Keywords

1 Introduction

The hypoplastic constitutive model has been utilized for all geotechnical applications of field studies since 1985. The original version of the hypoplastic model is coined by Kolymbas (1985) and is improved in further versions. The present improved version of the hypoplastic model is more advanced over the elasto-plastic models. P.‐A. von Wolffersdorff (1996) has described the mathematical formulations involved in this model. These formulations capture the influence of mean pressure and density along various deformation paths and the soil behavior bounded by asymptotic states including the widely accepted critical state, as stated by P.‐A. von Wolffersdorff (1996).

The hypoplastic constitutive relation requires eight parameters; the standard procedure required to determine these material parameters from real experimental data have been published by P.‐A. von Wolffersdorff (1996) and Herle (1997, 2008). This model requires three input data files including model parameters, initial soil state condition and test conditions for the analysis.

2 Hypoplastic Model

It is found that hypoplastic models are more advanced to elasto-plastic models for continuum modeling of granular materials. The first version of the hypoplastic constitutive model was proposed by Kolymbas (1985). It used a single-state variable, the current Cauchy stress Ts. Later another state variable, the void ratio (e), was added. The hypoplastic constitutive equation in general form is given by

$$T_{s} = F(T_{s} ,\,e,\,D)$$
(1)

Herein, Ts represents the objective stress rate tensor as a function of the current void ratio e, the Cauchy granulate stress tensor Ts and the stretching tensor of the granular skeleton D.

$$D = \frac{{L + L^{T} }}{2}$$
(2)

It is the symmetric part of the deformation gradient \(L = \frac{{{\text{d}}v\left( {x,t} \right)}}{{{\text{d}}x}}\), v being the velocity vector of the continuum representing the grain skeleton at a point x.

The present study has focused on performing the numerical simulations on silty sand under undrained cyclic loading.

2.1 Model Parameters

The hypoplastic constitutive relation requires eight parameters: the granular stiffness hs, the critical friction angle φc, the critical void ratio ec0 at zero pressure, the minimum and maximum void ratios ed0 and ei0 at zero pressure and the constants of n, α and β.

The silty sand used in this study is processed by mixing 40% quarry dust into the fine sand (herein, designated as SS40). The fine sand is collected from Cherthala, Kerala and quarry dust is procured from Blue Diamond M-sand manufacturers, Kattangal, Kerala. The details about the preparation of this combination and its basic properties are explained in Akhila et al. (2018). The model parameter is listed in Table 1.

Table 1 Hypoplastic model parameters
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3 Validation of the Model

The element test program has been prepared by Herle using mathematical formulations involved in hypoplastic soil model. The test program requires three input files, namely material parameters, initial state parameters and test conditions. The variation of pore water pressure with number of cycles for SS40 tested at CSR = 0.127 is shown in Fig. 1 (based on experimental test) and Fig. 2 (based on numerical simulation).

Fig. 1
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Experimental variation of pore water pressure with number of cycles for sand containing 40% fines for CSR = 0.127

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Fig. 2
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Numerical variation of pore water pressure with number of cycles for sand containing 40% fines for CSR = 0.127

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The number of cycles to reach the state of initial liquefaction is almost 55 for experimental and 60 for numerical simulations. The numerical and experimental results at other CSRs are also matching. The results of the same are shown in Fig. 3. Hence, the model simulations can be extended to find the effect of various parameters on the liquefaction susceptibility of silty sand.

Fig. 3
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Comparison of variation of pore water pressure with number of cycles for SS40

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4 Results and Discussion

The section discusses the numerical results obtained from the hypoplastic model. Initially, the effect of various parameters on liquefaction susceptibility of soil was studied by using hypoplastic model. The results are displayed in terms of stress paths, hysteresis loops, variation of pore water pressures and strain with no. of cycles, etc.

The parameters affecting soil liquefaction i.e., the magnitude of cyclic stress ratios, void ratio of soil and, consolidation pressures are studied with the help of element test program written using hypoplastic relations. The following sections discuss the effect of each parameter on liquefaction susceptibility.

4.1 Effect of Cyclic Stress Ratio

The silty sand in the present study is subjected to various cyclic stress ratios like 0.127, 0.152 and 0.178. The cyclic stress ratio is defined as the ratio of cyclic deviator stress to twice the applied effective consolidation pressure. The soil element having a void ratio of 0.74 is consolidated under the pressure of 100 kPa prior to applying the cyclic shear. The pore pressure variations with number of cycles, subjected to each CSR are overlapped in a single plot as shown in Fig. 4.

