Analysis of the Dynamic Stability of Tailing Dams: An Experimental Study on the Dynamic Characteristics of Tailing Silt

Abstract

With the improvement in tailing mining-grade requirements and in mineral processing technology, tailing materials tend to be fine-grained. Under the action of earthquakes, a tailing dam is prone to liquefaction, which endangers the safety and stability of the dam. To further explore the dynamic properties of tailing silt under cyclic stress, through a series of dynamic triaxial experiments, we investigated the growth of the hysteresis curve, the development of pore pressure, and the energy dissipation law of tailing silt. The experimental findings indicated that increasing the density of the sample significantly improves its liquefaction resistance and the pore pressure development curve can be fitted using the BiDoseResp function. At the same cyclic stress ratio, the sample’s anti-liquefaction strength did not rise monotonically with increasing confining pressure but changed variably at values near a specified low confining pressure; when the sample density rose under the same settings, the specific confining pressure reduced. We also further discussed the evolution law of the stress–strain curves of tailing silt. The results further explored the dynamic characteristics of tailing silt, which can provide some reference for the seismic design and reinforcement measures of many fine-grained tailing dams.

1. Introduction

Tailings are artificial soil particles discharged from ores after they are crushed, ground, and beneficiated. The particle composition, based on the particle size, is as follows: sand particles (>0.075 mm), powder particles (0.005–0.075 mm), and clay particles (<0.005 mm). According to the difference in the particle composition content and plasticity index, tailings can be classified as sand tailings, powder tailings, and viscous tailings. Tailing silt refers to powder tailings with a particle size greater than 0.074 mm, a particle mass less than 50% of the total mass, and a plasticity index less than 10. Owing to improvements in the mining-grade requirements and beneficiation process of tailings, the average particle size of tailing materials is expected to reduce. The primary components of many tailing ponds have been transformed from coarse-grained soil to fine-grained soil. Tailings discharged from the ore dressing plant are gradually deposited upstream of the tailings under hydraulic filling. After sorting, tailing silt forms the primary component of several dam structures. Compared with other tailing materials, tailing silt has poorer consolidation and drainage, lower permeability, and an uneven discharge. With an increase in the dam height, the infiltration line of the dam rises, which increases the susceptibility to liquefaction during an earthquake [1,2]. Data from various sources show that earthquakes are the second major cause of tailing dam accidents [3], and the dynamic instability of many fine-grained tailing dams during an earthquake has become an unavoidable issue for various mining corporations [4,5,6].

Scholars worldwide have conducted extensive research on the stability of rock and soil and the vibration liquefaction of soil particles and have achieved a lot of results. Hardin et al. [7] summarized the law of stress–strain development and proposed the Hardin–Drnevich hyperbolic model through the dynamic triaxial test of sand, which can reflect the dynamic stress–strain relationship of different types of sand with a high fitting degree. Wang et al. [8] summarized the pore pressure generation and diffusion mechanism of soil in the process of vibration liquefaction and put forward the corresponding liquefaction possibility evaluation criteria. Li et al. [9] carried out dynamic triaxial testing on silt with different dynamic stress amplitudes, summarized the trend of excess pore pressure in the process of vibration liquefaction, and analyzed the influencing factors and causes of pore pressure changes. Chen et al. [10] proposed a polygon approximation method to calculate the damping ratio of soil. This method can be directly used to calculate experimental data, and the algorithm can be reused, which considerably simplifies the calculation of the damping ratio. Zhu et al. [11] compared the dynamic strength curves of offshore silt under different failure criteria, discussed the existence of the critical value of the dynamic strength of the sample, and proposed a pore pressure development model suitable for offshore silt. James et al. [12] conducted dynamic tests on samples from a gold mine located in western Quebec and found that the CRR (which reflects the liquefaction resistance) of the samples was not significantly affected by the effective consolidation stress. Liu et al. [13] studied the influence of multiple factors on the dynamic strength of gravel soil through dynamic triaxial tests and explored the influence of various factors on the deformation of the dynamic strength of the sample. They found that the plastic strain increases with an increase in the initial static deviatoric stress and dynamic stress and decreases with an increase in the confining pressure, load frequency, and consolidation ratio. Liu et al. [14] found that there is a critical state between pore pressure and energy when the cyclic stress ratio is between 0.21 and 0.24, the pore water pressure–energy model of the cyclic stress ratio and consolidation ratio of tailing silt is pre-liminarily established. Mo et al. [15] studied the connection between the cumulative plastic strain development mechanism and the energy dissipation of tailing silt and found that with an increase in the cyclic stress ratio, the viscous cumulative energy dissipation rate gradually exceeds the plastic strain cumulative energy dissipation rate. They also analyzed the four types of strain growth modes. Through a dynamic test of silty clay, Jie et al. [16] combined various factors, such as confining pressure, dynamic stress, frequency, and consolidation ratio, and analyzed the influence of these factors and their coupling on the cumulative plastic strain of the sample. Gao et al. [17] considered the influence of an effective confining pressure and dynamic stress ratio on the dynamic strength of the sample through the indoor dynamic triaxial test. They found that at the same dynamic stress ratio, the liquefaction resistance of calcareous sand samples is obviously enhanced by increasing confining pressure. Zhang et al. [18] discovered that when the confining pressure and consolidation ratio rise, tailing silt’s dynamic elastic modulus rises, whereas the damping ratio decreases. A modified formula for the maximum dynamic shear modulus and modulus attenuation that can normalize the influence of the consolidation ratio was proposed. Zhao et al. [19] carried out dynamic triaxial experiments on silt under complex stress paths. Under the same dynamic shear strain conditions, it was discovered that the higher the moisture content, the lower the dynamic shear modulus; in addition, the higher the dry density and confining pressure, the higher the dynamic shear modulus. Wang et al. [20] studied the influence of the dynamic stress ratio and confining pressure on the dynamic characteristics of saturated gravel sand. They found that the cumulative strain and pore water pressure of the samples are different under the conditions of increasing dynamic stress ratio at the same confining pressure and increasing confining pressure at the same dynamic shear stress ratio.

