# Creep Fracture Characteristics and the Constitutive Model of Salt Rock under a Coupled Thermo-Mechanical Environment

Authors: Zhixia Wang; Xufeng Liang; Xue Guangwu; Tao Meng; Jigang Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The fracture extension characteristics of salt rocks under creep play an important role in the long-term safety and stability assessment of underground compressed air storage reservoirs. To study the fracture evolution characteristics of salt rock with time, the present study modified the Charles creep fracture intrinsic model and calibrated theA and n of the fracture extension coefficient parameters by creep fracture experiments with a constant stress rate loading method for salt rock under different temperature conditions. Finally, the modified Charles model was embedded into PFC (particle flow code) to simulate the fracture of salt rock numerically and the results were verified theoretically. The results show that (1) the fracture extension rate is related to the parameters A and n in addition to the stress intensity factor; (2) both n and A gradually decrease with the increase of temperature, and an decrease of n and A represents the decrease of fracture creep extension rate; and (3) the longer the initial fracture length of the specimen, the smaller the fracture time. The results of this study can provide theoretical guidance for evaluating the confinement of underground compressed air storage reservoirs.


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## Body

## 1. Introduction

With the economic development of the world, countries are actively developing and using renewable energy, among which electric energy has become the main source of energy for modern production, research, and life. However, solar and wind power generation has certain volatility and uncertainty, thus affecting the stable operation of power grids [1–6]. Power storage technology can effectively reduce the power fluctuation when converting renewable energy and electrical energy and improve the utilization rate of solar, wind, and other renewable energy. Among the power storage technologies, compressed air energy storage can meet the requirements of high efficiency, large capacity, and long service life at the grid level, so it is valued by countries around the world [7–10]. Salt caverns are considered as ideal sites for compressed air energy storage because of their high quality geological conditions such as wide distribution, their variety, and large-scale availability, and excellent physical and mechanical properties such as small permeability, low porosity, and compact structure [11–14].At present, the type of gas storage in China is vertical, that is, the vertical span of the solution chamber is greater than the horizontal span [15, 16]. Compared with the salt mounds in other countries which are particularly thick, the salt rock deposits in China are typically thin in single layers and many in interlayers [17] because in the underground storage of salt rock, the wall of the cavity is often under alternating and cyclic load; that is, the internal pressure of the reservoir fluctuates circularly between the minimum and maximum, and the fatigue microcracks can occur in the surrounding rock of cavities [18–20]. After that, the micro fractures gradually expanded, evolved, and penetrated with time, and macro fractures are formed finally. After when the surrounding rock gets cracked, the length of the crack will gradually increase due to the dislocation, slip, and dynamic diffusion of crystals at the crack tip. In this case, it will lead to the leakage of oil and gas in the storage. When the stress intensity factor reaches the fracture toughness value on the crack tip, the crack will expand rapidly. When the stress intensity factor is less than the fracture toughness, the crack will also expand. This phenomenon is defined as subcritical crack growth [21–24]. The mechanism of the phenomenon is stress corrosion; that is, brittle cracking of materials under the combined action of static stress and corrosive medium. Under the interaction of corrosion and stress-mechanical damage, the process of crack formation and propagation is accelerated [25–29]. To prevent air leakage in storage, it is very important to study the subcritical crack propagation of salt rock under different temperature and stress conditions.The experiment shows that the crack growth rate in water is higher than that in air [30, 31]. Nara et al. [32] measured the crack propagation rate of the Kumamoto andesite at different relative humidities and the same temperature, or at different temperatures and the same relative humidity and found that the increase in temperature and humidity will promote the propagation of cracks. Meng et al. [20] used the rock mechanics servo control experimental system to study the mechanical properties of rock under different temperature and stress conditions. It is considered that the elastic modulus of the specimen decreases with the increase of temperature. Therefore, it is very important to study the subcritical crack propagation mechanism of salt rock under different temperature and stress conditions.Lajtai and Bielus [33] measured the constant n of granite in air and water. It was proposed that the constant n becomes smaller in water. And the great difference of the constant n in granite may be due to the influence of surrounding environment and anisotropy of granite. Nara et al. [31] proposed that the constant n may depend on the temperature of the water. Therefore, in order to study the influence of the surrounding environment on the subcritical crack growth, it is also important to study the subcritical crack growth parameters A and n.Under the condition of constant stress velocity, the subcritical crack propagation was studied by the methods of double torsion test, Brazilian splitting, three-point bending test, slot-disk test, and single-edge notch bending test. It was found that the double torsion test was more suitable for the study of rock subcritical crack growth in the constant stress velocity test method [16, 34, 35]. Therefore, the crack growth rate of salt rock under different temperatures and stress conditions was measured by the constant stress loading rate method. Since only model I crack growth rate was considered by the Charles model and pure shear model II crack growth was not considered, the classical Charles model was modified in this paper. The modified creep fracture model was embedded into PFC simulation software, and then the creep fracture process of SCB specimen was numerically simulated and compared with the theoretical and experimental results.


## 2. Constitutive Model

### 2.1. Charles Creep Fracture Constitutive Model

When the rock is under a constant static load, the constitutive model of stress corrosion proposed by Charles was as follows [36]:(1)ν=dadt=Ae−H/RTΔKn,where V is the crack propagation velocity (mm/s), a is the length of newly formed crack (mm), t is the duration of crack propagation (s), H is the activation energy in (KJ/mol); R is the gas constant, T is the absolute temperature (K), A and n are the parameters of the subcritical fracture propagation, which depend on the rock type and external environment, and K is the stress intensity factor (MPa.m0.5).


