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Strength characteristics and damage constitutive model of sandstone under hydro-mechanical coupling

  • Qiang Liu , Yanlin Zhao EMAIL logo , Jian Liao , Tao Tan , Xiaguang Wang , Yang Li and Zhe Tan
Published/Copyright: December 14, 2023
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Applied Rheology
From the journal Applied Rheology Volume 33 Issue 1

Abstract

To study the mechanical properties of saturated sandstone, experiments were conducted under hydro-mechanical coupling on saturated sandstone. A damage constitutive model was established to describe the response of saturated sandstone under pore pressure, and its validity was verified using the results of the triaxial tests. The results indicate that the peak strength (σ p), effective peak strength (σ p′), residual strength (σ r), effective normal stress (σ n′), effective shear strength (τ n′), elasticity modulus (E), and rupture angle (θ) of sandstone are positively correlated with the confining pressure (σ 3) and negatively correlated with the pore pressure (P). Conversely, Poisson’s ratio (μ) exhibits an opposite relationship. The model parameters exhibit non-linear relationships with the confining pressure (σ 3), with the parameter m decreasing gradually as the confining pressure increases, and the parameter F 0 increasing with higher confining pressure (σ 3). Moreover, the pore pressure (P) and the confining pressure (σ 3) significantly affect the damage variables (D), with the stress value at the damage initiation point increasing with increasing confining pressure (σ 3), while the strain value at the damage initiation point decreasing with increasing pore pressure (P), indicating that pore pressure induces damage development in rocks.

Keywords: sandstone; triaxial compression test; strength characteristics; hydro-mechanical coupling; damage constitutive model

1 Introduction

Rock, being a complex natural geological material, exhibits characteristics such as micro-cracks, pores, cavities, joints, fissures, and other defects, which result in macroscopic discontinuous surfaces, inhomogeneity, and anisotropy [1–12]. These defects provide spaces for groundwater storage and transport, and the development of micro-cracks directly affects the permeability of rocks. Under in situ stress and loading, pore pressure alters the stress state of the rock, leading to complicated mechanical properties [13–18]. Therefore, studying the damaged mechanical characteristics of rocks under hydro-mechanical coupling is of great significance for the safety and stability evaluation of underground rock engineering.

The movement of water in rock is a complex and ancient subject. To explore the interaction mechanism between water rocks, numerous scholars have researched the hydro-mechanical coupling characteristics of various lithologies [19,20]. They have established the correlation between pore pressure and rock deformation under hydro-mechanical coupling and revealed the failure mechanism of confining pressure and pore pressure on rock deformation [21–28]. Rock permeability is as important as its mechanical properties under hydro-mechanical coupling [29]. Some scholars have studied the changing patterns of rock permeability and mechanical strength properties under hydro-mechanical coupling, establishing the correlation between rock stress–strain and permeability [30–36].

Since the proposal of the strain equivalence hypothesis by Lemaitre [37,38], scholars have combined statistical strength theory and continuous damage theory to investigate the damage ontological relationships of rock materials using random distribution functions [39]. The Weibull distribution is one of the most widely used statistical models for non-homogeneous brittle materials, and statistical homogeneity evaluation methods coupled with the Weibull distribution have been extensively applied to analyze the damage of brittle materials [40–43]. Building upon rock damage evolution equations, Tang [44] proposed a statistical damage constitutive model of rocks utilizing the Weibull distribution function and axial strain as rock micro-element strength. Cao et al. [45] presented a statistical damage constitutive model for strain-softening and hardening rocks under conventional triaxial stress conditions. Zhao et al. [46] established a statistical damage constitutive model for rocks considering residual strength. Pourhosseini and Shabanimashcool [47] proposed a constitutive model considering strain softening and dilatancy of rocks based on the Mohr–Coulomb criterion. Li et al. [48] developed a statistical damage constitutive model for rock softening behavior. Chen et al. [49] introduced a statistical damage constitutive model for rocks based on the Hoek–Brown criterion. Shen et al. [50] established a statistical damage principal structure equation for rocks based on the double shear unified strength theory. Zheng et al. [51] developed a statistical damage constitutive model for rocks considering pore pressure.

Although numerous studies have been on rock damage constitutive modeling, the consideration of pore-water pressure in these models remains insufficient and requires further investigation. Additionally, the fundamental mechanism of hydro-mechanical coupling has not been fully understood, and the variations in rock strength under hydro-mechanical coupling need to be further explored, particularly considering the diverse strength characteristics of different lithologies. Therefore, the conventional triaxial compression tests were conducted on sandstone specimens under different confining pressures and pore pressures using the MTS815 rock mechanics test system. The aim was to obtain deformation characteristic evolution curves throughout the complete stress–strain process, thereby revealing the mechanical behavior of sandstone under hydro-mechanical coupling. The study also investigates the change in rock strength characteristics. To achieve these objectives, a damage constitutive model of rock under hydro-mechanical coupling was constructed based on the Mohr–Coulomb damage criterion and the assumption that the strength of rock microelements follows a Weibull random distribution function. The rationality of the model was verified using the results obtained from the triaxial tests. Furthermore, the study analyzes the effects of damage weakening factors on the post-peak residual phase and examines the evolution of rock damage under varying confining pressures and pore pressures. A novel method for calculating rock microelement strength is also proposed.

2 Test investigations

2.1 Specimen preparation and porosity test

The sandstone specimens were taken from the Ma Cheng iron ore area in Hebei Province, China, at a depth of 420–500 m. The saturated density of sandstone is 2.20 g/cm3, and the saturated water absorption is 6.67%. According to the method suggested by the ISRM, all tested sandstone specimens are prepared as cylinders with a length of 100 and 50 mm in diameter, as shown in Figure 1. The porosity of the sandstone specimens was tested by using the AniMR-150 nuclear magnetic resonance test instrument. The test results show that the pore throat radius of the sandstone is mainly concentrated above 0.1 μm, the porosity ranges from 13.23 to 13.36%, and the mean porosity is 13.28%, which belongs to highly porous rocks. The basic information of the sandstone specimen is listed in Table 1.

Figure 1 Sandstone specimens.
Figure 1

Sandstone specimens.

Table 1

Basic physical and mechanical parameters of specimens

Lithology m sa (g) m w (g) ω sa (%) ρ d (kg/cm3) n (%) σ t (MPa) σ c (MPa)
Sandstone 429.92 458.61 6.67 2.20 13.28 0.48 13.75

Note: m dr dry quality, m sa saturated quality, ρ d dry density, ω sa saturated water absorption, n initial porosity, σ t tensile strength, σ c uniaxial compressive strength.

2.2 Testing apparatus and methods

The tests were carried out on the MTS815 rock mechanics test system at Hunan University of Science and Technology, which is equipped with independent servo-controlled systems for axial pressure, confining pressure, and pore pressure, as shown in Figure 2(a). The pore pressure is set at 0, 0.1, 0.2, 0.4, 0.6, 0.8, and 0.9 times the confining pressure. The confining pressures were set to 10, 20, and 30 MPa, respectively. Test procedures are as follows: (1) Heat-shrink tubing is used to wrap the rock specimen, which is placed inside the test system, while the water pipe, axial circ extensometer, and axial extensometer are installed, and the triaxial cell is dropped (Figure 2(b)). (2) At the beginning of the tests, the lower axial pressure of 1.0 MPa was pre-applied in the axial direction, and the confining pressure σ 3 and axial pressure σ 1 were sequentially loaded to the test design value at a loading rate of 0.05 MPa/s and is maintained (σ 1 = σ 3). (3) Applied water pressure at a loading rate of 0.05 MPa/s until a scheduled pore pressure P and ensuring that σ 3 > P. (4) The specimen was subjected to axial loading using a loading rate of 0.002 mm/s until the specimen was failure. The test loading process is shown in Figure 3.

