Visco-Elastoplastic Constitutive Fatigue Model for Rocks

'e study on the constitutive fatigue model for rocks under cyclic loading has an important significance in rock engineering. In order to study the fatigue properties of rocks under cyclic loading, according to the theory of rheological mechanics and the existing three basic one-dimensional fatigue elements, i.e., elastic, viscous, and plastic fatigue elements, the three-dimensional elastic, viscous, and plastic fatigue elements were constructed. Meanwhile, a fatigue yield criterion for rocks under cyclic loading was proposed, and a three-dimensional nonlinear visco-elastoplastic fatigue constitutive model (NVPFM) for rocks was established by using the flow criterion related to the proposed fatigue yield criterion. Compared with the test results of rocks under cyclic loading, the three-dimensional NVPFM could be used to describe the transient, steady-state, and tertiary creep phases of rocks under cyclic loading.


Introduction
When rocks are subjected to cyclic or dynamic loading, different types of rocks have different responses. Some rocks may become stronger under cyclic loading, while others become more fragile [1]. In geotechnical engineering, fatigue problems are often involved such as coal walls under mining stress [2][3][4][5], side slope of the dam under cyclic variation water level [6], tunnels under traffic loading, and the surrounding rocks under earthquake load [7][8][9][10][11][12]. e strength of rocks under long-term cyclic loading will be reduced; thus, fatigue damage of rocks is caused, and the engineering accidents may be caused. erefore, the study of fatigue characteristics of rocks under cyclic loading is of theoretical significance for the long-term stability evaluation and design of geotechnical engineering.
For rocks under long-term cyclic loading, even if the upper limit of cyclic loading is lower than the static strength of the material, the material can also be destroyed; this phenomenon is called fatigue [13]. ere is no exception for rocks [14][15][16][17]. e mechanical properties of rocks under cyclic loading are more complex than those under static load.
e mechanical properties of rocks under cyclic loading are affected by many factors. e fatigue life of rocks is related to the stress amplitude of cyclic loading [18][19][20][21], and the dynamic strength of rocks is related to the stress amplitude of cyclic loading and the stress upper limit of cyclic loading. Similarly, the dynamic strength of rocks is also affected by loading rate and the frequency of cyclic loading [22][23][24][25][26][27][28][29]. ere is a threshold value for rocks under cyclic loading, also known as critical strength. e deformation of rocks under cyclic loading is similar to the creep of rocks under static load [30]. When the upper limit of cyclic loading is higher than the critical strength, fatigue failure will occur [31][32][33]. However, when the upper limit of cyclic loading is less than the critical strength, fatigue failure does not occur. In order to study the fatigue deformation characteristics of rocks under cyclic loading, some scholars studied the damage process of rocks from the perspective of damage mechanics. e damage of rocks under cyclic loading is divided into three stages [34]. Xiao et al. proposed an inverted-S damage model to describe the damage process of rocks under cyclic loading [35,36]. Meanwhile, there are many constitutive models based on damage theory [37][38][39]. Additionally, internal variable fatigue constitutive model of rocks was also proposed [40]. However, in recent years, the component model based on rheological mechanics has been applied to build the constitutive model of rocks under cyclic loading as well. Wang et al. [41] established a burgers fatigue model under cyclic confining stress loading with low cyclic frequency. Guo and Huang [42] proposed three one-dimensional fatigue elements, i.e., elastic fatigue element, viscous fatigue element, and plastic fatigue element, and used creep theory to establish a nonlinear fatigue constitutive model of rocks under uniaxial cyclic loading by simplifying the cyclic loading reasonably. e research on the fatigue constitutive model for rocks is mainly based on the damage theory. Although the established constitutive models can effectively describe the rock fatigue deformation characteristic, most of the proposed models can only reflect the fatigue properties of rocks under uniaxial conditions. Moreover, some proposed models cannot reflect the influence of the properties of cyclic loading and strength of rocks on fatigue deformation of rocks. However, at present, the study of the three-dimensional fatigue constitutive model for rocks under cyclic loading is still lacking. Consequently, in this article, on the basis of rheological mechanics and the previous study [42], three three-dimensional fatigue elements, i.e., elastic, viscous, and plastic elements, are constructed. Moreover, the fatigue yield criterion for rocks under cyclic loading was proposed, and a three-dimensional nonlinear viscoelasticplastic fatigue model was established using the flow criterion related to the fatigue yield criterion. e rock fatigue constitutive equations under triaxial conditions were deduced. Finally, the model parameters were identified, and model applicability was verified.

