Creep Model and Experimental Verification of Sandstone under the Coupled Effect of Chemical Corrosion and Temperature

Featured Application

This concept could be used to guide the study of sandstone creep models under other conditions.

Abstract

The Nishihara creep model is an extremely effective method in the field of sandstone creep model research. However, the Nishihara creep model curve for sandstone under the marine environment (the coupled effect of chemical corrosion and temperature) does not fit the actual creep test data well. Based on the Nishihara creep model, we discovered that, on the one hand, the viscoelastic elements in the Nishihara model are replaced by the viscoelastic elements containing temperature and pH factors, which can accurately describe the influence of temperature and pH on rock creep characteristics; on the other hand, the viscoplastic elements in the Nishihara model are replaced by nonlinear viscoplastic elements, which can accurately describe the accelerated creep stage of the rock mass. After modifying Nishihara’s creep model twice, a new temperature–pH damage nonlinear creep model is established. The creep curve of the temperature–pH damage nonlinear creep model is compared with the creep test data of sandstone. The comparison results indicate that the creep curve of the nonlinear creep model has a high degree of fit with the creep test data of sandstone (accuracy > 92%). This validates the correctness of the newly established temperature–pH damage nonlinear creep model presented in this paper, demonstrating that the new model can effectively reflect the influence of temperature and pH on the creep characteristics of sandstone.

1. Introduction

Creep is a phenomenon that describes how the deformation of a rock changes over time under a constant load [1,2,3,4,5,6]. In marine engineering, people are more attentive to the time effect, and the stability of submarine tunnels and roadways after excavation has always been the focus of engineering research [7,8,9,10]. The problem of substantial deformation or cracking that can emerge after several years has become a major issue in marine engineering.

The creep properties of rocks are not only directly related to the properties of their materials but also their environment [11,12,13,14,15,16,17]. Several studies have shown that the creep characteristics of rocks are affected by chemical corrosion, temperature, stress, and other surrounding environmental factors [18,19,20,21,22,23,24,25]. Therefore, investigating the creep properties of sandstone under the coupled effect of chemical corrosion and temperature in marine environments is crucial.

Many scholars from home and abroad have put a lot of money and time into studying rock creep characteristics, and the findings have been impressive.

Wang Yongyan et al. [26,27] proposed a creep equation for sandstone under chemical corrosion based on the influence of effective stress and pH value of the solution and verified the rationality of the creep model. Wang Yanchun et al. [28,29] analyzed the creep mechanism of rock under chemical corrosion and established the creep equation of rock under chemical corrosion using a combination of experiment and simulation. Zhou Guanglei et al. [30] established a creep damage model for brittle rock under the coupled effect of temperature and stress based on the basic theories of rock deformation, thermodynamics, and damage mechanics. Wang Hongwei et al. [31] conducted multistage creep tests on rocks with cracks at different temperatures and confining pressures and studied the effects of temperature, confining pressure, and cracks on rock creep characteristics. Wang Yong et al. [32] used fractal theory to describe the pore development characteristics of white sandstone under the coupled action of chemical corrosion and temperature and investigated the relationship between the pore characteristics and mechanical properties of white sandstone under the coupled action of chemical corrosion and temperature. Jiang Zongbin et al. [33,34] investigated the creep test and model of slate under NaCl solution corrosion and proposed a nonlinear viscoelastic–plastic creep model considering chemical damage factors. Que Baoping et al. [35,36,37,38] studied the microstructure of rocks under thermomechanical coupling, conducted macromechanics experiments, and established a rheological model of rocks under thermomechanical coupling.

