SCIENCE CHINA
Technological Sciences
• ARTICLES •
UH model considering temperature effects
YAO YangPing*, YANG YiFan & NIU Lei
School of Transportation Science and Engineering, Beihang University, Beijing 100191, China
Received
; accepted
The influences of temperature on the mechanical behavior of saturated clays are discussed first. Based on the concept of true
strength and the revised calculation method of the potential failure stress ratio, the equation of the critical state stress ratio for
saturated clays under different temperatures is deduced. Temperature is introduced as a variable into the UH model
(three-dimensional elastoplastic model for overconsolidated clays adopting unified hardening parameter) proposed by Yao et al.
and then the UH model considering temperature effects is proposed. By means of the transformed stress method proposed by
Yao et al., the proposed model can be applied conveniently to three-dimensional stress states. The strain-hardening, softening
and dilatancy behavior of overconsolidated clays at a given temperature can be described using the proposed model, and the
volume change behavior caused by heating can also be predicted. Compared with the modified Cam-clay model, the proposed
model requires only one additional parameter to consider the behavior of the decrease of preconsolidation pressure with an increase of temperature. At room temperature, the proposed model can be changed into the original UH model and the modified
Cam-clay model for overconsolidated clays and normally consolidated clays, respectively. The considered temperature range
here is from the melting point to the boiling point of the pore water (e.g. the experimental temperatures (20℃~95℃) mentioned in this paper are within this range). Comparison with existing test results shows that the model can reasonably describe
the basic mechanical behavior of overconsolidated clays under various temperature paths.
clays, constitutive model, overconsolidation, temperature
Citation:
Yao Y P, Yang Y F, Niu L. UH model considering temperature effects. Sci China Tech Sci,
1 Introduction
The engineering properties of clays are affected a lot by
temperature change. The effects of temperature on the
stress-strain behavior of clays have been studied with more
attention in recent years, which has important theoretical
and practical significance. There are many applications
based on the thermomechanical behavior of clays, notably
for nuclear waste disposal [1], bearing capacity of clays
around zones of geothermic energy [2], exploitation of geothermic energy and geothermal structures [3]. Furthermore,
the tasks of heat storage, zones around buried high-voltage
cables, pavement-soil structure, transformation and stability
of furnace foundations [2,4,5] also need to be further studied.
The effects of temperature on the mechanical behavior of
clays have been noticed for a long time. Prior work of temperature influence on soil properties was carried out by
*Corresponding author (email: [email protected])
© Science China Press and Springer-Verlag Berlin Heidelberg 2010
Campanella and Mitchell [6]. Drained and undrained triaxial tests of three different clays performed by Hueckel et al.
showed that the shape of the yield surface was temperature
dependent, and thermal sensitivity was found to be different
in overconsolidated and normally consolidated clays. After
that, a thermomechanical constitutive model for saturated
clays was developed to consider temperature effects [7, 8].
Afterwards, Hucekel et al. [9] explained the mechanism of
thermal failure. Tanaka et al. [10] carried out tests on normally consolidated and slightly overconsolidated (OCR=2)
reconstituted illitic clay under different temperatures to
examine the influence of temperature on the mechanical
behavior of clays. In order to investigate the effect of temperature change on the preconsolidation pressure as well as
the effect of overconsolidation ratio on the thermal volume
change behavior of Boom clay, some tests were carried out
by Sultan et al. [11]. Based on the framework of the modified Cam-clay model, a model for saturated clays was put
forward by Cui et al. [12], which took into account the effect of temperature change on the volume change behavior
of saturated clays. By the framework of the modified
tech.scichina.com
www.springerlink.com
Cam-clay model, Graham et al. proposed a model to describe the transformation behavior during temperature
change for clays with different OCRs [13], where an associated flow rule was adopted and 11 parameters were used in
the model. Using a temperature-controlled triaxial apparatus,
Cekerevac and Laloui [14] carried out a study performed on
CM clay between 22℃ and 90℃ to study the thermal mechanical behavior of saturated clays. Based on considerations of the thermal effect on void ratio, Laloui et al. proposed a thermoplastic mechanism for isotropic thermomechanical paths including thermal hardening behavior [15,
16]. After the discussion on experimental phenomena, a
thermoplasticity stress-strain relationship was established
and the mathematical formulation of thermoplastic yield
function was introduced too. The work of Abuel-Naga et al.
included some isotropic and anisotropic consolidation tests
as well as drained and undrained triaxial tests of several
different saturated clays [17, 18], and then a thermomechanical model was built based on the modified Cam-clay
model [19]. Through application of a temperature-controlled
triaxial apparatus, triaxial tests through three different stress
paths under different temperatures were carried out by Chen
et al. [20] to demonstrate that the transformation and
strength characteristics of clays were affected a lot by temperature change.
