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Non-Associated Flow Rule-Based Elasto-Viscoplastic Model for Clay

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We develop a non-associated flow rule (NAFR) based elasto-viscoplastic (EVP) model for isotropic clays. For the model formulation, we introduce the critical state soil mechanics theory (CSSMT), the bounding surface theory and Perzyna's overstress theory. The NAFR based EVP model comprises three surfaces: the potential surface, the reference surface and the loading surface. Additionally, in the model formulation, assuming the potential surface and the reference surface are identical, we obtain the associated flow rule-based EVP model. Both EVP models require seven parameters and five of them are identical to the Modified Cam Clay model. The other two parameters are the surface shape parameter and the secondary compression index. Moreover, we introduce the shape parameter in the model formulation to control the surface shape and to account for the overconsolidation state of clay. Additionally, we incorporate the secondary compression index to introduce the viscosity of clay. Also, we validate the EVP model performances for the Shanghai clay,the San Francisco Bay Mud (SFBM) clay and the Kaolin clay. Furthermore, we use the EVP models to predict the long-term field monitoring measurement of the Nerang Broadbeach roadway embankment in Australia. From the comparison of model predictions, we find that the non-associated flow rule EVP model captures well a wide range of experimental results and field monitoring embankment data. Furthermore, we also observe that the natural clay exhibits the flow rule effect more compared to the reconstituted clay.
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Geosciences 2020, 10, doi: 10.3390/geosciences10060227 www.mdpi.com/journal/geosciences
Article
Non-associated Flow Rule-Based Elasto-Viscoplastic
Model for Clay
Mohammad N. Islam * and Carthigesu T. Gnanendran
School of Engineering and Information Technology, University of New South Wales, Canberra, ACT 2612,
Australia; r.gnanendran@adfa.edu.au
* Correspondence: Mohammad.Islam@netl.doe.gov
Received: 1 May 2020; Accepted: 4 June 2020; Published: date
Abstract: We develop a non-associated flow rule (NAFR)-based elasto-viscoplastic (EVP) model for
isotropic clays. For the model formulation, we introduce the critical state soil mechanics theory
(CSSMT), the bounding surface theory and Perzyna’s overstress theory. The NAFR based EVP
model comprises three surfaces: the potential surface, the reference surface and the loading surface.
Additionally, in the model formulation, assuming the potential surface and the reference surface
are identical, we obtain the associated flow rule-based EVP model. Both EVP models require seven
parameters and five of them are identical to the Modified Cam Clay model. The other two
parameters are the surface shape parameter and the secondary compression index. Moreover, we
introduce the shape parameter in the model formulation to control the surface shape and to account
for the overconsolidation state of clay. Additionally, we incorporate the secondary compression
index to introduce the viscosity of clay. Also, we validate the EVP model performances for the
Shanghai clay, the San Francisco Bay Mud (SFBM) clay and the Kaolin clay. Furthermore, we use
the EVP models to predict the long-term field monitoring measurement of the Nerang Broadbeach
roadway embankment in Australia. From the comparison of model predictions, we find that the
non-associated flow rule EVP model captures well a wide range of experimental results and field
monitoring embankment data. Furthermore, we also observe that the natural clay exhibits the flow
rule effect more compared to the reconstituted clay.
Keywords: elasto-viscoplastic; critical state; bounding surface theory; overstress theory; flow rule;
clay; viscosity
1. Introduction
In a saturated clay medium, the liquid phase occupies the interparticle void spaces of the solid
phase. When such a clay deposit experiences external loading, saturated soil either exhibits slow
loading conditions or fast loading conditions. The first one is also known as the drained behavior,
while the latter one represents undrained behavior. In most cases, the liquid removal of soft clays is
not instantaneous due to its low permeability, compressibility characteristics and the viscous nature
[1,2]. Therefore, depending on the loading state and the in situ condition, in many cases, the
deformation of clay may continue for a long time [3]. For example, in August 1173, the Leaning Tower
of Pisa in Italy was constructed on a highly compressible clay deposit, and it displaced horizontally
to the magnitude of 4.7 m in 1990, which is also increasing 1.5 mm/year [4]. A similar situation also
may happen in any geotechnical structure founded on soft clay (see also Brand and Brenner [2]). In
this regard, the viscosity of clay most often contributes to the long-term time-dependent creep of clay,
and subsequent damage to the structure, which requires billions of dollars in annual maintenance
costs [5].
Additionally, only in the USA, the sum of onshore abandoned wells is about 3 million, and every
year approximately 40,000 new deep boreholes are drilled [6]. The estimated cost using the bentonite
clay-based plug of those abandoned wells is about $160,000 per shallow borehole, which also
Geosciences 2020, 10, 10.3390/geosciences10060227 2 of 29
demonstrates a billion-dollar legacy (see also Islam et al. [6]). Furthermore, in an engineered barrier
system for hazardous wastes (e.g., nuclear waste), the application of bentonite clay is also common
and the clay type barrier requires 100’s of years of monitoring due to the sensitive nature of the
deposited materials [7]. Additionally, it is important to incorporate time-dependent viscous
responses of clay in a constitutive model formulation to obtain realistic hydro-mechanical behavior
[8]. Therefore, clay-based research is the active field of interest and the motivation of the present
paper.
From the early 19
th
century to the present time, to illustrate the clay behavior, a myriad of
coupled constitutive models have been developed, including viscous-inclusive (e.g., Adachi and
Okano [9]) and viscous-exclusive (e.g., Roscoe and Burland [10]) models. However, we limited our
discussion only to the first group of models. In this regard, among others, Liingaard et al. [11] and
Chaboche [12] provided literature reviews. Nevertheless, to avoid the mathematical formulation
complexity, in most cases, the time-dependent constitutive models are limited to the associated flow
rule (AFR). But, capturing the legitimate behavior of soft clay, the non-associated flow rule (NAFR)
is imperative (Zienkiewicz et al. [13]). In the literature, there are a couple of NAFR-based EVP models,
where the number of material parameters ranged between six to 44. In this regard, Islam et al. [14]
also presented a summary of NAFR-based elasto-viscoplastic (EVP) models. It is worth mentioning
that a constitutive model with too many model parameters may capture geomaterials’ behavior very
well, but most often, their practical applications are not convenient [15]. Additionally, for the
engineering application of any EVP model, the model formulation simplicity, its finite element
implementation and the objective determination of model parameters are essential.
In this paper, we develop a non-associated flow rule-based elasto-viscoplastic (EVP) model
considering the Modified Cam Clay (MCC) model [10] framework, Perzyna’s overstressed theory
[16], the Borja and Kavazanjian [17] concept, the bounding surface theory and the mapping rule (see
also Hashiguchi [18]). The EVP model here requires a total of seven parameters, and among them,
five parameters are identical to the MCC model. The other parameters are the secondary compression
index and the surface shape parameter. We also introduce a non-linear secondary compression index
to account for the viscosity of clay. Additionally, the shape of surfaces in this paper is different than
the original MCC model or EVP model with the MCC model equivalent surface (see also Islam et al.
[14]). We consider a non-circular surface in the π-plane. Also, we introduce a composite boundary
surface, and the shape parameter to control the bounding surfaces’ shape.
Furthermore, we also discuss the importance of the non-associated flow rule, the non-circular
shape surface and the composite bounding surface. For validation of the EVP model, we compare
numerical results with a wide variety of laboratory observed test data considering the Shanghai clay,
the San Francisco Bay Mud clay and the Kaolin Clay. Moreover, after validation of the developed
non-associated flow rule EVP model, we also implement it in a coupled finite element solver named
A Finite Element Numerical Algorithm (AFENA) [19]. Additionally, for a field application of the
developed EVP model, we compare the predicted response with the long-term monitoring measured
response of the Nerang Broadbeach Roadway (NBR) embankment in Australia [20].
2. Importance of the Non-Associated Flow Rule
In this paper, we formulate the non-associated flow rule elasto-viscoplastic model considering
the Modified Cam Clay (MCC) model framework, which was formulated considering the associated
flow rule. Therefore, at first, we discuss the importance of the non-associated flow rule in the context
of the MCC model as follows.
In the triaxial space, the incremental strain  is divided into the volumetric 
and the
deviatoric 
component, while each of them is also comprised of the elastic part and the inelastic
part (e.g., plastic or viscoplastic) as follows (see also Yu [15]; Roscoe and Burland [10]):

