WITHDRAWN: Study on accumulation deformation characteristics of silty clay based on dynamic triaxial tests

doi:10.21203/rs.3.rs-4886067/v1

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WITHDRAWN: Study on accumulation deformation characteristics of silty clay based on dynamic triaxial tests

https://doi.org/10.21203/rs.3.rs-4886067/v1

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Silty clay exhibits complex and significant deformation behavior under cyclic loading. This study systematically investigates the cumulative deformation characteristics of silty clay under varying moisture content, confining pressure, and cyclic stress ratio (CSR) using the GDS dynamic triaxial loading system. Through dynamic triaxial tests, the deformation patterns of silty clay under different conditions and the influencing factors were investigated. The results indicate that silty clay shows significant cumulative deformation characteristics under cyclic loading, where the rate of cumulative deformation decreases with increasing cycles until it eventually stabilizes. Moreover, moisture content, confining pressure, and CSR significantly impact the cumulative deformation of silty clay. Increases in moisture content, confining pressure, and CSR result in greater cumulative deformation. Specifically, a 50% increase in CSR results in a cumulative strain increment three times that caused by moisture content, whereas a 100% increase in moisture content yields a cumulative strain increment approximately four times greater than that caused by confining pressure. Additionally, after more than 1000 cycles, the cumulative deformation curve exhibits a distinct linear trend. This study offers essential experimental data and theoretical support for comprehending the cumulative deformation characteristics of silty clay under cyclic loading.

Physical sciences/Engineering/Civil engineering

Earth and environmental sciences/Solid earth sciences

Silty clay

Dynamic triaxial test

Accumulative deformation characteristics

Moisture content

Confining pressure

CSR

With the acceleration of urbanization and the expansion of engineering construction, the dynamic properties of soil and its behavior under engineering conditions have become a major focus of research in the field of engineering (Wang et al., 2023; De Morais et al., 2021). In engineering construction, silty clay, as a common foundation soil, plays a crucial role in the stability and safety of engineering structures due to its dynamic properties and deformation characteristics under cyclic loading (Xie et al., 2019; Yuan et al., 2023; Tiennot et al., 2020; Di Donna et al., 2015; Luo et al., 2022). Therefore, in–depth study of the cumulative deformation characteristics of silty clay under cyclic loading is of significant theoretical and practical importance for guiding engineering design and improving construction quality (Zhang et al., 2024).

In the field of soil dynamics, the cumulative deformation characteristics of soil under cyclic loading have always been a key area of interest (Zang et al., 2017; Lei et al., 2021). Cyclic loading refers to the phenomenon where soil undergoes persistent deformation due to repeated alternating loads, which can alter the internal structure of the soil and thus affect its mechanical properties. These mechanical properties are influenced by various factors such as confining pressure and temperature (Wang et al., 2023; Wu et al., 2023; Teng et al., 2023; Yang et al., 2023). Silty clay, as a typical compressible soil, exhibits complex and significant deformation behavior under cyclic loading, directly impacting the stability of foundations and imposing higher requirements on the safety of engineering structures (Liu et al., 2023; Dai et al., 2008; Tang et al., 2023). Researchers have conducted dynamic tests to explore the dynamic characteristics of soils, including the dynamic properties of cohesive soils and the dynamic modulus and damping ratio of saturated silty soils (Yu et al., 2019; Lai et al., 2016; Ma et al., 2005; Zhuang et al., 2023). Studies by Zhang and Sun on the stress-strain curve characteristics of frozen silty clay revealed that the failure process of the soil sample gradually exhibited brittle characteristics as the moisture content increased (Zhang et al., 2020; Sun et al., 2022). Cui's research on the hysteresis loop evolution of silty clay under cyclic loading found that the damage variable defined by the hysteresis loop was positively correlated with the number of cycles (Cui et al., 2020). Dynamic triaxial tests have been used to study the effects of freeze-thaw cycles on the damping ratio and dynamic nonlinear parameters of silty clay, revealing the changes in mechanical characteristics of silty clay under freeze-thaw cycles (Su et al., 2020; Su et al., 2021; Zhao et al., 2024; Zhang et al., 2024; Niu et al., 2024; Tao et al., 2022). To further investigate the mechanical properties of silty clay under different conditions, various indoor tests have been conducted to study the permeability, strength influencing factors, creep characteristics, and water absorption expansion stress coefficient of silty clay (Qian et al., 2016; Zhang et al., 2017; Gong et al., 2019; Wei et al., 2019; Guo et al., 2020).

