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基于非平衡热力学的超弹性塑性耦合土体模型及其应用
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内容概要：本文基于热力学原理提出了一种适用于粘结和非粘结土壤的超弹性和塑性耦合模型。重点在于探讨了超弹性势函数的概念并将其与塑性现象进行统一，涵盖了压力、密度依赖性、由应力诱导的各向异性和粘结效应以及它们对塑性的影响。这一新模型可以很好地模拟不同的土壤状态界限和破坏特性，并自然地从超弹性的角度预测土壤的状态边界表面而不需要额外定义或参数。 适用人群：岩土工程专家，土工材料科研工作者，机械工程设计师。 使用场景及目标：适用于土木工程建设中的不同饱和土质建模，特别关注非线性材料性质、不可逆形变特性的模拟及粘结强度的变化，为解释土壤在单轴和循环剪切条件下的关键性能行为提供科学依据。 其他说明：研究成果验证了模型在无粘结土壤到粘结土壤转变期间保持良好的准确度，在模拟小应变下剪切模量降低等复杂地质材料行为方面展现了潜力。
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International Journal of Plasticity 137 (2021) 102902
Available online 9 November 2020
07496419/© 2020 Elsevier Ltd. All rights reserved.
A thermodynamicsbased hyperelasticplastic coupled model
unied for unbonded and bonded soils
Zhichao Zhang
a
,
b
,
*
, Linhang Li
a
,
b
, Zhenglong Xu
a
,
b
a
Chongqing University, School of Civil Engineering, Chongqing, 400045, China
b
Key Laboratory of New Technology for Construction of City in Mountain Area (Chongqing University), Ministry of Education, Chongqing, 400045,
China
ARTICLE INFO
Keywords:
Elasticplastic coupling
Thermodynamics
Cohesion degradation
Stressinduced anisotropy
ABSTRACT
A hyperelasticplastic coupled constitutive model unied for bonded and unbonded soils is
developed in this paper based on thermodynamics. An elastic potential function applicable for
different kinds of soils is proposed to derive a hyperelastic model accounting for the pressure and
density dependency, the stressinduced anisotropy and the bonding effects as well as their
couplings with plasticity. From the perspective of elastic stability, state boundary and failure
surfaces of different soils can be naturally predicted by the hyperelasticity without any additional
denitions and parameters. Based on the classical nonequilibrium thermodynamics, novel plastic
constitutive relations are derived and naturally coupled with the hyperelasticity. As a result,
elastoplastic coupling features such as the dissipative history effect on elastic stiffness, the cyclic
shear behavior, the degradation of shear modulus under small strain conditions, the stress
induced anisotropy of plastic behavior and the cohesion degradation can be reproduced. The
model is well validated by predicting the undrained/drained monotonic and cyclic shear behavior
of unbonded and bonded sands, providing useful insights into their critical state behavior, irre

versible sheardilation/contraction and effects of bonding and cohesion degradation. It is also
shown that the cohesion degradation in different shearing stages to a large extent determines both
the monotonic and cyclic behavior of bonded soils.
1. Introduction
In the elds of geosciences and geotechnical engineering, it is important to study the nonlinear elasticplastic coupling behavior of
geomaterials such as granular soils, clays and rocks. Experimental results of small strain and bender element tests show that the elastic
moduli of soils are a function of both the conning pressure and the void ratio (Man et al., 2010; Gu et al., 2013). Laboratory study by
Giang et al. (2017) indicated that the smallstrain shear modulus of calcareous sand could also be signicantly inuenced by the
particle shape and gradation. State dependent elastic properties of articially bonded soils are also studied by Lee et al. (2011),
Morozov and Deng (2018), Nasi (2019) and so on. On the other hand, such statedependent elastic properties can be impacted by the
plasticity of soils and vice versa (Lashkari and Golchin, 2014) so that the coupling between elasticity and plasticity becomes essential
for soils. For example, Ezaoui and Benedetto (2009) measured the elastic stiffness evolution of Hostun sand, indicating signicant
couplings between the elastic stiffness and the plastic strain histories. More recently, experiment results by Khosravi et al. (2018)
* Corresponding author. Chongqing University, School of Civil Engineering, Chongqing, 400045, China.
Email addresses: zczhang15@cqu.edu.cn, zczhang15@cqu.edu.cn (Z. Zhang).
