STUDY OF THE BARCELONA BASIC MODEL. INFLUENCE
OF SUCTION ON SHEAR STRENGTH
Carlos Pereira
ABSTRACT
The Barcelona Basic Model, BBM, is one of the most used elasto-plastic models for unsaturated
soils. This summary presents and discusses some aspects of the BBM. Its mathematical formulation is
briefly presented and the influence that some parameters have on the model is evaluated. The integration of elasto-plastic models for unsaturated soils poses additional challenges associated to the presence of suction as an extra state variable. An explicit integration algorithm is adapted to BBM and implemented in MATLAB programming language. Some limitations, such as the stress state variables, the
hysteretic hydraulic behaviour and the shear strength increase with suction are discussed. At the end,
the evolution of shear strength along suction reduction paths, associated to the occurrence of collapse
phenomenon, is analyzed. A softening type behaviour is observed in some suction reduction paths.
Keywords: BBM; Unsaturated soils; Explicit integration algorithm; Elasto-plasticity; Softening.
1
INTRODUCTION
The classical Soil Mechanics was implicitly established for soils in dry or saturated conditions.
These two cases can be considered as two extreme and limiting soil conditions of a soil (Ng and Menzies, 2007). The unsaturated Soils Mechanics differentiates from the classical Soil Mechanics by the
existence of capillary forces related to the fact of the water pressure,
, being lower than the air
pressure,
. Suction, , is defined by the difference between
and
. The suction value is dependent on the amount of water in the soil voids, increasing with the decrease of the amount of water.
This relationship, characterized by the soil-water characteristic curve, SWCC, is specific to each type
of soil. In a cycle of drying/wetting paths, the SWCC shows a hysteretic behaviour. For the same value
of , the soils retain more water in the drying path than in the wetting path, existing several factors that
justify this behaviour (Lu and Likos, 2004).
The suction changes the way how the unsaturated soils behave to stress and strain variations,
when compared with the behaviour of saturated soils. Moreover, the suction variation produces a specific stress path. Given that the unsaturated soils are above the water table, they have a direct influence on shallow geotechnical structures.
Like saturated soils, there are several constitutive models with ability of modelling some aspects
of the behaviour of unsaturated soils. The BBM, presented by Alonso et al. (1990), was the driving
force behind the development of models for unsaturated soils and has been the most widely used over
the past years. Therefore, it is quite relevant to study the mathematical formulation of the BBM to assess its capabilities and limitations.
2
BARCELONA BASIC MODEL
The BBM uses the concepts of the plasticity theory, incorporating the critical state model, CSM,
and the Modified Cam Clay Model, MCCM, when the soil reaches the saturated state. The model was
developed to describe the behaviour of unsaturated soils, slightly or moderately expansive, such as
sands, silts and low plasticity clays.
1
The BBM considers that the behaviour of the unsaturated soils is represented by two stress
state variables, defined by equations (1) and (2), where
represents the total stress tensor, the
identity tensor and
the excess of total stress tensor over air pressure (net stress tensor).
(1)
(2)
The BBM has two yield surfaces, coupled in the stress space. The first yield surface, designated
by , is dependent on the value of yield stress for saturated conditions, , and the second yield surface, designated by , is dependent on the maximum historic suction value previously occurred, .
The yield surface
is represented by the equation (3) together with the equations (4) to (6),
where represents the deviatoric stress, the mean stress,
the slope of critical state line, the
parameter describing the increase in tension with suction,
the minimum mean stress (of tension)
corresponding to a given value of suction,
the yield stress for
,
the reference stress,
and
the stiffness parameter for net mean stress variations for soil virgin states (normal compression lines, NCL) for
and
, respectively, the elastic stiffness parameter for changes in net
mean stress, the parameter defining the maximum soil stiffness and the parameter controlling the
rate of soil stiffness increase with suction.
(3)
(4)
(5)
(6)
The yield surface
is represented by the equation (7). This yield surface intends to model the
occurrence of plastic volumetric strain due to suction increase beyond values occurred before.
(7)
Alonso et al. (1990) present a non-associated flow rule for the yield surface , according to the
equation (8), where is chosen in such a way that the flow rule predicts zero lateral strain for stress
states at rest. The value of is not an additional parameter of the model, since its value is determined
by the ratio between the earth pressure coefficient at rest, , and
and deduced by the imposition a
zero lateral plastic strain increment (Pereira, 2011). For , Alonso et al. (1990) present an associated
flow rule, according to equation (9).
