Introduction

Rocks are complex geological masses, containing extensive joints and fissures, because of the formation complexity. In addition, water frequently affects the underground rocks. Therefore, the mechanical properties of rocks, including the anisotropy, the heterogeneity, and the variation of strength resulting from the water-rock interaction, are complicated (Zhao et al. 2017a; Liste and Kerr 1991).

Many experimental and theoretical studies have investigated the influence of crevice and pore water pressure on rock/soil strength (Zhao et al. 2019; Wang et al. 2017; 2018, 2019a). Bidgoli and Jing (2015) used discrete element methods to simulate the rheological process of the fractured rock mass, and they found that water pressure has significant influence on the strength. Wong et al. (2015) studied the effects of water on rock strength, porosity, density, fabric of rocks, and external factors. Based on the rock fracture mechanics criterion, Liu et al. (2014) established a damage fracture mechanical model for the rock mass in the compression-shear state. Zhao et al. (2018) considered the penetration mechanism of cracked rock cracks under seepage pressure and proposed a corresponding failure criterion. Wei et al. (2018) developed a further improved maximum tangential stress criterion for estimating tensile fracture strength of cracked rock, and laboratory tests show that the criterion can predict experimental results well. Considering the actual effect of the water pressure on the stress field at crack tips, Tang et al. (2004) proposed a new criterion for the fracture strength of fractured rock under water.

The initiation, propagation, and coalescence of the internal defects, such as cracks, may lead to the deformation and the failure of rocks (Wei et al. 2015; Xu et al. 2016; Dai et al. 2016). Therefore, researchers have proposed extensive fracture criteria, using the fracture mechanics and the failure criterion for rock (Wei et al. 2017). For instance, Liu et al. (2004) investigated the relation between the Druker-Prager failure criterion and the fracture toughness (KI and KII) by substituting the total stresses at crack tips into the Druker-Prager failure criterion. Gao et al. (2017) proposed a new fracture criterion based on the energy principles, then, they studied the crack propagation process using this criterion. Zhou et al. (2007) established the criterion for compression-shear fracture using some failure criteria for geomaterials. Zuo et al. (2008) developed a nonlinear strength criterion for rock-like materials, then, they investigated the relation between the flaw inclination and the axial load. In addition, some researchers determine the peak strength of rock by introducing the GSI, which provides a basis for rock failure (Xuan and Zhang 2016; Jiang et al. 2018; Chu and Ji 2017). The above studies indicate that it is feasible to establish new fracture criteria, based on the macroscopic failure criterion.

The current studies have significantly contributed to understanding the fracture characteristics, and make the rock have a certain early warning before the fracture (Dai et al. 2015; Wei et al. 2016). However, few studies have incorporated the fracture theory and the macroscopic failure criteria to investigate the effect of the water on rock fracture strength, meanwhile, few analysis and summary, the influence of crack inclination angle and the internal friction angle on rock strength under the water-force action of rock compression and shear, has been done. Considering the effect of water on the stress intensity factors, this paper proposed a newly rock fracture criterion considering the influence of water by applying the Druker-Prager failure criterion and the fracture theory.

The stress intensity factors at crack tips considering the influence of water

The rock strength may be decreased by water pressure and hydro-chemical damage (Zhao et al. 2017b, c; Wang et al. 2019). Figure 1 shows the biaxial stress state of the rock, considering the influences of crack and water.

Fig. 1
figure 1

The biaxial stress state of the cracked rock influenced by water

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According to Fig. 1, the normal stress applied on the main crack surface σ and the shear stress applied on the main crack surface τ can be obtained (Guo et al. 2002). In this paper, the compressive stress is positive.

$$ \left.{\displaystyle \begin{array}{l}\sigma =\frac{1}{2}\left[\left({\sigma}_1+{\sigma}_3\right)+\left({\sigma}_1-{\sigma}_3\right)\cos 2\beta \right]-P\\ {}\tau =\frac{1}{2}\left({\sigma}_1-{\sigma}_3\right)\sin 2\beta \end{array}}\right\} $$
(1)

where σ1 and σ3 are the first and the third principal stresses of the main crack surface, respectively. β is the crack inclination angle; P is the water pressure, acting on the crack surface.

