Abstract
In order to reflect the creep characteristics of unsaturated silty clay, a triaxial compression consolidation drainage creep test was conducted under the condition of controlling the matric suction. According to the results of the creep test, combined with the empirical models, Mesri model and Logmodified model, the relationship between the initial tangent modulus and the matric suction was established, and two empirical models of unsaturated soil creep considering the effect of matric suction were constructed. The study confirmed the stress–strain through the ε/D–ε relationship curve, and determined the parameters F and n through power function. The methods for determining the strain–time relationship parameters of the two improved models are different. The improved Mesri model was obtained by fitting the ln ε–ln t relationship, while the improved Logmodified model was solved by the BFGS algorithm and the general global optimization method. By comparing the two improved models of unsaturated soil creep tests, it was found that the improved Mesri model can more accurately describe the creep characteristics of unsaturated soils, which confirms the rationality and feasibility of this model and method.
1 Introduction
Creep property is one of the most important parts of soil mechanics and it is closely related to the longterm stability of slope [1,2,3,4,5]. Unsaturated soil is a kind of threephase soil, which contains solid, liquid, and gas. Compared with saturated soil, the creep characteristics of unsaturated soil are more complex. In the reservoir bank slope, due to rainfall infiltration and the change in reservoir water level, the soil transforms between saturated and unsaturated states. The soil has unsaturated characteristics and its creep deformation gradually accumulates, posing a potential threat to the longterm stability of the reservoir bank slope [6,7,8,9]. Currently, some research has been made to study the creep characteristics of unsaturated soil. Di et al. [10] carried out soil–water characteristic curve tests in the compressive creep state for unsaturated accumulation soil, so as to establish a threephase (solid–liquid–gas) coupling model for unsaturated soil. Fan et al. [11] studied the creep characteristics of unsaturated soil and found that the increase in water content would cause rebound deformation and compression creep deformation, and the former deformation was larger than the latter. Wang et al. [12] carried out creep tests of unsaturated loess under the effect of matric suction, and results showed that the time required for the creep curve to reach stability with the smaller matric suction was longer, and when the stress level was low, the deformation of loess was small and had a certain degree of recoverability. Moradi et al. [13] studied the influence of strain rate on the deformation and strength of unsaturated soil. In the study, the yield stress was expressed as a function of strain rate, so as to derive the incremental equation of unsaturated soil under onedimensional conditions to describe the longterm deformation behavior of unsaturated soil over time. Regarding the use of constitutive model, Sun and Shen [14] and Sun and Sumelka [15] used fractional plastic flow rules to analyze granular soils, research the combined fractional plastic flow rules to demonstrate noncorrelated boundary surface models, and successfully discovered important features. Additionally, to capture the ratedependent stress–strain behavior of compressed soils, a stress fractional viscoplastic model was developed based on Perzyna. Liang et al. [16] established a fractional critical state constitutive model of soil, predicted the test results under typical stress path conditions, and found that the model built could reasonably describe the deformation and strength characteristics of the soil.
Currently, some achievements have been made in the study of soil creep models. The existing creep models can be divided into empirical models and component models, among which the empirical model has been widely used because of the advantages of flexibility. Typical empirical models include Singh–Mitchell model and Mesri model [17,18,19]. The Singh–Mitchell model adopted exponential function to describe the stress–strain relationship, while the Mesri model adopted hyperbolic function to express, and the strain–time relationship of two models is described by power function [20]. Some research also used Morgan Mercer Florin function to describe the string–time relationship, and some research improved the Mesri model based on the Logmodified function and proposed the Logmodified empirical model. However, most of the empirical creep models are based on the creep deformation characteristics of saturated soil and cannot reflect the influence of matric suction on creep characteristics of unsaturated soil.
Based on previous research, this research took the unsaturated silty clay as the object, and it considered the matric suction under the condition of consolidation creep test. Based on the Mesri model and the Logmodified model, the relationship between the initial tangent modulus and the suction was established to construct a creep empirical model that can reflect the change in the matric suction. A comparative analysis of the two models was also conducted to verify the feasibility of this model and method.
