# Visco-Elastoplastic Constitutive Fatigue Model for Rocks

Authors: Jianhao Liu; Shaoyun Pu; Junying Rao
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The 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.


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## Body

## 1. 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 [2–5], side slope of the dam under cyclic variation water level [6], tunnels under traffic loading, and the surrounding rocks under earthquake load [7–12]. The 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. Therefore, 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]. There is no exception for rocks [14–17]. The mechanical properties of rocks under cyclic loading are more complex than those under static load. The mechanical properties of rocks under cyclic loading are affected by many factors. The fatigue life of rocks is related to the stress amplitude of cyclic loading [18–21], 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 [22–29]. There is a threshold value for rocks under cyclic loading, also known as critical strength. The 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 [31–33]. 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. The 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 [37–39]. 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.The 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 viscoelastic-plastic fatigue model was established using the flow criterion related to the fatigue yield criterion. The rock fatigue constitutive equations under triaxial conditions were deduced. Finally, the model parameters were identified, and model applicability was verified.


## 2. Basic Theory of Rock Fatigue

### 2.1. Cyclic Loading

The common types of cyclic loading are sinusoidal wave, cosine wave, rectangular wave, and triangular wave cyclic loading. The cyclic loading can be expressed as(1)σ=σσ,maxσmin,t,f,A,where σmax is the upper limit of cyclic loading; σmin 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.


### 2.2. 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 Figures1(a) and 1(b)) [30–33, 44]. The 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].Figure 1
Typical creep characteristics of the rock under cyclic loading [43]. (a) Creep curves of the rock under cyclic loading. (b) Division of creep stages.
(a)(b)


### 2.3. Simplified Cyclic Loading

To use the component model based on rheological mechanics, assume thatσ1 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]:(2)σ1−σ3=σavexpσ1max−σ3−σsσcf,where σav is the average stress of cyclic loading, σ1t−σ3; σ1max is the upper limit of cyclic loading; σs is the critical strength of the rock under the complex stress state, which is the maximum stress value from the σ1−σ3−ε curve of the rock under cyclic loading when the rock happens to be destroyed; σc is the peak stress from the σ1−σ3−ε curve of the rock under static load; and σ3 is the small principal stress on rock specimens.According to equation (2), when the peak deviatoric stress of cyclic loading σ1max−σ3>σs, the equivalent deviatoric stress σ1−σ3>σav, and the fatigue breakdown of the rock will occur. When the peak deviatoric stress of cyclic loading σ1max−σ3≤σs, the equivalent deviatoric stress σ1−σ3≤σav, and the rocks do not undergo fatigue breakdown.When confining pressureσ3=0, the cyclic loading is simplified into the stress function shown in the following equation [42]:(3)σ=σavexpσmax−σsσcf,where σav is the average stress of cyclic loading; σmax is the upper limit of cyclic loading; σs is the critical strength of the rock under uniaxial cyclic loading; and σc is the rock uniaxial compressive strength.Similarly, according to equation (3), when σmax>σs, the simplified stress σ>σav, and the rock will undergo fatigue breakdown. However, when σmax≤σs, σ≤σav, and the rock does not undergo fatigue breakdown.


### 2.4. Fatigue Yield Criterion for Rocks

#### 2.4.1. Rock Fatigue Yield Criterion under Uniaxial Stress

Under uniaxial cyclic loading, whenσmax≤σs, namely, σ≤σav, the rock will undergo fatigue damage. When σmax≤σs, σ≤σav, and the rock will not undergo fatigue failure.


#### 2.4.2. 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 deviatorJ2. According to equation (2), the proposed fatigue yield criterion of the rock under cyclic loading is as follows:(4)FJ2,σav=J2−σav3,where J2 is the second invariant of the equivalent stress deviator, J2=1/6σ1−σ22+σ2−σ32+σ3−σ12.Whenσ2=σ3, FJ2,σav=σ1−σ3−σav/3. When σ1−σ3>σav, FJ2,σav>0, and the rock will undergo fatigue breakdown. Conversely, when σ1−σ3<σav, FJ2,σav<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 σav of cyclic loading and equivalent stress.


### 2.5. Basic One-Dimensional Fatigue Elements [42]

The one-dimensional 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. The plastic fatigue element is represented by friction sheets shown in Figure 2(c). The constitutive equations are as follows:Figure 2
One-dimensional fatigue element [42].
(a)(b)(c)


#### 2.5.1. Elastic Fatigue Element [42]

(5)σ=EεeN,where E is the elastic fatigue coefficient, which can be determined by the numerical fitting method. εeN is the fatigue strain.


#### 2.5.2. Viscous Fatigue Element [42]

(6)σ=ηdεvtdt=ηdεvNdNf,where η is the viscous fatigue coefficient, which can be determined by the numerical method, dεvt/dt is the strain increment of the viscous fatigue element within the time period t,t+Δt, and dεvN/dt is the fatigue strain increment of the element in each cycle of cyclic loading.


#### 2.5.3. Plastic Fatigue Element [42]

Whenσmax≤σs, that is, σ≤σav, even if the cycle number N tends to infinity, the fatigue strain of the element is still 0. When σmax≤σs, namely, σ≤σav, 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 . The equation of the state of the plastic fatigue element is(7)εpN⟶∞,σs>σmax,namely,σ>σav,εpN=0,σs≤σmax,namely,σ≤σav,where εpN 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.


