As shown in Figure 2, KTL-DYN10 dynamic triaxial equipment was selected for this experiment, and its hardware system consists of axial loading equipment, a controlling chamber of dynamic confining pressure, a back pressure controller, 8-channel high-speed control and acquisition equipment; The software system is a DSP high-speed digital control system with a maximum operating frequency of 10 Hz and a maximum dynamic stress amplitude of ±10 kN.

In this research, the tailings silty sand was taken from an iron tailings pond in Sichuan, and it is the main damming material of the tailings dam. Figure 3 shows the grading analysis of the tailings samples, the sample used in this test is Void ratio e =0.69, Specific Gravity G=2.725, The dry density of the sample after consolidation is controlled as ρ d =1.63 g/cm³.

In the process of sample preparation, according to the 《Code for Geotechnical Test: GBT 50123-2019》 (General Institute of Water Resources and Hydropower Planning and Design, 2019), the drying temperature was determined to be 110°C. In this test, a cylindrical sample with a diameter of 50 mm and a height of 105 mm was processed, which was compacted in four layers; at the same time, the sample density was controlled according to the dry density, and the dry density of samples was determined to be 1.63 g/cm³ according to the deposition law of the tailings pond. Then, prepared samples were installed as shown in Figure 4.

After the sample preparation, the first action was to use the back-pressure controller to test its air impermeability, and then the CO₂ saturation, water head saturation, and back-pressure saturation were subsequently performed to improve the sample saturation. After the sample saturation reached 98% (B>0.98), it was consolidated under the corresponding conditions; After that, the cyclic load with fixed frequency was applied for the test under the undrained condition.

In this test, the load frequency was set as 1 Hz, the dynamic stress was set as 70 kPa, the consolidation confining pressure was kept to be 100 kPa, and the cyclic load was subjected using the method of equal stress amplitude. For the isotropic consolidation ( K c = 1 ), the consolidation stress was loaded at one time; for the anisotropic consolidation ( K c > 1 ), the consolidation stress was loaded step-wise. The effect of different consolidation ratios on the dynamic characteristics of saturated tailing silty sand was studied by applying different axial static stresses, and the test scheme is shown in Table 1. In order to facilitate analysis and comparison, the failure strain was taken as the criterion for terminating the test in this paper: the failure strain of isotropic consolidation ( K c = 1 ) was defined as 5% of the dynamic strain ( ε d = 5 % ), and that of anisotropic consolidation ( K c > 1 ) ws defined as 5% of the total strain (including residual strain and dynamic strain)( ε = 5 % ) (Mei-Ying, 1980), while the limit equilibrium failure criterion was adopted in the theoretical analysis.

As shown in Figure 5, When K c = 1.0 ∼ 1.9 , the higher the K c , the more vibration times the sample needs to meet the failure standard; When K c = 1.9 ∼ 2.6 , the changing rule was opposite to the above; Therefore, it can be seen that the influence of the consolidation ratio K c on the dynamic strength of soil samples is very obvious. When K c = 1.0 ∼ 2.6 , the value of 1.9 was the turning point; when K c < 1.9 , dynamic strength gradually increased; when K c > 1.9 , dynamic strength first increased and then decreased. Thus, the existence of critical consolidation ratio is proved. When K c = 1 , the initial shear stress was zero, the reverse shear stress was the maximum, and the soil particle skeleton was in a certain equilibrium state. The cyclic load made the direction of the large principal stress axial rotated 90°, and the cyclic shear stress made the pore space uniformly distributed, which resulted in the occurrence of plastic compaction, and the increase in pore pressure due to the sliding of particle skeleton. When 1 < K c < 1.9 , the initial shear stress increased with K c , and the shear effect was induced during the test, which made the sand particles be rearranged and the deformation of the soil particle skeleton in a more stable state due to the disturbance of shear on the sand particle skeleton. The reverse shear stress slowly decreased with K c , its influence on the change of pore space and volume of samples decreased gradually during vibration, as well as the deformation rate and amplitude, thus the seismic stability and liquefaction resistance became higher. As shown in Figure 6, I, II and III represent stress circles under different consolidation ratios, ①, ②, ③ represents the distance between consolidation stress circle and strength envelope; It can be seen that ③<②<①. This indicates that when K c > 1.9 , the greater the difference between σ 1 and σ 3 , the larger the initial shear stress, the larger the diameter of the molar circle during static consolidation, and the less the distance from the Mohr-Coulomb envelope. For larger K c , even if a periodic load with small amplitude σ d was applied, the soil sample would also be deformed or damaged quickly; even without the dynamic load, the specimen would also occur shear failure as the static load was continuously applied by increasing σ 1 .

