Evolution of the Damping Ratio Considering Cyclic Confining Pressure Under Intermittent Cyclic Loading

Abstract

The damping ratio is essential to conducting dynamic analysis for underground engineering under traffic loading. Variations in the damping ratio are usually studied using cyclic triaxial tests with continuous cyclic loading; however, intermittent loading is observed under traffic loading. Moreover, both the deviator stress and confining pressure vary cyclically. So far, the development of the damping ratio under intermittent cyclic loading with cyclic confining pressure has rarely been studied. Thus, cyclic triaxial tests with continuous and intermittent cyclic loading were conducted. Unlike continuous loading, where the normalized damping ratio progressively decreases, the corresponding variations under intermittent cyclic loading showed a sudden increase in the initial damping ratio at each restart. Critically, the cyclic confining pressure significantly reduced the normalized damping ratio, with greater attenuation under intermittent loading at higher cyclic confining pressures. In addition, an empirical model incorporating these effects for the damping ratio under intermittent cyclic loading was developed.

1. Introduction

The damping ratio is crucial for the dynamic analysis of underground engineering under traffic loads. Moreover, the damping ratio is usually determined with laboratory tests, whereas traffic loads are simulated with cyclic triaxial tests with continuous cyclic loading [1,2,3,4,5]. The variations in the damping ratio in terms of physical indexes and loading parameters were analyzed [6]. Okur et al. [7] analyzed the development of the damping ratio under different consolidated confining pressures and concluded that both the physical index and consolidated confining pressure had a strong influence on the dynamic behaviors of fine-grained soils. Ling et al. [3] analyzed the dynamic properties of frozen clays under cyclic loading and concluded that the damping ratio was higher with lower confining pressure and higher moisture content. Based on that, they developed a damping ratio calculation model with normalized accumulated axial strains. Accompanied by test results, some empirical formulas were proposed to calculate the damping ratio [8]. For example, Hardin and Drnevich [9] supposed that the stress–strain relationship satisfied a hyperbolic function and proposed a model to calculate the damping ratio for clays. Subramaniam and Banerjee [10] developed a model to calculate the damping ratio with some correction factors, in which the effects of both the plasticity index and modulus degradation were considered. Based on the above research results, there was a limitation that only the cyclic deviator stress was employed during the application of cyclic loading. However, both the confining pressure and deviator stress varied cyclically under traffic loads [11,12]. This indicated that the simulation of traffic loads was not accurate enough because the cyclic confining pressure was not incorporated into the laboratory tests. The simulation of traffic loads needs further study.

Reviewing previous studies, cyclic triaxial tests with variable confining pressure were conducted, in which cyclical variations in deviator stress and confining pressure were imposed on soils simultaneously. The developments of strain under cyclic loading with and without cyclic confining pressure were studied, and the differences in strain between the two test conditions were analyzed [13]. Moreover, the development of strain was also affected by the drained conditions under cyclic loading; the strain decreased as the cyclic confining pressure increased under undrained conditions [14], whereas it increased under partially drained conditions [15]. Similar test results were observed by Sun and Wang [15]. Huang et al. [16,17] concluded that the damping ratio of clay decreased as the cyclic confining pressure increased and developed a model for the damping ratio in relation to accumulated axial strains. However, there was a phenomenon in which the interval period of adjacent trains existed, meaning that the subgrade underwent intermittent cyclic loads [18,19]. Therefore, simulations of traffic loads need to consider the effect of intermittent loading as much as possible, and the variations in the damping ratio under intermittent cyclic loading require further analysis.