### 2.6. Establishment of Three-Dimensional Fatigue Elements

#### 2.6.1. Three-Dimensional (3D) Elastic Fatigue Element

The three-dimensional elastic fatigue element is shown in Figure3, and its equation of state is(8)Sij=2G1eijeN,εkk=13Kσkk,where Sij is the equivalent deviatoric stress tensor Sij=σij−σmδij, in which δij is the Kronecker delta; eijeN is the deviatoric strain tensor of the fatigue element; σkk is the first invariant of the stress tensor; εkk is the first invariant of the strain tensor; and K is the fatigue volume modulus of the rock, and its physical dimension is stress.Figure 3
3D elastic fatigue element.Then, the total strainεije of the 3D elastic fatigue element under σij is(9)εije=12G1Sij+13Kσmδij=12G1Sij+19Kσkkδij,where σmδij is the spherical stress tensor, σm=σkk/3.


#### 2.6.2. Three-Dimensional (3D) Viscous Fatigue Element

For the 3D viscous fatigue element (see Figure4), without considering the volume strain of the element under the spherical stress tensor, the state equation of the element is(10)Sij=2η1e˙ijvt=2fη1e˙ijvN,where e˙ijvt is the increment of the deviatoric strain of the fatigue element in period t,t+Δt and e˙ijvN is the deviatoric strain increment of the viscous fatigue element in each cycle.Figure 4
3D viscous fatigue element.The following equation can be obtained by integrating equation (10):(11)eijvN=12η1NfSij,where eijvN is the deviatoric strain of the viscous fatigue element.


#### 2.6.3. Three-Dimensional (3D) Plastic Fatigue Element

Three-dimensional plastic fatigue element can be represented by the friction sheet shown in Figure5. When σmax−σ3≤σs, namely, FJ2,σav≤0, and even if the number of cycles N tends to infinity, the element fatigue strain is still 0. When σmax−σ3>σs, namely, FJ2,σav>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. The equation of state of the element is(12)εijpN⟶∞,σmax−σ3>σsorFJ2,σav>0,εijpN=0,σmax−σ3≤σsorFJ2,σav≤0,where εijpN is the plastic strain of the plastic fatigue element.Figure 5
3D plastic fatigue element.


## 2.1. Cyclic Loading

The common types of cyclic loading are sinusoidal wave, cosine wave, rectangular wave, and triangular wave cyclic loading. The cyclic loading can be expressed as(1)σ=σσ,maxσmin,t,f,A,where σmax is the upper limit of cyclic loading; σmin 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.


## 2.2. 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 Figures1(a) and 1(b)) [30–33, 44]. The 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].Figure 1
Typical creep characteristics of the rock under cyclic loading [43]. (a) Creep curves of the rock under cyclic loading. (b) Division of creep stages.
(a)(b)


## 2.3. Simplified Cyclic Loading

To use the component model based on rheological mechanics, assume thatσ1 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]:(2)σ1−σ3=σavexpσ1max−σ3−σsσcf,where σav is the average stress of cyclic loading, σ1t−σ3; σ1max is the upper limit of cyclic loading; σs is the critical strength of the rock under the complex stress state, which is the maximum stress value from the σ1−σ3−ε curve of the rock under cyclic loading when the rock happens to be destroyed; σc is the peak stress from the σ1−σ3−ε curve of the rock under static load; and σ3 is the small principal stress on rock specimens.According to equation (2), when the peak deviatoric stress of cyclic loading σ1max−σ3>σs, the equivalent deviatoric stress σ1−σ3>σav, and the fatigue breakdown of the rock will occur. When the peak deviatoric stress of cyclic loading σ1max−σ3≤σs, the equivalent deviatoric stress σ1−σ3≤σav, and the rocks do not undergo fatigue breakdown.When confining pressureσ3=0, the cyclic loading is simplified into the stress function shown in the following equation [42]:(3)σ=σavexpσmax−σsσcf,where σav is the average stress of cyclic loading; σmax is the upper limit of cyclic loading; σs is the critical strength of the rock under uniaxial cyclic loading; and σc is the rock uniaxial compressive strength.Similarly, according to equation (3), when σmax>σs, the simplified stress σ>σav, and the rock will undergo fatigue breakdown. However, when σmax≤σs, σ≤σav, and the rock does not undergo fatigue breakdown.


## 2.4. Fatigue Yield Criterion for Rocks

### 2.4.1. Rock Fatigue Yield Criterion under Uniaxial Stress

Under uniaxial cyclic loading, whenσmax≤σs, namely, σ≤σav, the rock will undergo fatigue damage. When σmax≤σs, σ≤σav, and the rock will not undergo fatigue failure.


### 2.4.2. 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 deviatorJ2. According to equation (2), the proposed fatigue yield criterion of the rock under cyclic loading is as follows:(4)FJ2,σav=J2−σav3,where J2 is the second invariant of the equivalent stress deviator, J2=1/6σ1−σ22+σ2−σ32+σ3−σ12.Whenσ2=σ3, FJ2,σav=σ1−σ3−σav/3. When σ1−σ3>σav, FJ2,σav>0, and the rock will undergo fatigue breakdown. Conversely, when σ1−σ3<σav, FJ2,σav<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 σav of cyclic loading and equivalent stress.


## 2.4.1. Rock Fatigue Yield Criterion under Uniaxial Stress

Under uniaxial cyclic loading, whenσmax≤σs, namely, σ≤σav, the rock will undergo fatigue damage. When σmax≤σs, σ≤σav, and the rock will not undergo fatigue failure.


## 2.4.2. 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 deviatorJ2. According to equation (2), the proposed fatigue yield criterion of the rock under cyclic loading is as follows:(4)FJ2,σav=J2−σav3,where J2 is the second invariant of the equivalent stress deviator, J2=1/6σ1−σ22+σ2−σ32+σ3−σ12.Whenσ2=σ3, FJ2,σav=σ1−σ3−σav/3. When σ1−σ3>σav, FJ2,σav>0, and the rock will undergo fatigue breakdown. Conversely, when σ1−σ3<σav, FJ2,σav<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 σav of cyclic loading and equivalent stress.