2 Methods
2.1 Test materials and sample preparation
The silty clay samples were taken from reservoir bank slope, and the basic physical and mechanical parameters are listed in Table 1. Soil samples in the experiment was taken after natural airdrying condition, and used 2 mm of screening. In order to facilitate the sample shape in the experiment, distilled water was used to prepare a soil sample with a water content of 20%, wrapped the soil sample with plastic wrap, and left it for 1 day to spread the water evenly. Finally, the test used a soil cutter to make a reshaped cylindrical sample Φ60 mm × 120 mm.
Gs  Water content (%)  Density ρ (g cm^{−3})  Void ratio (e)  Liquid limit ω _{L} (%)  Plastic limit ω _{P} (%)  Cohesion c (kPa)  Internal friction angle (°) 

2.64  14.6  1.97  0.54  30.2  13.9  8.87  16.82 
2.2 Test procedure and scheme
This research carried out unsaturated triaxial compression creep test under the control of matric suction. Before the creep test, the triaxial drainage shear test was carried out to determine the drainage shear strength τ _{f}. The confining pressure σ _{3} was set as 100 kPa, and the matric suction s was set as 100, 200, 300, and 400 kPa, respectively. The test adopted the hierarchical loading method, and the loading duration of each step was more than 200 h. The stress D started from 0.55, and the corresponding deviatoric stress difference was 0.55τ _{f}. The deviatoric stress increases by 0.05 at each stage until it reaches the failure state, and the failure deviatoric stress is (σ _{1}–σ _{3})_{f}. The triaxial drainage shear test results and the creep test loading scheme are shown in Table 2.
No.  s (kPa)  τ _{f} (kPa)  (σ _{1}–σ _{3})_{f} (kPa)  σ _{1}–σ _{3} (kPa) 

R1  100  590.11  472  325 354 384 413 443 
R2  200  658.39  523  362 395 428 461 494 
R3  300  812.28  609  447 487 528 569 
R4  400  1086.40  815  598 652 706 760 
Figure 1 shows the creep curve under graded loading (setting s to 400 kPa as an example), and the creep curves of different loadings treated by Boltzmann linear superposition principle [21,22] are shown in Figure 2. The deviatoric stress and strain at 9 time nodes of 1, 26, 51, 76, 101, 126, 151, 176, and 201 h, as shown in Figure 2, were selected to draw isochronous stress–strain curve, as shown in Figure 3. Figure 4 shows the isochronous stress–strain curves under different matric suction. Since the curves are relatively dense, only the curves with time nodes of 1 and 201 h are retained for the convenience of comparison.
According to Figures 1 and 2, the following analysis can be drawn: the soil sample shows a certain amount of elastic instantaneous strain at the moment of axial loading, and the creep deformation increases continuously with the accumulation of loading time. The higher the stress level, the greater the creep deformation. As can be seen from Figures 3 and 4, when the loading time exceeds 51 h, the isochronous stress–strain curve has certain nonlinear characteristics, and the curve cluster shows a trend of gradually deviating to the strain axis (horizontal axis). In fact, the creep curves of unsaturated silty clay have strong similarity under different stress levels and matric suction, so the same stress–strain–time relationship function can be used to describe the creep mechanical properties of unsaturated silty clay.
2.3 Empirical creep model
The creep model widely used now are mainly Singh–Mitchell model and Mesri model. Research proposed a model based on MesriLogmodified empirical model, and Singh–Mitchell model has proved unable to predict creep deformation under low stress level. Because Mesri model and Logmodified model do not consider the influence of matric suction on the creep characteristic, this research respectively improved Mesri model and Logmodified model. The identification ability of the two improved models for the creep characteristics of unsaturated soil was analyzed.
2.4 Improved MESRI model
2.4.1 Stress–strain–time relationship
The total strain is composed of instantaneous strain and drainage creep strain. If research does not consider the effects of thixotropy effect, consolidation ratio, aging, and other factors, the stress, strain, and time relationship of soil creep can be obtained using the following equation:
where ε is the strain, f _{1}(k) and f _{2}(k) are the stress–strain relationship function and strain–time relationship function, respectively.