Figure 7 shows development curves of dynamic pore water pressure of the sample under different consolidation ratios K c , it can be seen that when K c = 1 , the pore pressure would finally reach the confining pressure and realize complete liquefaction; When K c > 1 , the pore pressure would always be less than the confining pressure, and the initial liquefaction would not be caused; besides, the larger the consolidation ratio, the smaller the pore pressure.

In order to analyze the influence of different consolidation ratios on the evolution of dynamic pore pressure of tailing silty sand, this paper used the limit equilibrium criterion to analyze the pore water pressure of materials under the limit equilibrium state, namely, the critical pore water pressure u c r (Ding-Yi et al., 1981; Li et al., 2021a; Liu et al., 2022; Zhou et al., 2022). It was assumed that the static limit equilibrium condition was also applicable to the dynamic test, and their Mohr-Coulomb failure envelopes were the same, which means the dynamic effective cohesion c d ′ and internal friction angle φ d ′ was respectively equal to static effective cohesion c ′ and internal friction angle φ ′ as shown in Figure 7. The stress circle ① represents the stress state of the sample before vibration, and the stress circle ② represents the maximum stress circle during dynamic load application, that is, the dynamic stress was equal to the instantaneous stress circle with an amplitude of σ d . During the process of dynamic loading, the pore water pressure in the sample would develop continuously, and the stress circle ② would move towards the strength envelope; when the pore water pressure reached the critical value u c r , the stress circle was tangent to the strength envelope, suggesting that the tailing silty sand reached the failure state according to the limit equilibrium condition.

According to the geometric conditions shown in Figure 8, the pore water pressure at the limit equilibrium state can be deduced as follows. u c r = σ 1 + σ 3 2 + c ′ tan ⁡ φ ′ − σ 1 − σ 3 + σ d 1 − sin ⁡ φ ′ 2 sin ⁡ φ ′ (1)

Where: c ′ and φ ′ is the static effective cohesion and static effective internal friction angle of soil samples; σ d is the dynamic stress amplitude; σ 1 and σ 3 represents axial pressure and confining pressure respectively.

Setting σ 1 = K c σ 3 , it can be obtained: u c r = K c + 1 2 + K c − 1 2 sin ⁡ φ ′ σ 3 + c ′ tan ⁡ φ ′ + σ d sin ⁡ φ ′ − 1 2 sin ⁡ φ ′ (2) making λ = K c + 1 2 + K c − 1 2 sin ⁡ φ ′ (3) λ is the critical pore pressure ratio Δ σ = c ′ tan ⁡ φ ′ + σ d sin ⁡ φ ′ − 1 2 sin ⁡ φ ′ (4) then u c r = λ σ 3 + Δ σ (5)

For common tailings, when σ d = 30 k P a , Δ σ ≈ 0 ; when σ d > 30 k P a , Δ σ < 0 , and it decreased with the increase in σ d .

When K c = 1 , by substituting, u c r = σ 3 + Δ σ ; when K c > 1 , aftering substituting λ = 1 , the following can be gotten: λ = K c + 1 2 + K c − 1 2 sin ⁡ φ ′ = 1 + sin ⁡ φ ′ 2 sin ⁡ φ ′ − 1 − sin ⁡ φ ′ 2 sin ⁡ φ ′ K c (6)

Because 0 < sin ⁡ φ ′ < 1 , the critical pore pressure ratio λ decreased with the increase of consolidation stress ratio K c . After substituting Eq. 6, When K c > 1 , the critical pore pressure ratio λ < 1 , and u c r < σ 3 + Δ σ (7)

Therefore, for a given dynamic stress amplitude, the pore water pressure will approach the confining pressure during isobaric consolidation, and during anisotropic consolidation, the critical pore pressure will be e less than the confining pressure, and the larger the consolidation ratio, the smaller the critical pore pressure.

In this paper, PFC 3D was used to simulate the standard indoor undrained dynamic triaxial test of saturated tailing silty sand, and the simulated effective confining pressure was controlled to be 100 kPa; the test was carried out using the loading method of equal stress amplitude. The upper and lower walls are loaded uniformly by the control program, and when the bias stress reaches the set stress amplitude q_cyc, the upper and lower walls reverse and continue to move until they reach the set stress amplitude q_cyc in the opposite direction and then reverse again.