The theories about stress-strain behavior of overconsolidated saturated clays without consideration of temperature
effects have been well developed. The bounding surface
theory proposed by Dafalias can describe the plastic transformation caused by stress change inside the yield surface
[21]. The concept of subloading surface was presented by
Hashiguchi [22], and it was applied to the constitutive model of metal and soil. The concept of structural properties was
combined with the concepts of subloading yield surface by
Asaoka [23], which can describe the decay or collapse of
the soil structure, the loss of overconsolidation and the evolution of anisotropy. The concepts of subloading surface
was then introduced to the tij space by Nakai and Hinokio
[24], which made a better description for the
three-dimensional stress-strain behavior of overconsolidated
clays. The UH model (three-dimensional unified hardening
model for overconsolidated clays) proposed by Yao et al. is
a simple and practical model based on the theory of the
modified Cam-clay model and the subloading surface
framework [25, 26], which could better describe the
stress-strain behavior of overconsolidated clays without any
additional parameter, compared with the modified Cam-clay
model.
This paper summarizes the effects of temperature on the
mechanical behavior of clays first. The relationship between
preconsolidation pressure and temperature change, which is
called the LY (loading yield) curve, is presented. Based on
the revised calculation method of the potential failure stress
ratio for overconsolidated clays, the equation of the critical
Sci China Tech Sci
state stress ratio for saturated clays under different temperatures is deduced, which makes it possible to consider the
strength change behavior by only one material parameter
used in the LY curve. The stress-strain behavior of overconsolidated clays is studied and a three-dimensional critical state model for overconsolidated clays considering temperature effects is proposed. The proposed model inherits
and develops the UH model. The strain-hardening, softening
and dilatancy behavior of overconsolidated clays can be
described, and the effects of temperature change or high
temperature on the stress-strain behavior can also be described. The considered temperature range here is from the
melting point to the boiling point of the pore water (e.g. the
experimental temperatures (20℃~95℃) mentioned in this
paper are within this range).
2 Effects of temperature on the mechanical behavior of saturated clays
Triaxial compression tests were carried out by many researchers to investigate the effects of temperature on the
mechanical behavior of saturated clays. By the analysis of
some test results, three main properties are summarized as
follows:
2.1
Compressibility
By performing isothermal oedometer tests at different temperatures as shown in Figure 1, Campanella and Mitchell
found that the slopes of the loading and unloading lines
were independent of temperature [6], but the compressibility curves in the e-log p space was moved downward as
temperature rised. This phenomenon has been subsequently
confirmed by many investigators [14, 17].
1
T=24.7℃
0.96
T=37.7℃
0.92
T=51.4℃
0.88
e
YAO YangPing, et al.
0.84
0.8
0.76
100
log p/ kPa
1000
Figure 1 Effects of temperature on compressibility of clays.
2.2
Preconsolidation pressure
Figure 2 shows the isothermal oedometer test results deter-
YAO YangPing, et al.
mined by Eriksson [27], in which the points are the different
preconsolidation pressures at different temperatures and the
curve is a fitted curve. A decrease in the preconsolidation
pressure can be seen with an increase of temperature. Tidfors and Sällfors [28], Boudali et al. [29], Cekerevac and
Laloui [14] also carried out some tests, from which the
same phenomenon was observed. Based on the relationship
between preconsolidation pressure and temperature [15], as
well as the LC (loading collapse) curve defined by Alonso
et al. [30], the expression of LY curve is given as follows:
pxT  px  T ,
Sci China Tech Sci
strain along the loading-unloading path A  B  C at
initial temperature T0 is equal to the plastic volumetric
strain along the heating-loading-unloading-cooling path
A  D  E  C  C (where C  C is a cooling
course). As point C is coincide with point C , line BE is
the LY curve in e-ln p space.
100
90
(1)
80
70
T  1   ln
T/℃
where
T
,
T0
(2)
50
40
30
where T is the current temperature; T0 is the initial temperature; p xT is the preconsolidation pressure at the current temperature; px is the preconsolidation pressure at
the initial temperature;  is a parameter, which can be
obtained by isothermal oedometer tests at different temperatures. As px is different, the evolution law of LY curve
can be given by using eq. (1) as shown in Figure 3. Figure 4
shows different shapes of LY curves when a different  is
given.
60
20
0
200
400
p/kPa
600
800
Figure 3 Evolution law of LY curve as px is different.
100
90
0.15
80
 =0.05
0.1
T/℃
70
60
50
40
50
30
40
T/℃
60
20
500
30
700
p/kPa
800
900
20
Figure 4 Effect of  on shape of LY curve.
10
0
0
20
40
p/kPa
60
80
Figure 2 Effect of temperature on preconsolidation pressures of clays.
2.3
600
e
N (T0 )
N (T )
NCL(T0 )
A
D
1
C(C)
Shear strength
LY
E
B
As temperature rises, the stickiness of water decreases and
the permeability coefficient increases, which causes a reduction of the void ratio and an increment of the density and
strength. The shear strength of saturated clays is considered
to increase as temperature rises.
The plastic volumetric strains of each two points on the
LY curve are equal. As shown in Figure 5, if the volume
change during heating is irreversible, the plastic volumetric
1
NCL(T )
0
p0
Figure 5
pxT
1
px
Shape of LY curve in e–ln p space.


