# Study on the Creep Constitutive Model of a Sandstone Rock under the Water-Rock Interaction

Authors: Zuosen Luo; Zuoxiang Zhu; Jianlin Li; Lehua Wang; Qiao Jiang; Jingyu Zhang; Eleyas Assefa; Huafeng Deng
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

With the continuous construction of large-scale geotechnical engineering, more and more attention has been paid to the long-term stability of rock mass engineering, especially the problem of rock creep under the influence of water. Combined with the author’s previous research on the triaxial creep characteristics of sandstone under water-rock interaction, a nonlinear creep constitutive model was established to capture the degradation behavior of a sandstone rock due to cyclic wetting and drying of the reservoir water. Due to the limitations of the visco-elastoplastic model, a thorough modification was done to account the effect of the water-rock interaction on the mechanical degradation of the sandstone rock. Finally, the predicted results were proved to be in a good agreement with the experimental results. Moreover, the strong correlations between the predicted results and the experimental results show the effectiveness of the modified model to scrutinize the nonlinear creep behavior of sandstone rock. Relevant research results have important theoretical significance for the accurate prediction and effective control of the long-term stability of rock mass engineering under the influence of water-rock interaction.


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## Body

## 1. Introduction

After the completion of the Three Gorges Hydropower Project, the reservoir water level has been fluctuating by 30 m (between 145 m and 175 m). The Three Gorges reservoir has 600 km length. It inundates 4 counties in Hubei Province and 16 counties in Chongqing. The reservoir bank slope in the hydrofluctuation zone is subsequently altered by the wetting and drying cycles. This phenomenon has an adverse effect on the mechanical properties of the rock mass [1–5]. According to statistics, more than 1,190 landslides have occurred in the Three Gorges reservoir area [6]. After the commencement of the Three Gorges project, plenty of progressive deformations have been developing in the reservoir bank slopes. Typical landslides were reported in 2008 and 2009. For instance, the Goyang and the Liangshuijing landslides occurred in 2008 and 2009, respectively. The deformation and the failure of the reservoir bank slopes are mainly due to the effect of the water-rock interaction on the creep mechanical properties of the rock mass [7–10].There have been significant advances in developing constitutive models for the rock creep test. Rock creep phenomenon is getting a wide attention [11–13]. The deterioration of rock and soil properties effect of water is also a hot issue [14–16]. The effect of water on the rock creep behavior has been studied by different investigators. According to Wawersik and Brown, the effect of water on the time dependent deformation behavior of granite and sandstone rocks was increasing when the moisture content increased. Under uniaxial stress, the rate of steady-state creep for dry and saturated specimens differs by two orders of magnitude [17]. Zhu Hehua and Ye Bin [18] conducted uniaxial creep tests on dry and saturated tuff rock. According to their finding, water has a significant impact on the ultimate creep behavior of tuff rock. The ultimate creep values showed 5-6-fold differences in dry and saturated specimens. Under uniaxial creep test, the long-term strength of saturated granite was decreasing when the creep and the strain rates significantly increased [19]. Liu et al. [20] conducted both uniaxial and biaxial creep tests on soft conglomerates. In the above study, the rheological deformation of saturated conglomerates was about 10-fold of the saturated ones at the same stress levels. Liu Lang [21] conducted a creep test on the deep-saturated rock mass in Dongguashan Mountain using single-stage and gradient incremental cyclic loadings. The rock creep characteristics of both dry and saturated specimens were found to be high. Deng et al. [22] conducted conventional triaxial compression and creep tests on red-bed soft rocks. According to the above studies, water plays a significant role in the rock creep properties. Recently, many scholars have proposed a number of constitutive models based on the creep test results of different kinds of rocks [23–26]. Some researchers introduced the water content parameter into the creep constitutive models to scrutinize the effect of water on the degradation mechanism of rocks [27–29]. Further research is important to understand the long-term effect of water on the creep behavior of rocks.In this paper, the degradation effect of water-rock interaction on the mechanical properties of the rock was considered, the sandstone creep constitutive relationship considering the water-rock interaction was established, and the research results can provide references for specific engineering calculations and analysis.


