ORIGINAL PAPER

An Intrinsic Compressibility Framework for Clayey Soils

G. Belokas • M. Kavvadas

Received: 31 March 2009 / Accepted: 11 June 2011 / Published online: 28 June 2011

� Springer Science+Business Media B.V. 2011

Abstract A mathematical framework for the

description of structureless clay behaviour is pro-

posed in this paper. It is based on the observation that

there exists a biunique relationship between the radial

stress path direction and the compression curve on the

specific volume (v = 1 ? e)—mean effective stress

(r or p) plane. The projection on the v - r plane

defines the Intrinsic Compression Curve (ICC) of the

corresponding radial stress path, which results in

infinite possible ICC curves. Following a normaliza-

tion procedure, a general methodology is presented,

which can be applied to any mathematical formula-

tion of the ICC curve. The constants included in this

description are called intrinsic properties. This pro-

cedure can be easily implemented to anisotropic

constitutive models. Its importance is based on the

fact that modern constitutive models require a

definition of the structureless state, as this state is

the limiting reference state of the fully destructured

material. In addition, the paper presents a set of

correlations for the estimation of the intrinsic prop-

erties from the index properties, which helps on the

preliminary selection of the material constants.

Keywords Clay � Structure � Structureless �Intrinsic compressibility � Correlations � Stress

dilatancy

List of Symbols

eq Axisymmetric deviatoric strain

ev Volumetric strain

g Stress ratio tensor

gr Tensorial stress ratio

gq Axisymmetric stress ratio

k Intrinsic Compressibility

r Stress tensor

rij Stress tensor

r Mean or isotropic effective stress

rv Vertical effective stress

CSC Critical State Curve

CSE Critical State Envelope

cu Undrained shear strength

dq Axisymmetric dilatancy

e Voids’ ratio

I Unit tensor

ICC Intrinsic Compression Curve

ICC Intrinsic Compressibility Envelope

K Earth pressure coefficient

M Critical slope

Ng Position of ICC

p Mean or isotropic effective stress

G. Belokas (&) � M. Kavvadas

School of Civil Engineering, Geotechnical Department,

National Technical University of Athens, 5, Iroon

Polytechniou Street, 157 80 Zografou, Greece

e-mail: gbelokas@gmail.com

M. Kavvadas

e-mail: kavvadas@central.ntua.gr

123

Geotech Geol Eng (2011) 29:855–871

DOI 10.1007/s10706-011-9422-0

s Deviatoric stress tensor

v Specific volume

1 Introduction

Modern constitutive modelling of structured (e.g.

cemented, leached, aged, preconsolidated) soils

requires the knowledge of the structureless state of

the soil, which is a reference state. Following the

definitions of Vaughan (1988) and Leroueil and

Vaughan (1990) a soil material is found in a

structureless state when the only variables required

to describe its response are the current specific

volume (v = 1 ? e) and current effective stress state

(i.e. stress tensor r). This state has also been

described as intrinsic (Burland 1990) and requires

that the material has no bonding and has erased all its

past stress history memory.

A classic example of a structureless material is the

laboratory reconstituted clay at a moisture content of

about w = 1.0–1.5wL (wL the liquid limit) and

subsequently loaded in the oedometer device at

a large vertical effective stress (e.g. rv [ 1 MPa).

The state of this material is completely defined by the

earth pressure coefficient at rest Ko = rh/rv and the

vertical effective stress (rv). Moreover, this state lies

on the Intrinsic Compression Curve (ICC) proposed

by Burland (1990), which relates e with rv.

The locus of the structureless (or intrinsic) states

represents a reference limiting state of the structured

soils and is important for the constitutive modelling

of structured soils (e.g. Kavvadas 1998). Typical

examples of such models are the ones proposed by

Kavvadas (1995) and Wood (1995), which use

isotropic bounding surface, and Kavvadas and Amo-

rosi (2000), Gajo and Wood (2001), Kavvadas and

Belokas (2001), Belokas (2008) and Belokas and

Kavvadas (2010), which use anisotropic bounding

surface. The use of isotropic bounding surface

simplifies the mathematical definition of the struc-

tureless state, while the use of anisotropic bounding

surface can accommodate a more accurate represen-

tation of the structureless state. Earlier mathematical

assumptions of intrinsic states representation applied

to anisotropic constitutive models have been pre-

sented by Pender (1978), Stipho (1982) and Banerjee

et al. (1984) for unbonded materials, nevertheless

they were not calibrated on experimental data.

This paper proposes a mathematical framework for

the description of all structureless states of a soil

material and has been developed mainly from the

observed behaviour of imposed radial stress paths

under axisymmetric conditions (typical triaxial device)

on clayey soils. It is also presented in the tensorial

stress, which is applicable to constitutive modelling.

