Mechanical Properties and Constitutive Relationship of Cretaceous Frozen Sandstone under Low Temperature

Abstract

During the construction of coal mine shafts through Cretaceous water-rich stratum using the freezing method, the frozen shaft lining can break and lose stability. Hence, it is necessary to examine the mechanical properties and constitutive relationship of Cretaceous water-rich sandstone under the effect of surrounding rocks. To this end, in this work, the mechanical properties of red sandstone at different confining pressures and freezing temperatures were examined by using a ZTCR-2000 low-temperature triaxial testing system, wherein the 415–418 m deep red sandstone in the Lijiagou air-return shaft of Wenjiapo Mine was taken as the research object. The test results indicated that the stress–strain curves of rock under triaxial compression and uniaxial compression presented four stages: pore compaction, elastic compression, plastic yield, and post-peak deformation. The difference between the two cases was that the post-peak curve of the former was abrupt, while the latter exhibited a post-peak strain softening section. As the freezing temperature was constant, with the raise in the confining pressure, the elastic modulus and peak strength of the rock rose linearly, while the Poisson’s ratio decreased quadratically. As the control confining pressure was constant, the elastic modulus and rock’s peak strength increased with the decrease in the temperature, and under the condition of negative temperature, the two parameters were linearly correlated with the temperature, while the Poisson’s ratio showed the opposite trend. The two-part Hooke’s model and the statistical damage model based on Drucker–Prager (D-P) yield criterion were used to establish the stress–strain relationship models before and after the rock yield point, optimize the model parameters, and optimize the junction of the two models. The results revealed that the optimized model curve was in good agreement with the experimental curve, which suggests that the proposed model can accurately describe the stress–strain characteristics of rock under three-dimensional stress. This verified the feasibility and rationality of the proposed model for examining the constitutive relationship of rocks.

1. Introduction

Coal is still the largest and most reliable source of energy in the world. Since the majority of China’s coal resources are located in the western region, the coal mining operations are being gradually shifted to this region [1,2,3,4]. During the construction of vertical shafts in the western region, the Cretaceous water-rich bedrock section contains a large number of sandstones with low strength and weak cementation, which often leads to the fracture of frozen pipe due to the instability or excessive deformation of surrounding rocks, causing serious engineering accidents [5,6]. Hence, it is of great significance to examine the mechanical properties and constitutive relationship of Cretaceous water-rich sandstone under the effect of surrounding rocks.

To study the mechanical properties of ultra-deep sandstone reservoir rocks, Li Qinghui et al. [7] used a variety of laboratory tests to examine the relationships between the compressive strength, internal friction angle, shear strength, and other mechanical parameters of sandstone and the confining pressure and temperature and compared them with those of shallow sandstone. Tan Wenhui et al. [8] studied the mechanical properties of granite under uniaxial compression by using computerized tomography (CT) scanning technology because of the significant influence of joints and fissures and deduced the constitutive relationship closer to the actual stress state of the sample. Meng Zhaoping et al. [9] studied the impact of lateral pressure on porosity, permeability, deformation, the mechanical properties, and failure mechanism and obtained the fitting equation for the relation between the physical and mechanical properties of sandstone and confining pressure. Although several studies have been conducted on the mechanical properties of rocks under different conditions [10,11,12,13], the mechanical properties of the Cretaceous water-rich sandstone in the western region of China have been rarely examined, and the stress state of sandstone under complex stress conditions needs to be further studied.

The stress state of rock under the crustal stress is very complex, and the rock mechanics basically involves studying the complete stress–strain relationship to characterize the deformation and failure of rocks, i.e., establishing the constitutive relationship of rock [14]. The early studies on the constitutive relationship of rocks were mainly based on the classical continuum mechanics theory. For example, Li Wenting et al. [15] characterized the post-peak elastic modulus obtained during the compressive failure of rock as a function of strain and considered the internal friction angle as the intermediate variable to establish the post-peak nonlinear constitutive relationship on the basis of the Mohr–Coulomb strength criterion. The obtained model accurately described the post-peak mechanical behavior of marble under different confining pressures. Zhang Chunhui et al. [16] introduced the post-peak strength decline index to describe the influence of confining pressure on the post-peak residual strength and secant modulus and established a post-peak strain softening mechanical model of rock by considering the influence of confining pressure. The rationality of the established model was verified by comparing the experimental and numerical simulation results. However, rock is a coherent aggregate of one or more minerals and has complex mechanical properties. On the basis of the continuum theory, the precision of the constitutive model is restricted to the actual working conditions [17].