### 2.2. Charles Creep Fracture Constitutive Model Modified

Because the Charles model (PFC stress corrosion model) only considered the crack growth rate of the model I but did not consider the crack growth of the pure shear model II, it was assumed that the crack propagation model of the pure shear model II has the same equation form as the model I. Therefore, the following equation can be obtained [37].(2)vI=AIKIKICnI,vII=AIIKIIKIICnII..In order to embed equation (2) into PFC numerical simulation software, it is necessary to obtain the degradation equation of bonding strength of PFC particles. Because the particles in PFC were randomly distributed (Gaussian distribution), and the cementation length was also randomly distributed. According to the above assumptions and equation (2), the degradation equation of meso-particle cementation strength was the following equation [37].(3)dJiTdt=−λ1iAIKIiKICnIdJiCdt=−λ2iAIKIiKICnI,KIi=σ¯iπai2,σ¯i=FniLi−ai.(ForKI fracture)(4)dJiTdt==−λ1iAIIKIIiKIICnIIdJiCdt=−λ2iAIIKIIiKIICnII,KIIi=τ¯iπai2,τ¯i=FSiLi−ai.(ForKII fracture).According to the research, the degradation equation of cementation strength was obtained by superimposing the above equations (3, 4) for mixed fracture, as shown in equation (5) [37].(5)dJiTdt=−λ1iAIKIiKICnI+AIIKIIiKIICnIIdJiCdt=−λ2iAIKIiKICnI+AIIKIIiKIICnII..(For mixed fracture).In equation (5), the fracture toughness of model I and model II was obtained by experiment. However, the stress intensity factor KIi and KIIi for a particle was unknown. Liu et al. [38] deduced the relationship between fracture toughness and meso-parameters of the contact bonding model in the PFC. According to the idea and derivation process, the relationship between fracture toughness and bonding strength in the creep fracture model was as follows.(6)KI=σnRA+RB−12R¯βKII=τnkn/ksRA+RB−12βR¯.According to the equation (6), the equation of the stress intensity factor for mixed fracture is as follows [38]:(7)KIi=σ¯miRA+RB−12R¯βsin2βiKIIi=σ¯mikn/ksRA+RB−12βR¯sinβicosβiσ¯mi=Fni2+FSi2Li−ai..Stress corrosion is the main mechanical mechanism causing rock deformation and failure, so this model is suitable for the study of rock fracture, chemical corrosion, cyclic loading and unloading fatigue damage, and creep fracture damage. From equations (1)–(7), it can be concluded that the crack growth rate of the specimen is closely related to parameters A and n besides stress intensity factor. However, there is no real time and effective experimental test, so the creep fracture experiment is designed in this paper, the creep fracture growth rate of salt rock is measured and the parameters A and n are calibrated.


## 2.1. Charles Creep Fracture Constitutive Model

When the rock is under a constant static load, the constitutive model of stress corrosion proposed by Charles was as follows [36]:(1)ν=dadt=Ae−H/RTΔKn,where V is the crack propagation velocity (mm/s), a is the length of newly formed crack (mm), t is the duration of crack propagation (s), H is the activation energy in (KJ/mol); R is the gas constant, T is the absolute temperature (K), A and n are the parameters of the subcritical fracture propagation, which depend on the rock type and external environment, and K is the stress intensity factor (MPa.m0.5).


## 2.2. Charles Creep Fracture Constitutive Model Modified

Because the Charles model (PFC stress corrosion model) only considered the crack growth rate of the model I but did not consider the crack growth of the pure shear model II, it was assumed that the crack propagation model of the pure shear model II has the same equation form as the model I. Therefore, the following equation can be obtained [37].(2)vI=AIKIKICnI,vII=AIIKIIKIICnII..In order to embed equation (2) into PFC numerical simulation software, it is necessary to obtain the degradation equation of bonding strength of PFC particles. Because the particles in PFC were randomly distributed (Gaussian distribution), and the cementation length was also randomly distributed. According to the above assumptions and equation (2), the degradation equation of meso-particle cementation strength was the following equation [37].(3)dJiTdt=−λ1iAIKIiKICnIdJiCdt=−λ2iAIKIiKICnI,KIi=σ¯iπai2,σ¯i=FniLi−ai.(ForKI fracture)(4)dJiTdt==−λ1iAIIKIIiKIICnIIdJiCdt=−λ2iAIIKIIiKIICnII,KIIi=τ¯iπai2,τ¯i=FSiLi−ai.(ForKII fracture).According to the research, the degradation equation of cementation strength was obtained by superimposing the above equations (3, 4) for mixed fracture, as shown in equation (5) [37].(5)dJiTdt=−λ1iAIKIiKICnI+AIIKIIiKIICnIIdJiCdt=−λ2iAIKIiKICnI+AIIKIIiKIICnII..(For mixed fracture).In equation (5), the fracture toughness of model I and model II was obtained by experiment. However, the stress intensity factor KIi and KIIi for a particle was unknown. Liu et al. [38] deduced the relationship between fracture toughness and meso-parameters of the contact bonding model in the PFC. According to the idea and derivation process, the relationship between fracture toughness and bonding strength in the creep fracture model was as follows.(6)KI=σnRA+RB−12R¯βKII=τnkn/ksRA+RB−12βR¯.According to the equation (6), the equation of the stress intensity factor for mixed fracture is as follows [38]:(7)KIi=σ¯miRA+RB−12R¯βsin2βiKIIi=σ¯mikn/ksRA+RB−12βR¯sinβicosβiσ¯mi=Fni2+FSi2Li−ai..Stress corrosion is the main mechanical mechanism causing rock deformation and failure, so this model is suitable for the study of rock fracture, chemical corrosion, cyclic loading and unloading fatigue damage, and creep fracture damage. From equations (1)–(7), it can be concluded that the crack growth rate of the specimen is closely related to parameters A and n besides stress intensity factor. However, there is no real time and effective experimental test, so the creep fracture experiment is designed in this paper, the creep fracture growth rate of salt rock is measured and the parameters A and n are calibrated.