The complete mathematical formulation is developed

herein for linear Intrinsic Compression Curve (ICC) in

the v – ln r plane, which can be implemented to any

anisotropic constitutive model with rotational—dis-

tortional hardening. Variances of this Intrinsic Com-

pressibility Framework have been implemented to the

anisotropic elastoplastic constitutive model for natural

soils proposed by Kavvadas and Belokas (2001) and

Belokas (2008). In the final part of this work, a set of

equations that correlate the constants of the framework

(intrinsic properties) with the index properties is

presented. This set of equations is useful for the

preliminary estimation of the intrinsic properties.

The typical tensorial and triaxial (axisymmetric)

stress and strain measures are used throughout the

paper, which correspond to horizontal x-axis and

y-axis and vertical downwards z-axis. Boldface char-

acters are for tensorial quantities and a dot over a

character indicates incremental quantity. Stress tensor

equals to r = rI ? s (or rij = rdij ? sij), where r is

the mean effective stress, s or sij the deviatoric stress,

I the unit tensor and dij the kronecker delta. Useful

measures of stress are: (a) the second invariant

J02;r ¼ sijsji

� �=2 ¼ s : sð Þ=2, where sij, the deviatoric

stress tensor (b) the third invariant J03;r ¼ sijsjkski

� �=3

and (c) the Lode angle h ¼ 13

cos�1 3ffiffi3p

2

J03;r

J03=2

2;r

� �. Strain

tensor equals to e = eI/3 ? e, where e is the isotropic

strain and e the deviatoric strain. The triaxial or

axisymmetric conditions are described by p =

(rz ? 2rx)/3, q = rz - rx stress and ev = ez ? 2ex,

eq = 2(ez - ex)/3 strain measures.

2 Radial Stress Path Response of Structureless

Soils

At a radial stress path the soil is subject to a stress

path in which it is g = constant (where g = s/r), i.e.

a stress path of a constant slope (gr = ct in tensioral

representation, where gr = (s:s/r2)0.5, or gq = ct in

856 Geotech Geol Eng (2011) 29:855–871

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axisymmetric conditions, where gq = q/p) and a

constant third invariant (the projection in the deviatoric

space). A special case of this loading is the axisym-

metric conditions imposed by the typical triaxial

device, during which rx = ry(=rh) = rz(=rv) and

therefore the earth pressure coefficient K = rx/

rz = ry/rz is directly related to the stress ratio

gq = q/p = 3(rz–rx)/(rz ? 2rx) according to Eq. 1.

gq ¼ q=p ¼ 3 1� Kð Þ= 1þ 2Kð Þ ð1Þ

For a cross anisotropic material (i.e. same response in

x and y axes) the total dilatancy of the material is

defined by Eq. 2.

dq ¼ _eq=_ev ¼ 2 _ez � _exð Þ= 3ð_ez þ 2_exÞ½ � ð2Þ

Special cases of radial stress path axisymmetric

loading (i.e. rx = ry) are the isotropic (K = 1, gq = 0)

and one dimensional compression (K = Ko = (rh/rv)o,

gq = gq,o = 3(1 - Ko)/(1 ? 2Ko), dq = 2/3), which

for a structureless material would result to a dq = 0

and dq = 2/3 respectively.

Radial stress path tests under axisymmetric con-

ditions have led to the conclusion that as mean

effective stress (p) increases, the path in the specific

volume (v) mean effective stress (p) plane tends to

fall into separate but uniquely defined virgin com-

pression curves (e.g. Roscoe et al. 1963; Lewin and

Burland 1970, Fig. 1). It can be argued that at these

states the material has erased all its past memory as

the response is controlled by current state (i.e. stress

state, r, and specific volume, v) and, according to

Leroueil and Vaughan (1990), these states are

structureless. Therefore, these curves can be called

Intrinsic Compression Curves (ICC), similarly to

Burland’s (1990) Intrinsic Compression Line (Fig. 2).

The most common mathematical representation of

ICC curves is given by Eq. 3.

v ¼ Ng � k ln p ð3Þ

This equation includes constants: (a) Ng, which is the

specific volume for p = 1 kPa and controls the

position of ICC at the m – ln r plane and (b) k,

which is the compressibility coefficient and controls

the slope of ICC at the m – ln r plane.