Subsequently, it was found that the damage constitutive model established by introducing damage mechanics can more accurately describe the stress state of rocks. Cao Wengui et al. [18,19] derived a three-dimensional (3D) damage evolution equation and a damage-softening constitutive equation for rock considering that the micro-element strength of rock obeyed Weibull distribution and normal distribution. Huang Haifeng [20] assumed that the micro-element strength of rock obeyed the improved Harris probability density distribution, established a statistical damage softening model of rock considering the modification of damage variables, and compared the modeling results with the experimental data, proving the applicability and rationality of the proposed model. Considering that there is a compaction section before the peak of weakly cemented sandstone and a strain softening section after the peak, Zhang Weizhong et al. [21] established a parabolic curve linear elastic Duncan hyperbola plastic softening section residual ideal plasticity five-segment model for rock triaxial stress. Combined with the test curve, it was shown that the model could accurately reflect the deformation characteristics of sandstone under triaxial stress, but the multi-segment model had many nodes and complex forms, which is not suitable for engineering applications.

Although the mechanical properties and constitutive relationship of rocks have been extensively examined, there are few reports on the western Cretaceous water-rich sandstone under complex stress conditions. To this end, in this study, the freezing project of the Lijiagou air-return shaft in Wenjiapo Mine was taken as the research background, and a low-temperature triaxial test system was used to investigate the mechanical properties of red sandstone under different confining pressures and different freezing temperatures. On this basis, to accurately describe the obvious crack compaction and strain softening sections of the Cretaceous red sandstone under the action of 3D stress, the stress–strain relationship was established on the basis of the two-part Hooke’s model (TPHM) and the statistical damage model. The findings of this study provide useful insights on the mechanical properties of the Cretaceous sandstone and can serve as a theoretical reference for the constitutive modeling of western water-rich sandstone.

2. Test Plan

2.1. Sample Preparation

The rock samples in the test were taken from the Lower Cretaceous Luohe Formation of Lijiagou air-return shaft in Wenjiapo Mine, Shaanxi Province, and the burial depth of the rock samples was 415–418 m. A drilling sampler was used to sample the large rocks obtained from the site. After sampling, the rock samples were cut and polished. The diameter and height of the prepared samples were 50 and 100 mm, respectively. The verticality and flatness of the samples were checked on the basis of the relevant specifications of the International Society for Rock Mechanics [22].

The scanning electron microscopy (SEM) images of the rock sample are presented in Figure 1. It is clear from Figure 1a that there were numerous internal cracks in the rock. The rock could be easily broken, and the joints and fissures were relatively developed. When the rock sample was immersed in water, it disintegrated into granules, as depicted in Figure 1b. The particle size of rock was approximately 0.15–0.5 mm, and the particle size distribution was relatively uniform, but there were several pores and cracks in the rock, which corresponded to medium-grained rock characteristics. The rock primarily consisted of cement and mineral particles, wherein the mineral particles were wrapped by cement and connected with adjacent particles, as described in Figure 1c. The microscopic image of the outer cement is depicted in Figure 1d.

Through X-ray diffraction (XRD) analysis, it was found that the rock contained minerals such as quartz, calcite, pyrite, K-feldspar, and clay minerals. Figure 2 shows the X-ray diffraction (XRD) results of the test soil sample, with the main crystal in the soil sample being SiO₂. According to the rock classification standard, the rock was red sandstone. The basic physical parameters of the sandstone are shown in

Table 1. Basic physical parameters of sandstone.

Natural Moisture Content/%	Saturated Water Content/%	Dry Density/g·cm−3	Saturation Density/g·cm−3	Porosity/%
5.18	8.69	2.12	2.23	23.61

.

2.2. Test Equipment

The experiment equipment mainly included sample preparation and characterization instruments, including a rock drilling sampler (ZS-100), rock cutting machine (DJ-1), double face grinder (SHM-200), electric blast drying oven (DHG9075A), vacuum saturator (NEL-VJH), and non-metallic ultrasonic testing analyzer (NM-4B).

A low-temperature rock triaxial testing system (ZTCR-2000) at the Anhui University of Architecture and Construction was used as the main testing equipment, which included a servo oil source, temperature control system, confining pressure system, axial pressure system, and computer. Presently, it is the most advanced rock testing system in China. The sample installation and working diagram of the rock triaxial compression testing system are shown in Figure 3.