In the fracture mechanics, cracks are often classified into three types according to their forces and crack propagation paths. These cracks are the opening crack (the mode I crack), the sliding mode crack (the mode II crack), and the tearing mode crack (the mode III crack). The three-mode cracks are shown in Fig. 2.

Fig. 2
figure 2

The three-mode cracks

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For the mode I crack:

$$ {K}_I=\sigma \sqrt{\pi a} $$
(2)

where KΙ is the stress intensity factor of mode I crack; π is a constant and a is the half-length of the crack.

Considering the influence of water, the stress intensity factor is:

$$ {K}_I=\left\{\frac{1}{2}\left[\left({\sigma}_1+{\sigma}_3\right)+\left({\sigma}_1-{\sigma}_3\right)\cos 2\beta \right]-P\right\}\sqrt{\pi a} $$
(3)

For the mode II crack:

$$ {K}_{II}=\tau \sqrt{\pi a} $$
(4)

where KΙΙ is the stress intensity factor of mode II crack.

First, when the crack is open, the stress intensity factor of the mode II crack, influenced by water (the state of stress has been shown in Fig. 3), is:

$$ {K}_{II}=\frac{1}{2}\left({\sigma}_1-{\sigma}_3\right)\sin 2\beta \sqrt{\pi a} $$
(5)
Fig. 3
figure 3

The force state of fully closed crack

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Second, when the crack is fully closed, the stress intensity factor of the mode II crack, influenced by water, is:

$$ {K}_{II}=\frac{1}{2}\left({\sigma}_1-{\sigma}_3\right)\sin 2\beta \sqrt{\pi a}-\left(c+\sigma \tan \varphi \right)\sqrt{\pi a} $$
(6)

where c and φ are the cohesion and the internal friction angle of the rock, respectively.

The stress field at rock crack tips in the compressive-shear state

Figure 4 shows the stress and the strain fields at a certain point A near the crack tip. To facilitate the calculation the stress, we transformed the calculation system (Fig. 5):

Fig. 4
figure 4

The stress and the strain field at a certain point A near the crack tip

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Fig. 5
figure 5

New and old coordinate systems with the center and the top of the crack as the origin of the coordinate, respectively

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Using the superposition criterion, the stress field at the tip of the mode I–II cracks is (Yu et al. 1991):

$$ \left.{\displaystyle \begin{array}{l}{\sigma}_{\mathrm{x}}=\frac{K_I}{\sqrt{2\pi r}}\cos \frac{\theta }{2}\left(1-\sin \frac{\theta }{2}\sin \frac{3}{2}\theta \right)-\frac{K_{II}}{\sqrt{2\pi r}}\sin \frac{\theta }{2}\left(2+\cos \frac{\theta }{2}\cos \frac{3}{2}\theta \right)\\ {}{\sigma}_{\mathrm{y}}=\frac{K_I}{\sqrt{2\pi r}}\cos \frac{\theta }{2}\left(1+\sin \frac{\theta }{2}\sin \frac{3}{2}\theta \right)+\frac{K_{II}}{\sqrt{2\pi r}}\sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{3}{2}\theta \\ {}{\tau}_{\mathrm{x}\mathrm{y}}=\frac{K_I}{\sqrt{2\pi r}}\sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{3}{2}\theta +\frac{K_{II}}{\sqrt{2\pi r}}\cos \frac{\theta }{2}\left(1-\sin \frac{\theta }{2}\sin \frac{3}{2}\theta \right)\end{array}}\right\} $$
(7)

where θ and r are the angle and the distance between a certain point A near the crack tip and the polar axis, respectively.