, (1)

,
(2)
,
(3)
Geosciences 2020, 10, 10.3390/geosciences10060227 3 of 29
where, 
and
are the total volumetric strain and the total deviatoric strain, respectively; while
superscript and represent their corresponding elastic and plastic components. 
and
are
the incremental major strain and the incremental minor principal strain, respectively. Again, for the
1D consolidation by noting
in Equation (3), the ratio of the incremental volumetric strain to
the incremental deviatoric strain is obtained as:
. (4)
Additionally, by neglecting the relative magnitude of the elastic shear strain to the plastic shear
strain, Yu [15] presented Equation (4) as:
. (5)
Also, Yu [15] presented the stress dilatancy relation as follows:
. (6)
In Equation (6),  are the material constants. and are the critical state line (CSL) slope,
and the stress ratio, respectively (see also Yu [15]). Furthermore, for the MCC model, . In
Equation (5),

is a constant and assuming

 (see Schofield and Wroth [21]), the
magnitude of the incremental volumetric plastic strain rate ratio


becomes 1.2. By substituting


 in Equation (6), the stress ratio for the normal compression 

and the
conditions
become 0.40. However,



 (see also Yu [15]). Additionally, McDowell and his co-
workers also proposed that

 (see McDowell and Hau [22]). Therefore, it implies
that in the associated flow rule condition, the predicted

is too low.
Moreover, the yield surface of the MCC model is given by:

, (7)
where,  and
are the differential quantities of  and
respectively. It is to note that
during the undrained triaxial tests, the magnitude of  is a negative quantity for the normally
consolidated clay. Therefore, in the ‘wet’ part when  (see also Schofield and Wroth [21]):

. (8)
As a result, if the yield surface is not permitted to contract, as 
and , the strain-
softening phenomena will not occur.
In addition, when the stress state reaches the yield condition, a material is subjected to the plastic
deformation, which is known as the plastic flow (see also Hashiguchi [18]). The plastic potential
function illustrates the post-yield and the failure behavior of soils, and the plastic strain increment is
normal to it. If the plastic strain is calculated on any surface other than the potential surface as in the
associated flow rule condition, the predicted plastic shear strain is too high (see also Yu [15];
McDowell and Hau [22]).
It is worth mentioning that in the associated flow rule condition, the adopted yield surface over-
estimates failure stresses on the dry side, and the bifurcation is not possible in the hardening regime
(see also Yu [15]; Schofield and Wroth [21]). Additionally, in the associated flow rule (AFR), the yield
surface and the potential surface are the same. From the evidence of the triaxial test results in the
literature, it is observed that in the AFR condition, if the contraction of the yield surface with
hardening is not allowed, the deviatoric strain is suppressed. Therefore, strain hardening behavior in
the drained triaxial shearing entailed that the yield surface shrinkage with the hardening should not
be permitted. Hence, in the associated flow rule condition, the strain hardening in the drained triaxial
test and the strain softening in the undrained tests will not transpire. For this reason, to explain the
plastic volume expansion and the softening behavior of plastically compressible materials, e.g., soft
clay, a plastic potential surface rather than the yield surface is essential (see also Yu [15]).
To resolve the limitations mentioned above in the Modified Cam Clay (MCC) framework, we
introduce the non-associated flow rule-based elasto-viscoplastic (EVP) model. Additionally, the MCC
model is not capable of modeling the long-term viscous behavior of soft clay (see Islam and
Geosciences 2020, 10, 10.3390/geosciences10060227 4 of 29
Gnanendran [23]), which is also a motivation of the EVP model formulation herein. Moreover, it is
noteworthy that the MCC model yield surface (see also Equation (7)) assume the Von-Mises type
circular surface in the π-plane. Thereby, except in the triaxial compression state, in any other loading
state, the MCC model or equivalent EVP model with the circular type surface overestimate the stress
(see also Islam and Gnanendran [23]; Islam et al. [14]). Furthermore, from the literature, comparing
the single surface-based model with the composite surface-based model, it is observed that the first
one underpredicts the soil state in the overconsolidated state compared to the latter one.
Moreover, from experimental evidence, we find two phenomena (see Schofield and Wroth [21]).
The first one is for the overconsolidated clay, and it is evident that with increases in the
overconsolidation ratio (OCR), the strength locus for the overconsolidated clay approach to the zero-
tension line. The second case is for the normally consolidated clay where the strength locus intersects
the critical state line in the mean pressure-deviatoric (p-q) plane. In this paper, to resolve such
shortcomings, we also revise the MCC model’s single yield surface, and we discuss details of them
in the model formulation section.
3. Numerical Modeling
We assume that the porous media: (i) comprises of two phases (the solid phase and the liquid
phase (see also Figure 1), (ii) fully saturated, (iii) obeys the small deformation, (iv) fluids follow
Darcy’s law, (v) supports the isotropic state, the static equilibrium and the isothermal equilibrium
conditions, (vi) individual soil grains and the liquid phase are incompressible. Additionally, we
ignore the geochemical effect, interaction forces and dynamic actions. Thereby, we also assume each
phase density is constant. Furthermore, we consider the soil mechanics’ basic principles (see Terzaghi
[1]). First, deformations of solid-phase originate due to the liquid-phase removal and rearrangement
of the solid’s grain. Second, the total stress is the summation of the load carried by the soil (effective
stress) and the fluid (pore pressure) (see also Schofield and Wroth [21]). Introducing Terzaghi’s
effective stress concept [1], we couple the solid phase and the liquid phase.
Figure 1. Two phases of representative elementary volume.