Most of the aforementioned research results are aimed at sandy soil with a coarser particle size, while there are relatively few studies on soil particles with a finer particle size, and most tests only consider the influence of a single factor on the dynamic strength of soil (e.g., only the dynamic stress or confining pressure applied to the specimen is changed during the test without considering the interaction between the factors). So, to improve the dynamic stability of a tailing pond, it is necessary to clarify the influence of different factors on the dynamic strength of tailing materials under cyclic load. Therefore, based on the existing research, a series of dynamic tests at different densities and under different consolidation conditions were carried out on tailing silt. The test scheme is innovative and easy to implement. This article further discusses the dynamic characteristics of pore pressure development, softening failure, and energy dissipation of tailing silt, which can serve as a valuable reference for the seismic stability of numerous tailing dams.

2. Content and Methods of Testing

2.1. Test Soil Sample

This experiment used tailing silt acquired from an iron tailing pond in Sichuan, China. The material used was the primary dam-building material for tailing dams. Figure 1 shows the grading analysis results for the tailing silt sample. The physical properties were as follows: sand content, 22%; silt content, 67%; clay content, 11%; plasticity limit,${\mathrm{W}}_{\mathrm{p}}$, 18.2%; liquid limit, ${\mathrm{W}}_{\mathrm{L}}$, 27.3%; and plasticity index, ${\mathrm{I}}_{\mathrm{p}}$, 9.1. The particle size primarily ranged from 0.005 to 0.075 mm. Because the content of particles with a size greater than 0.075 mm was less than 50% and the plasticity index, ${\mathrm{I}}_{\mathrm{p}}$, was less than 10, the sample used in this experiment was defined as tailing silt. The uniformity coefficient, Cu, of the sample was 10.8, and the curvature coefficient, Cc, was 4.1. The analysis revealed that the particle size distribution of the sample was relatively concentrated, the particle distribution was not uniform, and the grading curve was not continuous, which is not conducive to the stability of the tailing pond [21].

2.2. Test Apparatus

This test used the KTL-DYN10 dynamic triaxial apparatus(Xian KTL Instruments Technology Company, Xian, China), as shown in Figure 2. The test instrument is mainly composed of a hardware system and a software system. The hardware system mainly includes data acquisition equipment, a pressure chamber, a back pressure controller, an eight-channel dynamic controller, axial loading equipment, and a dynamic confining pressure controller. The software system mainly includes GeoSmartLab software matched with the instrument. The device is capable of measuring the soil’s pore pressure and strain under cyclic stress. The sensors function normally, and the data are highly reliable.

