LEAP-2017 Centrifuge Test Simulation Using HiPER

Simulations by effective stress analysis were carried out for nine centrifuge experiments of a gentle sloped ground. The constitutive equation is a hyperbolic model as a stress–strain relationship and a bowl model as a strain–dilatancy relationship. The experimental results largely vary depending on density and input acceleration. The analyses results can explain such a tendency of variability.

function. The hyperbolic stress-strain model parameters are determined from dynamic deformation tests (G/G max -γ, h-γ relationships). The bowl model parameters are determined from liquefaction resistance tests (from the relationship between stress ratio and number of cycles).

Hyperbolic Model and Its Parameters
In multidimensionalizing the hyperbolic model, the shear stress versus shear strain relationships for the shear component and the axial difference component, respectively, are defined by the following equations.
When solving two-dimensional problems, the hyperbolic model is used for the shear components τ xy À γ xy and the axial difference components (σ x À σ y )/2 À ε x À ε y.
Here, G max is the initial shear modulus, and γ r is the reference strain. γ r is obtained from the shear strength τ f by the following equation.
The h-γ relationship is given by the following equation.
where, h is hysteretic damping parameter, h max is the maximum damping ratio and G is shear modulus. Three parameters are required to construct the hyperbolic model; G max , h max , and γ r . Of these, G max and γ r are functions of effective stress. If G max and γ r at a certain reference effective stress σ 0 mi are G maxi and γ ri, then G max and γ r satisfy the following equations.
As the effective stress varies, at each incremental calculation step, these parameters are calculated as they vary with time based on the above equations. At the same time, the shear stress versus shear strain relationship in Eq. 23.1 varies with time in accordance with the effective stress.
Applying the Masing rule to the hyperbolic model results in excessive hysteretic damping. Therefore, hysteretic damping h is adjusted using the method described by Ishihara et al. (1985).
The three parameters of the hyperbolic model (G maxi , h max , γ ri ) are defined in Table 23.1 and Fig. 23.1. As shown in Eq. 23.4 above, G max and γ r depend on the effective stress. The formulation G maxi , γ ri is used when the mean effective stress σ 0 m is 1.0 kN/m 2 . These values can be simply determined from the stiffness reduction curve (G/G max versus γ relationship) or the damping increase curve (h-γ relationship) obtained from the dynamic deformation tests.

Bowl Model (Dilatancy Model) and Its Parameters
To model deformation in three dimensions, in addition to the shear strains (γ zx , γ yx , γ xy ) representing simple shearing deformation and the axial shear strain differentials (ε x À ε y , ε y À ε z , ε z À ε x ) representing axial deformation differential, The bowl model (as proposed by Matsuoka 1989, 1993) focuses on the resultant shear strain Γ and the cumulative shear strain G Ã . Dilatancy ε s v results from a certain soil particle repeatedly dropping into the valley between other soil particles (negative dilatancy), and rising up on other soil particles (positive dilatancy). This mechanism is represented by the superposition of positive and negative dilatancy using the following equation.
Here A, C, and D are parameters. ε G is monotonic negative dilatancy (compressive strain). It is irreversible and represented as a hyperbolic function with respect to G Ã . ε Γ is cyclic positive dilatancy (swelling strain) and is reversible and represented as an exponential function with respect to Γ. The ε G component is the master curve and is the component that determines the basic dilatancy during cyclic shearing, while the ε Γ component is a oscillating component associated with it. 1/D is the asymptotic line to the hyperbolic curve, corresponding to a relative density of 100%.
The mechanism of such a bowl model is the movement of soil particles in sevendimensional strain space with γ xy , γ yz , γ zx , (ε x À ε y ), (ε y À ε z ), (ε z À ε x ), and ε s v as axes. Figure 23.2 illustrates the case of unidirectional cyclic shearing. Next, the consolidation term is taken into consideration in the stress-strain relationship and effective stress is modeled under undrained (constant volume) conditions.
The volumetric strain ε v s represented by Eq. 23.7 is the dilatancy component due to shearing, but an additional volumetric strain due to change in the effective stress σ 0 m has to be considered. The total volumetric strain increment of the soil dε v is then given by the following equation, where the shear component is dε v s and the consolidation component is dε v c .
The consolidation term dε v c is given by the following equations, assuming one-dimensional consolidation conditions.
Here, C s is the swelling index, C c is the compression index, and e 0 is the initial void ratio. If undrained conditions (dε v ¼ 0) are assumed in Eqs. 23.9 and 23.10, the following equation is obtained.
If the mean effective stress in the initial shearing stage is σ 0 m0 , and if the above equation is integrated under the initial conditions (σ 0 m ¼ σ 0 m0 ), the following equation is obtained.
From Eq. 23.12, the effective stress under undrained shear σ 0 m is given by the following equations.
Therefore, substituting α from the above equation, the effective stress reduction ratio is given by the following equation: In order to suppress the occurrence of dilatancy under small shear amplitude, a spherical region with shear strain radius Γ ¼ R e is considered within the strain space, and within this region there is no dε G . R e is given as follows, with reference to Fig. 23.3. Figure 23.3a shows the liquefaction resistance curve and the relationship between liquefaction resistance and the lower limit value X l . X l is the liquefaction resistance after a very large number of cycles, in other words, it represents the stress ratio at which liquefaction does not occur even however many cycles occur. Also, for simplicity, it is assumed that the excess pore water pressure does not arise (P w ¼ 0) for repeated stress ratios less than X l . From Fig. 23.3b, c, setting the shear strain when the stress ratio is X l equal to R e , then R e is given by the following equation.
Here σ 0 m0 is the mean effective stress in initial shear. When the amplitude of the stress ratio is X l or less, positive excess pore water pressure does not arise.
There are six parameters in the bowl model (A, C, D, C s /(1 + e 0 ), C c /(1 + e 0 ) and X l ) as shown in Table 23.2. The values of these parameters are determined from a liquefaction resistance curve. Table 23.2 and Fig. 23.4 give the meanings of the bowl model parameters. These parameters are determined by fitting to the liquefaction resistance curve.