During the loading process, the lateral strain ε 1 is equal to the axial strain − ε 2 by adjusting the movement rate of the lateral walls, at which time the specimen volume is kept constant, and this constant volume condition is comparable to the undrained condition in the conventional cyclic triaxial test.Based on the constant volume condition, the liquefaction characteristics of the sample were simulated (ZHOU et al., 2009). The advantage of particle flow numerical simulation is the ability to observe the evolution of the meso fabric parameters in the sample at different cyclic loading times while modelling macroscopic mechanical performance of sample liquefaction; thus, the internal relationship between the change of meso fabric and the macroscopic mechanical response in the process of sand liquefaction can be analyzed, so as to further explore the meso-mechanical mechanism of sand liquefaction (Yang et al., 2007; Liu et al., 2020). In this simuation, situations under K c = 1.0 , K c = 1.5 , K c = 1.9 , K c = 2.0 were taken as examples to analyze the coordination number and energy under different consolidation conditions. In this paper, the particle grading of the discrete element sample was determined first, as shown in Figure 9. After that, the stress-strain curve simulated by PFC triaxial shear test was comapred to the results of indoor triaxial shear test to calibrate the reasonable contact model parameters (AHMAD et al., 2019a; Yao-Hui et al., 2021), and these calibtared parameters were adopted for the subsequent dynamic triaxial test simulation and microscopic property analysis (JIANG, 2019). The comparison between numerical simulation and the indoor triaxial shear test is shown in Figure 10, and calibrated physical parameters of numerical model are shown in Table 2.

The mechanical coordination number refers to the average number of contacts per particle in the sample (MAHMOOD et al., 2020; AHMAD et al., 2021), excluding suspended particles with contact numbers less than or equal to 1. For granular materials, their mechanical properties are mainly affected by the density, which is affected by the indirect contact density of particles at the microscopic level (Thornton, 2000; Li et al., 2021b), which can be expressed by the mechanical coordination number of particles.

It can be seen from Figure 11 that with the increase of the consolidation ratio, the initial value of the mechanical coordination number became larger, as well as the number of particle contacts and the compactness of the sample, at the initial stage of cyclic loading, the mechanical coordination number of the sample continued to decrease. Figure 12 shows the changing curve of the instantaneous dynamic pore water pressure and corresponding mechanical coordination number curve while meeting the failure criteria under different consolidation ratios K c ; it can be seen from Figure 12A, when K c = 1 , when the “initial liquefaction” was initiated, and the mechanical coordination number decreased to about 3, which was reflected by the increase of pore water pressure under the undrained condition at the macroscopic level. At this time, the pore water pressure reached the confining pressure. According to Figures 12B–D, it can be seen that when K c = 1.5 , K c = 1.9 , K c = 2.0 , with the increase of the consolidation ratio, the decreasing amplitude of the mechanical coordination number decreased gradually during the cyclic loading of the sample. When the pore water pressure was stable, the mechanical coordination number did not fluctuate significantly, and the pore water pressure failed to reach the confining pressure. As the consolidation ratio increased from K c = 1 , K c = 1.5 to K c = 1.9 the time required for the sample to occur liquefaction became longer r; And from K c = 1.9 to K c = 2.0 , with the increase in consolidation ratio, the time required for the sample to reach the failure standard became shorter.

The energy storage and dissipation of particles during the loading process can represent their macroscopic mechanical response to a certain extent, and their failure actually experienced the evolution process of energy dissipation and instability (AHMAD et al., 2019b; AHMAD et al., 2020). The following will focus on the evolution law of The evolution law of elastic energy of particles (refers to the energy stored in the contact area of particles to deform the particles), particle friction (Bolton et al., 2008; Li et al., 2023), rolling energy dissipation (refers to the energy dissipated by sliding friction and anti-rolling between particles respectively) and boundary energy (refers to the energy generated by wall driving) (AHMAD et al., 2019c; Gao et al., 2023).

Figure 13 shows the energy evolution curves of the saturated tailing silty sand samples. it can be seen when the wall was subjected to cyclic loading and unloading, the wall would generate a certain amount of energy named boundary energy. At the same time, there was a certain amount of elastic energy stored at the contact of particles, and it increased and decreased successively with the loading and unloading process, but the decreasing amplitude was greater than the increasing amplitude so that the elastic energy was gradually released. In addition, under the effect of cyclic load, relative sliding (friction energy consumption) occurred between particles, which was followed by different degrees of rolling between particles (rolling energy consumption).

At the initial state, the elastic energy of particles increased with the increase of the consolidation ratio. As shown in Figure 13A, when K c = 1 , as the cyclic loading was subjected, the elastic energy of particles became 0, and the slope of particle friction dissipation energy and particle bending dissipation energy gradually decreased to approach 0; At this time, the sample began to occur initial liquefaction. As shown in Figures 13B–D, when K c = 1.5 , K c = 1.9 , K c = 2.0 , the particle elastic energy did not decrease to 0, while the particle friction dissipation energy and particle bending dissipation energy continued to increase, without showing signs of initial liquefaction. Moreover, with the increase of the consolidation ratio, the elastic energy of the remaining particles became larger, which was the liquefaction of the sample when K c = 1 on the macro level and pore water pressure reached confining pressure, and when K c > 1 , the sample did not occur liquefaction, and the pore water pressure decreased with the increase of consolidation ratio.