A New Creep–Fatigue Interaction Model for Predicting Deformation of Coarse-Grained Soil

Studying the creep–fatigue interaction of the coarse-grained soil (CGS) is very important for safety assessment and disaster prevention in subgrade engineering. Current research work is mainly focused on single creep or fatigue deformation. In this paper, a new creep–fatigue interaction model is established to predict the creep–fatigue interaction deformation of different gradation CGS based on the rheological mechanics and the interactive relationship between creep and fatigue complex compliance method. Triaxial creep–fatigue interaction tests of different gradations CGS under different average stresses and frequencies were conducted to verify the new creep–fatigue interaction model. Research results show that for the creep–fatigue and fatigue–creep interaction, the fatigue deformation is always larger than the creep deformation under the same stress level. For the creep–fatigue multi-interaction, the second creep and fatigue deformation are always smaller than the first creep and fatigue deformation. The results of the triaxial creep–fatigue interaction tests verify the validity of this new model.


Introduction
Coarse-grained soil (CGS in short) is a mixture of pebble, sand and clay, etc. [1,2], and is commonly used as a subgrade material in highway and railway engineering [3][4][5]. It is generally subjected to static or repeated dynamic load caused by transportation tools (vehicles and trains) [6,7]. Since pebble and sand usually show elasticity while clay has viscoelasticity, the CGS is considered to be of rheological properties [8]. Under longterm static-dynamic (i.e., creep-fatigue) interaction loading, the CGS may cause uneven settlement and potential safety hazards for the highway and railway [9].
Currently, there are two main creep-fatigue loading forms: the simultaneous superposition of creep and fatigue loads [10,11] and alternative creep and fatigue loads [12,13], among which the latter is a common case for the CGS of subgrade in highway and railway engineering since the vehicles and trains usually have an alternative state of stopping or moving (i.e., successive creep-fatigue loads). Therefore, it is of great significance to study the successive creep-fatigue interacting models and mechanisms for predicting the longterm deformation of CGS. Niya Dong [14] proposed a newly modified multi-sequenced repeated load test, measured the strain, resilient modulus, and creep compliance, proposed new factors to evaluate the viscoelasticity of different asphalt mixtures. V. P. Golub [15] proved that there exists a definite correlation between the processes of static and cyclic creep and proposed a relationship of creep-fatigue interaction. Yunliang Li [16] analyzed the cement emulsified asphalt composite binder evolution process of fatigue damage by fatigue test, and with the increase in fatigue damage, instantaneous elastic deformation and creep rate increase in steady creep stage during creep process. Linjian Ma [13] conducted the creep-fatigue interaction test of rock salt by trapezoidal wave, and the results showed that under combined creep and fatigue, the cycling life of the rock salt is remarkably lower compared with that observed in pure fatigue or creep tests and the life of creep-fatigue interaction can be alternatively predicted without rupturing the salt specimens. Hongfu Liu [11] established a fatigue-creep damage interaction model considering the interaction of fatigue and creep damage based on the principle of semi-sine fatigue stress decomposition. The results indicated that the damage of the asphalt mixture under the action of the semisine cyclic load was not a simple linear accumulation of fatigue damage and creep damage, but an overall result of the interaction of fatigue damage, creep damage, and fatigue-creep damage. Qian Li [17] researched and indicated that cyclic loading has a significant influence on the creep of high-strength high-performance concrete, especially during the early stage of creep development. Additionally, the modified creep model can be adopted to predict the long-term behaviors of concrete bridges, which are always subjected to cyclic traffic loading. Bara Wasfi Al-Mistarehi [18] considered that the accumulated strain which increased with time at different levels of temperature for all types of fillers was obtained from the creep test and fatigue test and between logarithm strain and logarithm number of cycles. As a result, the waste toner as filler was the best filler. Rafiqul Tarefder [19] showed that the increase in freeze-thaw cycles in asphalt concrete decreased the indirect tensile strength and tensile strength ratio of asphalt. Xiaozhao Li [20] proposed an analytical solution by coupling the confined cyclic static loading and unloading path, the Hooke-Kelvin viscoelastic model and the formulated micro-macro model which explains the total timedependent visco-elastic-plastic deformation caused by the microcracks variable during cyclic static compressive failure. Vitaliy M. Kindrachuk [21] developed a fatigue model for normal strength concrete under compressive loading conditions based on continuum damage mechanics which incorporates creep and fatigue phenomena in a single framework and postulates damage to be driven by inelastic deformations. Yu Su [5] developed a fatigue model of railway soil that considered the coarse grain content and the number of loading cycles. Abdulgazi Gedik [22] studied the fatigue and dynamic creep modulus of asphalt pavement with recycling fluorescent lamps and Yasser M [23] studied the dynamic rheological characteristics of asphalt pavement with two different recycled plastic wastes. Estelle Delfosse Ribay [24] presented the creep and fatigue behaviors of grouted sand to analyze the strain rate, creep slope or fatigue slope, and the creep limit strength or fatigue limit strength. Alireza Ameli [25] analyzed the resilient modulus, indirect tensile strength and dynamic creep of asphalt binders and mixtures. Very few studies are available on the creep-fatigue interaction of the CGS.
In this paper, two new interaction parameters are proposed to establish the new creepfatigue interaction model based on rheology mechanics and the relationship between the creep and fatigue complex compliance. The interaction parameters were determined by the triaxial creep and fatigue tests, and the new interaction model was verified by the predicted and tested results of the triaxial creep-fatigue interaction tests. Figure 1 shows the creep-fatigue interaction process of the CGS under triaxial compression (σ 1 > σ 3 , σ 1 and σ 3 are axial and confining stresses), where the interaction order can be the creep first followed by fatigue (creep-fatigue interaction, C-F for short), or the fatigue first followed by creep (fatigue-creep interaction, F-C in short). For each creepfatigue interaction process, the creep stress (σ 1 = σ c , σ c is the creep stress, σ c − σ 3 is the deviatoric stress of creep) is applied to the CGS for a period of time (t = t 1 ) and unloaded to be σ 1 = σ 3 , and then the fatigue stress in terms of semi-sine (σ 1 = σ f = 2σ c |sinωt|, ω is circular frequency, ω = 2πf, f is fatigue loading frequency) is applied to the CGS for a period of time (t = t 2 − t 1 ). Let the creep deviatoric stress (σ c − σ 3 ) be equal to the average deviatoric stress of the fatigue cycle (σ a = σ c − σ 3 , σ a is the average deviatoric stress of fatigue, σ a = (σ f − σ 3 )/2) since in order to establish a relationship under the same stress factor. deviatoric stress of the fatigue cycle (σa = σc − σ3, σa is the average deviatoric stress of fatigue, σa = (σf − σ3)/2) since in order to establish a relationship under the same stress factor.