## 2. Creep Characteristics of Sandstone under Water-Rock Interaction

Typical triaxial creep test results of sandstone rock are presented in Luo et al.’s research [30]. According to the analysis of the creep test results, it can be seen that the creep curve of sandstone has the following characteristics:(1)
The sandstone sample in the test has instantaneous elastic strain and instantaneous elastic deformation at the moment of stress loading. At this stage, a spring element can be used to simulate its deformation characteristics.(2)
After the sandstone sample completes the instantaneous deformation, creep deformation occurs. The amount of deformation slowly increases with time, and the strain rate slowly decreases. It enters the attenuation creep and stable creep stage of the rock, and the creep curve is concave upward. The deformation characteristics of the first stage can be simulated with viscoelastic elements.(3)
When the constant stress carried by the sandstone sample is smaller than the long-term strength, only the attenuation creep stage and the stable creep stage occur. At this time, the creep is stable creep; and when the constant load is greater than (or equal to) the long-term strength, the sandstone sample will develop in the direction of unstable creep. At this time, the rock will undergo irreversible plastic flow deformation. The creep at this stage can be simulated by viscous elements. Since there is a limit (long-term strength) from stable to unstable change, a switch element can be used to represent this limit, and a viscoplastic element with a switch can be used to represent the creep deformation characteristics at this stage.(4)
When the constant load is greater than its long-term strength, the sandstone sample enters the accelerated creep stage after a period of constant creep. The deformation characteristics of this stage are nonlinear. Consider using nonlinear elements to simulate these deformation characteristics of the stage.


## 3. Creep Constitutive Model

### 3.1. Advantages and Limitations of the Visco-Elastoplastic Model

The conventional Nishihara model (Figure1) consists of the Hokkaido (H), the viscoelastic (N/H), and the viscoplastic (N/St. V), where E1 is the elastic modulus, E2 is the viscoelastic modulus, and both η1 and η2 are the viscosity coefficients of the dashpots.Figure 1
Components of Nishihara model.The visco-elastoplastic model can be used to represent the creep characteristics of a sandstone rock. The visco-elastoplastic model has elastic-viscoelastic-viscoplasticity characteristics, which can comprehensively reflect the entire rock creep characteristics. Apart from that, it has the following limitations: ① it cannot reflect the nonlinearly accelerated rock creep stage and ② it cannot consider the effect of water on the mechanical degradation of the rock.


### 3.2. Modification of the Creep Stage

In order to establish a constitutive model that describes the nonlinear creep characteristics of sandstone rock, a nonlinear viscoplastic rheological element is introduced here, as shown in Figure2.Figure 2
Diagram of a nonlinearly accelerated creep stage.Under the action ofσ, the expression for the nonlinear viscoplastic element is(1)ε=ση3t−t0,where η1 stands for the viscosity of nonlinear viscous element and t0 represents the initial time. To address the mechanical damage that occurs during the nonlinear creep failure of the rock, another parameter η3 is introduced as shown in the following equation:(2)η3=ηs1−D,where ηs is the viscous coefficient of a nonlinear viscous component without the mechanical damage and D is the mechanical damage variable of rock. According to Wang Yu et al. [31], the internal mechanical damage variable of rock can be defined by using(3)D=1−e−t−t0n,t≥t0,0,t<t0,where n is a constant. When the time t<t0, the rock sample will be in a steady-state creep phase and the damage factor D will approach to zero. In the other case, when the time t≥t0, the rock will start a nonlinearly accelerated creep phase. At this stage, the damage variable D will gradually increase with time till it gets 1.To address the previously stated mechanical damage by using a viscoelastic model, the damage variable was introduced into the viscosity coefficient of the viscous element in the accelerated creep stage. A nonlinear viscous element was proposed. The stress and the time can be used as the triggering conditions for the nonlinear viscous element to form a new nonlinearly accelerated creep model. After combining this with the visco-elastoplastic model, a new nonlinear viscoelastic-plastic creep model is established, as shown in Figure3.Figure 3
Modified visco-elastoplastic model.The corresponding creep equation is given in equation (4) as shown below:(4)ε=σE1+σE21−e−E2/η1t,σ<σs,σE1+σE21−e−E2/η1t+σ−σsη2t,σ≥σs,t<t0,σE1+σE21−e−E2/η1t+σ−σsη2t0+σ−σsηs1−Dt−t0,σ≥σs,t≥t0,where E1 and E2 are the elastic modulus of elastic and viscoelastic body, η1, η2, and η3 are coefficients of viscosity for the viscoelastic body, the viscoplastic body, and the nonlinear viscous components, σs is the rock’s long-term strength, and t0 is the initial time.


