Experimental Study on Creep Characteristics of Saturated Q2 Loess

Catastrophic failures often occur to engineered infrastructure in areas underlain by saturated loess due to its high moisture content (degree of saturation >85%), low shear strength, and large deformation. Time effect (rheological property) and water are the most important factors affecting the mechanical properties of saturated loess. The rheological mechanism and characteristics of saturated loess are vital to understanding the interaction between infrastructure and foundation and the trigger of failures. Based on step-load testing, this study obtains the displacement-time relationship of saturated Q2 loess to analyze its creep behavior. Experimental results show that the creep strength of saturated Q2 loess is 75–80% of its unconfined compression strength, and the creep behavior can be simulated by the Burgers model. Additionally, the rheological parameters under different load conditions are obtained using the improved Marquardt iterative method. All these parameters can be used for numerical analysis of the rheological behavior of saturated Q2 loess.


INTRODUCTION
Loess occurs widely across China. The regions covered by loess account for 6.31 × 10 5 km 2 (Dijkstra et al., 1994), where it serves as the primary building material, building environment, or the foundation for the infrastructure. Due to the unique genesis of loess (Liu et al., 1985;Zhang et al., 1989;An et al., 1998;Sun, 2005;Chen et al., 2019;Fu et al., 2021), it possesses particular characteristics in deformation, strength, and stability (Wang, 1982;Feng and Zheng, 1982;Zheng, 1982;Lei, 1987), and abundant studies have been carried out on loess in the world. Loess below the groundwater table has high water content and a degree of saturation greater than 85%. Compared with unsaturated loess, saturated loess has lower strength, larger deformation, higher compressibility, and is more sensitive (Qiao and Li, 1990). These characteristics lead to geotechnical problems such as slope flow failure (Wen and Jiang, 2017;Xu et al., 2018;Xie et al., 2018), tunnel collapse, and piping in saturated loess areas (Peng et al., 2015).
The Loess Plateau is the core area of the Belt and Road Program and the primary site of Western Development. Many major projects, including Lanzhou-Chongqing railway, Zhengzhou-Xi'an highspeed railway, Xi'an-Lanzhou high-speed railway, Lanzhou-Xinjiang high-speed railway, and the Tao River Diversion Project, have traversed through saturated Q 2 loess areas to a certain extent. Along with these projects, increasingly more problems have emerged due to the seepage failure of saturated Q 2 loess. For example, six tunnels out of 15 in phase I of the Tao River Diversion Project passed through saturated loess. During the construction of Tunnel No. 15, flow failure ( Figure 1A) occurred in a segment of about 30 m long, leading to roof collapse ( Figure 1B) and construction delay of nearly 1 year. Another example is the Dayingliang tunnel through saturated Q 2 loess along the second track of Baoji-Lanzhou Railway, which is less than 2 km away from Tunnel No. 15 of the Tao River Diversion Project. It was seriously damaged due to seepage during the construction by traditional methods. A much more expensive construction method, i.e., the ground freezing by liquid nitrogen, was adopted to finish some tunnel sections.
The study of rock and soil creep is extensive. For example, Wang et al. (2020) used step-loading triaxial creep tests under different moisture content to reveal the relationships between the steady-state creep rate and the initial strain and shear modulus. Lin et al. (2020) introducd the nonlinear viscoplastic element into the classic Burgers model and verified the rationality of the model by the shear creep test results of rock discontinuity, the results show that the modified Burgers model can reflect the mechanical properties of rock in three creep stages. Xia et al. (2021) proposed an improved simulation method based on the classical Burgers model and the Parallel Bonded model in the Particle Flow Code (PFC) and applied it to simulate the full-stage creep process in soft rock. Tang et al. (2019) studied the creep behavior of loess under different moisture content and confining pressure conditions, they revealed that the creep behavior of loess is prominent at high moisture content and minor time-dependent deformation occurs at high confining pressures. Zhu et al. (2013) investigated the creep characteristics of red-bed sliding soil under different vertical loads and water content through direct shear creep tests. Other significant studies on the creep behavior of different geotechnical materials include Fabre and Pellet. (2006), Wang et al. (2014), and Ye et al. (2015). However, the studies on the creep characteristics of saturated Q 2 loess are rare.
Engineering incidents quickly increase along with major projects in loess areas (Palmer, 2017;Wang et al., 2018). The key to curbing these incidents and mitigating the risks related to saturated Q 2 loess is to understand its rheological mechanism and characteristics. This paper conducts creep tests of saturated Q 2 loess obtained from a tunnel construction site by step loading. Testing results, including the displacement-time relationship and creep strength, are presented. In addition, the Burgers model was used to simulate the creep behavior, and the rheological parameters under different load conditions are obtained using the improved Marquardt iterative method. At the same time, deformation prediction by using laboratory experimental results agrees favorably with field observations.