Figure 2 MTS815 rock mechanics test system at (a) testing devices and (b) specimen installation diagram.
Figure 2

MTS815 rock mechanics test system at (a) testing devices and (b) specimen installation diagram.

Figure 3 Schematic diagram of test force and loading paths. (a) Specimen loading force and (b) path of loading.
Figure 3

Schematic diagram of test force and loading paths. (a) Specimen loading force and (b) path of loading.

3 Results and discussion

3.1 Stress–strain curves

The results of the triaxial tests on the saturated sandstone are summarized in Table 2. Figure 4 presents the deviatoric stress–strain curves for the saturated sandstone under different confining pressures and pore pressures. ε 1 and ε 3 represent the axial strain and circumferential strain, respectively. σ 1–σ 3 represents the deviatoric stress, and P represents the pore pressure. Figure 4 demonstrates that the stress–strain curve of the saturated sandstone can be divided into five distinct stages. The crack compression closure stage (OA) is followed by the linear elastic stage (AB). The stable crack extension stage (BC) is characterized by the formation, extension, and penetration of secondary cracks within the specimen as the axial load increases. The unstable crack extension stage (CD) indicates severe damage to the rock sample and the unstable growth of microcracks. The post-peak stage is marked by a rapid decline in stress and exhibits brittle failure characteristics. However, in some specimens, a secondary drop in stress occurs during the post-peak stage. This phenomenon can be attributed to the frictional occlusion between crystals on the ruptured surface of the rock specimen under continuous confining pressure. As a result, the stress levels maintain a small range. When the axial stress exceeds the shear strength of the fracture surface, the stress experiences two drops with increasing strain until it reaches the residual strength.

Table 2

Mechanical parameters obtained from triaxial compression tests of saturated sandstone

Confining pressure σ 3 (MPa) Pore pressure P (MPa) Peak strength σ p (MPa) Elastic modulus E (GPa) Poisson’s ratio μ Residual strength σ r (MPa) Dilatancy stress σ d (MPa) Rupture angle θ (°)
10 0 110.16 9.90 0.11 45.21 79.58 70
20 0 153.35 12.41 0.15 68.41 120.68 63
30 0 179.33 12.48 0.14 93.69 148.21 59
10 1 115.66 11.74 0.15 48.52 83.61 74
2 103.98 10.81 0.22 35.73 63.87 60
4 74.38 8.67 0.32 31.49 44.57 67
6 63.18 7.96 0.36 27.56 33.85 73
8 55.33 8.26 0.43 18.40 18.70 76
9 48.76 7.41 0.49 13.47 13.28 76
20 2 142.80 11.42 0.19 61.25 104.75 66
4 123.19 10.90 0.23 54.41 91.41 67
8 108.05 10.87 0.27 43.25 70.05 71
12 98.64 9.03 0.25 38.72 70.04 72
16 88.87 10.38 0.29 23.5 59.03 82
18 71.61 9.25 0.49 22.58 33.79 76
30 3 180.28 12.71 0.13 87.74 151.50 60
6 148.77 11.34 0.18 75.75 119.20 66
12 145.53 12.07 0.18 67.87 115.98 69
18 121.2 10.78 0.27 51.36 87.95 72
24 98.85 12.28 0.36 33.11 62.61 74
27 80.39 10.18 0.46 31.74 42.25 79
Figure 4 Deviatoric stress–strain curves of sandstone under hydro-mechanical coupling. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.
Figure 4

Deviatoric stress–strain curves of sandstone under hydro-mechanical coupling. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.

For studies on the deformation characteristics of rocks under the hydro-mechanical coupling, the slope of the linear elastic stage in the range of 20–60% of the peak deviatoric stress is taken as the elastic modulus E, the elastic modulus was calculated as follows:

(1) E = σ B − σ A ε 1 B − ε 1 A ,

where σ A and σ B are the axial stresses corresponding to the beginning and end of the elastic stage, respectively, and the corresponding axial strains are ε 1A and ε 1B, respectively.

Figure 5 shows the relationship curve of Poisson’s ratio and elastic modulus of saturated sandstone with pore pressure under different confining pressures. It can be seen from Figure 5(a) that under the same confining pressure, the rock elastic modulus tends to decrease with the increase of pore pressure. For example, under the confining pressure of 20 MPa, the elastic modulus of rock decreases from 12.41 to 9.25 GPa when the pore pressure increases from 0 to 18 MPa, with a decrease of 3.16 GPa, reduced by 25.46%. On the other hand, under the same pore pressure, the elastic modulus of rock tends to increase with the increase of confining pressure. For instance, under the pore pressure of 12 MPa, the rock elastic modulus increases from 9.03 to 12.07 GPa when the confining pressure increases from 20 to 30 MPa, with an increase of 3.04 GPa, increased by 33.67%. It can be seen from Figure 5(b) that under the same confining pressure, with the increase of pore pressure, Poisson’s ratio of the rock shows an upward trend. For example, under the confining pressure of 20 MPa, Poisson’s ratio of the rock increases from 0.15 to 0.49 when the pore pressure increases from 0 to 18 MPa, with an increase of 0.34, increased by 226.67%. This can be attributed to the fact that under the triaxial compression tests, when the confining pressure remains constant, the compressive stress between particles reduces as the pore pressure increases, leading to an increase in the lateral deformation of rock, and Poisson’s ratio increases. Conversely, Poisson’s ratio of the rock decreases with the confining pressure increases under the same pore pressure. For example, under the pore pressure of 12 MPa, Poisson’s ratio of the rock decreases from 0.25 to 0.18 when the confining pressure increases from 20 to 30 MPa, a decrease of 0.07, which is a decrease of 28.00%. This is because under the same pore pressure, the lateral deformation of rock weakens, and Poisson’s ratio decreases when the confining pressure increases.

Figure 5 Relationship between elastic modulus and Poisson’s ratio and pore pressure. (a) E–P curves, and (b) µ–P curves.
Figure 5

Relationship between elastic modulus and Poisson’s ratio and pore pressure. (a) E–P curves, and (b) µ–P curves.

3.2 Strength characteristics analysis

3.2.1 Influence of confining pressure and pore pressure on peak strength

The peak strength σ p (maximum principal stress) of the sandstone is correlated significantly with the pore pressure and the confining pressure. The relationship between peak strength and pore pressure in a saturated sandstone is shown in Figure 6. It can be seen from Figure 6 that the strength characteristics of sandstone have an obvious confining pressure effect. Under the same pore pressure, the peak rock strength tends to increase with the increase of confining pressure. For example, under the pore pressure of 0 MPa, the peak strength of the rock increases from 110.16 to 179.33 MPa when the confining pressure increases from 10 to 30 MPa, with an increase of 69.17 MPa, and the peak strength increases by 62.79%. Similarly, under the pore pressure of 18 MPa, the peak rock strength increases from 71.61 to 121.20 MPa when the confining pressure increases from 20 to 30 MPa, with an increase of 49.59 MPa, and the peak strength increases by 69.81%. In addition, the strength characteristics of sandstone have an obvious pore pressure effect. Under the same confining pressure, the peak strength of rock tends to decrease with the increase of pore pressure. For example, under the confining pressure of 30 MPa, the peak strength of the rock decreases from 181.28 to 80.39 MPa when the pore pressure increases from 3 to 27 MPa, with a decrease of 99.89 MPa, and the peak strength decreases by 55.41%.