This paper aims to systematically study the cumulative deformation characteristics of silty clay under cyclic loading through dynamic triaxial tests, exploring the cumulative deformation patterns under different water contents, confining pressures, and CSR. The findings provide theoretical basis and technical support for engineering design and practical engineering applications. The research will contribute to a deeper understanding of the cumulative deformation characteristics of silty clay under cyclic loading, providing scientific basis and technical support for engineering practice, as well as offering new ideas and methods for research in the field of soil mechanics.

2.1. Experimental soil samples

The experimental soil samples were taken near Dashiban Tunnel in Ya'an City, Sichuan Province, China. The grain size distribution curve of the soil sample is shown in Fig. 1. The laboratory tests determined the plastic limit moisture content of the soil sample to be 16.6%, liquid limit moisture content to be 27.4%, and plasticity index to be 10.8. Based on these test data, the soil sample was identified as silty clay.

The field–collected soil samples were dried and crushed, and the crushed soil was sieved to obtain particles with a diameter ≤ 1mm. Dry soil samples were weighed with an electronic balance accurate to 0.01g, and corresponding amounts of distilled water were added to achieve moisture contents of 8%, 12%, and 16%. The soil samples with added distilled water were thoroughly mixed, wrapped in cling film, and sealed for 24 hours to ensure uniform moisture content of the soil samples.

Cylindrical specimens measuring 100mm in height and 50mm in diameter (Fig. 2) were prepared using a tri–part mold, compacted to a dry density of 2.4g/cm³. After specimen preparation, the specimens were sealed in cling film and cured in a constant temperature and humidity chamber for 14 days.

2.2. Test program

The cyclic loading tests in this study utilized the GDS dynamic triaxial testing system manufactured in the UK (Fig. 3), with a sine wave loading waveform at a frequency of 1Hz.

Table 1 shows the test plan for cyclic loading. Three variables were considered: moisture content, confining pressure, and cyclic stress ratio (CSR). During the tests, while keeping other variables constant, the effects of moisture content, confining pressure, and cyclic stress ratio (CSR) on cumulative deformation were studied.

Table 1

Test plan.
Sample number	Moisture content	Confining pressure (kPa)	CSR
1	8%	50	0.3
2	12%	50	0.3
3	16%	50	0.3
4	12%	50	0.4
5	12%	100	0.4
6	12%	200	0.4
7	8%	200	0.3
8	8%	200	0.4
9	8%	200	0.45

Definitions for CSR and confining pressure are as follows:

$${\sigma _{{\text{d}}0}}={\sigma _1} - {\sigma _3}$$

$${\sigma _{\text{d}}}=\sigma _{1}^{{\hbox{max} }} - \sigma _{3}^{{\hbox{min} }}$$

$$CSR=\frac{{{\sigma _{\text{d}}}}}{{2{\sigma _{\text{c}}}}}$$

where σ_d0 is initial deviatoric stress, σ_d is cyclic deviatoric stress amplitude (Fig. 4), σ_c is effective confining pressure, and CSR is cyclic stress ratio.

Figure 5 illustrates the strain curve under cyclic loading. Under cyclic loading, the soil specimen exhibits axial strain, which includes recoverable strain ε_c and non–recoverable cumulative strain ε_a, with total strain ε_m being the sum of these two strains.

3.1. Influence of different moisture contents on cumulative strain

Figure 6 shows the variation of cumulative strain with the number of dynamic stress loading cycles for different moisture contents. Analysis of the cumulative strain variation curve reveals that initially under dynamic stress loading, cumulative strain shows rapid growth with a vertical rise in the cumulative strain curve. After approximately 300 cycles of dynamic stress, the growth rate of cumulative strain begins to slow down, and by 1000 cycles, the cumulative strain curve enters a phase of slow growth. The overall trend of the curve indicates that initially under dynamic stress loading, the rate of cumulative strain is high. After 1000 cycles, the rate of cumulative strain begins to decrease and eventually stabilizes. Total cumulative strain is affected by moisture content, increasing as moisture content increases. At 300 cycles, cumulative strains for moisture contents of 8%, 12%, and 16% are 2.695%, 3.797%, and 5.507% respectively; at 1000 cycles, cumulative strains are 3.101%, 4.453%, and 6.609%; and at 10000 cycles, cumulative strains are 4.377%, 6.638%, and 8.870% respectively for moisture contents of 8%, 12%, and 16%.

In this study, observation of the cumulative strain variation curve reveals that after 1000 cycles of dynamic stress loading, the cumulative strain curve exhibits a distinct linear characteristic (Fig. 7). This finding is significant for understanding the strain behavior of materials under cyclic loading conditions.