Grgic, D. et al. [39,40] conducted uniaxial compression tests and long–term uniaxial creep tests on rocks in various chemical environments and analyzed the impacts of the various chemical environments on the uniaxial compressive strength, long–term strength, and elastic modulus of the rocks. Cekerevac, C. et al. [41] conducted a triaxial compressive test on water–saturated kaolin at different temperatures (22 °C and 90 °C) and analyzed the effects of different temperatures on the shear strength, elastic modulus, and triaxial compressive strength of water–saturated kaolin. Rybacki, E. et al. [42] investigated the mechanical performance of shale under different temperatures and discovered that the shale’s creep strain gradually increased as the temperature increased. HEAP M. J. et al. [43] conducted a triaxial creep study on three types of sandstones in different regions at different temperatures and discovered that the strength of each sandstone gradually decreased as the temperature increased.

Most earlier studies on the creep behavior of rock have focused on a single factor, such as temperature and chemical corrosion, and have seldom considered both influencing factors simultaneously. We discovered that the Nishihara model is an exceptionally effective way of studying sandstone creep. This study focuses on describing the accelerated creep stage by defining new viscoplastic elements and introducing damage variables. However, when this is compared with the actual creep data, there will be a significant difference.

Based on previous research, this study investigates the effect of temperature and chemical corrosion on the creep characteristics of sandstone and combines the evolution law of creep rate with temperature and chemical corrosion during the test. A viscoelastic element that describes the effects of temperature and chemical corrosion on creep rate is constructed by introducing damage factors that vary with temperature and pH. To describe the effect of temperature and pH on the creep properties of sandstone, the viscoelastic element in the Nishihara model is replaced with a newly constructed viscoelastic element and a nonlinear viscoplastic element is introduced in the Nishihara model to describe the accelerated creep stage of rock. A nonlinear creep model under the coupled effect of chemical corrosion and temperature is established after two revisions to the Nishihara creep model. The results of creep tests at different temperatures and chemical corrosions were compared with the fitting results of the new model to verify its accuracy.

2. Establishment and Analysis of Damaged Components

2.1. Test Scheme

In order to mimic the real environment of the sandstone, it was soaked in 20 L 0.1 mol/L H₂SO₄ solution, 20 L 0.1 mol/L NaOH solution, and distilled water, respectively. Then, the solution of the soaked specimen was placed into the PH–80 constant temperature and humidity test box, and heating was controlled to the corresponding temperature (0 °C, 25 °C, 50 °C, 75 °C, 100 °C) through the test box. The sandstone reacts with the chemical solution during the soaking process of the sample, resulting in a change in pH value. The pH value of the solution should be monitored every day, and the stability of the pH value should be ensured by replenishing the solution. Finally, when the pH value of H₂SO₄ solution or NaOH solution at 100 °C remains constant, the sandstone samples soaked in chemical solutions at different temperatures were taken out.

According to the uniaxial compressive strength of sandstone under different coupling conditions, the initial load (first stage load) is set to 8 kN (4 MPa). The incremental load of the stage loading method is 8 kN (4 MPa). The load control method was used to load the specimens, and the loading rate of each stage was set to 50 N/s. The loading time is 4 h, and the loading time is the same. Step–by–step loading is performed until the specimen is damaged, and then the test is over.

2.2. Components Damaged by Chemical Corrosion and Temperature

Experiments show that the deformation and creep rates of rocks vary under the coupled effect of varied chemical corrosion and temperature [44]. Notably, pH and temperature affect the creep model’s construction. Therefore, when constructing the creep model of rock under the coupled effect of varied chemical corrosion and temperature, it is necessary to consider the effect of pH and temperature on the rock’s creep rate. The Nishihara model can accurately describe the stages of primary creep and steady–state creep in rock. Figure 1 shows the Nishihara model.

The creep equation of the Nishihara model is given as follows:

$$\left\{\begin{array}{ll}\epsilon ={\displaystyle \frac{\sigma }{E_1}}+{\displaystyle \frac{\sigma }{E_2}}\left(1-e^{-{\displaystyle \frac{E_2}{{\eta }_2}}t}\right)& \left(\sigma \le {\sigma }_s\right)\\ \epsilon ={\displaystyle \frac{\sigma }{E_1}}+{\displaystyle \frac{\sigma }{E_2}}\left(1-e^{-{\displaystyle \frac{E_2}{{\eta }_2}}t}\right)+{\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_1}}t& \left(\sigma >{\sigma }_s\right)\end{array}\right.$$

(1)

where $\sigma$ and $\epsilon$ are the total stress and strain of the Nishihara model, respectively; $E_1$ and $E_2$ are the elastic modulus of parts 1 and 3, respectively; ${\eta }_1$ and ${\eta }_2$ are the viscosity damping coefficients of parts 2 and 3, respectively, which are constant; ${\sigma }_s$ is the maximum yield stress of the rock; t is the time.