### 3.3. Incorporating the Effect of Water into the Creep Model

#### 3.3.1. Definition of Damage Variable due to the Water-Rock Interaction

Cyclic water-rock interactions can affect the creep properties of sandstone rock. Wetting and drying cycles of the reservoir water can develop microcracks on the sandstone rock. Consequently, the cementation and the mineralogical compositions of the rock will be altered. As a result, there will be a significant change in the macroscopic mechanical parameters (the elastic modulus, the rate of deformation, and the peak intensity). According to the theory of damage mechanics, there are two main definitions of damage variables: the first one is the damage variable defined in terms of the geometric damage on the effective bearing area of the structure; and the other one is defined as the change in the elastic modulus with energy loss. This method considers the elastic property deterioration as the main cause of the failure. In this study, the damage variables were defined based on the latter definition. Here, both the elastic and the viscous coefficients were considered as the damage variables, and the macroscopic mechanical parameters (such as elastic modulus and viscous coefficient) were used to characterize the effect of water-rock interactions on the mechanical damage. The effect of water-rock interaction on the rock damage after theNth cycle is defined as follows:(5)dEN=1−ENE0,(6)dηN=1−ηNη0,where the parameters EN and E0 stand for the elastic modulus of the sandstone at the Nth and the initial cycles, respectively, and ηN and η0 are the viscosities of the sandstone at the Nth and the initial cycles, respectively.


#### 3.3.2. Establishing a Versatile Creep Constitutive Model

A damage variable was introduced into the earlier creep constitutive model by considering the effect of water on the degradation mechanism of the sandstone rock in the Three Gorges reservoir. The mechanical model is shown in Figure4.Figure 4
Constitutive model for a sandstone rock (considering the effect of water).Initially, the transient elastic strain was developed by the rock loading (Figure4), that is, the first part of the constitutive model, so that the damage due to the water-rock interaction can be defined by dEN1. When the stress level is low, the sample will exhibit creep deceleration and the rate of creep will gradually approach to zero. This phase corresponds to the second part of the constitutive model; hence the water-rock damage at this stage can be expressed as dEN2 and dηN1. When the stress level is greater than or equal to the long-term strength, the strain rate will gradually approach to a nonzero constant. The creep deceleration phase was followed by the steady creep phase. Similarly, this stage corresponds to the third part of the constitutive model; therefore, the water-rock damage was defined as dηN2. The progressive mechanical damage on the rock specimen resulted in failure. This stage corresponds to the fourth part of the constitutive model; here again the water-rock damage was introduced as dηN3.In summary, the one-dimensional creep constitutive model for the sandstone rock considering the effect of water-rock interactions can be expressed by using the following equation:(7)ε=σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t,σ<σs,σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t+σ−σsη21−dηN2t,σ≥σs,t<t0,σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t+σ−σsη21−dηN2t0+σ−σsηs1−D1−dηN3t−t0σ≥σs,t≥t0.Under normal circumstances, the rock is in a complex and three-dimensional stress state. Therefore, it is of great significance to establish a three-dimensional creep constitutive model. The one-dimensional creep equation can be transformed into a more general three-dimensional form, by using the normal bulk modulus and Poisson’s ratio methods. In this paper, the ordinary bulk modulus method is used to derive the one-dimensional creep equation.Under the condition of three-dimensional stress, the stress tensors can be decomposed into global stress tensorσm and the deviatoric stress tensor Sij, and the expressions are as follows:(8)σm=13σ1+σ2+σ3=σkk3,Sij=σij−δijσm=σij−13δijσkk.We have(9)σij=Sij+δijσm.The strain tensors can also be decomposed into the global strain tensorεm and the bias strain tensor eij, where(10)εm=13ε1+ε2+ε3=εkk3,eij=εij−δijεm=εij−13δijεkk.Similarly,(11)εij=eij+δijεm.Introducing the bulk modulusK and the shear modulus G,(12)K=E31−2v,G=E21+v,where E and v represent the elastic modulus and Poisson’s ratio of the rock, respectively. For three-dimensional state of stress, we have(13)σm=3Kεm,Sij=2Geij,where σm is the global stress tensor, Sij is the deviatoric stress tensor, εm is the strain tensor, and eij is the deviatoric strain tensor.Then the three-dimensional creep constitutive equation is summarized in the following equation:(14)ε1=σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t,σ1−σ3<σs,σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t+σ1−σ3−σs3η21−dηN2t,σ1−σ3≥σs,t<t0,σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t+σ1−σ3−σs3η21−dηN2t0+σ1−σ3−σs3ηs1−D1−dηN3t−t0,σ1−σ3≥σs,t≥t0.