2.4.2 Stress–strain relationship
As can be seen from Figures 3 and 4, the shape of the curve is similar to that of a hyperbola, so the hyperbolic stress–strain equation proposed by Kondner [23] is introduced. The equiaxial hyperbola can be written as:
where σ _{1} and σ _{3} are the maximum and minimum principal stresses, respectively, and a and b are the hyperbolic equation parameters.
By differentiating equation (2), the initial tangent modulus E _{ u } can be obtained as
When ε → ∞, the limit value of equation (2), which is the final deviatoric stress difference (σ _{1}–σ _{3})_{ult}, is:
In fact, the strain cannot reach infinity, and it breaks when the drainage shear strength τ _{f} is reached. (σ _{1}–σ _{3})_{f} is the actual failure shear stress of soil. In order to make the hyperbola pass through (ε _{f}, [σ _{1}–σ _{3}]_{f}), the failure ratio R _{f} is introduced:
Substitute equations (3) and (5) in equation (2) at the same time to obtain:
In the formula, D is the stress level, D = (σ _{1}–σ _{3})/(σ _{1}–σ _{3})_{f}.
In order to facilitate the calculation of parameters, equation (6) is transformed as follows:
2.4.3 Strain–time relationship
Strain–time function can take many forms, including hyperbolic function, power function, logarithmic function, etc. The MESRI model uses a power function:
In the formula, t _{1} is the reference time of initial creep. Since the creep of soil sample is not obvious at 1 h, in order to facilitate calculation, t _{1} is set as 1 h, ε _{1} is the initial creep strain at t = t _{1}, and m is the slope of curve: ln ε−ln t.
2.4.4 Improved MESRI model establishment
In order to make the empirical model of unsaturated soil reflect the quantitative action of water content and incorporate matric suction into the model as an independent variable, a stress–matric suction strain–time relationship model should be established. Janbu [24] found that the initial tangent model and the confining pressure σ _{3} were linearly correlated in log–log coordinate, the initial tangent modulus E _{u} was a power function of σ _{3}. In the previous analysis [25], the power function of σ _{3} was obtained by fixing the matric suction s. However, this study mainly analyzes the effect of matric suction, which fixed σ _{3} = 100 kPa. Therefore, the same function can be used between E _{u} and s.
where p _{a} is the atmospheric pressure (101.33 kPa), F and n are the material constants, and s is the matric suction. Set t = t _{1} in equation (6) to get ɛ _{1}. Substitute ɛ _{1} from equation (6) in equation (10) to obtain:
where D _{1} is the value of D when t = t _{1}. The values of failure deviatoric stress (σ _{1}–σ _{3})_{f}, initial tangent modulus E _{u}, and failure ratio R _{f} are independent of t.
By substituting equation (11) in equation (12), we can obtain
It is the improved MESRI model considering matric suction in this article.
2.5 Improved Logmodified model
The stress–strain relationship function of the Logmodified model is consistent with that of the Mesri model. The difference is that the strain–time relationship has been described by a new threeparameter power function, which is expressed as
where p, q, and r are the model parameters. Set t = t _{1} in equation (6) to get ɛ _{1}. Substitute ɛ _{1} from equation (6) in equation (14) to obtain:
By substituting equation (11) into equation (15), we can obtain
Equation (16) is the improved Logmodified model considering matric suction in this article.