Cyclic Loading.
e common types of cyclic loading are sinusoidal wave, cosine wave, rectangular wave, and triangular wave cyclic loading. e cyclic loading can be expressed as where σ max is the upper limit of cyclic loading; σ min is the lower limit of cyclic loading; f is the frequency of cyclic loading, f � 1/T; T is the period of cyclic loading; t is the time of cyclic loading, and the relationship between N and t is t � NT; N is a positive integer; and A is the stress amplitude of cyclic loading.

Fatigue
Behavior of Rocks. When the upper limit of cyclic loading is higher than the critical strength, fatigue failure will occur, and the creep behavior presents transient, steady-state, and tertiary creep phases (see II curve in Figures 1(a) and 1(b)) [30][31][32][33]44]. e creep phases of under cyclic loading are also called as deceleration, constant velocity, and acceleration creep. However, when the upper limit of cyclic loading is less than the critical strength, fatigue failure does not occur, and the creep behavior presents two creep stages, i.e., transient and steady-state (see I curve in Figure 1(a)) [43,45].

Simplified Cyclic Loading.
To use the component model based on rheological mechanics, assume that σ 1 is the direction of the cyclic loading and it is the first principal stress. Under confining pressure, the equivalent deviatoric stress acting on rocks under cyclic loading was simplified into the following stress function in our previous study [46]: where σ av is the average stress of cyclic loading, σ 1 (t) − σ 3 ; σ 1max is the upper limit of cyclic loading; σ s is the critical strength of the rock under the complex stress state, which is the maximum stress value from the (σ 1 − σ 3 ) − ε curve of the rock under cyclic loading when the rock happens to be destroyed; σ c is the peak stress from the (σ 1 − σ 3 ) − ε curve of the rock under static load; and σ 3 is the small principal stress on rock specimens. According to equation (2), when the peak deviatoric stress of cyclic loading (σ 1 max − σ 3 ) > σ s , the equivalent deviatoric stress (σ 1 − σ 3 ) > σ av , and the fatigue breakdown of the rock will occur. When the peak deviatoric stress of cyclic loading (σ 1 max − σ 3 ) ≤ σ s , the equivalent deviatoric stress (σ 1 − σ 3 ) ≤ σ av , and the rocks do not undergo fatigue breakdown.
When confining pressure σ 3 � 0, the cyclic loading is simplified into the stress function shown in the following equation [42]: where σ av is the average stress of cyclic loading; σ max is the upper limit of cyclic loading; σ s is the critical strength of the rock under uniaxial cyclic loading; and σ c is the rock uniaxial compressive strength.
Similarly, according to equation (3), when σ max > σ s , the simplified stress σ > σ av , and the rock will undergo fatigue breakdown. However, when σ max ≤ σ s , σ ≤ σ av , and the rock does not undergo fatigue breakdown.

Rock Fatigue Yield Criterion under Uniaxial Stress.
Under uniaxial cyclic loading, when σ max ≤ σ s , namely, σ ≤ σ av , the rock will undergo fatigue damage. When σ max ≤ σ s , σ ≤ σ av , and the rock will not undergo fatigue failure.