Geosciences 2020, 10, 10.3390/geosciences10060227 5 of 29
3.1. Governing Equations
Considering the assumptions above, we present governing equations as follows (see also Bear
and Bachmat [24]).
Momentum balance equation:

. (9)
Mass balance equation:

, (10)

,
(11)
where ‘’ is the divergence operator. is the total stress tensor. ,
and
are the total density
of the porous medium, the liquid phase density and the solid phase density. is the gravitational
acceleration acting along the z-axis (see also Figure 1). is the porosity of the porous medium.

is the time derivative.
and
are the solid phase and the liquid phase velocity, respectively.
Additionally,  is also known as the body force
, where ‘Trepresents transpose.
3.2. Constitutive Assumptions
In Equation (9), can be defined as (see Bear and Bachmat [24]):
.
(12)
Here, is the identity tensor.
is the effective stress tensor and
is the liquid pressure.
Additionally, in Equations (10) and (11), we assume
and
are constant. Hence, after the
summation of Equations (10) and (11), we obtain:


.
(13)
For the two-phases porous media, Bear and Bachmat [24] presented the relation between
and
using the relative velocity
term and the relative specific discharge (

) as follows:
,
(14)

,
(15)
,
(16)
where, is the gradient operator.

is the hydraulic conductivity which depends on the fluid
phase or fluidity
and the specific permeability tensor of soil . Additionally,
is the liquid
phase volumetric weight
, which also express as

, where T represents
transpose.
is the liquid phase viscosity and is the gravitational acceleration.
Again, in Equation (13), ‘
’ term is given by (see Bear and Bachmat [24]):

.
(17)
Here,
is the volumetric strain and reads:


.
(18)
Assuming the small deformation, we also obtain the strain-displacement relation as follows:

(19)
where u is the displacement component.
Substituting, Equations (16) to (18) into Equation (13), then rearranging, we obtain:

.
(20)
It is worth mentioning that in the non-associated flow rule EVP model formulation, we assume
that 

(see Equations (17) to (19)) consists of the elastic component and the viscoplastic component
as follows.
3.2.1. Strain Rate Tensor of the EVP Model
The total strain rate

in the non-associated flow rule-based EVP model which is given by
(see also Lubliner [25]):




,
(21)
Geosciences 2020, 10, 10.3390/geosciences10060227 6 of 29
where,

and


are the elastic strain rate tensor and the viscoplastic strain rate tensor,
respectively. We obtain

as follows (see also Lubliner [25]):



,
(22)
where,

is the effective stress tensor.

is the fourth-order compliance tensor and written as:







.
(23)
Here, is the Kronecker delta. is the modulus of elasticity and is the Poisson’s ratio.
Again, assuming Perzyna’s overstressed theory [16], we obtain


in Equation (21) as:



;
,
(24)
where is the rate sensitivity function; is the Macaulay’s bracket and is the overstress
function. Additionally,
,
and
are the potential surface, the loading surface and the reference
surface, which we discuss in the next section. It is worth noting that if
, the geomaterials
behave elastically, while
, similar material will experience the viscoplastic strain. Additionally,
we present the derivation of in Appendix A. Moreover, we also discuss details derivation of


for the finite element implementation in Appendix B.
3.2.2. Bounding Surfaces of the EVP Model
Constitutive models that adopt the classical plasticity theory, such as the MCC model, generally
consider a single yield surface (SYF). The limitations of the SYF can be summarized as follows [15,18]:
(i) The SYF separates the elastic domain from the plastic state and forms an elastic state boundary
within the yield surface. From the comparison of the experimental data and the model predictions, it
is evident that the predicted elastic domain is larger than the observed one.
(ii) For the SYF models, the observed transition from the elastic state to the plastic state is in
contrast to the experimentally observed gradual changes in the stiffness.
(iii) The SYF provides limited scopes to exemplify the plastic modulus in the loading direction.
(iv) The SYF model is usually incapable of capturing the proportional loadings.
During the last couple of decades, limitations of a single-surface model have opened up a more
comprehensive research area. There are several methods to overcome these shortcomings. However,
the two most popular theories are the multi-surface plasticity and the bounding surface model (see
also Hashiguchi [18]; Yu [15]).
In this paper, for any loading history, we consider three bounding surfaces in the EVP model
formulation (see also Equations (25) to (27) and Figure 2). Each surface has two ellipses: ellipse 1 (see
also Kaliakin and Dafalias [26]) and ellipse 2 (see also Kutter and Sathialingam [27]). In the composite
surface, two ellipses of each surface meet at common tangents and allow control of the shape of
surfaces. In Figure 3, we illustrate surfaces where is the slope of the surface at any point on the
potential surface. Additionally, as the deviatoric stress decreases with increases in the mean
effective pressure , the magnitude of is negative (see also Figure 3). To predict the
overconsolidation effect of clay, the surface shape in the wet side’ is higher than the ‘dry side’. In this
paper, assuming the ellipse shape parameter (R) is equal to two, we obtain the extended MCC model
equivalent ellipse shape (see also Islam et al. [14]).
Potential surface,





(25)
Reference surface,




(26)
Loading surface,





(27)
Geosciences 2020, 10, 10.3390/geosciences10060227 7 of 29
In Equations (25) to (27), the suffixes , and represent the potential surface, the reference
surface and the loading surface, respectively.

,

and

are the intersection of the
corresponding surface with the positive mean pressure axis (see also Figure 2). Additionally,

and


are the mean effective normal stress and the deviatoric stress, respectively
(see also Schofield and Wroth [21]). It is worth mentioning that we considered

as the total
mean stress. Also, represents the slope of the critical state line and is defined as (see also Prashant
and Penumadu [28]):


,
(28)
where is the maximum internal friction angle for any specific stress path. b represents
b-value (see also Islam and Gnanendran [23]; Islam et al. [14]). and designate the
triaxial compression and the triaxial extension test, respectively. By changing the b-value, we obtain
any stress path in the present model formulation. In Equations (25) to (27), we revise the expression
of to obtain the realistic surface in the π-plane. Additionally, in Figure 4, we present a comparison
of the MCC surface and proposed modification in the surfaces (see also Figure 2) with the true triaxial
test results on Kaolin clay (see also Prashant and Penumadu [28]). Moreover, Lade [29] presented the
relation between b-value and Lode angle as follows (see also Figure 5):

(29)
In Equation (29), and represent, the Lode angle and the b-value




respectively, while
and
are the major principal stress, the intermediate principal stress
and the minor principal stress, respectively, while
represents their corresponding effective stress
(see also Prashant and Penumadu [28]).
3.2.3. Image Parameters of the EVP Model
Using the triaxial test, we find the reference state for any clay and at any time. Additionally, the
current loading stress state
is associated with its image stress on the reference surface
and the potential surface
through the radial mapping rule (see also Hashiguchi [18]).
Moreover, similar to the Modified Cam Clay (MCC) model, we also assume that the projection center
is in the origin of the stress space (see also Figures 2 and 3). In this regard, Kutter and Sathialingam
[27] reported that separation of the projection center and the stress state origin results in an elastic
nucleus. Also, such a nucleus requires additional model parameters that split up the elastic domain
from the inelastic domain (see also, Hashiguchi [18], Chapter 7). Therefore, ignoring the elastic
nucleus, we obtain the image stress of the loading surface on the reference surface (see also
Hashiguchi [18]) as follows:
.
(30)
Now substituting Equation (30), into Equation (26) for Ellipse 1 () (see also Figure 3):


.
(31)
Similarly, for and Ellipse 2, we find:

.
(32)
Again, for the potential surface, we also find:


.
(33)