## 2.5. Basic One-Dimensional Fatigue Elements [42]

The one-dimensional 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. The plastic fatigue element is represented by friction sheets shown in Figure 2(c). The constitutive equations are as follows:Figure 2
One-dimensional fatigue element [42].
(a)(b)(c)


### 2.5.1. Elastic Fatigue Element [42]

(5)σ=EεeN,where E is the elastic fatigue coefficient, which can be determined by the numerical fitting method. εeN is the fatigue strain.


### 2.5.2. Viscous Fatigue Element [42]

(6)σ=ηdεvtdt=ηdεvNdNf,where η is the viscous fatigue coefficient, which can be determined by the numerical method, dεvt/dt is the strain increment of the viscous fatigue element within the time period t,t+Δt, and dεvN/dt is the fatigue strain increment of the element in each cycle of cyclic loading.


### 2.5.3. Plastic Fatigue Element [42]

Whenσmax≤σs, that is, σ≤σav, even if the cycle number N tends to infinity, the fatigue strain of the element is still 0. When σmax≤σs, namely, σ≤σav, 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 . The equation of the state of the plastic fatigue element is(7)εpN⟶∞,σs>σmax,namely,σ>σav,εpN=0,σs≤σmax,namely,σ≤σav,where εpN 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.


## 2.5.1. Elastic Fatigue Element [42]

(5)σ=EεeN,where E is the elastic fatigue coefficient, which can be determined by the numerical fitting method. εeN is the fatigue strain.


## 2.5.2. Viscous Fatigue Element [42]

(6)σ=ηdεvtdt=ηdεvNdNf,where η is the viscous fatigue coefficient, which can be determined by the numerical method, dεvt/dt is the strain increment of the viscous fatigue element within the time period t,t+Δt, and dεvN/dt is the fatigue strain increment of the element in each cycle of cyclic loading.


## 2.5.3. Plastic Fatigue Element [42]

Whenσmax≤σs, that is, σ≤σav, even if the cycle number N tends to infinity, the fatigue strain of the element is still 0. When σmax≤σs, namely, σ≤σav, 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 . The equation of the state of the plastic fatigue element is(7)εpN⟶∞,σs>σmax,namely,σ>σav,εpN=0,σs≤σmax,namely,σ≤σav,where εpN 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.


## 2.6. Establishment of Three-Dimensional Fatigue Elements

### 2.6.1. Three-Dimensional (3D) Elastic Fatigue Element

The three-dimensional elastic fatigue element is shown in Figure3, and its equation of state is(8)Sij=2G1eijeN,εkk=13Kσkk,where Sij is the equivalent deviatoric stress tensor Sij=σij−σmδij, in which δij is the Kronecker delta; eijeN is the deviatoric strain tensor of the fatigue element; σkk is the first invariant of the stress tensor; εkk is the first invariant of the strain tensor; and K is the fatigue volume modulus of the rock, and its physical dimension is stress.Figure 3
3D elastic fatigue element.Then, the total strainεije of the 3D elastic fatigue element under σij is(9)εije=12G1Sij+13Kσmδij=12G1Sij+19Kσkkδij,where σmδij is the spherical stress tensor, σm=σkk/3.


### 2.6.2. Three-Dimensional (3D) Viscous Fatigue Element

For the 3D viscous fatigue element (see Figure4), without considering the volume strain of the element under the spherical stress tensor, the state equation of the element is(10)Sij=2η1e˙ijvt=2fη1e˙ijvN,where e˙ijvt is the increment of the deviatoric strain of the fatigue element in period t,t+Δt and e˙ijvN is the deviatoric strain increment of the viscous fatigue element in each cycle.Figure 4
3D viscous fatigue element.The following equation can be obtained by integrating equation (10):(11)eijvN=12η1NfSij,where eijvN is the deviatoric strain of the viscous fatigue element.


### 2.6.3. Three-Dimensional (3D) Plastic Fatigue Element

Three-dimensional plastic fatigue element can be represented by the friction sheet shown in Figure5. When σmax−σ3≤σs, namely, FJ2,σav≤0, and even if the number of cycles N tends to infinity, the element fatigue strain is still 0. When σmax−σ3>σs, namely, FJ2,σav>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. The equation of state of the element is(12)εijpN⟶∞,σmax−σ3>σsorFJ2,σav>0,εijpN=0,σmax−σ3≤σsorFJ2,σav≤0,where εijpN is the plastic strain of the plastic fatigue element.Figure 5
3D plastic fatigue element.


## 2.6.1. Three-Dimensional (3D) Elastic Fatigue Element

The three-dimensional elastic fatigue element is shown in Figure3, and its equation of state is(8)Sij=2G1eijeN,εkk=13Kσkk,where Sij is the equivalent deviatoric stress tensor Sij=σij−σmδij, in which δij is the Kronecker delta; eijeN is the deviatoric strain tensor of the fatigue element; σkk is the first invariant of the stress tensor; εkk is the first invariant of the strain tensor; and K is the fatigue volume modulus of the rock, and its physical dimension is stress.Figure 3
3D elastic fatigue element.Then, the total strainεije of the 3D elastic fatigue element under σij is(9)εije=12G1Sij+13Kσmδij=12G1Sij+19Kσkkδij,where σmδij is the spherical stress tensor, σm=σkk/3.


## 2.6.2. Three-Dimensional (3D) Viscous Fatigue Element

For the 3D viscous fatigue element (see Figure4), without considering the volume strain of the element under the spherical stress tensor, the state equation of the element is(10)Sij=2η1e˙ijvt=2fη1e˙ijvN,where e˙ijvt is the increment of the deviatoric strain of the fatigue element in period t,t+Δt and e˙ijvN is the deviatoric strain increment of the viscous fatigue element in each cycle.Figure 4
3D viscous fatigue element.The following equation can be obtained by integrating equation (10):(11)eijvN=12η1NfSij,where eijvN is the deviatoric strain of the viscous fatigue element.