Under the plane stress condition, σ2 is equal to 0; therefore, the principal stresses are:

$$ {\displaystyle \begin{array}{l}{\sigma}_1=\frac{\sigma_{\mathrm{x}}+{\sigma}_{\mathrm{y}}}{2}+\sqrt{{\left(\frac{\sigma_{\mathrm{x}}-{\sigma}_{\mathrm{y}}}{2}\right)}^2+{\tau_{\mathrm{x}\mathrm{y}}}^2}\\ {}{\sigma}_3=\frac{\sigma_{\mathrm{x}}+{\sigma}_{\mathrm{y}}}{2}-\sqrt{{\left(\frac{\sigma_{\mathrm{x}}-{\sigma}_{\mathrm{y}}}{2}\right)}^2+{\tau_{\mathrm{x}\mathrm{y}}}^2}\end{array}}\Big\} $$
(8)

Equations (7) and (8) yield the principle stresses at the tips of the crack in the compressive-shear state (Yu et al. 1991):

$$ {\displaystyle \begin{array}{l}{\sigma}_1=\frac{1}{\sqrt{2\pi r}}\left({K}_{\mathrm{a}}+{K}_{\mathrm{b}}\right)\\ {}{\sigma}_3=\frac{1}{\sqrt{2\pi r}}\left({K}_{\mathrm{a}}-{K}_{\mathrm{b}}\right)\end{array}}\Big\} $$
(9)

where \( \Big\{{\displaystyle \begin{array}{l}{K}_{\mathrm{a}}={K}_I\cos \frac{\theta }{2}-{K}_{II}\sin \frac{\theta }{2}\\ {}{K}_{\mathrm{b}}=\frac{1}{2}\sqrt{{\left({K}_I\sin \theta +2{K}_{II}\cos \theta \right)}^2+{K_{II}}^2{\sin}^2\theta}\end{array}} \)

The D-P failure criterion

Extensive failure criteria, including the Griffith criterion, the modified Griffith criterion, the Druker-Prager failure criterion, the Coulomb-Mohr criterion, the Hoek-Brown criterion, and the Mises criterion, have been proposed. However, researchers failed to propose a universal criterion to predict the rock properties (Wu et al. 2019; Yang et al. 2008; Liu et al. 2017; Zhao et al. 2016; Wang et al. 2014), because of the various application conditions for the above criteria.

In 1952, Drucker and Prager proposed a failure criterion, based on the modification and development of the Mises criterion. By introducing the first stress invariant into the Mises criterion, this criterion is written as:

$$ f\left({I}_1,{J}_2\right)=\alpha {I}_1+\sqrt{J_2}-k=0 $$
(10)

where I1 is the first invariants of the stress tensor,

$$ {I}_1={\sigma}_1+{\sigma}_2+{\sigma}_3 $$
(11)

J2 is the second invariants of the stress deviator:

$$ {J}_2=\frac{1}{6}\left[{\left({\sigma}_1-{\sigma}_2\right)}^2+{\left({\sigma}_2-{\sigma}_3\right)}^2+{\left({\sigma}_3-{\sigma}_1\right)}^2\right] $$
(12)

α and k are constants which are written as:

$$ \Big\{{\displaystyle \begin{array}{l}\alpha =\frac{2\sin \varphi }{\sqrt{3}\left(3-\sin \varphi \right)}\\ {}k=\frac{6c\cos \varphi }{\sqrt{3}\left(3-\sin \varphi \right)}\end{array}} $$

The criterion for the compressive-shear fracture based on the Druker-Prager failure criterion

The criterion for the compressive-shear fracture

Based on the maximum tensile stress fracture criterion (Erdogan and Sih 1963), the initiation angles (θ0) of the central oblique crack in compressive-shear states are equal to 0 and 70.5° for the mode I crack and the mode II crack, respectively.

Equations (9), (11), and (12) yield:

$$ {I}_1=\frac{2}{\sqrt{2\pi r}}{K}_{\mathrm{a}} $$
(13)
$$ {J}_2=\frac{1}{6\pi r}\left(3{K_{\mathrm{b}}}^2+{K_{\mathrm{a}}}^2\right) $$
(14)

By substituting Eqs. (13) and (14) into Eq. (10), the compressive-shear fracture criterion can be written as:

$$ \alpha \frac{2}{\sqrt{2\pi r}}{K}_{\mathrm{a}}+\frac{1}{\sqrt{6\pi r}}\sqrt{3{K_{\mathrm{b}}}^2+{K_{\mathrm{a}}}^2}-k=0 $$
(15)