.
(34)
Geosciences 2020, 10, 10.3390/geosciences10060227 8 of 29
Figure 2. In the p-q plane, schematic representation of the potential surface, the reference surface and
the loading surface and the surface shape parameter effect.
Figure 3. Meridional section of the potential surface with ellipse 1 and ellipse 2.
D
e
v
i
a
t
o
r
i
c
S
t
r
e
s
s
(
q
)
cl
p
cr
p
cp
p
(
)
,
l l
p q
(
)
,
r r
p q
(
)
,
p p
p q
0
vp
v
d
=
ε
εε
ε
M
η
=
M
η
<
M
η
>
0
vp
v
d
>
ε
εε
ε
,
vp
p v
p
ε
εε
ε
,
vp
p q
q
ε
εε
ε
cp
p
1
tan
vp
v
vp
q
d
d
ψ
ψ
=
ε
εε
ε
ε
εε
ε
0
vp
v
d
<
ε
εε
ε
M
M
M
η
η
η
<
>
=
vp
q
d
ε
εε
ε
vp
q
d
ε
εε
ε
vp
q
d
ε
εε
ε
Geosciences 2020, 10, 10.3390/geosciences10060227 9 of 29
Figure 4. In the π-plane, comparison of the true triaxial test (TTT) results with the original Modified
Cam Clay (MCC) surface and the extended MCC surface (for TTT test results see also Prashant and
Penumadu [28]).
Figure 5. Relations of b-value-Lode Angle and b-value-M (see also Lade
[29]
; Islam and Gnanendran
[23]
; Ye et al.
[30]
).
It is noteworthy that in Equations (31) to (34) for the Modified Cam Clay (MCC) equivalent
surface (R = 2, see also Figure 2), we obtain


and


(see also Islam et al. [14]),
where,

. In Appendix A, we discuss the derivation of

and

(see also Figure
2).
3.3. Finite Element Implementation
3.3.1. Couple Finite Element Formulation
We assume that a soil mass occupies in a domain and its surface is , which is subdivided
into a subdomain and segment of the surface during the discretization of space, as  and ,
respectively. In the following sequences, to obtain a coupled solution for the governing equations
(see Equations (9) to (11), (20)), we use the weak form solution and the Galerkin weighted residual
1
p
σ
2
p
σ
3
p
σ
M
3
cos
θ
( )
3 2
3
22
1 2 3 3 2
321
b b b
cos
b b
θ
+
=
+
2 3
1 3
b
σ σ
σ σ
=
( )
2
6 1
3 2 1
sin b b
M
b sin
φ
φ
+
=+
Geosciences 2020, 10, 10.3390/geosciences10060227 10 of 29
method (see also Zienkiewicz et al. [31]). We use the effective stress relation (see Equation (12)) and
the Darcy’s law (see Equation (16)) to obtain the coupled hydro-mechanical relationship for the
elasto-viscoplastic model. In our model formulation, we have a total of 16 equations. One equation
for the equilibrium relation (see Equation (9)). Three equations for the mass balance relations (see
Equations (10) and (11)). Moreover, we present six equations for the stress-strain relationship (see
Equations (22) and (24)) and six equations for the strain displacement relation (see Equation (19)).
Additionally, we have 16 unknowns (six for the effective stress, six for the strain, three for the
displacement and one for the liquid pressure). Thereby, our coupled solutions represent a “well-
posed” problem definition (see also Bear and Bachmat [24])
Additionally, the element matrices for two phases porous media are given by (see also
Zienkiewicz et al. [31]):
(35)
We also find the global matrix by a sum over the number of elements in the element matrix.
Also, in a simplified form, we re-write Equation (35) as (see also Owen and Hinton [32]):

(36)
where
,
,
,
and
(37)
(38)
(39)
(40)


(41)

(42)
(43)
(44)

(45)
(46)



(47)

(48)
In the above equations,
is the tangential stiffness matrix.
is the coupling matrix.
is
the load vector.
is the flux matrix and
is the fluid conduction matrix. T and T represent
transpose and the traction force, respectively.
is the elastic constitutive matrix related to (see
also Equations (22) and (23)).

is the strain-displacement matrix, where is the tangential
operator.
and
are the displacement shape function and the pore pressure shape function,
respectively.
represents time.
demonstrates the integration parameter (see also Segerlind
[33]). is a mapping vector.
is the gradient matrix (see also Owen and Hinton [32]).
It is worth mentioning that for the isoparametric element, the shape function and the
interpolation function are identical (see Zienkiewicz et al. [31]; Potts and Zdravkovic [34]). This
simplification allows flexibility to consider any arbitrary shape of elements. Additionally, the shape
functions are defined in terms of the local coordinates  and the strain interpolation matrix
requires global derivatives, with respect to the global coordinates . To map both coordinates,
the chain rule can be applied as follows (see also Zienkiewicz et al. [31]):
Geosciences 2020, 10, 10.3390/geosciences10060227 11 of 29



 
 





 (49)
In Equation (49), is the Jacobian matrix.
is the shape function for nodal values, where
represents nodal points of elements. In the literature, there are two types of assumptions to account
for the shape function for the displacement and the pore pressure. In the first case, it is assumed that
both shape functions are identical, while in the second case, different shape functions are adopted for
variables groups. For example, the displacement and the pore pressure variables vary linearly in the
linear triangular element (e.g., three nodes) and the bilinear rectangular element (e.g., four nodes)
(Zienkiewicz et al. [31]; Potts and Zdravkovic [34]). However, in a six nodes triangular element and
an eight nodes rectangular element, the displacement and the pore pressure field change
quadratically. Additionally, when the displacement varies quadratically, the effective stress change
linearly (see also Zienkiewicz et al. [31]). Thereby, there is a variation between the pore pressure and
effective stress. To achieve the same order of variation in the primary variables, for the eight nodes
rectangular element, the degrees of freedom (DOF) for the pore pressure can be obtained at four
corners of the rectangular element. In contrast, for the triangular element, a similar DOF for the pore
pressure is calculated only from the apex of the triangle (see Potts and Zdravkovic [34]). Hence, the
shape function for the displacement and the pore pressure can be separated. Moreover, to account
for the large deformation for the homogeneous porous media (e.g., triaxial creep or relaxation test)
and heterogeneous porous media (e.g., embankment with multiple layers), special attention is
essential to consider the element type (see also Zienkiewicz et al. [31]). For example, to model the
consolidation behavior of porous media comprising sand deposit overlying clay deposit, the sand is
modeled assuming no pore pressure DOF at the nodes, which behaves like drained media or non-
consolidating elements. In contrast, clay deposit is modeled as a consolidating element considering
fluid pressure degrees of freedom at the nodes (see Potts and Zdravkovic [34]). Additionally,
Zienkiewicz et al. [31] demonstrated that when porous media is approaching the undrained limit
state, to satisfy the Babuska-Brezzi convergence condition (see Zienkiewicz et al. [31]; Potts and
Zdravkovic [34]), the shape function for the nodal displacement and the pore pressure need to be
separated. In such a case, the choice of element types is limited, and details can be found in
Zienkiewicz et al. [31]. In any other state than the undrained limit state, element type selection is
extensive in the finite element simulation practice.
In the Results and Discussion Section, we discuss both the drained and the undrained triaxial tests
considering the short term loading and the long-term loading. Additionally, in this paper, for
validation of the triaxial test, we use the first-order three nodes triangular element. Moreover, for the
long-term prediction (e.g., embankment performance estimation), we consider the second-order six
nodes triangular element (see also Zienkiewicz et al. [31]).
3.3.2. Time Integration
For finite element modeling (FEM) of time-dependent porous media (e.g., clay), there are several
approaches to discretize the time domain (see also Segerlind [33])). However, for the FEM solutions,
the -method is the simplest method (see Potts and Zdravkovic [34]) and in this paper we also use
this method. The value of ranged in between 0 to 1 while , 0.5, 1 represent the fully explicit
time integration (also known as the forward difference method), the implicit trapezoidal time
integration (also known as the Crank-Nikolson rule) and the fully implicit time integration (also
known as the backward difference method), respectively (see also Zienkiewicz et al. [31]; Owen and
Hinton [32]; Segerlind [33]; Potts and Zdravkovic [34]).
Again, nodal values of the interpolation polynomial
for variables (e.g., displacement, pore
pressure) at nodal points varies with respect to individual element’s nodal coordinates. For example,
Geosciences 2020, 10, 10.3390/geosciences10060227 12 of 29
for the linear triangular element
has three nodal points. Segerlind [33] presented, Equation (36)
for the interpolation polynomial of the nodal values as follows:
(50)
From Figure 6, for a given function
with time interval 