Contents lists available at ScienceDirect
International Journal of Plasticity
journal homepage: http://www.elsevier.com/locate/ijplas
https://doi.org/10.1016/j.ijplas.2020.102902
Received 15 June 2020; Received in revised form 4 November 2020; Accepted 4 November 2020
International Journal of Plasticity 137 (2021) 102902
2
showed the impacts of plastic compression on the smallstrain shear modulus of unsaturated soils. Zhuang et al. (2018) and Ueno et al.
(2019) reported the irreversible shear modulus variations of sands in response to undrained cyclic shearing, consolidation and
reconsolidation. The elasticplastic coupling of granular matters is also studied using DEM approach by Kuhn and Daouadji (2018).
However, even with clear proofs reported in laboratory tests, the elastoplastic coupling is usually ignored in the constitutive modeling
of soils.
Nonlinear elastic constitutive relations considering the effects of dissipative histories are essential for the modeling of elastoplastic
coupling behavior. One of the approaches for describing the nonlinear elasticity is to dene the nonlinear stressstrain relation directly,
leading to the Cauchy elasticity or hypoelasticity which is still widely used (Marinelli et al., 2018; Zhou and Ng, 2018). However, such
a kind of nonlinear elasticity may violate the laws of thermodynamics by generating abundant energy in a closed stress loop. This
problem is avoided in hyperelasticity that employs the concept of elastic potential or free energy function (Humrickhouse et al., 2010).
Representative hyperelastic models for soils include the ones for granular soils with stress ratio limits, e.g. models by Einav and Puzrin
(2004) and by Jiang and Liu (2007; 2009), and the ones without restrictions on the stress states, e.g. the model by Houlsby et al.
(2005). More recently, Xiao et al. (2020) proposed a hyperelastic model for granular solids to predict both the inherent and
stressinduced anisotropy of elastic properties under different stress conditions. Similar models considering the anisotropic hyper

elasticity of soils can also be found in Bennett et al. (2019) and Amorosi et al. (2020). However, these models are not applicable for
bonded soils.
The elastoplastic couplings are usually considered by coupling plasticity with the abovementioned nonlinear hypoelasticity or
hyperelasticity. Lashkari (2010) and Lashkari and Golchin (2014) proposed elasticplastic coupling models for sands, in which the
power coefcient for the relation between elastic shear modulus and conning pressure is dened as a function of a plastic hardening
parameter. More recently, Bennett et al. (2019) proposed a hyperelasticplastic model with a large strain elastic potential function
taking into account the fabric tensor of geomaterial. Similar efforts on the hyperelasticplastic modeling of soft matters subjected to
cyclic loadings with large strain are presented by Zhang and Montans (2019). Amorosi et al. (2020) also pioneered an elasticplastic
model based on the hyperelastic model proposed by Houlsby et al. (2005), incorporating the fabric tensor and the plastic work into the
free energy function to better describe the anisotropic elasticplastic coupling. However, there are still controversies on whether
process variables like plastic strains can be regarded as state variables in thermodynamics.
Moreover, most models discussed above cannot account for the distinctions of elasticplastic couplings between different geo

materials. For the naturally or articially bonded soils, the cohesion degradation should be considered according to the experimental
study by MendozaUlloa et al. (2020). The bonding effects and the cohesion degradation in bonded soils will not only affect their elastic
moduli and plastic deformation, but also lead to a cohesiondependent nonlinear strength criterion in stress space (Consoli et al., 2010;
Maghous et al., 2012). Based on these concepts, constitutive models of bonded soils are usually developed within the framework of
critical state model. Nguyen et al. (2014) proposed a constitutive model accounting for the effects of cohesion degradation on the
monotonic shear behavior of cemented clays. Similar models are also developed by Montoya and Dejong (2015) and Porcino and
Marcian
`
o (2017) for bonded sands. Rahimi et al. (2018) further incorporated the cementation effect into the bounding surface
plasticity to model the monotonic and cyclic behavior of loosely cemented sands and weak rocks. More recently, DEM simulations on
the elastic behavior of cementbonded sands were studied by Theocharis et al. (2020), which showed signicant dependency on
cementation morphologies and coordination numbers. However, the existing models for bonded soils seldom employ the plasticity
coupled with nonlinear hyperelasticity.