Reviewing previous studies, some scholars have conducted cyclic triaxial tests with intermittent cyclic loading [20,21,22]. Nie et al. [19] concluded that lower deformation of subgrade soil was obtained with greater intermittent time. Similar test results were found by Zheng et al. [21]. They considered that the adjustment of the soil skeleton was related to the duration of intermittent periods and analyzed the variation in excess pore water pressure. Moreover, the development of the resilient modulus was studied and increased with an increase in the duration of intermittent periods [23]. In addition, the effect of the drained conditions during intermittent periods was analyzed. The duration of intermittent periods was 60 min under partially drained conditions in the tests conducted by Yıldırım and Ersan [24]; however, the differences in deformation behaviors between undrained and partially drained conditions during intermittent periods were analyzed [25]. Although cyclic triaxial tests with intermittent loading were used to simulate the loading intermittence of traffic loads, and the mechanical properties of soil were studied under intermittent cyclic loading, this research mainly focused on the development of deformation and the variation in excess pore water pressure. There was little mention of the development of the damping ratio. Moreover, the cyclic confining pressure was not considered under intermittent cyclic loading.

Previous studies have concluded that intermittent loading and cyclic confining pressure need to be incorporated into the simulation of traffic loads. Moreover, the effects of intermittent loading and cyclic confining pressure under intermittent cyclic loading should not be ignored. As a result, research on the variation in the damping ratio under intermittent cyclic loading while considering cyclic confining pressure is required.

Two types of tests, i.e., cyclic triaxial tests with continuous and intermittent cyclic loading, were carried out; then, the differences in the damping ratio between the two types of tests were analyzed; moreover, the impact of cyclic confining pressure under intermittent cyclic loading was clarified; finally, an empirical model was used to calculate the damping ratio under intermittent cyclic loading with cyclic confining pressure.

2. Materials and Methods

2.1. Soil Descriptions

Undisturbed specimens were obtained with a thin-wall soil sampler in Zhuhai city with a burial depth of 12.0–14.0 m according to the Ministry of Water Resources of the People’s Republic of China [26]. The obtained specimens were stored in a cylindrical sample barrel and sealed with wax and plastic tape under constant temperature and humidity conditions. Furthermore, according to specifications of the Ministry of Water Resources of the People’s Republic of China [24], the fundamental physical–mechanical indices of the tested material were obtained. The unit weight was 17.60 kN/m³; the natural water content, the liquid limit, and the plastic limit were 48.60%, 51.90%, and 19.80%, respectively. The permeability of the specimens was poor, i.e., the coefficient of permeability was 2.26 × 10⁻⁷cm/s. Furthermore, the grain size distribution was obtained as follows: clay (diameter < 0.005 mm) and silt particles (0.005 mm < diameter < 0.075 mm) accounted for 35.1% and 64.9% of the overall composition, respectively. According to the specifications of ASTM [27], the tested soil was classified as CH.

2.2. Test Procedures

Cyclic triaxial tests were performed on the undisturbed specimens, in which advanced test equipment was used, i.e., a GDS dynamic triaxial system. The diagram of the GDS dynamic triaxial system is shown in Figure 1. The advanced device used here includes five parts, namely, an axial loading system, cell pressure controller, back pressure controller, data acquisition system, and measuring and controlling system. Here, the range and resolution of the axial displacement sensor are 100 mm and 0.208 μm, indicating that the advanced apparatus has high accuracy for measuring the deformation of soil. The axial loading system includes axial displacement sensors and an actuator. When the specimen is in contact with the axial load sensor, the initial value of the axial displacement sensor is cleared. Then, when this is combined with the upward movement of the base, the deformation of the specimen can be obtained by reading the axial displacement sensor.

To minimize the disturbance of the sample, specimens of 38 mm diameter and 76 mm height were hand-trimmed from the center of the cylindrical sample with the sample maker. Here, the excess soil samples could be removed using a wire saw by adjusting the size of the sample maker. Then, many cylindrical specimens were placed on the base of the apparatus for saturation, where both back pressure and confining pressure were applied simultaneously. To protect the stability of the sample, an effective consolidated confining pressure of 20 kPa was used. When the Skempton B-value exceeded 0.95, the saturation process of the specimen was completed. Then, when this was combined with the increasing confining pressure, all specimens were isotropically consolidated under a constant back pressure. When the specimen’s drainage rate was less than 100 mm³/h [15], the consolidation of the specimen was completed, and the corresponding effective consolidated confining pressure was 100 kPa. After that, cyclic loading was performed on the isotropically consolidated specimens.