2.3. Preparation of Samples

During sample preparation, the drying temperature was determined to be 110 °C according to the specification of the soil test (standard for the geotechnical testing method (GB/T 50123-2019)) [22]. A cylindrical sample was used in this test, and the density of the sample was controlled according to the dry density during sample preparation. The dry densities of the silty soil in the test tailings were determined to be 1.62 g/cm³, 1.67 g/cm³, and 1.70 g/cm³ according to the survey data of tailing dams.

2.4. Test Scheme

The air tightness of the sample was tested after it had been prepared, and then, CO₂ saturation, water head saturation, and back pressure saturation were used in turn to enhance the saturation of the sample. After the back pressure stage, the density of each saturated sample was tested to ensure that the saturation effect of the sample was good. After the consolidation of the sample, dynamic stress was applied to complete the liquefaction. At this stage, the pore pressure was defined to reach 95% of the confining pressure as the standard for the liquefaction of the sample.

In this test, the influence of the sample density and consolidation confining pressure on the dynamic characteristics of tailing silt was considered. The specific operation was to keep the cyclic stress ratio (CSR; $CSR={\sigma }_d/2{\sigma }_o^{\prime }=0.17$) constant while increasing both confining pressure and dynamic load. The molecules and denominators in the dynamic stress ratio represent the dynamic stress and the confining pressure, respectively, and their values directly reflect the dynamic strength of the sample, which is of great significance for determining the anti-liquefaction strength of tailings under different conditions. Therefore, the designed test scheme is shown in

Table 1. Experimental scheme.

Test Number	Density (g/cm3)	CSR	Consolidation Confining Pressure (kPa)	Termination of Cyclic Vibration (Nf)
1-1	1.62	0.17	50	460
1-2			100	190
1-3			150	97
1-4			200	35
1-5			250	165
1-6			300	295
1-7			400	510
1-8			600	980
1-9			1000	No liquefaction
2-1	1.67		50	670
2-2			100	220
2-3			150	380
2-4			200	650
2-5			300	1120
2-6			600	No liquefaction
3-1	1.70		100	483

to further study the interaction between different factors under the same CSR conditions.

3. Analysis of the Test Results

3.1. Theoretical Analysis of the Law of Pore Pressure Evolution of Tailing Silt

The tailing silt dynamic pore pressure evolution law is affected by various factors. To thoroughly examine the impact of the confining pressure, dynamic stress, and sample density on the evolution of tailing silt’s dynamic pore water pressure law, in this study, we adopted limit equilibrium theory to analyze the pore pressure value of the material under limit equilibrium (i.e., when the tailing silt is initially damaged), or, in other words, when the critical pore water pressure, ${\mu }_{cr}$, is affected by various factors [21].

The static limit equilibrium of the material is applicable to the dynamic test as well, and the Mohr–Coulomb failure envelope of the dynamic load and static load is the same. As shown in Figure 3, stress circle ① represents the stress state before vibration and stress circle ② represents the maximum stress circle in dynamic load application (i.e., the stress circle whose dynamic stress is equal to its amplitude). During the application of dynamic load, the pore pressure in the sample continues to develop and stress circle ② continues to move to the failure envelope. When the pore pressure reaches the critical value, stress circle ③ is tangential to the failure envelope. According to limit equilibrium theory, the tailing silt reaches the failure state in this condition.

According to the geometric conditions shown in Figure 3, the critical pore water pressure in the limit equilibrium state can be deduced as follows:

$$u_{cr}=\frac{{\sigma }_1+{\sigma }_3}{2}+\frac{c^{\prime }}{\tan {\phi }^{\prime }}-\frac{{\sigma }_1-{\sigma }_3+{\sigma }_d(1-\sin {\phi }^{\prime })}{2\sin {\phi }^{\prime }}$$

(1)

In Formula (1), $c^{\prime }$ and ${\phi }^{\prime }$ are the effective cohesion and effective internal friction angle of the tailing silt sample, respectively; ${\sigma }_d$ is the dynamic stress amplitude; and ${\sigma }_1$ and ${\sigma }_3$ represent the axial compression and confining pressure, respectively.