Figure 6 Relationship between peak strength and pore pressure of saturated sandstone.
Figure 6

Relationship between peak strength and pore pressure of saturated sandstone.

Furthermore, we have defined the ratio of the residual strength under pore pressure to the residual strength σ p without pore pressure as the residual strength σ p0 ratio (σ p/σ p0). Figure 7 shows the relationship curve between σ p/σ p0 and P/σ 3. It can be seen from Figure 7 that the σ p/σ p0 of the rock shows a linear decreasing trend with the increase of the P/σ 3. In order to further analyze the internal relationship between the parties, the σ p/σ p0 was used as the dependent variable, and the P/σ 3 was used as the independent variable to perform the linear fitting. The fitting results show that the residual strength ratio has an excellent linear relationship with the P/σ 3, the fitting effect is well, and the correlation coefficient R 2 exceeds 0.9.

Figure 7 Relationship between σ p0/σ p and P/σ 3.
Figure 7

Relationship between σ p0/σ p and P/σ 3.

The pore between the sandstone solid skeleton will be subjected to a combination of axial pressure, confining pressure, and pore pressure during compression under hydro-mechanical coupling. By introducing an effective peak strength σ p′ for further analysis of the hydro-mechanical coupling effect in the sandstone, the effective confining pressure (effective minimum principal stress) σ 3′ is expressed as follows:

(2) σ ′ p = σ p − P σ ′ 3 = σ 3 − P .

Figure 8 shows the relationship between effective peak stress and effective confining pressure under different confining pressures and pore pressures, according to the test data and equation (2). It can be obtained from Figure 8 that the effective peak strength increases with the increase in effective confining pressure. In other words, the effective peak strength is positively correlated with the effective confining pressure. In addition, the effective peak strength was fitted to the effective circumferential pressure in order to explore the inherent relationship between the effective peak strength and the effective confining pressure. The fitting results show that both have a well-linear relationship, and the fitted equation describing the relationship between σ p′ and σ 3′ can be expressed as follows:

(3) σ ′ p = k σ ′ 3 + d ,

where k and d are fitting parameters, respectively. The fitted curves for the test data are shown in Figure 8. The linear fitting equation with k = 4.72, and d = 47.590 is in good agreement with the test data with a correlation coefficient R 2 above 0.9.

Figure 8 Relationship between the effective peak stress and the effective confining pressure curves.
Figure 8

Relationship between the effective peak stress and the effective confining pressure curves.

According to the Mohr–Coulomb criterion, the relationship between the effective peak strength, σ p′, and the effective confining pressure, σ 3′, is expressed as follows:

(4) σ ′ p = 1 + sin φ ′ p 1 − sin φ ′ p σ ′ 3 + 2 c ′ p cos φ ′ p 1 − sin φ ′ p ,

where φ ′ p is the effective internal friction angle and c ′ p is the effective cohesion.

By combining equations (2)–(4), the effective cohesion c p′ = 10.95 MPa, and the effective internal friction angle φ p′ = 40.56°. This indicates that the sandstone test results confirm the validity of Mohr–Coulomb under hydro-mechanical coupling conditions. The unified effective strength parameters can be obtained by the Mohr–Coulomb criterion under hydro-mechanical coupling conditions.

3.2.2 Influence of confining pressure and pore pressure on residual strength

The variation curves of sandstone residual strength with pore pressure under different confining pressures are shown in Figure 9. It can be seen from Figure 9 that the change law of residual strength of sandstone with pore pressure under various confining pressures is similar to that of peak strength. Under the same confining pressure, the residual strength of rock shows a downward trend with the increase of pore pressure. For example, under the confining pressure of 30 MPa, the residual strength of the rock decreases from 87.74 to 31.74 MPa when the pore pressure increases from 3 to 27 MPa, with a decrease of 56.0 MPa, and the residual strength decreases by 63.82%. In addition, under the same pore pressure, the residual strength of the rock increases with the increase of confining pressure. For example, under the pore pressure of 18 MPa, the residual strength of rock increases from 22.58 to 51.36 MPa when the confining pressure increases from 20 to 30 MPa, an increase of 28.78 MPa, and the residual strength increases by 127.46%.

Figure 9 Relationship between residual strength and pore pressure of saturated sandstone.
Figure 9

Relationship between residual strength and pore pressure of saturated sandstone.

Furthermore, we have defined the ratio of the residual strength under pore pressure to the residual strength σ r without pore pressure as the residual strength σ r0 ratio (σ r/σ r0). Figure 10 shows the relationship curve between σ r/σ r0 and P/σ 3. It can be seen from Figure 10 that similar to the change law of the peak strength ratio, the σ r/σ r0 of the rock shows a linear decreasing trend with the increase of the P/σ 3. In order to further analyze the internal relationship between the parties, the σ r/σ r0 was used as the dependent variable, and the P/σ 3 was used as the independent variable to perform the linear fitting. The fitting results show that the residual strength ratio has an excellent linear relationship with the P/σ 3, the fitting effect is well, and the correlation coefficient R 2 exceeds 0.9.

Figure 10 Relationship between σ r/σ r0 and P/σ 3.
Figure 10

Relationship between σ r/σ r0 and P/σ 3.

In order to further understand the residual strength characteristics of rocks under hydraulic coupling, the concept of effective residual strength ( σ ′ r = σ r − P ) is introduced for analysis. Figure 11 shows the relationship between effective residual strength and effective confining pressure under different confining pressures and pore pressures. It can be obtained from Figure 11 that the effective residual strength increases with the increase in effective confining pressure. In addition, the effective peak strength was fitted to the effective confining pressure in order to explore the inherent relationship between the effective peak strength and the effective confining pressure. The fitting results show that both have a well-linear relationship and the test data with a correlation coefficient R 2 above 0.9.

Figure 11 Relationship between the effective peak stress and the effective confining pressure curves.
Figure 11

Relationship between the effective peak stress and the effective confining pressure curves.

According to the Mohr–Coulomb criterion, the relationship between the effective peak strength, σ r′, and the effective confining pressure, σ 3′, is expressed as follows:

(5) σ ′ r = 1 + sin φ ′ r 1 − sin φ ′ r σ ′ 3 + 2 c ′ r cos φ ′ r 1 − sin φ ′ r ,

where φ ′ r and c ′ r are the effective residual internal friction angle and the effective residual cohesion, respectively.

Combining equation (5) and fitting results, 1 + sin φ ′ r 1 − sin φ ′ r = 3.043 , 2 c ′ r cos φ ′ r 1 − sin φ ′ r = 3.021 , the effective cohesion c′r = 0.87 MPa, and the effective internal friction angle φ′r = 30.36° are obtained.

3.2.3 Relationship between shear strength and normal stress on the rupture surface

To compare and analyze the evolution characteristics of rock shear strength under different confining pressures and different pore pressures, the relationship between effective normal stress σ n′ and effective shear strength τ n′ and their corresponding peak strength σ p, confining pressure σ 3, and pore pressure P can be expressed as follows [2]:

(6) σ ′ n = 1 2 ( σ p + σ 3 − 2 P ) + 1 2 ( σ p − σ 3 ) cos 2 θ τ ′ n = 1 2 ( σ p − σ 3 ) sin 2 θ ,

where θ is the rock rupture angle.