, using the mean value
theorem, we obtain:

,
(51)
where represents time (see also Figure 6 and Segerlind [33])
Additionally, from Figure 6 and Equation (51), the value of
can be obtained as:
.
(52)
Assuming, the integration parameter


, Equation (52) becomes:
.
(53)
Again, at (see Figure 6), in Equation (50) is given by (see Segerlind [33]):
.
(54)
Substituting Equations (51) to (54), into Equation (50), we obtain:
.
(55)
By changing
with corresponding primary variables,
(see Equation (36)), we obtain
a coupled two-phase solution.
For 0, 0.5 and 1, we also find a simplified form of Equation (55) (see also Segerlind [33]).
Additionally, to demonstrate the effect of  and in solutions, for simplicity, we assume an
ordinary differential equation as follows:
,
,
,
(56)
where and are the total time and time step incremental number, respectively. The exact solution
of Equation (56) is given by:
(57)
Moreover, we also find the solution of Equation (57) assuming the backward Euler or implicit
Euler and the forward Euler or the explicit Euler as follows, respectively:
(58)
(59)
Figure 6. An approximation of
for given time increment ∆t =

.



Geosciences 2020, 10, 10.3390/geosciences10060227 13 of 29
From Equations (58) and (59), we find that for any value , there will be notable oscillation
in the forward Euler or the explicit Euler solution. In Figure 7, we present the effect of the selection
of  and in solutions.
Again, the solution of the coupled Equation (55) for the non-associated flow rule-based EVP
model herein is more complicated compared to Equation (57). However, the solution scheme of
Equation (55) holds a similar degree of convergence challenge to select  and as that evident in
Figure 7. Thereby, for the solution of the hydro-mechanical coupled equation, careful consideration
is essential to choose  for the corresponding -method (see also Owen and Hinton [32]). For our
solution, we assume  and we obtain critical values of  following Potts and Zdravkovic
[34].
Figure 7. Effect of ∆t for the solution using θ-method (a) Explicit and (b) Implicit.
3.3.3. Incremental Stress and Strain
In Equation (55), replacing
with the viscoplastic strain rate


, we obtain:



(60)
Again, using the Taylor series expansion and ignoring the higher-order of


, Equation (60)
is given by (see also Owen and Hinton [32]):


(61)
We presented
in Equation (45). Now, substituting Equation (61) into Equation (60), we
obtain:


(62)
Combining Equations (22) and (24), the incremental stress can be rewritten as (see also Owen
and Hinton [32], p. 272):

(63)
where, the total incremental strain, 

and 
is the incremental displacement, while
is the strain-displacement matrix. Additionally, from Equations (61) and (62), we find

in
Equation (63). Moreover, we obtain from Equation (43) (see also Owen and Hinton [32], p.274).
3.4. Initial and Boundary Conditions
For two-phase coupled porous mediums, we find initial conditions of primary variables as
follows:
in
, (64)
(a) (b).
Geosciences 2020, 10, 10.3390/geosciences10060227 14 of 29
in
(65)
Additionally, the boundary conditions are given by:
on
, (66)
on
(67)
on
, (68)
on
, (69)
where,
and
are the initial displacement and the initial porewater pressure.
represents the
unit normal vector.
denotes the traction force. We have two types of boundary conditions. They
are the Neumann boundary conditions (e.g.,
) and the Dirichlet boundary condition (e.g.,
).
In this paper, for the validation of the non-associated flow rule-based elasto-viscoplastic model,
we use the conventional triaxial test (e.g., cylindrical specimen). We discuss the details of the
validation in the Results and Discussion Section. For validation, we consider different clay samples.
Thereby, the initial conditions of those triaxial samples are also different, which we obtain from the
published literature. Depending on the clay sample, the initial stress state (e.g.,
), the initial pore
pressure
, the loading rate (e.g., stress-controlled or strain-controlled test) and the confining
pressure are different. We assume that in the initial state, all clay samples are fully saturated.
Additionally, we consider an axisymmetric section of the triaxial sample for the EVP model
validation. In Figure 8, we present a representative illustration of the initial and the boundary
condition of the conventional triaxial sample.
Figure 8. The Cambridge stress state in a conventional triaxial test (a) a cylindrical sample, (b) an
axisymmetric section, (c) a finite element mesh and (d) p-q diagram for loading state (see also Lade
[29]
), p. 62).
Geosciences 2020, 10, 10.3390/geosciences10060227 15 of 29
Again, both the drained and the undrained tests consist of two stages. They are the isotropic
consolidation stage and the shearing stage (see also Figure 8d). Additionally, for both tests, the
isotropic consolidation procedure is identical. During the first stage of the triaxial test, on the bottom,
we assume that the horizontal and the vertical displacement components are fixed. Additionally, due
to the symmetry of the axis, we also restrict the horizontal displacement along the radial or x-axis in
the left side boundary. We also apply the confining pressure along the right boundary. In addition,
depending on the stress-controlled or the strain-controlled test, we also introduce the corresponding
value at the top. Moreover, in the initial state, we allow the drainage at the top boundary. Also, we
invoke the “geostatic” stress condition considering the body force of the sample. The “geostatic”
condition ensures the equilibrium state of the sample. Furthermore, we consider the stress ratio along
the horizontal direction to the vertical direction is 1. Then, we run the axisymmetric triaxial sample
for one incremental step, which we assume as the initial state for the shearing stage. The initial state
makes sure that the triaxial sample satisfies the initial yield surface of the elasto-viscoplastic model.
Additionally, in the shearing stage, there are two essential criteria for the drained and the
undrained tests. The first one is the boundary condition, and the latter one is the loading rate. For the
drained test, the top and the bottom boundaries are permeable, while these boundaries are
impermeable for the undrained test. Additionally, for the drained test, the loading rate is slow to
avoid the development of the excess pore water pressure, while in the undrained test, the loading
rate is fast. It is noteworthy that for the drained condition, selection of the time increment is crucial,
and it needs to be small (see also Figure 7). After the selection of the optimal time step for the drained
test, we plot the excess pore water (EPW) pressure with respect to time, confirming that the EPW is
approximately zero. Moreover, in the undrained test, immediately after the incremental load
application, the excess pore water pressure builds up. Therefore, initially, we consider a small-time
step for the undrained test, and then we increase the time step incrementally.
In Figure 8d, we discuss two stress paths for the cylindrical sample. They are the triaxial
compression (TC) and the triaxial extension (TE). After the isotropic consolidation state, we apply
,
and
for corresponding TC ad TE tests. It is worth mentioning that the EVP model presented
in this paper is not limited to any specific geometry shape or the specific stress paths, which are in
most cases, the limitations of many elasto-viscoplastic models in the literature. In the present paper,
by changing the
-value, we will able to obtain any stress path in the stress space. Additionally, we
will able to introduce a specialized stress path too. For example, the constant mean pressure test or
the constant
test (see also Figures 4, 5 and Lade [29]). For such a special loading condition, we
need to revise the corresponding
,
and
with respect to the mean pressure, the deviatoric
pressure and the
-value. From three equations, we obtain three unknowns,
,
and
.
3.5. Model Parameters
In this paper, the non-associated flow rule-based elasto-viscoplastic model requires a total of
seven parameters. They are divided into (i) consolidation parameters, (ii) strength parameter, (iii)
elastic property, (iv) state parameter, (v) creep parameter and (vi) surface shape parameter. The
consolidation properties are the normal consolidation line gradient