ln p
YAO YangPing, et al.
Corresponding to p xT , point E lies on the unloading line
BE at T0, which means that high temperature can be regarded as an overconsolidation state. Therefore, the strength
equation at a given temperature T can be obtained using the
revised calculation method of the potential failure stress
ratio M f presented by Yao et al. [25,26]. As shown in eq.
(3), the equation of the critical state stress ratio M T at a
temperature T is similar to the equation of the potential failure stress ratio mentioned in Appendix A:
   
M T  6  0 1  0
 T  T
 0 
  ,
 T 
(3)
where
0 
M 02
,
12  3  M 0 
(4)
where M 0 is the critical state stress ratio at initial temperature T0 . If the critical state stress ratio M 0  1.0 at initial
temperature 20℃ and   0.15 , the relationship between T
and critical state stress ratio M T can be obtained by eq. (3),
as shown in Figure 6.
3
Thermoelastoplastic stress-strain relationship
for overconsolidated clays
Based on the classical elastoplastic theory and the consideration of the effects of temperature on the mechanical behavior of saturated clays, a thermoelastoplastic constitutive
model is built in the framework of critical state soil mechanics to describe the effects of temperature on the
stress-strain behavior of normally consolidated and overconsolidated clays, respectively.
3.1
Yield function
The effects of temperature on the mechanical behavior of
saturated clays are combined with the modified Cam-clay
model [31], and the yield surface at a given temperature T is
an ellipse as shown in eq. (5), where an associated flow rule
is adopted:
f  q 2  M T 2 p( pxT  p)  0,
(5)
Compared with the yield function of the modified
Cam-clay model, the critical state stress ratio M T and
preconsolidation pressure p xT are adjusted, but the form is
not changed.
3.2
1.12
Current yield surface and reference yield surface
As shown in Figure 7, the current yield surface at a given
temperature T passes through the current stress point
A ( p, q) , which is similar to the yield surface described by
eq. (5), but a hardening parameter H is adopted (see in section 3.6). The current yield function can be expressed as
1.1
1.08
MT
Sci China Tech Sci
1.06
1.04
1.02
f  ln
1
20
30
40
50
60
T/℃
70
80
90
100
Figure 6 Relationship between temperature T and critical state stress
ratio MT.
The effects of temperature on the mechanical behavior of
saturated clays are introduced in this section. The slopes of
the loading line and unloading line in the e  ln p space
are the same at different temperatures, that is to say,  and
 are temperature independent, but the lines move downward when heating happens. The preconsolidation pressure
is decreased as temperature rises, whose evolution law is
described by the LY curve. The critical state stress ratio is
increased after heating, the value of which can be obtained
through the revised calculation method of the potential failure stress ratio.
p
q2
1
 ln(1  2 2 )  ln T  H  0,
px 0
cp
MT p
(6)
where p and q are the current stresses; px 0 is the initial
mean principal stress at T0 ; cp       1  e0  ;  and
 are the slopes of the normal compression line (NCL)
and the unloading line, respectively, and e0 is the initial
void ratio.
The reference yield surface passes through the reference
stress point B ( p, q ) , where the plastic volumetric strain
 vp is chosen as the hardening parameter of the reference
yield surface. The reference stress point is defined so that
point B has the same stress ratio as point A. After the reference yield surface is introduced, the degree of overconsolidation can be described using the relationship between the
current yield surface and the reference yield surface. The
effects of the degree of overconsolidation on the hardening
parameter H and potential failure stress ratio M fT can also
YAO YangPing, et al.
Sci China Tech Sci
be described by the overconsolidation parameter RT ( RT
can be seen in section 3.4). The reference yield surface can
be written as
RT 
(7)
where px 0 is the initial mean principal stress at T0 corresponding to the reference stress point, which is the initial
value of px .
  vp
 MT 2 
p  px 0T 
exp

2
2 
 MT  
 cp
RT 
1
A
B
Reference
yield surface
( p, q )
( p, q)
q
Current LY
curve
pxT
Reference
LY curve
1
RT 
T0
px
  vp
p 
2 
exp
 
1 

px 0T  M T 2 
 cp

 ,

(11)
where  vp   d vp    d v  dp K  ; K  E 3 1  2  is the
3.5
p q

p q
Potential failure stress ratio MfT
A potential failure stress ratio M fT is introduced to de-
px
0
(10)
elastic bulk modulus. The value of RT is less than 1 in the
overconsolidation state, which keeps increasing up to the
critical state ( RT =1) during loading.
pxT
M0

 ,

where   q p is the stress ratio.
Substituting eq. (10) into eq. (9) gives
MT
Current yield
surface
(9)
From eq. (7), p can be solved as
p
p
q2
f  ln
 ln(1  2 2 )  ln T  v  0,
px 0
cp
MT p
T
p q
 .
p q
scribe the potential capacity of overconsolidated clays in
resisting shear failure at the current temperature, stress condition and density. The revised M fT (see Appendix A) can
p
Figure 7 Current yield surface and reference yield surface.
be written as
3.3
Current LY curve and reference LY curve
It can be seen from Figure 7 that the current LY curve and
the reference LY curve are the intersecting lines of the current yield surface and reference yield surface with p  T
plane respectively. The expression of the current LY curve is
the same as eq. (1), where p xT and px are the points in
      
M fT  6  T 1  T   T  ,
 RT  RT  RT 
where
T 
the current LY curve corresponding to T and T0 , respectively. The reference LY curve can be written as
pxT  px  T ,
(8)
where p xT and px are the points in the reference LY
curve corresponding to T and T0 , respectively. The current
LY curve coincides with the reference LY curve for normally consolidated clays.
3.4
Overconsolidation parameter RT
An overconsolidation parameter RT is defined as the ratio
of the current stress to its corresponding reference stress at
the same stress ratio:
(12)
3.6
MT 2
.
12  3  M T 
(13)
Unified hardening parameter H
Yao et al. have proposed a unified hardening parameter related to the potential failure stress ratio and critical state
stress ratio [25], which can be written as
H   dH  
M 4fT   4
M T4   4
1
d vp   d vp ,