Specifically, under different moisture content conditions, through parameter fitting calculations, relationships (Eq. 4) between cumulative strain and number of cyclic loading cycles were obtained. Table 2 presents the fitting parameters for different moisture contents. The results indicate high fitting degrees ranging from 0.96132 to 0.99395, demonstrating a high fitting accuracy. This suggests that the adopted fitting formula can effectively reflect the relationship between cumulative strain and number of cyclic loading cycles under different moisture content conditions. The high fitting degrees indicate that despite the significant influence of moisture content changes on material strain behavior, accurate predictions of strain variations under different moisture content conditions can still be achieved through appropriate fitting models. This provides a reliable theoretical basis for further research and practical applications of the material in engineering.

In summary, through analysis of experimental data and establishment of fitting models, this study elucidates the cumulative strain characteristics of silty clay under dynamic stress loading conditions at different moisture contents. The high fitting degrees validate the effectiveness of the fitting formulas and provide important reference for the engineering application of the material.

$${\varepsilon _{\text{a}}}=g{\text{+}}hS$$

Here, g and h are fitting parameters.

Table 2

Linear fitting parameters for different moisture contents.
Moisture content	g	h	R²
8%	3.08593 ± 0.04024	1.31697E-4 ± 6.59603E-6	0.96132
12%	4.28654 ± 0.03143	2.33571E-4 ± 4.87006E-6	0.99395
16%	6.55952 ± 0.07009	2.45672E-4 ± 1.11441E-5	0.97194

3.2. Influence of different confining pressures on cumulative strain

Based on the results of moisture content influence on cumulative strain, we conducted a detailed analysis of the relationship between cumulative strain and the number of dynamic stress loading cycles under different confining pressure conditions. The variation curves in Fig. 8 show distinct phase characteristics of cumulative strain throughout the loading process.

Firstly, during cycles 0 to 300 of dynamic stress loading, the cumulative strain curve rapidly rises, indicating a high rate of cumulative strain. Specifically, at confining pressures of 50 kPa, 100 kPa, and 200 kPa, the rates of cumulative strain are 6.923E-3ε_a/S, 1.290E-2ε_a/S, and 1.058E-2ε_a/S, respectively. This rapid growth phase suggests that silty clay is highly sensitive to dynamic stress during initial loading, resulting in rapid accumulation of strain.

In the subsequent cycles from 300 to 1000, the curve transitions from rapid growth to slow growth, with the growth rate of cumulative strain slowing down. During this phase, the rates of cumulative strain at confining pressures of 50 kPa, 100 kPa, and 200 kPa decrease to 2.183E-3ε_a/S, 2.646E-3ε_a/S, and 3.311E-3ε_a/S, respectively. This transitional phase reflects the stabilization of strain mechanisms of silty clay under dynamic stress.

After exceeding 1000 cycles, the cumulative strain curve enters a phase of stable growth, maintaining a low growth rate, indicating a slow and stable increase. This suggests that under long–term cyclic loading, cumulative strain of silty clay stabilizes, with further dynamic stress loading having diminishing effects on cumulative strain.

Comparison and analysis of cumulative strain curves under different confining pressure conditions reveal that increasing confining pressure significantly affects the total cumulative strain. Specifically, at confining pressures of 50 kPa, 100 kPa, and 200 kPa, the total cumulative strains are 3.036%, 3.933%, and 4.496%, respectively. This indicates that with increasing confining pressure, the total cumulative strain of silty clay under dynamic stress loading also increases.

In conclusion, this study elucidates the cumulative strain characteristics of silty clay under dynamic stress loading at different confining pressure conditions. The phase changes in cumulative strain curves and the significant influence of confining pressure on total cumulative strain provide important experimental basis for further understanding the mechanical properties of silty clay under complex stress conditions.

Through analysis of strain curves under different confining pressure conditions, it was found that after a certain number of cycles, strain curves exhibit linear characteristics. Specifically, fitting analysis of data after exceeding 1000 cycles is shown in Fig. 9, with the linear relationship described by Eq. (5). This equation effectively reflects the relationship between cumulative strain and number of cycles under different confining pressures.

$${\varepsilon _{\text{a}}}=a{\text{+}}bS$$

In Eq. (5), parameters a and b represent the intercept and slope of the fitting curve, with specific values listed in Table 3. The fitting degrees range from 0.99111 to 0.99496, indicating a high degree of fit with actual data. This high fitting degree implies that Eq. (5) accurately captures the linear relationship between cumulative strain and number of cycles under different confining pressure conditions, applicable across various scenarios.

Further analysis of fitting parameters a and b reveals that variations in confining pressure significantly affect these parameters. With increasing confining pressure, parameter a (slope) also increases, indicating a greater rate of cumulative strain increase with cycle number under higher confining pressures. This finding is consistent with the earlier observation of confining pressure's impact on total cumulative strain, further validating the influence of confining pressure on strain behavior of silty clay under dynamic stress loading.