## 3.1. Advantages and Limitations of the Visco-Elastoplastic Model

The conventional Nishihara model (Figure1) consists of the Hokkaido (H), the viscoelastic (N/H), and the viscoplastic (N/St. V), where E1 is the elastic modulus, E2 is the viscoelastic modulus, and both η1 and η2 are the viscosity coefficients of the dashpots.Figure 1
Components of Nishihara model.The visco-elastoplastic model can be used to represent the creep characteristics of a sandstone rock. The visco-elastoplastic model has elastic-viscoelastic-viscoplasticity characteristics, which can comprehensively reflect the entire rock creep characteristics. Apart from that, it has the following limitations: ① it cannot reflect the nonlinearly accelerated rock creep stage and ② it cannot consider the effect of water on the mechanical degradation of the rock.


## 3.2. Modification of the Creep Stage

In order to establish a constitutive model that describes the nonlinear creep characteristics of sandstone rock, a nonlinear viscoplastic rheological element is introduced here, as shown in Figure2.Figure 2
Diagram of a nonlinearly accelerated creep stage.Under the action ofσ, the expression for the nonlinear viscoplastic element is(1)ε=ση3t−t0,where η1 stands for the viscosity of nonlinear viscous element and t0 represents the initial time. To address the mechanical damage that occurs during the nonlinear creep failure of the rock, another parameter η3 is introduced as shown in the following equation:(2)η3=ηs1−D,where ηs is the viscous coefficient of a nonlinear viscous component without the mechanical damage and D is the mechanical damage variable of rock. According to Wang Yu et al. [31], the internal mechanical damage variable of rock can be defined by using(3)D=1−e−t−t0n,t≥t0,0,t<t0,where n is a constant. When the time t<t0, the rock sample will be in a steady-state creep phase and the damage factor D will approach to zero. In the other case, when the time t≥t0, the rock will start a nonlinearly accelerated creep phase. At this stage, the damage variable D will gradually increase with time till it gets 1.To address the previously stated mechanical damage by using a viscoelastic model, the damage variable was introduced into the viscosity coefficient of the viscous element in the accelerated creep stage. A nonlinear viscous element was proposed. The stress and the time can be used as the triggering conditions for the nonlinear viscous element to form a new nonlinearly accelerated creep model. After combining this with the visco-elastoplastic model, a new nonlinear viscoelastic-plastic creep model is established, as shown in Figure3.Figure 3
Modified visco-elastoplastic model.The corresponding creep equation is given in equation (4) as shown below:(4)ε=σE1+σE21−e−E2/η1t,σ<σs,σE1+σE21−e−E2/η1t+σ−σsη2t,σ≥σs,t<t0,σE1+σE21−e−E2/η1t+σ−σsη2t0+σ−σsηs1−Dt−t0,σ≥σs,t≥t0,where E1 and E2 are the elastic modulus of elastic and viscoelastic body, η1, η2, and η3 are coefficients of viscosity for the viscoelastic body, the viscoplastic body, and the nonlinear viscous components, σs is the rock’s long-term strength, and t0 is the initial time.