This test is isobaric consolidation ($K_c=1.0$), and ${\sigma }_1={\sigma }_3$; hence, Formula (1) can be simplified as:

$$u_{cr}={\sigma }_3+\frac{c^{\prime }}{\tan {\phi }^{\prime }}+\frac{{\sigma }_d(\sin {\phi }^{\prime }-1)}{2\sin {\phi }^{\prime }}$$

(2)

According to Formula (2), the critical pore water pressure is linked to the effective shear strength indices, $c^{\prime }$ and ${\phi }^{\prime }$$;$ the dynamic stress, ${\sigma }_d$; and the confining pressure, ${\sigma }_3$, during the test. Therefore, when the influence of a single factor on the critical pore water pressure is analyzed, other factors are assumed to remain unchanged.

(1)

Confining pressure’s effect on critical pore pressure

Formula (2) can be obtained as follows: Under the condition that other factors remain unchanged, the critical pore pressure, ${\mu }_{cr}$, is proportional to the confining pressure, ${\sigma }_3$. The greater the confining pressure applied to the sample, the greater the critical pore pressure, ${\mu }_{cr}$, and the stronger the liquefaction resistance of tailing silt.

(2)

Dynamic stress’s effect on critical pore pressure

As evident from Formula (2), the critical pore pressure, ${\mu }_{cr}$, is linearly related to the dynamic stress, ${\sigma }_d$; the linear proportional coefficient is $\frac{(\sin {\phi }^{\prime }-1)}{2\sin {\phi }^{\prime }}$; and the coefficient is always negative. Therefore, the larger the dynamic force provided by the sample, the lower the critical pore pressure, and the weaker the anti-liquefaction ability of tailing silt.

(3)

Sample density’s effect on critical pore pressure

With an increase in sample density, the angle of internal friction increases, the connection between tailing silt particles becomes stronger, the dynamic strength is bound to increase, the critical pore pressure (${\mu }_{cr}$) increases, and the liquefaction resistance of tailing silt increases.

3.2. Law of Dynamic Pore Pressure Development for Tailing Silt under the Same CSR Conditions

3.2.1. Pore Pressure Development Law for Tailing Silt at the Same CSR and Varied Confining Pressures

As shown in Section 3.1, the sample’s anti-liquefaction performance significantly rose as the consolidation pressure rose, and the anti-liquefaction strength gradually diminished as the dynamic load increased. However, if the test dynamic stress and consolidation confining pressure increase at the same time, the influence on the dynamic strength of the sample is still unclear. In this section, the combined impact of increasing confining pressure and dynamic load on tailing silt under the same CSR conditions is studied and the rule of pore pressure development is analyzed.

As shown in Table 1, when the CSR was 0.17, and a confining pressure of 50, 100, 150, 200, 250, 300, 400, 600, or 1000 kPa was applied to the sample, the termination of cyclic vibrations was considered as 460, 190, 97, 35, 165, 295, 510, and 980, respectively, and the sample did not attain liquefaction at ${\sigma }_o^{\prime }=100 \mathrm{kPa}$. Figure 4 shows the cumulative pore water pressure growth curve of tailing silt at the same CSR and different confining pressures. The trend in pore pressure development in the tailing silt under different confining pressures was approximately the same, and at low confining pressures (${\sigma }_o^{\prime }\le 200 \mathrm{kPa}$), the rising dynamic load was the major factor in tailing silt vibration liquefaction. This is because when the confining pressure is low and the dynamic stress is small, the sample exhibits a certain strength; hence, the number of vibrations for the sample is relatively greater. As the confining pressure continued to increase, the applied dynamic stress also increased. At this stage, the gradually increasing dynamic stress continued to overcome the smaller binding effect of lateral pressure on soil particles under the lower confining pressure, resulting in an increase in dynamic stress as the dominant factor in sample liquefaction at this stage and the achievement of a critical state under ${\sigma }_o^{\prime }=200 \mathrm{kPa}$. In this phase, the rise in dynamic load has a larger effect on the sample’s liquefaction resistance than the confining pressure does, and the sample’s liquefaction rate steadily increases. As the confining pressure increased, the soil particles were compressed under a greater confining pressure (${\sigma }_o^{\prime }\ge 200 \mathrm{kPa}$). The growing dynamic load at this stage was unable to overcome the binding effect of the lateral pressure on the soil particles under a greater confining pressure, resulting in the confining pressure increasing as the dominating factor of sample liquefaction at this stage.