(8)
(9)
The hardening laws associated with the yield surfaces
and
are obtained by the equations
(10) and (11), respectively, where
represents the plastic volumetric strain increment, the specific volume,
the stiffness parameter for changes in suction for soil virgin states,
the elastic stiffness parameter for changes in suction and
the atmospheric pressure.
(10)
(11)
The plastic multipliers
and
are obtained from the consistency conditions associated to the
yield surfaces
and , respectively. Three types of elasto-plastic behaviour can be distinguished
associated to (Pereira, 2011): (i) the yield surface , (ii) the yield surface
and (iii) both yield surfaces. The latter situation only occurs when the stress state is located at the intersection of the yield surfaces and .
For the elasto-plastic behaviour type (i), the plastic multiplier
is obtained by equation (12),
together with equations (13) and (14), where represents the bulk modulus, the shear modulus,
the increment of the deviatoric strain tensor and the deviatoric stress tensor. For the elasto-plastic
behaviour type (ii), the plastic multiplier
is obtained by equation (15).
2
(12)
(13)
(14)
(15)
For the elasto-plastic behaviour type (iii), nothing is mentioned in Alonso et al. (1990) about its
mathematical solution. The existence of simultaneous increments of stress tensor and suction can
lead to three distinct situations. The final stress state can be (a) on the yield surface , (b) on the yield
surface , or (c) at the intersection of the yield surfaces and , satisfying, simultaneously, the consistency conditions of both yield surfaces.
A methodology is proposed to assess the elasto-plastic behaviour type (iii) based on sequential
application of the increments of the strain tensor and suction. Where an increment of the strain tensor is
considered before the suction increment, the final stress state will be in the elastic domain. To avoid this
situation, the suction increment must be applied before the increment of the strain tensor.
The equation (16) represents the incremental constitutive law of the BBM for a problem controlled by increments of the strain tensor and suction. The complete mathematical formulation of the
BBM for increments of the strain or stress tensors with suction is available in Pereira (2011).
(16)
The variables that define the initial state and the parameters of the BBM are represented in Table
1. The six parameters required to the mathematical description of the LC curve (intersection of the yield
surface with the plane
, for positive values of ) have the restrictions presented in Table 2.
Table 1 – Variables defining the initial state and parameters of the BBM.
Group
Initial state
Parameters related with LC curve (equation (4))
Parameters related with suction changes
Parameters related with changes in the shear
stress and the shear strength
Initial state / Parameters
, , , ,
e
,
, , e
e
,
e
Table 2 – Restrictions to the parameters values of the LC curve.
Parameters
Restrictions
;
;
The equation (17) defines the condition for the existence of positive concavity at the yield surface
. The positive concavity implies the non-convexity of the yield surface . Due to the non-convexity of
the yield surface, a stress increment that starts and ends inside the yield surface may still cross it more
3
than once. According to Sheng et al. (2004), this aspect requires a different treatment in explicit resolution schemes and can lead to convergence problems in implicit resolution schemes. Pereira (2011)
adapted two different explicit integration algorithms to BBM and implemented one of them in MATLAB
programming language.
The and
parameters have direct implications in the geometry of the yield surface
and,
therefore, in elastic domain dimension. The shear strength, , is linear relation of and , as show in
equation (18).
(17)
3
(18)
BBM CAPABILITIES AND LIMITATIONS ANALYSIS
According to Alonso et al. (1990), the BBM adequately reproduces the main phenomena of unsaturated soil mechanics, such as: (i) the stiffness and the shear strength changes induced by suction
changes, (ii) the elasto-plastic behaviour of the soil due to stress and suction changes, and (iii) the
collapse phenomenon, characterized by the existence of plastic volumetric strain associated with
some suction decrease paths. The main capabilities of the BBM were illustrated by Alonso et al. (1990)
through the presentation of some cases studies. Some of these cases of study were successfully reproduced by Pereira (2011), through the explicit integration algorithm implemented in MATLAB programming language and the CODE_BRIGHT program.
As mentioned in the chapter 2, the BBM uses the concepts of the plasticity theory, incorporating
the CSM and the MCCM when the soil reaches the saturation state. Therefore, this model has all the
limitations of the MCCM, such as, the inability to represent the anisotropic behaviour, the viscous behaviour, the dependence of the shear strength with the loading direction, etc (Potts and Zdravković, 1999).