The Nishihara model’s variation law of creep rate with time is obtained by deriving Equation (1) as follows:

$$\left\{\begin{array}{ll}\dot{\epsilon }={\displaystyle \frac{\sigma }{{\eta }_2}}{e}^{-{\displaystyle \frac{E_2}{{\eta }_2}}t}& \left(\sigma \le {\sigma }_s\right)\\ \dot{\epsilon }={\displaystyle \frac{\sigma }{{\eta }_2}}{e}^{-{\displaystyle \frac{E_2}{{\eta }_2}}t}+{\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_1}}& \left(\sigma >{\sigma }_s\right)\end{array}\right.$$

(2)

The Nishihara creep model cannot describe the accelerated creep stage of rock, according to the analysis of Equation (2). The creep rate of the rock material when the applied load is less than its maximum yield stress is only related to $E_2$, ${\eta }_2$, and $\sigma$. To overcome the problem that the Nishihara model cannot depict the influence of pH and temperature on the creep rate, a viscoelastic element under the coupled effect of pH and temperature (Figure 2) is constructed to replace the Kelvin body in the Nishihara model. This study replaces the viscoplastic elements in the Nishihara creep model with nonlinear viscoplastic elements to improve the description of the accelerated creep stage using the new creep model.

2.3. Effects of Chemical Corrosion and Temperature on Viscoelastic Strain

According to the creep test of rock under the coupled action of different chemical corrosion and temperatures, the rock’s creep rate and variable changes with the pH and temperature; damage variables, including pH and temperature, are established based on the variation law of rock creep with pH and temperature under different coupling conditions. Formula (3) shows the damage variable’s expression.

$$D=1-e^{{\displaystyle \frac{-a\left(T-T_0\right)}{\left|pH-p{H}_0\right|}}}$$

(3)

where $T$ is the temperature of the rock mass during the test; $T_0$ is the room temperature during the test (the room temperature in this test is 20 °C); $pH$ is the $pH$ of the rock mass during the test; $p{H}_0$ is the $pH$ of the comparative solution (the $pH$ of the comparative solution in this test is 6.7); α is a model parameter (related to $pH$ and others).

Figure 2 shows the viscoelastic element constructed in this study under the coupled effect of temperature and pH. Its composition is made up of elastic and viscous elements connected in series. According to the theory of the combined model, the stress expression of the viscoelastic element under the coupled action of temperature and pH is given as follows:

$${\sigma }_{TC}=E_2{\epsilon }_3+{\eta }_2\left(T,pH\right){\dot{\epsilon }}_3$$

(4)

${\eta }_2\left(T,pH\right)$ is the viscosity damping coefficient affected by temperature and pH, ${\eta }_2\left(T,pH\right)={\eta }_2\left(1-D\right)$.

After combining Equations (3) and (4), the strain of the viscoelastic element under the joint action of temperature and pH can be obtained as follows:

$${\epsilon }_3={\displaystyle \frac{{\sigma }_{TC}}{E_2}}\left(1-e^{-{\displaystyle \frac{E_2{e}^{\frac{a\left(T-T_0\right)}{\left|pH-p{H}_0\right|}}}{{\eta }_2}}t}\right)$$

(5)

After deriving Formula (5), the strain rate of the viscoelastic element under the coupled action of temperature and pH can be obtained as follows:

$${\dot{\epsilon }}_3=e^{{\displaystyle \frac{a\left(T-T_0\right)}{\left|pH-p{H}_0\right|}}}\times {\displaystyle \frac{{\sigma }_{TC}}{{\eta }_2}}\times e^{-{\displaystyle \frac{E_2{e}^{\frac{a\left(T-T_0\right)}{\left|pH-p{H}_0\right|}}}{{\eta }_2}}t}$$

(6)

According to the boundary conditions and test data of this test, take ${\sigma }_{TC}$ = 16 MPa, $E_2$ = 2500 MPa, ${\eta }_2$ = 2800 GPa·h, $a$ = 0.15, $T_0$ = 20 °C, and $p{H}_0$ = 6.7.

