A damage-based analytical model to evaluate seepage pressure effect on rock macro mechanical behaviors from the perspective of micro-fracture

Abstract

The deterioration of rock material properties induced by seepage pressure is a serious danger to the stability of geotechnical engineering. The formation and propagation of microcracks is the primary cause of rock macro failure. This work proposes an damage-based analytical model to assess the impact of seepage pressure on the macro mechanical behaviors of rocks from the standpoint of micro fracture. A wing crack model serves as the foundation for the analytical model. This model has taken into account the impact of seepage pressure on the initiation and growth of wing cracks. The constitutive relation is constructed based on the equivalency connection of damage defined by strain and wing crack length. A comparison between the analytical results and the reported experimental data confirms the reasonableness of the analytical model. Investigations are conducted on the relationship between the macro mechanical behavior of rocks and micro fracture under various seepage pressures, confining pressures, and microscopic parameters. The findings demonstrate that the cracks growth is initially steady before becoming unstable. The growing process of wing cracks stops when they connect with one another, and friction between the crack surfaces takes over. The initiation and growth of wing cracks may be aided by the seepage pressure. As the wing crack propagates, the seepage pressure effect initially increases, then decreases, and eventually has practically no impact. The influence of seepage pressure on rock macro mechanical behavior is that with seepage pressure increasing, the initiation stress and peak stress decrease, but the residual stress is basically a constant. The rock micro fracture process is significantly influenced by confining pressures and microscopic factors, which in turn affect the macro mechanical behavior. The study’s findings offer a micro fracture foundation for comprehending how seepage pressure affects the macro mechanical behaviors of rocks.

Keywords

Analytical model wing crack model seepage pressure micro-fracture damage model

Introduction

Due to long-term geological activity, there are a lot of cracks in the rock. The seepage pressure induced by water or gas will drive the initiation and growth of cracks, and then weaken the rock mechanical properties. Therefore, the seepage pressure significantly affects the stability of geotechnical engineering, such as dam (Rice and Duncan, 2010; Wu et al., 2016), slope (Saada et al., 2012; S. Wang et al., 2021; Yu et al., 2022) and deep buried tunnel (Perazzelli et al., 2014; Y. Wang et al., 2021; Wu et al., 2023). The investigation on mechanical model to reflect the seepage pressure effect on rock strength, deformation and failure is necessary for the design and analysis of geotechnical engineering (Liu et al., 2019; Wang and Zhang, 2020; Zhou et al., 2020).

The seepage pressure can reduce the normal stress on crack surface, and make the rock strength decrease and deformation increase. The effective stress principle was first proposed by Terzaghi (1923) to describe the quantitative relationship between effective stress, external stress and seepage pressure. However, this principle is only applicable in saturated porous media. The concept of effective stress coefficient was proposed by Biot (1941) to extended the effective stress principle in the application of unsaturated porous media. Brace and Martin (1968) verified the applicability of effective stress principle in low porosity rocks through laboratory test. Zimmerman et al. (1986) developed a micromechanical model based on the elastic theory and the effective stress principle to investigate the volumetric behavior of porous rocks under compression. However, this micromechanical model is only applicable in elastic process. The underlying reason is that the weakening of material mechanical properties induced by crack growth has not been considered in this model.

The damage mechanics is an effective method to reflect the influence of micro fracture on the weakening of macro mechanical behavior (Hou et al., 2019; Li et al., 2019; J. Wang et al., 2021). Recently, the seepage pressure effect has been embedded into damage models to investigate the deformation and strength characteristics of rocks (Jiang et al., 2021; Xiao et al., 2020; Zhao et al., 2019). The nonlinear stress-strain curve before peak stress and peak strength can be obtained by these damage models. However, the residual stress after rock failure can not be reflected effectively. The phenomenological nature of the these damage models is the main explanation. These damage models have not taken into account the actual micro fracture, which includes the crack’s initiation, growth, and interaction as well as the friction between its surfaces.

Under the framework of fracture mechanics, Ashby and Hallam (1986) and Ashby and Sammis (1990) proposed a wing crack model to consider the influence of micro-fracture process on the macro mechanical behaviors of brittle material under compression. The relationship between external force and wing crack length was verified by the experiments (Ashby and Hallam, 1986). The wing crack model has been improved and adopted in creep (Brantut et al., 2012; Li and Shao, 2016), high strain loading (Bhat et al., 2012; Li et al., 2020; Ravichandran and Subhash, 1995), shear fracture (Li et al., 2018; Moore and Lockner, 1995) and so on. However, the investigation about the wing crack model to consider the seepage pressure effect is rare.