(14)
where

M T4   4
.
M 4fT   4
(15)
YAO YangPing, et al.
3.7
Thermoelastic strain increment
If the elastic properties are temperature independent, then
the thermoelastic strain increment can be calculated by
1 

dεije 
dσ ij  dσ mm δij ,
E
E
Sci China Tech Sci
A2 , A3 , B1 and B2 can be derived from the equations of
the proposed model as
A1 
(16)
A2 
where  is the Poisson’s ratio, and the elastic modulus E
can be expressed as
E
3.8
3(1  2 )(1  e0 )

p.
A3 
(17)
(18)
where

2 pq
    c p dp  2 2
dq 
2
M

T p q
4
fT
 M 4fT
M
2
p
4
2
p
p
2
T
2 2
p
2
T
2
p
2
p
4
p
4
2
T
2 2
2
p
K ( M T2   2 )  c p p





, (22)
2 



2 

2
T
2
4
4
fT
4
fT
4
p
2
T
2


M T T T

2
2
2

 M T ( M T   )  2 T T

4
4
2
2
2 2
( M fT   ) p  12Gc p  Kc p ( M T   ) 
.
6G  c p p

B2 


M T T T


 M T ( M T2   2 )  2 T 2T
4
4
2
2
2 2 
( M fT   ) p  12Gc p  Kc p ( M T   ) 
The thermoplastic strain increment can be expressed as
f
,
σ ij
4
fT
B1 
Thermoplastic strain increment
dεijp  
M
 M    p  12Gc 
   p  12Gc   Kc ( M   )
2c  M   
   p  12Gc   Kc ( M   )
 M    p  Kc (M   )
   p  12Gc   Kc ( M   )

(23)
(19)
From eq. (21), it can be seen that if temperature is not
changed, the proposed model can be changed into the original UH model and the modified Cam-clay model for overconsolidated clays and normally consolidated clays, respectively. Suppose dpT  B1dT is the equivalent mean prin-
3 T  2 0  0 T   0   2 0 T   0 
 
. (20)
T
T 2 T   0 
cipal stress increment during heating and dqT  B2 dT is
the equivalent deviatoric stress increment during heating,
respectively. The change of temperature can be equivalent
to stresses to consider the effects of temperature on the mechanical behavior of clays.
  p( M T2 p 2  q 2 )
 
2 pq 2

dT  ,

2 2
2
2 2
2 
 T T ( M T p  q ) M T ( M T p  q )  
where
Compared with the UH model, dT is added to the plastic factors in eq. (19) to consider the plastic transformation
during heating. When temperature is invariable (dT  0) ,
the effects of temperature on the stress-strain behavior is
caused only by the change of the critical state stress ratio
MT .
3.9 Thermoelastoplastic stress-strain relationship in
p--q space
If temperature change is regarded as an equivalent loading,
then the stress-strain relationships in p  q space are expressed as follows:
dp  B1dT   K  A1


dq  B2 dT  3KG  A2
3KG  A2   d v 

,
3G  A3  d d 
(21)
where G  E 3 1   is the elastic shear modulus; A1 ,
A2 and A3 are three different plastic affecting factors;
B1 and B2 are different thermal affecting factors. A1 ,
4
Application of transformed stress method in
the model
The essence of the transformed stress method based on the
SMP criterion [32] is to transform the SMP criterion in the
ordinary stress space into the extended Mises criterion in
the transformed stress space. Hence, the stress-induced anisotropy of clays can be changed into isotropy, and any constitutive model can be applied conveniently to
three-dimensional stress states.
4.1 Introduction of transformed stress space and
transformed stress tensors
In Figure 8, the SMP criterion can be drawn a convex curve
in the ordinary  plane of the three-dimensional stress
space. However, in the  plane of transformed stress
space, the SMP criterion is considered as a circle with a
YAO YangPing, et al.
radius of r . r can be expressed as
 ~ 
1
1
effects can be expressed as
dσij  dσijT  Dijkl dεkl ,
Mises
B
stress increment tensor caused by temperature change;
Dijkl is the three-dimensional thermoelastoplastic tensor by
~
r  r0

(28)
where dσ ij is the stress increment tensor; dσijT is the
SMP
A
Sci China Tech Sci
using the transformed stress method; dεkl is the strain increment tensor. Eq. (28) will be deduced in Appendix B.
0
5
 3 ~3 
 2 ~2 
r  r0 
The proposed model can describe the effects of temperature
on the transformation behavior of clays. At a given temperature, the strength and stress-strain behavior in different
stress paths for both normally consolidated and overconlidated clays can be described. Material parameters as listed
in Table 1 are used in this section.
SMP criterion in  - plane.
Figure 8
2 I1
2
,
3 3 ( I1 I 2  I 3 ) /( I1 I 2  9 I 3 )  1
2
qc 
3
Fundamental feature of the model
(24)
Table 1 Material parameters of Kaolin
where r0 is the stress radius in the original stress space
Material parameters
Value

0.104

0.026
The same as the
M0
0.82
parameters used in the
e0
0.94
modified Cam-clay model
In the condition that the principal directions of  1 ,  2 ,
v
0.3
 3 and  1 ,  2 ,  3 are coaxial axes, the transformation

0.15
when the Lode’s angle   0 ; I1 , I 2 and I 3 are the
three stress invariants. From eq. (24), we can obtain
qc 
2I1
3 ( I1 I 2  I 3 ) /( I1 I 2  9I 3 )  1
.
(25)
from A( p, q, ) to B( p, q, ) can be realized from equations:
p  p

q  qc  .
   