2.3. Test Design

According to the designed burial depth and freezing temperature of rock samples, the test temperatures were 25, −5, −10, and −15 °C, and four confining pressures were considered: 0 (no confining pressure), 2, 4, and 6 MPa. To mark the tested rock samples, the temperature and confining pressure were abbreviated as T and W, respectively. For example, W0T25 implies that the sample was tested at 0 MPa and 25 °C. The specific test scheme is shown in

Table 2. Test scheme and number.

Experimental Temperature	Confining Pressure
	0 MPa (No Confining Pressure)	2 MPa	4 MPa	6 MPa
25 °C	W0T25	W2T25	W4T25	W6T25
−5 °C	W0T-5	W2T-5	W4T-5	W6T-5
−10 °C	W0T-10	W2T-10	W4T-10	W6T-10
−15 °C	W0T-15	W2T-15	W4T-15	W6T-15

.

The selected rock samples were placed in a curing box for water retention, and then they were frozen at low temperature. The loading test was only able to be conducted when the temperature of the samples reached the design value and remained stable for at least 24 h. Before the loading test, the test box needed to be cooled to ensure that the ambient temperature of the samples met the test temperature requirements.

3. Mechanical Test Results of Cretaceous Red Sandstone

3.1. Uniaxial Compression Test of Cretaceous Red Sandstone

To examine the mechanical properties of Cretaceous red sandstone under complex crustal stress and different low temperature environments, firstly, uniaxial compression tests were conducted under different ambient temperatures [23,24]. The stress–strain curve of red sandstone under uniaxial compression is depicted in Figure 4. It can be seen that under different temperatures, the stress–strain curve of sandstone included four stages: pore compaction, elastic compression, plastic yield, and post-peak deformation, which were essentially consistent with the typical stress–strain curve of rocks. The difference was that at room temperature (25 °C), the sandstone was destroyed beyond the peak strength point, and the bearing capacity changed suddenly. Under the condition of negative temperature, the sandstone had a residual strength after reaching the peak strength through plastic deformation, which is called the “strain softening stage.” Comparing the stress–strain curves of rock samples under room temperature and three negative temperatures, it was found that with the decrease in the temperature, the elastic modulus and peak strength of the rock increased, but the length of the crack compaction section decreased.

3.2. Triaxial Compression Test of Cretaceous Red Sandstone

The stress–strain curves of red sandstone under triaxial compression at different negative temperatures are provided in Figure 5, and the red sandstone sample under triaxial compression at room temperature was considered the control group. It was evident that under different temperature conditions, the stress–strain curve of red sandstone under triaxial compression was essentially consistent with that under uniaxial compression, conforming to the typical stress–strain curve of rock. Figure 4a demonstrates that at room temperature, with the increase in the confining pressure, the elastic modulus and peak strength were improved. The rock did not have residual strength after the peak strength point, but exhibited a sudden change in the strength, causing rock failure. Under the condition of negative temperature, the triaxial compression strength of sandstone increased with the raise in the confining pressure. Furthermore, with the raise in the confining pressure and the decrease in the temperature, the length of the compaction section decreased. This was because the increase in the confining pressure limited the compression effect of the pores. With the decrease in the temperature, the content of unfrozen water in the rock decreased, and the water in the pores condensed into ice, reducing the pores in the rock.

3.3. Analysis of Mechanical Properties of Rock

Generally, the ground stress of rock in the stratum is relatively complex, usually in the state of three-way loading, and obvious differences also exist due to the influence of temperature during freezing construction. On the basis of the Mohr–Coulomb strength criterion, the peak strength of frozen sandstone under different low temperatures and confining pressures can be obtained from triaxial tests using the values of σ_max, Poisson’s ratio μ, and rock’s elastic modulus E. The test results are shown in

Table 3. Test results.

Specimen Number	σmax/MPa	E/MPa	μ
W0T25	12.242	2407.36	0.249
W0T-5	20.689	2973.21	0.239
W0T-10	27.122	3728.39	0.230
W0T-15	31.812	4947.37	0.210
W2T25	16.827	3026.18	0.234
W2T-5	25.557	3802.13	0.226
W2T-10	33.277	4507.64	0.216
W2T-15	38.981	5807.88	0.197
W4T25	23.611	3713.07	0.222
W4T-5	31.276	4478.78	0.216
W4T-10	38.704	5292.02	0.204
W4T-15	49.109	6723.26	0.187
W6T25	32.784	4376.91	0.213
W6T-5	41.582	5439.73	0.207
W6T-10	51.706	6390.49	0.196
W6T-15	58.642	7507.74	0.180

.