and the swelling line
gradient

, where
and
are the compression index and the swelling index, respectively
(see also Figure A1 in Appendix A). The slope of the critical state line
is considered as the
strength parameter and related to the angle of internal friction at failure (see Figures 2, 5 and Equation
(28)). We consider the Poisson’s ratio
as the elastic property. Additionally, at any reference time
and unit mean pressure, the void ratio
is the state parameter (see also Figure A1 in Appendix
A). Moreover, to account for the time-dependent behavior of clay, we introduce the secondary
compression index
as the creep parameter. Finally, we also introduce a surface shape
parameter
to control the shape of the bounding surface (see Figures 2 and 3 and Equations (25)
to (27)). In Table 1, we present a summary of the non-associated flow rule-based elasto-viscoplastic
(EVP) model parameters in this paper and their determination method.
Table 1. Model parameters in the elasto-viscoplastic model.
Geosciences 2020, 10, 10.3390/geosciences10060227 16 of 29
Parameters
Meaning
Method of
determination
Slope of NCL
Triaxial or oedometer test
Slope of SL
Triaxial or oedometer test
Slope of CSL
Triaxial test
Poisson’s ratio
Assumed
Void ratio at
p
= 1 with NCL at
Triaxial or oedometer test
Creep parameter
Triaxial or oe
dometer test
Shape parameter
Undrained triaxial test
Note: NCL = the normal consolidation line, SL = the swelling line, CSL = the critical state line,
is the reference time. Also,

(see also Islam et al. [14]),

It is noteworthy that among seven parameters, five parameters are identical to the Modified Cam
Clay (MCC) model. The other two parameters are
and

Additionally, the MCC model
parameters determination processes are well documented in many soil mechanics textbooks (see also
Schofield and Wroth [21]; Roscoe and Burland [10]). Moreover, there are three approaches available
in the literature to obtain
. The first one is the empirical relation with
(see also Mesri and Castro
[35]) which is assumed constant over time. The second one is a direct calculation from the oedometer
test or the triaxial test. This approach is also divided into the void ratio-based method



and the strain-based method





(see also, Liingaard et al. [11]). Additionally, using
the cone-penetration test field data, we also may obtain
(see also Tonni and Simonini [36]).
Moreover, in the literature, there are two concepts regarding the secondary compression index:
constant or linear function and non-linear function, while in both cases, strong opinions are available.
In this paper, we use the void ratio-based non-linear
which is neither tied to any specific clay nor
requires any fittings parameters (see also Islam and Gnanendran [23]; Islam et al. [14]) and reads:


(70)
where
and
represent the present step and the previous time step, respectively.
Additionally, to obtain the surface shape function
, there are several methods. For example,
Dafalias and Herrmann [37] proposed a fitting method using the undrained triaxial stress path as
follows:





(71)
It is worth mentioning that to predict the pore pressure for the undrained test, we also need the
permeability (see Equation (16)), which is the material parameter of clay and not relevant to the EVP
model parameters. Additionally, in Table 2, we summarize EVP model parameters for different clays.
Table 2. Elasto-viscoplastic (EVP) model parameters for different clays.
Clay
EVP Model Parameters in This Paper
Shanghai clay
[38]
0.22
0.046
1.28
---
0.30
2.2
3
0.016
2.00
SFBM clay
[39]
0.37
0.054
1.40
---
G
3.17
0.053
2.10
Kaolin clay
[40]
0.15
0.018
1.25
0.95
0.30
1.51
0.014
2.50
Note: SFBM = San Francisco Bay Mud clay; G = 23540 kPa (see Kaliakin and Dafalias [26]); M
c
and M
e
represent the triaxial compression and triaxial extension test, respectively.
Geosciences 2020, 10, 10.3390/geosciences10060227 17 of 29
4. Results and Discussion
4.1. Shanghai Clay
From Huang et al. [38], we obtain the Shanghai soft clay properties and model parameters. It is
an undisturbed soft sensitive clay (sensitivity = 4.86). The water content, the liquid limit and the
plastic limit of this clay sample are 51.80%, 44.17% and 22.40%, respectively. Additionally, the specific
gravity of the Shanghai clay is 2.74. Moreover, the clay fraction