The relationship curves between effective shear stress and effective normal stress under hydro-mechanical coupling in saturated sandstone were plotted (Figure 12), according to equation (6). As is easily seen from Figure 12, the effective shear strength increases with the increase of the effective normal strength while decreasing with the increase of the pore pressure under the same confining pressure. For example, with the confining pressure of 20 MPa, the effective normal stress decreases from 38.32 to 5.02 MPa when the pore pressure increases from 2 to 18 MPa, with a decrease of 86.90% and the effective shear strength decreases from 45.63 to 12.11 MPa, with a decrease of 73.46%. Additionally, the effective normal stress and the effective shear strength both increase gradually with the increase in the confining pressure. The effective normal stress increases from 4.65 to 21.33 MPa, with an increase of 358.71%, and the effective shear strength increases from 10.34 to 27.10 MPa, with an increase of 162.09% at the confining pressure increases from 10 to 20 MPa. To further explore the inherent relationship between effective shear strength and effective normal stress, a linear fitting is performed on the two variables using test data (Figure 12). The fitting results indicate a strong linear and well-fitted relationship between effective shear strength and effective normal stress, with the fitting correlation coefficient R 2 above 0.99. This shows that the τ n′−σ n′ failure strength curve of the saturated sandstone under hydro-mechanical coupling satisfies the Coulomb criterion.

Figure 12 Relationship between effective shear strength and effective normal stress of saturated sandstone under hydro-mechanical coupling.
Figure 12

Relationship between effective shear strength and effective normal stress of saturated sandstone under hydro-mechanical coupling.

4 Damage constitutive model

4.1 Establishment of the constitutive model

Following the Lemaitre theory of strain equivalence [37,39], the effect of damage on strain behavior is reflected through the effective stress, and the damage constitutive relationship of the rock can be expressed as follows:

(7) σ i j = σ ˜ i j ( 1 − D ) ,

where σ ˜ i j is the effective stress tensor of the undamaged part of the material, σ i j is the stress tensor, and D is the damage variable. When D = 0, the rock is undamaged, and when D = 1, the rock is in a fully damaged state. For hydro-mechanical coupling problems, the effective stress principle is usually used for analysis, and Biot Maurice [52] modified the effective stress principle as follows:

(8) σ ′ ˜ i j = σ i j − ξ P δ i j ,

where σ ′ ˜ i j is the effective stress tensor, P is the pore pressure, δ ij is the unit second order tensor δ ij = 1 (i = j), otherwise δ ij = 0(i ≠ j), and ξ is the Biot coefficient, determined by the following:

(9) ξ = 1 − K K s ,

where K and K s are the bulk modulus of the solid particles and the bulk modulus of the skeleton in the rock mass, respectively. For most materials, the compressibility of the skeleton is much smaller than that of the solid particles themselves, that is, K ≤ K s, so ξ = 1 is taken to facilitate the investigation. Assuming that the rock material is isotropic, that is, the damage variables, the modulus elasticity, and the Biot coefficient are isotropic. Combining equations (7) and (8) can obtain the effective stress tensor under the hydro-mechanical coupling can be expressed as follows:

(10) σ ′ ˜ i j = σ i j − p w δ i j 1 − D .

The results of extensive triaxial tests indicate that part of the compressive and shear stresses are also transferred after the failure of rock microelements during the compression of the rock. Since the damaged area has a certain strength (e.g., residual strength), the damage weakening factor λ associated with the residual strength is introduced as follows [53]:

(11) λ = 1 − σ r σ 1 c , λ ∈ ( 0 , 1 ) ,

where λ is the damage weakening factor, σ r is the residual strength, and σ 1c is the peak stress.

Substituting equation (11) into equation (10), the effective stress after correction of the damage variable can be expressed as follows:

(12) σ ′ ¯ i j = σ i j − P δ i j 1 − λ D .

A large number of fractures and micro-cracks exist in the rock material, and their distribution, size, shape, and penetration length all show significant randomness, resulting in large differences between the individual micro-units of the rock. Assuming that the number of micro-elements damaged under a certain load is n and the total number of micro-elements is N, the damage variable D can be defined as the ratio of the number of damaged micro-elements to the total number. Then, the statistical damage variable can be expressed as follows:

(13) D = n N .

Assuming that the damage of rock microelements obeys the Weibull distribution, its probability density functions as follows:

(14) P ( F ) = m F 0 F F 0 m − 1 exp − F F 0 m ,

where P(F) is the Weibull distribution function, F is the Weibull distribution variable, and F 0 and m are the Weibull distribution parameters.

The deformation damage of rocks is a continuous process of continuous damage within the material, and the damage variable D is related to the strength probability of the micro-element as follows:

(15) D = ∫ 0 F P ( F ) d F .

Substituting equations (14) and (15) into equation (13) yields

(16) D = ∫ 0 F N P ( F ) d F N = 1 − exp − F F 0 m .

Considering the damage threshold for the damage variable as follows:

(17) D = 1 − exp − F F 0 m ( F ≥ 0 ) 0 ( F < 0 ) .

Since the Mohr–Coulomb criterion is established based on tests, the expression form is simple and has better rationality, and it can reflect the bearing capacity of the rock more accurately; thus, it has been more widely applied. Considering the rock as a series of defective micro-elements, the rock micro-element strength criterion can be written as follows [2]:

(18) F = f ( σ ′ ˜ i j ) = ( 1 − α ) I ′ ˜ 1 3 + α J ′ ˜ 2 sin θ σ + ( 2 − α ) J ′ ˜ 2 3 cos θ σ = σ t ,

where I 1 is the first invariant of the stress tensor, J 2 is the second invariant of the stress bias, θ σ is the Loder angle, σ t a is the tensile strength, and α is the parameters of the corresponding functions, with

(19) I ′ ˜ 1 = σ ′ ˜ 1 + σ ′ ˜ 2 + σ ′ ˜ 3 ,

(20) J ′ ˜ 2 = 1 6 [ ( σ ′ ˜ 1 − σ ′ ˜ 2 ) 2 + ( σ ′ ˜ 2 − σ ′ ˜ 3 ) 2 + ( σ ′ ˜ 3 − σ ′ ˜ 1 ) 2 ] ,

(21) α = 1 − sin φ 1 + sin φ σ t = 2 c cos φ 1 − sin φ .

Assuming that the stress–strain relationship of the rock obeys the generalized Hooke’s law, considering the stress–strain relationship in the principal direction can be expressed as follows:

(22) ε ′ ˜ 1 = 1 E [ σ ′ ˜ 1 − μ ( σ ′ ˜ 2 + σ ′ ˜ 3 ) ] .

In the conventional triaxial test, the stress state at this point satisfies σ 1＞σ 2 = σ 3， θ σ = 30°. Considering the conditions of coordination of material deformation of the damaged and undamaged parts of the rock material, we can obtain the axial strain ε 1 = ε ′ ˜ 1 . Substituting equation (12) into equation (22), the relationship of axial stress–strain under the hydro-mechanical coupling can be obtained as follows:

(23) σ 1 = E ε 1 ( 1 − λ D ) + 2 μ σ 3 + ( 1 − 2 μ ) P ,

where σ 1 and σ 3 are, respectively, the maximum (axial stress) and minimum principal stresses (confining pressure).