3.3.1. Relationship between the Freezing Temperature/Confining Pressure and the Peak Intensity of Sandstone

The variation in the sandstone peak intensity as a function of the temperature and confining pressure is depicted in Figure 6. It can be seen in Figure 6a that the peak strength of sandstone was linearly proportional to the confining pressure. At room temperature, as the confining pressure of sandstone rose from 0 MPa to 2, 4, and 6 MPa, the corresponding peak strengths increased by 37.45%, 92.87%, and 167.80%, respectively. When the rock temperature was −15 °C, as the confining pressure increased from 0 MPa to 2, 4, and 6 MPa, the peak strengths of sandstone rose by 22.54%, 54.37%, and 84.34%, respectively. When the temperatures of sandstone were 25, −5, 10, and −15 °C, and the applied confining pressure increased from 0 to 6 MPa, the peak strengths increased by 167.80%, 96.33%, 90.64%, and 84.34%, respectively. This indicated that the peak strength of sandstone can be improved by increasing the confining pressure. This is because at a high confining pressure, the lateral deformation of sandstone is constrained, the rock particles are continuously compressed and compacted in the axial direction, and the compacted particles are subjected to axial compression until the rock is sheared.

According to Figure 6b, as the confining pressure was constant, the peak strength linearly increased with the decrease in the negative temperature, but the peak strength at room temperature was different from that at negative temperature. This was because under the condition of negative temperature, the bearing capacity of the rock not only depends on the rock particles, but also on the ice–water mixture generated by the temperature reduction in the rock. The water in the rock can be divided into unfrozen water and ice. The ice not only has a certain strength, but also contributes to the cementation between rock particles, greatly improving the bearing capacity of the rock. As depicted in Figure 6b, as the temperature dropped from room temperature to −5 °C, the peak strengths at the confining pressures of 0, 2, 4, and 6 MPa increased by 40.83%, 34.16%, 24.51%, and 26.84%, respectively. As the confining pressure was 0 MPa, and the temperature dropped from −5 to −15 °C, the growth rates of rock peak strength were 40.83%, 31.09%, and 17.29%, respectively. When the confining pressure was 6 MPa, the corresponding values were 26.84%, 24.35%, and 13.41%, respectively. This shows that under triaxial compression, the strength of rock decreased with the decrease in the negative temperature. This was because with the decrease in the temperature, the content of unfrozen water in the rock decreased gradually, but the water content in the rock was fixed, and the water that can be frozen in the rock gradually decreased, leading to a decrease in the growth rate of rock peak strength.

3.3.2. Relationship between the Freezing Temperature/Confining Pressure and the Elastic Modulus of Sandstone

Elastic modulus reflects the ability of rock to withstand elastic deformation. The elastic modulus is essentially the bonding strength between micro atoms or molecules. Any change in the temperature and confining pressure obviously affects the elastic modulus. The variations in the elastic modulus of sandstone as a function of the temperature and confining pressure are depicted in Figure 7. The elastic modulus of rock was linearly proportional to the confining pressure. As the confining pressure of rock increased from 0 to 6 MPa, the elastic modulus measured at 25, −5, −10, and −15 °C increased by 123.04%, 108.19%, 64.89%, and 45.88, respectively. This was because as the confining pressure increased, the lateral constraint increased, and the lateral strain of rock decreased, which undoubtedly had a restraining effect on the destruction of rock and improved its resistance to deformation.

It can be seen in Figure 7a that with the decrease in the temperature, the variation trend of the elastic modulus was similar to that of the peak strength, i.e., the elastic modulus of rock increased as the temperature decreased. As the test temperature of rock decreased from room temperature to negative temperature, the elastic modulus corresponding to the confining pressures of 0, 2, 4, and 6 MPa increased by 23.51%, 25.64%, 10.88%, and 15.28%, respectively. When the test temperature dropped from −5 to −15 °C, the elastic modulus of rock under different confining pressures increased to a higher extent, and the corresponding growth rates from low to high confining pressures were 73.09%, 52.75%, 72.21%, and 39.82%, respectively. The above results are mainly attributed to the fact that a part of pore water in the rock under negative temperature froze into ice, which had a certain strength, and its cementation improved the ability of rock mass to resist deformation. Therefore, the elastic modulus of rock was not only determined by the rock skeleton but also by the ice in the pores.

