature. The model predictions are

quite comparable and model parameters are easily determinable.

Keywords: Tropical residual soils, hyperbolic model, stress-strain-pore pressure response,

yield stress, confining pressure.

INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND

TECHNOLOGY (IJCIET)

ISSN 0976 – 6308 (Print)

ISSN 0976 – 6316(Online)

Volume 4, Issue 3, May - June (2013), pp. 121-133 © IAEME: www.iaeme.com/ijciet.asp

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1. INTRODUCTION

Soils are very complicated engineering materials, whose constitutive response

depends on many compositional and environmental factors. The availability of high-speed

computers and powerful numerical techniques (such as the finite element method) makes it

possible to incorporate the non-linear behaviour of materials into the analysis of soil systems

and soil structure interaction problems. Some advanced soil models have been proposed for

the non-linear stress-strain behaviour of soils, including the hypo elastic models, the hyper

elastic models and the plasticity models. However, these models require the determination of

many parameters for the investigated soils.

The explicit nature of stress-strain response of tropical residual soil mostly depends

on fabric and nature of cement bonding in addition to the usual factors such as current state,

stress history, stress path and drainage conditions. Despite the availability of quite a good

number of constitutive relations concerning the behaviour of clays, there are still a large

number of problems which have not been satisfactorily tackled. Among these, strain

softening behaviour of tropical residual soils during deformation process is of importance.

Strain softening is an important phenomenon causing concern as regards the design problems

associated with estimation of bearing capacity, stability and deformation.

Most often tropical residual soils are treated as overconsolidated soils because they

also exhibit similar features like strain softening, higher initial stiffness, etc. But a closer

study of the test results of different tropical residual soils found in literature would reveal that

the behaviour of these soils in undrained shear is very much different from that of

uncemented overconsolidated soils. Most important difference is that softening here is

associated with continued positive pore pressures whereas softening in overconsolidated soils

is associated with negative pore pressures. Probably, this is because there is an additional

component of resistance from cementation bonds.

Thus there is a need for development of a realistic and simple model comprising of

easily determinable constitutive parameters which is capable of capturing the most important

aspects of the behaviour. For instance, the hyperbolic elastic models (Duncan and Chang,

1970) are still widely used in the non linear finite element analysis of uncemented soils is one

such example. The reasons for using hyperbolic models are ascribed to its simplicity and well

defined constants associated with the model. It is well known that hyperbolic model was

originally formulated to fit the undrained triaxial test results with only two constants to be

defined. It subsequently grew in strength and came to be applied to realistic boundary value

problems involving both drained and undrained conditions with corresponding modifications

using incremental approach. The inherent capability of the hyperbolic form to capture the

softening behaviour of tropical residual soils has not been attempted in the past (Nagendra

Prasad et al. 1999).

2. BACKGROUND INFORMATION

In a tropical region, residual soil layers can be very thick, sometimes extending to

hundreds of meters before reaching un-weathered rock. Unlike the more familiar transported

sediment soil, the engineering properties and behaviour of tropical residual soils may vary

widely from place to place depending upon the rock of origin and the local climate during

their formation; and hence are more difficult to predict and model mathematically. Despite

their abundance and significance, our knowledge and understanding of these soils is not as



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extensive as that of transported sediment soil (Huat et al. 2012). However, with respect to

residual soil, both its interaction mechanism and its failure behaviour in soil composites are

not well understood due to limited study (Mofiz et al. 2010).

Tropical soils appear in large regions of the world and have been less studied than

soils from temperate climates, particularly with respect to critical state and limit state

conditions. Most geo-materials are structured in nature and this natural structure affects the

behavior of tropical soils. Structural features affecting soil behavior include soil cementation

and soil fabric.

Sarma et al. (2008) observed that the consolidation properties of soils indicate an

insight on the compressibility behaviour of soils with associated expulsion of water.