Hyperbolic Model
The initial shear stiffness G max is calculated from the following equation based on research by Liu et al. (2017). Here γ r and h max are standard values for sand based on the shear strain dependence of stiffness ratio and damping for Toyoura sand (G/G max -γ, h-γ relationship). Since the experiment of Ottawa sand was not carried out, it replaced it with Toyoura sand experiment. Figure 23.5 shows the G/G max -γ, h-γ relationship used in calculations. The larger the absolute value of A, the greater the cyclic mobility C, D Parameters representing the compression component ε G of the dilatancy components. 1/C is the slope of the dilatancy in the initial stage of shear. 1/D is calculated from the minimum void ratio e min on the hyperbolic asymptotic line (maximum amount of compression) C s /(1 + e 0 ) C s is the swelling index; e 0 is the initial void ratio C c /(1 + e 0 ) C s is the compression index; e 0 is the initial void ratio X l The lower limit value of the liquefaction resistance. In the relationship between stress ratio τ/σ 0 m0 and the number of cycles N c , it is represented by τ/σ 0 m0 when N c is sufficiently large. When τ/σ 0 m0 > X l excess pore water pressure arises

Bowl Model
The parameter D is determined from the following equation since 1/D is the asymptote of dilatancy. The average of e min is 0.49.

1
D ¼ e 0 À e min 1 þ e 0 ð23:17Þ The lower limit value of liquefaction resistance X l is determined based on the liquefaction resistance curve. The liquefaction resistance curve is calculated using values for standard sand for the other parameters. Parameter C is adjusted while comparing the test results and the calculation results so as to match the whole liquefaction resistance curve with the measured values.
The parameters for Ottawa sand set by this procedure are given in Table 23.3. Figure 23.6 shows the simulations of liquefaction resistance curves. They explain the experiment results for the three selected densities. Figure 23.7 shows the effective stress path and stress-strain relationship at these three different densities. In the dense case, strong cyclic mobility is present.

Determination of Parameters
The model parameters (that is to say the G/G 0 -γ, h-γ relationship and the liquefaction resistance curve) are determined by element test simulations of Phase I except for G 0 and D. G 0 and D are reset by the following procedure for each centrifuge test. From the density of sand (Table 23.4), the relative density D r is obtained, and the void ratio e is calculated from D r . The initial shear stiffness G 0 is calculated from Eq. 23.16 based on the research of Liu et al. 2017. G 0i is the value at unit mean stress (σ m ¼ 1.0 kN/m 2 ). Parameter D is determined from Eq. 23.17 since 1/D is the asymptote of dilatancy. The average of e min is 0.49. Table 23.5 shows the parameters set by this method. The Poisson ratio of the ground is set to 0.33. The bulk modulus of water is K w ¼ 2.2E + 6 kN/m 2 . For soil permeability, k ¼ 0.015 cm/s is used from experimental results.

Summary of the Numerical Simulations
The FEM mesh is shown in Fig. 23.8. The FEM model has 2013 nodes and 1920 elements. A plane strain four-noded quadrilateral element with displacements and pore pressure degrees of freedom at each node is employed. Four integration points   are used in each element. The u-p formulation is used. The excess pore water pressure is evaluated at the nodes (the Sandhu method). The nodes located at the base of the model are fully constrained in the x and y directions while the nodes on the side walls are constrained laterally. The nodes on the free surface allow full drainage and have a fixed pore water pressure. Figure 23.9 shows the initial stress calculated by linear analysis. The initial stress is determined by linear self-weight analysis. For stiffness, the value at 4 m of depth is used.
In the dynamic analysis, horizontal acceleration and vertical acceleration were input simultaneously.

Results of Centrifuge Simulations
Horizontal displacement time histories of the ground surface at the center are shown in Fig. 23.10. The amount of lateral spreading of the ground surface was of the same  order in experiment and analysis. For UCD3, Ehime2, and CUD1, the simulated values and experimental values show good correspondence. The analysis values are rather smaller for CU 2, ZJU 2, and NCU 3. The experimental show large variation depending on density and input acceleration. The simulations successfully explain this tendency. Figure 23.11 shows fivefold enlarged deformation plots and contours of excess pore water pressure ratio. For KAIST1, KyU3, and UCD1, liquefaction does not occur and deformations are small. In the other six cases, the ground liquefies and deforms in the downstream direction.  Fig. 23.12, shear strain accumulates in the direction in which the initial shear stress acts. The reverse warping tendency of the stress-strain relationship due to cyclic mobility is not noticeable.
Generally, horizontal deformation is large in the upper layer at the center of the slope. Vertical deformation is dominated by sinking on the upper side of the slope. Deformation peaks at the end of the excitation, and then recovers slightly during the process of dissipating excess pore water pressure. Excess pore water pressure is fully dissipated after 600 s.

Conclusion
We simulated the lateral spreading of sloping ground using a hyperbolic model and a bowl model for comparison against experimental results. The parameters of the constitutive equation were determined based on simulation of element tests. Experimental results show large variation with density and input acceleration. The simulations successfully explain this tendency to variability. The lateral spreading of the ground surface was of the same order in the experiments and simulations.