Furthermore, in most elasticplastic models for unbonded and bonded soils, the state boundary are dened by denitions of yield
surface or bounding surface that require additional equations as well as certain mapping rules for the plastic ow. For example, Lee
et al. (2017) proposed a yield criterion accounting for anisotropic hardening by the coupling of quadratic and nonquadratic yield
functions. However, It is worth noting that the state boundary of geomaterials can be intrinsically linked with the nonlinear hyper

elasticity (Zhang, 2018; Xiao et al., 2020), which should be one of the main sources of elasticplastic coupling. This fact is ignored by
most existing elastoplastic models in which the state boundary or other similar concepts are completely independent of the elastic
behavior. Thus, the main purpose of this paper is thus to develop a nonlinear hyperelasticplastic coupled model unied for different
soils covering the abovementioned mechanical behavior. Novel thermodynamicsbased plastic constitutive relations without the
concepts of yield surface and plastic potential are derived and then naturally coupled with a nonlinear hyperelasticity that considers
the cohesion degradation and predicts the state boundary of different soils theoretically. As a result, both monotonic and cyclic
behavior of bonded and unbonded soils can be predicted within a unied theory.
2. Theory development
In this Section, the nonequilibrium thermodynamics of saturated soils will be rst presented to describe both the reversible and
irreversible energy processes of different soils, based on which the nonlinear hyperelastic and plastic constitutive relations are derived
from thermodynamic principles and coupled with each other.
2.1. Thermodynamics of saturated bonded and unbonded soils
Saturated soils in this paper are classied into three categories including granular soils, unbonded clayey soils and bonded soils.
Only the bonded soils, e.g. articially bonded sands/clays and naturally structured clays, are considered to be with true cohesion
induced by the bond between soil particles (Xiao et al., 2017). Dependent on the overconsolidation state and the stress path to failure,
the state boundary line (also the failure line) of such saturated unbonded clays may be nonlinear in effective stress space. However, it
Z. Zhang et al.
International Journal of Plasticity 137 (2021) 102902
3
should be noted that there is no true cohesion in most saturated clays, and thus the shear strengths of them should completely vanish
under a zero conning pressure. This feature is considered in most clay models (e.g. CamClay model) with a yield surface passing
through the origin in stress space (Xiao et al., 2017). On the contrary, a tensile effective stress region is allowed for bonded geo

materials with a curved stress boundary line (Consoli et al., 2010). Moreover, the bonding in soils and the shearinduced degradation of
cohesion will change both the elastic and plastic properties of soils signicantly. The cohesion degradation will also lead to the release
of a certain portion of the elastic energy stored within soil minerals. Therefore, the bondinginduced cohesion should be considered in
the thermodynamics developed below for bonded soils. To this end, a cohesion index, denoted as c, is dened here to quantify the
current bonding degree in soils and its effects on the energy evolutions. For simplication, the inherent anisotropy of soils is not
considered here.
2.1.1. State equations for reversible thermodynamic processes
A set of independent state variables is rst needed to describe the thermodynamic potentials such as elastic energy, pressure
potential and thermal potential. For saturated soils, the elastic strain
ε
e
ij
, the cohesion index c, the dry density
ρ
, the intrinsic pore water
density
ρ
w
and the entropy density s can be dened as the independent thermodynamic state variables. According to the thermody
namics, there should be complete differential relations between the internal energy and the independent thermodynamic state vari
ables as well as their conjugate state variables, denoted as
π
ij
, D
c
,
μ
s
,
μ
w
and T, respectively. Therefore, for a soil element without heat
and mass exchanges with external environment, the innitesimal increment of internal energy can be expressed by
d
ω
=
π
ij
d
ε
e
ij
+ D
c
dc +
μ
s
d
ρ
+
μ
w
d
ρ
w
+ Tds (1)
where
ω
is the internal energy density; the Cauchy stress
π
ij
is called elastic stress as a major part of the effective stress (Zhang and
Cheng, 2014); D
c
is an internal stress forming within the bonding areas between soil particles;
μ
s
and
μ
w
are associated with the
pressure potentials of solid and water phases under the pressure of pore water; and T is the temperature. Eq. (1) can also be trans

formed into the following form in terms of derivatives with respect to time:
d
t
ω
=
π
ij
d
t
ε
e
ij
+ D
c
d
t
c +
μ
s
d
t
ρ
+
μ
w
d
t
ρ
w
+ Td
t
s (2)
where d
t
=
∂
t
+ v
i
∇
i
is the material derivative with respect to the solid phase with a velocity denoted as v
i
.
On the righthand side of Eq. (1) or Eq. (2), the rst to third terms represent the elastic potential increments stored or released due
to the straining, the cohesion degradation and the change in dry density, respectively; the fourth term represents the increment of
pressure potential stored in the volumetric deformation of pore water compressed under the pore water pressure; and the last term just
represents the increment of thermal energy. Correspondingly, for saturated soils,
ω
includes the elastic potential, the thermal potential
and the pore water pressure potential, i.e.
