6th International Conference on Earthquake Geotechnical Engineering
1-4 November 2015
Christchurch, New Zealand
Modeling of Volumetric Change Behavior
of Sand Subject to Large Cyclic Loading
T. Namikawa 1, J. Koseki 2, S. Tsukuni3
ABSTRACT
During an earthquake, cyclic shear loading induces the phenomenon of liquefaction in saturated
sand deposits, resulting in undesirable ground settlement and severe damage to structures. In order
to predict quantitatively the settlement of liquefiable soil deposits during an earthquake, it is
important to evaluate the volumetric strain change of sand subject to cyclic loadings. This paper
presents the modeling of behavior of saturated sand subject to large cyclic loading. A three
dimensional elasto-plastic model is proposed on the basis of the cyclic shear torsional test results.
In the proposed model, the linear stress-dilatancy relationship is defined from the experimental
relationship between the ratio of the volumetric strain increment to the shear strain increment and
the stress ratio. The simulation of the experimental results demonstrates that the proposed model
can describe the drained and undrained cyclic torsional shear behavior of the saturated sand.
Introduction
A lot of small buildings and houses suffered severe damage such as settlement and tilt owing to
liquefaction phenomenon occurring at the Tokyo Bay area during the 2011 Great East Japan
Earthquake. Countermeasure methods against liquefaction for such small structures are often
required not to prevent liquefaction perfectly but to reduce the settlement and tilt to allowable
level. Finite element analysis is a powerful tool to evaluate the displacement of the structures and
ground. Then the constitutive equation incorporated in the finite element analysis is required to
describe properly the mechanical behavior of saturated sands under cyclic shear loadings.
Especially the ground settlement occurring after liquefaction depends on the volumetric change
of sand caused under a drained condition by the cyclic loading. In order to predict quantitatively
the ground settlement of liquefiable sand deposits after an earthquake, it is important to evaluate
the volumetric strain change of sand subject to cyclic loadings.
The authors have developed elasto-plastic models describing cyclic shear deformation behavior
of sand on the basis of cyclic torsional shear test results. De Silva (2008) proposed a onedimensional model which can simulate the cyclic torsional shear test results. Namikawa et al.
(2011) extended the one-dimensional model to a three-dimensional elasto-plastic model by
adopting the concept of the model with infinite number of nesting surfaces (INS model) (Mroz et
al., 1981). This paper provides an extension of work by Namikawa et al. (2011) on modeling
inelastic sand response under cyclic stress conditions. The model is developed to describe the
1
Professor, Department of Civil Engineering, Shibaura Institute of Technology, Tokyo, Japan, [email protected]
2
Professor, Department of Civil Engineering, University of Tokyo, Tokyo, Japan, [email protected]
3
General Manager, Takenaka Civil Engineering & Construction Co., Ltd, Tokyo, Japan, [email protected]
cyclic shear and volumetric strain changes of sand subjected to large cyclic loading. On the basis
of the drained cyclic torsional shear test results, a simple stress-dilatancy rule is defined to
describe the volumetric strain changes during cyclic shear loading. The proposed model is
validated by the simulation of the drained and undrained cyclic torsional shear tests.
Elasto-plastic Model
Model with infinite number of nesting surfaces
The infinite number of nesting surfaces model (INS model) (Mroz et al., 1981) is adopted to
extend a one-dimensional model (De Silva, 2008) with the Masing’s rule (Namikawa et al.,
2011). Yield surface, reversal surface and active loading surface are defined to express the cyclic
loading behavior in the INS model. During the primary loading process, the skeleton curve could
be described by the hardening rule applied to the yield surface. After the stress reverses, the
hysteresis curve that follows the Masing’s rule could be described by the hardening rule applied
to the active loading surface. More information on the INS model adopted in this study is given
in Namikawa et al. (2011).
Basic formulation
It is first postulated that the total strain increment εij is a sum of the elastic component eije and
plastic component εijp . The elastic strain increment is linearly related to the effective stress
increment σ ij as
−1 
eije = Eijkl
σ kl
(1)
in which E ijkl is the matrix of elastic constant. The dependency of the elastic modulus E on the
effective mean stress p is assumed of the following form.
E = E0
p
po
(2)
where p 0 is the reference effective mean stress and E 0 is the elastic modulus at p 0 . The plastic
strain increment is expressed as the following equation,
∂F
σ
∂
σ kl kl ∂Q
p
εij =
∂σ ij
H
(3)
in which F is the loading function, Q is a plastic potential and H is the plastic modulus. F
corresponds to the yield function f y during the primary loading process and the active loading
function f a during the reverse loading process.
Yield and active loading function
A conical yield surface in the stress space is assumed to describe the shear plastic strain. The
yield function f y is given as :
fy =
1 s s s s − pk y = 0
2 ij ij
(4)
in which s ijs is the effective deviator stress and k y is the internal variable for the yield function.
The active loading function similar to the yield surface is given as :
fa =
(
)(
)
1 s ijs − pa ij s ijs − pa ij − pk a = 0
2
(5)
in which aij is the center of the active loading surface and k a is the internal variable for the active
loading function. Assuming that the deviator plastic flow is associative, the derivative of the
plastic function in Eq.(3) is given as :
∂f y
∂Q
=
(yield surface) ,
∂s ij ∂s ijs
∂f
∂Q
= a (active loading surface)
∂s ij ∂s ijs
(6)
A non-associative flow rule is adopted for the component of expressing the volumetric strain
induced by shear stresses. The detail of the stress-dilatancy relationship will be discussed later.
Hardening rule
The modified general hyperbolic equation is adopted for the monotonic loading stress-strain
relationships (Tatsuoka and Shibuya,1992; De Silva, 2008). The internal variables k y and k a are
given as :
ky =
ε
ε ap
p
1
εp
+
DC1 SC 2
,
ka =
2
,
εp
1
+ a
DC1 2 SC 2
εp =
1 sp sp
εij εij
2
, εap = 1 εijspa εijspa
2
(7)
in which ε ijsp is the deviator plastic strain occurring on the primary loading process and ε ijspa is
the deviator plastic strain occurring after the stress reverses. For the cyclic loading, Eq.(7) could
be used to describe the skeleton curve in the Masing’s rule. In Eq.(7), C 1 and C 2 are the variable
coefficients. C 1 and C 2 vary with the strain level, given as
C1 =
C2
{C1 (0) + C1 (∞ )} + {C1 (0) − C1 (∞ )} cos
2