It makes sense to investigate at how seepage pressure affects a material’s macromechanical characteristics from the standpoint of microfracture. In order to assess the impact of seepage pressure on the macro mechanical behaviors of rocks from the standpoint of micro fracture, an analytical model based on the wing crack model is presented in this study. This model takes into account the impact of seepage pressure on the initiation and growth of wing cracks.

Based on the equivalence relation of damage defined by strain (Wong et al., 2006) and wing crack length (Ashby and Hallam, 1986), the constitutive relation is derived. The stress-strain curves calculated by the constitutive relation can effectively reflect the main characteristics of experimental results (Xiao et al., 2020) under the conditions of various seepage pressures. Due to the crack surface friction being considered in this model, the residual stress after rock failure can be reflected effectively. The influences of in-situ stress, seepage pressure and microscopic parameters on the rock macro mechanical behavior and micro-fracture are investigated, respectively. Finally, the relation between rock macro mechanical behavior and micro-fracture is discussed. The results show that the micro-fracture process can well explain the rock macro mechanical behavior by the analytical model.

Theoretical model

An analytical model to incorporate seepage pressure effect in rock failure is proposed based on a wing crack model (Ashby and Hallam, 1986; Ashby and Sammis, 1990), as shown in Figure 1. The model is under biaxial stress state, i.e., the axial stress σ₁ and the confining pressure σ₃. When the stress is tensile, it is treated as a positive value. Conversely, when the stress is compressive, it is treated as a negative value. In this model, the microcracks are assumed to be uniformly distributed. The initial length of cracks is 2a, and the included angle between the tangent direction of cracks and the direction of axial stress is φ. The distance between two adjacent cracks is S. The seepage pressure is P.

Wing crack initiation

The original wing crack model (Ashby and Hallam, 1986; Ashby and Sammis, 1990) only considers the effect of in-situ stress, i.e., σ₁ and σ₃. Based on the original wing crack model, the seepage pressure effect on rock failure is considered. Because the effect of seepage pressure on the tangential stress of crack is small (Eid et al., 2015; Liu et al., 2014), only the effect of seepage pressure on the normal stress of crack is considered in the proposed model. Considering the effect of remote stress field and seepage pressure simultaneously, as shown in Figure 2, the tangential and normal stress on crack surface can be expressed as

\begin{array}{l} τ = \frac{σ_{3} - σ_{1}}{2} \sin 2 φ \\ σ_{n} = \frac{σ_{3} + σ_{1}}{2} + \frac{σ_{3} - σ_{1}}{2} \cos 2 φ + P \end{array}

(1)

where τ is the tangential stress on crack surface, σ_n is the normal stress on crack surface, P is the seepage pressure, whose sign definition is the same with the in-situ stress, i.e., its value is negative when it is compressive.

A normal stress will provide a friction force on the crack surface to stop it from sliding, whereas tangential stress causes the crack surface to move. The effective sliding stress is

τ^{'} = τ + μ σ_{n}

(2)

where

τ^{'}

is the effective sliding stress, μ is the friction coefficient of crack surface.

The tensile stress near the initial crack tip is

σ_{θ θ} = \frac{3}{2} \frac{τ^{'} \sqrt{π a}}{\sqrt{2 π r}} \sin θ \cos \frac{θ}{2}

(3)

where σ_θθ is the tensile stress, r is the distance between measuring point and crack tip, θ is the angle of monitoring point deviating from the tangential direction of crack surface.

According to Cotterell and Rice (1980), the mode I stress intensity factor near the initial crack tip can be approximated by $σ_{θ θ} \sqrt{π l}$ , where l is the length of a very small wing crack. Take the measuring point at r = l/2, then

K_{I} = \frac{3}{2} τ^{'} \sqrt{π a} \sin θ \cos \frac{θ}{2}

(4)

where K_I is the mode I stress intensity factor.