3.3.3. Relationship between the Freezing Temperature/Confining Pressure and Poisson’s Ratio of Sandstone

The variations in the Poisson’s ratio of sandstone as a function of the confining pressure and temperature are depicted in Figure 8. It is clear that the Poisson’s ratio of rock quadratically decreased with the rise in the confining pressure. This was because the lateral deformation was restrained with the rise in the confining pressure. According to the formula of Poisson’s ratio, the lateral strain decreases, and it is more influenced by the confining pressure than the axial strain. Therefore, the Poisson’s ratio decreases in a quadratic manner with the in the confining pressure [25,26]. As displayed in Figure 8b, the Poisson’s ratio of rock decreased as the temperature was reduced at the same confining pressure, which was quite different from the relation between temperature and the rock’s elastic modulus.

4. Constitutive Relationship

4.1. Constitutive Relationship before Yield Point

According to the above tests, the room-temperature triaxial compression test was selected, and the stress–strain curve before the yield point was obtained, as depicted in Figure 9. It can be seen that the pores and cracks in the red sandstone were compacted at the onset of compression, and the strain vs. stress curve had a concave parabolic shape. The compaction of pores and cracks led to a large degree of deformation of the rock. Then, the internal defects were compacted, and the rock particles bore the 3D stress. The rock skeleton was able to bear a small deformation, and the stress–strain curve in this phase showed a linear growth.

For accurately describing the constitutive relationship in the two stages, the TPHM [27,28] was used. Hooke’s law based on natural strain was used to describe the crack compaction section, while Hooke’s law based on engineering strain was used to describe the elastic stage.

$$\left.\begin{array}{l}{\epsilon }_1=\frac{3-{\gamma }_h}{3E_r}[{\sigma }_1-\mu ({\sigma }_2+{\sigma }_3)]+\frac{{\gamma }_h}{3}[1-\exp (-\frac{{\sigma }_1}{E_h})]\\ {\epsilon }_2=\frac{3-{\gamma }_h}{3E_r}[{\sigma }_2-\mu ({\sigma }_1+{\sigma }_3)]+\frac{{\gamma }_h}{3}[1-\exp (-\frac{{\sigma }_2}{E_h})]\\ {\epsilon }_3=\frac{3-{\gamma }_h}{3E_r}[{\sigma }_3-\mu ({\sigma }_1+{\sigma }_2)]+\frac{{\gamma }_h}{3}[1-\exp (-\frac{{\sigma }_3}{E_h})]\end{array}\right\}$$

(1)

where ε₁, ε₂, and ε₃ are the strains measured in the triaxial test of rock. σ₁, σ₂, and σ₃ are the corresponding nominal principal stresses. E_r and E_h represent the modulus of elasticity of the hard and soft components, respectively (soft components refers to the pores and fractures in the rock, and hard components corresponds to the rock particle skeleton, except the soft components), and γ_h is the proportion of soft components in the rock, and μ represents the Poisson’s ratio of the hard components.

Under the triaxial stress σ₂ = σ₃, ε₂ = ε₃ = 0, the principal strain of rock under triaxial confining pressure can be obtained as follows:

$${\epsilon }_1=\frac{3-{\gamma }_h}{3E_r}[{\sigma }_1-\mu ({\sigma }_2+{\sigma }_3)]+\frac{{\gamma }_h}{3}[1-\exp (-\frac{{\sigma }_1}{E_h})]$$

(2)

Equation (2) is the constitutive equation of sandstone before the yield point. The axial strain and stress can be measured by the test. The crucial step is the determination of three parameters: γ_h, E_r, and E_h. To determine each parameter, the ratio ${\gamma }_r{}^{\prime }$ of the hard components to the whole rock mass and the ratio ${\gamma }_h{}^{\prime }$ of the soft components to the whole rock mass are introduced. The sum of the two is 1, and the proportional relationship between ${\gamma }_r{}^{\prime }$ and ${\gamma }_h{}^{\prime }$ can be expressed as

$${\gamma }_h{}^{\prime }=\frac{{\gamma }_h}{3}$$

(3)

On the basis of the stress–strain curve acquired from this test, the value of $\frac{3E_r}{3-{\gamma }_h}$ in Equation (2) can be determined from the slope of the straight line in the elastic stage of the rock. The intersection point of the extension of the straight line in the elastic section and the strain axis is ${\gamma }_h{}^{\prime }$. After determining the parameters γ_h and E_r, the value of E_h can be obtained according to the low stress stage, i.e., the stress and strain data points of the compaction section are substituted in Equation (2). The stress–strain curve and the parametric solution of the TPHM model are depicted in Figure 10.