Rock Fatigue Yield Criterion under Confining Pressure.
Regardless of the effect of spherical stress tensor on fatigue deformation, it is assumed that fatigue deformation of the rock is related only to the second invariant of stress deviator J 2 . According to equation (2), the proposed fatigue yield criterion of the rock under cyclic loading is as follows: where J 2 is the second invariant of the equivalent stress deviator, . When (σ 1 − σ 3 ) > σ av , F(J 2 , σ av ) > 0, and the rock will undergo fatigue breakdown. Conversely, when (σ 1 − σ 3 ) < σ av , F(J 2 , σ av ) < 0, and the rock does not undergo fatigue breakdown. Consequently, the established equation (4) can be used as a fatigue yield function for rocks under cyclic loading, and the yield criterion is a function of σ av of cyclic loading and equivalent stress. [42].

Basic One-Dimensional Fatigue Elements
e onedimensional fatigue model consists of three basic fatigue elements [42], i.e., elastic, viscous, and plastic fatigue elements. For the elastic fatigue element (see Figure 2(a)), the stress acting on the material is proportional to the strain. For the viscous fatigue element (see Figure 2(b)), the stress acting on the material is proportional to the strain rate. e plastic fatigue element is represented by friction sheets shown in Figure 2(c). e constitutive equations are as follows: [42].

Elastic Fatigue Element
where E is the elastic fatigue coefficient, which can be determined by the numerical fitting method. ε e (N) is the fatigue strain. [42].

Viscous Fatigue Element
where η is the viscous fatigue coefficient, which can be determined by the numerical method, dε v (t)/dt is the strain increment of the viscous fatigue element within the time period [t, t + Δt], and dε v (N)/dt is the fatigue strain increment of the element in each cycle of cyclic loading. [42]. When σ max ≤ σ s , that is, σ ≤ σ av , even if the cycle number N tends to infinity, the fatigue strain of the element is still 0. When σ max ≤ σ s , namely, σ ≤ σ av , rocks will undergo fatigue breakdown, and the fatigue strain of rocks will continue to increase with the increase in the number of cycles of cyclic loading . e equation of the state of the plastic fatigue element is

Plastic Fatigue Element
where ε p (N) is the plastic strain of the plastic fatigue element. Based on the above establishment process of one-dimensional fatigue elements and rheological mechanics theory, the three-dimensional fatigue elements under confining pressure can be established in this article.

Establishment of ree-Dimensional Fatigue Elements
ree-Dimensional (3D) Elastic Fatigue Element. e three-dimensional elastic fatigue element is shown in Figure 3, and its equation of state is where S ij is the equivalent deviatoric stress tensor S ij � σ ij − σ m δ ij , in which δ ij is the Kronecker delta; e e ij (N) is the deviatoric strain tensor of the fatigue element; σ kk is the first invariant of the stress tensor; ε kk is the first invariant of the strain tensor; and K is the fatigue volume modulus of the rock, and its physical dimension is stress. en, the total strain ε e ij of the 3D elastic fatigue element under σ ij is where σ m δ ij is the spherical stress tensor, σ m � σ kk /3.

ree-Dimensional (3D) Viscous Fatigue Element.
For the 3D viscous fatigue element (see Figure 4), without considering the volume strain of the element under the spherical stress tensor, the state equation of the element is where _ e v ij (t) is the increment of the deviatoric strain of the fatigue element in period [t, t + Δt] and _ e v ij (N) is the deviatoric strain increment of the viscous fatigue element in each cycle. e following equation can be obtained by integrating equation (10): where e v ij (N) is the deviatoric strain of the viscous fatigue element.

2.6.3.
ree-Dimensional (3D) Plastic Fatigue Element. ree-dimensional plastic fatigue element can be represented by the friction sheet shown in Figure 5. When σ max − σ 3 ≤ σ s , namely, F(J 2 , σ av ) ≤ 0, and even if the number of cycles N tends to infinity, the element fatigue strain is still 0. When σ max − σ 3 > σ s , namely, F(J 2 , σ av ) > 0, fatigue failure can occur in rocks, and the fatigue strain of rocks will still increase with the increase in the number of cycles of cyclic loading. e equation of state of the element is where ε p ij (N) is the plastic strain of the plastic fatigue element.