## 3.3. Incorporating the Effect of Water into the Creep Model

### 3.3.1. Definition of Damage Variable due to the Water-Rock Interaction

Cyclic water-rock interactions can affect the creep properties of sandstone rock. Wetting and drying cycles of the reservoir water can develop microcracks on the sandstone rock. Consequently, the cementation and the mineralogical compositions of the rock will be altered. As a result, there will be a significant change in the macroscopic mechanical parameters (the elastic modulus, the rate of deformation, and the peak intensity). According to the theory of damage mechanics, there are two main definitions of damage variables: the first one is the damage variable defined in terms of the geometric damage on the effective bearing area of the structure; and the other one is defined as the change in the elastic modulus with energy loss. This method considers the elastic property deterioration as the main cause of the failure. In this study, the damage variables were defined based on the latter definition. Here, both the elastic and the viscous coefficients were considered as the damage variables, and the macroscopic mechanical parameters (such as elastic modulus and viscous coefficient) were used to characterize the effect of water-rock interactions on the mechanical damage. The effect of water-rock interaction on the rock damage after theNth cycle is defined as follows:(5)dEN=1−ENE0,(6)dηN=1−ηNη0,where the parameters EN and E0 stand for the elastic modulus of the sandstone at the Nth and the initial cycles, respectively, and ηN and η0 are the viscosities of the sandstone at the Nth and the initial cycles, respectively.


### 3.3.2. Establishing a Versatile Creep Constitutive Model

A damage variable was introduced into the earlier creep constitutive model by considering the effect of water on the degradation mechanism of the sandstone rock in the Three Gorges reservoir. The mechanical model is shown in Figure4.Figure 4
Constitutive model for a sandstone rock (considering the effect of water).Initially, the transient elastic strain was developed by the rock loading (Figure4), that is, the first part of the constitutive model, so that the damage due to the water-rock interaction can be defined by dEN1. When the stress level is low, the sample will exhibit creep deceleration and the rate of creep will gradually approach to zero. This phase corresponds to the second part of the constitutive model; hence the water-rock damage at this stage can be expressed as dEN2 and dηN1. When the stress level is greater than or equal to the long-term strength, the strain rate will gradually approach to a nonzero constant. The creep deceleration phase was followed by the steady creep phase. Similarly, this stage corresponds to the third part of the constitutive model; therefore, the water-rock damage was defined as dηN2. The progressive mechanical damage on the rock specimen resulted in failure. This stage corresponds to the fourth part of the constitutive model; here again the water-rock damage was introduced as dηN3.In summary, the one-dimensional creep constitutive model for the sandstone rock considering the effect of water-rock interactions can be expressed by using the following equation:(7)ε=σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t,σ<σs,σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t+σ−σsη21−dηN2t,σ≥σs,t<t0,σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t+σ−σsη21−dηN2t0+σ−σsηs1−D1−dηN3t−t0σ≥σs,t≥t0.Under normal circumstances, the rock is in a complex and three-dimensional stress state. Therefore, it is of great significance to establish a three-dimensional creep constitutive model. The one-dimensional creep equation can be transformed into a more general three-dimensional form, by using the normal bulk modulus and Poisson’s ratio methods. In this paper, the ordinary bulk modulus method is used to derive the one-dimensional creep equation.Under the condition of three-dimensional stress, the stress tensors can be decomposed into global stress tensorσm and the deviatoric stress tensor Sij, and the expressions are as follows:(8)σm=13σ1+σ2+σ3=σkk3,Sij=σij−δijσm=σij−13δijσkk.We have(9)σij=Sij+δijσm.The strain tensors can also be decomposed into the global strain tensorεm and the bias strain tensor eij, where(10)εm=13ε1+ε2+ε3=εkk3,eij=εij−δijεm=εij−13δijεkk.Similarly,(11)εij=eij+δijεm.Introducing the bulk modulusK and the shear modulus G,(12)K=E31−2v,G=E21+v,where E and v represent the elastic modulus and Poisson’s ratio of the rock, respectively. For three-dimensional state of stress, we have(13)σm=3Kεm,Sij=2Geij,where σm is the global stress tensor, Sij is the deviatoric stress tensor, εm is the strain tensor, and eij is the deviatoric strain tensor.Then the three-dimensional creep constitutive equation is summarized in the following equation:(14)ε1=σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t,σ1−σ3<σs,σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t+σ1−σ3−σs3η21−dηN2t,σ1−σ3≥σs,t<t0,σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t+σ1−σ3−σs3η21−dηN2t0+σ1−σ3−σs3ηs1−D1−dηN3t−t0,σ1−σ3≥σs,t≥t0.