Experiments and Results
Undisturbed saturated loess samples were obtained at section km105 + 240 ( Figure 1C), the pressure and deformation were monitored by pressure boxes and convergence meter at 100 m away from the flow failure location in km 105 + 410, as shown in ( Figure 1D). This section describes its grain size distribution, uniaxial compressive strength, and creep test scheme.

Grain Size Distribution
The particle size distribution of Q 2 loess was obtained by a laser particle size analyzer, and the results are listed in Table 1. The Q 2 loess samples have relatively uniform particle sizes, consisting of 5.2-8.0% of sand content, 28.5-31.8% of clay content, and more than 60% of silt content. The clay content of the Q 2 loess is high and can be classified as silty clay.

Uniaxial Compressive Strength
The compressive strength is one of the essential mechanical properties of geomaterials, including the Q 2 loess. Additionally, it serves as a crucial reference for determining the step loads at various stages of the creep test. This study conducted uniaxial compression tests to obtain the unconfined compressive strength, σ c , of saturated Q 2 loess samples, as presented in Table 2. The results show that the values of σ c range between 0.5 and 0.6 MPa, and the strength decreases sharply with increasing water content.

Creep Test and Results
Loess is a complex porous geomaterial with a unique sensibility to water. The influence of pore water on its mechanical properties, such as deformation, strength, and constitutive behavior, has always been at the center of the study (Xie and Qi, 1999). Both the stress and deformation of loess are closely related to time (Huang, 1983;Qian and Yin, 1996). Time effect ( Engineering practice has demonstrated that the results of the uniaxial creep test can meet the requirements of design and construction (Zhou, 1990). Therefore, this study adopted the uniaxial creep test to analyze the creep properties of saturated Q 2 loess. Specifically, a CSS-44100 electronic universal testing machine with a closed-circuit control system was used. It is capable of applying a load ranging from 400 N to 100 kN with an error of less than 0.1%. Six to seven testing specimens of 5 cm in diameter and 10 cm in height were prepared for each of the two groups of undisturbed saturated Q 2 loess samples.
The step loading protocol was adopted for the creep test. Specifically, six to seven gradually increasing loading steps were applied on the same specimen, with the maximum load controlled at 75% σ c (Refer to Table 2). Uniform loading was adopted to reach the first load level in 600 s. Each loading step was terminated when the displacement increment (or the maximum value when the loading time reaches 24 h) was less than 0.001 mm/h. The stress and strain data was obtained every 30 s and 200 s in the loading stage and stable stage after loading. Figure 2 depicts representative deformation vs. time data from the creep tests of saturated Q 2 loess samples.
The following observations can be made for saturated Q 2 loess from Figure 2: a) A large instantaneous deformation occurs when the saturated Q 2 loess specimen is subjected to a step load, accounting for more than 50% of the total deformation of each loading stage. However, with the increase in load, the proportion of the instantaneous deformation out of the total gradually decreases. The deformation accumulates with time at decreasing rates until it plateaus. This pattern can be classified as the attenuated or damped rheological stage or initial rheological stage. b) Following the initial rheological stage, there is a segment where the deformation accumulates at a constant rate for all stages of loading except for the last one. The rate at which the deformation occurs, or the creep rate, in this segment is almost constant under the same level of load, is very close even under different levels of load. This pattern can be categorized into the constant rheological stage or the steady rheological stage. c) When the stress level is high (e.g., the last load step), if the deformation increases sharply, the specimen usually undergoes quick failure, i.e., entering into accelerated creep stage without the constant rheological stage. It often takes a  very short time from accelerated deformation to specimen collapse. As illustrated in Figure 3, one can see that cracks appear outside the specimen and rapidly expand throughout the entire specimen, forming an inclined slip surface. Therefore, the creep failure under uniaxial compression load is essentially a shear failure accompanied by noticeable lateral bulging. For sample No.1, the accelerated creep trend appeared at load step six, but failure did not occur within the loading time of this stage. It is presumed that creep failure would occur if the loading time is sufficient. When the load exceeds 0.45 MPa, brittle shear failure occurred instantly on sample No. 1 ( Figure 3A). For sample No.3, the accelerated creep occurred at the load step six (i.e., 0.40 MPa), and creep failure occurred 2 hours after the loading was applied. d) Theoretically, the stress corresponding to the asymptotic line at t ∞ is its long-term strength. According to the deformation vs. time curves in Figure 2, the long-term strength values of specimens No. 1 and No. 3 are 0.45 MPa and 0.40 MPa, respectively. It can be seen from that the failure stress of saturated Q 2 loess is generally about 75%-80% of its uniaxial compressive strength. In engineering application, it is suggested to take 75% of the uniaxial compressive strength as the long-term strength of saturated Q 2 loess.