Three-Dimensional Nonlinear Visco-Elastoplastic Fatigue Constitutive
Model for Rocks e Burgers fatigue model consists of a series of Maxwell and Kelvin fatigue models, which can describe the transient and steady-state creep phases of the rock but cannot describe the tertiary creep phases of the rock. To fully describe the law of creep of the rock under cyclic loading, a nonlinear viscoelastoplastic fatigue model is connected to the burgers model in series, which is a one-dimensional nonlinear visco-elastoplastic fatigue constitutive model (see Figure 6) proposed by Guo and Huang [42].
Schiessel et al. [47] considered the time accumulation of the viscous elements (η K ) in the Kelvin body and defined the fractional viscous element by fractional derivatives. e creep compliance J(t) of the fractional element is proportional to t β ; β is a constant, and t is the rheological time. According to the aforementioned and Guo and Huang's study [42], the viscous coefficient of the series-connected viscous fatigue element in the nonlinear visco-elastoplastic fatigue model, shown in Figure 6, is where n is the parameter that reflects the strain rate of the tertiary creep phases of the rock under cyclic loading, n ≠ 1. η 30 is the initial value of the viscous coefficient, and η 30 ≠ 0; its physical dimension is stress·time n. Equation (13) can also be further expressed as

Establishment of the Nonlinear Viscoplastic Fatigue
Model. Under confining pressure stress, the fatigue mechanics parameters of the rock are different from those under one-dimensional stress. e three-dimensional nonlinear viscoelastic-plastic fatigue model consists of a three-dimensional elastic fatigue element, a viscous fatigue element, a viscoelastic fatigue model, and a nonlinear viscoplastic fatigue model (see Figure 7). In the model, G 1 is the elastic shear fatigue coefficient, G 2 is the viscoelastic shear fatigue coefficient, η 1 is the viscous shear fatigue coefficient, η 2 is the viscoplastic shear fatigue coefficient, and η 30 is the initial value of the viscoplastic shear fatigue coefficient. e physical dimension of model parameters G 1 and G 2 is stress.

Viscoelastic Fatigue Model.
For the viscoelastic fatigue model shown in Figure 8, the effect of the spherical stress tensor on volumetric strain is not considered. e state σ σ E ε e (N) Figure 2: One-dimensional fatigue element [42].  Figure 8, is where _ _ e  (15):

Nonlinear Viscoplastic Fatigue Model.
For the viscoplastic fatigue model shown in Figure 9, the viscoplastic shear fatigue coefficient η 3 (N) in the model is the same as equation (14), and the plastic strain rate of the fatigue model according to the rheologic basic principle can be written as where e • nl ij (t) is the deviator strain increment of the fatigue element in the time period [t, t + Δt]; e • nl ij (N) is the deviator strain increment of the fatigue element in each cycle; F is the fatigue yield function of the rock; and Q is the plastic potential function of rock fatigue. e following fatigue constitutive equation can be obtained by substituting equation (14) into equation (17) and integrating N: where e nl ij (N) is the plastic deviator strain tensor of the nonlinear viscoplastic fatigue model. Among them, In equation (20) [48], B m , m, and M are constants. In this article, we used M � 1 and B 1 � 1 for convenience.
When F(J 2 , σ av ) ≤ 0, the rock does not undergo fatigue breakdown. When F(J 2 , σ av ) > 0, the following fatigue constitutive equation of the fatigue element can be obtained by using the flow rules related to the fatigue yield criterion:

Constitutive Equation of the Nonlinear Visco-Elastoplastic
where ε ij (N) is the total strain tensor of the fatigue model. When the rock fatigue yield function F(J 2 , σ av ) > 0, the rock will undergo fatigue failure. At this time, the model is a