Figure 2 shows the schematic diagram of cyclic triaxial tests with intermittent cyclic loading; Figure 2a presents the variations in the cyclic confining pressure, and Figure 2b shows the cyclic deviator stress under intermittent cyclic loading. Here, four loading stages (i.e., loading stages AC, CE, EG, and GI) were performed on the tested specimens to shorten the test duration and ensure stable operation of the testing system based on previous studies [18,19,24]. Each loading stage was divided into cyclic loading and intermittent periods. During cyclic loading periods, i.e., stages AB, CD, EF, and GH, the cyclic stresses were applied in a stress-controlled mode. Moreover, both stress waveforms were half-sine waves. To depict the amplitude values of both the cyclic deviator stress and cyclic confining pressure, two parameters, i.e., the inclination of the stress path (η) and the cyclic stress ratio (CSR), were used and defined as p^ampl/q^ampl and q^ampl/2P_o′, respectively [2,28], in which p^ampl and q^ampl are the amplitude of the cyclic mean principle total stress and cyclic deviator stress, and P_o′ is the effective mean principle total stress after isotropic consolidation.

Recalling Figure 1, there is a rubber sealing ring between the axial loading system and the base. Based on previous studies [16,29,30], greater cyclic loading can wear the rubber sealing ring, affecting the stable functioning of the test equipment. Thus, the amplitudes of cyclic stress should be reduced. In this study, three values of η = 1/3, 1.0, and 2.0 were used to study the effect of the cyclic confining pressure. η = 1/3 represents the test condition with constant confining pressure, whereas the other η values represent cyclic loading with cyclic confining pressure. Then, due to the limited number of specimens, the effect of the cyclic deviator stress was not considered. Thus, the value of CSR was constant; CSR = 0.20 was used (i.e., q^ampl = 40 kPa) based on the study of Huang et al. [29].

Moreover, an excessive number of cycles can wear the equipment sealing ring. Thus, the number of cycles employed should also be reduced. According to previous studies [22,30], the number of cycles during a cyclic loading period was 1000 cycles.

Recalling Figure 1, measuring excess pore water pressure accurately in low-permeability soil proved problematic because the pore water pressure was only measured at the bottom of the sample [14,31]. The mechanical behaviors of soils are affected significantly by excess pore water pressure. Therefore, a smaller loading frequency was chosen for the variation in excess pore water with enough time, and a loading frequency of 0.1 Hz was utilized [32].

After that, considering the test results of Yıldırım and Ersan [24], the duration of the intermittent period, i.e., in stages BC, DE, FG, and HI, was 3600 s because the physical properties of soil in the literature are close to those of the soil tested in this study. When a train passes quickly, there is not enough time to discharge pore water because of the poor permeability of soft soil and the short duration of cyclic loading. Thus, the specimens are considered to be in an undrained condition during the cyclic loading period, and this condition can be replicated by closing the drainage valve. After the train passes, the excess pore water pressure caused by cyclic loading gradually decreases over a long period as the pore water is drained. Because the permeability of clayey soil is low, the pore water in specimens cannot be fully discharged during intermittent periods. Consequently, the clayey subsoils are deemed to be in a partially drained state [2,12]. The partially drained condition can be realized by opening the drainage valve in tests [28,33]. Therefore, the drainage valves were opened during intermittent periods.