DISCUSSION
Rheological constitutive models and appropriate parameters based on experimental data are essential for gaining an indepth understanding of the creep behavior of geomaterials. Representative rheological models include those consisting of a combination of elements such as springs and dashpots, empirical models, integral models, damage-based models, and models based on classic elastic-viscoplastic potential theory (Xia and Sun, 1996;Zheng et al., 1997;Wang et al., 2004). Among these models, the element model is simple to construct, has clear physical meanings, and can comprehensively reflect various rheological properties of geomaterials such as creep and stress relaxation, and is selected for modeling the creep behavior of saturated Q 2 loess. Typical element models include the Burgers model, the Maxwell model, the Generalized Maxwell model, the Kelvin model, the Kelvin-Voigt model, the Bingham model, the Poynting-Thomson model, the Nishihara model, etc. Table 3.
The creep test data in Figure 2 demonstrate that saturated Q 2 loess exhibits obvious three-stage creep characteristics. Based on the observed deformation vs. time behavior and the features of various element models, it is found that the Burgers model can better describe the creep characteristics of saturated Q 2 loess (Figure 4).
The creep equation of the Burgers model is: Where E 1 , E 2 are elastic modulus; η 1 , η 2 are viscosity coefficients; σ 0 is the step load.  Because of the complex constitutive equation of the creep model, it is difficult to obtain the model parameters directly according to the test results. Yang et al. (2004) provided a method to obtain creep model parameters based on the axial strain, instantaneous elastic strain, and other parameters. Since the instantaneous elastic strain is often not available in practical applications, the least square fitting method is generally used to determine the parameters in the element model. However, the least square method usually requires the selection of initial parameter values, and improper initial parameter values can easily lead to divergence during iteration. Marquardt damped least-squares method is an extremely powerful tool for the iterative solution of nonlinear problems. In this study, the modified Marquardt method was used to determine the rheological parameters based on the experimental data, and the results are listed in Tables 4, 5.
The results in Tables 4, 5 show that the values of elastic modulus and viscosity coefficient range in 10-70 MPa and 1.0×10 9 -9.0 × 10 13 Pa·s, respectively. The two main indexes reflecting the creep characteristics are far lower than those of hard rock, i.e., 50-100 GPa and 1.0×10 16 -1.0 × 10 17 Pa·s, indicating that saturated Q 2 loess has a more prominent creep tendency. It can be seen from Tables 4, 5 that, as the loading stress rises, E 1 decreases but E 2 increases. Meanwhile, η 1 grows first, reaches a maximum, then falls as the stress level increases. However, η 2 consistently increases with the rise in the stress level and surges when creep failure occurs. The viscosity coefficients reflect the increase in the viscosity of saturated loess specimen as the stress rises. The results indicate that the creep property of saturated Q 2 loess becomes more and more evident with the increase in load, and saturated Q 2 loess is a nonlinear viscoelastic material.
Taking Table 4 as an example, draw the curves of η 1 and η 2 vs E 1 and E 2 . It can be seen from Figure 5 that the relationship curves of η 1 and η 2 vs. E 1 and E 2 are almost symmetrical. η 1 first increases and then decreases with the increase in E 1 and E 2 ; η 2 decreases with the rise in E 1 and increases with the increase in E 2 .

Comparison of Model-Simulated and Experimental Creep Curves of Saturated Q 2 Loess
To verify the applicability of the Burgers model of saturated Q 2 loess, the creep parameters of saturated loess (obtained from Table 4 and Table 5) were substituted into the Burgers model to calculate and obtain the simulated creep curve. Comparing the simulated curve with the experimental one in Figure 6, it is found that the Burgers model can be used to simulate the experimental data. The model-simulated and the experimental curves almost coincide under the first four loadings. When the load continues to increase, the model-simulated curve is slightly higher than the experimental one, indicating that the trend of the modelsimulated accelerated stage is more obvious at the last two loading levels.
It can be seen from the comparison between the theoretical simulation curve and the experimental curve that the fourparameter Burgers model can well simulate the rheological curve of saturated Q 2 loess, and the model parameters can be used for the elastic-viscoplastic analysis of saturated Q 2 loess.