The number of vibrations necessary for liquefaction significantly rose as the confining pressure rose. When ${\sigma }_o^{\prime }$ was 600 kPa, the number of vibrations necessary for liquefaction was 980, and the sample did not liquefy at ${\sigma }_o^{\prime }=100 \mathrm{kPa}$. This demonstrates that the impact of consolidation confining pressure on the sample’s anti-liquefaction ability always dominates at high confining pressure and the anti-liquefaction potential of the sample gradually increases with the continuous increase in confining pressure.

3.2.2. Pore Pressure Development Law for Higher Density Tailing Silt at the Same CSR and Varied Confining Pressures

With the CSR and other test conditions remaining the same, and only an increase in sample density $\rho =1.67 \mathrm{g}/{\mathrm{cm}}^3$, a consolidation confining pressure of 50, 100, 150, 200, 300, or 600 kPa was applied to the high-density tailing silt samples. The required number of vibrations for liquefaction was 670, 220, 380, 650, or 1120, respectively, and the sample did not achieve liquefaction at ${\sigma }_o^{\prime }=600 \mathrm{kPa}$. It can be seen from Figure 5 that the pore pressure growth trend of tailing silt under two different density conditions in this experiment was similar. As shown in Figure 6, under higher density, with an increase in confining pressure, the number of vibrations required for the liquefaction of the sample showed a trend of first decreasing and then increasing, and the number of vibrations required for the liquefaction of the sample was the lowest at ${\sigma }_o^{\prime }=100 \mathrm{kPa}$, with a value of 220. Therefore, the dynamic strength of the sample also has a critical state at higher density, and the number of vibrations corresponding to different test conditions in Figure 6 shows that the dynamic strength significantly increases with an increase in sample density. The time-critical confining pressure of the tailing silt was close to 200 kPa at $\rho =1.62 \mathrm{g}/{\mathrm{cm}}^3$ and close to 100 kPa at $\rho =1.67 \mathrm{g}/{\mathrm{cm}}^3$. Therefore, increasing the density of tailing silt at the same CSR significantly advances the liquefaction resistance critical point, as shown in the text explanation of Figure 6.

This is because when the confining pressure is low (${\sigma }_o^{\prime }=50 \mathrm{kPa}$), the effect of increasing dynamic load on the dynamic strength of the sample is somewhat larger than the effect of confining pressure and the rise in dynamic load still dominates sample liquefaction. As shown in Section 3.1, the soil particles of the tailing silt sample were more closely connected under high density, and the liquefaction resistance of the sample increased with an increase in critical pore pressure, ${\mu }_{cr}$. Therefore, when the confining pressure was greater than 100 kPa, the liquefaction resistance of the sample significantly improved at $\rho =1.67 \mathrm{g}/{\mathrm{cm}}^3$. The gradually increasing dynamic stress at this stage cannot overcome the binding effect of the lateral confining pressure on the soil particles under high density. Therefore, under the same conditions, the critical confining pressure significantly increases with an increase in sample density. If the confining pressure continues to increase, the vibration frequency required for liquefaction of the sample increases rapidly and liquefaction gradually becomes more difficult to achieve. This conclusion is consistent with the findings under low-density conditions and conforms with the conclusion reported by Gao et al. [17].

3.3. Model of Pore Pressure Growth for Typical Tailing Silt

For the cumulative pore pressure growth model of saturated soil, several scholars have performed extensive research and suggested corresponding dynamic pore pressure rise models. Among these, the inverse sine function model suggested by Seed et al. [23] is one of the more typical models. However, Seed’s proposed development model is only applicable to the cumulative pore pressure growth model in a single development stage, and the description of the cumulative pore pressure growth in the multiple stages of tailing silt is not ideal. In addition, Seed and others have generally used the vibration ratio as an independent variable to study the cumulative pore pressure development law, which cannot better reflect the difference in pore pressure curves. Therefore, this paper used the vibration number N as an independent variable to summarize the development law of pore pressure and found that its growth trend is consistent with the h₁–A₂ segment in the BiDoseResp function (as shown in Figure 7 and Formula (3)).

$$y=\frac{\mu }{{\sigma }_0^{\prime }}=A_1+(A_2-A_1)\times \left[\frac{c}{1+{10}^{(v_1-x)h_1}}+\frac{1-c}{1+{10}^{(v_2-x)h_2}}\right]$$

(3)