To analyze the effect of intermittent loading, cyclic triaxial tests with continuous cyclic loading were conducted. Figure 3 presents a schematic diagram of the cyclic triaxial tests with continuous cyclic loading; Figure 3a presents the variations in cyclic confining pressure, and Figure 3b shows the cyclic deviator stress. Here, cyclic variations in both the confining pressure and deviator stress were imposed on the isotropically consolidated specimens under partially drained conditions, in which the opening drainage valve represented a partially drained condition. The amplitudes of cyclic stresses were consistent with those under intermittent cyclic loading. The total number of cycles was 4000, and these were divided into a loading stage every 1000 cycles. Moreover, the loading frequency of the tests was 0.1 Hz. The test parameters for each specimen are shown in

Table 1. Summary of the cyclic triaxial tests.

No. of Tests	Load Modes	Effective Confining Pressure Po′ (kPa)	η	${\mathit{\sigma }}_3^{\mathbf{ampl}}$ (kPa)	CSR	qampl (kPa)	Loading Cycles, N	Intermittent Time △t (s)	Drained Conditions
C01	Continuous cyclic loading	100	0.33	0	0.20	40	4000	/	Partially drained conditions
C02			1.00	27
C03			2.00	67
I01	Intermittent cyclic loading		0.33	0			1000 × 4	3600	Cyclic loading period: undrained conditions Intermittent period: partially drained conditions
I02			1.00	27
I03			2.00	67

.

3. Test Results

3.1. Determination of the Damping Ratio Under Cyclic Loading

A diagram of hysteresis loops under cyclic loading is shown in Figure 4. According to the results of Lin et al. [34], the damping ratio is calculated as follows:

$$D_N={\displaystyle \frac{A_{\mathrm{loop}}}{\pi A_{\mathrm{triangle}}}}$$

(1)

where D_N is the damping ratio at cycle N; A_loop and A_triangle represent the dissipated energy and total applied energy during one cycle, respectively. Furthermore, A_loop and A_triangle can be calculated with the area of the enclosed hysteresis loop and the area of the triangle ABC, respectively.

Recalling Figure 4, the above parameters A_loop and A_triangle can be presented as follows:

$$A_{triangle}={\displaystyle \frac{1}{2}}{q}^{\mathrm{ampl}}\mathrm{\Delta }{\epsilon }_N$$

(2)

$$A_{loop}={\displaystyle \sum _{i=1}^{n-1}\frac{1}{2}\left(q_{i+1}+q_i\right)}\left({\epsilon }_{i+1}-{\epsilon }_i\right)$$

(3)

where $\mathrm{\Delta }{\epsilon }_N={\epsilon }_{\mathrm{d},\max }-{\epsilon }_{\mathrm{d},\min }$ represents the variation in axial strain during one cycle. Furthermore, ε_d,max and ε_d,min are the maximum and minimum axial strain during one cycle; q_i and ${\epsilon }_i$ are the deviator stress and axial strain at each point, respectively; n is the number of points recorded during one cycle and is set to 50.

Taking the test results of cyclic triaxial tests in the continuous cyclic loading mode as an example, Figure 5 shows the relationships between the deviator stress and axial strain at 10, 100, 500, 1000, 2000, 3000, and 4000 cycles. Here, the residual axial strain produced in each cycle gradually decreases with an increasing number of cycles, implying that the hysteresis loop becomes flatter after several cycles. The obtained test results were consistent with previous studies conducted by Ling et al. [3], Sun and Wang [15], and Huang et al. [16,33]. As the hysteresis loop becomes flatter, the corresponding area of the hysteresis loop decreases. Recalling Figure 5, the A_loop values are 13.84, 7.09, 3.31, 1.87, 0.84, 0.60, and 0.31 J/m³ at 10, 100, 500, 1000, 2000, 3000, and 4000 cycles, respectively, based on Equation (3); the corresponding A_triangle values are 29.62, 26.29, 20.49, 18.61, 16.93, 16.18, and 16.05 J/m³ using Equation (2). Thus, the damping ratios are 0.148, 0.086, 0.051, 0.032, 0.016, 0.012, and 0.006 according to Equation (1), meaning that a greater damping ratio can be obtained with fewer cycles.