## 3. Experimental Preparation

### 3.1. Specimen Preparation and Description

The salt rock specimens used in the test were taken from the Pakistan deposit, and the salt rock used in the test was mainly composed of NaCl (96%), K2SO4 (3%), and CaMg5 (CO3) (PO4)3 (OH) (1%) with a density of 2120 kg·m−3. The ore coring machine is a YQZ-30 hydraulic servo portable drilling rig. After the coring is completed, the two ends of the specimen are flattened by a small drill to meet the machining size standard recommended by the Institute of Mechanics. In this experiment, mode I and mode II fracture toughness tests (SCB specimen) were carried out using the three-point bending semidisk specimen with a central straight crack recommended by the International Society of Rock Mechanics. The diameter of the specimen is 75 mm, the height is 37.5 mm, the thickness is 20 mm, and the artificial notch length is 15 mm. The prefabricated crack angle is 0° (corresponding to pure mode I fracture) and 50° (corresponding to pure mode II fracture), respectively. The SCB sample after processing is shown in Figure 1.Figure 1
Geometry of the SCB specimen (R-specimen radius; B-specimen thickness; a-artificial notch length; S-support span; P-load;α-prefabricated crack angle; D-specimen diameter).
(a)(b)


### 3.2. Experimental Instruments and Procedures

Kemeny et al. [39] found that the fracture toughness of the specimen under different loading rates can be used to indirectly obtain the crack growth rate by experimental comparative analysis. Therefore, four different temperature settings were employed in the experiment which are (20°C), 50°C, 80°C, and 110°C.The equipment WDW-200 universal testing machine of Taiyuan University of Science and Technology was used in the three-point bending test, and its professional software was used to automatically test the data of maximum force and tensile strength. The experiment system included controller and all-round modular computer test software. The advantages of the system were reasonable structure, stable loading, and accurate control. The maximum test force can reach 500 kN, and the test force measurement range can reach 0.4% ∼ 100% F.S./1% ∼ 100% F.S. The specimen is heated with a muffle furnace as shown in Figure2.Figure 2
Test equipment.According to the proposed method, three-point bending experiments were performed to the specimens on a universal testing machine. The different stress loading rates (0.01 MPA/s, 0.1 MPa/s, 1 MPa/s, 10 MPa/s) were used to test the specimens at different temperatures [36]. Six specimens were tested under each condition in the experiment. Test results were automatically recorded by a computer system.


### 3.3. Theory

Kemeny derived the relationship between crack growth rate and fracture toughness according to Charles equation [39] as the following equations (9)∼ (27).Charles equation was simplified as the following [36, 39].(8)v=daat=AKIKICn.The equation for calculating the stress intensity factor was as the following [36, 39].(9)KI=YIσa,KII=YIIσa.By taking the model I stress intensity factor as an example, the following equation was obtained by substituting equation (9) into equation (8) [36, 39].(10)v=dadt=AYσaKICn=AKICnYnσnan/2.By separating variables [36, 39],(11)a−n/2da=AYKICnσndt.The following equation was provided by integrating both sides of the above equation at the same time.(12)∫aiafa−n/2da=AYKICn∫0tσtndt,where ai was the length of the initial crack, af was the length of the final crack, and t was time variable. Then, the following three equations were obtained by calculating and arranging of the above equation.(13)2n−2ai2−n/2−af2−n/2=AYnKICn∫0tσtndt,KIC=Yσiai,KIC=Yσfaf.Equation (13) are the inertia strength obtained after testing under inert conditions (representing no crack growth). The inert strength is the ultimate strength of the material which is not affected by the external environment. The most feasible way to measure this value is to apply a high-speed loading rate to the specimen. Therefore, in this paper, the strength of the specimen is tested by applying a loading rate of 10 MPa/s and using this value as the inertia strength.Transforming equation (13), we get(14)ai=KICPY2,af=KICPY2.Then the substituting equation (14), we get(15)2KICnYn−2AYnn−2KICn−2σin−2−σfn−2=∫0tσtndt,2KICnAY2n−2σin−2−σfn−2=∫0tσtndt..In the experiment of loading constant stress, sinceσt=σ˙t1, the above equation can be expressed by the following equation [36, 39].(16)2KICnAYnn−2σin−2−σfn−2=σ˙n∫0ttndt=σ˙nn+1tn+1=σ˙nn+1σfσ˙n+1=1n+1σfn+1σ˙−1,2KICnAYnn−2σin−21−σfσin−2=1n+1σfn+1σ˙−1.Because the parametern of rock is relatively large, it was assumed that σf/σin−2≪1. The above equation can be expressed as the following equation:(17)σfn+1=2KIC2n+1AY2n−2σin−2σ˙.Taking logarithm on both sides of the equation (17), it was further expressed as the follows [36, 39]:(18)logσf=1n+1logσ˙+1n+1log2KIC2n+1AY2n−2σin−2,logσf=1n+1logσ˙+logD,logD=1n+1log2KIC2n+1AY2n−2σin−2.When the relationship between fracture toughness and loading rate was linearly regressed, the slope coefficientα and intercept coefficient β were obtained. Therefore, the crack growth parameters A and n were obtained from the slope coefficient α and intercept coefficient β. The equation for calculating parameter n is as follows.(19)n=1α−1.Therefore, the equation for calculating parametersD and A is as follows.(20)D=10βA=2KIC2n+110n+1βY2n−2σin−2.Equation (20) was also applicable to calculate the fracture’s propagation of the pure model II.