However, determination of such properties involves considerable time, cost and rigorous

testing process. Further, natural state of partial saturation and soil-moisture is not simulated

in the standard consolidation procedures. The sampling technique is also not specific for the

Oedometer tests and sampling disturbance influences the results considerably. As such,

modified methodologies of Odometer test for field simulation as well as simple correlations

of the consolidation parameters with fundamental properties are always preferred by

practising engineers.

Karmakar et al. (2004) brought out that the soil undergoes both elastic and plastic

deformation when subjected to loading. The basic requirement for integrated analyses of

movements and failure of a soil mass is a constitutive relationship capable of modelling

stress-strain behaviour of soil up to and beyond failure. Development of such a relationship

generally involves separating the elastic and plastic behaviour. This is achieved using a well-

defined curve known as the yield locus located in a shear stress-normal stress space. If the

stress state of a soil plots inside the yield locus, it is considered to be elastic and undergoes

recoverable deformation. On the other hand, if a particular stress path puts the stress state of

the soil on or outside the yield locus, plastic or irrecoverable deformation of soil occurs.

Elasto-plastic constitutive models help to distinguish between the recoverable and

irrecoverable deformations for understanding the stress strain behaviour of soil during

loading and unloading. In order to develop a simple framework, a mechanistic approach is

needed based on well planned experimental investigation.

3. SCOPE OF THE PAPER

Particularly soils in the Southern Indian Region are residual in nature (those derived

by in-situ weathering of rocks). In residual soils the particles and their arrangement would

have evolved progressively as a consequence of physical and chemical weathering. Although

the geological study of the formation and structure of in-situ residual soils is well advanced,

the simple and rapid methods to analyze and assess the engineering properties of these soils

have not received the same level of attention. This is in contrast to the situation while

sedimentary soil deposits are encountered. Quite often cementation in rock would be left

behind due to varied degrees of weathering.

The objective of this paper is to develop a simple practical procedure for representing

the nonlinear, stress dependent, inelastic stress-strain behaviour of tropical residual soil

during undrained shear. Accordingly, the relationship described has been developed in such a

way that values of the required parameters may be derived from the results of the standard

laboratory triaxial tests. The formulations are proposed within the framework of a hyperbolic

relation of stress ratio (q/p) and also of effective mean principal stress (p) with strain. The



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formulation incorporates stress dependency, nonlinearity, strain softening aspects of the

behaviour quite effectively. It must be pointed out that this is an attempt to circumvent the

difficulty associated with the choice of complicated constitutive models while dealing with

undrained field situations.

4. EXPERIMENTAL INVESTIGATIONS

4.1 Residual Soils Tested In order to understand the mechanisms involved in shear and compression behaviour

in relation to naturally sedimented soils, a detailed experimental program has been

undertaken on undisturbed soil samples extracted from regional soil deposits in Tirupati and

its surroundings. These soil deposits are residual in nature which has been subjected to a

number of wetting and drying cycles. Owing to increase in construction activity concerning

these soils, there is a need to comprehensively understand the mechanisms involved in shear

and compression behaviour. The investigation considers the laboratory testing on

representative field samples (both undisturbed and remolded) which have been extracted

from the bottom of test pits of depths ranging from 1.8m to 3.5m. It may be seen that these

soils represent wide spectrum of residual soils encountered in practice in the region. The

liquid limit values range from 27-92 and fine fraction ranging from 32-79. The basic soil

properties of the soils considered are shown in table 1.