## 3.3.1. Definition of Damage Variable due to the Water-Rock Interaction

Cyclic water-rock interactions can affect the creep properties of sandstone rock. Wetting and drying cycles of the reservoir water can develop microcracks on the sandstone rock. Consequently, the cementation and the mineralogical compositions of the rock will be altered. As a result, there will be a significant change in the macroscopic mechanical parameters (the elastic modulus, the rate of deformation, and the peak intensity). According to the theory of damage mechanics, there are two main definitions of damage variables: the first one is the damage variable defined in terms of the geometric damage on the effective bearing area of the structure; and the other one is defined as the change in the elastic modulus with energy loss. This method considers the elastic property deterioration as the main cause of the failure. In this study, the damage variables were defined based on the latter definition. Here, both the elastic and the viscous coefficients were considered as the damage variables, and the macroscopic mechanical parameters (such as elastic modulus and viscous coefficient) were used to characterize the effect of water-rock interactions on the mechanical damage. The effect of water-rock interaction on the rock damage after theNth cycle is defined as follows:(5)dEN=1−ENE0,(6)dηN=1−ηNη0,where the parameters EN and E0 stand for the elastic modulus of the sandstone at the Nth and the initial cycles, respectively, and ηN and η0 are the viscosities of the sandstone at the Nth and the initial cycles, respectively.


## 3.3.2. Establishing a Versatile Creep Constitutive Model

A damage variable was introduced into the earlier creep constitutive model by considering the effect of water on the degradation mechanism of the sandstone rock in the Three Gorges reservoir. The mechanical model is shown in Figure4.Figure 4
Constitutive model for a sandstone rock (considering the effect of water).Initially, the transient elastic strain was developed by the rock loading (Figure4), that is, the first part of the constitutive model, so that the damage due to the water-rock interaction can be defined by dEN1. When the stress level is low, the sample will exhibit creep deceleration and the rate of creep will gradually approach to zero. This phase corresponds to the second part of the constitutive model; hence the water-rock damage at this stage can be expressed as dEN2 and dηN1. When the stress level is greater than or equal to the long-term strength, the strain rate will gradually approach to a nonzero constant. The creep deceleration phase was followed by the steady creep phase. Similarly, this stage corresponds to the third part of the constitutive model; therefore, the water-rock damage was defined as dηN2. The progressive mechanical damage on the rock specimen resulted in failure. This stage corresponds to the fourth part of the constitutive model; here again the water-rock damage was introduced as dηN3.In summary, the one-dimensional creep constitutive model for the sandstone rock considering the effect of water-rock interactions can be expressed by using the following equation:(7)ε=σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t,σ<σs,σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t+σ−σsη21−dηN2t,σ≥σs,t<t0,σE11−dEN1+σE21−dEN21−e−E2⋅1−dEN2/η1⋅1−dηN1t+σ−σsη21−dηN2t0+σ−σsηs1−D1−dηN3t−t0σ≥σs,t≥t0.Under normal circumstances, the rock is in a complex and three-dimensional stress state. Therefore, it is of great significance to establish a three-dimensional creep constitutive model. The one-dimensional creep equation can be transformed into a more general three-dimensional form, by using the normal bulk modulus and Poisson’s ratio methods. In this paper, the ordinary bulk modulus method is used to derive the one-dimensional creep equation.Under the condition of three-dimensional stress, the stress tensors can be decomposed into global stress tensorσm and the deviatoric stress tensor Sij, and the expressions are as follows:(8)σm=13σ1+σ2+σ3=σkk3,Sij=σij−δijσm=σij−13δijσkk.We have(9)σij=Sij+δijσm.The strain tensors can also be decomposed into the global strain tensorεm and the bias strain tensor eij, where(10)εm=13ε1+ε2+ε3=εkk3,eij=εij−δijεm=εij−13δijεkk.Similarly,(11)εij=eij+δijεm.Introducing the bulk modulusK and the shear modulus G,(12)K=E31−2v,G=E21+v,where E and v represent the elastic modulus and Poisson’s ratio of the rock, respectively. For three-dimensional state of stress, we have(13)σm=3Kεm,Sij=2Geij,where σm is the global stress tensor, Sij is the deviatoric stress tensor, εm is the strain tensor, and eij is the deviatoric strain tensor.Then the three-dimensional creep constitutive equation is summarized in the following equation:(14)ε1=σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t,σ1−σ3<σs,σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t+σ1−σ3−σs3η21−dηN2t,σ1−σ3≥σs,t<t0,σ1−σ33G11−dEN1+σ1+2σ39K1−dEN1+σ1−σ33G21−dEN21−e−G21−dEN2/η11−dηN1t+σ1−σ3−σs3η21−dηN2t0+σ1−σ3−σs3ηs1−D1−dηN3t−t0,σ1−σ3≥σs,t≥t0.