## 3.1. Specimen Preparation and Description

The salt rock specimens used in the test were taken from the Pakistan deposit, and the salt rock used in the test was mainly composed of NaCl (96%), K2SO4 (3%), and CaMg5 (CO3) (PO4)3 (OH) (1%) with a density of 2120 kg·m−3. The ore coring machine is a YQZ-30 hydraulic servo portable drilling rig. After the coring is completed, the two ends of the specimen are flattened by a small drill to meet the machining size standard recommended by the Institute of Mechanics. In this experiment, mode I and mode II fracture toughness tests (SCB specimen) were carried out using the three-point bending semidisk specimen with a central straight crack recommended by the International Society of Rock Mechanics. The diameter of the specimen is 75 mm, the height is 37.5 mm, the thickness is 20 mm, and the artificial notch length is 15 mm. The prefabricated crack angle is 0° (corresponding to pure mode I fracture) and 50° (corresponding to pure mode II fracture), respectively. The SCB sample after processing is shown in Figure 1.Figure 1
Geometry of the SCB specimen (R-specimen radius; B-specimen thickness; a-artificial notch length; S-support span; P-load;α-prefabricated crack angle; D-specimen diameter).
(a)(b)


## 3.2. Experimental Instruments and Procedures

Kemeny et al. [39] found that the fracture toughness of the specimen under different loading rates can be used to indirectly obtain the crack growth rate by experimental comparative analysis. Therefore, four different temperature settings were employed in the experiment which are (20°C), 50°C, 80°C, and 110°C.The equipment WDW-200 universal testing machine of Taiyuan University of Science and Technology was used in the three-point bending test, and its professional software was used to automatically test the data of maximum force and tensile strength. The experiment system included controller and all-round modular computer test software. The advantages of the system were reasonable structure, stable loading, and accurate control. The maximum test force can reach 500 kN, and the test force measurement range can reach 0.4% ∼ 100% F.S./1% ∼ 100% F.S. The specimen is heated with a muffle furnace as shown in Figure2.Figure 2
Test equipment.According to the proposed method, three-point bending experiments were performed to the specimens on a universal testing machine. The different stress loading rates (0.01 MPA/s, 0.1 MPa/s, 1 MPa/s, 10 MPa/s) were used to test the specimens at different temperatures [36]. Six specimens were tested under each condition in the experiment. Test results were automatically recorded by a computer system.


## 3.3. Theory

Kemeny derived the relationship between crack growth rate and fracture toughness according to Charles equation [39] as the following equations (9)∼ (27).Charles equation was simplified as the following [36, 39].(8)v=daat=AKIKICn.The equation for calculating the stress intensity factor was as the following [36, 39].(9)KI=YIσa,KII=YIIσa.By taking the model I stress intensity factor as an example, the following equation was obtained by substituting equation (9) into equation (8) [36, 39].(10)v=dadt=AYσaKICn=AKICnYnσnan/2.By separating variables [36, 39],(11)a−n/2da=AYKICnσndt.The following equation was provided by integrating both sides of the above equation at the same time.(12)∫aiafa−n/2da=AYKICn∫0tσtndt,where ai was the length of the initial crack, af was the length of the final crack, and t was time variable. Then, the following three equations were obtained by calculating and arranging of the above equation.(13)2n−2ai2−n/2−af2−n/2=AYnKICn∫0tσtndt,KIC=Yσiai,KIC=Yσfaf.Equation (13) are the inertia strength obtained after testing under inert conditions (representing no crack growth). The inert strength is the ultimate strength of the material which is not affected by the external environment. The most feasible way to measure this value is to apply a high-speed loading rate to the specimen. Therefore, in this paper, the strength of the specimen is tested by applying a loading rate of 10 MPa/s and using this value as the inertia strength.Transforming equation (13), we get(14)ai=KICPY2,af=KICPY2.Then the substituting equation (14), we get(15)2KICnYn−2AYnn−2KICn−2σin−2−σfn−2=∫0tσtndt,2KICnAY2n−2σin−2−σfn−2=∫0tσtndt..In the experiment of loading constant stress, sinceσt=σ˙t1, the above equation can be expressed by the following equation [36, 39].(16)2KICnAYnn−2σin−2−σfn−2=σ˙n∫0ttndt=σ˙nn+1tn+1=σ˙nn+1σfσ˙n+1=1n+1σfn+1σ˙−1,2KICnAYnn−2σin−21−σfσin−2=1n+1σfn+1σ˙−1.Because the parametern of rock is relatively large, it was assumed that σf/σin−2≪1. The above equation can be expressed as the following equation:(17)σfn+1=2KIC2n+1AY2n−2σin−2σ˙.Taking logarithm on both sides of the equation (17), it was further expressed as the follows [36, 39]:(18)logσf=1n+1logσ˙+1n+1log2KIC2n+1AY2n−2σin−2,logσf=1n+1logσ˙+logD,logD=1n+1log2KIC2n+1AY2n−2σin−2.When the relationship between fracture toughness and loading rate was linearly regressed, the slope coefficientα and intercept coefficient β were obtained. Therefore, the crack growth parameters A and n were obtained from the slope coefficient α and intercept coefficient β. The equation for calculating parameter n is as follows.(19)n=1α−1.Therefore, the equation for calculating parametersD and A is as follows.(20)D=10βA=2KIC2n+110n+1βY2n−2σin−2.Equation (20) was also applicable to calculate the fracture’s propagation of the pure model II.