Fatigue Constitutive Equation for
Rocks under Triaxial Compression e dynamic triaxial test is an indoor method for studying the fatigue properties of rocks under confining pressure stress. e adaptability of the proposed model can be verified by dynamic triaxial test data. Consequently, on the basis of three-dimensional rock fatigue constitutive equation (23), the fatigue constitutive equation under triaxial condition is deduced.
According to equation (2), the equivalent stress state of the rock under dynamic triaxial condition is where σ 3 is the confining pressure acting on specimens.
Deviatoric stress under the simplified stress state is as follows: where δ ij is the Kronecker symbol, σ m is the average stress, σ m � (σ 1 + 2σ 3 )/3, and σ m δ ij is the spherical stress tensor. e total strain of the rock under dynamic triaxial condition based on the 3D NVPFM is as follows: where ε e v , ε v v , ε ve v , and ε nl v are rock elastic, viscous, viscoelastic, and viscoplastic volume strains, respectively.
Without considering the viscous, viscoelastic, and viscoplastic volume strains of the rock and considering only the elastic volume strain, then ε v v � ε ve v � ε nl v � 0. Consequently, the following equation can be obtained from equation (26): where e ij is the total deviatoric strain tensor caused by deviatoric stress. When F(J 2 , σ av ) ≤ 0, the rock will not undergo fatigue failure, and circumferential constitutive equation (28) and axial constitutive equation (29) of rocks under dynamic triaxial condition can be obtained by substituting equations (24)- (27) into equation (22): When F(J 2 , σ av ) > 0, the rock will undergo fatigue damage, and circumferential constitutive equation (30) and axial constitutive equation (31) under dynamic triaxial condition can be obtained by substituting equations (24)- (27) into equation (23):

Parameter Determination and the Curve Fitting Method for Fatigue Constitutive Equations.
For equations (28)-(31), the model parameters can be obtained by [48] regression analysis [49,50]. 1stOpt is world's leading software for nonlinear curve fitting and the comprehensive optimization analysis of nonlinear equation parameters. It has been widely recognized in the field of nonlinear regression, nonlinear curve fitting, and parameter estimation of the nonlinear equation. e genetic algorithm (GA) method is a built-in nonlinear optimization control algorithm in 1stOpt, which can solve the parameter value of the multiparameter nonlinear equation and fit the nonlinear curve according to the corresponding data [45]. In this article, the GA method in 1stOpt was used to solve the parameters of the proposed model and fit the curve of test data. e proposed three-dimensional NVPFM can describe the creep of the rock under cyclic loading. e parameters of the three-dimensional NVPFM equation can be solved. According to the previous study [42], the detailed process of obtaining model parameters and fitting test data is as follows. According to the stress condition of the sample during the test, when rocks undergo fatigue breakdown, equation (30) is selected. However, when rocks do not undergo fatigue breakdown, equation (28) is selected. en, the test data from the literature and the selected constitutive equation were input into software in the form of a program. Afterwards, the GA method was used to fit the fatigue curve of the rock under cyclic loading. After the program runs, the parameter values and the curve of the model can be obtained. In this study, only the constitutive equation of the rock under fatigue breakdown is studied.

Model Parameters and the Curve of Red Sandstone.
Zhang et al. [51] used sinusoidal wave cyclic loading with 0.2 Hz frequency to conduct triaxial tests on red sandstone under confining pressure. e fatigue curve of red sandstone under 15 MPa confining pressure is shown in Figure 10 Table 1 and Figure 11, respectively.

