The compression-shear mixed crack model with hydraulic pressure is shown in Figure 1. σ 1 and σ 3 are far field compression stress and σ 3 = λ σ 1 ( 0 ≤ λ ≤ 1 ); α is the angle between the crack and the σ 3 direction; p is the internal water pressure of the crack; the crack tip is reduced to two curvature radii of ρ semicircle; the crack length is 2a and θ is the crack initiation angle.

According to the fracture mechanics, stress concentration exists at crack tip. Before the crack closure, it is generally judged whether it is cracked by the

stress distribution at the edge of the crack tip. The stress component of any section can be obtained by the unit body balance equation as the far field stress ( σ 1 , σ 3 ) is known [7] .

Surface stress of crack surface:

σ N = 1 2 [ ( σ 1 + σ 3 ) + ( σ 1 − σ 3 ) cos ( 2 α ) ] − p . (1)

σ T = 1 2 [ ( σ 1 + σ 3 ) − ( σ 1 − σ 3 ) cos ( 2 α ) ] − p . (2)

τ = 1 2 ( σ 1 − σ 3 ) sin ( 2 α ) . (3)

σ N is the normal stress perpendicular to the crack surface; τ is the shear stress on the crack surface, and σ T is the vertical direction stress at the crack tip. According to the fracture mechanics [8] , the mode I stress intensity factor produced by the normal compression stress σ N on the fissure surface is:

K I N = − σ N π a = − { 1 2 [ ( σ 1 + σ 3 ) + ( σ 1 − σ 3 ) cos ( 2 α ) ] − p } π a . (4)

σ T has a tension effect on the edge of the crack, the maximum tensile stress appears in the crack tip and the size is equal to σ T [9] , as shown in Figure 1. Then the mode I stress intensity factor produced by the maximum tensile stress σ ′ T at the crack tip is:

K I T = σ T ρ / a π a = { 1 2 [ ( σ 1 + σ 3 ) − ( σ 1 − σ 3 ) cos ( 2 α ) ] − p } ρ / a π a . (5)

According to the superposition principle of linear elastic fracture mechanics, the mode I stress intensity factor of crack is: K Ι = K Ι T + K Ι N

It can be seen that the initiation and expansion of compression-shear mode I cracks depend on the normal stress of two vertical directions, the radius of curvature of crack tip and the circumference angle. The normal compressive stress σ N inhibits the expansion of compression-shear fracture, and σ T promotes the expansion. The seepage water pressure p has a promotion effect on the initiation and expansion of compression-shear fracture.

The mode II fracture of I-II mixed mode crack under uniform shear stress is independent of the internal osmotic pressure of the crack. The analytic solution for the mode II crack stress intensity factor of composite crack is given [6] :

K Ι Ι = − τ π a = − 1 2 π a ( σ 1 − σ 3 ) sin ( 2 α ) . (6)

In the experiment, the vertical pressure, lateral pressure, water pressure and crack thickness were controlled as follows: σ 1 = 20   MPa , σ 3 = 0   MPa , p = 1   MPa , ρ = 1.5   mm . The dip angle of pre-exciting crack was changed and set the dip angle to 15˚, 30˚, 45˚, 60˚, 75˚. The analytical solution of stress intensity factor and crack initiation angle was calculated by substituting the experimental parameters into the above derivation formulas. The results are as follows (Table 1).

It is intuitive to see the variation of the stress intensity factor and the crack initiation angle with the dip angle of pre-exciting crack. The mode I stress intensity factor decreases with the increase of the dip angle, and the minimum value is obtained when the dip angle is 60 degrees. The mode II stress intensity factor is symmetrically distributed with the dip angle of pre-exciting crack, and the maximum value is obtained when the dip angle is 45 degrees.

As shown in Figure 2, it was observed in the experiment that the crack initiation angle decreases with the increase of the dip angle of pre-exciting crack which is consistent with the theoretical derivation.

In the experiment, the dip angle of pre-exciting crack was controlled to β = 45 ∘ . The water pressure was changed and set the water pressure to 0 MPa, 0.5 MPa, 1 MPa. The analytical solution of stress intensity factor and crack initiation angle was calculated by substituting the experimental parameters into the above derivation formulas. The results are as follows:

It can be seen from the table (Table 2) that the mode I stress intensity factor decreases with the increase of water pressure. And mode II stress intensity factor is independent of water pressure. The crack initiation angle decreases with the increase of water pressure.

In the experiment, the dip angle of pre-exciting crack and water pressure were controlled as follows: The crack width was changed and set the crack width to 1 mm, 1.5 mm, and 2 mm. The analytical solution of stress intensity factor and crack initiation angle was calculated by substituting the experimental parameters into the above derivation formulas. The results are as follows:

It can be seen from the table (Table 3) that the mode I stress intensity factor decreases with the increase of crack width. And mode II stress intensity factor is independent of crack width. The crack initiation angle decreases with the increase of crack width.

In the experiment, the crack initiation strength was recorded when the lateral pressure was controlled to σ 3 = 0   MPa and set the water pressure to p = 0 MPa, 0.5 MPa and 1 MPa; the crack initiation strength was also recorded when the water pressure was controlled to p = 0 MPa and set the lateral pressure to σ 3 = 0   MPa , 1   MPa , 5   MPa , 10   MPa .

The analytical solution of crack initiation strength was calculated by substituting the experimental parameters into the above derivation formulas. By comparing the analytical values with the experimental values, we can see the variation of crack initiation strength with the water pressure and lateral pressure. The results are as follows:

It can be seen from the tables (Table 4 and Table 5) that the crack initiation

strength decreases with the increase of water pressure and increases with the increase of lateral pressure. The experimental results were consistent with the theoretical results.