The dependence of the position of the ICC on the

stress ratio (g) has been investigated by various authors

(e.g. Bishop and Eldin 1953; Broms and Ratnam 1963;

Roscoe et al. 1963, Lewin and Burland 1970; Donaghe

and Townshed 1978; Gens 1982; Georgiannou 1988;

Rampello et al. 1997; Belokas et al. 2005), which leads

to the conclusion that as stress ratio increases, constant

Ng reduces. The maximum value of Ng is the Niso,

which corresponds to the ‘‘Isotropic Intrinsic Com-

pression Curve’’ (ICCiso), while the minimum value of

Ng is the C, which corresponds to the ‘‘Critical State

Curve’’ (CSC). It also holds Niso [ NKo [ C, where

NKo is for the ‘‘One Dimensional Intrinsic Compres-

sion Curve’’ (ICCKo).

It has therefore been established that there exists a

biunique relationship between the compression curve

Fig. 1 Dependence of compression curves on earth pressure coefficient during an axisymmetric radial compression by Lewin and

Burland (1970)

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(v - p plot) and the stress ratio (gq) of a radial stress

path (e.g. Figs. 1, 3, 4 and 5). Moreover, it appears

that the dilatancy (dq ¼ _eq=_ev) stabilizes after a

certain level of stress (e.g. Gens 1982; Belokas

2008; Fig. 5) as material anisotropy stabilizes. Under

these conditions the behaviour depends only on

current soil’s state (i.e. specific volume v, and stress

state p and q) and the soil is in a structureless state.

The Critical State Line (CSL) represents the lower

boundary of any possible compression line, which

corresponds to a radial stress path of a slope equal to the

critical slope (gq = M, e.g. Gens 1982). According to

Gens (1982), the critical slope also represents the

maximum stress ratio of a continuously hardening

radial stress path (i.e. it does not fail). This slope can

vary only with respect to the third stress invariant.

3 Stress—Dilatancy Relationship Under Radial

Stress Paths

Each soil has a different stress—dilatancy relation-

ship g ¼ q=p versus dq ¼ _eq=_ev under radial com-

pression loading. As already mentioned, the total

dilatancy tends to stabilize for a radial stress path as

the mean effective stress increases. This is attributed

to the fact that the material achieves a certain degree

of anisotropy. A plot of the normalized stress ratio g/

M (where M is the critical slope) with respect to

dilatancy d leads to a unique curve for each soil

(Fig. 6). The resulting curve has two properties: (a) it

starts from (gq/L = 0, eq/ev = 0) and (b) it tends

asymptotically to gq/M = 1 for eq/ev ? ?. An

exponential curve of the form of Eq. 4, where a is

a constant, represents well the stress—dilatancy

relationship (see Fig. 6).

g=M¼ 1� e�adq ) dq ¼� 1=að Þ ln ð1� g=MÞ ð4Þ

Constant a varies from 0.32 to 1.30 for the 10

different materials examined in Fig. 6.

In incremental plasticity the total incremental

strain is decomposed to the elastic and the plastic

component (i.e. _ev ¼ _eev þ _ep

v and _eq ¼ _eeq þ _ep

q in

axisymmetric conditions) and the total dilatancy is

dq ¼ ð_eeq þ _ep

qÞ=ð_eev þ _ep

vÞ. Moroever, the plastic dilat-

ancy dpq ¼ _ep

q=_epv is an inseparable constituent of the

flow rule. Therefore, the plastic dilatancy can be

related to the total dilatancy according to Eq. 5.

dpq ¼

dq _ev � _epq

_ev � _eev

ð5Þ

In Eq. 5 the total volumetric increment for a radial

structureless stress path and an ICC linear in v – ln p

plane is given by Eq. 6.

_ev ¼ k_p

vpð6Þ

Assuming isotropic poroelasticity for the elastic

component, the elastic strain increment components

Fig. 2 Intrinsic Compression Line (Burland 1990)

Fig. 3 Radial stress paths projection on e–ln p0 space for

reconstituted Lower Cromer Till (Gens and Potts 1982)

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are given by Eq. 7a and 7b (where 2Ge/Ke a constant

that depends on Poisson’s ratio).

_eev ¼ j

_p

vp¼ Ke _p ð7aÞ

_eeq ¼

_q

2GeGe; Ge ¼ 1=2ð Þ 2Ge=Keð ÞKe ð7bÞ

Applying Eqs. 6, 7a into 5 results in Eq. 8 for the

plastic dilatancy for a radial structureless stress path,

where dq is given by Eq. 4.

dpq ¼

dqk_p

vp�_q

2Ge

_pvp ðk� jÞ

¼�ð1=aÞ lnð1� gÞk _p

vp�_q

2Ge

_pvp ðk� jÞ

ð8Þ

It is evident that the plastic dilatancy depends on the

formulation of the elastic component and of the

Intrinsic Compression Curves. In an incremental

bounding surface plasticity model, this procedure

can be used for the calibration of the flow rule or the

derivation of the plastic potential function.