Table 1: Soil Properties

S.No. Description

Vinayaka

Nagar

Gayathri

Nagar Renigunta

Muni Reddy

Nagar Tiruchanur

Depth of Sampling, m

2.70 3.50 2.50 1.80 2.70

1 % Gravel 3.00 1.00 6.60 0.50 7.00

2 %Sand

46.00 43.00 63.40 57.50 48.00

3 % Silt + Clay 51.00 56.00 30.00 42.00 45.00

4 Liquid limit (%) 42.00 33.00 92.00 27.00 55.00

5 Plastic limit (%) 30.10 22.17 45.30 20.75 35.24

6 Plasticity index (%) 11.90 10.83 46.70 6.25 19.76

7 Void ratio (eo) 0.610 0.601 0.620 0.605 0.550

8 Percent < 425µ 68 79 32 77 50

9 Modified liquid

limit, (WL )M % 28.56 26.00 29.44 21.00 27.50

10 eo/eLM 1.080 0.800 0.785 0.750 0.850

11 IS Classification CI CL SC SC SC

12 Field density, kN/m3 19.58 19.85 19.62 19.18 19.54

13 Natural moisture

content, % 16.70 17.73 19.86 16.25 14.92

14 Yield stress σy in

kPa 80 76 67 64 80



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4.2 Analysis of Test Data

The test results are analysed using the effective mean normal and deviatoric stress

parameters p and q as given by:

3

2 31 σσ +=p (1)

31 σσ −=q (2)

Where σ1 = axial stresses on a cylindrical sample.

σ3 = radial stresses on a cylindrical sample.

The deviatoric strain is expressed by:

)2(3

231 εεε −=s (3)

31 2εεε +=v (4)

Where ε1 = axial strains

ε3 = radial strains.

For undrained tests, εv = 0 and hence εs = ε1. Axial strains were measured externally and the

deviatoric stresses were calculated from the readings of pressure controller and the current

sample area using conventional area correction.

Figures 1 and 2 shows the stress-strain-pore pressure response of tropical residual

soils for two confining pressures (50kPa and 100kPa) of two soils. From the figures 1 and 2,

it may be observed that the strain softening is associated with positive pore pressure. The

other soils are also following the same trend. It is well known that it is not possible to get a

unique plot of q/po versus εs for tropical residual soils while it is possible in the case of

normally consolidated clays. This may be attributed to the fact that the evolution of

cementation bond resistance and subsequent softening during deformation process is not

proportional to the initial confining pressures, there by being more predominant for low

confining pressures in comparison to the equivalent unbonded response.

Effective stress paths of the two samples are presented in figures 3 and 4 for the

confining pressures tested. The other soils are also following the same trend. These stress

paths indicate that mean effective stress decreases during strain hardening and strain

softening process. The specimens tested under different confining pressures tend to reach the

critical states corresponding to remoulded situation if cementation bonds were not present. It

turns out that critical state is approached only slowly at large strains. The results indicate that

it is the type of soil that determines the critical state parameters and not the initial state or

cementation bonding. The results show that the value of stress ratio (η=q/p) upon reaching

respective peak values remains nearly constant for two confining pressures as indicated in

figures 3 and 4. It may be further observed from figures 1 and 2 that the pore water pressure

continues to be positive even in the softening region indicating that the behaviour is not

similar to that of overconsolidated soils as is frequently reported. Strain softening associated

with positive pore pressures is perhaps peculiar feature concerning the behaviour of tropical

residual soils. This may be ascribed to the additional stress transfer on to the pore pressure as

a consequence of debonding with progressing shearing. This stress transfer seems to occur in

such a way that the value of η remains fairly a constant with distortional strain.



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Fig.2 Stress-strain-pore pressure response of

Gayathri nagar soil

Fig.4 Effective stress paths of

Gayathri nagar soil

Fig.3 Effective stress paths of

Vinayaka nagar soil

Fig.1 Stress-strain-pore pressure response

of Vinayaka nagar soil



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An examination of data obtained from experimental results of tropical residual soils

indicates that p versus εs (figures 5 and 6) and η versus εs (figures 7 and 8) relations are

hyperbolic. Other soils are also following the same trend. These observations form the basis

for the formulations proposed in this paper. The two constant hyperbolic relations have been

utilized with advantage to analyze the consolidated undrained triaxial test results. Variation

of stress ratio (q/p) and the mean principal stress with deviatoric strain in terms of hyperbolic

relation takes the form as

)( 22 s

s

Xba ε

εη

+= (5)