Creep Fatigue
Unloading Unloading In order to study the creep-fatigue interaction for the CGS deformation, it is necessary to seek a creep-fatigue interaction parameter to link up the two loading processes.
In rheologic mechanics, there exists a relationship of Fourier equivalent transformation between the creep compliance J(t) and fatigue compliance J(ω), where  is Fourier transformation and  −1 is inverse Fourier transformation. Since test results show that the CGS has larger fatigue deformation than creep deformation under σa = σc [26], J(t) and J(ω) cannot be equivalently transformed by Equation (1). For modifying Equation (1) to obtain the actual equivalent transform relationship between J(t) and J(ω) of the CGS, J(ω) is divided into a real part Jr(ω) and imaginary part Ji(ω), and J(t) can also be in terms of its real part Jr(t) and imaginary part Ji(t) by Fourier transform.
Considering that both the creep and fatigue compliances have real and imaginary parts, define two new creep-fatigue interaction factors k and g, define the matrices A1×2 = (Jr(t) Ji(t)), B1×2 = (Jr(ω) Ji(ω)) and M2×2 = k 0 0 g , and then establish the relationship between the real and imaginary parts of the creep (Jr(t), Ji(t)) and fatigue (Jr(ω), Ji(ω)) compliances, i.e., the new creep-fatigue interaction model: For simplifying the expression, that is Equation (  In order to study the creep-fatigue interaction for the CGS deformation, it is necessary to seek a creep-fatigue interaction parameter to link up the two loading processes. In rheologic mechanics, there exists a relationship of Fourier equivalent transformation between the creep compliance J(t) and fatigue compliance J(ω), where F is Fourier transformation and F −1 is inverse Fourier transformation.
Since test results show that the CGS has larger fatigue deformation than creep deformation under σ a = σ c [26], J(t) and J(ω) cannot be equivalently transformed by Equation (1). For modifying Equation (1) to obtain the actual equivalent transform relationship between J(t) and J(ω) of the CGS, J(ω) is divided into a real part J r (ω) and imaginary part J i (ω), and J(t) can also be in terms of its real part J r (t) and imaginary part J i (t) by Fourier transform.
Considering that both the creep and fatigue compliances have real and imaginary parts, define two new creep-fatigue interaction factors k and g, define the matrices A 1×2 = (J r (t) J i (t)), B 1×2 = (J r (ω) J i (ω)) and M 2×2 = k 0 0 g , and then establish the relationship between the real and imaginary parts of the creep (J r (t), J i (t)) and fatigue (J r (ω), J i (ω)) compliances, i.e., the new creep-fatigue interaction model: For simplifying the expression, that is Equation (5): Since J r (t), J i (t) and J r (ω), J i (ω)) are obtained by the following creep and fatigue tests, respectively, k and g can be calculated by Equation (4).