At the crack tip, the crack will propagate along the direction with the highest K_I. Take the derivative of K_I with respect to θ, it is found that when θ = 70.5°, K_I is maximum. The maximal K_I near the initial crack tip is

K_{I} = \frac{2}{\sqrt{3}} τ^{'} \sqrt{π a}

(5)

Substitute equations (1) and (2) into equation (5), and get

K_{I} = \frac{2}{\sqrt{3}} (\frac{σ_{3} - σ_{1}}{2} \sin 2 φ + μ \cdot (\frac{σ_{3} + σ_{1}}{2} + \frac{σ_{3} - σ_{1}}{2} \cos 2 φ + P)) \sqrt{π a}

(6)

When the K_I reaches the mode I fracture toughness K_IC, the wing crack will initiate. The critical axial stress for wing crack initiation, σ_i, is

σ_{i} = \frac{A_{1}}{A_{2}} σ_{3} + \frac{2 μ \cdot P \sqrt{π a} - \sqrt{3} K_{IC}}{A_{2} \sqrt{π a}}

(7)

where

\begin{matrix} A_{1} = μ \cos 2 φ + \sin 2 φ + μ \\ A_{2} = μ \cos 2 φ + \sin 2 φ - μ \end{matrix}

(8)

Under a certain in-situ stress condition, the critical seepage pressure for wing crack initiation, P_i, is

P_{i} = \frac{1}{μ} (\frac{\sqrt{3}}{2} \frac{K_{IC}}{\sqrt{π a}} - \frac{σ_{3} - σ_{1}}{2} \sin 2 φ) - (\frac{σ_{3} + σ_{1}}{2} + \frac{σ_{3} - σ_{1}}{2} \cos 2 φ)

(9)

Wing crack growth and interaction

Based on the experimental observations, Ashby and Hallam (1986) found that once the wing crack initiates, it would basically grow along the direction of σ₁. In the process of wing crack growth, the adjacent wing cracks will interact. There are four forces influencing the growth and interaction of wing cracks, namely the effective sliding stress on initial crack surface $τ^{'}$ , the confining pressure σ₃, the seepage pressure P and the interaction force σ_int of adjacent wing cracks.

The effective force F₃ to drive wing crack growth induced by $τ^{'}$ is its component on the σ₃ direction, which is

F_{3} = (τ + μ σ_{n}) \cdot 2 a \cdot \sin φ

(10)

The contribution of F₃ to K_I at the tip of wing crack is

{(K_{I})}_{1} = \frac{F_{3}}{\sqrt{π (l + β a)}}

(11)

where (l + βa) is the effective crack length, β is a coefficient to overcome the singularity of wing crack at the initial crack tip.

The σ₃ not only acts on the initial crack surface but also on the wing crack surface. Its contribution to K_I at the crack tip (Tada et al., 1973) is

{(K_{I})}_{2} = σ_{3} \sqrt{π l}

(12)

Once the wing crack initiation, the seepage pressure will act on the wing crack surface. The contribution of P to K_I at the tip of wing crack is

{(K_{I})}_{3} = P \sqrt{π l}

(13)

The distance between two adjacent cracks, S, is

S = \frac{1}{\sqrt{N_{A}}}

(14)

where N_A is the crack number per unit area.

The interaction force σ_int of adjacent wing cracks is

σ_{int} = \frac{F_{3}}{S - 2 (l + a \cdot \cos φ)}

(15)

The contribution of σ_int to K_I at the tip of wing crack is

{(K_{I})}_{4} = σ_{int} \sqrt{π l}

(16)

The K_I at the tip of wing crack is the sum of four contributions

K_{I} = {(K_{I})}_{1} + {(K_{I})}_{2} + {(K_{I})}_{3} + {(K_{I})}_{4}

(17)

Substitute equations (11) to (13) and (16) into equation (17), and get

K_{I} = σ_{1} C_{2} \frac{B_{1} \sqrt{π l} + B_{2}}{B_{1} B_{2}} + σ_{3} \frac{C_{3} (B_{1} \sqrt{π l} + B_{2}) + B_{1} B_{2} \sqrt{π l}}{B_{1} B_{2}} + 2 P C_{1} \frac{B_{1} \sqrt{π l} + B_{2}}{B_{1} B_{2}}

(18)

where

\begin{array}{l} B_{1} = 2 \sqrt{π (l + β a)} \\ B_{2} = 2 \cdot (S - 2 (l + a \cdot \cos φ)) \\ C_{1} = μ \cdot 2 a \cdot \sin φ \\ C_{2} = (1 - \cos 2 φ) \cdot C_{1} - \sin 2 φ \\ C_{3} = (1 + \cos 2 φ) \cdot C_{1} + \sin 2 φ \end{array}

(19)

When K_I reaches K_IC, the wing crack will grow. The critical axial stress for wing crack growth, σ_g, is

σ_{g} = - \frac{σ_{3} \cdot (C_{3} (B_{1} \sqrt{π l} + B_{2}) + B_{1} B_{2} \sqrt{π l}) + P (2 C_{1} (B_{1} \sqrt{π l} + B_{2}) + B_{2}) - B_{1} B_{2} K_{I C}}{C_{2} (B_{1} \sqrt{π l} + B_{2})}