## 4. Parameter Identification and Verification

The Levenberg–Marquardt method for nonlinear least-squares curve fitting was used to verify the nonlinear creep model (considering the effect of the water-rock interaction). Besides, the sandstone rock data were identified by using the software called 1stOpt as shown in Table1. The results are shown in Table 1.Table 1
Identified parameters for the nonlinear creep constitutive model.
CyclesDirectionLoad levelG1/GPaG2/GPaη1/GPa⋅h−1η2/GPa⋅h−1η3/GPa⋅h−1R20AxialFirst step2.197.1021.44--0.99Second step2.824.8324.49--0.99Third step1.884.1324.06--0.99Fourth step10.414.5410.17--0.99Fifth step12.586.818.842.085.450.98LateralFirst step14.452.4465.83--0.99Second step27.372.7155.31--0.99Third step3.951.9028.00--0.99Fourth step4.111.9011.70--0.99Fifth step4.116.799.470.5210.620.982AxialFirst step0.825.212.8396.032.40.96Second step0.983.99.40.97Third step5.72.88.70.97Fourth step1.81.36.30.99Fifth step9.35.0763.30.98LateralFirst step3.92.747.5111.97.60.98Second step2.52.422.60.99Third step1.61.510.20.98Fourth step0.970.453.30.97Fifth step450.63.4516.80.996AxialFirst step54.50.682.143.963.60.99Second step0.862.612.80.98Third step1.01.118.90.99Fourth step2.73.620.91.00LateralFirst step26.41.911.373.51.90.98Second step4.41.113.70.98Third step1.30.246.50.99Fourth step11.34.6677.51.0010AxialFirst step0.613.04.0246.329.70.99Second step1.43.59.60.98Third step0.781.29.10.99Fourth step63.91.217.31.00LateralFirst step0.883.82.662.16.80.99Second step5.11.49.70.98Third step18.50.333.60.98Fourth step0.9711.114.81.00From Table1, one can see a strong correlation between the experimental and the predicted values due to the modification of a nonlinear constitutive model. The relationships between the experimental and the computed values are presented in Figures 5 and 6.Figure 5
Relationships between the experimental and the calculated values at the low stress level. (a) 0th cycle. (b) 2nd cycle. (c) 6th cycle. (d) 10th cycle.
(a)(b)(c)(d)Figure 6
Relationships between the experimental and the calculated values at the failure stress level. (a) 0th cycle. (b) 2nd cycle. (c) 6th cycle. (d) 10th cycle.
(a)(b)(c)(d)


## 5. Conclusion

(1)
The creep test results of sandstone under water-rock interaction show that the mechanical properties of sandstone continue to deteriorate with the increase of the water-rock interaction period. For the evaluation of the long-term strength of the rock mass under the action of water, it is very important to consider the damage of the water-rock interaction. Therefore, this paper introduces the water-rock interaction damage variable and establishes the consideration of the water-rock interaction damage creep model.(2)
When the stress level is higher than the failure stress level, the model can describe the nonlinearly accelerated creep characteristics of the rock. The Levenberg–Marquardt method for nonlinear least-squares curve fitting was used to identify the creep parameters of the sandstone rock. The predicted values were in a very good agreement with the experimental values.(3)
The strong correlations between the predicted and the experimental results show the effectiveness of the introduced constitutive model to capture the degradation behavior of a sandstone rock under the water-rock interactions.(4)
The introduced creep constitutive model of a sandstone rock is a powerful tool to scrutinize the rheological characteristics of the rock mass in the reservoir bank slope.



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*Source: 6648421-2021-03-17.xml*