Determination of Creep-Fatigue Interaction Factor
In order to determine the creep-fatigue interaction parameters k and g: (1) conduct the triaxial creep tests (creep stress σ c ) and the fatigue tests (fatigue average stress σ a ) under the same average stress condition (σ c − σ 3 = σ a ) to determine creep compliance and fatigue compliance; (2) calculate the creep and fatigue real and imaginary parts of compliance; and (3) compare the real parts and imaginary parts of the creep and fatigue compliance to determine the k and g.

Scheme of Triaxial Creep and Fatigue Tests
According to the Code for the Design on Subgrade of Railway in China (TB10001-2005), three different gradations of CGS were prepared by different mass ratios of pebble (γ) (with high elastic modulus), sand and clay (with rheological properties), as listed in Table 1. Figures 2 and 3 showed their grain-size distribution curves and water content (w)-dry density (ρ d ) curves for three different gradations of CGS, respectively. The CGS specimens are cylinders of Φ300 mm × 600 mm and prepared by the layered compacting method ( Figure 4).
Since Jr(t), Ji(t) and Jr(ω), Ji(ω)) are obtained by the following creep and fatigue tests, respectively, k and g can be calculated by Equation (4).

Determination of Creep-Fatigue Interaction Factor
In order to determine the creep-fatigue interaction parameters k and g: (1) conduct the triaxial creep tests (creep stress σc) and the fatigue tests (fatigue average stress σa) under the same average stress condition (σc − σ3 = σa) to determine creep compliance and fatigue compliance; (2) calculate the creep and fatigue real and imaginary parts of compliance; and (3) compare the real parts and imaginary parts of the creep and fatigue compliance to determine the k and g.

Scheme of Triaxial Creep and Fatigue Tests
According to the Code for the Design on Subgrade of Railway in China (TB10001-2005), three different gradations of CGS were prepared by different mass ratios of pebble (γ) (with high elastic modulus), sand and clay (with rheological properties), as listed in Table 1. Figures 2 and 3 showed their grain-size distribution curves and water content (w)-dry density (ρd) curves for three different gradations of CGS, respectively. The CGS specimens are cylinders of Φ300 mm × 600 mm and prepared by the layered compacting method (Figure 4).