4 Intrinsic Compressibility Framework

According to the above observations a structure-

less state corresponds to a radial stress path

gq ¼ q=p or g ¼ s=r� �

, in which the soil has erased

all past memory (stress induced and bond induced)

and its current memory is solely described by its

current state (v – p - q) in triaxial space or m - r

(i.e. (m, r, s) in complete stress tensor). Moreover, a

structureless soil state always lies on an Intrinsinc

Compression Curve (ICCg),1 which is defined on the

v - r plane. The Intrinsic Compressibility Frame-

work describes the biunique relationship between the

radial stress path (described by the stress ratio

gq = q/p or gs = s/r) and the projection of the ICCg

in the m - r plane via a normalization procedure.

This relationship is portrayed schematically in Fig. 7,

for ICCg linear in the v – ln r plane.

It has also been stated that during a radial stress

path and with increasing mean effective stress the

dilatancy of the soil tends to stabilize, which is

attributed to the stabilization of material anisotropy.

This happens because the particle orientation within

the soil is affected by the stress ratio and therefore the

stress ratio also describes the inherent anisotropy of

the soil. When the dilatancy becomes constant, the

material has eventually reached a structureless state

and the soil is at the maximum mean effective stress

(rcurrent = rmax) of its current stress history, which is

greater than the maximum preconsolidation stress at

the beginning of loading.

As there exist infinite possible radial stress paths

(even in axisymmetric conditions) the biunique

relationship ensures the existence of infinite ICCg.

Fig. 4 Radial stress paths on reconstituted Vallericca clay (Rampello et al. 1997)

1 The subscript g stands for the dependence of the ICC on the

radial stress path.

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This is a generalization of the Intrinsic Compression

Line for the one dimensional compression conditions

proposed by Burland (1990). The biunique descrip-

tion can be generalized to include the complete stress

tensor so as to describe any radial stress path

(irrespectively of the projection on the deviatoric

plane).

The above description is independent of the

mathematical formulation of the ICCg. Figure 7

presents a graphical representation for the case that

the ICCg are linear and parallel in m – ln r plane

(a classic assumption in critical state soil mechanics).

However, these curves could theoretically coincide

under two conditions:

• At the loosest condition, which corresponds to a

state where r = 0. This could be the case in

which the moisture content is above the liquid

limit (i.e. semi liquid state with zero shear

strength).

• At the densest condition, during which the void

ratio tends to zero (e ? 0). This could only exist

Fig. 5 Response of reconstituted Vallericca clay under radial stress path (data from Belokas et al. 2005)

860 Geotech Geol Eng (2011) 29:855–871

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for extremely high stress level, which would also

include particle breakage.

A general description of all structureless states

irrespectively of the ICCg formulation is depicted in

Fig. 8. In the v - r(i.e. v – s - r) hyper space, all

ICCg lie on the Intrinsic Compressibility Envelope

(ICE). The ICE is a kind of Roscoe–Rendulic surface

as it delimits all possible radial stress path structureless

states. This envelope is neither the State Boundary

Surface (SBS) of Critical State Soil Mechanics

(following Lewin and Burland (1970) observations),

nor the modified SBS proposed by Gens (1982), as ICE

describes only structureless states. It has been found

that the stress path traces of v = ct for different

loading conditions (e.g. radial stress path—ICE,

undrained shearing, drained shearing) do not coincide

to define a single SBS (e.g. Lewin and Burland 1970).

On the other hand, the SBS proposed by Gens (1982) is

the bounding envelope of all the v = ct stress paths

Fig. 6 Application of the

stress dilatancy relationship

during radial stress paths for

different soil materials

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that result from shear tests starting from different

stress ratios g.

The ICE has been described through a normaliza-

tion procedure (Belokas, 2008), which is general and

leads to a single equation. This procedure can be

applied to any mathematical representation of the

ICCg and an application is presented in the following

paragraph.

4.1 Mathematical Formulation for ICCg Linear

in v – ln r Plane

The mathematical formulation of the Intrinsic Com-

pressibility Framework is presented for ICCg linear in

the v – ln r plane (Fig. 7). In Fig. 7, constant Ng of

Eq. 3 is a function of the radial path stress ratio. The

equations presented herein cover both the axi-

symmetric and tensorial stress measures. The gener-

alization to the tensorial space includes the following

assumptions: (a) the ICCg position in the m - rplane depends only on the normalized stress ratio

(gr/M(0)), (i.e. it is independent of the Lode angle 0or equivalently on the third invariant) and (b) the

shape of ICCg in the m - r plane is independent of

Lode angle (0) and stress ratio.