According to the generalized Hooke’s law and the principle of effective stress, the effective stress under the action of hydro-mechanical coupling can be obtained as follows:

(24) σ ′ ˜ 1 = ( σ 1 − P ) E ε 1 σ 1 − 2 μ σ 3 + ( 2 μ − 1 ) P σ ′ ˜ 3 = ( σ 3 − P ) E ε 1 σ 1 − 2 μ σ 3 + ( 2 μ − 1 ) P .

Substituting equation (24) into equations (19) and (20), we obtain

(25) I 1 = E ε 1 ( σ 1 + 2 σ 3 − 3 P ) σ 1 − 2 μ σ 3 + ( 2 μ − 1 ) P ,

(26) J 2 = E ε 1 ( σ 1 − σ 3 ) 3 [ σ 1 − 2 μ σ 3 + ( 2 μ − 1 ) P ] .

In the conventional triaxial test, the axial deviatoric stress σ 1t recorded in the test is the difference between the axial stress σ 1 and the confining pressure σ 3:

(27) σ 1 = σ 1 t + σ 3 .

The true axial strain ε 1 is the sum of the test-measured strain ε 1t and the initial strain ε 10:

(28) ε 1 = ε 1 t + ε 10 = ε 1 t + 1 − 2 μ E ( σ 3 − P ) .

Substituting equations (27) and (28) into equations (25) and (26), we obtain

(29) I 1 = [ E ε 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] ( σ 1 t + 3 σ 3 − 3 P ) σ 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ,

(30) J 2 = [ E ε 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] σ 1 t 3 [ σ 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] .

Combining equations (17) and (23), the statistical damage principal structure of the rock considering the damage threshold under hydro-mechanical coupling can be obtained as follows:

(31) σ 1 t = [ E ε 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] 1 − λ + λ exp − F F 0 m + ( 2 μ − 1 ) ( σ 3 − P ) ( F ≥ 0 ) [ E ε 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] + ( 2 μ − 1 ) ( σ 3 − P ) ( F < 0 ) ,

with

(32) F = [ E ε 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] [ σ 1 t + ( 1 − α ) ( σ 3 − P ) ] [ σ 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] .

4.2 Determination of constitutive model parameters

Since the peak point method uses the stress at the peak point to get the maximum value and the slope of zero at the peak point to establish a system of equations to solve the model parameters, which has the advantage of being able to fit well at the peak point, this article uses the peak point method to solve the model parameters. The typical stress–strain curve of strain-softened rock material is shown in Figure 13. The point Q(ε 1c,σ 1c) is the peak point of the full stress–strain curve, and according to the geometric condition of the complete stress–strain curve of the rock, the following two boundary conditions can be obtained:

ε 1 t = ε 1 c , σ 1 t = σ 1 c ,

σ 1 t = σ 1 c , ∂ σ 1 c ∂ ε 1 c = 0 ,

where σ 1c is the peak stress and ε 1c is the strain relevant to the peak stress.

Figure 13 Typical stress–strain curves of strain-softening rock materials.
Figure 13

Typical stress–strain curves of strain-softening rock materials.

Substituting equation (31) into the boundary condition (1) to yield.

(33) λ exp − F c F 0 m = σ 1 c + ( 1 − 2 μ ) ( σ 3 − P ) E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) + λ − 1 ,

(34) F c F 0 m = ln λ [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] σ 1 c + ( 1 − 2 μ ) ( σ 3 − P ) + ( λ − 1 ) [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] ,

where F c is the peak micro-element strength.

Substituting equation (31) into the boundary condition (2) to yield

(35) ∂ σ 1 t ∂ ε 1 t = E 1 − λ + λ exp − F c F 0 m + [ E ε 1 t + ( 1 − 2 μ ) ( σ 3 − P ) ] λ exp − F c F 0 m − m F 0 × F c F 0 m − 1 ∂ F ∂ ε 1 t = 0 .

By further simplifying and organizing equation (35), it yields

(36) F c F 0 m = ln λ [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] σ 1 c + ( 1 − 2 μ ) ( σ 3 − P ) + ( λ − 1 ) [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] ,

(37) m F c m − 1 F 0 m = E 1 − λ + λ exp − F c F 0 m [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] λ exp − F c F 0 m ∂ F c ∂ ε 1 t .

Finally, the parameters m and F 0 can be obtained as follows:

(38) m = F c E A 1 [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] A 3 ln ( A 2 ) ∂ F c ∂ ε 1 t ,

(39) F 0 = F c [ ln ( A 2 ) ] 1 m ,

with

A 1 = σ 1 c + ( 1 − 2 μ ) ( σ 3 − P ) E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ,

A 2 = λ [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] σ 1 c + ( 1 − 2 μ ) ( σ 3 − P ) + ( λ − 1 ) [ E ε 1 c + ( 1 − 2 μ ) ( σ 3 − P ) ] ,

A 3 = A 1 + λ − 1 ,

∂ F c ∂ ε 1 t = E σ 1 t + ( 1 − α ) ( σ 3 − P ) σ 1 t + ( 1 − 2 μ ) ( σ 3 − P ) .

5 Verification of model and parameter analysis

5.1 Verification of model

To verify the feasibility of the proposed damage constitutive model of rock considering the pore pressure, the stress–strain curve was calculated by the triaxial compression test, and the internal friction angle φ = 41.1° was obtained by processing the test data. The model parameters are listed in Table 3. According to the test results in Figure 14, the theoretical curves of the rock damage constitutive model under hydro-mechanical coupling can be calculated by equation (31). The theoretical curves for different pore pressures under three confining pressures are shown in Figure 14 and compared with the test curves. It can be seen from Figure 14 that the proposed damage constitutive model can reflect the stress–strain relationship of the rock specimens at different stages during the loading process, while the post-peak stage can reflect the softening characteristics and residual strength of the rock, and the theoretical curve matches well with the indoor test curve, which verifies the feasibility of the damage constitutive model of sandstone under hydro-mechanical coupling.

Table 3

Model parameters

σ 3 (MPa) P (MPa) E (GPa) σ 1c (MPa) λ m F 0 (MPa)
10 0 9.90 100.17 0.81 6.21 164.13
1 11.74 106.66 0.80 4.85 183.87
2 10.81 93.98 0.85 5.42 159.40
4 8.67 65.40 0.82 5.56 109.90
6 7.96 54.35 0.82 5.51 90.29
8 8.26 46.33 0.90 9.49 66.69
9 7.41 38.76 0.95 7.79 58.11
20 0 12.41 133.35 0.80 5.70 243.23
2 11.42 122.80 0.81 5.39 228.35
4 10.90 103.19 0.82 5.22 197.07
8 10.87 88.05 0.86 6.45 160.26
12 9.03 78.64 0.87 5.45 148.06
16 10.38 68.87 0.97 6.38 124.35
18 9.25 51.61 0.97 5.69 98.28
30 0 12.48 149.33 0.76 5.13 272.03
3 12.71 150.28 0.78 5.02 273.51
6 11.34 118.78 0.78 5.64 213.01
12 12.07 115.53 0.82 5.27 206.19
18 10.78 91.20 0.88 6.72 132.88
24 12.28 68.85 0.98 3.20 143.54
27 10.18 50.39 0.98 4.60 91.00
Figure 14 Comparison of theoretical and test curves. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.
Figure 14

Comparison of theoretical and test curves. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.