## 2.6.3. Three-Dimensional (3D) Plastic Fatigue Element

Three-dimensional plastic fatigue element can be represented by the friction sheet shown in Figure5. When σmax−σ3≤σs, namely, FJ2,σav≤0, and even if the number of cycles N tends to infinity, the element fatigue strain is still 0. When σmax−σ3>σs, namely, FJ2,σav>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. The equation of state of the element is(12)εijpN⟶∞,σmax−σ3>σsorFJ2,σav>0,εijpN=0,σmax−σ3≤σsorFJ2,σav≤0,where εijpN is the plastic strain of the plastic fatigue element.Figure 5
3D plastic fatigue element.


## 3. Three-Dimensional Nonlinear Visco-Elastoplastic Fatigue Constitutive Model for Rocks

The 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 visco-elastoplastic fatigue model is connected to the burgers model in series, which is a one-dimensional nonlinear visco-elastoplastic fatigue constitutive model (see Figure6) proposed by Guo and Huang [42].Figure 6
One-dimensional NVPFM of the rock [42].Schiessel et al. [47] considered the time accumulation of the viscous elements (ηK) in the Kelvin body and defined the fractional viscous element by fractional derivatives. The creep compliance J(t) of the fractional element is proportional to tβ; 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(13)η3t,n=η30t1−n,where n is the parameter that reflects the strain rate of the tertiary creep phases of the rock under cyclic loading, n≠1. η30 is the initial value of the viscous coefficient, and η30≠0; its physical dimension is stress·time n. Equation (13) can also be further expressed as(14)η3N=η30N/f1−n.


### 3.1. 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. The 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 Figure7). In the model, G1 is the elastic shear fatigue coefficient, G2 is the viscoelastic shear fatigue coefficient, η1 is the viscous shear fatigue coefficient, η2 is the viscoplastic shear fatigue coefficient, and η30 is the initial value of the viscoplastic shear fatigue coefficient. The physical dimension of model parameters G1 and G2 is stress.Figure 7
Three-dimensional NVPFM for the rock.


### 3.2. Viscoelastic Fatigue Model

For the viscoelastic fatigue model shown in Figure8, the effect of the spherical stress tensor on volumetric strain is not considered. The state equation of the viscoelastic fatigue model, shown in Figure 8, is(15)Sij=2G2eijvet+2η2e˙˙ijvet=2G2eijveN+2fη2e˙˙ijveN,where e˙˙ijvet is the deviator strain increment of the fatigue element in the time period t,t+Δt, e˙˙ijvN is the deviator strain increment of the fatigue element in each cycle, and eijveN is the deviator strain tensor of the fatigue element.Figure 8
Three-dimensional viscoelastic fatigue model.The following fatigue constitutive equation of the fatigue element can be obtained from equation (15):(16)eijveN=12G21−exp−G2η2NfSij.


### 3.3. Nonlinear Viscoplastic Fatigue Model

For the viscoplastic fatigue model shown in Figure9, the viscoplastic shear fatigue coefficient η3N 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(17)e•ijnlt=fe•ijnlN=1η3NφF∂Q∂σij,where e•ijnlt is the deviator strain increment of the fatigue element in the time period t,t+Δt; e•ijnlN 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.Figure 9
Three-dimensional nonlinear viscoplastic fatigue model.The following fatigue constitutive equation can be obtained by substituting equation (14) into equation (17) and integrating N:(18)eijnlN=1η30nφF∂Q∂σijNfn,where eijnlN is the plastic deviator strain tensor of the nonlinear viscoplastic fatigue model. Among them,(19)φF=0,F<0,φF,F>0,(20)φF=∑m=1MBmFm.In equation (20) [48], Bm, m, and M are constants. In this article, we used M=1 and B1=1 for convenience.WhenFJ2,σav≤0, the rock does not undergo fatigue breakdown. When FJ2,σav>0, the following fatigue constitutive equation of the fatigue element can be obtained by using the flow rules related to the fatigue yield criterion:(21)eijnlN=1η30nF∂F∂σijNfn.


### 3.4. Constitutive Equation of the Nonlinear Visco-Elastoplastic Fatigue Model

When the rock fatigue yield functionFJ2,σav≤0, the rock will not be subjected to fatigue failure. At this time, the model degenerates into a burgers model which can describe the two creep stages of the rock under cyclic loading, i.e., transient and steady-state phases. From equations (8)–(20), the constitutive equation of the fatigue model can be obtained as(22)εijN=12G1Sij+13Kσmδij+σ2η1NfSij+12G21−exp−G2η2NfSij,where εijN is the total strain tensor of the fatigue model.When the rock fatigue yield functionFJ2,σav>0, the rock will undergo fatigue failure. At this time, the model is a 3D nonlinear visco-elastoplastic fatigue model which can describe the three creep stages of the rock under cyclic loading, i.e., transient, steady-state, and tertiary creep phases. From equations (8)–(21), the following constitutive equation of the fatigue model can be obtained:(23)εijN=12G1Sij+σ2η1NfSij+12G21−exp−G2η2NfSij+13Kσmδij+1η30nF∂F∂σijNfn.


## 3.1. 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. The 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 Figure7). In the model, G1 is the elastic shear fatigue coefficient, G2 is the viscoelastic shear fatigue coefficient, η1 is the viscous shear fatigue coefficient, η2 is the viscoplastic shear fatigue coefficient, and η30 is the initial value of the viscoplastic shear fatigue coefficient. The physical dimension of model parameters G1 and G2 is stress.Figure 7
Three-dimensional NVPFM for the rock.