Using the normalized specific volume Ng,norm

(Eq. 9) and the normalized stress ratio gnorm

(Eq. 10) it is possible to represent all possible

structureless states into one diagram.

Ng;norm ¼Niso � Ng

Niso � C; 0�Ng;norm� 1;

Niso�Ng�Cð9Þ

gnorm ¼ffiffiffiffiffiffiffiffiffig : gp

=Mð#Þ ð10Þ

These states are presented in Fig. 9 for seven

different materials under axisymmetric conditions.

Niso, C and Ng have already been defined, while

s

σ1

2

3

4

5

CSE, η5 → σηη M/: =1

η4

η3

η2Isotropic

compression

lnσ1kPa

1

λ

λλλλ

2

3

4 5

N1=Niso

N2

N3

N4

N5=Γ

ICCη5: Critical StateCurve (CSC)

ICCη4

ICCη3

ICCη2

ICCη1: Isotropic Intrinsic

Compression Curve = ICCiso

vFig. 7 Intrinsic

Compressibility Framework

for m – ln r0 ICCg curves

Critical State Curve – CSC

Critical State Envelope – CSECritical State Line – CSL

σ

s

v

Intrinsic Compressibility Envelope – ICE

Isotropic Intrinsic Comrpession CurveICCiso

Fig. 8 General

representation of the

structureless states

862 Geotech Geol Eng (2011) 29:855–871

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Ng,norm = 0 corresponds to ICCiso and Ng,norm = 1 to

CSC.

All available data of Fig. 9 correspond to

gnorm \ 0.85 jai Mnorm \ 0.5. Higher values of gnorm

are not common, as they correspond to stress paths

that approach closely the critical state slope. The

representation of Fig. 9 can be approximated by

Eqs. 11a or 11b.

Ng;norm ¼Niso � Ng

Niso � C¼ 1� 1�

gq

Mq

� �n

or

Ng ¼ Cþ Niso � Cð Þ 1�gq

Mq

� �n

ð11aÞ

Ng;norm ¼Niso � Ng

Niso � C¼ 1� 1�

ffiffiffiffiffiffiffiffiffig : gp

Mð#Þ

� �n

or

Ng ¼ Cþ ðNiso � CÞ 1�ffiffiffiffiffiffiffiffiffig : gp

Mð#Þ

� �n

ð11bÞ

Equation 11b has been directly applied to the aniso-

tropic bounding surface plasticity constitutive model

for structured soils proposed by Belokas (2008) and

Belokas and Kavvadas (2010), improving greatly the

radial stress path response. This equation can also be

applied to other anisotropic models, like the ones

proposed by Pender (1978), Stipho (1982), Banerjee

et al. (1984) and Dafalias (1987).

For the available test data, a regression analysis

gives n = 0.33 (with R2 = 0.734) with a range of

n = 0.15–0.60. That means that n should be cali-

brated separately for each soil material (see Table 1).

A manipulation of Eq. 11a, 11b gives Eq. 12a, 12b,

which represent the Intrinsic Compressibility Enve-

lope (ICE).

1

M2q

q2 � 1� mþ k ln p� CNiso � C

� �1=n" #2

p2 ¼ 0 ð12aÞ

1

M2ð#Þ s : s� 1� mþ k ln r� CNiso � C

� �1=n" #2

r2 ¼ 0

ð12bÞ

5 Estimation of Intrinsic Properties from Index

Properties

The intrinsic properties are the constants of Eq. 11a

or 12a. Wood (1990) presented a methodology for the

estimation of some intrinsic properties from the index

properties for Intrinsic Compression Curves (ICCg)

linear in the m – ln r plane. This methodology does

not cover all the intrinsic properties. Some additional

semi-empirical correlations of the intrinsic properties

with the index properties are presented in the current

paper.

For a saturated soil, the undrained shear strength

equals to cu = qcs/2 = Mpcs (where qcs and pcs

correspond to critical state conditions) and it is

v = 1 ? Gsw (where Gs is the specific gravity).

Wood assumed that: (a) at the liquid limit (w = wL)

the undrained shear strength is cuL & 2 kPa

Fig. 9 Normalized specific

volume versus normalized

stress ratio

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(Terzaghi et al. 1996, mention a range of 1.5 kPa to

2.5 kPa) and (b) at the plastic limit (w = wP) it is

cuP & 200 kPa (the same value is mentioned by

Mitchell 1992), which means that cuP/cuL & 100. In

the following, the cuP/cuL & 100 assumption is re-

examined. It has been portrayed (Belokas, 2008) that

Wood’s methodology is easily adjusted to any

mathematical formulation of the Intrinsic Compres-

sion Curves.