(20)

Under a certain in-situ stress condition, the critical seepage pressure for wing crack growth, P_g, is

P_{g} = \frac{K_{I C} B_{1} B_{2} - σ_{1} C_{2} (B_{1} \sqrt{π l} + B_{2}) - σ_{3} (C_{3} (B_{1} \sqrt{π l} + B_{2}) + B_{1} B_{2} \sqrt{π l})}{2 C_{1} (B_{1} \sqrt{π l} + B_{2}) + B_{2}}

(21)

Constitutive relation

Damage is the degradation of the mechanical characteristics of a material brought on by the formation and propagation of microcracks. Damage factor is a quantitative mechanical parameter that describes damage, which can be defined by strain, energy, crack length, and so on. A damage factor is defined by Wong et al. (2006) based the axial strain to describe the rock damage evolution process, which can be expressed as

D = 1 - \exp [- {(ε / ε_{0})}^{m}]

(22)

where D is the damage factor, m and ε₀ are material parameters, ε is axial strain.

On the other hand, a another damage factor is defined by Ashby and Hallam (1986) based the wing crack length, which can be expressed as

D = π {(l + λ a)}^{2} N_{A}

(23)

where λ = cosφ.

Determining damage based on axial strain and wing crack length is like narrating the same event from two different perspectives. Therefore, the damage factors in equation (22) and equation (23) are equivalent. Combining equations (22) and (23), and the relation between wing crack length and axial strain can be obtained

ε = ε_{0} {[- \ln (1 - \frac{{(l + λ a)}^{2} D_{0}}{λ^{2} a^{2}})]}^{\frac{1}{m}}

(24)

l = λ a [D_{0}^{- \frac{1}{2}} {(1 - \exp [- {(ε / ε_{0})}^{m}])}^{\frac{1}{2}} - 1]

(25)

where D₀ is the initial damage factor, D₀ = π(λa)²N_A.

Substituting equation (25) into equation (20), the constitutive relation can be obtained

σ_{1} = - \frac{σ_{3} \cdot (C_{3} (B_{1} \sqrt{π l} + B_{2}) + B_{1} B_{2} J_{1}) + P (2 C_{1} (B_{1} J_{1} + B_{2}) + B_{2}) - B_{1} B_{2} K_{I C}}{C_{2} (B_{1} J_{1} + B_{2})}

(26)

where

\begin{array}{l} J_{1} = {(π α a J_{2})}^{\frac{1}{2}} \\ J_{2} = D_{0}^{- \frac{1}{2}} {(1 - \exp [- {(ε / ε_{0})}^{m}])}^{\frac{1}{2}} - 1 \end{array}

(27)

Experimental verification

To verify the ability of the proposed analytical model to reflect the seepage pressure effect on rock macro mechanical behavior, a seepage pressure test (Zhao et al., 2017) is analyzed. In this test, σ₃ = 17 MPa, P = 2 MPa, 8 MPa and 14 MPa. The microscopic parameters of red sandstone in the seepage pressure test are calibrated as K_IC = 1.27 MPa·m^1/2, μ = 0.31, a = 0.33 mm, φ = 46.4°, S = 5.2 mm, m = 30 and ε₀ = 0.017.

The fundamental features of actual stress-strain curves may be effectively reflected in theoretical results, as demonstrated by the comparison of theoretical stress-strain curves with experimental data (Figure 3). The theoretical peak stress and residual stress closely match the experimental results. It also should be noticed that there are some differences between the theoretical results and experimental one. The underlying reason is that actual crack arrangement is random, i.e., the crack length, crack angle and the distance between two adjacent cracks are all random. Therefore, under the action of external load, the initiation and growth processes of different cracks are not synchronous. However, the crack arrangement is assumed to be uniform in this analytical model, which leads to the simultaneous initiation and growth processes of different cracks. Nevertheless, this model can reflect the main characteristics of experimental results, and is helpful to understand the seepage pressure effect on rock macro mechanical behaviors from the perspective of micro-fracture.

Analysis of wing crack initiation and growth

The initiation and growth of wing cracks is determined by the in-situ stress, seepage pressure and microscopic parameters of the micro-cracks. The interaction of these influence factors is investigated in this section. The microscopic parameters calibrated by the seepage pressure test (Zhao et al., 2017) is adopted in this analysis.