By extension, the BBM has a set of limitations specifically related to the unsaturated soils behaviour, as
listed briefly below.
3.1 Stress state variables
The choice of appropriated stress variables to represent the unsaturated soils behaviour has
had an extensive discussion in the scientific community (Gens et al., 2006). It is generally accepted
the idea that no single stress variable, that replaces the concept of effective stress of saturated soil
mechanics, can describe, by itself, all the aspects of the unsaturated soils mechanical behaviour
(Jommi, 2000). According to Gens et al. (2006), the two stress variables used by most unsaturated constitutive models can be expressed by the equations (19) and (20), where
and
are functions of
and, sometimes, of other variables.
(19)
(20)
Depending on the expression for , the generality of the unsaturated constitutive models can be
divided into three classes: (i)
, (ii)
as function of and (iii)
dependent of and the
degree of saturation, . Table 3 presents a qualitative comparison between the three classes of the
unsaturated constitutive models (Nuth and Laloui, 2008). The first class corresponds to the pair of state
variables used by the BBM. In any case, all constitutive models, belonging to these classes, share the
same core of assumptions: (i) the use of two independent stress variables, (ii) the formulation of some
type of LC yield surface, and (iii) the use of a saturated model as a limiting case.
4
Table 3 – Qualitative comparison between the three classes of the unsaturated constitutive models.
Classes
Stress paths
representation
+
-
Saturated-unsaturated
transition
+
+
Hysteresis and
hydraulic effects
+
Direct accounting of
increase in strength
+
+
3.2 Hysteretic hydraulic behaviour
The relationship between
and is different for increasing and decreasing suction paths. This
difference leads to a hysteretic hydraulic response in suction cycles, resulting in a volumetric plastic
strain increase. The BBM does not have the capability to model such volumetric plastic strain due to
the hysteretic hydraulic behaviour.
3.3 Shear strength increase with suction
Guan et al. (2010) indicate that the pattern of the shear strength evolution with suction can be
qualitatively related with the SWCC. For values of suction below the air-entry value, AEV, the shear
strength increases linearly with suction. For values of suction greater than AEV, a non-linear relation
is found.
The linear relationship considered by BBM (equation (18)) can only represent adequately the
shear strength for values of suction lower than AEV. For values of higher suction, the values of
shear strength determined by BBM are higher than real values, increasing the difference with the
increasing of suction.
Based on the typical SWCC of sand, silt and clay (Lu and Likos, 2004), it is possible to define,
approximately, the application limits of the BBM, being according with values of AEV for sands, silts and
clays of, approximately,
,
and
.
3.4 Soil collapse
Alonso et al. (1990) assumed, based on experimental results, that
decreases with suction increase (
). However, based on experimental results presented by Wheeler and Sivakumar (1995),
it is possible to observe that some soils show an increase of
with suction. As mentioned by
Wheeler et al. (2002), it is probably for many soils that, for a wide range of ,
diverge and converge for lower and higher values of , respectively. This means that the maximum potential for collapse phenomenon occurs for an intermediate value of . The most practical applications have a restricted range for the values of , making valid the hypothesis of straight NCL, which may diverge or
converge with increasing of .
In order to include in the BBM the convergence of
with increase of , the possibility of
increase with suction is required, i.e., assuming
, which implies (i) a sufficiently higher value for
, with
, to ensure a curve development of the LC, (ii)
, to achieve the relationship
, and (iii) the change in the definition, not considering as the parameter which
multiplied by
leads
.
3.5 Strain-softening soils
The BBM assumes that
is always crescent and the yield surfaces
and
are coupled.
Taken together, these two hypotheses give contradictory results in situations of plastic behaviour on
dry side (dilated soils), characterized by strain-softening with
reduction. Nothing is mentioned by
Alonso et al. (1990) about this contradiction.
5
In the author’s opinion, it is possible to consider, without changing the model, two possibilities for
the solution of this problem: (i) the decreasing the
value or (ii) to assume that the movements of the
yield surfaces
and
is couple only for increasing values of . In the mathematical formulation of the
BBM, the hypothesis (i) may lead to negative values of
for significant variations of . The hypothesis
(ii) prevents this phenomenon, although it is questionable the consideration that
does not change
when the soil particles arrangement suffers irreversible changes.