 α + 1 
 x 
2
π
{C (0) + C2 (∞ )} + {C2 (0) − C2 (∞ )} cos
= 2
2
2


β


+1
 x

π
(8)
in which C 1 (0), C 1 (∞), C 2 (0), C 2 (∞), a and β are the material parameters. In Eq.(7), D and S is
variable coefficients for the effect of the cyclic damage and hardening. D is assumed to be
defined by an exponential function of the total plastic strain that is accumulated up to the current
turning point ∑ ε p as the following equation.
( ∑ ε )
D = εxp − d1
p
(9)
in which d 1 is the material parameter. D represents the ratio of the plastic modulus to the initial
plastic modulus.
S is assumed to be defined by a hyperbolic function of ∑ ε p as the following equation.
S = 1+
∑ ε
p
(10)
∑ ε
S1 +
S ult − 1
p
in which S ult is the maximum value of S after applying infinite cycle loading and S 1 is the
material parameter.
Stress-dilatancy relationship
The stress-dilatancy relationships of the drained cyclic torsional shear loading tests on saturated
Toyoura sand (SAT11 and SAT12) are shown in Figure 1 (De Silva et al., 2014). In this figure,
p
τ rev is the shear stress at the reversal loading point, ε vol
is the plastic volumetric strain and ε p is
the amount of deviator strain. This result indicates that the dilatancy ratio varies almost linearly
with the stress ratio. De Silva et al. (2014) proposed a bi-linear stress-dilatancy model based on
these experimental results. For the sake of simplicity, a linear relationship between the dilatancy
ratio and the stress ratio is adopted in this study. The linear stress-dilatancy relationship on the
yield surface takes the form
p
 εvol

k y = − A1
 ε p


 + B1


(11)
in which A 1 , B 1 are the material parameters. k y corresponds to the ordinate value with τ rev = 0 in
Figure 1. The stress-dilatancy relationship on the active loading surface takes the form
 ε p
2k a − k r = − A1  volp