For ICCg linear in the m - ln r plane (i.e. Eq. 3

holds) Wood’s methodology gives Eqs. 13–16. Wood

(1990) simplified them by further assuming that

cuL & 2 kPa, cuP & 200 kPa and cuP/cuL & 100.

cu ¼M

2exp

C� mk

� �¼M

2exp

C� 1�Gsw

k

� �ð13Þ

k ¼ Gs wL � wPð Þln cuP=cuLð Þ ¼

GsIp

ln cuP=cuLð Þ ð14Þ

M ¼ 2cuL

exp C� 1� GswLð Þ=k½ � ; ð15aÞ

M ¼ 2cuP

exp C� 1� GswPð Þ=k½ � ð15bÞ

C ¼ 1þ GswL þ k ln 2cuL=Mð Þ; ð16aÞC ¼ 1þ GswP þ k ln 2cuP=Mð Þ ð16bÞ

All regression analysis data presented in the following

section correspond to typical triaxial test conditions.

5.1 Intrinsic Compressibility

The application of Eq. 14 on 54 different soil data

assuming cuP/cuL & 100 gives Fig. 10, which

compares the predicted compressibility, kpredicted (by

Eq. 14), with the experimentally estimated compress-

ibility, kexperiment. The data come from Lewin and

Burland (1970), Gens (1982), Wood (1990), Allman

and Atkinson (1992), Canestrari and Scarpelli (1993),

Rampello and Silvestri (1993), Rampello et al.

(1993), Cotecchia and Chandler (1997), Rampello

et al. (1997), Lehane and Faulkner (1998), d’Onofrio

et al. (1998), Santucci et al. (1998), Amorosi et al.

(1999), Chandler (2000), Cafaro and Cotecchia

(2001), Koninis et al. (2001) and Amorosi and

Rampello (2007). The line kexperiment = kpredicted

provides a correlation coefficient of R2 = 0.5926.

However, Fig. 11 shows a clear correlation

between k and Gs�IP. The regression analysis is

summarized in Table 2, which shows that the power

law gives a high correlation coefficient (R2 = 0.859),

meaning that the assumption that cuP/cuL = 100

needs reappraisal. This leads to Eq. 17, which results

in a considerable reduction of the scatter of the data

with respect to kexperiment = kpredicted line (Fig. 12).

k ¼ 0:1849 Gs � IPð Þ0:8021 ð17ÞMoreover, the ratio cuP/cuL is given by Eq. 18

(applying Eqs. 14 and 17), which gives a range of

about cuP/cuL = 50–500 for most of the data used in

the regression analysis.

cuP ¼ cuL exp GsIp

� �0:1979=0:1849

h ið18Þ

5.2 Critical State Curve Position

Constant C defines the position of the critical state

curve, which is experimentally more difficult to

Table 1 Estimation of parameter n

n Material No points R2a Reference

0.3045 Lower Cromer till 4 0.9739 Gens (1982)

0.3577 Kaolin 3 0.9865 Lewin and Burland (1970)

0.3346 Bothkennar clay 1 – Allman and Atkinson (1992)

0.3342 Pappadai clay 1 – Cotecchia and Chandler (1997)

0.5120 Sand 1 – Coop (1990)

0.1662 Vallericca Clay 2 0.7782 Rampello et al. (1993)

0.6057 Clay 1 – Lehane and Faulkner (1998)

a Correlation is omitted when there is only one data for g = 0.0, 1.0

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define than the critical slope L. Equations 16a, b

were applied to the experimental data of 9 different

materials from Allman and Atkinson (1992), Canes-

trari and Scarpelli (1993), Cotecchia and Chandler

(1997), Gens (1982), Lehane and Faulkner (1998),

Lewin and Burland (1970), Rampello et al. (1993,

1997) and Santucci de Magistris et al. (1998)

assuming that cuL = 2 kPa, cuP = 200 kPa and that

cuP = 2 kPa(cuP/cuL) (using Eq. 18). A reasonable

agreement between the experimental and the pre-

dicted values of C was achieved (see Fig. 13).

Moreover, it seems that the application of cuP gives

the best prediction.