Fig.6 Mean principal stress-strain

response of Gayathri Nagar soil

Fig.7 Stress ratio-strain response

of Vinayaka Nagar soil

Fig.8 Stress ratio-strain response

of Gayathri Nagar soil

Fig.5 Mean principal stress-strain

response of Vinayaka Nagar soil



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)()(

33 s

s

oXba

ppε

ε

+=− (6)

Where η = q/p

εs = shear strain

po = Initial mean principal stress

Equations 5 and 6 can be transformed into the linear form as presented below in order to be

able to make them suitable for experimental verification.

)( 22 s

s Xba εη

ε+= (7)

)()(

33 s

o

s Xbapp

εε

+=−

(8)

The experimental data of five soils is plotted in the form represented by equations 7

and 8 and are shown in figures 9 and 10. A good straight line can be fitted to the experimental

data between εs/η versus εs and εs/(po-p) versus εs for all the soils of selected confining

pressures. This is a good indication of the applicability of the form proposed to represent the

stress-strain response of tropical residual soils.

Elimination of εs in equations 5 and 6 yields:

)1()(1

)(

2

2

3

3

b

a

ppb

ppa

o

o

−=

−−

− η (9)

which describes the undrained stress path of tropical residual soil.

Fig.9 Transformed stress ratio-

strain curves

Fig.10 Transformed mean

principal stress-strain curves



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For meaningful application of the relations proposed it is necessary to determine the

exact nature of the parameters a2, b2, a3 and b3 in relation to the confining pressure

normalized with yield stress. Figures 11-14 shows the variation of these constants with the

confining pressure normalized with yield stress.

Experimental results indicate that it is convenient to express the parameters a3 and b3

in terms of initial confining pressures normalized with yield stress in the form of a power

function and a2 and b2 in the form of linear relationship with confining pressure normalized

with yield stress on log scale (Equations 10 and 11).

Fig.12 Variation of parameter b2 with

confining pressure normalized with yield

stress

Fig.13 Variation of parameter a3 with

confining pressure normalized with yield stress

Fig.14 Variation of parameter b3 with

confining pressure normalized with yield stress

Fig.11 Variation of parameter a2 with

confining pressure normalized with yield

stress



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488.0ln326.02 +

=

y

opa

σ (10a)

481.0ln143.02 +

=

y

opb

σ (10b)

28.1

3 109.0

−

=

y

opa

σ (11a)

87.1

3 011.0

−

=

y

opb

σ (11b)

The hyperbolic constants a2, b2, a3 and b3 as obtained by the equations mentioned

above are used to compute the stress-strain response. In computing the above parameters, in

addition to the experimental results, data from the literature related to tropical residual soils is

also used. The test data of reddish lateritic soil (Futai et al. 2004) which is sampled at 1m

depth and having yield strength of 100kPa. This soil is tested under different confining

pressures ranging from 25kPa to 400kPa. The experimental data is plotted in the transformed

hyperbolic form with εs / η and εs / (po-p) on y-axis and deviatoric strain on x-axis and are

presented in figures 15 and 16. The computed and the observed plots of q versus εs of soils

tested are presented in figures 17 and 18 respectively. The close agreement between the

computed and experimental results seems to confirm the applicability of the hyperbolic

model for the tropical residual soils in undrained shear. Other soils are also following the

same trend. However, these formulations may not be applicable for very low confining

pressures where the pore pressure response does not follow a hyperbolic variation with strain.