When using $T$ = 50 °C, substitute $pH$ = 1 and 13 into Formula (5) to obtain a set of creep curves with the same temperature but different $pH$.

Figure 3 shows the effect of temperature on the strain of viscoelastic elements in an acidic environment. The figure shows that when other parameters are the same, the creep strain and rate of the viscoelastic element gradually increase as the temperature increases.

Figure 4 shows the effect of pH on the strain of the viscoelastic element at the same temperature. The figure shows that when other parameters are the same, the creep strain and rate of the viscoelastic element in an acidic environment exceed those in an alkaline environment.

The viscoelastic element constructed in this study under the coupled action of temperature and pH can accurately describe the effects of temperature and pH on rock creep strain and rate.

3. Establishment of a Nonlinear Creep Model Considering Temperature–pH Damage

The Nishihara creep model can accurately explain the two stages of primary creep and steady–state creep in rock, but it cannot accurately describe the accelerated creep stage of rock. To describe the influence of temperature and pH on the creep characteristics of rock, this study replaces the viscoelastic element in the Nishihara creep model with a viscoelastic element that includes the influence factors of temperature and pH based on the Nishihara creep model. To describe the accelerated creep stage of rock mass, this study replaces the viscoplastic elements in the model with nonlinear viscoplastic elements. A nonlinear creep model under the coupled effect of chemical corrosion and temperature is constructed after two revisions to the Nishihara creep model, as shown in Figure 5.

Let the total stress of the creep model be 1 and the total strain be 2.

When $\sigma \le {\sigma }_s$ in the creep model, only the viscoelastic and elastic elements participate in the creep, and the nonlinear viscoplastic element 2 does not deform. The equation of states is given as follows:

$$\left\{\begin{array}{l}{\sigma }_1=E_1{\epsilon }_1\\ {\sigma }_3=E_2{\epsilon }_3+{\eta }_2\left(T,pH\right){\dot{\epsilon }}_3\\ \sigma ={\sigma }_1={\sigma }_3\\ \epsilon ={\epsilon }_1+{\epsilon }_3\end{array}\right.$$

(7)

When $\sigma >{\sigma }_s$ in the creep model, elastic element 1, nonlinear viscoplastic element 2, and viscoelastic element 3 participate in creep. The equation of states is given as follows:

$$\left\{\begin{array}{l}{\sigma }_1=E_1{\epsilon }_1\\ {\sigma }_2={\sigma }_s+{\eta }_1\left(n,t\right){\dot{\epsilon }}_2\\ {\sigma }_3=E_2{\epsilon }_3+{\eta }_2\left(T,pH\right){\dot{\epsilon }}_3\\ \sigma ={\sigma }_1={\sigma }_2={\sigma }_3\\ \epsilon ={\epsilon }_1+{\epsilon }_2+{\epsilon }_3\end{array}\right.$$

(8)

${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ are stresses of elastic element 1, nonlinear viscoplastic element 2, and viscoelastic element 3, respectively. ${\epsilon }_1$, ${\epsilon }_2$, and ${\epsilon }_3$ are the strains of elastic element 1, nonlinear viscoplastic element 2, and viscoelastic element 3, respectively. ${\eta }_1\left(n,t\right)$ is the viscosity damping coefficient of the nonlinear viscoplastic element 2.