ω
=
ω
s
+
ω
w
,
ω
s
=
ω
e
+
ω
s
T
+
ω
s
p
,
ω
w
=
ω
w
T
+
ω
w
p
(3ac)
where
ω
e
is the elastic potential density;
ω
s
and
ω
w
are the internal energy densities of solid and water phases, respectively;
ω
s
T
and
ω
s
p
are the thermal energy density and the pore water pressure potential density of solid phase, respectively;
ω
w
T
and
ω
w
p
are those of the
pore water phase.
The elastic potential density
ω
e
is a function of the elastic strain tensor. It had been widely observed that the elastic stiffness of soils
is also a function of the dry density and the cohesion. Therefore,
ω
e
should depend on
ε
e
ij
,
ρ
and c, and thus
ω
e
=
ω
e
(
ε
e
ij
,
ρ
, c). The
functional form of
ω
e
will be developed in Sect. 2.2. The thermal energy densities can be simply dened by
ω
s
T
=
ρ
C
s
T,
ω
w
T
=
ρ
w
1 −
ρ
ρ
C
w
T (4)
where C
s
and C
w
are the specic thermal capacities of mineral solid and pore water, respectively; and
ρ
is the intrinsic density of
mineral solid. Ignoring the compressibility of soil particles (
ω
s
p
= 0), the pressure potential density induced by the pore water pressure
can be dened as
ω
w
p
=
1
2
1 −
ρ
ρ
c
w
u
2
, u =
1
c
w
ln
ρ
w
ρ
w0
(5ab)
where u is the pore water pressure; c
w
is the intrinsic compressibility of pore water;
ρ
w0
is the intrinsic density of pore water at 1atm
pressure.
From Eq. (1), the elastic stress and the cohesionrelated internal stress are
π
ij
=
∂ω
∂ε
e
ij
=
∂ω
e
∂ε
e
ij
, D
c
=
∂ω
∂
c
=
∂ω
e
∂
c
(6ab)
It will be clear that, strictly speaking, the hyperelastic relation Eq. (5a) gives the major part but not the whole of effective stress for
soils. Similarly, the thermodynamic conjugate variables
μ
s
and
μ
w
are derived from Eqs. (1 and 35) as follows:
Z. Zhang et al.
International Journal of Plasticity 137 (2021) 102902
4
μ
s
=
∂ω
∂ρ
=
∂ω
e
∂ρ
+ C
s
T −
ρ
w
ρ
C
w
T −
1
2
ρ
c
w
u
2
(7a)
μ
w
=
∂ω
∂ρ
w
=
1 −
ρ
ρ
u
ρ
w
+C
w
T
(7b)
μ
s
and
μ
w
are thus related to the water pore pressure and will be used to get the effective stress principle of saturated soils in Sect. 2.3.2.
2.1.2. Irreversible thermodynamics and energy dissipations
Under isothermal conditions, the elastic stress
π
ij
, the cohesionrelated internal stress D
c
and the deformation rate of soil skeleton
are the main dissipative forces triggering the deviation of soils from thermodynamic equilibrium state. Accordingly, ignoring the heat
and mass exchanges with external environment, the energy dissipation of a soil element induced by irreversible deformation, cohesion
degradation and viscosity can be described by
Td
t
s =R = f
ij
π
ij
+ f
c
D
c
+
σ
v
ij
v
ij
≥ 0 (8)
where v
ij
= − (v
i,j
+v
j,i
)/2 is the deformation rate (or strain rate under small strain conditions); R is the energy dissipation rate; f
ij
, f
c
and
σ
v
ij
are the dissipative ows for dissipations induced by irreversible deformation, cohesion degradation and viscosity. Eq. (8) just gives
the expression for the last term on the righthand side of Eq. (2).
σ
v
ij
is also known as the viscous stress of soil skeleton, and only the
viscosity of solid phase is considered here. Different from ideal solids, the behavior of saturated soils can be transited from solidlike
states to uidlike states when they lose the resistance to shearing (e.g., in sand liquefaction). In such a case, dissipative ows f
ij
and f
c
will vanish and the viscosity will be dominant. From the thermodynamic perspective, it depends on the coupling between viscosity and
other dissipations.