3.2. Evolution of the Damping Ratio Under Continuous Cyclic Loading

To clarify, the damping ratio of each cycle is normalized with the corresponding damping ratio of the first cycle, i.e., D_N/D₁. Here, the parameter D₁ was the damping ratio in the first cycle. Figure 6 shows the variation in D_N/D₁ under continuous cyclic loading. Here, the variations in D_N/D₁ under different cyclic confining pressure conditions are consistent; D_N/D₁ rapidly decays at small strains, and then the decay rate of D_N/D₁ decreases. This phenomenon was also observed by Huang et al. [16,33].

Moreover, a greater normalized damping ratio is obtained with greater cyclic confining pressure, implying that the reduction in the damping ratio increases with a decrease in cyclic confining pressure. However, the phenomenon is opposite to that obtained by Huang et al. [16], in which a lower normalized damping ratio can be obtained under conditions with a higher cyclic confining pressure. The possible reason for this phenomenon is that the drained conditions of the specimens are inconsistent under cyclic loading. For example, the D_N/D₁ values with η = 0.33 and 2.00 are 0.181 and 0.278, respectively, at the accumulated axial strain of 1.00%.

Furthermore, the damping ratio is lower under test conditions with η = 0.33 than that in test conditions with η = 2.00, implying that more energy is dissipated with greater cyclic confining pressure. Recalling previous studies, Ling et al. [3] considered that the variations in dynamic behaviors of soils were related to the generated accumulated axial strains under cyclic loading. Thus, the reason for the differences in the energy loss of specimens with different cyclic confining pressures may be related to the deformation behaviors under corresponding conditions; greater accumulated axial strains are generated with greater cyclic confining pressures under partially drained conditions. Figure 7 presents the differences in the hysteresis loops between η = 0.33 and 2.00 at 100 cycles. As seen, the areas of the hysteresis loops and the corresponding generated accumulated axial strains are different. The accumulated axial strains are 0.629% and 1.145% for the specimens with η = 0.33 and 2.00, respectively. Moreover, A_loop increases from 5.74 to 7.09 J/m³, and A_triangle increases from 21.79 to 26.29 J/m³; consequently, the damping ratio increases from 0.083 to 0.085.

3.3. Evolution of the Damping Ratio Under Intermittent Cyclic Loading

Figure 8 presents the development of D_N/D₁ for soft clay with different η-values. Figure 8a–c present the test conditions with η = 0.33, 1.00, and 2.00, respectively. Here, the trend of all curves is similar to that shown in Figure 6, as well as that shown in studies by some scholars [16,33,35]. Moreover, a greater D_N/D₁ value is obtained in the first loading stage, and D_N/D₁ decreases with the increase in loading stages.

The variations in the normalized damping ratio in each loading stage under continuous and intermittent cyclic loading are counted, as shown in Figure 9. Divisions of 1000 cycles are considered to constitute each loading stage under continuous cyclic loading, i.e., cycle numbers of 1000, 2000, 3000, and 4000 represent the first, second, third, and fourth loading stages. As seen, the evolutions of the variation in the normalized damping ratio (i.e., ΔD_N/D₁) are different between the two loading modes; a greater value of ΔD_N/D₁ is observed in the first loading stage, and then it gradually decreases under continuous cyclic loading. Most of the variations are developed in the first loading stage, reaching 0.918; however, ΔD_N/D₁ is greater in the first loading stage, while the corresponding variations of the other loading stages are similar under intermittent cyclic loading. This indicates that the attenuation degrees of the normalized damping ratios are different in each loading stage. For example, the values of ΔD_N/D₁ for the specimen with η = 1.00 are 0.728, 0.665, 0.674, and 0.684 with the increase in the loading stage under intermittent cyclic loading, and the corresponding values are 0.918, 0.047, 0.036, and 0.013 under continuous cyclic loading. The above test results demonstrate that the effect of intermittent loading cannot be ignored.