(26)
The relationship between transformed stress tensors and
original stress tensors is expressed as
σ ij  pδij 
σ ij  σ ij ,
qc
(σ ij  pδij ),
q
 q  0

 q  0 
,


(27)
where δij is Kronecker’s delta.
5.1
Comment
Thermal parameter
Isotropic heating
As shown in Figure 9, the sample is consolidated from point
O to point A and then unloaded to point B. If the sample is
heated to a temperature T, the current stress point will
evolve from point B to point C, and the reference stress
point will evolve from point A to point D along a certain
path. It can be seen that the value of current stress is not
changed, but the value of reference stress is decreased.
From the definition of overconsolidation parameter, we
know the value of RT increases continuously, which
means that heating can cause a reduction or even a loss of
the OCR. The current LY curve and reference LY curve
evolve from the curve that passes through point A and point
B (the thick curve) to the curve that passes through point C
and point D (the thin curve).
4.2 Three-dimensional stress-strain relationship for
overconsolidated clays considering temperature effects
5.2
The tensor form of three-dimensional stress-strain relationship for overconsolidated clays considering temperature
In Figure 10, a sample of normally consolidated clays is
sheared from point A (corresponding to p) to point C at
Shearing at a different constant temperature
YAO YangPing, et al.
20℃. It can be seen from the geometrical relationship of the
elliptic yield surface shown in Figure 11 that point F corresponds to 2p, so eAC＝      ln 2 . The sample is then
heated from the initial temperature T0 (point A) to a given
temperature T (point B). As  and  are temperature
independent, the NCL which passes point B is parallel to the
NCL which passes point A. As p is a constant, during the
shearing B→D at temperature T, the variation of void ratio
eBD is equal to eAC , whose value is also       ln 2 .
Sci China Tech Sci
dated clays is higher than that of normally consolidated
clays, which also shows a character of shear dilatancy and
strain softening. High temperature causes an increase of the
critical state stress ratio, so the residual stress ratio at a
higher temperature is a little higher than the residual stress
ratio at a lower temperature. The peak stress ratio at a higher temperature is also higher than the peak stress ratio at a
lower temperature.
q
Therefore, the critical state lines (CSLs) at different temperatures are parallel too.
M0
C
T
Current LY
curve
1
Reference LY
curve
dp  0
O
Figure 11
D
C
p
F
A
Evolution law of yield surface during loading.
T
1.2
p
A
B
O
q/p
1
T0
Figure 9 Stress paths and LY curves during isotropic heating.
20℃
90℃
OCR=10
0.8
OCR=1
0.6
OCR=10
0.4
0.2
(a)
0
e
-0.2 0
1
B
1
C
εv /%
A

10
-0.6

40
50
OCR=1
F
1.2
NCL
OCR=10
1
p
2p
ln p
NCLs and CSLs at different temperatures in e-ln p space.
q/p
0.8
CSL
Figure 10
30
ε1 /%
D
0
20
-0.4
OCR=1
0.6
20℃
90℃
0.4
0.2
(b)
0
Drained and undrained triaxial test paths were carried out
on normally consolidated and overconsolidated (OCR=10)
clays at a temperature of 20℃ and 90℃, respectively. As
shown in Figure 12, the thick curves are the stress-strain
curves at 20℃ and the thin curves are the stress-strain
curves at 90℃. Normally consolidated clays and overconsolidated clays reach the same residual stress ratio at the
same temperature, but the peak stress ratio of overconsoli-
0
10
20
30
40
50
ε1 /%
Figure 12 Stress-strain relationships in drained and undrained triaxial
test paths at different temperatures with different OCRs: (a)drained;
(b)undrained.
Shearing-heating-shearing
Drained triaxial test paths were first simulated to overconsolidated clays with different OCRs at a temperature of 20℃.
When stress ratio  reaches 0.65, 0.85 and 1, respectively,
stop shearing and heat the sample to 90℃. After that,
drained shearing is continued. As shown in Figure 13, the
strain of normally consolidated clays (OCR=1) is hardened
without softening, and lines AB, CD and EF indicate that
the clays are being heated. It can be seen that although the
stress is not increased during heating, the strain is increased
obviously.
5.4
section 5.2, when p is a constant, the volume change behavior during shearing is temperature independent. Therefore, there is a superposition of the volumetric strain curves
at different temperatures.
1.2
ε2、ε3
q/p
1
OCR=5
0.8
C D
0.6
A
OCR=10
0.2
OCR=5
0
-0.2 0
10
Figure 13
Stress-strain relationships of shearing-heating-shearing path.
q/p
ε3
0
(b)
5
10
15
-10
-5
0
ε1
10
0.8
ε3
15
q/p
1
ε2
5
εv /%
q/p
0.6
ε1、ε2
0.6
20℃
90℃
0.4
(c)
90℃
0
5
εv /%
10
0.2
(d)
εi /%
0
20℃
0.4
0.2
Figure 14
-0.2
-0.4
0.8
-0.4
εi /%
0
εv /%
1
-0.2
20℃
90℃
0.4
0.2
-0.4
-5
ε1
0.6
0
-10
ε2
0.8
20℃
90℃
εi /%
ε3
50
1
0.2
-0.2
40
ε1 /%
ε1
0.4
-5
30
OCR=1
-0.6
0.6
-10
20
-0.4
0.8
(a)
OCR=1
B
0.4
True triaxial test paths at different temperatures
With application of the transformed stress method mentioned above, true triaxial test paths of different Lode’s angles for normally consolidated clays (OCR=1) were simulated. As shown in Figure 14, the thick curves are the
stress-strain curves at 20℃ and the thin curves are the
stress-strain curves at 90℃, where p=600kPa. As stated in
OCR=10
E F
1
εv /%
5.3
Sci China Tech Sci
q/p
YAO YangPing, et al.
εi /%
0
15
-10
-5
-0.2
-0.4
0
5
10
εv /%
Stress-strain relationships of true triaxial test paths at different temperatures: (a)  =0°; (b)  =30°; (c)  =45°; (d)  =60°.
15
YAO YangPing, et al.
6
6.1
 
