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. Namikawa1, J. Koseki2, 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 one-dimensional 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

1Professor, Department of Civil Engineering, Shibaura Institute of Technology, Tokyo, Japan, namikawa@shibaura-it.ac.jp 2Professor, Department of Civil Engineering, University of Tokyo, Tokyo, Japan, koseki@civil.t.u-tokyo.ac.jp 3General Manager, Takenaka Civil Engineering & Construction Co., Ltd, Tokyo, Japan, tsukuni-s@takenaka-doboku.co.jp

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 ijε is a sum of the elastic component e

ijε and plastic component p

ijε . The elastic strain increment is linearly related to the effective stress increment ijσ as

klijkleij E σε 1−= (1)

in which Eijkl 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. (2) where p0 is the reference effective mean stress and E0 is the elastic modulus at p0. The plastic strain increment is expressed as the following equation, (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 fy during the primary loading process and the active loading function fa during the reverse loading process.

oppEE 0=

ij

klklp

ijQ

H

F

σ

σσ

ε∂∂∂

∂

=

Yield and active loading function A conical yield surface in the stress space is assumed to describe the shear plastic strain. The yield function fy is given as :

021 =−= y

sij

sijy pkf σσ

(4) in which s

ijσ is the effective deviator stress and ky is the internal variable for the yield function. The active loading function similar to the yield surface is given as :

( )( ) 021 =−−−= aij

sijij

sija pkppf aσaσ

(5) in which aij is the center of the active loading surface and ka 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 :

sij

y

ij

fQσσ ∂

∂=

∂∂ (yield surface) , s

ij

a

ij

fQσσ ∂

∂=

∂∂ (active loading surface) (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 ky and ka are given as : , , (7) in which sp

ijε is the deviator plastic strain occurring on the primary loading process and spaijε 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), C1 and C2 are the variable coefficients. C1 and C2 vary with the strain level, given as

( ) ( ){ } ( ) ( ){ }

( ) ( ){ } ( ) ( ){ }

+

∞−+

∞+=

+

∞−+

∞+=

120

20

120

20

22222

11111

xcosCCCCC

xcosCCCCC

βπ

aπ

(8)

spaij

spaij

pa εε=ε

21sp

ijspij

p εε=ε21

2121 21

2,1

SCDC

k

SCDC

k pa

pa

ap

p

y ε

ε

εε

+=

+=

in which C1(0), C1(∞), C2(0), C2(∞), 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 ∑ pε as the following equation.

( )∑−= pdD ε1exp (9) in which d1 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 ∑ pε as the following equation.

1

1

1 −∑+

∑+=

ult

p

p

SS

Sε

ε

(10)

in which Sult is the maximum value of S after applying infinite cycle loading and S1 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, τrev is the shear stress at the reversal loading point, p

volε is the plastic volumetric strain and pε 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

11 BAk p

pvol

y +

−=

εε

(11)

in which A1, B1 are the material parameters. ky corresponds to the ordinate value with τrev = 0 in Figure 1. The stress-dilatancy relationship on the active loading surface takes the form

112 BAkk p

pvol

ra +

−=−

εε

(12)

in which kr is the internal variable for the reversal surface. ka and kr in Eq. (12) correspond to

prev 2ττ − and prevτ 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.

(a) SAT11 (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 stress-dilatancy 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 A1 and B1 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 E0 and Poisson’s ratio ν are determined from the initial part of the stress-strain relationship of tests SAT11 and SAT12. The values of C1(0), C1(∞), C2(0), C2(∞), a and β are set by fitting the initial stress-strain curve obtained in test SAT11. The values of d1 , S1 and Sult are determined from the test SAT11 and SAT12 results. The determination procedure of the values of A1 and B1 are mentioned in the previous section.

-0.8-0.6-0.4-0.20.00.20.40.60.8

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(| τ −

τre

v|-| τ

rev|)

/ p

ppvol εε

A1=0.6 B1=0.43

A1=6.0 B1=0.43

-0.8-0.6-0.4-0.20.00.20.40.60.8

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

(| τ -

τ rev

|-|τr

ev|)

/ p

ppvol εε

A1=0.6 B1=0.43

A1=6.0 B1=0.43

-80

-60

-40

-20

0

20

40

60

80

-1.0 -0.5 0.0 0.5 1.0

τ (k

N/m

2 )

γ (%)

Simulation

Experiment

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-80 -60 -40 -20 0 20 40 60 80

ε v(%

)

τ (kN/m2)

Simulation

Experiment

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.

E0 (MPa) p0 (kPa) ν C1(0)

C1(∞) C2(0)

C2(∞)

a β d1 S1 Sult

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 stress-strain 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.

(a) Stress-strain relationship (b) Volumetric strain

-80

-60

-40

-20

0

20

40

60

80

-2.0 -1.0 0.0 1.0 2.0

τ(k

N/m

2 )

γ (%)

Simulation

Experiment

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-80 -60 -40 -20 0 20 40 60 80

ε v(%

)

τ (kN/m2)

Simulation

Experiment

Figure 2. Comparison between measured and computed results (SAT11)

(a) Stress-strain relationship (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

-20

-10

0

10

20

30

40

0 20 40 60 80 100 120

τ(k

N/m

2 )

p (kN/m2)

-40

-30

-20

-10

0

10

20

30

40

-6 -4 -2 0 2 4 6

τ(k

N/m

2 )

γ (%)

-40

-30

-20

-10

0

10

20

30

40

0 20 40 60 80 100 120

τ(k

N/m

2 )

p (kN/m2)

-40

-30

-20

-10

0

10

20

30

40

-6 -4 -2 0 2 4 6

τ(k

N/m

2 )

γ (%)

(a) Stress-strain relationship (b) Stress pass

Figure 4. Experiment result (SAT28)

(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

De Silva LIN. Deformation characteristics of sand subjected to cyclic drained and undrained torsional loadings and 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. Geotechnique 1981; 31(4): 451-469.

Namikawa T, Koseki J, De Silva LIN. Three-dimensional modeling of stress-strain relationship of sand subject to large cyclic loading. 5th International Conference on Earthquake Geotechnical Engineering 2011;No.TMSN, CD-ROM.

Shiomi T, Shigeno Y, Zienkiewicz OC. Numerical prediction for model No.1. Verification of numerical procedures for the analysis of soil liquefaction problems, Balkema: 213-219.

Tatsuoka F, Shibuya S. Deformation characteristics of soils and rocks from field and laboratory tests, Keynote Lecture (Session 1). Proceedings of 9th Asian Region Conference on SMFE 1992: 101-170.