No. Pebble (γ)
Triaxial compressive creep and fatigue tests testing system ( Figure 5). Table 2 lists their loadi dations of CGS specimens, where the confining average stress of fatigue test σa = σc − σ3. There w stress) and 27 fatigue tests (3 specimen × 3 stress tical specimens were prepared and there were 1 tests, the strain was recorded every 2 min to ob tests, the strain and stress were recorded 10 time Triaxial compressive creep and fatigue tests were conducted on the TAJ-2000 triaxial testing system ( Figure 5). Table 2 lists their loading conditions for the three different gradations of CGS specimens, where the confining pressure σ 3 was 0.2 MPa (σ 3 < σ 1 ). Let the average stress of fatigue test σ a = σ c − σ 3 . There were a total of 9 creep tests (3 specimen × 3 stress) and 27 fatigue tests (3 specimen × 3 stress × 3 frequency). For each test, three identical specimens were prepared and there were 108 specimens in total. During the creep tests, the strain was recorded every 2 min to obtain the creep curves. During the fatigue tests, the strain and stress were recorded 10 times in one cycle.    In this study, the Burgers creep model ( Figure 6, Equation (5)) was adopted to describe the creep phenomenon of the different gradations of CGS and the creep compliance J(t) can be obtained by fitting the tested creep curves:

022, 15, 3904
where E 1 , E 2 , η 1 , η 2 are creep parameters of the Burgers model. In this study, the Burgers creep model ( Figure 6, Equation (5)) was adopte scribe the creep phenomenon of the different gradations of CGS and the creep com J(t) can be obtained by fitting the tested creep curves:     (14)). Table 3 lists the fitted Burgers model parameters with high fitting precision (R 2 > 0.995). It is seen that all the creep curves have three stages: the instantaneous deformation stage (t = 0), attenuation deformation stage (with decreasing slope), and stable creep deformation stage (with a constant slope). Obviously, for the same gradation of the CGS specimen, ε c is increased with the increase in σ c . While for the different gradations of the CGS specimen, ε c is increased with the decrease in pebble content γ since a smaller γ means more clay content with stronger rheological properties. 0.995). It is seen that all the creep curves have three stages: the instantaneous deformation stage (t = 0), attenuation deformation stage (with decreasing slope), and stable creep deformation stage (with a constant slope). Obviously, for the same gradation of the CGS specimen, εc is increased with the increase in σc. While for the different gradations of the CGS specimen, εc is increased with the decrease in pebble content γ since a smaller γ means more clay content with stronger rheological properties.     (2) Unified creep parameters for different creep stresses and gradations It was found in Table 3 that the Burgers creep parameters depend not only on the creep stress (σ c ) or the deviatoric stress (σ = σ c − σ 3 ), but also on the gradations (mass ratios γ, water content w). In order to obtain a unified creep parameter relationship (related to σ, γ, and w) to describe the creep behaviors for different gradations of CGS under different stresses, the nonlinear regression is needed. Since the Burgers model ( Figure 6) is the Maxwell model connected to the Kelvin model in series, E 1 and η 1 are independent parameters, while E 2 and η 2 are coupling parameters. Therefore, the unified creep parameter relationship can be determined by nonlinear regression with high precision (R 2 > 0.989). Table 4 lists the nonlinearly regressed and tested values of Burgers creep parameters for comparison. They fit well, mostly with the error E < 10%, which proves the validity of Equation (6). Based on Burgers creep model and its parameter (Table 4), Figure 8 gives the tested and nonlinearly regressed creep curves (ε c -t) under different creep stresses for three gradations of CGS. It is seen that they are almost the same at the instantaneous deformation stage. In the attenuation deformation stage, their differences increase with time but tend to be unchanged in the stable deformation stage. In a word, they are approximately close to each other, which again proves the validity of Equation (6). gradations of CGS. It is seen that they are almost the same at the instantaneous deformation stage. In the attenuation deformation stage, their differences increase with time but tend to be unchanged in the stable deformation stage. In a word, they are approximately close to each other, which again proves the validity of Equation (6).