## 4. Experimental Results

### 4.1. Experimental Force-Displacement Curves

The relationship between mode I fracture toughness and mode II fracture toughness and the stress loading rate of salt rock treated at different temperatures is shown in Figures3–6, where the dashed line represents the best linear fit line. When the loading rate was lowest, under the conditions of 20°C, 50°C, 80°C, and 110°C, the model I fracture toughness were in proper order 0.07 MPa⋅m0.5, 0.213 MPa⋅m0.5, 0.324 MPa⋅m0.5, and 0.356 MPa⋅m0.5. When the loading rate was highest, the model I fracture toughness under the above conditions were in proper order 0.107 MPa⋅m0.5, 0.276 MPa⋅m0.5, 0.511 MPa⋅m0.5, and 0.522 MPa⋅m0.5, and the values also gradually increased with the increase in temperature. According to the equation (20), the parameters of n, D, and A were carried out. Under the condition of the specimen at 20°C, the calculation was n = 6.4, D = 50.8, and A = 3.59 × 10−15. Under the condition of the specimen at 50°C, the calculation was n = 4.9, D = 22.7, and A = 6.68 × 10−20. Under the condition of the specimen at 80°C, the calculation was n = 2.7, D = 39, and A = 5.27 × 10−22. Under the condition of the specimen at 110°C, the calculation was n = 3.4, D = 33.7, and A = 4.13 × 10−25. When the loading rate was lowest, under the conditions of 20°C, 50°C, 80°C, and 110°C, the model I fracture toughness were in proper order 0.053 MPa⋅m0.5, 0.145 MPa⋅m0.5, 0.226 MPa⋅m0.5, and 0.353 MPa⋅m0.5, When the loading rate was highest, the fracture toughness of model II under the above conditions were in proper order 0.098 MPa⋅m0.5, 0.215 MPa⋅m0.5, 0.473 MPa⋅m0.5, and 0.557 MPa⋅m0.5. The corresponding calculation of n, D, and A were in different specimens, such as n = 3.7, D = 51.7, A = 7.79 × 10−20 for the 20°C, n = 5.4, D = 62.6, A = 5.26 × 10−22 for the 50°C, n = 7.1, D = 80.6, A = 3.65 × 10−23 for the 80°C, and n = 6.7, D = 47.5, A = 2.08 × 10−26 for the 110°C. The above results are shown in Table 1.Figure 3
The relationship between the model I fracture toughness of salt rock and loading rate under different conditions: (a) 20°C, (b) 50°C, (c) 80°C, and (d) 110°C.
(a)(b)(c)(d)Figure 4
The relationship between the model II fracture toughness of salt rock and loading rate under different conditions: (a) 20°C, (b) 50°C, (c) 80°C, and (d) 110°C.
(a)(b)(c)(d)Figure 5
The mode I fracture toughness of salt rock at different load rates. (a) 20°C. (b) 50°C. (c) 80°C. (d) 110°C.
(a)(b)(c)(d)Figure 6
The mode II fracture toughness of salt rock at different load rates: (a) 20°C. (b) 50°C. (c) 80°C. (d) 110°C.
(a)(b)(c)(d)Table 1
The parameters of creep fractureA and n.
Temperature (°C)AInIAIInII203.59 × 10−156.47.79 × 10−203.7506.68 × 10−204.95.26 × 10−225.4805.27 × 10−222.73.65 × 10−237.11104.13 × 10−253.42.08 × 10−266.7From the creep fracture parameters of specimens I and II, it can be concluded thatn and A gradually decreases with the increase in temperature, and the decrease of n and A means the decrease in crack creep propagation rate. This is because when the salt rock specimen is anisotropic after heating treatment, the rock particles undergo thermal expansion, resulting in an increase in volume. Due to the anisotropy of volume change and thermal expansion, the thermal stress effect will occur within the rock particles and between particles and particles, so the creep fracture parameters will gradually decrease with the increase of temperature.


### 4.2. PFC Simulation

#### 4.2.1. Simulation Results

In order to ensure the accuracy of the above experiments, PFC numerical simulation was used to compare with the experimental data.It had been studied that the modified shear creep fracture model and mixed creep fracture model were embedded into the PFC, which need not be repeated in this paper [37, 40]. According to equations (5)–(7), if the stress intensity factor at the crack tip was lower than the fracture toughness, the crack propagation rate increased slowly. If it was close to the limit, the crack growth rate quickly reaches infinity. In the calculation of PFC, the crack creep fracture can be calculated by the reduction of the cementation strength length between particles. The cementation length decreased gradually if the contact force between particles was less than the tensile strength or shear strength. If the contact force between particles was close to the tensile strength or shear strength, the particles brake instantaneously. Moreover, if the stress intensity factor reached the fracture toughness, the particles got broken too. Then, the calculation stopped, and equations (5)–(7) were no longer called. If the particles were still connected, the degradation equations (5)–(7) were being calculated and the cementation length was decreased as before. In the process of calculation, the real time span was embedded in each time step. That is, by multiplying the current crack growth rate and the real time span corresponding to each time step, the latest crack length can be obtained. Then the stress and crack conditions were updated to the program automatically. And the next step of calculation was continued and iterated in turn.Due to the irregularity of salt rock crystals, it was assumed that the particles distribution in the model followed the law of Gaussian distribution. It was also assumed that each particle represented a crystal and the particles were incompressible. In addition, it was assumed that cracks in the model only occurred between particles. The microscopic parameters of model particles were shown in Table2, and the model is shown in Figure 7.Table 2
Micromechanical parameters of the numerical model.
ParameterValueParameterValuekn27384 GPa/mJrC0 MPaks6957 GPa/mρ2120 kg/m³Φ39°Gb24.0 GPaφr19.5°Kb25.8 GPaJC45 MPaμb0.13JT16 MPad10.002 mJrT0 MPad20.0012 mFigure 7
The PFC numerical model of the mode I and II.According to the above experiments, theAI of the specimen after 110°C treatment was between 3.89 × 10−15 and 5 × 10−17, and the AII was 3.36 × 10−20. The range of n is between 4.1 and 6.3, and the nII is 6.2. In the calculation, AI = 6.27 × 10−17, AII = 7.36 × 10−20, and nI = 6.3, nII = 3.2. In order to reduce the calculation time and ensure the calculation accuracy, the automatic time step adjustment program is called in the simulation process. The program is as follows [37]:(21)Δtmin≤Δtb≤ΔtmaxifFu<100N,Δtb=2∗Δtb−1ΔtbifFu≥100N,Δtb=Δtmin.In this calculation, the maximum time step was set to 1 × 102, and the minimum time step was set to 1 × 10−5. The simulation results are as follows.The number curve of particle contact failure of specimen can be observed from Figures8 and 9. If the fracture reaches 80% of the peak value, the load of model I fracture was 2.33 kN and the load of model II fracture was 1.91 kN. The initial crack length is 15 mm. According to the figure, the number of cracks increases slowly with time. The phenomenon would remain for a long time. It showed that the crack propagated slowly in the subcritical state. When it was close to the critical value, the specimen broke instantaneously and the cracks were completely connected in series, which was consistent with the previous experimental results.Figure 8
The number of cracks of mode I of SCB specimen versus time.Figure 9
The number of cracks of mode II of SCB specimen versus time.