3 Result
3.1 Parameters of stress–strain relationship
Since the stress–strain functions of the Mesri model and the Logmodified model show similar properties, the stress–strain relationship parameters of the two models are also the same. It can be seen from equation (9) that ε/D is linearly correlated with ε, R _{f} is the slope, and (σ _{1}–σ _{3})_{f}/E _{u} is the intercept. The creep data of 9 time nodes from 1 to 201 h were selected. Taking s equal to 400 kPa as an example, the relation curves of ε/d−ε at different times were plotted (as shown in Figure 5). In order to facilitate observation, some nodes were omitted. It can be seen from Figure 5 that ε/D has obvious linear correlation with ε, and (σ _{1}–σ _{3})_{f}/E _{u} and R _{f} have no relation to stress and time. Due to the limitation of space, only the values of parameters under s of 400 kPa are given, as shown in Table 3.
s (kPa)  t (h)  (σ _{1}–σ _{3})_{f} (E _{u})  R _{f}  R ^{2} 

400  1  0.5258  1.3686  0.9953 
26  1.1332  1.3003  0.9972  
51  1.2975  1.2886  0.9969  
76  1.3563  1.2844  0.9972  
101  1.3922  1.2833  0.9969  
126  1.4317  1.2801  0.9978  
151  1.4703  1.2764  0.9984  
176  1.4943  1.2745  0.9967  
201  1.5104  1.2734  0.9965 
It can be seen from Table 3 that R _{f} decreases with time, while (σ _{1}–σ _{3})_{f}/E _{u} increases with time. Since the Mesri model and the Logmodified model assume that the model parameters are independent of stress and time, the (σ _{1}–σ _{3})_{f}/E _{u} and R _{f} at different matric suction are averaged at 9 time nodes, as shown in Table 4.
s (kPa)  (σ _{1}–σ _{3})_{f} (E _{u})  R _{f}  E _{u} (kPa) 

100  1.5953  2.3894  483.95 
200  1.5087  1.8975  527.66 
300  1.3425  1.5843  580.14 
400  1.2902  1.2922  631.69 
It can be seen from Table 4 that the mean value of (σ _{1}–σ _{3})_{f}/E _{u} and E _{u} increases with the increase in the matric suction s, while the mean value of R _{f} decreases with the increase in the matric suction s. With the decrease in s, the initial tangent modulus of unsaturated soil in the reservoir bank slope decreases continuously and the soil becomes soft, indicating that the creep deformation of soil is more significant with the increase in water content in the reservoir bank slope.
3.2 Determination of parameters F and n
Both the modified Mesri model and the Logmodified model are based on equation (11) to establish the relationship between the suction force, where equation (11) is the power relationship between E _{u} and parameters F and n. Therefore, the parameters F and n of the two improved models are also consistent. s/p _{a} was fitted through E _{u} in Table 4, as shown in Figure 6.
It can be seen from Figure 6 that the fitting effect of the power of the curve is good, R ^{2} reaches 0.9602, F is 477.47, and n is 0.1882.
4 Discussion
4.1 Parameter solution of strain–time relationship
4.1.1 Improved MESRI model
In equation (10), ln ε is linearly related to ln t and m is its inclination. The curve of ln ε−ln t (setting s as 400 kPa) is shown in Figure 7. The strain–time relationship parameters of MESRI model are listed in Table 5.
s (kPa)  D  m  R ^{2} 

400  0.55  0.1545  0.9627 
0.6  0.1064  0.9904  
0.65  0.0821  0.9736  
0.7  0.0696  0.9700 
It can be seen from Figure 7 that the linear correlation of ln ε−ln t matches well, with an average R ^{2} of 0.9742. As can be seen from equation (10), parameter m has nothing to do with stress and time, but is completely determined by the slope of ln ε−ln t curve. Therefore, the average value of m at different stress levels is taken as the parameter of the MESRI model under the matric suction condition, as shown in Table 6.
s (kPa)  m  R ^{2} 

100  0.0712  0.9635 
200  0.0794  0.9812 
300  0.0935  0.9607 
400  0.1032  0.9741 
The failure deviatoric stress (σ _{1}–σ _{3})_{f} in Table 2, the average value of R _{f} in Table 4, the average value of parameter m in Table 6, and the parameters F and n are substituted into equation (13) to obtain
Equation (17) is an improved empirical Mesri creep model of unsaturated silty clay under σ _{3} = 100 kPa and s = 400 kPa. Due to the limitation of space, only the model under s = 400 kPa is taken into account.