Test results and model predictions
Test results and model predictions I
Isotropic heating-cooling test results of Boom clay were
quoted by refs. [12] and [18]. The samples were first consolidated to px 0 =6MPa. After that, the samples were unloaded to 3MPa and 1MPa, respectively, so the samples of
OCRs=1, 2, 6 were obtained. Afterwards, the samples were
heated from 21.5℃ to 95℃ and then cooled to 21.5℃ when
the loads were not changed. Model parameters  =0.178,
 =0.046 and  =0.135 can be obtained from ref. [18];
e0 =0.72 and M 0 =0.87 can be obtained from ref. [12]. After the same temperature cycle, the volume change behavior
is different for clays with different OCRs. The proposed
model can describe the volume change behavior of Boom
clay, as shown in Figure 15.
6.2
Sci China Tech Sci
1  pxT px
.
ln T T0 
(29)
Therefore, the value of  can be obtained from eq. (29),
where the preconsolidation pressures at different temperatures were obtained from the oedometer test results at different temperatures given by ref. [14]. From the figures in
ref. [14] we can get the preconsolidation pressure
px =600kPa at initial temperature T0 =22℃; p xT =533kPa
at T=60℃; p xT =518kPa at T=90℃. The mean value of 
is obtained by using eq. (29), and  =0.1 is given. The
proposed model can reasonably describe the drained triaxial
stress-strain behavior at different temperatures for Kaolin as
shown in Figure 16, in which the thick curves are the predicted curves at 22℃ and the thin curves are the predicted
curves at 90℃.
OCR=6
100
Test results and model predictions II
OCR=2
OCR=1
90
70
60
50
40
30
20
-0.5
Figure 15
800
800
600
600
400
0
0.5
1
1.5
Predicted and test results in isotropic heating-cooling tests for
Boom clay.
400
22℃
200
22℃
200
90℃
0
5
10
15
-2005
20
25
30
0
5
10
15
-2005
εv /%
εv /%
90℃
0
0
-400
10
2
εv /%
q /kPa
q /kPa
e0  0.94 can be obtained from ref. [14]; suppose Poisson’s ratio v =0.3;   0.021 is confirmed by empirical
equation   0.2 [33], and  is determined as follows.
From eq. (1) and eq. (2) we can obtain
80
T /℃
Predictions here are compared with the drained triaxial tests
in ref. [9]. The samples were first consolidated to
px 0 =600kPa at 22℃. After that, the samples were unloaded
to 500kPa, 400kPa, 300kPa and 200kPa, respectively. Two
groups of samples with OCRs =1, 1.2, 1.5, 2 and 3 were
obtained. One group was sheared at 22℃, the other was
heated to 90℃ and then sheared. Model parameter
M 0  0.82 can be obtained from ref. [9];   0.104 and
(a)
ε1 /%
-400
10
(b)
ε1 /%
20
25
30
Sci China Tech Sci
600
600
450
450
q /kPa
q /kPa
YAO YangPing, et al.
300
300
22℃
150
22℃
150
90℃
0
0
0
5
10
15
20
25
30
0
-1505
5
10
15
20
25
30
-1505
εv /%
εv /%
90℃
(c)
-300
10
(d)
-300
10
ε1 /%
ε1 /%
300
q /kPa
225
150
22℃
75
90℃
0
0
5
10
15
20
25
30
εv /%
-755
(e)
-150
10
Figure 16
Predicted and test results in drained triaxial tests at different temperatures for Kaolin:
(a)OCR=1; (b)OCR=1.2; (c)OCR=1.5; (d)OCR=2; (e)OCR=3.
Test results and model predictions III
Predictions here are compared with the test results of MC
clay in ref. [19]. The samples were first consolidated to
px 0 =196kPa. After that, undrained triaxial tests were carried out. Material parameters are obtained from ref. [19],
including  =0.304,  =0.07, M 0 =0.675; e0 =1.5 is obtained from the e-log p curve for normally consolidated MC
clay in ref. [34];  =0.065 is obtained from ref. [18]; suppose v =0.3. It can be seen in Figure 17 that the proposed
model can describe the undrained triaxial stress-strain behavior of normally consolidated MC clay well.
100
75
q /kPa
6.3
ε1 /%
50
22℃
90℃
25
(a)
0
0
5
10
ε1 /%
15
20
YAO YangPing, et al.
Appendix A Revised calculation method of the
potential failure stress ratio
100
q /kPa
75
50
22℃
90℃
25
(b)
0
0
50
100
p /kPa
150
200
Figure 17 Predicted and test results in undrained triaxial tests at different
temperatures for MC clay: (a)stress-strain relationships; (b)stress paths.
7
Sci China Tech Sci
Conclusions
By analyzing the effects of temperature on the mechanical
behavior of saturated clays, temperature variable is introduced into the framework of classical elastoplastic theory.
Hence, a new thermoelastoplastic model is built based on
the UH model.
(1) The effects of temperature on the compressibility, preconsolidation pressure and shear strength of saturated clays
are analyzed. The slopes of the loading line and unloading
line in e-ln p space are temperature independent. The preconsolidation pressure is decreased with an increase of
temperature, whose evolution law is described by the LY
curve. The critical state stress ratio is increased after heating,
where the equation of the critical state stress ratio for saturated clays under different temperatures is deduced based on
the concept of true strength and the revised calculation
method of the potential failure stress ratio.
(2) By taking temperature as a variable to introduce into the
UH model proposed by Yao et al., a thermoelastoplastic
model is represented to consider temperature effects. The
strain-hardening, softening and dilatancy behavior of overconsolidated clays at some temperature can be described
using the proposed model, and the volume change behavior
caused by heating can be predicted too.
(3) Comparison with existing experimental results shows
that the proposed model has a better description of the basic
mechanical behavior after considering temperature effects.
Compared with the modified Cam-clay model, the proposed
model requires only one additional material parameter,
which can be conveniently obtained by isothermal oedometer tests at different temperatures.
This work was supported by the National Natural Science Foundation of
China (Grant No. 50879001, 90815024, 10872016, 11072016) and the National Basic Research Program of China (Grant No. 2007CB714103).
The reasonable form of strength envelope for overconsolidated clays should consist of the zero extension envelop
OD and the Hvorslev line CD, but the intersection of the
two lines appears as a cuspidal point, which is not convenient for numerical calculations. In order to overcome this
disadvantage, a parabola was adopted to replace the strength
envelope of OD and CD [26], as shown in Figure 18. By the
revised Hvorslev envelope, the connection of the zero extension envelop and the Hvorslev line is getting smoother
and more continuous. For clays of a high OCR, the disadvantage of overestimating the undrained shear strength before modified is also overcome. Meanwhile, a parameter Mh
is reduced.
q
CSL
Hvorslev envelope
C
D
B
Revised Hvorslev
envelope
O
pe
p
Figure 18 Revised Hvorslev envelope.
Finally, we can write M f as
   