It can be seen in Figure 8 that the slope of the elastic stage of sandstone obviously increased with the increase in the confining pressure. Further, the value of ${\gamma }_h{}^{\prime }$ decreased. Therefore, it can be concluded that the parameters of TPHM model were related to the stress state of sandstone. The constitutive equation established by using the above single test curve is not universal. Therefore, to reduce the error of the constitutive equation, the triaxial stress–strain curve under different temperatures was obtained on the basis of the parametric solution of the TPHM model. The variations in the γ_h, E_r, and E_h as a function of the confining pressure are presented in Figure 11.

Further, the corresponding fitting equations are obtained by linear fitting:

$$E_r=329.78{\sigma }_3+2389.26$$

(4)

$$E_h=-0.16849{\sigma }_3+1.21486$$

(5)

$${\gamma }_h=-0.000207{\sigma }_3+0.00273$$

(6)

By substituting the fitting equations into Equation (2), a double-strain Hooke constitutive model of red sandstone with only three basic mechanical parameters can be obtained as follows:

$$\begin{array}{c}{\epsilon }_1=\frac{0.000207{\sigma }_3+2.99727}{989.34{\sigma }_3+7167.78}({\sigma }_1-2\mu {\sigma }_3)+\hfill \\ \hspace{1em}\frac{-0.000207{\sigma }_3+0.00273}{3}[1-\exp (-\frac{{\sigma }_1}{-0.16849{\sigma }_3+1.21486})]\hfill \end{array}$$

(7)

The stress–strain curves under the confining pressures of 0, 2, 4, and 6 MPa were obtained by using the above constitutive equations, as shown in Figure 12. Compared with the other models, the optimized TPHM was simpler and allowed for higher-level simulation. It was able to reflect the crack compaction and elastic deformation phases before the yield point more effectively. However, it is a macro-scale approximation, and it is based on Hooke’s laws of natural and engineering strains. Moreover, it cannot reflect the stress–strain relationship beyond the yield point.

4.2. Constitutive Relationship after the Yield Point

On the basis of the Lemaitre strain equivalence hypothesis [29,30,31,32,33,34], the rock damage constitutive equation can be established as follows:

$$[\sigma ]=[E][1-D][\epsilon ]=[1-D][{\sigma }^{*}]$$

(8)

where [σ] and [σ*] are the stress matrix and the effective stress matrix, respectively. [E] is the elastic matrix, [ε] is the axial strain matrix, (1 − D) is the relative area of the rock that can effectively bear the internal force during compression, and D is the rock damage parameter.

Assuming that the number of damaged micro-elements during rock compression is N_h, the damage parameter D is defined as the ratio of N_h to the total number of micro-elements N [35,36,37], i.e.,

$$D=\frac{N_h}{N}$$

(9)

According to Figure 1, the red sandstone contains many internal pores and fissures, and there are weak layers with different micro-element strengths in the rock. Considering the continuous damage of rock during triaxial compression, it is assumed that the micro-element strength and micro-element force are isotropic and obey Hooke’s Law. Before rock failure, the distribution variable F of the micro-element intensity follows Weibull distribution, and the probability density function is defined as follows:

$$P(F)=\frac{m}{F_0}{(\frac{F}{F_0})}^{m-1}\exp [-{(\frac{F}{F_0})}^m]$$

(10)

where m and F₀ represent the Weibull distribution parameters.

Hence, the number of destroyed micro-elements in the interval [F, F + dF] can be expressed as NP(y)dy. When the rock is loaded to some degree F, the number of damaged micro-elements can be obtained as follows:

$$N_h(F)={\displaystyle {\int }_0^{F}NP(y)dy}=N\{1-\exp [-(\frac{F}{F_0})]\}$$

(11)

Substituting the above equation into Equation (9), the rock damage evolution equation can be derived as

$$D=\frac{N_h}{N}=1-\exp [-{(\frac{F}{F_0})}^m]$$

(12)

According to Equation (12), the micro-element strength F must be calculated to determine the damage parameter D. The traditional method is to use uniaxial strain as the distribution variable, which cannot describe the influence of rock stress state on the micro-element strength. Therefore, the distribution variable of the micro-element intensity is used to establish the failure criterion of rock. On the basis of the simple parametric form of Drucker–Prager (D-P) criterion, the infinitesimal strength can be obtained.