Furthermore, recalling Figure 5, during a cycle at the end of each loading stage (i.e., N = 1000, 2000, 3000, and 4000 cycles), both A_loop and A_triangle can be obtained under continuous cyclic loading. A_loop decreases from 1.87 to 0.31 J/m³, and A_triangle decreases from 18.61 to 16.05 J/m³; thus, the damping ratio drops from 0.032 to 0.006. Similarly, Figure 10 illustrates the variations in hysteresis loops at the end of each loading stage. Herein, both A_loop and A_triangle vary with the increase in the loading stage; the values of A_loop are 3.14, 2.02, 1.49, and 0.95 J/m³, and the corresponding A_triangle values are 23.72, 21.18, 19.62, and 18.88 J/m³. Consequently, the corresponding damping ratios are 0.042, 0.030, 0.024, and 0.016, respectively. The above test results mean that most of the energy is consumed in the first loading stage, regardless of the different types of tests.

The values of D₁ are 0.339, 0.147, 0.106, and 0.083 for each loading stage under intermittent cyclic loading with η = 2.00, meaning that the energy consumption during the first cycle of each loading stage decreases. Compared to the final damping ratio of the previous loading stage, the initial damping ratio of the current loading stage increases. Figure 11 shows the variations in the hysteresis loop at N = 1000 cycles for the first loading stage and N = 1 cycle for the second loading stage. Herein, A_loop increases from 3.14 to 11.39 J/m³, and A_triangle increases from 23.72 to 24.61 J/m³. Thus, the damping ratio increases from 0.042 to 0.147 when starting the second loading stage. This means that more energy is consumed, accompanied by an increase in the accumulated axial strain for the discharge of pore water during intermittent periods.

3.4. Evolution of the Damping Ratio with Various Cyclic Confining Pressures

Figure 12 shows the evolution of the normalized damping ratio with various cyclic confining pressures. Here, the attenuation degrees of D_N/D₁ are different; the normalized damping ratio for the specimen with η = 0.33 decays by 60.0%, while the corresponding attenuation of the normalized damping ratio for the specimen with η = 2.00 reaches 77.2%. Moreover, a lower normalized damping ratio can be obtained under conditions with higher cyclic confining pressures; the corresponding normalized damping ratios are 0.411, 0.326, and 0.228 with η = 0.33, 1.00, and 2.00. Figure 13 shows the variations in hysteresis loops under different η-values at the end of the third loading stage (i.e., 3000 cycles). Here, the hysteresis loops become flatter under test conditions with a greater cyclic confining pressure. Moreover, both A_loop and A_triangle decrease as the cyclic confining pressure increases; A_loop decreases from 7.39 to 1.49 J/m³, and A_triangle drops from 29.31 to 19.62 J/m³.

There is less energy dissipated during one cycle under test conditions with cyclic confining pressure compared to those without cyclic confining pressure. The reason may be related to the generated accumulated axial strain of specimens with different cyclic confining pressure; a greater accumulated axial strain is generated with lower cyclic confining pressure during cyclic loading periods under undrained conditions. Based on that, the accumulated axial strains decrease from 2.097% to 1.284% as η increases from 0.33 to 2.00, and the damping ratio decreases from 0.080 to 0.024. Consequently, a lower cyclic confining pressure corresponds to a larger damping ratio.

Furthermore, the variations in the normalized damping ratios in each loading stage (i.e., ΔD_N/D₁) for the specimens with η = 0.33, 1.00, and 2.00 are counted and shown in Figure 14. Here, a greater value of ΔD_N/D₁ is obtained in the first loading stage, while the differences in ΔD_N/D₁ in other loading stages can be ignored in all cases. For example, the ΔD_N/D₁ values are 0.728, 0.665, 0.674, and 0.684 in each loading stage for the specimen with η = 1.00. The presence of cyclic confining pressure leads to a greater value of ΔD_N/D₁. For example, in the first loading stage, the value of ΔD_N/D₁ for the specimen with η = 0.33 was 0.649, while the corresponding values were 0.728 and 0.875 at η = 1.00 and 2.00, respectively.