The high accuracy of the linear model through high fitting degrees enables more precise prediction of cumulative strain behavior of materials under long–term cyclic loading conditions under different confining pressures. This holds significant implications for assessing the performance of silty clay in practical engineering applications. Understanding the strain characteristics of silty clay under different confining pressures can assist researchers in predicting material performance over extended periods, thereby enabling more informed design and selection decisions.

In summary, this study reveals the linear relationship between cumulative strain and number of cycles under different confining pressure conditions through fitting analysis of strain curves. The high accuracy of fitting Eq. (5) validates its applicability, providing a reliable theoretical basis for predicting strain behavior of silty clay under complex stress environments.

Table 3

The linear fitting parameters for different confining pressures.
σ_c(kPa)	a	b	R²
50	2.10322 ± 0.01301	9.62396E-5 ± 2.02367E-6	0.99341
100	2.53772 ± 0.01718	1.38623E-4 ± 2.84752E-6	0.99496
200	3.14484 ± 0.01554	1.31635E-4 ± 2.72027E-6	0.99111

3.3. Influence of different CSR on cumulative strain

In this study, we also analyzed the relationship between cumulative strain and the number of loading cycles under different cyclic stress ratios (CSR), as shown in Fig. 10. Strain curves under different CSR conditions exhibit distinct phase characteristics, and the cycle numbers of these phase transitions correspond to those observed under different moisture content and confining pressure conditions.

Specifically, during cycles 0 to 300, the cumulative strain curves show a rapid growth trend. During this phase, the rates of cumulative strain for CSR values of 0.3, 0.4, and 0.45 are 1.019E-2ε_a/S, 1.304E-2ε_a/S, and 1.848E-2ε_a/S, respectively. This indicates that during initial cyclic loading, the strain accumulates rapidly, demonstrating high sensitivity of silty clay to initial loading.

From cycles 300 to 1000, the cumulative strain curves transition from rapid growth to slow growth, gradually stabilizing. During this phase, the rates of cumulative strain for CSR values of 0.3, 0.4, and 0.45 decrease to 3.617E-3ε_a/S, 4.993E-3ε_a/S, and 7.770E-3ε_a/S, respectively. This change suggests that after a certain number of loading cycles, the strain rate of silty clay significantly slows down, and cumulative strain tends towards stability.

After exceeding 1000 cycles, the cumulative strain curves exhibit a slow and stable growth state. At this point, the total cumulative strains for CSR values of 0.3, 0.4, and 0.45 are 5.488%, 9.275%, and 13.179%, respectively. This indicates that under long–term cyclic loading conditions, higher CSR values lead to greater total cumulative strain, implying that silty clay is more prone to accumulating larger deformations under higher stress ratios.

Through comparative analysis of cumulative strain curves under different CSR conditions, several patterns are observed:

Distinct phase characteristics: Cumulative strain curves under different CSR conditions exhibit significant phase characteristics, consistent with the transition points observed under previous moisture content and confining pressure conditions. This consistency indicates consistent behavior of silty clay under strain in different environmental conditions.

High initial sensitivity: During cycles 0 to 300, the strain rates are high, indicating high sensitivity of silty clay to cyclic stress during initial loading, resulting in rapid accumulation of strain.

Tendency towards stability in cumulative strain: From cycles 300 to 1000, the strain rates slow down significantly, indicating that the strain of silty clay has approached its peak, and cumulative strain tends towards stability.

Significant influence of CSR: Higher CSR values lead to greater total cumulative strain, indicating that under long–term cyclic loading conditions, an increase in cyclic stress ratio significantly affects the cumulative deformation of silty clay.

In summary, this study reveals the cumulative strain characteristics of silty clay under cyclic loading processes under different CSR conditions. The significant phase characteristics and the impact of CSR on total cumulative strain provide important experimental basis for understanding the mechanical behavior of silty clay under complex stress environments.

Through analysis of the variation curves of cumulative strain with loading cycles under different cyclic stress ratios (CSR), it was found that after exceeding 1000 cycles, the cumulative strain curves exhibit good linear characteristics. This linear feature is particularly pronounced for CSR values of 0.3 and 0.4. Therefore, data after 1000 cycles were selected for fitting analysis, with results shown in Fig. 11. The fitting curve is described by Eq. (6), which reflects the linear relationship between cumulative strain and number of loading cycles under different CSR conditions.

$${\varepsilon _{\text{a}}}=e{\text{+}}fS$$

In Eq. (6), parameters e and f represent the intercept and slope of the fitting curve, with specific fitting parameters for different CSR conditions detailed in Table 4. The fitting degrees range from 0.90990 to 0.99801, indicating high fit with the data. Although the fitting degree for the CSR 0.45 curve is relatively lower, it still exceeds 0.9, demonstrating the high accuracy and reliability of the fitting model.