## 3.2. Viscoelastic Fatigue Model

For the viscoelastic fatigue model shown in Figure8, the effect of the spherical stress tensor on volumetric strain is not considered. The state equation of the viscoelastic fatigue model, shown in Figure 8, is(15)Sij=2G2eijvet+2η2e˙˙ijvet=2G2eijveN+2fη2e˙˙ijveN,where e˙˙ijvet is the deviator strain increment of the fatigue element in the time period t,t+Δt, e˙˙ijvN is the deviator strain increment of the fatigue element in each cycle, and eijveN is the deviator strain tensor of the fatigue element.Figure 8
Three-dimensional viscoelastic fatigue model.The following fatigue constitutive equation of the fatigue element can be obtained from equation (15):(16)eijveN=12G21−exp−G2η2NfSij.


## 3.3. Nonlinear Viscoplastic Fatigue Model

For the viscoplastic fatigue model shown in Figure9, the viscoplastic shear fatigue coefficient η3N 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(17)e•ijnlt=fe•ijnlN=1η3NφF∂Q∂σij,where e•ijnlt is the deviator strain increment of the fatigue element in the time period t,t+Δt; e•ijnlN 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.Figure 9
Three-dimensional nonlinear viscoplastic fatigue model.The following fatigue constitutive equation can be obtained by substituting equation (14) into equation (17) and integrating N:(18)eijnlN=1η30nφF∂Q∂σijNfn,where eijnlN is the plastic deviator strain tensor of the nonlinear viscoplastic fatigue model. Among them,(19)φF=0,F<0,φF,F>0,(20)φF=∑m=1MBmFm.In equation (20) [48], Bm, m, and M are constants. In this article, we used M=1 and B1=1 for convenience.WhenFJ2,σav≤0, the rock does not undergo fatigue breakdown. When FJ2,σav>0, the following fatigue constitutive equation of the fatigue element can be obtained by using the flow rules related to the fatigue yield criterion:(21)eijnlN=1η30nF∂F∂σijNfn.


## 3.4. Constitutive Equation of the Nonlinear Visco-Elastoplastic Fatigue Model

When the rock fatigue yield functionFJ2,σav≤0, the rock will not be subjected to fatigue failure. At this time, the model degenerates into a burgers model which can describe the two creep stages of the rock under cyclic loading, i.e., transient and steady-state phases. From equations (8)–(20), the constitutive equation of the fatigue model can be obtained as(22)εijN=12G1Sij+13Kσmδij+σ2η1NfSij+12G21−exp−G2η2NfSij,where εijN is the total strain tensor of the fatigue model.When the rock fatigue yield functionFJ2,σav>0, the rock will undergo fatigue failure. At this time, the model is a 3D nonlinear visco-elastoplastic fatigue model which can describe the three creep stages of the rock under cyclic loading, i.e., transient, steady-state, and tertiary creep phases. From equations (8)–(21), the following constitutive equation of the fatigue model can be obtained:(23)εijN=12G1Sij+σ2η1NfSij+12G21−exp−G2η2NfSij+13Kσmδij+1η30nF∂F∂σijNfn.


## 4. Fatigue Constitutive Equation for Rocks under Triaxial Compression

The dynamic triaxial test is an indoor method for studying the fatigue properties of rocks under confining pressure stress. The 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(24)σij=σ1=σavexpσ1max−σ3−σsσcf+σ3000σ3000σ3,σ1>σ3≥0,where σ3 is the confining pressure acting on specimens.Deviatoric stress under the simplified stress state is as follows:(25)Sij=σij−σmδij=σij−13σkkδij,where δij is the Kronecker symbol, σm is the average stress, σm = (σ1 + 2σ3)/3, and σmδij is the spherical stress tensor.The total strain of the rock under dynamic triaxial condition based on the 3D NVPFM is as follows:(26)εijN=eije+eijv+eijve+eijnl+13εve+εvv+εvve+εvnlδij,where εve, εvv, εvve, and εvnl 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, thenεvv=εvve=εvnl = 0. Consequently, the following equation can be obtained from equation (26):(27)εijN=eij+13εveδij=eij+13Kσmδij=eij+19Kσkkδij,where eij is the total deviatoric strain tensor caused by deviatoric stress.WhenFJ2,σav≤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):(28)ε11N=σ1+2σ39K+σ1−σ33G1+σ1−σ33η1Nf+13σ1−σ3G21−exp−G2η2Nf,(29)ε33N=σ1+2σ39K+σ3−σ16G1+σ3−σ16η1Nf+σ3−σ16G21−exp−G2η2Nf.WhenFJ2,σav>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):(30)ε11N=σ1+2σ39K+σ1−σ33G1+σ1−σ33η1Nf+13σ1−σ3G21−exp−G2η2Nf+13σ1−σ3−σav1η30nNfn,(31)ε33N=σ1+2σ39K+σ3−σ16G1+σ3−σ16η1Nf+σ3−σ16G21−exp−G2η2Nf−13σ1−σ3−σav1η30nNfn.


## 5. Validation of NVPFM Adaptability

### 5.1. 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. The 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.The proposed three-dimensional NVPFM can describe the creep of the rock under cyclic loading. The 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. Then, 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.