ε

 + B1


(12)
in which k r is the internal variable for the reversal surface. k a and k r in Eq. (12) correspond to
τ − τ rev 2 p and τ rev p in Figure 1, respectively. By using Eq. (11) and Eq. (12), the plastic
volumetric strain increment can be calculated form the plastic deviator strain increment.
0.8
0.6
0.4
(|τ - τrev|-|τrev|) / p
(|τ − τ rev|-|τ rev|) / p
0.8
0.6
A1=0.6
B1=0.43
0.2
0.0
-0.2
-0.4
A1=6.0
B1=0.43
-0.6
-0.8
0.4
0.2
0.0
-0.2
A1=0.6
B1=0.43
-0.4
-0.6
A1=6.0
B1=0.43
-0.8
-0.5
0.0
0.5
1.0
1.5
p
εvol
ε p
2.0
2.5
3.0
-0.5
0.0
(a) SAT11
0.5
1.0
1.5
p
εvol
ε p
2.0
2.5
3.0
(b) SAT12
Figure 1. Stress-dilatancy relationship
De Silva et al. (2014) suggested that the stress-dilatancy relationship during the virgin loading is
different from that within the phase transformation stress state. The inclination of the stressdilatancy relationship varies with the amplitude of the shear loading in SAT12 result in Figure 1.
In this study, two linear relationships are assumed for the stress-dilatancy relationship. The A 1
and B 1 values for the stress-dilatancy relationship before the phase transformation stress state are
assumed to be 6.0 and 0.43 and those values for the stress-dilatancy relationship after the phase
transformation stress state are assumed to be 0.6 and 0.43.
Numerical Example
Numerical analysis method
A three dimensional finite element analysis is performed to simulate the laboratory tests. A finite
element analysis code MuDIAN (Shiomi et al., 1993) involving the developed elasto-plastic
model is adopted in the simulation. In this study, the torsional shear loading condition is assumed
to correspond practically to the simple shear condition. Thus the specimen modelled as an
element and the boundary condition is set as the simple shear condition. One phase formulation
is adopted in the drained test simulations by assuming the perfectly drained condition. One phase
analysis in which the boundary condition is set as no volumetric change condition is conducted
in the undrained test simulations.
The parameters used in the simulations are shown in Table 1. The elastic modulus E 0 and
Poisson’s ratio ν are determined from the initial part of the stress-strain relationship of tests
SAT11 and SAT12. The values of C 1 (0), C 1 (∞), C 2 (0), C 2 (∞), a and β are set by fitting the
initial stress-strain curve obtained in test SAT11. The values of d 1 , S 1 and S ult are determined
from the test SAT11 and SAT12 results. The determination procedure of the values of A 1 and B 1
are mentioned in the previous section.
Simulation of drained cyclic torsional tests
The drained cyclic torsional shear loading tests (De Silva et al., 2014) are simulated by the
proposed model. Two tests, SAT11 and SAT12 are selected in the simulation. In these tests, the
specimens were subjected to isotropic compression up to 400 kPa and isotropic unloading to 100
kPa followed by drained large cyclic torsional shear loading. The initial relative density of
specimens is around 78%.
Table 1. Material parameters.
E 0 (MPa)
p 0 (kPa)
ν
C 1 (0
)
C 1 (∞)
C 2 (0
)
C 2 (∞
)
a
β
d1
S1
S ult
295
100
0.18
3200
3200
0.58
0.75
0.0
0.007
6.0
0.1
1.1
Before phase transformation stress state
After phase transformation stress state
A1
B1
A1
B1
6.0
0.43
0.6
0.43
The model predictions for test SAT11 are shown in Figure 2. Figure 2(a) shows the shear stressstrain response. Although there is some difference in strain amplitude between the experimental
and numerical results, the proposed model simulates reasonably the hysteresis curve observed in
the experimental result. Figure 2(b) shows the volumetric strain change during the shear cyclic
loading process. Despite adopting the simplified linear dilatancy relationship, the proposed
model captures the feature of the volumetric strain change occurring under the large cyclic
loading. However the volumetric strain change in the simulation result is larger than that in the
experimental result. Figure 2(a) shows that the simulation overestimates the shear strain. It seems
that such overestimate of the shear strain leads to the overestimate of the volumetric strain in the
simulation results.
0.40
80
Simulation
Simulation
60
0.35
Experiment
40
0.25
20
0.20
εv (%)