Triaxial Fatigue Test and Analyze
(1) Fatigue compliance formulae When the sine wave is adopted as the fatigue load: The strain response is where σ f and ε f are the peak stress and strain of fatigue, δ is the phase angle and |J(ω)| is the module of fatigue compliance. Equations (7) and (8) can be rewritten as Substituting Equation (9) into Equation (10) yields the strain-stress relationship, i.e., the elliptic Equation: σ Since this elliptic Equation is obtained by fitting the data of a one-cycle fatigue test, the phase angle δ can be determined by Equation (11).
Therefore, the real parts J r (ω) and imaginary parts J i (ω) of the fatigue compliance can be calculated by Equation (8) and the relationship of the phase angle δ and the fatigue compliance [23] are as follows: (2) Experimental determination of fatigue compliance Dynamic loading adopts the stress-controlled cyclic loading method, applying semisine wave axial cyclic stress, and the fatigue load was applied on the specimen after the balance of σ 1 = σ 3 , and recorded 10 points of the stress and strain in one cycle by the test system. Figure 9 shows some specimens after the fatigue test of different gradations of CGS. Since this elliptic Equation is obtained by fitting the data of a one-cycle fatigue test, the phase angle δ can be determined by Equation (11).
Therefore, the real parts Jr(ω) and imaginary parts Ji(ω) of the fatigue compliance can be calculated by Equation (8) (2) Experimental determination of fatigue compliance Dynamic loading adopts the stress-controlled cyclic loading method, applying semisine wave axial cyclic stress, and the fatigue load was applied on the specimen after the balance of σ1 = σ3, and recorded 10 points of the stress and strain in one cycle by the test system. Figure 9 shows some specimens after the fatigue test of different gradations of CGS.
(a) (b) (c) Figure 9. Deformation of CGS after fatigue tests: (a) S1; (b) S2; and (c) S3. Figures 10 and 11 show the tested and fitted hysteretic curves of the S1 specimen (at different frequency f for the same average stress σa) and the S2 specimen (under different σa for the same f) for a one-cycle load for example. Table 5 lists the areas of the hysteretic circles (A represents the energy dissipation in one cycle, Equation (13) [27]) obtained by Matlab with its own written code. It can be seen that A slightly decreased with the increase in f, that is, the frequency effect is much smaller, while A was almost multiplied as it increased with the multiplied increasing of σa for the same gradation CGS. Additionally, A increased with the decrease in pebble content (γ) for different gradations of CGS, because a greater γ means that the CGS specimen had weaker rheological properties; therefore, the smaller γ represents the stronger rheological properties which could dissipate more  Figures 10 and 11 show the tested and fitted hysteretic curves of the S1 specimen (at different frequency f for the same average stress σ a ) and the S2 specimen (under different σ a for the same f ) for a one-cycle load for example. Table 5 lists the areas of the hysteretic circles (A represents the energy dissipation in one cycle, Equation (13) [27]) obtained by Matlab with its own written code. It can be seen that A slightly decreased with the increase in f, that is, the frequency effect is much smaller, while A was almost multiplied as it increased with the multiplied increasing of σ a for the same gradation CGS. Additionally, A increased with the decrease in pebble content (γ) for different gradations of CGS, because a greater γ means that the CGS specimen had weaker rheological properties; therefore, the smaller γ represents the stronger rheological properties which could dissipate more energy in one cycle. The fatigue compliance (J r (ω), J i (ω), J(ω) and δ) can be calculated by Equations (7)- (13).