4.1.2 Improved Logmodified model
Substitute the failure deviatoric stress (σ _{1}–σ _{3})_{f} in Table 2, the average value of R _{f}, F, and n in Table 4 to equation (16), and use mathematical optimization software 1stOpt, which is based on BFGS algorithm and general global optimization method, to calculate the parameters for each stress level of p, q, and r, take the average value. The Logmodified model parameters p, q, and r are listed in Table 7.
s (kPa)  P  Q  r  R ^{2} 

100  0.7209  0.0203  4.1462  0.9698 
200  0.6184  0.0244  3.9894  0.9672 
300  0.4685  0.0352  3.9326  0.9853 
400  0.3330  0.0385  3.9107  0.9660 
By substituting the model parameters in Tables 4 and 7 as well as parameters F and n in equation (16), the following empirical creep equation can be obtained:
Equation (18) is the improved logmodified creep empirical model of unsaturated silty clay under σ _{3} = 100 kPa and s = 400 kPa. Due to the limitation of space, only the model under s = 400 kPa is given.
4.2 Model validation
Taking the test data of s = 300 and 400 kPa as examples, the two improved models were compared and verified, respectively, as shown in Figure 8. For the convenience of observation, the improved Mesri model and the Logmodified model were, respectively, abbreviated as M model and L–M model.
The average R ^{2} of the modified MESRI model identification test data in Figure 8 is 0.9912, indicating that the modified MESRI model has a good fitting effect and can better reflect the creep characteristics of unsaturated silty clay. However, the average R ^{2} of the identification test data of the improved Logmodified model is 0.9634, and the identification ability of the attenuated creep stage is poor. The theoretical curve of this section is always lower than the test curve, and the theoretical curve of the part near the end of the stable creep stage is higher than the test curve.
In the improved Mesri model, the strain–time relationship has only one unknown parameter m, which can be obtained by linear fitting. The strain–time relationship in the improved Logmodified model contains three unknown parameters, which are obtained by mathematical software 1stOpt based on a certain algorithm. Compared with the former model, the strain–time relationship in the improved Logmodified model has certain uncertainty.
Based on the comprehensive analysis, the modified MESRI model has a good fitting effect and can describe the creep characteristics of unsaturated silty clay more accurately.
5 Conclusion
This research took the unsaturated silty clay as the object, and it considered the matric suction under the condition of consolidation creep test. Based on the Mesri model and the Logmodified model, the relationship between the initial tangent modulus and the suction was established to construct a creep empirical model that can reflect the change in the matric suction. The main contribution is:
(1) A consolidated drainage triaxial compression creep test of unsaturated silty clay was carried out. Result found that the creep characteristics of the soil sample were significant, and the creep curves were relatively similar under different matric suction and stress levels. Therefore, two empirical models considering the matric suction were established to describe the creep characteristics under different stress environments.
(2) Matric suction is inversely proportional to moisture content. The decrease in matric suction results from an increase in water content, an increase in water content results in a decrease in shear strength, and a decrease in shear strength will induce a bigger creep. At the same stress level, the creep deformation of unsaturated soil increased with the decrease in matric suction. The change in water content had a significant effect on the aging deformation of unsaturated soil, and the increase in water content aggravated the creep deformation.
(3) Based on the relationship between the initial tangent modulus and matric suction, the improved Mesri model and the Logmodified model were established. Based on the experimental data, the modified MESRI model can better describe the creep characteristics of unsaturated soil.
Since the research results are based on the creep characteristics of unsaturated silty clay, the applicability of the empirical creep model established in this research to other soils remains to be further studied. In addition, due to the limit of the equipment, this study is carried out based on the empirical creep model, which is mainly aimed to multiply stress–strain relationship functions by strain–time relationship functions, and it cannot be verified within the scope of the 3D model. But in the future, research suggests that strain gages can be attached to the sides of triaxial instruments to capture lateral creep to sort out the 3D situation.

Conflict of interest: The author declares that they have no conflict of interest.
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