M f  6
1     ,
 R  R  R 
(30)
where

M2
.
12  3  M 
(31)
In Figure 19, the horizontal dash line is the zero extension envelop, and the calculated curves of M f are very
close before and after revision. When R=1, the two curves
coincide with each other, and the critical state is reached.
However, when approaching the zero extension envelop, the
revised M f gets a peak value of 3, but the original M f
continues to rise up after passing through the zero extension
envelop. According to the concept of the zero extension
envelop, the revised equation of M f is obviously more
reasonable.
YAO YangPing, et al.
Sci China Tech Sci
where
X
6
Before
revised
5
3
by temperature, and Dijkl is the thermoelastoplastic consti-
Revised
tutive tensor. Their expressions are given as follows
2
Critical state
1
Figure 19
between potential failure
and
0 Relationships 0.5
1 stress ratio Mf 1.5
overconsolidatedRparameter R.
Appendix B Derivation of thermoelastoplastic
constitutive tenser Dijkl
e
dσ ijT  Dijkl
f
f p f q 1 f
f qc



δij 
,
σij p σij q σij 3 p
q σij
f
dεijp  
.
σ ij
 dε
3
qc
q I
 c m .
σ ij m 1 I m σ ij
(33)
The stress increment and elastic strain increment can be
given by use of the generalized Hooke’s law:
dσ ij  D dε  D
kl
 dε
,
1
2
(34)
3
where dε is the plastic strain increment, and the elastic
constitutive tensor can be expressed by
p
kl
e
Dijkl
 Kδij δkl  G  δik δ jl  δil δ jk  .
4
(35)
5
By substituting eq. (27) into the corresponding yield
functions expressed by f  f  p, q, T   H  0 shown in
6
eq. (32), the yield function is written as
7
f  f  σ ij , T   H  0,
(36)
Differentiating eq. (36) and then substituting eqs. (33)
and (34) into it, we can obtain
 f




e
σij Dijkl
dεkl  f T dT
X
(42)
where
The plastic strain increment can be given by
p
kl
(40)
cp
 M T2 p 2  q 2

f
 2 2
δij  3  σ ij  pδij   , (41)
2 
σ ij M T p  q 
3p

(32)
p
q2
T
1
ln
 ln(1  2 2 )  ln(1   ln )  H  0.
px 0
T0
cp
MT p
e
ijkl
f f
e
Dstkl
X,
σ mn σ st
(39)
where
f g
e
kl
f f
dT X ,
σ kl T
e
e
Dijkl  Dijkl
 Dijmn
In the transformed stress space, the current yield surface and
potential surface can be expressed as
e
ijkl
(38)
Substituting eqs. (33) and (37) into eq. (34), the tensor
form of stress-strain relationship for overconsolidated clays
considering temperature effects can be obtained as shown in
eq. (28), where dσijT is the stress increment tensor caused
Zero extension
Envelop
4
Mf
f
f
1 f
e
Dijkl