5.2 Parameter sensitivity analyses

5.2.1 Damage weakening factor influence on the present model

To analyze the effect of the damage weakening factor on the residual softening stage of the stress–strain curve, the test data were selected for confining pressures of 10, 20, and 30 MPa and pore pressure of 0 MPa, keeping other parameters constant, and the damage weakening factor λ increases from 0.7 to 1.0, as shown in Figure 15. It is seen from Figure 15 that the damage weakening factor has an insignificant effect before the peak stress–strain and a more significant effect during the residual softening stage of the post-peak curve, with the residual strength of the rock gradually decreasing as the damage weakening factor increases. The closer the damage weakening factor is to 1, the less significant the increase in residual rock strength and the greater the discrepancy between the theoretical and test values.

Figure 15 Relationship between the damage weakening factor and the stress–strain curves. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.
Figure 15

Relationship between the damage weakening factor and the stress–strain curves. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.

5.2.2 Relationship between confining pressure and model parameters

The confining pressure and the model parameters in Table 3 were selected for the one-dimensional quadratic fitting, and the results are shown in Figure 16. The results show that the two have a well nonlinear relationship, and the fitting correlation coefficients R 2 of 1.0. Additionally, the model parameter m decreases with increasing confining pressure, while the model parameter F 0 increases with increasing confining pressure. This implies a negative correlation between confining pressure and the model parameter m, and a positive correlation between confining pressure and the model parameter F 0. These findings indicate that confining pressure enhances the strength of micro-elements in the rock and reduces brittleness.

Figure 16 Relationship between (a) σ 3∼m and (b) σ 3∼F 0.
Figure 16

Relationship between (a) σ 3∼m and (b) σ 3∼F 0.

6 Damage evolution characteristics analysis

6.1 Evolution of damage variables under different confining pressure

To investigate the influence of confining pressure on the damage variable, the variation curve of D∼σ 1t under different confining pressures was obtained using equation (31), as shown in Figure 17. Due to the consideration of the damage threshold effect, the damage variable D is 0 when the stress level is relatively low. As the stress increases to a certain level, the damage variable increases with the stress, approaching the peak strength of the specimen, and rapidly increases until complete damage (D = 1). Additionally, the damage starting point varies under different confining pressures, with the damage starting point increasing as the confining pressure increases, resulting in a rightward shift of the curve. This indicates that using the Mohr–Coulomb strength criterion as the random distribution variable for damage, the stress value corresponding to the yield point increases with the increase of confining pressure, which is consistent with the actual situation.

Figure 17 Relationship curve between damage variable and axial deviatoric stress under different confining pressures.
Figure 17

Relationship curve between damage variable and axial deviatoric stress under different confining pressures.

6.2 Evolution of damage variables under different pore pressure

To explore the influence of pore pressure by damage variables, the damage values of the stress–strain process were calculated based on the test data in Table 3 and the model parameters using equation (31), and the D∼ε 1t curves were obtained for different confining pressures and pore pressures, as shown in Figure 18. As can be seen from Figure 18, the damage evolution graphs are “S,” and the rock damage curve undergoes a gentle-steep-smooth process, which reflects well the various stages of rock compression deformation: (1) The crack compression closure and elastic deformation stage. It corresponds to the near-horizontal phase of the damage evolution curve at the early stage of loading, where the damage variable is almost constant and tends to zero. In this stage, the rock cracks gradually compacted and closed but did not expand, and the microelements produce elastic deformation. (2) The stage of crack stable and unstable extended deformation. This corresponds to the upper concave phase of the damage evolution curve, where the damage variable increases significantly. At this point, the rock cracks begin to expand rapidly, and plastic yielding begins. (3) The post-peak deformation stage (strain softening and residual deformation stage). The strain softening deformation phase corresponds to the upper convex section of the damage evolution curve, with a decreasing slope of the curve. The sandstone specimen has an expanding internal crack, penetration, and gradual losses of strength. In the residual deformation stage, the rock is destroyed by the appearance of macroscopic crack surfaces inside the rock, and the damage variable tends to 1 and remains stable.

Figure 18 Relationship between damage variables and strain under hydro-mechanical coupling. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.
Figure 18

Relationship between damage variables and strain under hydro-mechanical coupling. (a) σ 3 = 10 MPa, (b) σ 3 = 20 MPa, and (c) σ 3 = 30 MPa.

The damage variables corresponding to the same strain become smaller sequentially with the increase of the confining pressure when the pore pressure is constant and gradually approaches the direction of the coordinate axis of zero. This indicates that the confining pressure can inhibit the crack extension of the rock and thus inhibit the evolution of the damage, which the macroscopic shows an increase in the strength of the rock. The damage variable corresponding to the same strain value increases with the increase of pore pressure when the confining pressure is kept constant. This indicates that the pore pressure has a significant cleavage effect on the crack tip inside the specimen and promotes the expansion of the rock cracks and the penetration of the rock cracks so that the damage deterioration is intensified, which the macroscopic shows a decrease in the strength of the rock. Furthermore, the strain value of the damage threshold decreases with the increase of confining pressure and pore pressure under the common action of confining pressure and pore pressure, which indicates that the effective confining pressure of the rock is decreasing, which will then accelerate the damage failure of the rock specimen.

7 Conclusion

  1. From the analysis of the mechanical properties of sandstone under hydro-mechanical coupling, the peak strength, effective peak strength, residual strength, effective residual strength, effective normal stress, effective shear strength, elastic modulus, and fracture angle are positively correlated with confining pressure, and negatively correlated with pore pressure. However, Poisson’s ratio exhibits the opposite relationship.

  2. To establish a constitutive model for sandstone damage, considering factors such as pore pressure, damage threshold, and residual strength. The effectiveness of the model was validated using experimental data, and it accurately assesses the stress–strain relationship of sandstone under the coupling effect of water and stress.

  3. Regarding the analysis of the sandstone damage process, the damage starts at a certain stress level and increases rapidly as it approaches the peak strength. The starting point of damage varies under different confining pressures and pore pressures, with the stress value of the starting point increasing with the increase of confining pressure, and the strain value of the starting point decreasing with the increase of pore pressure. This indicates that pore pressure tends to induce rock damage.

  4. It was found that the rock becomes more brittle and less ductile as the value of parameter m increases, while the average macroscopic strength of the rock increases with the increase of parameter F 0. The value of parameter m gradually decreases with the increase of confining pressure, while the value of parameter F 0 increases with the increase of confining pressure.

The model can predict the stress–strain relationship of sandstone under hydro-mechanical coupling, evaluate its mechanical properties and stability, and thus has wide application value in geotechnical engineering (improving engineering safety), petroleum engineering (optimizing oil and gas extraction schemes), hydrogeology (optimizing underground water resource management), and other fields. It can provide a scientific basis for decision-making in related areas.

Acknowledgments

This research was funded by the National Natural Science Foundation of China (No. 52274118) and the Postgraduate Scientific Research Innovation Project of Hunan Province (CX20221042). All authors have read and agreed to the published version of the manuscript.

  1. Author contributions: Yanlin Zhao and Qiang Liu – conceptualization; Qiang Liu, Jian Liao Tao Tan, Xiaguang Wang, Yang Li, and Zhe Tan – investigation; Qiang Liu – writing – original draft.