$$F=f(\sigma )=\alpha I_1+\sqrt{J_2}$$

(13)

where α is the cohesion, and φ is the internal friction angle of rock. Further, the first and second invariants of the stress tensor are represented by I₁ and J₂, respectively.

$$\left.\begin{array}{l}\alpha =\frac{\sqrt{3}\sin \phi }{3\sqrt{3+{\sin }^2\phi }}\\ I_1={\sigma }^{*}{}_x+{\sigma }^{*}{}_y+{\sigma }^{*}{}_z={\sigma }^{*}{}_1+{\sigma }^{*}{}_2+{\sigma }^{*}{}_3\\ J_2=\frac{1}{6}[{({\sigma }^{*}{}_1-{\sigma }^{*}{}_2)}^2+{({\sigma }^{*}{}_2-{\sigma }^{*}{}_3)}^2+{({\sigma }^{*}{}_3-{\sigma }^{*}{}_1)}^2]\end{array}\right\}$$

(14)

where σ₁^*, σ₂^*, and σ₃^* are the effective stress values of rock under compression, and the corresponding nominal stress values are σ₁, σ₂, and σ₃. The nominal stress of rock can be measured by triaxial test, and σ₁^* = σ₂^*, σ₁ = σ₂. According to the Hooke’s Law and the concept of stress tensor,

$$\left.\begin{array}{l}{I}_1=\frac{E{\epsilon }_1({\sigma }_1+2{\sigma }_3)}{{\sigma }_1-2\mu {\sigma }_3}\\ \sqrt{J_2}=\frac{E{\epsilon }_1({\sigma }_1-{\sigma }_3)}{\sqrt{3}({\sigma }_1-2{\sigma }_3)}\end{array}\right\}$$

(15)

Combining Equations (8), (12), and (13) and Hooke’s law, and using the D-P strength criterion, the damage constitutive equation of rock is derived as follows:

$${\sigma }_1=E{\epsilon }_1(1-D)+2\mu {\sigma }_3=E{\epsilon }_1\cdot \exp [-{(\frac{\alpha I_1+\sqrt{J_2}}{F_0})}^m]+2\mu {\sigma }_3$$

(16)

It can be seen from the established damage constitutive relationship (16) that the Weibull distribution parameters F₀ and m need to be determined in advance, and the peak point of the stress–strain curve under triaxial compression can be used to obtain the parameter m, while the function F₀ is related to F [38,39,40,41,42], and the specific expression is

$$m=\frac{1}{\ln \frac{E{\epsilon }_1{}^0}{{({\sigma }_1)}_{\max }-\mu ({\sigma }_2+{\sigma }_3)}}$$

(17)

$$F_0={\left(\frac{F_1{}^m}{\ln \frac{E{\epsilon }_1{}^0}{{({\sigma }_1)}_{\max }-\mu ({\sigma }_2+{\sigma }_3)}}\right)}^{\frac{1}{m}}$$

(18)

where (σ₁)_max is the peak stress of the stress–strain curve, ε₁⁰ is the strain value corresponding to the peak stress, and F₁⁰ is the micro-element strength at the peak point.

To prove the validity of the constitutive model, the internal friction angle of sand is considered to be 28.56° according to the triaxial freezing test. Using Equations (17) and (18), the values of model parameters m and F₀ can be determined. The constitutive model equation of soft rock damage is derived by substituting m and F₀ into Equation (16). The constitutive model curve is compared with the test curve in Figure 13.

The following inferences can be drawn by comparing the test results with the modeling results.

• On the basis of the D-P strength criterion, the damage-softening constitutive model of rock is more suitable for describing the stress–strain curve of rock, especially beyond the yield point, with a high fitting degree.
• The stress before the yield point described by the constitutive relation is generally high, and it cannot be used to accurately describe the crack compaction and elastic deformation stages of Cretaceous red sandstone under 3D stress.
• Compared with the test curve under high confining pressure, the model curve is more consistent with the test stress–strain curve obtained under low confining pressure, and under the negative temperature state, the fitting degree of the model curve rises with the decrease in the temperature, which may be ascribed to the increase in the elastic modulus of rock as the temperature decreases.