For the pure mode I crack with the θ0 of 0, Ka and Kb are equal to KΙ and 0, respectively. Then, Eq. (15) yields:

$$ \frac{2\alpha {K}_I}{\sqrt{2\pi r}}+\frac{K_I}{\sqrt{6\pi r}}=k $$
(16)

When the stress intensity reaches the fracture toughness, then it yields:

$$ {K}_{IC}=\frac{k\sqrt{6\pi r}}{2\sqrt{3}\alpha +1} $$
(17)

By substituting Eq. (17) into Eq. (15), the criterion is written as:

$$ \frac{2\alpha {K}_{\mathrm{a}}\sqrt{3}+\sqrt{3{K_{\mathrm{b}}}^2+{K_{\mathrm{a}}}^2}}{\left(2\sqrt{3}\alpha +1\right){K}_{\mathrm{IC}}}-1=0 $$
(18)

Feasibility verification of the criterion without the influence of the water

To verify the feasibility of the proposed criterion, we compared the results of this criterion with those from the maximum-tangential-stress theory. The special cases for the maximum tangential stress theory are:

First, for the mode I crack, KΙΙ and θ0 are equal to 0. Therefore, KΙ is equal to KΙC.

Additionally, for the mode II crack, KΙ and θ0 are equal to 0 and − 70.5°, respectively. Therefore, according to the maximum tensile stress criterion (Liu et al. 2017), when the material parameter of the rock, Poisson’s ratio ν is equal to 0, then, KΙΙC is equal to 0.866 KΙC.

For the proposed criterion, Eq. (18) yields:

For the mode I crack, KΙΙ and θ0 are equal to 0. Thus, the fracture toughness in Eq. (17) is equal to KΙ.

For the mode II crack, KΙ and θ0 are equal to 0 and − 70.5°, respectively. The fracture toughness in Eq. (15) is:

$$ {K}_{\mathrm{IIC}}=\frac{k\sqrt{\pi r}}{0.4715+0.8161\alpha } $$
(19)

Based on Eqs. (17) and (19), we can obtain that:

$$ \frac{K_{II\mathrm{C}}}{K_{\mathrm{IC}}}=\frac{2\sqrt{3}\alpha +1}{\sqrt{6}\left(0.4715+0.8161\alpha \right)} $$
(20)

where α determines \( \frac{K_{II\mathrm{C}}}{K_{I\mathrm{C}}} \). In addition, when φ and α are equal to 0, \( \frac{K_{II\mathrm{C}}}{K_{I\mathrm{C}}} \) is equal to 0.866, the same value to that of the maximum-tangential-stress theory. Thus, this criterion based on the Druker-Prager failure criterion is reasonable.

The criterion of the compressive-shear fracture considering water action

Extensive studies (Zhao et al. 2017d; Huang et al. 2017; Wang et al. 2019b, c) have investigated the strength and the failure mode of the crack in compressive-shear state. However, the strength of the rock, subjected to the compressive-shear stress and the water action, lacks detailed investigations (Zhang et al. 2011; Zhao et al. 2017e). Thus, based on the above criterion in the compressive-shear state, this paper proposed a strength criterion by introducing the stress intensity factors of the mode I crack and the mode II crack, subjected to the compressive-shear stress and the water action. According to the failure forms of the rock, containing a crack, the following three conditions are considered:

For the closed crack, the maximum values of the stress intensity factor of the mode I crack and the mode II crack are equal to 0 (Guo et al. 2002). The above results show that when the crack propagates in a tensile manner, KΙΙ is equal to 0. However, KΙ is equal to 0 when the crack propagates in a shear manner, in other words, cracks propagate on the plane of the main crack.