Detailed analysis of fitting parameters e and f reveals the cumulative strain behavior under different CSR conditions:

Significant linear characteristics: After exceeding 1000 cycles, cumulative strain curves under different CSR conditions exhibit clear linear characteristics. This indicates that under long–term loading conditions, the relationship between cumulative strain of silty clay and number of loading cycles can be well described by a linear model.

High fitting accuracy: The fitting parameters range from 0.90990 to 0.99801, showing high fitting degrees, validating the effectiveness of Eq. (5) in describing the relationship between cumulative strain and number of loading cycles under different CSR conditions. Even though the CSR 0.45 curve has a slightly lower fitting degree, it still falls within the high fitting degree range, further confirming the applicability of the fitting model.

Impact of CSR: Different CSR conditions lead to variations in fitting parameters e and f, indicating significant effects of CSR on the rate of cumulative strain and initial strain amount. With increasing CSR, the rate of cumulative strain increases, suggesting that under higher CSR conditions, the material undergoes greater deformation.

The high accuracy of the linear model with high fitting degrees allows for more precise prediction of cumulative strain behavior of materials under long–term cyclic loading conditions under different CSR conditions. This holds significant implications for assessing the performance of silty clay. In practical engineering applications, understanding the strain characteristics of silty clay under different CSR conditions can assist engineers in better predicting the performance of silty clay over extended periods, thereby enabling more rational design and selection decisions.

In conclusion, this study reveals the linear relationship between cumulative strain and number of loading cycles under different CSR conditions through fitting analysis of strain curves. The high accuracy of fitting Eq. (5) validates its applicability, providing a reliable theoretical basis for predicting strain behavior of silty clay under complex stress environments.

Table 4

The linear fitting parameters for different CSR values.
CSR	e	f	R²
0.3	3.32621 ± 0.01497	2.16119E-4 ± 2.27483E-6	0.99801
0.4	5.09748 ± 0.04461	4.15707E-4 ± 6.73733E-6	0.99607
0.45	9.41171 ± 0.22293	4.10381E-4 ± 3.43928E-5	0.90990

4.1. Sensitivity of cumulative strain to different moisture contents

In this study, the variation of cumulative strain with the number of dynamic stress loading cycles under different moisture content conditions was systematically analyzed. The cumulative strain variation curves in Fig. 6 reveal a rapid initial growth trend of cumulative strain under dynamic stress loading, gradually transitioning into a phase of slow growth around 1000 cycles. Overall, the influence of different moisture contents on cumulative strain is significant, showing pronounced sensitivity throughout the dynamic stress loading process.

4.1.1. Phase changes in cumulative strain

Observation of the cumulative strain variation curves shows that during the initial dynamic stress loading (cycles 0 to 300), cumulative strain exhibits a rapid growth trend with the curve almost vertically ascending. During this phase, the rate of cumulative strain is high, indicating the material's high sensitivity to dynamic stress during initial loading. As the number of cycles increases to around 1000, the rate of cumulative strain growth gradually slows down, entering a phase of slow growth, and ultimately stabilizes.

4.1.2. Influence of moisture content on cumulative strain

Figure 12 illustrates the cumulative strain corresponding to different moisture contents at various cyclic loading stages. The overall trend in cumulative strain under different moisture content conditions indicates that increasing moisture content significantly increases cumulative strain. Specific data are as follows:

At cycle number S = 300, cumulative strains for moisture contents of 8%, 12%, and 16% are 2.695%, 3.797%, and 5.507%, respectively. Increasing moisture content from 8–12% and 16% results in increases of cumulative strain by 40.89% and 104.34%, respectively. At cycle number S = 1000, cumulative strains for moisture contents of 8%, 12%, and 16% are 3.101%, 4.453%, and 6.609%, respectively. Increasing moisture content from 8–12% and 16% results in increases of cumulative strain by 43.60% and 113.12%, respectively. At cycle number S = 10000, cumulative strains for moisture contents of 8%, 12%, and 16% are 4.377%, 6.638%, and 8.870%, respectively. Increasing moisture content from 8–12% and 16% results in increases of cumulative strain by 51.66% and 102.65%, respectively.