.
σij
σkl  σmm
8
9
10
,
(37)
(43)
Laloui L, Modaressi H. Modelling of the thermo-hydroplastic behaviour of clays. Hydromechanical and Thermohydromechanical Behaviour of Deep Argillaceous Rock, 2002: 161–170.
Bai B, Zhao C G. Temperature effects on mechanical characteristics
of clay soil (in Chinese). Rock and Soil Mechanics, 2003, 24(4):
533-537.
Laloui L, Nuth M, Vulliet L. Experimental and numerical investigations of the behavior of a heat exchanger pile. Int. J. Numer. Anal.
Meth. Geomech., 2006, 30: 763-781.
Mitchell J K, McMillan J, Green S, et al. Field testing of cable
backfill systems. Underground Cable Thermal Backfill, 1982: 19–33.
Laloui L, Moreni M, Fromentin A, et al. In-situ thermo-mechanical
load test on a heat exchanger pile. 4th International Conference on
Deep Foundation Practice. Singapore, 1999: 273-279.
Campanella R G, Mitchell J K. Influence of temperature variations on
soilm behavior. Journal of the Soil Mechanics and Foundations Division, ASCE, 1968, 94: 709-734.
Hueckel T, Borsetto M. Thermoplasticity of saturated soils and shales:
Constitutive equations. J. Geotech. Engrg., ASCE, 1990,116(12),
1765-1777.
Hueckel T, Baldi G. Thermoplasticity of saturated clays: Experimental constitutive study. J. Geotech. Engrg., ASCE, 1990, 116(12):
1778-1796.
Hueckel T, Francois B, Laloui L. Explaining thermal failure in saturated clays. Geotechnique, 2009, 3: 197-212.
Tanaka N, Graham J, Crilly T N. Stress–strain behaviour of reconstituted illitic clay at different temperatures. Engineering Geology, 1997,
47: 339–350.
YAO YangPing, et al.
11
12
13
14
15
16
17
18
19
20
21
22
23
Sultan N, Delage P, Cui Y J. Temperature effects on the volume
change behavior of boom clay. Engineering Geology, 2002, 64:
135–145.
Cui Y J, Sultan N, Delage P. A thermo-mechanical model for saturated clays. Can. Geotech. 2000, 37(3): 607–620.
Graham J, Tanaka N, Crilly T, et al. Modified Cam-Clay modeling of
temperature effects in clays. Can. Geotech, 2001, 38(3): 608–621.
Cekerevac C, Laloui L. Experimental study of thermo effects on the
mechanical behavior of a clay. International Journal for Numerical
Analytical Methods Geomechanics, 2004, 28: 209–228.
Laloui L, Cekerevac C. Thermoplasticity of clays: an isotropic yield
mechanism. Computer and Geotechnics, 2003, 30(8): 649–660.
Laloui L, Francois B. ACMEG-T: soil thermo-plasticity model. Journal of Engineering Mechanics, ASCE, 2009, 135(9): 932-944.
Abuel-Naga H M, Bergado D T, Lim B F. Effect of temperature on
shear strength and yielding behavior of soft Bangkok clay. Soils
Found, 2007, 47(3): 423–436.
Abuel-Naga H M, Bergado D T, Bouazza A, et al. Volume change
behavior of saturated clays under drained heating conditions: experimental results and constitutive modeling. Can. Geotech. J. 2007,
44(8): 942– 956
Abuel-Naga H M, Bergado D T, Bouazza A, et al. Thermomechanical
model for saturated clays. Geotechnique, 2009, 3: 273-278.
Chen Z H, Xie Y, Sun S G, et al. Temperature controlled triaxial apparatus for soils and its application (in Chinese). Chinese Journal of
Geotechnical Engineering, 2005, 27(8): 928－933.
Dafalias Y F. Bounding surface plasticity. I: Mathematical foundation
and hypoplasticity. Journal of Engineering Mechanics, 1986, 112(9):
966-987.
Hashiguchi K. Subloading surface model in unconventional plasticity.
International Journal of Plasticity. 1989, 25(8): 917-945.
Asaoka A. Consolidation of clay and compation of sand-an elasto-plastic. 12 Asian Regional Conference on Soil Mechanics and Ge-
Sci China Tech Sci
24
25
26
27
28
29
30
31
32
33
34
otechnical Engineering. World Scientific Publishing Company.
2004,1157–1195.
Nakai T, Hinokio M. A simple elastoplastic model for normally and
over consolidated soils with unified material parameters. Soils and
Foundations, 2004, 44 (2):53–70.
Yao Y P, Hou W, Zhou A N. UH model: three-dimensional unified
hardening model for overconsolidated clays. Geotechnique, 2009,
59(5): 451-469.
Yao Y P, Li Z Q, Hou W, et al. Constitutive model of
over-consolidated clay based on improved Hvorslev envelope (in
Chinese). Journal of Hydraulic Engineering, 2008, 39(11):
1244-1250.
Eriksson L G. Temperature effects on consolidaton properties of sulphide clays. 12th International Conference on Soil Mechanics and
Foundation Engineering, 1989: 2087-2090.
Tidfors M, Sällfors S. Temperature effect on preconsolidation pressure. Geotechnical Testing Journal, 1989, 12(1):93-97.
Boudali M, Leroueil S, Sinivasa Murthy B R. Viscous behaviour of
natural clays. 13th International Conference on Soil Mechanics and
Foundation Engineering, 1994:411-416.
Alonso E E, Gens A, Josa A. A constitutive model for partially saturated soils. Geotechnique, 1990, 40(3): 405-430.
Roscoe K H, Burland J B. On the generalized stress-strain behavior
of ‘wet’ clay, Engineering Plasticity. Cambrige University Press,
1968: 535-609.
Matsuoka H, Yao Y P, Sun D A. The cam-clay models revised by the
SMP criterion. Soils and Foundations, 1999, 39(1): 81-95.
Yao Y P, Sun D A. Application of Lade's criterion to Cam-clay model.
ASCE, J. Engrg. Mech., 2000, 126(1): 112-119.
Towhata I, Kuntiwattanakul P, Seko I, et al. Volume change of clays
induced by heating as observed in consolidation test. Soils Found,
1993, 33(4):170-183.