2.1 Similar materials and size

The large size and poor integrity of columnar jointed basalt make it difficult and costly to conduct in situ tests, while laboratory tests provide an effective means for studying the mechanical behavior of CJRM. The determination of similar materials and column sizes is the basis for the success of physical model tests [32]. In this study, the similarity relationship between the prototype and the model adopted by QUE et al [28] was followed, where the similarity ratios for geometry, density, strength, deformation and dimensionless parameters were 6, 2.5, 15, 15 and 1, respectively.

A mixture of gypsum, fine sand and water was selected to simulated columns, and the corresponding mass ratio of each material was determined as 3:1:2.4 [23, 24]. As shown in Table 1, the mechanical properties of the material for simulating columns are listed. Furthermore, joint similar materials used in existing studies included resin for 3D printing [7], white latex [33] and cement slurry [21, 23, 24]. After applying these three materials, resin and white latex were found not to be conducive to the generation and expansion of cracks at joints. When the cement slurry is used as the jointed surface, its thickness should be controlled thin to ensure that it does not enhance the strength of the sample. Therefore, in this study, cement slurry was used to simulate joint planes and the water-cement ratio was determined as 0.4 [24]. The friction angle φ_j and cohesion c_j were 32° and 1.23 MPa, respectively.

The field investigation results of the Baihetan CJRM showed that the diameter of column cross section is 13-25 cm [34]. The representative elementary volume size of CJRM is approximately 4 times the average side length of the column cross-section [28]. To satisfy the size effect and geometric similarity ratio, the sample was determined a standard cube with a side length of 10 cm, and the average length of the maximum diagonals of the sections was 3 cm.

2.2 Sample preparation

Figure 2 shows the preparation process of ICJRM samples. A 3D irregular jointed network was first established with the help of the Voronoi procedure in MATLAB software, and the average length of the maximum polygon diagonals was set as 3 cm. Subsequently, the intersection between the 3D jointed network and a cube with a side length of 12 cm was obtained. Before the intersection was performed, the cube was rotated in the vertical plane to obtain digital jointed models with different inclination angles. The digital models were exported to a 3D printer, and the actual jointed molds were printed using white photosensitive resin (Figure 2(a)). Next, similar material was prepared and poured into a combined mold (Figures 2(b) and (c)), including an acrylic mold with an internal side length of 12 cm and a printed jointed mold. After 20 min, the columns were removed from the molds and placed in sequence. The columns were bonded with joint filler material to form cubic blocks (Figure 2(d)). After curing for 20 d, each surface of the cubic blocks was first cut by a cutting machine and then smoothed by a grinder (Figures 2(e) and (f)). Finally, standard ICJRM samples with various inclination angles were obtained (Figure 2(g)).

Figure 2全屏查看

Preparation process of ICJRM samples: (a) Printed jointed molds; (b) Combined mold; (c) Similar material; (d) Cubic blocks; (e) Cutting machine; (f) Grinder; (g) Standard ICJRM samples

2.3 Test system and procedures

Conventional triaxial compression tests of the ICJRM samples with different inclination angles were conducted on a triaxial electrohydraulic servo rock test system, including a loading system, control system and oil source. Three mutually perpendicular loading heads independently applied loads to the sample, and the minimum (σ₃), intermediate (σ₂) and maximum (σ₁) principal stresses were loaded along the X, Y and Z loading heads, respectively.

Before testing, each surface of the sample was evenly oiled and covered with polyethylene film to reduce the end effect. The horizontal geostress of the Baihetan CJRM is 15.0-17.0 MPa [27]; therefore, the minimum initial confining pressure was determined to be 1 MPa in the model test.

Figure 3 shows the loading steps. First, three loading heads were controlled to apply a stress of 0.1 MPa to the sample at 0.05 MPa/s, thus keeping the sample in a stable state. Then, the load was applied to the preset value of the confining pressure at the same rate in three directions. The preset pressure values used in this study were 1, 2, 3 and 4 MPa. After maintaining the initial stress state for 2 min, σ₁-head was controlled to apply the load at 0.3 mm/min until the sample damaged.