  2. Conflict of interest: All the authors state that there is no conflict of interest.

  3. Ethical approval: The conducted research is not related to either human or animal use.

  4. Data availability statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

[1] Goodman RE. Introduction to Rock Mechanics. Press: Wiley; 1989.Search in Google Scholar

[2] Cai MF, He MC, Liu DY. Rock Mechanics and Engineering. Beijing: Science Press; 2002.Search in Google Scholar

[3] Chen W, Wan W, Zhao YL, Peng WQ. Experimental study of the crack predominance of rock-like material containing parallel double fissures under uniaxial compression. Sustainability. 2020;12(12):1–19.10.3390/su12125188Search in Google Scholar

[4] Wu H, Jia Q, Wang WJ, Zhang N, Zhao YM. Experimental test on nonuniform deformation in the tilted strata of a deep coal mine. Sustainability. 2021;13:1–14.10.3390/su132313280Search in Google Scholar

[5] Yuan ZG, Zhao JT, Li SQ, Jiang ZH, Huang F. A unified solution for surrounding rock of roadway considering seepage, dilatancy, strain-softening and intermediate principal stress. Sustainability. 2022;14(13):8099.10.3390/su14138099Search in Google Scholar

[6] Zhao YL, Zhang LY, Wang WJ, Liu Q, Tang LM, Cheng GM. Experimental study on shear behavior and a revised shear strength model for infilled rock joints. Int J Geomech. 2020;20(9):4020141.1–4020141.17.10.1061/(ASCE)GM.1943-5622.0001781Search in Google Scholar

[7] Zhao YL, Liu Q, Zhang CS, Liao J, Lin H, Wang YX. Coupled seepage-damage effect in fractured rock masses: model development and a case study. Int J Rock Mech Min Sci. 2021;144:104822.10.1016/j.ijrmms.2021.104822Search in Google Scholar

[8] Zhao YL, Zhang LY, Wang WJ, Tang JZ, Lin H, Wan W. Transient pulse test and morphological analysis of single rock fractures. Int J Rock Mech Min Sci. 2017;91:139–54.10.1016/j.ijrmms.2016.11.016Search in Google Scholar

[9] Feng T, Chen HS, Wang KL, Nie YP, Zhang XB, Mo HW. Assessment of underground soil loss via the tapering grikes on limestone hillslopes. Agric Ecosyst Environ. 2020;297:106935.10.1016/j.agee.2020.106935Search in Google Scholar

[10] Peng YX, Liu GJ, Wu L, Zou QJ, Liu YS, Zhang C. Comparative study on tunnel blast-induced vibration for the underground cavern group. Environ Earth Sci. 2021;80(2):68.10.1007/s12665-020-09362-zSearch in Google Scholar

[11] Zheng CS, Jiang BY, Xue S, Chen ZW, Li H. Coalbed methane emissions and drainage methods in underground mining for mining safety and environmental benefits: A review. Process Saf Environ. 2019;127:103–24.10.1016/j.psep.2019.05.010Search in Google Scholar

[12] Zhao YL, Liu Q, Lin H, Wang YX, Tang WY, Liao J, et al. A review of hydromechanical coupling tests, theoretical and numerical analyses in rock materials. Water. 2023;15(13):2309.10.3390/w15132309Search in Google Scholar

[13] Louis C. Rock hydraulics. Muller Led. Wein-New York: Springer; 1974.Search in Google Scholar

[14] Liu JH, Zhao YL, Tan T, Zhang LY, Zhu ST, Xu FY. Evolution and modeling of mine water inflow and hazard characteristics in southern coalfields of China: A case of Meitanba mine. Int J Min Sci Technol. 2022;32(3):513–24.10.1016/j.ijmst.2022.04.001Search in Google Scholar

[15] Zhao YL, Wang YX, Wang WJ, Tang LM, Liu Q, Cheng GM. Modeling of rheological fracture behavior of rock cracks subjected to hydraulic pressure and far-field stresses. Theor Appl Fract Mech. 2016;73(9):2039–51.Search in Google Scholar

[16] Li YC, Hu XX, Ren BZ. Treatment of antimony mine drainage: challenges and opportunities with special emphasis on mineral adsorption and sulfate reducing bacteria. Water Sci Technol. 2016;2039–51.10.2166/wst.2016.044Search in Google Scholar PubMed

[17] Zhang HQ, He YN, Tang CA, Ahmad B, Han LJ. Application of an improved flow-stress-damage model to the criticality assessment of water inrush in a mine: a case study. Rock Mech Rock Eng. 2009;42(6):911–30.10.1007/s00603-008-0004-2Search in Google Scholar

[18] Xue TY, Xue XL, Long SC, Chen QN, Lu SD, Zeng CF. Effect of pre-existing underground structures on groundwater flow and strata movement induced by dewatering and excavation. Water. 2023;15(4):814.10.3390/w15040814Search in Google Scholar

[19] Zhao YL, Luo SL, Wang YX, Wang WJ, Zhang LY, Wan W. Numerical analysis of karst water inrush and a criterion for establishing the width of water-resistant rock pillars. Mine Water Environ. 2017;36:508–19.10.1007/s10230-017-0438-4Search in Google Scholar

[20] Yu WJ, Li K, Liu Z, An BF, Wang P, Wu H. Mechanical characteristics and deformation control of surrounding rock in weakly cemented siltstone. Environ Earth Sci. 2021;80:337.10.1007/s12665-021-09626-2Search in Google Scholar

[21] Chen ZQ, Ma C, Li TB, He C. Experimental investigation of the failure mechanism of deep granite under high seepage water pressure and strong unloading effect. Acta Geotech. 2022;17(11):5009–30.10.1007/s11440-022-01665-8Search in Google Scholar

[22] Tan T, Zhao YL, Zhao X, Chang L, Ren S. Mechanical properties of sandstone under hydro-mechanical coupling. Appl Rheol. 2022;32(1):8–21.10.1515/arh-2022-0120Search in Google Scholar

[23] Liu Q, Zhao YL, Tang LM, Liao J, Wang XG, Tan T, et al. Mechanical characteristics of single cracked limestone in compression-shear fracture under hydro-mechanical coupling. Theor Appl Fract Mech. 2022;119:103371.10.1016/j.tafmec.2022.103371Search in Google Scholar

[24] Fan SY, Song ZP, Wang HZ, Zhang YW, Zhang Q. Influence of the combined action of water and axial pressure on the microscopic damage and mechanical properties of limestone. Geoenergy Sci Eng. 2023;228:212027.10.1016/j.geoen.2023.212027Search in Google Scholar

[25] Fan SY, Song ZP, Xu T, Zhang YW. Investigation of the microstructure damage and mechanical properties evolution of limestone subjected to high-pressure water. Constr Build Mater. 2022;316:125871.10.1016/j.conbuildmat.2021.125871Search in Google Scholar

[26] Song ZP, Wang T, Wang JB, Xiao KH, Yang TT. Uniaxial compression mechanical properties and damage constitutive model of limestone under osmotic pressure. Int J Damage Mech. 2022;31(4):557–81.10.1177/10567895211045430Search in Google Scholar

[27] Zhou HW, Wang ZH, Ren WG, Liu WG, Liu ZL, Liu JF. Acoustic emission based mechanical behaviors of Beishan granite under conventional triaxial compression and hydro-mechanical coupling tests. Int J Rock Mech Min Sci. 2019;123:104125.10.1016/j.ijrmms.2019.104125Search in Google Scholar

[28] Wang SS, Xu WY, Wang W. Experimental and numerical investigations on hydro-mechanical properties of saturated fine-grained sandstone. Int J Rock Mech Min Sci. 2020;127:104222.10.1016/j.ijrmms.2020.104222Search in Google Scholar