4.3. Model Validation

Cretaceous rock is different from ordinary rocks. It contains a large number of pore structures. The stress–strain curve under triaxial stress had an obvious crack compression section, and a strain softening section was observed under negative temperature. The double strain Hooke constitutive model can effectively describe the stress and strain state before the yield point, but it cannot reveal these characteristics after the yield point. However, the stress–strain curve obtained by the damage constitutive model on the basis of the D-P criterion was in good agreement with the test stress-strain curve after the rock yield point. Therefore, the constitutive relationship of red sandstone can be more accurately described by establishing the constitutive model in sections. However, there is no unified standard for determining the rock yield point. The peak point of the stress–strain curve, namely, the peak stress, is generally regarded as the yield stress of rock in classical rock theory, which can not only lead to the underestimation of rock plastic deformation, but also affect the calculation of hardening parameters [43,44,45,46,47,48,49].

In fact, the rock is damaged only when it reaches a certain stress state under the 3D compression process. According to previous studies, the yield stress value of sandstone under various conditions is 73–78% of the peak stress value before the peak point. For simplifying the calculations, 78% of the peak stress value before the peak point was selected as the yield point in this study. On the basis of the experimental results under different confining pressures at room temperature, the established constitutive relationship before and after the yield point was compared with the test results, as shown in Figure 14.

It is clear from Figure 14 that the established model can competently fit the stress–strain curve under different confining pressures. However, the model did not predict the sudden drop of stress at the point of rock failure, and there was no good contact between the joints in the segmented model, especially the fitting curve when the confining pressure was 2 MPa. The difference between the theoretical values of the two models at the joint was 1.4 MPa, which was strongly related to the failure of the established damage constitutive model to better fit the elastic stage. To quantify the fitting degree of the model, the model deviation was analyzed, and the specific expressions are as follows:

$$\delta =\frac{{\displaystyle \sum _{i=1}^n\left|{\sigma }_i{}^{*}-\left.{\sigma }_i\right|\right.}}{n}$$

(19)

$$\overline{\delta }=\frac{{\displaystyle \sum _{i=1}^n\left|{\sigma }_i{}^{*}-\left.{\sigma }_i\right|\right.}}{{\displaystyle \sum _{i=1}^n{\sigma }_i}}\times 100$$

(20)

where $\delta$ and $\overline{\delta }$ are the absolute deviation and relative deviation between the modeling and experimental results, respectively, and ${\sigma }_i$ and ${\sigma }_i{}^{*}$ are the corresponding test stress and theoretical stress values, respectively (i = 1, 2, 3…). n represents the number of data points. The model parameters obtained from the above equation are listed in

Table 4. Results of model deviation.

Confining Pressure/MPa	Absolute Deviation $\mathit{\delta }$ /MPa	$\mathbf{Relative} \mathbf{Deviation} \overline{\mathit{\delta }}$ /%
2	0.312	3.28
4	0.526	3.82
6	0.706	3.68

.

According to the above results, the absolute deviation of the established rock constitutive model did not exceed 1 MPa, and the relative deviation was less than 5%, which indicates that the model was able to accurately describe the stress–strain behavior of Cretaceous red sandstone under triaxial stress.

5. Conclusions

In this study, the mechanical properties and constitutive relationship of Cretaceous water-rich sandstone at different freezing temperatures and confining pressures were examined under the effect of surrounding rocks. The main results of the study are summarized as follows:

• Triaxial compression tests were conducted on the frozen samples of weakly cemented red sandstone in the Luohe Formation of Lower Cretaceous in Lijiagou air-return shaft of Wenjiapo Mine under different confining pressures. The stress–strain curves of rock subjected to triaxial compression exhibited four phases: crack compaction, elastic deformation, yielding phase, and failure phase. Different from the deformation characteristics under confining pressure, there was no strain-softening stage behind the peak of rock under uniaxial compression, which manifested as a sudden stress drop and rock failure.
• The peak strength and the elastic modulus of rock increased with the decrease in the temperature when the confining pressure was constant and showed a linear growth trend under negative temperature conditions, but the rate of growth gradually decreased. When the test temperature was constant, as the confining pressure increased, the peak strength and the elastic modulus increased, while the Poisson’s ratio decreased.
• The TPHM was able to effectively describe the constitutive relationship of rock before the yield point. On the basis of the Lemaitre strain equivalence hypothesis, the micro-elements in the sandstone were assumed to obey Weibull distribution. According to the D-P criterion, the damage constitutive relationship of rock was established to describe the stress–strain relationship after the yield point.
• The results based on the constitutive relationship were compared with the test results. The absolute error between the two was not more than 1 MPa, and the relative error was less than 5%. This indicated that the established constitutive model was not only better able to describe the rock fracture compression and elastic stages before the yield point, but also revealed the post-peak strain softening section, verifying the applicability and rationality of the model.