τ (kN/m2)
Experiment
0.30
0
0.15
0.10
-20
0.05
-40
0.00
-60
-80
-1.0
-0.05
-0.10
-0.5
0.0
0.5
γ (%)
(a) Stress-strain relationship
1.0
-80
-60
-40
-20
0
20
40
τ (kN/m2)
(b) Volumetric strain
60
80
Figure 2. Comparison between measured and computed results (SAT11)
80
0.50
Simulation
60
Simulation
0.40
Experiment
Experiment
0.30
40
0.10
εv (%)
τ (kN/m2)
0.20
20
0
0.00
-0.10
-20
-0.20
-40
-0.30
-60
-0.40
-80
-2.0
-0.50
-1.0
0.0
1.0
2.0
γ (%)
(a) Stress-strain relationship
-80
-60
-40
-20
0
20
40
60
80
τ (kN/m2)
(b) Volumetric strain
Figure 3. Comparison between measured and computed results (SAT12)
The model predictions for test SAT12 are shown in Figure 3. The amplitude of the applied shear
stress increases in test SAT12. Figure 3(a) shows that the simulated shear stress-strain response
agrees reasonably with the experimental response. Figure 3(b) shows that the proposed model
describes reasonably the volumetric strain induced the cyclic shear stress. These results indicate
that the numerical analysis can simulate the saturated sand behavior under the drained condition.
Simulation of undrained cyclic torsional tests
The undrained cyclic torsional shear loading tests on saturated Toyoura sand (De Silva, 2008)
are simulated by the proposed model. Test SAT28 is selected in the simulation. The specimens
was consolidated isotropically at the effective confining pressure of 100 kPa and applied by the
cyclic torsional shear loading under an undrained condition. The initial relative density of
specimens was around 75%. The amplitude of the cyclic shear stress was set to be around 30kPa.
The test SAT28 result is shown in Figure 4. The typical liquefaction behavior is observed in this
result. The mean effective stress decreases with the cyclic loading and the amplitude of the shear
strain increases drastically after the stress reaches the phase transformation stress state.
Subsequently, the cyclic mobility behavior appears and the amplitude of the shear strain
increases gradually with the cyclic loading. The model predictions for test SAT28 are shown in
Figures 5. The numerical analysis can appropriately describe the shear stress-strain response and
stress-pass observed in the experiment result, indicating that the adopted stress-dilatancy
relationship is suitable for describing the undrained cyclic shear behavior obtained from the
laboratory test. Figure 4 and Figure 5 suggest that the linear stress-dilatancy relationship defined
based on the drained cyclic loading test results could be applied to the behavior occurring under
the undrained cyclic loading condition.
40
30
30
20
20
10
10
τ (kN/m2)
τ (kN/m2)
40
0
-10
0
-10
-20
-20
-30
-30
-40
-40
-6
-4
-2
0
2
4
6
0
20
40
γ (%)
60
p
(a) Stress-strain relationship
80
100
120
(kN/m2)
(b) Stress pass
40
40
30
30
20
20
10
10
τ (kN/m2)
τ (kN/m2)
Figure 4. Experiment result (SAT28)
0
-10
0
-10
-20
-20
-30
-30
-40
-40
-6
-4
-2
0
2
4
6
0
γ (%)
20
40
60
80
100
120
p (kN/m2)
(a) Stress-strain relationship
(b) Stress pass
Figure 5. Simulation result (SAT28)
Conclusions
A three dimensional elasto-plastic model is proposed on the basis of the cyclic shear torsional
loading test results. In the proposed model, the linear stress-dilatancy relationship is defined
from the drained cyclic shear loading test result. The finite element analysis incorporating the
proposed model is conducted to simulate the laboratory cyclic loading tests. The simulation of
the experimental results demonstrates that the proposed model could describe the behavior of the
saturated sand under the undrained cyclic loading condition.
References
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their modelling. Ph.D. thesis, The University of Tokyo 2008.
De Silva LIN, Koseki J, Wahyudi S, Sato T. Stress-dilatancy relationships of sand in the simulation of volumetric
behavior during cyclic torsional shear loadings. Soils and Foundations 2014; 54(4): 845-858.
Mroz Z, Norris VA, Zienkiewicz OC. An anisotropic critical state model for soils subject to cyclic loading.
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Namikawa T, Koseki J, De Silva LIN. Three-dimensional modeling of stress-strain relationship of sand subject to
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