For the mode I crack propagation, KΙΙ is equal to 0, Eq. (18) yields:

$$ {K}_I=\frac{\left(2\sqrt{3}\alpha +1\right){K}_{\mathrm{IC}}}{2\sqrt{3}\alpha \cos \frac{\theta }{2}+\sqrt{\frac{3}{4}{\sin}^2\theta +{\cos}^2\frac{\theta }{2}}} $$
(21)

By substituting Eq. (21) into Eq. (3), the initiation criterion influenced by water is:

$$ {\sigma}_1=\frac{2P}{1+\cos 2\beta }+\frac{\left(\cos 2\beta -1\right){\sigma}_3}{1+\cos 2\beta }+\frac{2\left(2\sqrt{3}\alpha +1\right){K}_{IC}}{\left(2\sqrt{3}\alpha \cos \frac{\theta }{2}+\sqrt{\frac{3}{4}{\sin}^2\theta +{\cos}^2\frac{\theta }{2}}\right)}\frac{1}{\left(1+\cos 2\beta \right)\sqrt{\pi a}} $$
(22)

Under certain conditions, the variables, excluding t σ1 and P, are known. σ1 linearly increases with the increase of P.

To express the relationship of σ1 and P, we supposed \( {m}_1=\frac{2}{1+\cos 2\beta } \), \( {n}_1=\frac{\left(\cos 2\beta -1\right)}{1+\cos 2\beta } \), \( {t}_1=\frac{2\left(2\sqrt{3}\alpha +1\right){K}_{I\mathrm{C}}}{\left(2\sqrt{3}\alpha \cos \frac{\theta }{2}+\sqrt{\frac{3}{4}{\sin}^2\theta +{\cos}^2\frac{\theta }{2}}\right)}\frac{1}{\left(1+\cos 2\beta \right)\sqrt{\pi a}} \),

Then, Eq. (22) can be written as:

$$ {\sigma}_1={m}_1P+{n}_1{\sigma}_3+{t}_1 $$
(23)

Under certain conditions, Eq. (23) shows that the variables, except σ3, P, and β, are known, β directly determines the effect of P on σ1. Obviously, \( \mid \frac{m_1}{n_1}\mid \) and m1 increase with the increase of β. This indicates that P plays a leading role on the rock strength, and when β increases, the rock strength will be significantly promoted.

This phenomenon is reasonable because the water pressure leads to the decreases in the effective stress on the crack surfaces and KI. Then, the delay of the crack initiation, resulting from the decrease in KI, the initiation strength of rock blocks, is increased.

For the mode II crack propagation, KΙ is equal to 0, Eq. (18) yields:

$$ {K}_{II}=\frac{\left(2\sqrt{3}\alpha +1\right){K}_{IC}}{\sqrt{\frac{3}{4}+\frac{9}{4}{\cos}^2\theta +{\sin}^2\frac{\theta }{2}}-2\sqrt{3}\alpha \sin \frac{\theta }{2}} $$
(24)

By substituting Eq. (24) into Eq. (6), the crack rock initiation strength criterion, considering water action, is:

$$ {\displaystyle \begin{array}{l}{\sigma}_1=\frac{\left[\sin 2\beta +\left(1-\cos 2\beta \right)\tan \varphi \right]{\sigma}_3-2\tan \varphi P}{\sin 2\beta -\left(1+\cos 2\beta \right)\tan \varphi }+\frac{-2c}{\sin 2\beta -\left(1+\cos 2\beta \right)\tan \varphi}\\ {}\kern0.84em +\frac{2\left(2\sqrt{3}\alpha +1\right){K}_{IC}}{\left[\sin 2\beta -\left(1+\cos 2\beta \right)\tan \varphi \right]\left(\sqrt{\frac{3}{4}+\frac{9}{4}{\cos}^2\theta +{\sin}^2\frac{\theta }{2}}-2\sqrt{3}\alpha \sin \frac{\theta }{2}\right)\sqrt{\pi a}}\end{array}} $$
(25)

In order to express the relationship of σ1 and P, suppose \( {m}_2=\frac{-2\tan \varphi }{\sin 2\beta -\left(1+\cos 2\beta \right)\tan \varphi } \), \( {n}_2=\frac{\sin 2\beta +\left(1-\cos 2\beta \right)\tan \varphi }{\sin 2\beta -\left(1+\cos 2\beta \right)\tan \varphi } \), \( {t}_2=\frac{-2c}{\sin 2\beta -\left(1+\cos 2\beta \right)\tan \varphi }+\frac{2\left(2\sqrt{3}\alpha +1\right){K}_{I\mathrm{C}}}{\left[\sin 2\beta -\left(1+\cos 2\beta \right)\tan \varphi \right]\left(\sqrt{\frac{3}{4}+\frac{9}{4}{\cos}^2\theta +{\sin}^2\frac{\theta }{2}}-2\sqrt{3}\alpha \sin \frac{\theta }{2}\right)\sqrt{\pi a}} \) then Eq.(25) can be written as:

$$ {\sigma}_1={m}_2P+{n}_2{\sigma}_3+{t}_2 $$
(26)