In Formula (3), A₁, A₂, c, v₁, v₂, h₁, and h₂ are the model parameters, where A₁ represents the initial value of the function development, A₂ represents the final value of the function, h₁ represents the slope of the rapid growth stage of pore pressure, h₂ represents the slope of the instantaneous failure stage, and c, v₁, and v₂ represent other parameters of the fitting function. To fit the pore pressure ratio–vibration number curves of saturated silty soil at different densities, the BiDoseResp function was used. Owing to space limitation, Figure 8 only shows the pore pressure ratio–vibration number curves at $\rho =1.62 \mathrm{g}/{\mathrm{cm}}^3$ and ${\sigma }_o^{\prime }=250 \mathrm{kPa}$, $\rho =1.67 \mathrm{g}/{\mathrm{cm}}^3$ and ${\sigma }_o^{\prime }=150 \mathrm{kPa}$, and $\rho =1.70 \mathrm{g}/{\mathrm{cm}}^3$ and ${\sigma }_o^{\prime }=100 \mathrm{kPa}$. The specific fitting parameters under other test conditions are shown in

Table 2. Fitting results for the model parameters.

Test Number	A1	A2	C	h1	h2	V1	V2	R2
1-1	−0.114	1.077	0.751	0.003	0.009	73.56	49.5	0.992
1-2	−0.173	1.302	0.503	0.006	0.017	10.12	202.3	0.993
1-3	−0.334	1.464	0.475	0.014	0.030	111.23	532.42	0.990
1-4	−0.203	1.112	0.521	0.026	0.14	72.64	80.37	0.994
1-5	−0.295	0.997	0.759	0.015	0.207	20.30	142.52	0.995
1-6	−0.286	1.413	0.533	0.026	0.322	283.57	421.42	0.994
1-7	−0.301	1.018	0.698	0.031	0.457	468.38	459.39	0.996
1-8	−0.072	1.213	0.701	0.044	0.422	540.58	354.85	0.991
2-1	−0.322	1.206	0.363	0.014	0.006	64.57	50.72	0.992
2-2	−0.271	1.023	0.664	0.042	0.018	226.65	193.97	0.997
2-3	−0.155	1.200	0.609	0.153	0.032	424.57	301.25	0.991
2-4	−0.204	1.197	0.563	0.168	0.178	677.57	186.89	0.990
2-5	−0.121	1.316	0.691	0.188	0.544	355.87	542.07	0.995
3-1	−0.103	1.143	0.642	0.342	0.159	674.5	348.4	0.990

. The fluctuation ranges of the test parameters v₁ and v₂ in each group were large, the other parameters’ fluctuation range was quite limited, and R² surpassed 0.990. The fitting effect achieved was good. As a result, the model may be deemed applicable to the cumulative pore pressure development of tailing silt at different densities. In addition, as shown in Table 2, at $\rho =1.67 \mathrm{g}/{\mathrm{cm}}^3$ and $\rho =1.70 \mathrm{g}/{\mathrm{cm}}^3$, the parameter h₁ was greater than that at $\rho =1.62 \mathrm{g}/{\mathrm{cm}}^3$, indicating that the pore pressure development rate increases with an increase in sample density when the cyclic load begins to be applicable; this is because the particle connection is relatively stronger and the pore is small at high density. When cyclic load is applied, the volume of pore pressure dissipation is limited, causing the pore pressure to rise fast at first.

4. Hysteresis Curve Analysis for Tailing Silt under Cyclic Loading

4.1. Hysteresis Curve Analysis of Typical Tailing Silt

The hysteresis curve shows the stress–strain relationship of a sample under cyclic loading, and its horizontal and vertical coordinates represent the axial and dynamic stresses of the sample, respectively, which can better indicate the dynamic strength and energy dissipation characteristics of soil particles. In this section, we examine the hysteresis curve of tailing silt at different confining pressures.