3.5. Damping Ratio Model Under Intermittent Cyclic Loading

Based on the above test results, the effects of both intermittent loading and cyclic confining pressure on the evolution of the damping ratio cannot be ignored. Thus, regarding previous studies [16], the following model is employed to calculate the damping ratio:

$${\displaystyle \frac{D_N}{D_1}}={\displaystyle \frac{1}{1+{\left(a{\epsilon }_p\right)}^b}}$$

(4)

where a and b are fitting parameters related to the cyclic confining pressure and intermittent loading.

Recalling Figure 8, the variations in the normalized damping ratio are fitted using Equation (4).

Table 2. Fitting parameters of Equation (4).

No. of the Loading Stage	η	a	b	D1	R2
First loading stage	0.33	2.715	0.592	0.381	0.935
	1.00	4.116	0.743	0.347	0.972
	2.00	6.111	1.237	0.339	0.998
Second loading stage	0.33	6.316	0.675	0.239	0.976
	1.00	10.127	0.643	0.203	0.989
	2.00	20.233	0.705	0.147	0.988
Third loading stage	0.33	12.805	0.585	0.200	0.989
	1.00	18.742	0.550	0.166	0.994
	2.00	40.548	0.551	0.106	0.977
Fourth loading stage	0.33	19.113	0.535	0.168	0.992
	1.00	27.613	0.530	0.137	0.997
	2.00	50.117	0.591	0.083	0.965

lists the fitting parameters for many cases. As seen, Equation (4) is better for calculating the damping ratio. Furthermore, an analysis is conducted on the correlation between the fitting parameters of Equation (4) and the factors (i.e., cyclic confining pressure and intermittent loading). Recalling Table 2, the variations in fitting parameters (i.e., a and D₁) versus the cyclic confining pressure show the opposite trends; the parameter a increases with an increase in the cyclic confining pressure, while the parameter D₁ decreases in each loading stage. Figure 15 depicts the relationships between D₁ and η in each loading stage. As seen, the values of D₁ are greater in test conditions without cyclic confining pressure than the corresponding values in test conditions with cyclic confining pressure. Moreover, the variations in D₁ concerning η can be depicted by a linear function as follows:

$$D_1=\alpha \eta +\beta$$

(5)

where α and β are fitting parameters.

Table 3. Fitting parameters of Equation (5).

No. of the Loading Stage	α	β	R2
First loading stage	−0.024	0.382	0.805
Second loading stage	−0.055	0.257	0.999
Third loading stage	−0.057	0.220	0.997
Fourth loading stage	−0.052	0.187	0.998

lists the fitting parameters of Equation (5) using regression analysis.

Figure 16 depicts the relationship between a and η. Here, the relationships for different loading stages can be depicted by a linear function, as shown in Equation (6).

$$a=c\eta +d$$

(6)

where c and d are regression parameters.

Furthermore, Figure 17 shows the variations in the fitting parameters in Equations (5) and (6) versus the loading stages; Figure 17a depicts the variations in parameters c and d, and Figure 17b presents the evolution of parameter β. Here, fitting parameters c and d show a linear increase with the loading stage, whereas parameter β decreases. The above relationships can be depicted as follows:

$$c=5.91n-3.17$$

(7)

$$d=3.01n-2.19$$

(8)

$$\beta =-0.06n+0.42$$

(9)

where n represents the loading stage index.