Critical in-situ stress for wing crack initiation

The critical in-situ stress for wing crack initiation can be obtained through equation (7). Firstly, the influence of P and σ₃ on σ_i is analyzed. The analytical results (Figure 4(a)) show that under the conditions different P, with σ₃ increasing, σ_i increases linearly. Under the conditions different σ₃, as shown in Figure 4(b), with P increasing, σ_i decreases linearly. It can be concluded that the seepage pressure can promote the wing crack initiation, while the confining pressure can restrain wing crack initiation.

To further investigate the interaction of microscopic parameters and P on wing crack initiation, the influences of microscopic parameters on σ_i under the conditions of different P are analyzed. Firstly, the influence of initial crack length is analyzed. In these cases, a changes from 0.1 mm to 0.5 mm with the increment of 0.1 mm. The analytical results (Figure 5(a)) show that for a certain a, with P increasing, σ_i decreases linearly. Under the condition of a certain P, with a increasing, σ_i decreases nonlinearly, and the decreasing rate is smaller. Then, the influence of friction coefficient is analyzed. In these cases, μ changes from 0.1 to 0.5 with the increment of 0.1. The analytical results (Figure 5(b)) show that for a certain μ, with P increasing, σ_i decreases linearly. When μ is larger, the decreasing rate of σ_i is larger. Under the conditions of different P, the influences of μ on σ_i are different. There is a critical P, which is about 5.5 MPa. When P is smaller than the critical value, with μ increasing, σ_i increases linearly. When P is larger than the critical value, with μ increasing, σ_i decreases linearly. Finally, the influence of initial crack angle is analyzed. In these cases, φ changes from 15° to 75° with the increment of 15°. The analytical results (Figure 5(c)) show that for a certain φ, with P increasing, σ_i decreases linearly. When φ is closer to 45°, the decreasing rate of σ_i is smaller. Under the condition of a certain P, with φ increasing from 15° to 75°, σ_i decreases firstly and then increases. When φ is closer to 45°, σ_i is smaller.

Critical seepage pressure for wing crack initiation

To investigate the critical seepage pressure for wing crack initiation under the conditions of different in-situ stresses, σ₁ changes from 0 MPa to 120 MPa with the increment of 10 MPa, and σ₃ changes from 0 MPa to 40 MPa with the increment of 5 MPa. The critical seepage pressure for wing crack initiation can be obtained through equation (9). The analytical results (Figure 6) show that under the condition of a certain σ₃, when σ₁ is large enough, the critical seepage pressure for wing crack initiation, P_i, is 0 MPa. This means that under these in-situ stress conditions, the in-situ stress can drive wing cracks initiation without considering the seepage pressure effect. Under the condition that the in-situ stress cannot drive wing cracks initiation, only when the seepage pressure reaches the critical value, the wing cracks initiates. Under the condition of a certain σ₃, with σ₁ increasing, P_i decreases linearly. Under the condition of a certain σ₁, with σ₃ increasing, P_i increases linearly.

The influences of microscopic parameters on P_i under different in-situ stress conditions are investigated. For the cases with different σ₁, keep σ₃ = 20 MPa, σ₁ changes from 0 MPa to 120 MPa with the increment of 10 MPa. For the cases with different σ₃, keep σ₁ = 60 MPa, σ₃ changes from 0 MPa to 20 MPa with the increment of 2 MPa.

Firstly, the influence of initial crack length is investigated. In these cases, a changes from 0.1 mm to 0.5 mm with the increment of 0.1 mm. As shown in Figure 7(a), under the condition of a certain a, with σ₁ increasing, P_i decreases linearly. For cases with different a, the decreasing rates of P_i are the same. Under the condition of a certain σ₁, with a increasing, P_i decreases nonlinearly, and the decreasing rate is smaller. As shown in Figure 7(b), under the condition of a certain a, with σ₃ increasing, P_i increases linearly. For cases with different a, the increasing rates of P_i are the same. Under the condition of a certain σ₃, with a increasing, P_i decreases nonlinearly, and the decreasing rate is smaller.

Then, the influence of friction coefficient is investigated. In these cases, μ changes from 0.1 to 0.5 with the increment of 0.1. As shown in Figure 8(a), under the condition of a certain μ, with σ₁ increasing, P_i decreases linearly. When μ is larger, the decreasing rate of P_i is smaller. Under the condition of a certain σ₁, with μ increasing, P_i decreases nonlinearly, and the decreasing rate is smaller. As shown in Figure 8(b), under the condition of a certain μ, with σ₃ increasing, P_i increases linearly. When μ is larger, the increasing rate of P_i is smaller. Under the condition of a certain σ₃, with μ increasing, P_i decreases nonlinearly, and the decreasing rate is smaller.