Determination of Creep-Fatigue Interaction Parameters
The creep-fatigue interaction parameters k and g can be determined by Equation (4) when the real part J r (t) and imaginary part J i (t) of the creep compliance J(t) is calculated by substituting Equation (5) with Table 4 into Equation (2), and the real part J r (ω) and imaginary part J i (ω) of the fatigue compliance J i (ω) is calculated by Equation (12). Tables 6-8 list the calculation results of creep and fatigue compliance for different gradations of CGS under different stress levels (σ a ). Table 6. Calculation results of creep-fatigue interaction parameters k and g for different gradations of CGS (σ a = 0.1 MPa).  Table 7. Calculation results of creep-fatigue interaction parameters k and g for different gradations of CGS (σ a = 0.2 MPa).
No.  Table 8. Calculation results of creep-fatigue interaction parameters k and g for different gradations of CGS (σ a = 0.3 MPa).
No. γ f /Hz It can be seen that with the increasing frequency f that the real part J r (t) of creep compliance (representing the energy storage compliance) is unchanged while the imaginary part J i (t) (representing the energy dissipation compliance) is deceased, which is the same as analytic results [28]. With the decreasing pebble content γ, J r (t) is decreased but J i (t) is increased, since a smaller value of γ means a greater clay content of the CGS specimen and thus stronger rheological properties with more energy dissipation in one cycle.
Similarly, with the increase in frequency f, the real part J r (ω) of fatigue compliance is unchanged while the imaginary part J i (ω) is deceased. With the decrease in γ, J r (ω) is decreased but J i (ω) is increased.
Obviously, the creep-fatigue interaction parameter k (corresponding to J r (t) and J r (ω)) is also unchanged while g (corresponding to J i (t) and J i (ω)) is increased with the increase in frequency f. With the decrease in γ, k changes non-monotonically while g is increased.