Fig.15 Transformed stress ratio-strain

curves of lateritic soil

Fig.16 Transformed mean principal

stress-strain curves of lateritic soil



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Fig.17 Experimental and predicted stress-

strain curves of Vinayaka Nagar soil

Fig.18 Experimental and predicted stress-

strain curves of Gayathri Nagar soil

Fig.19 Experimental and predicted stress-strain curves of saprolitic soil



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4.3 Application to other experimental investigations

It is desirable to consider the proposed mathematical form in relation to other

published literature in order to assess its general applicability. The test data of tropical

residual saprolitic soil (Futai et al. 2004) which is sampled at 5m depth and having yield

strength of 260kPa is examined in this connection. This soil is tested under different

confining pressures ranging from 25kPa to 690kPa. The equations 10 and 11 are used to

predict the stress-strain characteristics. The close agreement between predicted and

experimental values of saprolitic soil (Futai et al. 2004) is once again well demonstrated by

comparative plot shown in figure 19.

5. CONCLUDING REMARKS

Based on the analysis of test results of carefully planned experimental programme, the

following concluding remarks may be made.

1. The strain softening behaviour associated with positive pore water pressures noticed

in the residual soils can be captured using hyperbolic approach with appropriate

modifications.

2. The combination of stress ratio (η=q/p) and mean principal stress (p) is used to

represent the non-linear stress dependent behaviour of residual soils.

3. Four parameters are involved in the proposed hyperbolic model which can be

determined from simple consolidated undrained triaxial tests and one dimensional

compression tests.

4. The model parameters are found to have functional relationship with the yield stress

value (σy) in one dimensional oedometer compression.

5. The model developed has been applied to other soil data and the applicability is

evidenced from the model predictions being in close agreement with observed

behaviour.

REFERENCES

1. Bujang B.K. Huat, David G. Toll, Arun Prasad (2012) - Handbook of Tropical Residual

Soils Engineering- Published 24th May 2012 by CRC Press.

2. Duncan J.M. and Chang C.Y. (1970) - Nonlinear analysis of stress and strain in soils.

Journal of the Soil Mechanics and Foundations Division, ASCE, 1970, 96, No. SM5,

1629-1653.

3. Karmakar1, S., Sharma.J. and Kushwaha.R.L.(2004),” Critical state elasto-plastic

constitutive models for soil failure in tillage – A review”, Canadian biosystems

engineering, volume 46. 2004.

4. M. M. Futai, M. S. S. Almeida, and W. A. Lacerda, (2004) Yield, Strength, and

Critical State Behavior of a Tropical Saturated Soil, Journal Of Geotechnical And

Geoenvironmental Engineering © ASCE / November 2004, 1169 -1179

5. Mofiz M. and Mohammad Nurul Islam M. (2010)- Assess the Stress-Strain and

Interfacial Frictional Behaviour of Nonwoven Geotextile Reinforced Residual Soils-

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133

6. Nagendra Prasad, K., Srinivasa Murthy, B.R., Sitharam, T.G., and Vatsala, A. (1999)-

Hyperbolic stress-strain and pore pressure response of sensitive clays – Indian

Geotechnical Journal, 29 (3), 221-241.

7. Sarma, M.D. & D. Sarma,D. (2008) “Prediction of Consolidation Properties of

Partially Saturated Clays” The 12th

International Conference of International

Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6

October, 2008 Goa, India

8. Nagendra Prasad.K, Manohara Reddy.R, Chandra.B and Harsha Vardhan Reddy.M,

“Compression Behaviour of Natural Soils”, International Journal of Civil Engineering

& Technology (IJCIET), Volume 4, Issue 3, 2013, pp. 80 - 91, ISSN Print:

0976 – 6308, ISSN Online: 0976 – 6316.

9. Nagendra Prasad.K, Sivaramulu Naidu.D, Harsha Vardhan Reddy. M and Chandra.B,

“Framework for Assessment of Shear Strength Parameters of Residual Tropical

Soils”, International Journal of Civil Engineering & Technology (IJCIET), Volume 4,

Issue 2, 2013, pp. 189 - 207, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316.

Hyperbolic constitutive model for tropical residual soils

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