According to the nonequilibrium thermodynamics (Groot and Mazur, 2013), dissipative ows can be dened as a function of the
corresponding dissipative forces or their conjugate state variables. Further considering the coupling between f
ij
and
σ
v
ij
, one can dened
that
f
ij
=m
ijkl
π
kl
+ m
v
ijkl
v
kl
,
σ
v
ij
= − m
v
ijkl
π
kl
+
η
v
ijkl
v
kl
(9ab)
where m
ijkl
, m
v
ijkl
and
η
v
ijkl
are symmetric and positivedenite migration tensors.
η
v
ijkl
is also the viscosity tensor of soils. Eq. (9) is just the
application of Onsager’s reciprocity relation (Groot and Mazur, 2013; Moyne and Murad, 2006) in soil mechanics. It will be clear
below that when the soil deforms like a uid, the term m
v
ijkl
π
kl
→
π
ij
(see Sect. 2.3.2). To this end, one may simply dene that
m
v
ijkl
=ϑ
δ
ik
δ
jl
+ δ
il
δ
jk
2
(10)
where ϑ is ranged from 0 (for ideal solidlike states) to 1 (for uidlike states). From the microscopic point of view, ϑ should depend on
the evolution of soil particle conguration and thus can be dened as a kind of conguration index.
Noting that the nonlinear hyperelasticity is dened in elastic strain space, it is more convenient to dene m
ijkl
π
kl
in Eq. (9a) in terms
of elastic strains. Denoting
π
kl
= C
s
klst
ε
e
st
and dening λ
ijst
= m
ijkl
C
s
klst
, where C
s
klst
is the secant elastic stiffness tensor, we have
f
0
ij
=m
ijkl
π
kl
= λ
ijst
ε
e
st
(11)
A simple application of Eq. (11) is to divide f
0
ij
into a volumetric dissipative ow and a shear dissipative ow. Further considering
the strain rate dependency and the coupling between volumetric and shear dissipative ows, this reads
f
0
v
δ
ij
f
*
ij
=Θ
3λ
v
δ
ij
δ
st
−
βλ
v
ε
e
v
e
e
ij
δ
st
+ e
e
st
δ
ij
γλ
v
ε
e
v
e
e
ij
δ
st
+ e
e
st
δ
ij
λ
s
δ
ij
δ
st
ε
e
v
3
δ
st
e
e
st
(12)
Θ =
v
2
kk
+ v
*
ij
v
*
ij
(13)
where f
0
v
= f
0
kk
and f
*
ij
= f
ij
− f
0
v
δ
ij
/3; v
*
ij
= v
ij
− v
kk
δ
ij
/3; λ
v
and λ
s
are volumetric and shear dissipative coefcients, respectively; γ and β
are coefcients determining the volumetricshear coupling through irreversible sheardilation. Eq. (12) corresponds to a denition of
λ
ijst
for Eq. (11) as follow:
λ
ijst
=
λ
v
1 − β
ε
e
s
ε
e
v
2
−
λ
s
+ γλ
v
3
Θδ
ij
δ
st
+
λ
s
+ γλ
v
2
Θ
δ
is
δ
jt
+δ
it
δ
js
(14)
Thus, from Eqs. (9) and (10) and (12), f
0
ij
, f
ij
and
σ
v
ij
can be rewritten as
Z. Zhang et al.
International Journal of Plasticity 137 (2021) 102902
5
f
0
ij
=λ
v
Θ
ε
e
v
1 − β
ε
e
s
ε
e
v
2
δ
ij
+ (λ
s
+γλ
v
)Θe
e
ij
(15)
f
ij
=f
0
ij
+ ϑv
ij
,
σ
v
ij
= − ϑ
π
ij
+
η
v
ijkl
v
kl
(16ab)
Similarly, the dissipative ow of cohesion degradation, f
c
, can be dened as a function of the cohesionrelated internal stress D
c
. It
can also be expected that the cohesion degradation could be slower under a larger conning pressure and at the early stage of shearing.
Moreover, the cohesion can be fully destroyed for bonded soils subjected to shear loads. Therefore, one can dene f
c
as
f
c
=Θk
c
c
a −
c
c
0
2
D
p
′
(17)
where c
0
is the initial cohesion; k
c
is a degradation coefcient and a > 1 is a parameter controlling the initial cohesion degradation
rate.