Prediction Method and Test Scheme
Take the creep-fatigue interaction deformation of CGS as an example (the fatiguecreep interaction deformation is vice versa): the strain-time (ε-t) curve can be calculated by the following steps: (1) calculate J r (t) and J i (t) by Equation (2) to obtain the ε c -t curve; (2) calculate J r (ω) and J i (ω) by Equation (12) to obtain the ε f -t curve; (3) according to the creep-fatigue interaction factors k and g in Tables 6-8 to calculate the creep-fatigue interaction ε-t curve.
To verify the validity of the calculated ε-t curve based on the new creep-fatigue interaction model, triaxial creep-fatigue interaction tests, triaxial fatigue-creep interaction tests and the triaxial multi-interaction of three gradations of CGS specimens (Table 1) were conducted by triaxial test system ( Figure 5). Table 9 lists the loading conditions of creepfatigue interaction tests for different gradations of CGS under the same confining pressure (0.2 MPa).  Figure 12 shows the predicted and tested curves of creep-fatigue (C-F) interaction deformation for S1 specimens under different stress levels. It is seen that all creep curves (the first part of C-F interaction) and fatigue curves (the second part of C-F interaction) have three stages: (1) instantaneous deformation stage; (2) attenuation deformation stage; and (3) stable deformation stage. The predicted creep and fatigue curves are close to the tested curves, which can prove the validity of the new creep-fatigue interaction model. The difference between the predicted and tested fatigue curves might be caused by the impact effects of fatigue loads and the discreteness of geotechnical materials.  Figure 12 shows the predicted and tested curves of creep-fatigue (C-F) interaction deformation for S1 specimens under different stress levels. It is seen that all creep curves (the first part of C-F interaction) and fatigue curves (the second part of C-F interaction) have three stages: (1) instantaneous deformation stage; (2) attenuation deformation stage; and (3) stable deformation stage. The predicted creep and fatigue curves are close to the tested curves, which can prove the validity of the new creep-fatigue interaction model. The difference between the predicted and tested fatigue curves might be caused by the impact effects of fatigue loads and the discreteness of geotechnical materials.  Figure 13 shows the predicted and tested ε-t curves of fatigue-creep interaction for S2 specimens. It can be seen that all of the creep curves and all of the fatigue creep curves also have three deformation stages. In the fatigue stage, there is no situation in which the tested strain curve is much larger than the predicted strain curve compared with Figure  11, because the creep complex compliance is calculated based on the fatigue complex compliance and the predicted and tested curves are close to each other. In the creep stage, the predicted and tested curves are in good agreement. This again verifies the validity of the new creep-fatigue interaction model.  Figure 13 shows the predicted and tested ε-t curves of fatigue-creep interaction for S2 specimens. It can be seen that all of the creep curves and all of the fatigue creep curves also have three deformation stages. In the fatigue stage, there is no situation in which the tested strain curve is much larger than the predicted strain curve compared with Figure 11, because the creep complex compliance is calculated based on the fatigue complex compliance and the predicted and tested curves are close to each other. In the creep stage, the predicted and tested curves are in good agreement. This again verifies the validity of the new creep-fatigue interaction model.  Figure 14 shows the predicted and tested curves of creep-fatigue multi-interaction for S3 specimen. It is seen that the predicted curves are in good agreement with the tested curves for two creep stages, while the predicted curves are larger than the tested curves for two fatigue stages. The reason is the dynamic load make a greater deformation for the same average stress. For the two fatigue stages, the strain produced from 120 min to 240  Figure 14 shows the predicted and tested curves of creep-fatigue multi-interaction for S3 specimen. It is seen that the predicted curves are in good agreement with the tested curves for two creep stages, while the predicted curves are larger than the tested curves for two fatigue stages. The reason is the dynamic load make a greater deformation for the same average stress. For the two fatigue stages, the strain produced from 120 min to 240 min is approximately 0.8%, while the strain produced from 360 min to 480 min is approximately 0.6%; the strain of the second fatigue stage is smaller than that of the first fatigue stage, because the rough pores of the CGS have been irreversibly reduced in the first fatigue stage, and after the first fatigue stage, the internal particles of the CGS will form a more stable force chain; hence, the strain generated in the second fatigue stage is less than the first fatigue stage.  Figure 14 shows the predicted and tested curves of creep-fatigue multi-interaction for S3 specimen. It is seen that the predicted curves are in good agreement with the tested curves for two creep stages, while the predicted curves are larger than the tested curves for two fatigue stages. The reason is the dynamic load make a greater deformation for the same average stress. For the two fatigue stages, the strain produced from 120 min to 240 min is approximately 0.8%, while the strain produced from 360 min to 480 min is approximately 0.6%; the strain of the second fatigue stage is smaller than that of the first fatigue stage, because the rough pores of the CGS have been irreversibly reduced in the first fatigue stage, and after the first fatigue stage, the internal particles of the CGS will form a more stable force chain; hence, the strain generated in the second fatigue stage is less than the first fatigue stage.

Conclusions
(1) New transformation parameters (k and g) are proposed to establish a new creep-fatigue interaction model based on both the rheologic mechanics and the interactive relationship between creep and fatigue complex compliance, in order to predict the creep-fatigue interaction deformation of CGS.

Conclusions
(1) New transformation parameters (k and g) are proposed to establish a new creepfatigue interaction model based on both the rheologic mechanics and the interactive relationship between creep and fatigue complex compliance, in order to predict the creep-fatigue interaction deformation of CGS. (2) The creep-fatigue interaction factor k is almost unchanged with the frequency (f ) and changed non-monotonic with the pebble content (γ). The creep-fatigue interaction factor g is increased with both f and γ. The creep strain and fatigue strain of the creep-fatigue and fatigue-creep interaction are increased with the increase in average stress (σ a ) (3) For the different interaction orders (creep-fatigue interaction or fatigue-creep interaction), the fatigue deformation is always larger than the creep deformation under the same stress level because of the dynamic effect. (4) For the creep-fatigue multi-interaction, the second creep and fatigue deformation is always smaller than the first creep and fatigue deformation because of the compaction effect.  Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Conflicts of Interest:
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.