Figure 3全屏查看

Test system and plan

3.1 Stress-strain curves

Figure 4 depicts the stress-strain curves of ICJRM samples under various stress conditions. The ICJRM samples exhibit obvious strain softening and brittle characteristics when σ₂=σ₃=1 MPa. With increasing confining pressure, the post-peak curves corresponding to different inclination angles gradually rise, and the strain softening behavior weakens with the appearance of strain hardening. The ε₂ and ε₃ curves of the ICJRM samples with 0° inclination angle under different confining pressures almost overlap, showing transverse isotropy. For samples with other inclination angles, as the confining pressure increases, the ε₂ and ε₃ curves gradually converge and almost completely coincide when σ₂=σ₃=4 MPa. This indicates that with the increase of confining pressure, the influences of column inclination angle and the joint weakening effect both decrease, which is reflected in the weakening of anisotropy in horizontal plane of the ICJRM samples. Except for the sample with an inclination angle of 0°, the strain in 3-direction is greater than that in 2-direction, indicating that the failure mainly occurs along the column axis direction.

Figure 4全屏查看

Stress-strain curves corresponding to different inclination angles: (a) β=0°; (b) β=15°; (c) β=30°; (d) β=45°; (e) β=60°; (f) β=75°; (g) β=90°

3.2 Strength and deformation properties

As shown in Figure 5, the elastic moduli and failure strengths of the ICJRM samples under different stress conditions are calculated according to Figure 4. Figure 5 also shows the uniaxial compression test results in Ref. [28] of the ICJRM samples.

Figure 5全屏查看

Variation of E_j and σ₁-σ₃ with inclination angle: (a) Elastic modulus, E_j; (b) Deviatoric stress, σ₁-σ₃

Figure 5(a) shows the change of modulus E_j with inclination angle β. As the confining pressure increases, the inclination angle corresponding to the minimum modulus gradually increases from 30° (σ₂=σ₃=0 MPa) to 45° (σ₂=σ₃=1, 2, 3 MPa) and finally becomes 60° (σ₂=σ₃=4 MPa). The shapes of all five curves are concave, and they are less affected by the confining pressure. The curves gradually rise with increasing confining pressure, which indicates that the deformation resistances of the samples with different inclination angles are enhanced. Figure 5(b) shows the changes of deviatoric stress with inclination angle. All five curves obtain the maximum value when β=0°, and the inclination angle corresponding to the minimum deviatoric stress increases from 30° (σ₂=σ₃=0, 1 MPa) to 45° (σ₂=σ₃=2, 3, 4 MPa). With increasing confining pressure, the failure strength of ICJRM samples with different inclination angles increases, and the curves gradually become flat. This indicates that the effect of inclination angle on failure strength of the ICJRM gradually weakens as the confining pressure increases. Especially when σ₂=σ₃=4 MPa, the difference between the failure strengths of ICJRM samples with various inclination angles is small. Considering the field stress environment, special attention should be paid to engineering parts with an inclination angle of 30°-45° due to their weak resistance to damage and deformation.

3.3 Anisotropic characteristics

To reflect the anisotropic characteristics of the ICJRM samples under different confining pressures, the anisotropy ratio presented by SINGH et al [35] is used:

(1)

(2)

where R_cd and R_cs are the deformation and strength anisotropy ratios, respectively; E_j,max and E_j,min are the maximum and minimum elastic moduli under the same stress conditions, respectively; σ_1,max and σ_1,min are the maximum and minimum failure strengths under the same stress conditions, respectively. The larger the R_cd or R_cs value, the more significant the deformation or strength anisotropy.

The test results are substituted into Eqs. (1) and (2), and the calculated results are plotted in Figure 6. According to the anisotropy classification system proposed by RAMAMURTHY et al [31], Figure 6 is divided into four areas.