As shown in Figure 9, at $\rho =1.62 \mathrm{g}/{\mathrm{cm}}^3$ and ${\sigma }_o^{\prime }=200 \mathrm{kPa}$, the hysteresis curves of typical tailing silt at each stage showed axial compression as a whole cyclic loading and the strain increased with an increase in cyclic vibrations, which was followed by plastic failure; this reflects the deformation accumulation characteristics of tailing silt. With the application of cyclic load, the hysteresis curves of typical tailing silt can be divided into the elastic strain stage, plastic failure stage, and softening stable stage. As shown in Figure 9a, at the elastic strain stage, the overall shape of the hysteresis curves was relatively regular and symmetrical about the origin, indicating that the strain produced by tailing silt with an increase in the number of vibrations in the elastic strain stage is small and it can withstand high dynamic load and exhibit high liquefaction resistance. Additionally, this stage accounted for the largest number of vibrations required for the liquefaction of the sample. At this stage, the soil particles were primarily staggered and the deformation was small. Figure 9b depicts the plastic failure stage. At this stage, the maximum value of the hysteresis curves steadily diminished, the sample’s capacity to endure dynamic stress gradually declined, and the strain gradually increased. In addition, the tailing silt exhibited irreversible plastic deformation and obvious damage. The effective tension between many particles eventually passed to the pore water pressure, reducing the duration, and the sample entered the softening stable stage (as shown in Figure 9c) after a given number of vibrations. At this stage, the tailing silt accumulated multiple irreversible plastic deformations and the sample structure was completely deformed and showed softening. In addition, the dynamic stress of the sample remained unchanged with an increase in the number of vibrations and the dynamic strain increased steadily. This shows that the sample exhibits a certain cyclic activity.

4.2. Energy Dissipation Analysis of Typical Tailing Silt

As shown in Figure 9, the overall development of the hysteresis curves of tailing silt was relatively regular, and the figure is closed. The area enclosed by the hysteresis curves reflects the energy dissipated by the sample under a dynamic cycle. Under dynamic load, the energy dissipated by the sample was closely related to the strain failure of the sample and the liquefaction mechanism of the sample. Therefore, starting with the dynamic stress–dynamic strain relationship, we analyzed the energy dissipation characteristics of tailing silt under different conditions by calculating the area of the hysteretic curves, and the failure mechanism of the tailing silt under cyclic loads was also studied.

The calculation of the energy dissipation area primarily adopts the principle of spatial analytic geometry. The area of the triangle sandwiched by vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is 1/2 of the vector modulus of their cross-product, as shown in Formula (4). When the two-dimensional area is calculated, the first two values are 0.

$$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{cc}{a}_y& a_z\\ b_y& b_z\end{array}\right|i-\left|\begin{array}{cc}{a}_x& a_z\\ b_x& b_z\end{array}\right|j+\left|\begin{array}{cc}{a}_x& a_y\\ b_x& b_y\end{array}\right|k$$

(4)

In this experiment, 20 sets of dynamic stress–strain points were collected per vibration and corresponding polygons were formed. When the load frequency is 1 Hz, the independence of the area calculation of each vibration can be guaranteed. The area of dynamic stress–strain hysteresis curves can be calculated by calculating the polygon area. The calculation of the area of triangle OAB in Figure 10 is shown in Formula (5).

$$S_{OAB}=\frac{1}{2}\left|OA\times OB\right|=-\frac{1}{2}\left|\begin{array}{cc}{\epsilon }_{Ad}& {\sigma }_{Ad}\\ {\epsilon }_{Bd}& {\sigma }_{Bd}\end{array}\right|$$

(5)

Here, ${\epsilon }_{Ad}$ represents the dynamic strain corresponding to point A, ${\sigma }_{Ad}$ represents the dynamic stress corresponding to point A, ${\epsilon }_{Bd}$ represents the dynamic strain corresponding to point B, and ${\sigma }_{Bd}$ represents the dynamic stress corresponding to point B. Since the data collected in this experiment were clockwise, Formula (5) should be multiplied by −1/2. According to the method of superposition of data points and 20 datasets of dynamic stress–strain points, data points were collected for each vibration frequency; the area of the dynamic stress–strain hysteresis curve at vibration number i is shown in Formula (6).

$$Si=-\frac{1}{2}\left(\left|\begin{array}{cc}{\epsilon }_{1d}& {\sigma }_{1d}\\ {\epsilon }_{2d}& {\sigma }_{2d}\end{array}\right|+\left|\begin{array}{cc}{\epsilon }_{2d}& {\sigma }_{2d}\\ {\epsilon }_{3d}& {\sigma }_{3d}\end{array}\right|+\cdots +\left|\begin{array}{cc}{\epsilon }_{19d}& {\sigma }_{19d}\\ {\epsilon }_{20d}& {\sigma }_{20d}\end{array}\right|+\left|\begin{array}{cc}{\epsilon }_{20d}& {\sigma }_{20d}\\ {\epsilon }_{1d}& {\sigma }_{1d}\end{array}\right|\right)$$

(6)

As is evident from the derivation process, the coordinate quadrant and coordinate origin of the data point in Formula (6) are independent of the relative position of the hysteresis curve; thus, the stress–strain data can be used in the calculation of the energy dissipated by a certain cyclic vibration.