### 5.2. Verification of the Fatigue Model under Cyclic Loading

#### 5.2.1. 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. The fatigue curve of red sandstone under 15 MPa confining pressure is shown in Figure 10. It can be obtained that the static strength σc of red sandstone under 15 MPa confining pressure was 104.653 MPa, and its critical strength σs was 98.22 MPa by Origin software. The deviatoric stress acting on the RS-4-3 specimen under cyclic loading was in the range from 51.8 to 99.1 MPa. Constitutive equation (30) based on the proposed fatigue model was used to fit the fatigue curve of the RS-4-3 specimen. The obtained parameters and the curve of the NVPFM are shown in Table 1 and Figure 11, respectively.Figure 10
Fatigue curve and static curve of red sandstone under 15 MPa confining pressure [51].Table 1
NVPFM parameters of red sandstone under 15 MPa confining pressure.
K (MPa)G1 (MPa)G2 (MPa)η1 (MPa·s)η2 (MPa·s)η30 (MPa·sn)nR21.759 × 1033.439 × 1091.070 × 1043.922 × 1042.613 × 1044.788 × 1040.8720.990Figure 11
Fatigue deformation of red sandstone [51] fitted by the NVPFM.


#### 5.2.2. 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 ηd (see equation (32)) by using the loading method shown in Figure 12. The effects of dynamic stress amplitude σd and static deviatoric stress σst on the fatigue characteristics of the water-rich soft rock were studied by Ding [52]. The critical stress ratio ηd of the water-rich soft rock under 200 kPa confining pressure, 180 kPa static deviatoric stress σst, 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 σdc is 240 kPa. The cyclic loading σ1t is expressed by equation (33). Then, the upper limit σlmax (34) of cyclic loading can be obtained from equation (33). The critical strength of the water-rich soft rock σs was σst+σdc/2. For the dynamic strength σc 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. Therefore, the static strength σc of the water-rich soft rock was taken as σs/0.85 in this study. The static strength σc of the water-rich soft rock under 200 kPa confining pressure was 353 kPa, and its critical strength σs 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. The fitting results and model parameters are shown in Table 2 and Figure 13.(32)ηd=σdσcua,where σcua is the uniaxial compressive strength of the water-rich soft rock, and its value is 0.6 MPa.(33)σ1t=σcp+σst+σd2sinωt,(34)σ1max=σcp+σst+σd2.Figure 12
Stress state acting on the specimen and loading waveform [52].Table 2
NVPFM parameters of the water-rich soft rock under 200 kPa confining pressure whenηd is 0.5.
K (kPa)G1 (kPa)G2 (kPa)η1 (kPa·s)η2 (kPa·s)η30 (kPa·sn)nR25.830 × 1046.899 × 1071.320 × 1043.037 × 1072.023 × 1051.7516 × 10166.5590.994Figure 13
NVPFM curve of the water-rich soft [52] rock under 200 kPa confining pressure.


#### 5.2.3. Model Parameters and the Curve of the Water-Rich Soft Rock When Fatigue Breakdown Does Not Occur

In order to analyze the effect of dynamic stress amplitude on creep characteristics of the water-rich soft rock, Ding [52] conducted comparative tests on rocks under cyclic loading with dynamic stress amplitudes of 30, 60, 90, 120, and 180 kPa, respectively. The corresponding dynamic stress ratios under the five working conditions were 0.05, 0.1, 0.15, 0.2, 0.3, and 0.5. During testing, the loading frequency of cyclic loading was 3 Hz, the static deviatoric stress was 180 kPa, and the applied confining pressure was 200 kPa. From the data obtained from the test, the model parameters and model curves of the proposed fatigue constitutive equation (28) can be obtained. The obtained parameters and curves of the NVFPM are shown in Table 3 and Figure 14, respectively.Table 3
NVPFM parameters of the water-rich soft rock under 20 kPa confining pressure whenηd is 0.5.
ηdσ1−σ3 (kPa)K (kPa)G1 (kPa)G2 (kPa)η1 (kPa·s)η2 (kPa·s)R20.0573.744−5.894 × 1092.229 × 1043.227 × 1041.026 × 1074.100 × 1050.9920.183.7715.694 × 103-2.259 × 1032.070 × 1043.808 × 1061.634 × 1050.9990.1595.1602.780 × 104-2.030 × 1041.906 × 1043.744 × 1062.233 × 1050.9990.2108.0988.404 × 104-2.643 × 1081.417 × 1044.463 × 1062.078 × 1040.9970.3232.2736.493 × 104-1.131 × 1091.656 × 1043.416 × 1062.030 × 1050.999Figure 14
NVPFM curve of the water-rich soft rock [52] under 200 kPa confining pressure when fatigue breakdown does not occur.It can be seen from Figure14 that fatigue constitutive equation (28) established in this paper can reflect the transient, steady-state, creep phases of rocks under cyclic loading. The correlation between model and test data was good. As can be seen from Table 3, when σ1−σ3<σav, the rock does not suffer from fatigue failure, and some of the obtained parameters of the model showed apparent regularity. The values of viscoelastic shear fatigue parameters G2 and viscoelastic shear fatigue parameters η1 decay exponentially with the increase in the dynamic strain ratio ηd. The 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.Figure 15
Relationship betweenG2 and ηd.Figure 16
Relationship betweenη1 and ηd.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. Whenσ1−σ3≤σav, rocks do not undergo fatigue breakdown, and the proposed model would degenerate into the burgers model, which can describe the transient, steady-state, and creep phases of the rock. When σ1−σ3≤σav, rocks undergo fatigue breakdown, and the proposed NVPFM can reflect the transient, steady-state, and tertiary creep phases of the rock under cyclic loading.


## 5.1. 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. The 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.The proposed three-dimensional NVPFM can describe the creep of the rock under cyclic loading. The 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. Then, 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.