, the silt fraction and the
sand fraction of this clay are 26.60%, 63.40% and 10.00%, respectively. Furthermore, Huang et al.
(2011) demonstrated that diameter and length of triaxial samples were 39.10 mm and 80.00 mm,
respectively, which we use for the preparation of the numerical model geometry.
In Figures 9, we compare numerical results with laboratory-measured stress controlled, isotropic
and consolidated undrained triaxial tests (see also Huang et al. [38]). Additionally, we also compare
the EVP model predictions with the Kutter and Sathialingam [27] proposed EVP model. From the
comparison, we find that in the small strain zone and after 14% axial strain, both the EVP models
over predict the deviatoric stress. It is to note that in the EVP model formulation, we ignore the
hysteretic response of the clay (see also Whittle and Kavvadas [41]), which results in the
overprediction in the early stage. Also, in the present model formulation, we did not include the
destructured behavior of clay (see also Liu and Carter [42]). Therefore, we observe a slight
overprediction at the higher axial strain. Inevitably, in the present EVP model formulation, the
incorporation of the hysteretic response and the destructured response require extra model
parameters.
In Figure 9a, we compare the associated flow rule and the non-associated flow rule EVP models
predicted stress-strain responses with the Shanghai clay observed results. We find that for 150 kPa
mean pressure up to 1% axial strain; both models capture experimental responses well. Then, before
the measured peak deviatoric stress, we also observe overprediction. Afterward, under-prediction in
the associated flow rule-based EVP model continues. We also observe similar results for 200 kPa.
Comparing both the flow rule EVP model and the Kutter and Sathialingam [27] model, we also find
that the non-associated flow rule model well captures the experimental results.
Furthermore, we present the stress path responses in Figure 9b. We observe that after attaining
the peak deviatoric stress, the EVP model prediction gradually follows a ‘narrow region’ (see also
Islam et al. [14])) and such a phenomenon is common in the natural clay than the reconstituted clay
(see also Islam and Gnanendran [23]). In general, the MCC model (Roscoe and Burland [10]) and the
associated flow rule EVP models based on the MCC framework (see also Kutter and Sathialingam
[27]) are incapable of capturing the ‘narrow region’. To predict such a behavior, incorporation of the
additional model parameters in the MCC framework are also frequent (see also Liu and Carter [42]).
However, in this paper, we obtained such a ‘narrow region’ of the natural clay without any extra
model parameters.
4.2. San Francisco Bay Mud Clay
From Lacerda [39]; Kaliakin and Dafalias [26], we obtain EVP model parameters for the San
Franciso Bay Mud (SFBM) natural clay. In this section, we compare the EVP models’ predictions with
the experimentally observed undrained triaxial test results for the relaxation test. Additionally, we
demonstrate herein SR-I-5 test data (see also Lacerda [39]). The water content, the liquid limit, the
plastic limit, and the plasticity index of this clay sample are 88–93%, 88.4–90%, 35–44% and 45–55 %,
respectively. Moreover, the specific gravity of the SFBM clay is 2.66–2.75. Also, the isotropic
consolidation pressure of the clay sample is 78.4. Additionally, the initial void ratio and its equivalent
mean pressure are 1.30 and 156.9 kPa, respectively (see also Kaliakin and Dafalias [26]). In Table 3,
we present the shear and the relaxation phase, axial strain, strain rate and the duration of the test.
Geosciences 2020, 10, 10.3390/geosciences10060227 18 of 29
(a)
(b)
Figure 9. Comparison of observed and predicted triaxial compression test results of the Shanghai Clay
(a) deviatoric stress-axial strain and (b) stress path.
In Figure 10, we present a comparison of the measured data, and EVP models predicted
responses. For the EVP models, we also compare the flow rule effect considering the associated flow
rule (AFR) and the non-associated flow rule (NAFR). We observe that the NAFR based EVP model
capture well the experimental response. It is worth mentioning that Islam et al. [14] also find an
identical flow rule effects for the extended Modified Cam Clay equivalent surface.
0 3.5 7 10.5 14 17.5 21
Axial strain,
a
(%)
0
40
80
120
160
D
e
v
i
a
t
o
r
i
c
s
t
r
e
s
s
,
q
(
k
P
a
)
Triaxial Compression
Observed_150 kPa
Non associated flow rule_150 kPa
Associated flow rule_150 kPa
Kutter and Sathialingam model (1992)_150 kPa
Observed_200 kPa
Non associated flow rule_200 kPa
Associated flow rule_200 kPa
Kutter and Sathialingam model (1992)_200 kPa
Undrained triaxial test observed results
Shanghai clay (Huang et al. 2011)
Geosciences 2020, 10, 10.3390/geosciences10060227 19 of 29
(a)
(b)
Figure 10. Comparison of observed and predicted stress relaxation triaxial test results of the San
Figure 14. also find identical flow rule effects for the extended Modified Cam Clay equivalent surface.
0246
Axial Strain, a(%)
0
20
40
60
80
San Francisco Bay Mud (Lacerda 1976)
Observed
Non associated flow rule
Associated flow rule
Kutter and Sathialingam model (1992)
Geosciences 2020, 10, 10.3390/geosciences10060227 20 of 29
Table 3. Stress relaxation test on the San Francisco Bay Mud clay.
Phase
1
2
3
4
5
6
7
8
Test
Shear
*
Relax.
Shear
*
Relax.
Shear
*
Relax.
Shear
*
Relax.
1.5
0
1.5
0
0.0162
0
0.00081
0
0
0.38
0.38
0.38
2.3
2.3
2.3
3.94
3.94
3.94
5.3
5.3
Time
0.25
3070
1.28
1320
101.24
2700
1679
8370
Note: Time = minutes,
*
Relax. = Relaxation,

In Figure 11, we present a comparison of the EVP models’ predicted response with the observed
results of the drained triaxial compression test on the San Francisco Bay Mud clay. The triaxial sample
was isotropically consolidated to the confining pressure 156.9 kPa. Additionally, the axial strain rate
was 0.0031%/minute. From Figure 11, we observe that the non-associated flow rule EVP model well
captured the experimentally observed results.
4.3. Kaolin Clay
In this section, we present the over consolidation ratio effect of the Kaolin clay (see also
Herrmann et al. [40]) for the undrained triaxial compression and extension tests. It is a reconstituted
clay and comprised of Snow-Cal 50 Kaolin and 5% bentonite mixture. The specific gravity of this clay
is 2.64. The liquid limit and the plasticity index of this clay are 47.0% and 20.0%, respectively (see also
Herrmann et al. [40]).
The initial isotropic consolidation pressure of the triaxial Kaolin clay sample was 392.2, and
corresponding the initial void ratio for the unit over consolidation ratio (OCR) was 0.613 (see also
Islam and Gnanendran [23]; Islam et al. [14]). We calculate the confining pressure for different OCR’S
considering the initial state condition for both the triaxial compression and extension tests.
From Figure 12 and for OCR = 1, we observe that the non-associated flow rule-based EVP model
prediction of the stress-strain response is satisfactory before attaining the peak. Then, after 14.0%
axial strain, the under-prediction in the non-associated flow rule-based EVP model is 1.05%.
Additionally, such a magnitude comparing with experimental results for the associated flow rule are
3.5% and 1.40%. We also observe a similar prediction for the undrained triaxial compression with
OCR = 2, 4 and 6 for the stress-strain responses. For the triaxial extension test, we present a similar
comparison for OCR = 1 and 2. From such a contrast of EVP models with the measured experimental
results illustrate the effect of flow rule in the EVP models predictions.
Additionally, for OCR = 1 and 2, in Figures 12 b and 12 c, we present the pore pressure response
and the stress path response, respectively considering the triaxial compression and the triaxial
extension test. We also find that the non-associated flow rule prediction is close to the experimental
responses.
Geosciences 2020, 10, 10.3390/geosciences10060227 21 of 29
(a)
(b)
Figure 11. Comparison of the measured and the predicted consolidated drained triaxial compression
tests on the San Francisco Bay Mud clay: (a) the deviatoric stress vs. the axial strain and (b) the
volumetric strain vs. the axial strain.
0 5 10 15 20 25 30
Axial strain ( a,%)
0
50
100
150
200
250
San Francisco Bay Mud (Lacerda 1976)
Observed
Non associated flow rule
Associated flow rule
Geosciences 2020, 10, 10.3390/geosciences10060227 22 of 29
(
a
)
(
b
)
D
e
v
i
a
t
o
r
i
c
S
t
r
e
s
s
,
q
(
k
P
a
)
Geosciences 2020, 10, 10.3390/geosciences10060227 23 of 29
(c)
Figure 12. Comparison of observed and predicted triaxial compression test results of Kaolin Clay (a)
Deviatoric stress-axial strain and (b) stress path and (c) pore pressure.
5. Application of the EVP Models
We use the EVP models in this paper to capture the long-term behavior of the Nerang-
Broadbeach Roadway (NBR) embankment in Australia. Islam et al. [20] presented details of
geotechnical properties, subsurface and geology, the geometry of the NBR embankment sections,
measured instrumentation data for the settlement plates and piezometers, methodologies to
determine the model parameters and laboratory, as well as field measured values. In this paper, we
compare 590 days measured data of a surcharged-preloading section with the EVP models’ predicted
response considering the associated and the non-associated flow rule to illustrate the flow rule effect.
Additionally, we also compare the Modified Cam Clay (MCC) predicted results with the EVP models’
response to demonstrate the viscous response of the clay.
We present the geometry of the finite element section for the NBR embankment in Figure 13.
Additionally, to reduce the boundary effect in numerical simulations, we extend the width and depth
of the embankment. Moreover, to avoid the instabilities in the finite element simulations, we
implement surcharge preloading incrementally. Furthermore, 3.0 m preloading was applied in 370
days. Then, 1.0 m additional surcharged load added and monitored for 220 days. We demonstrate
the model parameter in Table 4. Moreover, in Figure 14, we present a comparison of the observed
and FE predicted responses. We observe that the non-associated flow rule model well captured the
measured settlement plate response compared to the associated flow rule model. Additionally, from
the comparison of the MCC model with the EVP models, it is evident that the MCC model
underpredicts the long-term behavior of the embankment. Such underprediction in the MCC model
developed due to the exclusion of the viscous behavior of clays. Additionally, we also observe that
for the long-term monitoring of EVP models, the flow rule effect is significant. We also noticed a
similar response in the Results and Discussion Section for natural clays.
Geosciences 2020, 10, 10.3390/geosciences10060227 24 of 29
Figure 13. Finite element geometry of the Nerang Broadbeach Roadway embankment.
Table 4. Model parameters for the Nerang Broadbeach Roadway embankment.
Soil Layer
(kPa)
Fill
000 kPa,
30
0
,
5.0 kPa
---
Silty sand
5,000 kPa,
35
0
,
2.5 kPa
---
Loose sand
7,000 kPa,
33
0
,
1.5 kPa
---
Silty clay 1
1.28
0.36
0.060
2.10
159.52
0.029
2.10
Silty clay 2
1.25
0.42
0.043
3.73
105.36
0.033
2.10
Silty clay 3 1.20 0.29 0.030 2.61 132.20 0.023 2.10
Sand lense
,000 kPa,
35
0
,
5.0 kPa
---
Silty clay 3
1.20
0.29
0.030
2.61
287.18
0.023
2.10
Bedrock