4
SUCTION INFLUENCE ON SHEAR STRENGTH
In shallow geotechnical structures, the suction variation can happen, fundamentally, by the
changes of atmospheric conditions, i.e., precipitation and/or relative humidity variation, and due to the
movement of groundwater. In the following sections, the evolution of shear strength, for the case of suction reduction with the occurrence of the collapse phenomenon, is analyzed.
4.1 Shear strength evolution for suction decrease path: single element analysis
The pattern of shear strength evolution with decrease suction, obtained by the BBM, is analyzed considering a single element, i.e., assuming a constant stress, strain and suction state in a
single soil element.
The unsaturated soils have a characteristic behaviour designated by collapse, when certain
stress-strain conditions are observed. In the BBM, the beginning of the collapse phenomenon corresponds to the intersection of the yield surface
when following a suction decrease path. Physically, the
plastic volumetric strain associated to collapse phenomenon corresponds to the reorganization of the
initial structure of the soil particles motivated by the decrease in the magnitude of capillary forces, in
unstable arrangement of the soil particles, for lower values of suction.
In isotropic stress states, the reorganization of the solid particles reaches always a stable situation
for any variation in suction. The magnitude of the volumetric plastic strains is proportional to the installed
stress state, reaching, in some cases, significant values (Pereira, 2011). However, when combining a
shear stress with the suction decrease, the reorganization of the soil particles may not reach a stable
situation, depending on the initial stress state, because the shear strength decreases with decreasing
suction. According to BBM, the occurrence of instability during a suction decrease path corresponds to
the intersection of the stress path with a critical state in an intermediate suction level.
In order to illustrate the occurrence of a critical state by suction reduction during the collapse
phenomenon, the results of two suction decreasing paths are represented in Figure 1 – paths EF
(
) and GH (
) – preceded by a shear path with constant – paths AE
(
) and AG (
) – with an initial value
. The shear paths AB, AC
and AD correspond to the increase of until the critical state, with
and
,
and 3
, respectively. Figure 1(a) and Figure 1(b) represent the relationship of and with deviatoric strain, , respectively. Figure 1(c) and Figure 1(d) represent the relationship of the specific volume
with and , respectively. The results represented by lines for increments of the stress tensor and
suction were obtained by the explicit integration algorithm (Pereira, 2011) and the points were determined by the CODE_BRIGHT program. The values of BBM parameters are given by Alonso et al.
(1990) in the study cases presented. The GH path achieves a stable state for the suction reduction,
with a positive variation of the specific volume. The EF path, with
, does not converge
to a stable state. For
(
), the soil element shows a typical behaviour of
the critical state (Figure 1(d)). This means that for this stress state
, the critical state at
is intercepted by EF path.
6
D
D
C
C
F
H
H
s (kPa)
s (kPa)
q (kPa)
q (kPa)
350
350
300
300 G
250
E
250 G
200 E
200
150
150
100
100
50
A
50
0
A
0 0
(a)
0
(a)
B F
B
sAB = 100 kPa
ssAC
= 200 kPa
AB = 100 kPa
ssAD
= 300 kPa
AC = 200 kPa
sAD = 300 kPa
0,1
0,1
0,2
0,2
εs
εs
0,3
0,3
0,4
0,4
350
E G
350
300
A E G
300
250 A
250
200
H
200
150
H
150
100
100
50
50
0
0 0
0,1
(b)
0
0,1
(b)
F
F
0,2
0,2
εs
εs
0,3
0,3
0,4
0,4
v v
v v
1,92
1,92
A
A
1,92
1,92
1,90
1,90 A
A
1,90
1,90
1,88
1,88
1,88
1,88
1,86
1,86
E
1,86
1,86
1,84
1,84
E
G
E
1,84
1,84
1,82
1,82
G
E
G
1,82
1,82
1,80
1,80
G
H
F
F
H
1,80
1,80
1,78
1,78
H
F
F
H
1,78 50 100 150 200 250 300 350
1,78 0
0,1
0,2
0,3
0,4
(c)
(d)
50 100 150 200 250 300 350
0
0,1
0,2
0,3
0,4
s (kPa)
εs
(c)
(d)
(kPa)
Figure 1 – Representations of
results of two decrease suction paths, preceded εby
the increasing of
s
shear stress path with
constant and
.