## 4.1. Experimental Force-Displacement Curves

The relationship between mode I fracture toughness and mode II fracture toughness and the stress loading rate of salt rock treated at different temperatures is shown in Figures3–6, where the dashed line represents the best linear fit line. When the loading rate was lowest, under the conditions of 20°C, 50°C, 80°C, and 110°C, the model I fracture toughness were in proper order 0.07 MPa⋅m0.5, 0.213 MPa⋅m0.5, 0.324 MPa⋅m0.5, and 0.356 MPa⋅m0.5. When the loading rate was highest, the model I fracture toughness under the above conditions were in proper order 0.107 MPa⋅m0.5, 0.276 MPa⋅m0.5, 0.511 MPa⋅m0.5, and 0.522 MPa⋅m0.5, and the values also gradually increased with the increase in temperature. According to the equation (20), the parameters of n, D, and A were carried out. Under the condition of the specimen at 20°C, the calculation was n = 6.4, D = 50.8, and A = 3.59 × 10−15. Under the condition of the specimen at 50°C, the calculation was n = 4.9, D = 22.7, and A = 6.68 × 10−20. Under the condition of the specimen at 80°C, the calculation was n = 2.7, D = 39, and A = 5.27 × 10−22. Under the condition of the specimen at 110°C, the calculation was n = 3.4, D = 33.7, and A = 4.13 × 10−25. When the loading rate was lowest, under the conditions of 20°C, 50°C, 80°C, and 110°C, the model I fracture toughness were in proper order 0.053 MPa⋅m0.5, 0.145 MPa⋅m0.5, 0.226 MPa⋅m0.5, and 0.353 MPa⋅m0.5, When the loading rate was highest, the fracture toughness of model II under the above conditions were in proper order 0.098 MPa⋅m0.5, 0.215 MPa⋅m0.5, 0.473 MPa⋅m0.5, and 0.557 MPa⋅m0.5. The corresponding calculation of n, D, and A were in different specimens, such as n = 3.7, D = 51.7, A = 7.79 × 10−20 for the 20°C, n = 5.4, D = 62.6, A = 5.26 × 10−22 for the 50°C, n = 7.1, D = 80.6, A = 3.65 × 10−23 for the 80°C, and n = 6.7, D = 47.5, A = 2.08 × 10−26 for the 110°C. The above results are shown in Table 1.Figure 3
The relationship between the model I fracture toughness of salt rock and loading rate under different conditions: (a) 20°C, (b) 50°C, (c) 80°C, and (d) 110°C.
(a)(b)(c)(d)Figure 4
The relationship between the model II fracture toughness of salt rock and loading rate under different conditions: (a) 20°C, (b) 50°C, (c) 80°C, and (d) 110°C.
(a)(b)(c)(d)Figure 5
The mode I fracture toughness of salt rock at different load rates. (a) 20°C. (b) 50°C. (c) 80°C. (d) 110°C.
(a)(b)(c)(d)Figure 6
The mode II fracture toughness of salt rock at different load rates: (a) 20°C. (b) 50°C. (c) 80°C. (d) 110°C.
(a)(b)(c)(d)Table 1
The parameters of creep fractureA and n.
Temperature (°C)AInIAIInII203.59 × 10−156.47.79 × 10−203.7506.68 × 10−204.95.26 × 10−225.4805.27 × 10−222.73.65 × 10−237.11104.13 × 10−253.42.08 × 10−266.7From the creep fracture parameters of specimens I and II, it can be concluded thatn and A gradually decreases with the increase in temperature, and the decrease of n and A means the decrease in crack creep propagation rate. This is because when the salt rock specimen is anisotropic after heating treatment, the rock particles undergo thermal expansion, resulting in an increase in volume. Due to the anisotropy of volume change and thermal expansion, the thermal stress effect will occur within the rock particles and between particles and particles, so the creep fracture parameters will gradually decrease with the increase of temperature.