## 5.2. Verification of the Fatigue Model under Cyclic Loading

### 5.2.1. 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. The fatigue curve of red sandstone under 15 MPa confining pressure is shown in Figure 10. It can be obtained that the static strength σc of red sandstone under 15 MPa confining pressure was 104.653 MPa, and its critical strength σs was 98.22 MPa by Origin software. The deviatoric stress acting on the RS-4-3 specimen under cyclic loading was in the range from 51.8 to 99.1 MPa. Constitutive equation (30) based on the proposed fatigue model was used to fit the fatigue curve of the RS-4-3 specimen. The obtained parameters and the curve of the NVPFM are shown in Table 1 and Figure 11, respectively.Figure 10
Fatigue curve and static curve of red sandstone under 15 MPa confining pressure [51].Table 1
NVPFM parameters of red sandstone under 15 MPa confining pressure.
K (MPa)G1 (MPa)G2 (MPa)η1 (MPa·s)η2 (MPa·s)η30 (MPa·sn)nR21.759 × 1033.439 × 1091.070 × 1043.922 × 1042.613 × 1044.788 × 1040.8720.990Figure 11
Fatigue deformation of red sandstone [51] fitted by the NVPFM.


### 5.2.2. 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 ηd (see equation (32)) by using the loading method shown in Figure 12. The effects of dynamic stress amplitude σd and static deviatoric stress σst on the fatigue characteristics of the water-rich soft rock were studied by Ding [52]. The critical stress ratio ηd of the water-rich soft rock under 200 kPa confining pressure, 180 kPa static deviatoric stress σst, 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 σdc is 240 kPa. The cyclic loading σ1t is expressed by equation (33). Then, the upper limit σlmax (34) of cyclic loading can be obtained from equation (33). The critical strength of the water-rich soft rock σs was σst+σdc/2. For the dynamic strength σc 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. Therefore, the static strength σc of the water-rich soft rock was taken as σs/0.85 in this study. The static strength σc of the water-rich soft rock under 200 kPa confining pressure was 353 kPa, and its critical strength σs 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. The fitting results and model parameters are shown in Table 2 and Figure 13.(32)ηd=σdσcua,where σcua is the uniaxial compressive strength of the water-rich soft rock, and its value is 0.6 MPa.(33)σ1t=σcp+σst+σd2sinωt,(34)σ1max=σcp+σst+σd2.Figure 12
Stress state acting on the specimen and loading waveform [52].Table 2
NVPFM parameters of the water-rich soft rock under 200 kPa confining pressure whenηd is 0.5.
K (kPa)G1 (kPa)G2 (kPa)η1 (kPa·s)η2 (kPa·s)η30 (kPa·sn)nR25.830 × 1046.899 × 1071.320 × 1043.037 × 1072.023 × 1051.7516 × 10166.5590.994Figure 13
NVPFM curve of the water-rich soft [52] rock under 200 kPa confining pressure.


### 5.2.3. Model Parameters and the Curve of the Water-Rich Soft Rock When Fatigue Breakdown Does Not Occur

In order to analyze the effect of dynamic stress amplitude on creep characteristics of the water-rich soft rock, Ding [52] conducted comparative tests on rocks under cyclic loading with dynamic stress amplitudes of 30, 60, 90, 120, and 180 kPa, respectively. The corresponding dynamic stress ratios under the five working conditions were 0.05, 0.1, 0.15, 0.2, 0.3, and 0.5. During testing, the loading frequency of cyclic loading was 3 Hz, the static deviatoric stress was 180 kPa, and the applied confining pressure was 200 kPa. From the data obtained from the test, the model parameters and model curves of the proposed fatigue constitutive equation (28) can be obtained. The obtained parameters and curves of the NVFPM are shown in Table 3 and Figure 14, respectively.Table 3
NVPFM parameters of the water-rich soft rock under 20 kPa confining pressure whenηd is 0.5.
ηdσ1−σ3 (kPa)K (kPa)G1 (kPa)G2 (kPa)η1 (kPa·s)η2 (kPa·s)R20.0573.744−5.894 × 1092.229 × 1043.227 × 1041.026 × 1074.100 × 1050.9920.183.7715.694 × 103-2.259 × 1032.070 × 1043.808 × 1061.634 × 1050.9990.1595.1602.780 × 104-2.030 × 1041.906 × 1043.744 × 1062.233 × 1050.9990.2108.0988.404 × 104-2.643 × 1081.417 × 1044.463 × 1062.078 × 1040.9970.3232.2736.493 × 104-1.131 × 1091.656 × 1043.416 × 1062.030 × 1050.999Figure 14
NVPFM curve of the water-rich soft rock [52] under 200 kPa confining pressure when fatigue breakdown does not occur.It can be seen from Figure14 that fatigue constitutive equation (28) established in this paper can reflect the transient, steady-state, creep phases of rocks under cyclic loading. The correlation between model and test data was good. As can be seen from Table 3, when σ1−σ3<σav, the rock does not suffer from fatigue failure, and some of the obtained parameters of the model showed apparent regularity. The values of viscoelastic shear fatigue parameters G2 and viscoelastic shear fatigue parameters η1 decay exponentially with the increase in the dynamic strain ratio ηd. The 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.Figure 15
Relationship betweenG2 and ηd.Figure 16
Relationship betweenη1 and ηd.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. Whenσ1−σ3≤σav, rocks do not undergo fatigue breakdown, and the proposed model would degenerate into the burgers model, which can describe the transient, steady-state, and creep phases of the rock. When σ1−σ3≤σav, rocks undergo fatigue breakdown, and the proposed NVPFM can reflect the transient, steady-state, and tertiary creep phases of the rock under cyclic loading.