4.1.3. Sensitivity analysis of moisture content

From the data in Fig. 12, it can be observed that with every 4% increase in moisture content, the increase in cumulative strain is nearly doubled. This indicates that cumulative strain is highly sensitive to changes in moisture content. The impact of increased moisture content on cumulative strain is significant during both the initial and stable phases of dynamic stress loading, and this impact strengthens with higher moisture content. This significant sensitivity reveals that moisture content is a crucial parameter controlling cumulative strain and plays a decisive role in the deformation behavior of silty clay.

4.1.4. Validation of fitting models

Through fitting analysis of the relationship between cumulative strain and number of cyclic loading cycles under different moisture content conditions (Fig. 7 and Eq. 4), the fitted parameters exhibit high degrees of fit, ranging from 0.96132 to 0.99395. This indicates that despite the significant influence of moisture content on the strain behavior of silty clay, the adopted fitting model accurately reflects the variation pattern of cumulative strain under different moisture content conditions. This high degree of fit not only validates the effectiveness of the fitting equation but also provides a reliable theoretical basis for further research on the performance of silty clay in practical engineering applications.

Overall, this study systematically reveals the variation pattern of cumulative strain of silty clay under dynamic stress loading conditions at different moisture contents through detailed analysis of experimental data and establishment of high–precision fitting models. The high sensitivity of cumulative strain to moisture content underscores the need for careful consideration of the effects of moisture content variations on material performance in practical engineering applications.

4.2. Sensitivity of cumulative strain to different confining pressures

This study systematically analyzed the relationship between cumulative strain of silty clay under dynamic stress loading and the number of cycles under different confining pressure conditions, revealing the significant influence of confining pressure on cumulative strain. Detailed observation and analysis of the cumulative strain variation curves in Fig. 8 demonstrate distinct phase characteristics throughout the loading process.

4.2.1. Phase characteristics of cumulative strain

During cycles 0 to 300 of dynamic stress loading, the cumulative strain curve rapidly rises, showing a high rate of cumulative strain. This rapid growth phase indicates that silty clay is highly sensitive to dynamic stress during initial loading, where dynamic stress application leads to rapid accumulation of strain.

From cycles 300 to 1000, the cumulative strain curve transitions from a rapid growth trend to a slow growth trend, with a significant slowdown in the rate of cumulative strain growth. This transition phase reflects the gradual stabilization of the strain mechanism of silty clay under dynamic stress.

After exceeding 1000 cycles, the cumulative strain curve enters a state of stable growth, maintaining a low level of strain rate, and the curve shows a slow and stable growth trend. This indicates that under long–term cyclic loading, cumulative strain of silty clay tends to stabilize, with further dynamic stress loading having a gradually diminishing effect on cumulative strain.

4.2.2. Influence of confining pressure on cumulative strain

Figure 13 shows the cumulative strain corresponding to different confining pressures at various cyclic loading stages. A comparative analysis of the cumulative strain curves under different confining pressures reveals that an increase in confining pressure significantly affects the total cumulative strain.

At a cycle count S = 300, the cumulative strains for confining pressures of 50 kPa, 100 kPa, and 200 kPa are 1.966%, 2.284%, and 3.007%, respectively. When the confining pressure increases from 50 kPa to 100 kPa and 200 kPa, the cumulative strain increases by 16.17% and 52.95%. At S = 1000, the cumulative strains for confining pressures of 50 kPa, 100 kPa, and 200 kPa are 2.183%, 2.646%, and 3.311%, respectively. When the confining pressure increases from 50 kPa to 100 kPa and 200 kPa, the cumulative strain increases by 21.21% and 51.67%. At S = 10000, the cumulative strains for confining pressures of 50 kPa, 100 kPa, and 200 kPa are 3.306%, 3.933%, and 4.496%, respectively. When the confining pressure increases from 50 kPa to 100 kPa and 200 kPa, the cumulative strain increases by 18.97% and 36.00%.

4.2.3. Sensitivity analysis of confining pressure

The data in Fig. 13 show that when the confining pressure increases by 50 kPa from 50 kPa, the cumulative strain increases by 16.17–21.21%. When the confining pressure increases by 150 kPa, the cumulative strain increases by 36.00–52.95%. The increase in cumulative strain is not linearly proportional to the increase in confining pressure, indicating that changes in low confining pressure have a higher sensitivity to cumulative strain than changes in high confining pressure. During both the initial and stable phases of dynamic stress loading, the increase in confining pressure significantly impacts cumulative strain. The relationship between confining pressure and cumulative strain is linear at S = 300 and S = 1000. This significant sensitivity highlights confining pressure as an important parameter controlling cumulative strain.

4.3. Sensitivity of cumulative strain to different CSR

We explored the relationship between cumulative strain and the number of cyclic loadings of silty clay under different cyclic stress ratios (CSR). The experimental results show that the cumulative strain curves under different CSR conditions exhibit distinct phase characteristics, and the cycle counts at these phase transitions are consistent with the transition times under various moisture content and confining pressure conditions.