2.2. Hyperelasticity for different saturated soils
As discussed in Sec. 2.1.1, the denition of elastic potential
ω
e
=
ω
e
(
ε
e
ij
,
ρ
, c) is needed for the hyperelastic relation in Eq. (6a). The
following three invariants of
ε
e
ij
are usually used for this purpose:
ε
e
v
=
ε
e
kk
,
ε
e
s
=
2
3
e
e
ij
e
e
ij
,
ε
e
t
=
e
e
ij
e
e
jk
e
e
ki
3
(18)
where
ε
e
v
is the elastic volumetric strain; e
e
ij
=
ε
e
ij
−
ε
e
v
δ
ij
/3 is the elastic deviatoric strain;
ε
e
s
and
ε
e
t
are the second and third invariants of
e
e
ij
, respectively. Hereinafter, all stresses and strains are taken as positive under compression.
Without loss of generality, the hyperelasticity can be established rstly for bonded soils with true cohesion and then naturally
reduced to other cases under certain simplications. The elastic potential function can be derived by considering the relation between
elastic stiffness and soil deformation. In the Hertz particlecontact model, the normal contact stiffness between two sphere particles K∝
Δ
1/2
, where Δ is the normal overlap value. Similarly, in consideration of the bonding between particles, K∝(Δ +C)
1/2
can be assumed
to allow the tensile contact force, where C is the maximum allowable normal tensile gap. At macroscopic scale, similar nonlinear
relations can be dened for soils between the elastic moduli and the volumetric elastic strain
ε
e
v
as well as the cohesion index c. Thus,
we assume that, under isotropic stress conditions (
ε
e
s
= 0), K
e
= Bf(
ρ
)(
ε
e
v
+ c)
m
and G
e
= Af(
ρ
)(
ε
e
v
+ c)
n
, where K
e
and G
e
are the elastic
bulk and shear moduli, respectively; m and n are two nonlinear indexes; A and B are two stressdimension parameters; the cohesion
index c just represents the maximum allowable tensile volumetric elastic strain; and f(
ρ
) is an increasing function determining the
effect of dry density
ρ
on the elastic stiffness. Thus, in the resulting hyperelasticity, the elasticity and the plasticity are coupled in two
ways, i.e. the change in
ρ
induced by irreversible shearcontraction or dilation and the degradation of c induced by irreversible
deformation.
Noting that the mean stress p =
∂ω
e
/
∂ε
e
v
= K
e
ε
e
v
and taking
ε
e
s
= 0 with other state variables unchanged, the volumetric part of the
elastic potential density can thus be derived as follow:
ω
ev
=
ε
e
v
0
K
e
ε
e
v
d
ε
e
v
=Bf (
ρ
)
ε
e
v
+ c
m
ε
e
v
+ c
2
m +2
−
c
ε
e
v
+ c
2
m + 1
+
Bf (
ρ
)c
m+2
(m + 2)(m +1)
(19)
Eq. (19) leads to K
e
∝f(
ρ
)
1/(m+1)
p
m/(m+1)
under isotropic stress conditions, which is widely supported in experimental studies. Without
the third elastic strain invariant
ε
e
t
, the shear part of
ω
e
,
ω
es
, can be derived from the relation q =
∂ω
e
/
∂ε
e
s
=3G
e
ε
e
s
, where q =
3s
ij
s
ij
/2
is the deviatoric stress invariant (s
ij
=
π
ij
− pδ
ij
). This reads
ω
es
= 3Af(
ρ
)(
ε
e
v
+ c)
n
(
ε
e
s
)
2
/2. Therefore, the complete function for the
elastic potential density unied for different saturated soils is expressed as follow:
ω
e
=
ω
ev
+
3
2
Af (
ρ
)
ε
e
v
+ c
n
ε
e
s
2
(20)
In order to describe the stressinduced anisotropy of elastic shear modulus, the elastic potential should also be a function of
ε
e
t
. This
can be simply considered by dening A as a function of Lode angle as follow:
A =A
0
1 − (1− cos3θ)
η
ε
e
s
ε
e
v
+ c
, cos3θ =
4
3
ε
e 3
t
ε
e 3
s
(21)
where A
0
is a material constant, θ is the Lode angle dened in elastic strain space and
η
is a constant parameter for stressinduced
anisotropy. Taking c = 0, Eq. (20) is reduced to the case for unbonded soils. It can be reduced to the case of linear elasticity when
m = n = 0 and
η
= 0. In the elastic potential proposed above, the elastic volumetric and shear deformations are coupled with each
other and it naturally restricts the stress state within a state boundary. These features will be discussed in detail in Sect. 3.
Substituting Eqs. (20) and (21) into Eq. (6a), the elastic stress of saturated soils can be derived as follow:
Z. Zhang et al.
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