[29] Zhao YL, Liu JH, Zhang CS, Zhang HQ, Liao J, Zhu ST, et al. Mechanical behavior of sandstone during post-peak cyclic loading and unloading under hydromechanical coupling. Int J Min Sci Technol. 2023;33:927–47. 10.1016/j.ijmst.2023.05.004.Search in Google Scholar

[30] Jia CJ, Xu WY, Wang HL, Zhang Q. Experimental investigation of the mechanical and permeability characteristics of volcanic breccia. Bull Eng Geol Environ. 2021;80:599–610.10.1007/s10064-020-01949-xSearch in Google Scholar

[31] Alam A, Niioka M, Fujii Y, Fukuda D, Kodama J. Effects of confining pressure on the permeability of three rock types under compression. Int J Rock Mech Min Sci. 2014;65:49–61.10.1016/j.ijrmms.2013.11.006Search in Google Scholar

[32] Chen X, Yu J, Tang CA, Li H, Wang SY. Experimental and numerical investigation of permeability evolution with damage of sandstone under triaxial compression. Rock Mech Rock Eng. 2017;50(6):1529–49.10.1007/s00603-017-1169-3Search in Google Scholar

[33] Jiang T, Shao JF, Xu WY, Zhou CB. Experimental investigation and micromechanical analysis of damage and permeability variation in brittle rocks. Int J Rock Mech Min Sci. 2010;47(5):703–13.10.1016/j.ijrmms.2010.05.003Search in Google Scholar

[34] Xu P, Yang SQ. Permeability evolution of sandstone under short-term and long-term triaxial compression. Int J Rock Mech Min Sci. 2016;85:152–64.10.1016/j.ijrmms.2016.03.016Search in Google Scholar

[35] Zhang R, Jiang Z, Sun Q, Zhu S. The relationship between the deformation mechanism and permeability on brittle rock. Nat Hazards. 2013;66:1179–87.10.1007/s11069-012-0543-4Search in Google Scholar

[36] Zhao YL, Tang JZ, Chen Y, Zhang LY, Wang WJ, Liao JP. Hydromechanical coupling tests for mechanical and permeability characteristics of fractured limestone in complete stress-strain process. Environ Earth Sci. 2017;76(24):1–18.10.1007/s12665-016-6322-xSearch in Google Scholar

[37] Lemaitre J. How to use damage mechanics. Nucl Eng Des. 1984;80(2):233–45.10.1016/0029-5493(84)90169-9Search in Google Scholar

[38] Lemaitre J. A continuous damage mechanics model for ductile fracture. J Eng Mater Tech. 1985;107(1):83–9.10.1115/1.3225775Search in Google Scholar

[39] Krajcinovic D, Manuel AGS. Statistical aspects of the continuous damage theory. Int J Solids Struct. 1982;18(7):551–62.10.1016/0020-7683(82)90039-7Search in Google Scholar

[40] Deng J, Gu D. On a statistical damage constitutive model for rock materials. Comput Geosci. 2011;37(2):122–8.10.1016/j.cageo.2010.05.018Search in Google Scholar

[41] Chen K. Constitutive model of rock triaxial damage based on the rock strength statistics. Int J Damage Mech. 2020;29(10):1487–511.10.1177/1056789520923720Search in Google Scholar

[42] Cao WG, Li X, Zhao H. Damage constitutive model for strain-softening rock based on normal distribution and its parameter determination. J Cent South Univ Technol. 2007;14(5):719–24.10.1007/s11771-007-0137-6Search in Google Scholar

[43] Zhang H, Chen S, Wang L, Cheng S. A dimensionless rock damage constitutive model under an improved harris distribution. Soil Mech Found Eng. 2022;59:338–46.10.1007/s11204-022-09820-9Search in Google Scholar

[44] Tang CA. Catastrophe in the rock failure process. Beijing: China Coal Industry Publishing House; 1993.Search in Google Scholar

[45] Cao WG, Zhao H, Li X, Zhang YJ. Statistical damage model with strain softening and hardening for rocks under the influence of voids and volume changes. Can Geotech J. 2010;47(8):857–71.10.1139/T09-148Search in Google Scholar

[46] Zhao H, Shi CJ, Zhao MH, Li XB. Statistical damage constitutive model for rocks considering residual strength. Int J Geomech. 2016;17(1):04016033.10.1061/(ASCE)GM.1943-5622.0000680Search in Google Scholar

[47] Pourhosseini O, Shabanimashcool M. Development of an elasto-plastic constitutive model for intact rocks. Int J Rock Mech Min Sci. 2014;66:1–12.10.1016/j.ijrmms.2013.11.010Search in Google Scholar

[48] Li X, Cao WG, Su YH. A statistical damage constitutive model for softening behavior of rocks. Eng Geol. 2012;143–144:1–17.10.1016/j.enggeo.2012.05.005Search in Google Scholar

[49] Chen YF, Lin H, Wang YX, Xie SJ, Yong WX. Statistical damage constitutive model based on the Hoek–Brown criterion. Archiv Civ Mech Eng. 2021;21(3):117.10.1007/s43452-021-00270-ySearch in Google Scholar

[50] Shen PW, Tang HM, Ning YB, Xia D. A damage mechanics based on the constitutive model for strain-softening rocks. Eng Fract Mech. 2019;216:106521.10.1016/j.engfracmech.2019.106521Search in Google Scholar

[51] Zheng Z, Xu HY, Wang W, Mei GX, Liu ZB, Zheng H, et al. Hydro-mechanical coupling characteristics and damage constitutive model of low-permeability granite under triaxial compression. Int J Damage Mech. 2023;32(2):235–61.10.1177/10567895221132868Search in Google Scholar

[52] Biot Maurice A. General theory of three-dimensional consolidation. J Appl Phys. 1941;12(2):155–64.10.1063/1.1712886Search in Google Scholar

[53] Bishop AW. Progressive failure with special reference to the mechanism causing it. Oslo: Proceedings of the Geotechnical Conference; 1967. p. 142–50.Search in Google Scholar
Received: 2023-09-11
Revised: 2023-10-29
Accepted: 2023-11-07
Published Online: 2023-12-14

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  28. Seismic response and damage mechanism of tunnel lining in sensitive environment of soft rock stratum
  29. Correlation analysis of physical and mechanical parameters of inland fluvial-lacustrine soft soil based on different survey techniques
  30. An effective method for real-time estimation of slope stability with numerical back analysis based on particle swarm optimization
  31. An efficient method for computing slope reliability calculation based on rigorous limit equilibrium
  32. Mechanical behavior of a new similar material for weathered limestone in karst area: An experimental investigation
  33. Semi-analytical method for solving stresses in slope under general loading conditions
  34. Study on the risk of seepage field of Qiantang River underground space excavated in water-rich rheological rock area
  35. Numerical analysis of the impact of excavation for undercrossing Yellow River tunnel on adjacent bridge foundations
  36. Deformation rules of deep foundation pit of a subway station in Lanzhou collapsible loess stratum
  37. Development of fiber compound foaming agent and experimental study on application performance of foamed lightweight soil
  38. Monitoring and numerical simulation analysis of a pit-in-pit excavation of the first branch line of Lanzhou Metro
  39. CT measurement of damage characteristics of meso-structure of freeze-thawed granite in cold regions and preliminary exploration of its mechanical behavior during a single freeze-thaw process
https://doi.org/10.1515/arh-2023-0112
Keywords for this article
sandstone; triaxial compression test; strength characteristics; hydro-mechanical coupling; damage constitutive model
Creative Commons
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