Finally, the influence of initial crack angle is investigated. In these cases, φ changes from 15° to 75° with the increment of 15°. As shown in Figure 9(a), under the condition of a certain φ, with σ₁ increasing, P_i decreases linearly. When φ is closer to 45°, the decreasing rate of P_i is larger. There is a critical σ₁, which is about 20 MPa. Under the condition of σ₁ smaller than the critical value, when φ is closer to 45°, P_i is larger. Under the condition of σ₁ larger than the critical value, when φ is closer to 45°, P_i is smaller. As shown in Figure 9(b), under the condition of a certain φ, with σ₃ increasing, P_i increases linearly. When φ is closer to 45°, the increasing rate of P_i is larger. Under the condition of a certain σ₃, when φ is closer to 45°, P_i is smaller.

Critical in-situ stress for wing crack growth

The critical in-situ stress for wing crack growth can be obtained through equation (20). Under the condition of P = 10 MPa, the influence of σ₃ on the relation between σ_g and l are analyzed. As shown in Figure 10, with l increasing, σ_g increases firstly and then decreases. When σ₃ is larger, σ_g is larger. Under the condition of σ₃ = 20 MPa, the influence of P on the relation between σ_g and l are analyzed. The analytical results (Figure 11) show that with P increasing, σ_g is smaller. With l increasing, the influence of P on σ_g becomes larger firstly and then becomes smaller.

The influences of microscopic parameters on the relation between σ_g and l under condition of P = 10 MPa and σ₃ = 20 MPa are investigated. As shown in Figure 12(a), when a is longer, σ_g is smaller. As shown in Figure 12(b), with μ increasing, σ_g increases. As shown in Figure 12(c), when φ is closer to 45°, σ_g is smaller. As shown in Figure 12(d), S has almost no influence on σ_g in the original stage of wing crack growth. With l increasing, the influence of S on σ_g is larger. When S is larger, σ_g is larger.

The influences of microscopic parameters on the peak value of σ_g, σ_P, under the conditions of different seepage pressures are investigated. The analytical results (Figure 13) show that under the conditions of different microscopic parameters, with P increasing, σ_P all decreases linearly. As shown in Figure 13(a), under the condition of a certain P, with a increasing, σ_P decreases linearly. As shown in Figure 13(b), under the condition of a certain P, with μ increasing, σ_P increases linearly. As shown in Figure 13(c), when φ is closer to 45°, σ_P is smaller. As shown in Figure 13(d), under the condition of a certain P, with S increasing, σ_P increases nonlinearly, and the increasing rate is smaller.

Critical seepage pressure for wing crack growth

The critical seepage pressure for wing crack growth can be obtained through equation (21). Under a certain in-situ stress condition, the critical seepage pressure for wing crack growth, P_g, is investigated. Firstly, the influence of σ₁ on P_g is analyzed. In these cases, σ₃ = 20 MPa, σ₁ changes from 25 MPa to 45 MPa with the increment of 5 MPa. As shown in Figure 14(a), with σ₁ increasing, P_g becomes smaller. Then, the influence of σ₃ on P_g is analyzed. In these cases, σ₁ = 35 MPa, σ₃ changes from 10 MPa to 30 MPa with the increment of 5 MPa. As shown in Figure 14(b), with σ₃ increasing, P_g becomes larger. The analytical results (Figure 14) show that σ₁ can promote the wing crack growth, while σ₃ can restrain wing crack growth. With l increasing, P_g increases firstly and then decreases.

Under the condition of σ₁ = 35 MPa and σ₃ = 20 MPa, the influences of microscopic parameters on the relation between P_g and l are investigated. As shown in Figure 15(a), when a is longer, P_g is smaller. As shown in Figure 15(b), with μ increasing, P_g becomes larger. As shown in Figure 15(c), when φ is closer to 45°, P_g is smaller. As shown in Figure 15(d), S has almost no influence on P_g in the original stage of wing crack growth. With l increasing, the influence of S on P_g becomes larger. When S is larger, P_g is larger.

Analysis of constitutive relation

Based on the relationship between wing crack length and axial strain (equation (24) and (25)), the constitutive relation (equation (26)) is derived. In this section, the influences of confining pressure, seepage pressures and microscopic parameters on constitutive relation are investigated.