4.3.1. Phase characteristics of cumulative strain

During the cycle count range of 0 to 300, the cumulative strain curve shows a rapid growth trend. At this stage, the cumulative strain rates for CSR of 0.3, 0.4, and 0.45 are 1.019E-2ε_a/S, 1.304E-2ε_a/S, and 1.848E-2ε_a/S, respectively. In the subsequent cycle range of 300 to 1000, the cumulative strain curve transitions from rapid growth to slow growth, gradually entering a stable state. During this stage, the cumulative strain rates for CSR of 0.3, 0.4, and 0.45 decrease to 3.617E-3ε_a/S, 4.993E-3ε_a/S, and 7.770E-3ε_a/S, respectively. After exceeding 1000 cycles, the cumulative strain curve shows a slow and stable growth state. At this point, the total cumulative strains for CSR of 0.3, 0.4, and 0.45 are 5.488%, 9.275%, and 13.179%, respectively.

The above characteristics of cumulative strain changes indicate that silty clay is highly sensitive to CSR during cyclic loading, with an increase in CSR leading to a significant increase in cumulative strain rate. When the CSR exceeds 0.4, even slight increases in CSR result in a notable rise in cumulative strain rate. This suggests that under long–term cyclic loading conditions, higher CSR lead to greater total cumulative strains, meaning that silty clay tends to accumulate more significant deformations under higher CSR conditions.

4.3.2. Influence of CSR on cumulative strain

Figure 14 shows the cumulative strain corresponding to different CSR at various cyclic loading stages. The overall trend in cumulative strain under different CSR conditions indicates that an increase in CSR significantly increases both the cumulative strain amount and rate. Specific data are as follows:

At cycle count S = 300, the cumulative strains for CSR of 0.3, 0.4, and 0.45 are 2.589%, 4.329%, and 6.995%, respectively, with cumulative strain rates of 1.019E-2ε_a/S, 1.304E-2ε_a/S, and 1.848E-2ε_a/S. When CSR increases from 0.3 to 0.4 and 0.45, cumulative strain increases by 67.21% and 170.18%, and cumulative strain rates increase by 27.97% and 81.35%.

At cycle count S = 1000, the cumulative strains for CSR of 0.3, 0.4, and 0.45 are 3.440%, 5.372%, and 8.966%, respectively, with cumulative strain rates of 3.617E-3ε_a/S, 4.993E-3ε_a/S, and 7.770E-3ε_a/S. When CSR increases from 0.3 to 0.4 and 0.45, cumulative strain increases by 56.16% and 160.64%, and cumulative strain rates increase by 38.04% and 114.82%.

At cycle count S = 10000, the cumulative strains for CSR of 0.3, 0.4, and 0.45 are 5.488%, 9.275%, and 13.179%. When CSR increases from 0.3 to 0.4 and 0.45, cumulative strain increases by 69.01% and 140.14%.

4.3.3 Sensitivity analysis of CSR

The data in Fig. 14 show that when CSR increases by 0.1 from 0.3, cumulative strain increases by 56.16–69.01%, and cumulative strain rates increase by 27.97–38.04%. When CSR increases by 0.15, cumulative strain increases by 140.14–170.18%, and cumulative strain rates increase by 81.35–114.82%. The increase in cumulative strain and strain rate is not linearly proportional to the increase in CSR. The analysis indicates that changes in high CSR are more sensitive to cumulative strain and strain rates than changes in low CSR. Throughout the dynamic stress loading process, the increase in CSR significantly impacts cumulative strain."

4.4 Study on the influence of moisture content, confining pressure, and CSR on cumulative strain

When the moisture content increases by 50%, the cumulative strain increases by 40.89%, 43.60%, and 51.65% at cycle counts of 300, 1000, and 10000, respectively. When the moisture content increases by 100%, the cumulative strain increases by 104.34%, 113.12%, and 102.65% at the same cycle counts.

When the confining pressure increases by 100%, the cumulative strain increases by 16.17%, 20.20%, and 29.55% at cycle counts of 300, 1000, and 10000, respectively. When the confining pressure increases by 300%, the cumulative strain increases by 52.95%, 51.67%, and 48.09% at the same cycle counts.

When the CSR increases by 33.33%, the cumulative strain increases by 67.21%, 56.16%, and 69.01% at cycle counts of 300, 1000, and 10000, respectively. When the CSR increases by 50%, the cumulative strain increases by 170.18%, 160.64%, and 140.14% at the same cycle counts.