## 4.2. PFC Simulation

### 4.2.1. Simulation Results

In order to ensure the accuracy of the above experiments, PFC numerical simulation was used to compare with the experimental data.It had been studied that the modified shear creep fracture model and mixed creep fracture model were embedded into the PFC, which need not be repeated in this paper [37, 40]. According to equations (5)–(7), if the stress intensity factor at the crack tip was lower than the fracture toughness, the crack propagation rate increased slowly. If it was close to the limit, the crack growth rate quickly reaches infinity. In the calculation of PFC, the crack creep fracture can be calculated by the reduction of the cementation strength length between particles. The cementation length decreased gradually if the contact force between particles was less than the tensile strength or shear strength. If the contact force between particles was close to the tensile strength or shear strength, the particles brake instantaneously. Moreover, if the stress intensity factor reached the fracture toughness, the particles got broken too. Then, the calculation stopped, and equations (5)–(7) were no longer called. If the particles were still connected, the degradation equations (5)–(7) were being calculated and the cementation length was decreased as before. In the process of calculation, the real time span was embedded in each time step. That is, by multiplying the current crack growth rate and the real time span corresponding to each time step, the latest crack length can be obtained. Then the stress and crack conditions were updated to the program automatically. And the next step of calculation was continued and iterated in turn.Due to the irregularity of salt rock crystals, it was assumed that the particles distribution in the model followed the law of Gaussian distribution. It was also assumed that each particle represented a crystal and the particles were incompressible. In addition, it was assumed that cracks in the model only occurred between particles. The microscopic parameters of model particles were shown in Table2, and the model is shown in Figure 7.Table 2
Micromechanical parameters of the numerical model.
ParameterValueParameterValuekn27384 GPa/mJrC0 MPaks6957 GPa/mρ2120 kg/m³Φ39°Gb24.0 GPaφr19.5°Kb25.8 GPaJC45 MPaμb0.13JT16 MPad10.002 mJrT0 MPad20.0012 mFigure 7
The PFC numerical model of the mode I and II.According to the above experiments, theAI of the specimen after 110°C treatment was between 3.89 × 10−15 and 5 × 10−17, and the AII was 3.36 × 10−20. The range of n is between 4.1 and 6.3, and the nII is 6.2. In the calculation, AI = 6.27 × 10−17, AII = 7.36 × 10−20, and nI = 6.3, nII = 3.2. In order to reduce the calculation time and ensure the calculation accuracy, the automatic time step adjustment program is called in the simulation process. The program is as follows [37]:(21)Δtmin≤Δtb≤ΔtmaxifFu<100N,Δtb=2∗Δtb−1ΔtbifFu≥100N,Δtb=Δtmin.In this calculation, the maximum time step was set to 1 × 102, and the minimum time step was set to 1 × 10−5. The simulation results are as follows.The number curve of particle contact failure of specimen can be observed from Figures8 and 9. If the fracture reaches 80% of the peak value, the load of model I fracture was 2.33 kN and the load of model II fracture was 1.91 kN. The initial crack length is 15 mm. According to the figure, the number of cracks increases slowly with time. The phenomenon would remain for a long time. It showed that the crack propagated slowly in the subcritical state. When it was close to the critical value, the specimen broke instantaneously and the cracks were completely connected in series, which was consistent with the previous experimental results.Figure 8
The number of cracks of mode I of SCB specimen versus time.Figure 9
The number of cracks of mode II of SCB specimen versus time.


## 4.2.1. Simulation Results

In order to ensure the accuracy of the above experiments, PFC numerical simulation was used to compare with the experimental data.It had been studied that the modified shear creep fracture model and mixed creep fracture model were embedded into the PFC, which need not be repeated in this paper [37, 40]. According to equations (5)–(7), if the stress intensity factor at the crack tip was lower than the fracture toughness, the crack propagation rate increased slowly. If it was close to the limit, the crack growth rate quickly reaches infinity. In the calculation of PFC, the crack creep fracture can be calculated by the reduction of the cementation strength length between particles. The cementation length decreased gradually if the contact force between particles was less than the tensile strength or shear strength. If the contact force between particles was close to the tensile strength or shear strength, the particles brake instantaneously. Moreover, if the stress intensity factor reached the fracture toughness, the particles got broken too. Then, the calculation stopped, and equations (5)–(7) were no longer called. If the particles were still connected, the degradation equations (5)–(7) were being calculated and the cementation length was decreased as before. In the process of calculation, the real time span was embedded in each time step. That is, by multiplying the current crack growth rate and the real time span corresponding to each time step, the latest crack length can be obtained. Then the stress and crack conditions were updated to the program automatically. And the next step of calculation was continued and iterated in turn.Due to the irregularity of salt rock crystals, it was assumed that the particles distribution in the model followed the law of Gaussian distribution. It was also assumed that each particle represented a crystal and the particles were incompressible. In addition, it was assumed that cracks in the model only occurred between particles. The microscopic parameters of model particles were shown in Table2, and the model is shown in Figure 7.Table 2
Micromechanical parameters of the numerical model.
ParameterValueParameterValuekn27384 GPa/mJrC0 MPaks6957 GPa/mρ2120 kg/m³Φ39°Gb24.0 GPaφr19.5°Kb25.8 GPaJC45 MPaμb0.13JT16 MPad10.002 mJrT0 MPad20.0012 mFigure 7
The PFC numerical model of the mode I and II.According to the above experiments, theAI of the specimen after 110°C treatment was between 3.89 × 10−15 and 5 × 10−17, and the AII was 3.36 × 10−20. The range of n is between 4.1 and 6.3, and the nII is 6.2. In the calculation, AI = 6.27 × 10−17, AII = 7.36 × 10−20, and nI = 6.3, nII = 3.2. In order to reduce the calculation time and ensure the calculation accuracy, the automatic time step adjustment program is called in the simulation process. The program is as follows [37]:(21)Δtmin≤Δtb≤ΔtmaxifFu<100N,Δtb=2∗Δtb−1ΔtbifFu≥100N,Δtb=Δtmin.In this calculation, the maximum time step was set to 1 × 102, and the minimum time step was set to 1 × 10−5. The simulation results are as follows.The number curve of particle contact failure of specimen can be observed from Figures8 and 9. If the fracture reaches 80% of the peak value, the load of model I fracture was 2.33 kN and the load of model II fracture was 1.91 kN. The initial crack length is 15 mm. According to the figure, the number of cracks increases slowly with time. The phenomenon would remain for a long time. It showed that the crack propagated slowly in the subcritical state. When it was close to the critical value, the specimen broke instantaneously and the cracks were completely connected in series, which was consistent with the previous experimental results.Figure 8
The number of cracks of mode I of SCB specimen versus time.Figure 9
The number of cracks of mode II of SCB specimen versus time.