Influences of confining pressure on constitutive relation

To investigate the influence of confining pressure on constitutive relation, keep P = 10 MPa, the in-situ stress conditions of σ₃ = 0 MPa, 5 MPa, 10 MPa and 20 MPa are analyzed. As shown in Figure 16, with ε increasing, σ₁ increases firstly and then decreases, finally becomes a constant value. With σ₃ increasing, the elastic modulus E, the wing crack initiation stress σ_i, peak stress σ_P and residual stress σ_R all increase linearly. As shown in Figures 4(a) and 10, σ₃ can restrain the wing crack initiation and growth, which leads to the increase of E, σ_i and σ_P. Due to σ₃ making the friction force on crack surface increase, σ_R increases.

Influences of seepage pressures on constitutive relation

To investigate the seepage pressures on constitutive relation, keep σ₃ = 10 MPa, the cases of P = 0 MPa, 5 MPa, 10 MPa and 20 MPa are analyzed. As shown in Figure 17, with P increasing, E, σ_i and σ_P decrease linearly, σ_R are basically a constant value. As shown in Figures 4(b) and 11, P can promote the wing crack initiation and growth, which leads to the decrease of E, σ_i and σ_P. In the later stage of wing crack growth, the influence of P on wing crack growth becomes slight. Therefore, σ_R are basically a constant value under the conditions of various P.

Influence of microscopic parameters on constitutive relation

Under the condition of P = 10 MPa and σ₃ = 10 MPa, the influences of microscopic parameters on constitutive relation are investigated. As shown in Figures 18(a) and 19(a), with a increasing, E, σ_i, σ_P and σ_R all increases. As shown in Figures 18(b) and 19(b), with μ increasing, E is a constant value, σ_i, σ_P and σ_R all increase linearly. As shown in Figures 18(c) and 19(c), when φ increases from 15° to 45°, E decreases obviously, and when φ increases from 45° to 75°, E increases slightly. The influence φ of on σ_i is slight. With φ increasing, σ_P decreases firstly and then increases. With φ increasing, σ_R increases. As shown in Figures 18(d) and 19(d), with S increasing, E and σ_i remain unchanged, σ_P and σ_R increase. As shown in Figures 18(e) and 19(e), with ε₀ increasing, E and σ_i decrease, σ_P and σ_R remain unchanged. As shown in Figures 18(f) and 19(f), with m increasing, E, σ_i, σ_P and σ_R all basically remain unchanged, the stress attenuation rate after peak stress becomes larger.

Relation between rock macro mechanical behaviors and micro-fracture

The macro mechanical behavior of rock is essentially determined by the micro-fracture. Take the seepage pressure test (Xiao et al., 2020) with P = 2 MPa as an example, the relation between rock macro mechanical behavior and micro-fracture is discussed. Displacement loading is adopted in the test, and the strain rate is 1 × 10⁻⁴/s. As shown in Figure 20, the microcracks are assumed to be uniformly distributed in the original state. With the increase of axial strain, the K_I at the crack tip increases. Once the K_I reaches K_IC, the wing crack initiates and grows. In the process of wing crack growth, the wing crack growth is stable firstly and then becomes unstable (Figures 10 and 11). The stable wing crack growth corresponds to the pre peak stage of stress-strain curve, i.e., with the increase of axial strain, axial stress increases. The unstable wing crack growth corresponds to the post peak stage of stress-strain curve, i.e., with the increase of axial strain, axial stress decreases. When the adjacent wing cracks connect with each other, as shown in C of Figure 20(b), the wing crack growth process ends, and the friction between crack surface plays a dominant role. The rock macro mechanical behavior induced by the friction between crack surface is the residual strength. In the process of wing crack growth, the wing crack growth rate increases firstly and then decreases, and the largest value appears in the post peak stage. In rock mechanics tests, acoustic emission (AE) technology is often adopted to monitor the fracture process. By comparing the analytical result of micro-fracture process and the AE characteristics during the failure process of pre-cracked specimen under compression (Liu et al., 2015; Xu et al., 2021), it can be found that the analytical result is consistent with the experimental observation.

From the perspective of micro-fracture, the confining pressure can restrain wing crack growth, while the seepage pressure can promote the wing crack growth. As shown in Figure 10, with σ₃ increasing, σ_g is larger. In the whole process of wing crack growth, σ₃ has a significant influence on σ_g. Therefore, with σ₃ increasing, σ_i, σ_P and σ_R all increase (Figure 16). As shown in Figure 11, with P increasing, σ_g is smaller. With the wing crack growing, the influence P of on σ_g becomes smaller. In the later stage of wing crack growth, P has almost no influence on σ_g. Therefore, as shown in Figure 17, with P increasing, σ_i and σ_P increase, and σ_R basically a constant.