Application of Burgers Model to project(Case Study)
Tunnel No. 15 of the Tao River Diversion Project adopts two-way excavation. The tunnel is surrounded by saturated soft and plastic Q 2 loess with an average water content of more than 28%. When one side of the tunnel was excavated to km105 + 410, the palm surface moved outward, and the arch frame sank and collapsed. On the 10th day, flow failure suddenly occurred. Saturated loess slid nearly 30 m from the palm surface within a few minutes, resembling toothpaste being squeezed out from a tube. After about 3 hours, the flow failure in the tunnel was stable. And 4 hours later, the tunnel roof at about 30 m above the tunnel base collapsed.
On the other side of the tunnel, the pressure and deformation at 100 m away from the flow failure location in km 105 + 410 section were monitored by pressure boxes and convergence meter, as shown in Figure 1D. The pressure boxes and the convergence meter are synchronously arranged, and the monitoring time of both is 170 d. From the pressure monitoring curve ( Figure 7A), it can be seen that an inflection point appeared on the surrounding rock pressure of the tunnel at 110 kPa after about 10 days and then the pressure rose slowly, reaching the maximum of 160 kPa during the monitoring period. Meanwhile, an inflection point also occurred on the deformation of baselines 1-3 of the monitoring section as shown in Figure 7B at 18.649 mm on the 10th day, with the maximum reaching 22.6 mm during the monitoring period.
According to the results of pressure and deformation monitoring, combined with the creep test load gradient, select Burgers model creep parameters at 150 kpa as the simulation parameters of the monitoring section. The indoor creep test accelerated the stress loading process, and the completion time of samples No.1 and No.3 under the first stage load was 600 s. Using the equivalent time method, the actual observed stress inflection point time (10 d) is equivalent to the first stage loading completion time (600 s) in the laboratory, and the residual stress monitoring 160 d is equivalent to the stable loading time after the first stage loading. Under the equivalent time, according to Burgers model and combined with the tunnel diameter, the surrounding rock deformation of the tunnel is simulated. Also, the trend of the simulation deformation results are in good agreement with that of the monitored, with the simulated deformation being about 5 mm larger. The discrepancy can be attributed to the following reasons: 1) the selected Burgers model creep parameters are the creep parameters obtained at 150 kPa, but the in-situ pressure is 110-160 kPa; and 2) the tunnel surrounding rock is supported after excavation, hence reducing the actual deformation.

CONCLUSION
The uniaxial compressive strength of saturated Q 2 loess is about 0.5-0.6 MPa, and decreases sharply with the increase in moisture content. Creep testing results of saturated Q 2 loess show that the proportion of instantaneous deformation at each loading stage decreases with the increase in loading level, and its rheological curve consists of three typical phases, including the initial, steady, and accelerated creep stages.
The creep failure of saturated Q 2 loess belongs to shear failure. Specimen failure is often accompanied by obvious lateral bulging. The failure stress of saturated Q 2 loess is generally about 75-80% of uniaxial compressive strength. In engineering applications, it is suggested to take 75% of uniaxial compressive strength as the long-term strength of saturated Q 2 loess.
The elastic modulus and viscosity coefficient of saturated Q 2 loess are far lower than those of hard rock, revealing its more prominent creep tendency obvious. The parameters obtained for the Burgers model can be used for rheological numerical analysis of saturated Q 2 loess.
It can be seen from the case study and comparison between the theoretical simulation curve and the experimental curve that the four-parameter Burgers model is suitable for saturated Q 2 loess, which can well simulate the rheological curve of saturated Q 2 loess, and the model parameters can be used for the elasticviscoplastic analysis of saturated Q 2 loess.
The experimental results shed light on the creep characteristics of saturated Q 2 loess. However, in practice, rheological failures often occur several years or even decades after constructing the projects. In addition, uniaxial creep tests were conducted with relatively short durations due to test equipment limitations. Therefore, the creep parameters obtained need further experimental and numerical verification.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

AUTHOR CONTRIBUTIONS
XL drafted the manuscript. XZ finished the comparison of modelsimulated and experimental creep curves of saturated Q 2 loess. XF prepared the data analysis. TY and ZS prepared the data collection. All authors have read and approved the final manuscript.

ACKNOWLEDGMENTS
The first author would like to express warm gratitude to the Fundamental Research Funds for the Central Universities of Lanzhou University. We thank Professor Wenwu Chen for his helpful suggestions and Dr. Pengbo Yuan for his experimental help. The authors would also like to thank the reviewers for reviewing the draft version of the manuscript.