Obviously, m2 and n2 will directly determine the effect of P and σ3 on σ1. Moreover, the parameters m2 and n2 are related with β and φ. Therefore, we considered the change of \( \mid \frac{m_2}{n_2}\mid \) to determine whether P or σ3 significantly affect the rock strength.

First, when β changes, φ remains unchanged, other parameters are known, and according to the calculation parameters in Table 1, the curve of \( \mid \frac{m_2}{n_2}\mid \) and β under different φ can be obtained (Fig. 6).

Table 1 Calculation parameter
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Fig. 6
figure 6

The curves of \( \mid \frac{m_2}{n_2}\mid \) versus β under different φ

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When β increases to φ, \( \mid \frac{m_2}{n_2}\mid \) sharply decrease from about 9.5, 10.5, and 11 to 1 (Fig. 6). Then these values slightly decrease to about 0.73, 0.84, and 0.88, respectively. Finally, these values increase to 1 when β increases to 90°.

The above descriptions show that when β is lower than φ, the increase in P can significantly decrease the rock strength. However, when β is higher than φ, the rock strength can be significantly increased by the increase of σ3 instead of that of P. In addition, when β is lower than φ, greater influence of P on σ1 is observed by the increase in φ.

Second, when φ varies, β remains unchanged, other parameters are known. According to the calculation parameters in Table 2, we obtained the curve of \( \mid \frac{m_2}{n_2}\mid \) and φ under different β (Fig. 7).

Table 2 Calculation parameter
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Fig. 7
figure 7

The curves of \( \mid \frac{m_2}{n_2}\mid \) versus φ under different β

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In this paper, according to the actual situation, the internal friction angle of the rock was 10°~75°.

For the β of 15°, 30°, 45°, 60°, and 75°, Fig. 7 shows that the increase of φ leads to the increase of \( \mid \frac{m_2}{n_2}\mid \). When the β increases to φ, \( \mid \frac{m_2}{n_2}\mid \) increases to 1. Particularly, for the β of 90°, \( \mid \frac{m_2}{n_2}\mid \) is 1.

The above results show that σ3 significantly increases the rock strength when φ is lower than β. However, when φ is higher than β, the increase of P dramatically decreases the rock strength. When φ is equal to β, the influences of P and σ3 on σ1 are the same. In addition, when β is lower than 30° and φ, σ3 greatly influences σ1.

For the open crack, the mode I crack propagation occurs instead of the mode II crack propagation. Correspondingly, the maximum stress intensity factor for the mode I crack corresponds to the condition of KΙΙ = 0. Thus, the crack propagation path is in accordance with the path for the KΙΙ = 0, and the crack initiation criterion can be presented by Eq. (22).

Conclusions

  1. (1)

    Based on the Druker-Prager failure criterion and the theory of fracture mechanics, a newly proposed criterion for the compressive-shear fracture is proven reasonable.

  2. (2)

    The results show that the water pressure improves the rock strength for the mode I crack, whereas for the mode II crack, the water pressure has a very complicated effect on the rock strength for the closed crack. And the water pressure decreases the rock strength for the open crack.

  3. (3)

    Moreover, the results show that the third principal stress significantly increases the rock strength when the internal friction angle of the rock is lower than the crack inclination angle. However, when the internal friction angle is higher than the crack inclination angle, the increase of water pressure dramatically decreases the rock strength. When the internal friction angle is equal to crack inclination angle, the influences of water pressure and the third principal stress on the rock strength are the same. In addition, when crack inclination angle is lower than 30°, the third principal stress influences the rock strength greatly. Generally, the water pressure decreases the rock strength.