As seems to be evident by the results shown in Figure 1(a), the occurrence of instability is linked
to cases in which the shear stress installed in the soil is higher than the shear strength corresponding to
the final suction. According to BBM, the shear strength has a linear relationship with and (equation
(18)) and defines a plane in the space
. In planes of constant suction, the relationship between
shear strength and is reduced to a line. If a decrease in suction, from
to , is imposed, all the
stress states between
and
, i.e., for
, with constant, will intersect the critical
state in an intermediate suction (
), reaching a ultimate state before completing the entire
variation of suction.
4.2 Shear strength evolution for suction decrease path: general situations
The shear strength evolution with suction reduction paths in a single element was analyzed in
the previous section. Therefore, an ultimate state in a geotechnical structures occur when all the single elements along a surface or a zone reach the critical state, with different values of stress and suction. The occurrence of critical state along a surface or a zone does not happen in a single instant.
This type of phenomenon is characterized by the progressive occurrence of the critical states characterized by stress transference between areas, due to softening of some points of the soil with the beginning of yield.
In contrast with Figure 1, the decrease in suction can cause softening due to the stress redistribution, from elements of lower to higher stiffness, induced by the continuous decreasing of the suction.
How the stress redistribution occurs in time and space, changes from case to case. During the suction
7
decrease path, the soil elements that have reached the critical state can no longer withstand stress increases, redistributing those stresses by the other adjacent soil elements that are not yet in critical state.
With the continued decrease of suction, the shear stress supported by the soil elements, that have reach
the critical state, have to be reduced for the corresponding shear strength values at suction levels installed, as illustrated by Figure 2.
To illustrate the stress redistribution phenomenon, induced by suction decrease, a non-uniform
suction decrease was applied to the stress state E, of the stress path AE (Figure 1(a)), assuming
between the top and the base. Figure 3 represents the numerical model scheme and
the calculation steps considered in the CODE_BRIGHT program. Figure 4 presents the relationships
between and
at the points a, b and c, at different scales.
As can be observed, during the suction decrease path, the stress state at the points a, b and c
have different values, due to the difference in suction values. The point a is the one closest to the top,
with lowest suction and shear strength, and the point c is the farthest to the top, with highest suction and
shear strength. The model reaches the ultimate state with
and
.
Critical state
to
Critical state paths
Figure 2 – Suction-softening schematic representation.
Point a
Point b
Point c
Step 1:
and
Step 2:
(AE path)
Step 3: Decrease of the
Step 4: Decrease of the
(point A)
to
and
in the
same quantity to the ultimate state.
Figure 3 – Numerical model representation and the computation stages performed.
In section 4.1, the soil element reaches the critical state to
. However, when the
soil element, corresponding to point a, reaches
the numerical model does not reach an
ultimate state. Instead, it starts stress redistribution to further decrease of suction. This stress redistribution process has the effect of increasing in the points below of the model, with special emphasis on the point c, by the fact it has the higher values of suction and shear strength. The model
only reaches an ultimate state when all the soil elements along a surface or a zone achieve a critical
state (stress state F).
8
350
240
300
235
E
Fc
230
q (kPa)
q (kPa)
250
Point c
200
150
E
Point b
225
Fa
220
100
Fb
Point a
215
50
A
0
210
0
0,1
0,2
εs
Figure 4 – Relationship between
0,3
and
0,4
0
0,1
0,2
0,3
0,4
εs
in the points a, b and c for the steps listed in Figure 3.
In conclusion, the suction decrease path on geotechnical structures creates, by one side, a
progressive occurrence of the critical states along a surface or a zone due to the stress redistribution from weak to stiff zones and, by the other side, a soil softening due to shear strength decrease
with suction decrease path (Pereira, 2011). The way how these factors develop in time and space
do not have to follow similar patterns between geotechnical structures. This development is dependent of some variables, such as the geometry of the geotechnical structure, its boundary conditions,
the stress state installed, etc.
5
CONCLUSIONS
The BBM uses the concepts of the plasticity theory, incorporating the CSM and the MCCM
when the soil reaches the saturation state. It considers that the behaviour of the unsaturated soils is
represented by two stress state variables, and , and has two yield surfaces,
and , dependent
on the values of
and , respectively. The movements are coupled in the stress space and dependent on the value of
. The following points stand out as important aspects resulting from the
analysis of the BBM (Pereira, 2011):
(i)
The existence of the elasto-plastic behaviour associated with simultaneous increments of
the strain tensor and suction, in stress states located at the intersection of the yield surfaces
and
(missing aspect in Alonso et al., 1990). To solve this particular case, a methodology based on sequential application of suction and strain tensor, in that order, is proposed.