The relation between the rock macro mechanical behavior and micro-fracture under the conditions of different microscopic parameters is also investigated. As shown in 12a, when a is longer, σ_g is smaller. The influence of a on rock macro mechanical behavior is that when a is longer, E, σ_i, σ_P and σ_R all decreases (Figures 18(a) and 19(a)). As shown in 12 b, with μ increasing, σ_g become larger. The corresponding rock macro mechanical behavior is that σ_i, σ_P and σ_R all increase (Figures 18(b) and 19(b)). As shown in Figure 12(c), with the wing crack growing, the influence of φ on σ_g becomes larger firstly and then becomes smaller. When φ is closer to 45°, σ_g is smaller. The influence of φ on rock macro mechanical behavior is that when φ is closer to 45°, σ_P is smaller (Figures 18(c) and 19(c)). With φ increasing, the friction between crack surface is larger, which leads to the increase of σ_R (Figures 18(c) and 19(c)). As shown in Figure 12(d), in the original stage of wing crack growth, S has almost no influence on σ_g. With the wing crack growing, the influence of S on σ_g becomes larger. When S is larger, σ_g is larger. Therefore, the influence of S on rock macro mechanical behavior is that with S increasing, σ_i basically remains unchanged, σ_P and σ_R increase (Figures 18(d) and 19(d)).

Conclusions

To assess the impact of seepage pressure on the macro mechanical behaviors of rocks from the standpoint of micro fracture, an damage-based analytical model is proposed. A wing crack model serves as the foundation for the analytical model. This model has taken into account the impact of seepage pressure on the initiation and growth of wing cracks. The constitutive relation is established based on the equivalency connection of damage defined by axial strain and wing crack length. The relation between rock macro mechanical behavior and micro fracture under the conditions of various seepage pressures, confining pressures and microscopic parameters is investigated. The detailed conclusions are as follows:

In the process of wing crack growth, the wing crack growth is stable firstly and then becomes unstable. The stable and unstable stages of wing crack growth correspond to the pre and post peak stages of the stress-strain curve, respectively. When the wing cracks of adjacent cracks connect with each other, the wing crack growth process ends, and the friction between crack surface plays a dominant role. The rock macro mechanical behavior induced by the friction between crack surface is the residual strength.

From the perspective of micro-fracture, in the whole process of wing crack growth, σ₃ has a significant influence on σ_g. With σ₃ increasing, σ_g is larger. Therefore, with σ₃ increasing, σ_i, σ_P and σ_R in stress-strain curves all increase. The seepage pressure can promote the initiation and growth of wing cracks. When P is larger, σ_g is smaller. With the wing crack growing, the influence of P on σ_g becomes smaller. In the later stage of wing crack growth, P has almost no influence on σ_g. Therefore, with P increasing, σ_i and σ_P decrease, and σ_R basically a constant.

When a is longer, σ_g is smaller. The influence of a on rock macro mechanical behavior is that when a is longer, σ_i, σ_P and σ_R all decreases. With μ increasing, σ_g become larger. The rock macro mechanical behavior is that σ_i, σ_P and σ_R all increase. With the wing crack growing, the influence of φ on σ_g becomes larger firstly and then becomes smaller. When φ is closer to 45°, σ_g is smaller. The rock macro mechanical behavior is that when φ is closer to 45°, σ_P is smaller. In the original stage of wing crack growth, S has almost no influence on σ_g. With the wing crack growing, the influence of S on σ_g becomes larger. When S is larger, σ_g is larger. Therefore, the influence of S on rock macro mechanical behavior is that with S increasing, σ_i basically remains unchanged, σ_P and σ_R increase.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by the National Natural Science Foundation for Young Scientists of China (No. 12202334), the China Postdoctoral Science Foundation (No. 2023MD744236), the Natural Science Basic Research Program of Shaanxi Province (No. 2024JC-YBQN-0061), the Postdoctoral Research Project of Shaanxi Province (No. 2023BSHEDZZ270) and the Special Scientific Research Plan Project of Education Department of Shaanxi Provincial Government (No. 23JK0509) and the Scientific Research Foundation for Excellent Returned Overseas Chinese Scholars funded by Shaanxi Provincial Government (No. 2023-021).

ORCID iDs

Zhushan Shao

Kui Wu

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