,000 kPa,
36
0
,
50.0 kPa
---
Figure 14. Comparison of observed and predicted settlement response of the Nerang Broadbeach
Roadway embankment’s surcharge-preloading.
6. Conclusions
In this paper, we presented a non-associated flow rule (NAFR)-based three surfaces elasto-
viscoplastic model. Assuming the potential surface and the reference surface are identical, the NAFR
S
e
t
t
l
e
m
e
n
t
(
m
m
)
Geosciences 2020, 10, 10.3390/geosciences10060227 25 of 29
model is also reduced to the two-surfaces associated flow rule mode. Both EVP models’ surfaces
comprises two ellipses, while the surface shape parameter
controls each surface shape. Again,
considering
, we also find the extended Modified Cam Clay equivalent surface shape.
Additionally, to obtain a realistic non-circular surface in the
-plane, we also included the
-value
to define the slope of the critical state line. Thereby, by changing the
-value, any stress path can also
be attained.
It is worth mentioning that EVP models in this paper require only seven parameters. Also, all
model parameters are well known in textbooks, and all of them can be deduced from simple
laboratory tests. Additionally, in the present models' formulations, we did not consider any fittings
parameters. It is to note that EVP models in this paper are formulated for the isotropic clay, and any
other state additional model parameters with associated modification are essential. Moreover, EVP
models herein are not capable of capturing the structured behavior of highly sensitive clay, which
require further changes with extra model parameters. Also, the present model may not be the best fit
for the dynamic performance of clay. Furthermore, the EVP model herein is limited to the Darcy type
and immiscible fluid flow.
For validation of EVP models, we considered a wide range of triaxial tests for natural and
reconstituted clays. From comparisons of observed and predicted responses, we found that the non-
associated flow rule EVP model well captured experimental results. Also, we found that the flow rule
effect is noticeable for the natural clays than the reconstituted clay. Additionally, we implemented
EVP models in a coupled consolidated finite element solver. Then, we used EVP models and the
Modified Cam Clay (MCC) model to predict long-term monitoring data of the Nerang Broadbeach
Roadway embankment. We also found that the non-associated flow rule EVP model well captured
the settlement plate measured response compared to the associated flow rule EVP model and the
MCC model.
Author Contributions: Conceptualization, M.N.I & C.T.G.; Investigation, M.N.I.; Methodology, M.N.I.;
Supervision, C.T.G.; Validation, M.N.I.; Writing original draft, M.N.I.; Writing – review & editing, M.N.I &
C.T.G.
Funding: This research received no external funding.
Acknowledgments: The first author was financially supported during his research at the University of New
South Wales, Canberra, Australia.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A: Derivation of
Islam and Gnanendran [23] presented a derivation of
for the associated flow rule-based EVP
model. In this paper, we demonstrated the derivation of
for the non-associated flow rule
considering the composite surface obtaining from the undrained triaxial compression test.
Additionally, Islam et al. [14] also illustrated
for the Modified Cam Clay equivalent single surface
model. Also, we discussed the limitations of the single surface over the composite surface base model
in Section Bounding surfaces of the EVP model.
Considering the consolidation test of clay, we present the void ratio and the natural logarithm
of mean pressure space relation (see Schofield and Wroth [21]) in Figure A1. Additionally, we obtain
the change of void ratio with respect to time (after Schofield and Wroth [21] and considering viscosity
(Islam and Gnanendran [23])



(A1)
In Equation (A1),
represents the arbitrary time, which is not a model parameter. Following
Borja and Kavazanjian [17], we introduced
in the model formulation (see also Kutter and
Sathialingam [27]). It is worth mentioning that in our model formulation, the
-line (see also Figure
A1) at the reference time represents the initial bounding surface (IBS) while due to creep at time
increment (∆t), the
-line will gyrate the IBS and a new bounding surface will generate.
Geosciences 2020, 10, 10.3390/geosciences10060227 26 of 29
Again, considering the viscoplasticity, we obtain the volumetric component of the viscoplastic
strain rate

as:



(A2)
Figure A1. Undrained triaxial test path for the virgin consolidated clay for the normal consolidation
line and the swelling consolidation line representing the initial surfaces.
From Figure A1, we also find (see also Roscoe and Burland [10]):



(A3)
Now, combining Equations (A1) to (A3), we find:



.
(A4)
In the triaxial stress space,

also can be written as:
.
(A5)
By comparing Equations (A4) and (A5), then separating
for calculation of the viscoplastic
strain component results in the flow rule independent elasto-viscoplastic model, which violates the
flow rule theory (see also Islam and Gnanendran [23]). Therefore, for the non-associated flow rule, to
obtain
, we find


considering the one-dimensional compression test criterion for relevant flow
rule as follows:



(A6)

(A7)
9

9
4

(A8)
Again, from Figure A1, we also find expression for

and

(see also Islam et al. [14]):



(A9)



(A10)
line
λ
line
κ
t
e
e
ln
p
e
cl
p
cr
p
cp
p
1
p
=
p
N
e
A
B
C
D
D
C
Geosciences 2020, 10, 10.3390/geosciences10060227 27 of 29
Appendix B: Derivation of


In Equation (24), we defined






while in Appendix A presented expression of

. Again, using the chain rule, we find



as follows:




(A11)

(A12)
(A13)
In Islam and Gnanendran [23]; Islam et al. [14] we presented



;



;