Figure 11 shows the dissipated energy, Si, per cycle at ${\mathsf{\sigma }}_0^{\prime }$ = 200 kPa, 400 kPa, and 600 kPa. As the number of cyclic vibrations grew, the energy dissipated with each vibration of the tailing silt gradually increased and tended to stabilize. Based on the size and trend of energy dissipation, the process can be divided into three stages: AB, BC, and CD. In Figure 11, section AB corresponds to the elastic strain stage at the early stage of cyclic loading. At this stage, the hysteresis curves gradually shifted and diffused based on the initial shape, and the maximum positive strain increased marginally. The area of the hysteresis curves remained unchanged or increased slightly, and the energy dissipation in the corresponding sample increased slightly, which manifested as the early vibration density of the tailing silt. Section BC corresponds to the plastic failure stage, at which point the area of the hysteresis curves increased rapidly and the corresponding energy dissipation also increased rapidly. This stage began to progress rapidly from point B to C at the softening stable stage of the sample, and the deformation resistance of the sample weakened considerably. Section CD corresponds to the softening stable stage. At this stage, the sample exhibited complete liquefaction, the area of the hysteresis curves decreased marginally and tended to stabilize, and the energy dissipation in the corresponding sample decreased marginally and tended toward constancy, exhibiting a certain cyclic tendency.

As shown in Figure 11, as the confining pressure applied to the sample increased, the final energy dissipation stability value (point D) of the sample increased gradually. This is because under the same CSR condition, the dynamic stress increases gradually, and the absolute value of the ordinate of the hysteresis curves increases such that the calculated energy dissipation area of the hysteresis curves also increases gradually. In addition, point B at the beginning of the plastic failure stage of tailing silt also increased gradually as the confining pressure increased. In addition to the rise in dynamic stress, the absolute value of the ordinate of the hysteresis curves increased, and the strain value of the abscissa was similarly affected by an increase in confining pressure. As the confining pressure increased, the lateral restraint force on the sample also increased, the degree of particle breakage increased when the sample produced the same axial strain, and more internal voids were released to provide greater dissipation space for the pore pressure. Therefore, additional deformation needs to be incorporated at the elastic strain stage to gradually accumulate the pore pressure to the plastic failure stage. The abscissa (strain value) of point B corresponding to plastic failure increased with a rise in confining pressure, resulting in an increase in the area of the hysteresis curves and energy dissipation.

5. Conclusions

In this study, a series of dynamic triaxial tests with the same CSR and different confining pressures were carried out on tailing silt from a tailing dam in Sichuan, China. The main conclusions are as follows:

(1)

Limit equilibrium theory was used to investigate the effect of various conditions on the sample’s critical pore pressure, ${\mu }_{cr}$, under limit equilibrium. With an increase in confining pressure, the increase in sample density and the decrease in dynamic load can lead to an increase in ${\mu }_{cr}$ and enhancement of the liquefaction resistance of the sample.

(2)

At the same CSR, the liquefaction resistance of the sample does not rise monotonically as confining pressure rises over time, but changes variably at values near a specific low confining pressure. The specific confining pressure value dramatically reduces when the sample density is merely raised under the same conditions, and the anti-liquefaction strength improves significantly. Therefore, in engineering, the dynamic stability of a tailing pond can be improved by increasing the compactness of tailing silt.

(3)

The pore pressure rising trend of the sample is the same under different confining pressures and densities, it can be matched using the BiDoseResp function, and the fitting effect is good.

(4)

The development of hysteresis curves of tailing silt can be divided into the elastic strain stage, plastic failure stage, and softening stable stage. The area of each hysteresis curve can be accurately calculated using the matrix formula. With an increasing number of vibrations, the area of the hysteresis curves and the dissipated energy increase gradually. With an increase in confining pressure, the overall dissipation area of the hysteresis curves gradually increases and shows cyclic activity.