Fig. 4
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Effect of CSR on pore pressure variation with no. of cycles

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Figure 4 illustrates the effect of CSR on pore pressure buildup. The soil model subjected to CSR of 0.127 develops the gradual buildup of pore pressure and the 100% pore pressure build occur at about 60 cycles of loading. However, the pore pressure buildup is very rapid in the case of soil model subjected to high CSR values and 100% pore pressure buildup occur at a low number of cycles about 18 and 6 corresponding to CSR values of 0.152 and 0.178, respectively. The liquefaction resistance curve is shown in Fig. 5.

Fig. 5
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Liquefaction resistance curve for silty sand

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4.2 Effect of Void Ratio

To find the effect of void ratio on undrained response under cyclic loading, the simulations were performed under various void ratios like 0.6, 0.74 and 0.9. The soil model is consolidated under an effective pressure of 100 kPa and subjected to different cyclic loadings of 0.127, 0.152 and 0.178. The pore pressure variation with number of cycles for soil model analyzed at different void ratios and subjected to each CSR is overlapped in a single plot as shown in Figs. 6, 7 and 8.

Fig. 6
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Effect of void ratio on pore pressure ratio with no. of cycles for CSR = 0.127

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Fig. 7
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Effect of void ratio on pore pressure ratio with no. of cycles for CSR = 0.152

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Fig. 8
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Effect of void ratio on pore pressure ratio with no. of cycles for CSR = 0.178

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The soil model with a low void ratio of 0.6 develops the gradual buildup of pore pressure and the 100% pore pressure build occur at large cycles of loading irrespective of applied CSR values. However, the pore pressure buildup is very rapid in the case of the soil model with high void ratios and 100% pore pressure buildup occurs at a low number of cycles. It indicates that loose samples are causing a rapid buildup of pore pressure that leads to liquefaction in short duration and causes large deformations. The effect of void ratio on liquefaction resistance curves is shown in Fig. 9.

Fig. 9
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Liquefaction resistance curves with different void ratios for silty sand

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4.3 Effect of Consolidation Pressure

To study the effect of consolidation pressure on liquefaction susceptibility of silty sand, soil model is consolidated with different pressures of 100, 150 and 200 kPa. The soil model with the void ratio of 0.74 is subjected to different cyclic stress ratios of 0.127, 0.152 and 0.178 to construct the liquefaction resistance curves.

The soil model consolidated with low pressure of 100 kPa develops the gradual buildup of pore pressure and the 100% pore pressure build occur at large cycles of loading irrespective of applied CSR values. However, the pore pressure buildup is very rapid in the case of the soil model consolidated with high pressures and 100% pore pressure buildup occurs at a low number of cycles.

The pore pressure variation with a number of cycles for soil model analyzed at different consolidation pressure (100, 150 and 200 kPa) and subjected to each CSR is overlapped in a single plot as shown in Figs. 10, 11 and 12. The liquefaction resistance curve for the varying void ratio is shown in Fig. 13.

Fig. 10
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Effect of consolidation pressure on pore pressure ratio for CSR = 0.127

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Fig. 11
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Effect of consolidation pressure on pore pressure ratio for CSR = 0.152

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Fig. 12
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Effect of consolidation pressure on pore pressure ratio for CSR = 0.178

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Fig. 13
figure 13

Liquefaction resistance curves with different consolidation pressures

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5 Summary and Conclusions

Based on the discussion of numerical results, the following conclusions can be drawn with regards to the effect of various parameters on the liquefaction susceptibility of silty sands:

  • The pore pressure buildup and strain propagations are very rapid in the case of soil model subjected to higher magnitudes of cyclic stress ratios. It is interesting to note that the strain variation is insignificant before triggering the liquefaction and becomes rapid during and after triggering of liquefaction. At high CSR values, the soil model will liquefy in short duration of less number of load cycles.

  • The pore pressure buildup and strain accumulations are very rapid in the case of the soil model with high void ratios/low relative densities. It indicates, the loose soil samples are causing a rapid buildup of pore pressure and large deformations lead to liquefaction in short duration.

  • The required cyclic stress amplitude causing initial liquefaction is decreasing with increase in the void ratio at any particular number of load cycles. It indicates that the loose soils are liquefiable at lesser amplitudes of cyclic loads.

  • The pore pressure buildup and strain accumulations are very rapid in the case of soil model consolidated with high pressures. It indicates the consolidated samples at large pressures are causing a rapid buildup of pore pressure that leads to liquefaction in short duration and large deformations.

  • The required cyclic stress amplitude causing initial liquefaction is decreasing with an increase in the consolidation pressure at any particular number of load cycles. It indicates that the soils consolidated at large pressures are liquefiable at lesser amplitudes of cyclic loads.