By arranging Equations (7) and (8), the creep equation of the nonlinear creep model considering temperature–pH damage is obtained as follows:

$$\left\{\begin{array}{ll}\epsilon ={\displaystyle \frac{\sigma }{E_1}}+{\displaystyle \frac{\sigma }{E_2}}\left(1-e^{-{\displaystyle \frac{E_2{e}^{\frac{a\left(T-T_0\right)}{\left|pH-p{H}_0\right|}}}{{\eta }_2}}t}\right)& \left(\sigma \le {\sigma }_s\right)\\ \epsilon ={\displaystyle \frac{\sigma }{E_1}}+{\displaystyle \frac{\sigma }{E_2}}\left(1-e^{-{\displaystyle \frac{E_2{e}^{\frac{a\left(T-T_0\right)}{\left|pH-p{H}_0\right|}}}{{\eta }_2}}t}\right)+{\displaystyle \frac{\sigma -{\sigma }_s}{{\eta }_1}}{t}^n& \left(\sigma >{\sigma }_s\right)\end{array}\right.$$

(9)

4. Experimental Verification of Creep Models

The creep test of the sandstone specimens under the coupled effect of different chemical corrosions and temperatures was conducted using a TAW–200–type Multifold test machine.

Because of time and space constraints, this study uses Python 3.10.2 software to verify the creep model by combining only a part of the creep data from sandstone soaked in alkaline solution at 100 °C (the creep test curve is shown in Figure 6). In this study, the creep curve when the axial pressure is 8 MPa is selected to verify the creep model when $\sigma <{\sigma }_s$, and the creep curve when the axial pressure is 16 MPa is selected to verify the creep model when $\sigma >{\sigma }_s$.

The Trust–Region method is used in this study to determine the model parameters in the nonlinear creep model considering temperature–pH damage based on the creep test results of sandstone under the coupled action of temperature and $pH$. The values are shown in

Table 1. Model parameters.

Axial Pressure/MPa	E1 /GPa	E2 /GPa	η1 (×103 GPa·h)	η2 (×103 GPa·h)	α	n	The Fitting Degree
8	1.294	3.146	\	1.520	0.212	\	0.95
16	1.352	4.524	0.0001	2.458	0.102	10.869	0.92

.

Figure 7 compares the creep test data of sandstone soaked in alkaline solution at 100 °C when the axial pressure is 8 and 16 MPa with the creep curve of the nonlinear creep model considering temperature–pH damage.

Figure 7 shows that the creep curve of the temperature–pH damage nonlinear creep model established in this study has a good fit with the creep test data of sandstone under the coupled effect of varied chemical corrosion and temperature. The validity of the newly established nonlinear creep model has been established, and it can accurately describe the influence of temperature and pH on sandstone creep characteristics.

5. Conclusions

In conclusion, we have developed a novel nonlinear creep model that considers temperature–pH damage to address the issue that the Nishihara creep model curve for sandstone under the coupled effect of chemical corrosion and temperature is poorly fitted with actual creep test data.

(1)

Based on the Nishihara creep model, the viscoelastic elements in the Nishihara model are replaced with viscoelastic elements incorporating the temperature and pH influence factors, and the new viscoelastic element is discovered to accurately describe the influence of temperature and pH on rock creep characteristics. The viscoplastic elements in the Nishihara model are replaced with nonlinear viscoplastic elements, and the nonlinear viscoplastic element is discovered to accurately describe the accelerated creep stage of the rock mass. Further, a nonlinear creep model under the coupled effect of chemical corrosion and temperature is established after two revisions to the Nishihara creep model, and it can well describe the influence of temperature and pH on the rock’s creep characteristics.

(2)

In this study, the creep test of sandstone specimens is conducted under the coupled effect of chemical corrosion and temperature, and a portion of the sandstone creep test data is compared with the creep curve of the nonlinear creep model considering temperature–pH damage. The comparison results show that the creep curve of the nonlinear creep model established in this study fits well with the sandstone creep test data. The validity of the newly developed nonlinear creep model has been established, and it can accurately describe the effects of temperature and pH on sandstone creep characteristics.