## 5.2.1. 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. The fatigue curve of red sandstone under 15 MPa confining pressure is shown in Figure 10. It can be obtained that the static strength σc of red sandstone under 15 MPa confining pressure was 104.653 MPa, and its critical strength σs was 98.22 MPa by Origin software. The deviatoric stress acting on the RS-4-3 specimen under cyclic loading was in the range from 51.8 to 99.1 MPa. Constitutive equation (30) based on the proposed fatigue model was used to fit the fatigue curve of the RS-4-3 specimen. The obtained parameters and the curve of the NVPFM are shown in Table 1 and Figure 11, respectively.Figure 10
Fatigue curve and static curve of red sandstone under 15 MPa confining pressure [51].Table 1
NVPFM parameters of red sandstone under 15 MPa confining pressure.
K (MPa)G1 (MPa)G2 (MPa)η1 (MPa·s)η2 (MPa·s)η30 (MPa·sn)nR21.759 × 1033.439 × 1091.070 × 1043.922 × 1042.613 × 1044.788 × 1040.8720.990Figure 11
Fatigue deformation of red sandstone [51] fitted by the NVPFM.


## 5.2.2. 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 ηd (see equation (32)) by using the loading method shown in Figure 12. The effects of dynamic stress amplitude σd and static deviatoric stress σst on the fatigue characteristics of the water-rich soft rock were studied by Ding [52]. The critical stress ratio ηd of the water-rich soft rock under 200 kPa confining pressure, 180 kPa static deviatoric stress σst, 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 σdc is 240 kPa. The cyclic loading σ1t is expressed by equation (33). Then, the upper limit σlmax (34) of cyclic loading can be obtained from equation (33). The critical strength of the water-rich soft rock σs was σst+σdc/2. For the dynamic strength σc 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. Therefore, the static strength σc of the water-rich soft rock was taken as σs/0.85 in this study. The static strength σc of the water-rich soft rock under 200 kPa confining pressure was 353 kPa, and its critical strength σs 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. The fitting results and model parameters are shown in Table 2 and Figure 13.(32)ηd=σdσcua,where σcua is the uniaxial compressive strength of the water-rich soft rock, and its value is 0.6 MPa.(33)σ1t=σcp+σst+σd2sinωt,(34)σ1max=σcp+σst+σd2.Figure 12
Stress state acting on the specimen and loading waveform [52].Table 2
NVPFM parameters of the water-rich soft rock under 200 kPa confining pressure whenηd is 0.5.
K (kPa)G1 (kPa)G2 (kPa)η1 (kPa·s)η2 (kPa·s)η30 (kPa·sn)nR25.830 × 1046.899 × 1071.320 × 1043.037 × 1072.023 × 1051.7516 × 10166.5590.994Figure 13
NVPFM curve of the water-rich soft [52] rock under 200 kPa confining pressure.


## 5.2.3. Model Parameters and the Curve of the Water-Rich Soft Rock When Fatigue Breakdown Does Not Occur

In order to analyze the effect of dynamic stress amplitude on creep characteristics of the water-rich soft rock, Ding [52] conducted comparative tests on rocks under cyclic loading with dynamic stress amplitudes of 30, 60, 90, 120, and 180 kPa, respectively. The corresponding dynamic stress ratios under the five working conditions were 0.05, 0.1, 0.15, 0.2, 0.3, and 0.5. During testing, the loading frequency of cyclic loading was 3 Hz, the static deviatoric stress was 180 kPa, and the applied confining pressure was 200 kPa. From the data obtained from the test, the model parameters and model curves of the proposed fatigue constitutive equation (28) can be obtained. The obtained parameters and curves of the NVFPM are shown in Table 3 and Figure 14, respectively.Table 3
NVPFM parameters of the water-rich soft rock under 20 kPa confining pressure whenηd is 0.5.
ηdσ1−σ3 (kPa)K (kPa)G1 (kPa)G2 (kPa)η1 (kPa·s)η2 (kPa·s)R20.0573.744−5.894 × 1092.229 × 1043.227 × 1041.026 × 1074.100 × 1050.9920.183.7715.694 × 103-2.259 × 1032.070 × 1043.808 × 1061.634 × 1050.9990.1595.1602.780 × 104-2.030 × 1041.906 × 1043.744 × 1062.233 × 1050.9990.2108.0988.404 × 104-2.643 × 1081.417 × 1044.463 × 1062.078 × 1040.9970.3232.2736.493 × 104-1.131 × 1091.656 × 1043.416 × 1062.030 × 1050.999Figure 14
NVPFM curve of the water-rich soft rock [52] under 200 kPa confining pressure when fatigue breakdown does not occur.It can be seen from Figure14 that fatigue constitutive equation (28) established in this paper can reflect the transient, steady-state, creep phases of rocks under cyclic loading. The correlation between model and test data was good. As can be seen from Table 3, when σ1−σ3<σav, the rock does not suffer from fatigue failure, and some of the obtained parameters of the model showed apparent regularity. The values of viscoelastic shear fatigue parameters G2 and viscoelastic shear fatigue parameters η1 decay exponentially with the increase in the dynamic strain ratio ηd. The 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.Figure 15
Relationship betweenG2 and ηd.Figure 16
Relationship betweenη1 and ηd.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. Whenσ1−σ3≤σav, rocks do not undergo fatigue breakdown, and the proposed model would degenerate into the burgers model, which can describe the transient, steady-state, and creep phases of the rock. When σ1−σ3≤σav, rocks undergo fatigue breakdown, and the proposed NVPFM can reflect the transient, steady-state, and tertiary creep phases of the rock under cyclic loading.


## 6. 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 related to the proposed fatigue yield criterion. Finally, the rationality of the model is verified. The following conclusions can be drawn from this study.The 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 rockFJ2,σav≤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 FJ2,σav>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.The parameters of the NFVPM can be easily solved using mathematical methods. The parametersG2 and η1 of the NFVPM decreased with the increase in dynamic stress ratio ηd 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.



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*Source: 4292043-2020-06-11.xml*