The above analysis indicates that a 33.33% increase in CSR causes a 56.16–69.01% increase in cumulative strain, while a 50% increase in CSR leads to a 140.14–170.18% increase in cumulative strain. A 50% increase in moisture content results in a 40.89–51.65% increase in cumulative strain, and a 100% increase in moisture content causes a 102.65–113.12% increase in cumulative strain. A 100% increase in confining pressure leads to a 16.17–29.55% increase in cumulative strain, while a 300% increase in confining pressure results in a 48.09–52.95% increase in cumulative strain. Thus, a 50% increase in CSR causes about three times the cumulative strain increase compared to a 50% increase in moisture content, and a 100% increase in moisture content causes about four times the cumulative strain increase compared to an increase in confining pressure. These findings show that the influence of the three factors on cumulative strain varies: CSR has the strongest influence, followed by moisture content, with confining pressure having the least effect. This indicates that the cumulative strain of silty clay is most sensitive to changes in CSR, with even small changes causing significant variations in cumulative deformation. The sensitivity of cumulative strain to moisture content is lower than to CSR, and confining pressure has the weakest influence among the three factors.

This study conducted cyclic load tests on silty clay from the Dashiban Tunnel in Ya'an City, Sichuan Province, using a GDS dynamic triaxial loading system. The cumulative strain characteristics under different moisture content, confining pressure, and CSR conditions were investigated, leading to the following conclusions:

(1) The cumulative strain rate decreases as the number of dynamic load cycles increases, eventually stabilizing. The cumulative strain curve exhibits a three–stage characteristic: rapid increase in the initial stage, slow increase in the middle stage, and stability in the later stage.

(2) Increases in moisture content, confining pressure, and CSR all lead to an increase in total cumulative strain. For every 4% increase in moisture content, the corresponding cumulative strain approximately doubles. The incremental increase in cumulative strain is greater between curves with low confining pressure than those with high confining pressure, and the increase is greater between curves with high CSR than those with low CSR.

(3) After 1000 cycles of dynamic stress, the cumulative strain curve shows a clear linear change characteristic. The linear correlation is best at 12% moisture content and 100 kPa confining pressure, with smaller CSR having better linear correlations.

(4) A 50% increase in CSR causes about three times the cumulative strain increase compared to a 50% increase in moisture content, and a 100% increase in moisture content causes about four times the cumulative strain increase compared to an increase in confining pressure. CSR has the strongest influence on cumulative strain, followed by moisture content, with confining pressure having the least effect.

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Correspondence and requests for materials should be addressed to Y.P.

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Author Contribution

G.Q.C: conceptualization, investigation, writing and preparing the original draft; W.J.Y: methodology, formal analysis, resource, funding acquisition, supervision; Y.G.P: writing, reviewing and editing; Y.C.Z: supervision, writing, reviewing and editing; H.J.Z: resource, funding acquisition, supervision; S.Y.S: writing, reviewing and editing; F.L: writing, reviewing and editing; C.H.Y: writing, reviewing and editing; L.L.G: writing, reviewing and editing.

Acknowledgements

This research was supported by the authors appreciate the financial support of the National Natural Science Foundation of China (Grant No.: 52078431), Supported by Sichuan Science and Technology Program (2024NSFSC0832).

Data Availability

The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

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WITHDRAWN: Study on accumulation deformation characteristics of silty clay based on dynamic triaxial tests

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Editorial Note

Abstract

Figures

1. Introduction

2. Experimental soil samples and test program

2.1. Experimental soil samples

2.2. Test program

3. Analysis of cumulative deformation characteristics of silty clay

3.1. Influence of different moisture contents on cumulative strain

3.2. Influence of different confining pressures on cumulative strain

3.3. Influence of different CSR on cumulative strain

4. Discussion

4.1. Sensitivity of cumulative strain to different moisture contents

4.1.1. Phase changes in cumulative strain

4.1.2. Influence of moisture content on cumulative strain

4.1.3. Sensitivity analysis of moisture content

4.1.4. Validation of fitting models

4.2. Sensitivity of cumulative strain to different confining pressures

4.2.1. Phase characteristics of cumulative strain

4.2.2. Influence of confining pressure on cumulative strain

4.2.3. Sensitivity analysis of confining pressure

4.3. Sensitivity of cumulative strain to different CSR

4.3.1. Phase characteristics of cumulative strain

4.3.2. Influence of CSR on cumulative strain

4.3.3 Sensitivity analysis of CSR

4.4 Study on the influence of moisture content, confining pressure, and CSR on cumulative strain

5. Conclusions

Declarations

Competing interests

Additional information

Author Contribution

Acknowledgements

Data Availability

References

Additional Declarations

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