5.3 Critical Slope

The application of Eq. 16 to predict constant C requires

the knowledge of the critical slope M. The investiga-

tion for an empirical correlation of M with index

properties led to the conclusion that the ratio of M/kseems to correlate well enough with the void’s ratio at

the plastic limit (again using the values at the liquid

limit reduces the correlation degree). Figure 14 has

been produced from the experimental data of Lewin

and Burland (1970), Gens (1982), Wood (1990),

Allman and Atkinson (1992), Canestrari and Scarpelli

(1993), Rampello et al. (1993), Cotecchia and Chan-

dler (1997), Rampello et al. (1997) Lehane and

Faulkner (1998), Cafaro and Cotecchia (2001) and

Cotecchia et al. (2007) which include 11 different types

Fig. 10 Comparison of the experimentally estimation intrinsic

compressibility with the compressibility estimated from Eq. 14

for cuP/cuL = 100

Fig. 11 Plot of the experimentally estimated k and Gs�IP

Table 2 Regression analysis results for the estimation of

compressibility k

Equation Constants R2

Linear: k = a (Gs�IP) a = 0.1719 0.803

Power: k = a (Gs�IP)b a = 0.1849 b = 0.8021 0.859

Fig. 12 Comparison of the experimentally estimation intrinsic

compressibility with the compressibility estimated from Eq. 13

Geotech Geol Eng (2011) 29:855–871 865

123

of soil materials under axisymmetric compression.

Equation 19 represents a power relationship, which

gives the best correlation coefficient (R2 = 0.7391).

M=k ¼ 4:369ðePLÞ�1:187 ð19Þ

5.4 Intrinsic Compression Curve Position

The position of the Intrinsic Compression Curve

(ICCg) in the v – ln r is defined by parameter Mg,

which corresponds to the specific volume, m = 1 ? e,

for mean effective stress equal to r = 1 kPa and

depends on the stress ratio g = q/p. Burland (1990)

and Chandler (2000) have proposed equations for the

e100 but this corresponds to e–logrv plane. The

e - rv representation is not applicable to the math-

ematical description of the Intrinsic Compressibility

Framework and to general constitutive modelling,

which adopt a m - r representation of the ICCg

curves in order to describe any radial compression

stress path.

Isotropic and one–dimensional compressions rep-

resent the most commonly experimentally examined

conditions. In the following, parameters Niso and NKo,

which respectively correspond to gq = 0 (i.e. isotro-

pic compression) and to gq = go (i.e. one dimen-

sional compression) are correlated with the index

properties of the soil.

The available data for NKo correspond to 28

different materials (source data: Burland 1990;

Rampello et al. 1993; Cotecchia and Chandler

1997; Cecconi et al. 1998; Kavvadas and Anagnos-

topoulos 1998; Amorosi et al. 1999; Cafaro and

Cotecchia 2001) that cover a wide range of Atterberg

limits. The plots of NKo - eL, NKo - eP jai NKo

- GsIP (Fig. 15) show that the best correlation holds

for the NKo - eL representation. The regression

analyses gave a high correlation degree for linear

and power functions of NKo - eL (Fig. 16).

The data for Niso correspond to 14 different

materials (source data: Roscoe et al. 1963; Lewin

and Burland 1970; Gens 1982; Allman and Atkinson

1992; Canestrari and Scarpelli 1993; Coop and

Atkinson 1993; Rampello and Silvestri 1993; Ramp-

ello et al. 1993; Cotecchia and Chandler 1997;

Rampello et al. 1997; Santucci de Magistris et al.

1998; Cafaro and Cotecchia 2001; Amorosi and

Fig. 13 Comparison of the

experimentally estimated

with the predicted values

of C

Fig. 14 Correlation of M/k with the void’s ratio at plastic limit

for different soils

866 Geotech Geol Eng (2011) 29:855–871

123

Rampello 2007). The functions chosen for the

regression analyses ensured that the condition of

Niso [ NKo is always satisfied. The correlation degree

is now lower but it is considered acceptable as the

data for Niso are less and cover a narrower range of

Atterberg limits (Fig. 16). Therefore, for the estima-

tion of parameters NKo and Niso Eqs. 20 and 21 are

chosen.

NKo ¼ 2:1011ðeLÞ0:6343 ð20Þ

Niso ¼ 2:1574ðeLÞ0:6343 ð21Þ

6 Application to Vallericca Clay

The mathematical description of the Intrinsic Com-

pressibility Framework and the correlation of the

intrinsic properties with the index properties pre-

sented above are directly applied to experimental data

of tests performed on Vallericca clay. These tests

were performed under axisymmetric compression

conditions (i.e. triaxial compression) and have been

presented in Belokas et al. (2005) and Belokas

(2008).

6.1 Intrinsic Compressibility Framework

The radial stress path and undrained shear data result

in the constants of Table 3. The ICCg curves fitted to

experimental data are presented in Fig. 17. The

normalized plot of these curves is presented in

Fig. 18, which gives constant n = 1.0.

6.2 Estimates of the Intrinsic Properties

from the Index Properties

Vallericca clay has the index properties of Table 4

and the intrinsic properties of Table 5 (from Belokas

2008). Table 4 results in eLL = Gs�wLL = 1.519 and

ePL = Gs�wPL = 0.692, while the application of

Eq. 18 for cuL = 2 kPa gives cuP = 365.55 kPa.