The variations in parameter b are not further studied due to their lack of clear representation through regression analysis. However, based on Table 2, the standard deviation and average value of parameter b under all test conditions are 0.194 and 0.661, respectively. Consequently, for simplicity, the value of parameter b in Equation (4) is considered constant at 0.661. The purpose of doing this is to facilitate the use of Equation (12). Similarly, the variation in parameter α in Equation (5) also cannot be described considering the above impact factors; thus, the average value of parameter α is taken as −0.047, and the standard deviation is 0.015. Taking Equations (7)–(9) into Equations (5) and (6), the formulas of fitting parameters a and D₁can be modified as follows:

$$a=(5.91n-3.17)\cdot \eta +3.01n-2.19$$

(10)

$$D_1=-0.047\eta -0.06n+0.42$$

(11)

Here, both Equations (5) and (6) are modified using Equations (10) and (11), considering the effects of the cyclic confining pressure and loading stage. Finally, combining Equations (10) and (11) and b = 0.661, Equation (4) is modified into Equation (12), which is related to the cyclic confining pressure and loading stage, as follows:

$$D_N=\left(-0.047\eta -0.06n+0.42\right)\cdot {\displaystyle \frac{1}{1+{\left\{\left[(5.91n-3.17)\cdot \eta +3.01n-2.19\right]\cdot {\epsilon }_p\right\}}^b}}$$

(12)

The application of Equation (4) is validated. The damping ratios at different cycles of each loading stage in Tests I01–I03, i.e., N = 1–10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, and 1000, are calculated using Equation (4). Figure 18 shows a comparison of the predicted results against test data. Herein, all data points are closely clustered around a line. Moreover, continuous cyclic loading could be regarded as a type of intermittent cyclic loading when the intermittent time equals zero. Moreover, the results of Tests C01–C03 and similar tests conducted by Liu et al. [35] and Huang et al. [33] are also used to validate the application of Equation (4).

Table 4. Fitting results.

No. of Tests	Soil	Reference	η	a	b	R2
C01	Clay	Present study	0.33	6.036	0.975	0.987
C02			1.00	12.377	0.777	0.972
C03			2.00	2.404	1.209	0.987
Y01		[33]	0.33	31.501	0.767	0.991
Y02			1.50	8.272	0.595	0.985
Y03			1.50	48.471	0.674	0.983
Y04			0.33	11.441	0.569	0.987
S4		[35]	0.33	12.744	1.345	0.984
S5			0.33	8.581	1.44	0.991

lists the fitting parameters. The comparison between the predicted results and the experimental results from the literature is shown in Figure 19. Herein, the predicted results align well with the measured data. Thus, combining Figure 18 and Figure 19, the proposed empirical model can also predict the damping ratio well under intermittent cyclic loading with cyclic confining pressure.

4. Conclusions

Two types of cyclic triaxial tests were carried out to study the evolution of the damping ratio, in which continuous and intermittent cyclic loadings were applied to undisturbed soft clay. The effects of intermittent loading and cyclic confining pressure were analyzed. Additionally, an empirical model was used to calculate the damping ratio under intermittent cyclic loading. The main findings of this research are presented in the following.

(1)

Unlike continuous loading, where the normalized damping ratio progressively decreases, the corresponding variations, except those in the first loading stage, showed similar variation across later stages under intermittent cyclic loading. The normalized damping ratio showed a sudden increase in the initial damping ratio at each restart under intermittent cyclic loading.

(2)

Higher cyclic confining pressure reduced the normalized damping ratio. Under intermittent loading, the attenuation was significantly greater at higher cyclic confining pressures; the normalized damping ratio for the specimen with η = 0.33 decayed by 60.0%, while the corresponding attenuation for the specimen with η = 2.00 reached 77.2%.

(3)

The developed model accurately predicted the damping ratio under intermittent loading, and it was validated against test and literature data. Fitting parameters linking the effects of cyclic confining pressure and intermittent loading were analyzed.

Due to the limited number of specimens and to ensure the normal operation of the test system, the loading parameters in this study were insufficient. Thus, the evolution of the damping ratio was not investigated deeply enough, and the fitting functions employed were only applicable to the tested soil under the loading conditions employed in this study. However, this research approach can be used in the study of other soils. Microscopic tests will be performed on specimens after the application of intermittent cyclic loading with different cyclic confining pressures to discuss the mechanisms underlying the phenomena. The obtained test results will be assessed with supplementary test results while considering more factors.