Model Parameters and the Curve of the Water-Rich Soft
Rock. In order to study the dynamic characteristics and structural safety of the soft rock in the foundation of the high-speed railway tunnel, Ding [52] used the MTS rock dynamic system to carry out fatigue tests on the water-rich soft rock under different dynamic stress ratios η d (see equation (32)) by using the loading method shown in Figure 12. e effects of dynamic stress amplitude σ d and static deviatoric stress σ st on the fatigue characteristics of the water-rich soft rock were studied by Ding [52]. e critical stress ratio η d of the water-rich soft rock under 200 kPa confining pressure, 180 kPa static deviatoric stress σ st , 3 Hz loading frequency, and 50 kPa groundwater pressure is 0.3-0.5. In this paper, the critical dynamic stress ratio of the water-rich soft rock is 0.4, and the critical dynamic stress amplitude σ dc is 240 kPa. e cyclic loading σ 1 (t) is expressed by equation (33). en, the upper limit σ lmax (34) of cyclic loading can be obtained from equation (33). e critical strength of the water-rich soft rock σ s was σ st + (σ dc /2). For the dynamic strength σ c rock, Ge [53] found that the critical strength of the rock under confining pressure is 0.85 to 0.9 times of the static strength of the rock. erefore, the static strength σ c of the water-rich soft rock was taken as σ s /0.85 in this study. e static strength σ c of the water-rich soft rock under 200 kPa confining pressure was 353 kPa, and its critical strength σ s was 300 kPa. Constitutive equation (28) based on the NVFPM proposed was used to fit the fatigue curve of the water-rich soft rock under cyclic loading with 300 kPa dynamic stress amplitude and dynamic stress ratio of 0.5. e fitting results and model parameters are shown in Table 2 and Figure 13.
where σ cua is the uniaxial compressive strength of the waterrich soft rock, and its value is 0.6 MPa.    Table 3 and Figure 14, respectively.
It can be seen from Figure 14 that fatigue constitutive equation (28) established in this paper can reflect the transient, steady-state, creep phases of rocks under cyclic loading. e correlation between model and test data was good. As can be seen from Table 3, when σ 1 − σ 3 < σ av , the rock does not suffer from fatigue failure, and some of the obtained parameters of the model showed apparent regularity. e values of viscoelastic shear fatigue parameters G 2 and viscoelastic shear fatigue parameters η 1 decay exponentially with the increase in the dynamic strain ratio η d . e specific attenuation law is shown in Figures 15 and 16. For the specific change of other parameters that cannot be determined for the time being, further study needs to be made in combination with more tests.
From the above regression analysis results, it can be seen that the fatigue constitutive model proposed in this paper can describe the creep characteristics of the rock under cyclic loading. When (σ 1 − σ 3 ) ≤ σ av , rocks do not undergo fatigue breakdown, and the proposed model would degenerate into the burgers model, which can describe the transient, steadystate, and creep phases of the rock. When (σ 1 − σ 3 ) ≤ σ av , rocks undergo fatigue breakdown, and the proposed NVPFM can reflect the transient, steady-state, and tertiary creep phases of the rock under cyclic loading.

Conclusions
According to the rheological mechanics, three-dimensional elastic, viscous, and plastic fatigue elements were constructed. Moreover, a fatigue yield criterion for the rock under cyclic loading was proposed based on the simplified cyclic loading in the previous study. A three-dimensional nonlinear visco-elastoplastic fatigue model (NFVPM) was established under confining pressure by using the flow rules   Advances in Civil Engineering related to the proposed fatigue yield criterion. Finally, the rationality of the model is verified. e following conclusions can be drawn from this study. e established NFVPM can describe the transient, steady-state, and tertiary creep phases of the rock under cyclic loading. Under confining pressure stress state, when the proposed fatigue yield criterion of the rock F(J 2 , σ av ) ≤ 0, the model degenerates into a burgers model that describes the transient and steady-state creep phases of the rock under cyclic loading. However, when F(J 2 , σ av ) > 0 and the parameter n ≠ 1, the model is a NFVPM that describes the transient, steady-state, and tertiary creep phases of the rock under cyclic loading. e parameters of the NFVPM can be easily solved using mathematical methods. e parameters G 2 and η 1 of the NFVPM decreased with the increase in dynamic stress ratio η d in the way of power function. Moreover, the curves of the NFVPM were in good agreement with the experimental results, and the correlation was above 0.990.

Data Availability
e data used to support the findings of this study are available within this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.