The application of Eq. 14 (Wood 1990) for

cuP/cuL = 100 gives k = 0.1795 and applying this

into Eq. 19 results in M = 1.204. However, the

application of Eq. 17 gives k = 0.1587 and applying

this into Eq. 19 results in M = 1.065; values closer to

the experimentally derived ones (Table 5). Utilizing

kexp = 0.153 of Table 5 results in M = 1.024, which

is again a better prediction than the previous one and

shows the importance of a good estimate of k on

M prediction.

A critical slope of M = 1.065 corresponds to a

critical state angle of shearing resistance equal to

/cs = 26.89 and according to Jaky’s equation, the

earth pressure coefficient at rest is Ko = 1 - sin/= 0.5479. This corresponds to a stress ratio of

go = 0.6472, while the values of M = 1.026 and

Mexp = 0.98 give go = 0.6188 and go = 0.5857

respectively. The experimentally derived value is

go = 0.613 (Table 5 and Belokas and Kavvadas M

Amorosi 2005).

A utilization of Eqs. 16 for k = 0.1587,

M = 1.065, cuL = 2 kPa and cuP = 365.55 kPa both

lead to C = 2.7291, which is a fairly good estimate

with respect to the experimentally estimated

Cexp = 2.70. Applying the same equations for

kexp = 0.153 and Mexp = 0.98 results in C = 2.734

(using cuL) and C = 2.705 (using cuP). That means

that using good estimates of k and M and cuP value

for cuL = 2 kPa resulted in a satisfactory prediction

of C.

Moreover, the application of Eqs. 20 and 21 result

in Ngo = NKo = 2.743 and Niso = 2.778, which are

also good estimates. Having all these it is possible to

calibrate constant n (see Eq. 8) of the Intrinsic

Compressibility Framework. Therefore, the correla-

tions presented herein can be used to give estimates

of all the intrinsic properties of a soil and can be

applied directly to any Cam—Clay kind of a consti-

tutive model.

Fig. 15 Plots of NKo with respect to eL, eP and GsIP

Geotech Geol Eng (2011) 29:855–871 867

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7 Conclusions

An Intrinsic Compressibility Framework for the

mathematical description of structureless clay behav-

iour has been proposed in this paper, based on the

Critical State Soil Mechanics principles and on past

experimental evidence. Structureless states correspond

Fig. 16 Linear and power

correlations of NKo - eL

and of NIso - eL

Table 3 Intrinsic compressibility constants for Vallericca clay

g 0 0.3 0.5 0.7 0.98 (=M)

Ng 2.810 2.772 2.752 2.742 2.710

k 0.153 0.153 0.153 0.153 0.153

868 Geotech Geol Eng (2011) 29:855–871

123

to radial stress paths during which soil has erased all its

past memory and the behaviour of the soil depends only

on current state (v - r).

The Intrinsic Comrpession Curves (ICC) portray

the structureless states on the v – ln r plane and there

is a biunique relationship between the ICC curve and

the direction of the radial stress path. This has been

illustrated via a normalization procedure, which is the

basis of the mathematical description of the frame-

work. This procedure results in an equation that

describes all structureless states and therefore defines

the Intrinsic Compressibility Envelope. The proposed

procedure is general and can be applied to any

mathematical formulation of the ICC curves. An

application to linear ICC curves on the v – ln r plane

has been presented herein.

Moreover, a simple stress dilatancy relationship

for the radial stress paths has been presented. It is

based on experimental data and has one parameter

that depends on the soil material. This relationship

fits well in the available data and can be used to

calibrate a constitutive model under radial stress

paths. It is deduced that during a structureless radial

stress path, the soil dilatancy has been stabilized. The

stress dilatancy relationship can be used together with

the elastic component and the ICC curve for the

calibration of the plastic dilatancy and the derivation

of the plastic potential function.

The constants that describe the Intrinsic Com-

pressibility Framework are called intrinsic properties

and a set of equations that correlate the intrinsic

properties with the index properties has been pro-

duced starting from Wood’s (1990) methodology. His

assumption that cuL = 2 kPa was verified. However,

it was shown that the relationship cuP/cuL & 100 is

not accurate and depends on the index properties. The

produced correlations can provide an estimate of all

intrinsic properties solely from the index properties.

All proposed correlations were applied to estimate

the intrinsic properties of the Vallericca Clay data of

Belokas et al. (2005) and were successfully compared

to the experimentally derived ones.

Acknowledgments The authors wish to thank the reviewers

for their constructive comments, which helped us to improve

the manuscript.

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