## 5. Discussion of the Comparison of Results

In order to verify the accuracy of the experimental and numerical simulation results, the creep fracture time of SCB specimens was calculated theoretically in this paper. Chen et al. [37] deduced the calculation equation for recognizing creep fracture of rectangular specimens under different loads and different precrack lengths. However, the calculation equation of semi-circular SCB specimens is not deduced in their research. According to the research idea and derivation process about the rectangular specimen, the creep fracture time of semi-circular SCB specimens is theoretically calculated in this paper, as is shown in equations (22) to (28). Therefore, the equation for the calculating model I stress intensity factor of semi-circular SCB specimen is shown in the following equation [34].(22)KI=YI·Pπam2RtYI≈4.47+7.40amR−106.0amR2+433.3amR3.When the specimen is loaded with a constant load, the crack length will gradually increase, which can be derived according to from equation (22) [37].(23)damdt=vI=AIYI·Pπam/KIC2RtnI.By separating variables [37]:(24)am−nI/2·dam=AIYInI·PnI·πnI/22RtKICnIdt.When the time change wasΔti and the crack length changes from ami−1t0ami, the following equation was obtained by integrating both sides at the same time [37].(25)∫am\i−1amiam−nI/2·dam=∫ti−1tiAIYInI·PnI·πnI/22RtKICnIdt.The parameterYI can be calculated by the equation (22). The relationship between parameter YI and the crack length was shown in Figure 10. The parameter YI can be temporarily constant assuming that the ami was infinitely close to ami−1. Based on this assumption, equation (26) can be further derived from equation (25) [37].(26)Δti=2RtKICnI1−πnIami1−nI/2AIYIi−1nIPnI1−nI/2,nI≠2YIi≈4.47+7.40amiR−106.0amiR2+433.3amiR3.Figure 10
The relationship between precrack length andYi.When the stress intensity factor is equal to the fracture toughness, the corresponding crack length isamcr. Then, the specimen was crossed by the crack at the growth rate measured in above experiments, and the specimen broke instantly. In order to calculate the time from the initial precrack to the critical crack, the crack critical under different loads can be calculated according to equation (27) [37].(27)KIC=4.47+7.40amcrR−106.0amcrR2+433.3amcrR3Pπamcr2Rt.As shown in Figure11, the initial length and critical length were divided into n parts. Then the total time tf can be obtained [37].(28)tf=∑i=1i=n2RtKICnI1−πnIami1−nI/2−ami−11−nI/2AIYIi−1nIPnI1−nI/2,nI≠2YIi≈4.47+7.40amiR−106.0amiR2+433.3amiR3..Figure 11
The precrack length and the critical length.According to Tables3, 4, and Figure 12, it can be concluded that for the creep fracture time obtained by theoretical calculation, experiment is basically consistent. The fracture time of salt rock specimen decreases with the increase of initial crack length and external load. When the length of the initial crack is different, the fracture time of the specimen is also different. Under the same external load, the fracture time decreases with the increase of the initial crack length. When the initial crack length of salt rock specimen is smaller, the specimen is more stable. The results of theoretical calculation, test, and numerical simulation of creep fracture time are basically consistent.Table 3
Salt rock at 25°C specimen under different loads,am0 = 15 mm.
Load (kN)Y0Crack critical length (mm)Theoretical time (s)Experiment time (s)Numerical simulation time (s)1.2017.921.24.972 × 107—4.577 × 1071.6017.919.79.163 × 106—9.451 × 1061.8017.918.61.582 × 106—1.568 × 1061.9117.917.93.277 × 1051.133 × 1053.119 × 1052.0017.917.42.057 × 105—2.278 × 1052.2017.916.71.086 × 105—2.164 × 1052.3317.916.56.163 × 1044.185 × 1048.539 × 1042.4017.916.24.428 × 104—5.472 × 1042.5217.915.63.667 × 1042.166 × 1044.339 × 1042.6017.915.22.184 × 104—3.279 × 1042.8017.914.88.946 × 103—1.131 × 104Table 4
Salt rock specimen under 2.33 kN load for differentam0.
Initial crack length (mm)Y0Critical fracture length (mm)Theoretical results (s)Numerical simulation time (s)15.017.916.44.278 × 1043.528 × 10415.520.416.41.682 × 1041.149 × 10416.021.616.47.371 × 1039.307 × 10316.523.716.43.608 × 1037.629 × 10317.026.416.41.594 × 1035.935 × 10317.529.116.49.397 × 1022.604 × 10318.030.816.44.445 × 1029.338 × 102Figure 12
Simulation and theoretical results of crack growth of the SCB specimen under various initial crack length and load; load = 2.33 kN and initial crack lengtham0 = 15 mm.
(a)(b)


## 6. Conclusion

The creep fracture characteristics and constitutive model of salt rock are studied in this paper. The relationship between the crack growth rate of the specimen and the related parameters is explored by modifying the Charles creep constitutive model. The constant stress rate loading method is used to calibrate the parameters of creep fracture experiment and the model of salt rock. The numerical simulation and theoretical verification of creep fracture were carried out by the PFC. The main conclusions were summarized as follows:(1)
By analyzing and correcting the Charles creep-fracture constitutive model, it is found that the crack growth rate of the salt rock specimen is related to various parameters. Besides the stress intensity factor, it is also closely related to the parametersA and n.(2)
From the creep fracture parameters of the specimen mode I and mode II fracture, it can be concluded that then and A decrease with the increase of temperature, and the increase of n and A means the decrease of crack creep propagation rate.(3)
After embedding the modified Charles stress corrosion constitutive model into the PFC numerical simulation software, the creep fracture of salt rock SCB specimens were numerically simulated and theoretically calculated. It was concluded that the results of numerical simulation, theoretical calculation, and experimental test are basically consistent.



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*Source: 4926917-2022-08-24.xml*