(ii)
The contradiction between the
definition, which assumes that its value only increase,
and the coupling of the movements of the yield surfaces
and . The plastic behaviour
of dry side (dilated soils) leads to strain-softening with
reduction, which would lead to
reduced .
(iii) The possibility of the yield surface
to assume concave forms for low values of suction
for certain combinations of the LC curve parameters. This property can cause problems
of convergence in implicit integration algorithms.
The BBM adequately reproduces the main phenomena of unsaturated soil mechanics, such as:
(i) the stiffness and the shear strength changes of the soil induced by suction changes, (ii) the elastoplastic behaviour of the soil due to stress and suction changes, and (iii) the collapse phenomenon,
characterized by the existence of plastic volumetric strain associated with some suction decrease
paths. However, there are characteristics of unsaturated soils that are not adequately modeled by
BBM, especially: (i) the incapacity to model volumetric plastic deformations due to hysteretic hydraulic
behaviour, in cycles of increasing and decreasing suction, and (ii) the incapacity to represent the non-
9
linear increase of shear strength with suction, to suction values higher than AEV. In this range, the
BBM predicts values of shear strength values higher than the “real” values.
Finally, it was verified, through a single element analysis, that the critical state can be achieved
on a decrease suction path, since
. However, the generalization of this behaviour pattern
to geotechnical structures is dependent of some variables, such as the geometry of the geotechnical
structure, its boundary conditions, the stress installed, etc. The suction decrease creates, by one hand,
a progressive occurrence of the critical states along a surface or a zone due to the stress redistribution
from weak to stiffer zones and, by the other hand, an induced soil softening due to shear strength decrease with suction decrease path (Pereira, 2011).
REFERENCES
Alonso, E. E., Gens, A. and Josa, A. (1990). A constitutive model for partially saturated soils.
Géotechnique, 40(3), 405-430.
Gens, A., Sánchez, M. and Sheng, D. (2006). On constitutive modeling of unsaturated soils.
Acta Geotechnica, 1, 137-147.
Guan, G. S., Rahardjo, H. and Choon, L. E. (2010). Shear strength equations for unsaturated
soil under drying and wetting. Journal of Geotechnical and Geoenvironmental Engineering, ASCE,
136(4), 594-606.
Jommi, C. (2000). Remarks on the constitutive modelling of unsaturated soils. Experimental
evidence and theoretical approaches in unsaturated soils, Balkema, Rotterdam, 139-153.
Lu, N. and Likos, W. J. (2004). Unsaturated soil mechanics. John Wiley & Sons, Inc., New Jersey.
Ng, C. W. W. and Menzies, B. (2007). Advanced unsaturated soil mechanics and engineering.
Taylor & Francis, London.
Nuth, M. and Laloui, L. (2008). Effective stress concept in unsaturated soils: clarification and
validation of a unified framework. International Journal for Numerical and Analytical Methods in Geomechanics, 32, 771-801.
Pereira, C. (2011). Study of the Barcelona Basic Model. Influence of suction on shear strength.
Master Thesis, IST, Lisbon (in Portuguese).
Potts, D. M. and Zdravković, L. (1999). Finite element analysis in geotechnical engineering.
Theory. Thomas Telford Limited.
Sheng, D., Fredlund D. G. and Gens A. (2008a). A new modelling approach for unsaturated
soils using independent stress variables. Canadian Geotechnical Journal, 45, 511-534.
Sheng, D., Gens, A., Fredlund, D. G. and Sloan, S. W. (2008b). Unsaturated soils: From constitutive modelling to numerical algorithms. Computers and Geotechnics, 35, 810-824.
Sheng, D., Sloan, S. W. and Gens, A. (2004). A constitutive model for unsaturated soils: thermomechanical and computational aspects. Computational Mechanics, 33, 453-465.
Wheeler, S. J., Gallipoli, D. and Karstunen, M. (2002). Comments on use of the Barcelona Basic Model for unsaturated soils. International Journal for Numerical and Analytical Methods in Geomechanics, 26, 1561-1571.
Wheeler, S. J. and Sivakumar, V. (1995). An elasto-plastic critical state framework for unsaturated soil. Géotechnique, 45(1), 35-53.
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