Figure 6全屏查看

Variation in anisotropy ratio R_cs and R_cd with confining pressure σ₃

As confining pressure increases, the black curve shows that the deformation anisotropy ratio roughly decreases with increasing confining pressure. When σ₂=σ₃=0 MPa, the deformation anisotropy ratio reaches the maximum value, which is 5.418. This ratio corresponds to a high deformation anisotropy. The curve reaches the minimum value of 2.170 when the confining pressure is 3 MPa. Except for uniaxial compression, all other calculated deformation anisotropy ratios show medium deformation anisotropy. The red curve shows that the strength anisotropy ratio exhibits an obvious exponential downward trend with increasing confining pressure. When the confining pressures are 0 and 4 MPa, the strength anisotropy ratios are the maximum (4.338) and minimum (1.051), respectively. The degrees of strength anisotropy of the ICJRM samples under five stress conditions, from 0 to 4 MPa, are high anisotropy, medium anisotropy, low anisotropy, low anisotropy and isotropy. When the confining pressure is greater than 3.5 MPa, the strength of the ICJRM samples is isotropic, and the effect of inclination angle on strength can be ignored.

3.4 Failure modes

Figure 7 shows the appearances of the failed samples. Four typical failure modes include tensile and rotational failure along columnar joints (TRA), shear failure along columnar joints (SA), tensile failure through columns (TT) and shear failure through columns (ST).

Figure 7全屏查看

Appearances of failed ICJRM samples under different confining pressures: (a) σ₃=σ₂=1 MPa; (b) σ₃=σ₂=2 MPa; (c) σ₃=σ₂=3 MPa; (d) σ₃=σ₂=4 MPa

Mode TRA: The reason for this failure is that the angle between joint direction and σ₁-direction is small, and the effect of confining pressure is weak. Under vertical loading, tensile cracks first appear on the joint plane. The separated columns rotate under loading, forming horizontal cracks.

Mode SA: The damaged sample is relatively complete, and almost no fracture of the column is observed. The failure of the sample is due to the shear stress acting on joint being greater than the shear strength; eventually, sliding failure occurs.

Mode TT: Several nearly vertical cracks are observed on the surface of the damaged sample. This is because the angle between joint direction and σ₁-direction is large, and the joint plane is less affected by shear stress. Tensile cracks developed in column material lead to the ultimate failure.

Mode ST: Under the action of a large confining pressure, it is difficult for the ICJRM samples with low inclination angles to split along joints. The vertical load causes the formation of shear cracks that intersect the columns.

The failure modes of ICJRM samples with the same inclination angle are different under different stress conditions. Due to the combined effects of column direction and stress condition, some samples have experienced failure that has not clearly identified as a single failure mode. For example, under confining pressures of 2, 3 and 4 MPa, both sliding along columnar joints and shearing through columns are observed for the ICJRM sample with an inclination angle of 15°. Overall, the failure appearance of the sample with an inclination angle of 15° is relatively fragmented, which is similar to the failure mode exhibited by the field CJRM [26, 34].

4.1 Joint factor method

The distribution of joints significantly weakens intact rocks. RAMAMURTHY et al [31] presented a traditional joint factor J_f to quantify this weakening based on extensive experimental data:

(3)

where J_n is the traditional frequency coefficient, which is equal to the joints number per unit length in σ₁-direction; n is the direction coefficient, which can be obtained from Ref. [31]; r is the roughness coefficient, which can be replaced by tanφ_j.

Based on this, to make the traditional joint coefficient more suitable for describing the structure of CJRMs, QUE et al [28] derived an improved joint factor:

(4)

where J_sn is the improved frequency coefficient, which is obtained by calculating the sections number per unit area in σ₁-direction.

To facilitate the application of the IJFM, the improved frequency coefficient is represented by the average length of the maximum diagonals D and the inclination angle β [28]:

(5)

JADE et al [36] established an empirical relationship between the strength and traditional joint factor:

(6)

where σ_cj is the failure strength under uniaxial stress; a, b and c are constant coefficients in the empirical relationship.

To predict the deformation of the jointed rock mass under triaxial stress, ZHU et al [24] proposed the following empirical equation:

(7)

where E_j(σ₃) is the elastic modulus under different confining pressures σ₃; E_j(σ₃=0) is the elastic modulus under uniaxial stress; L_j and K_j are the constant coefficients.

RAMAMURTHY et al [31] used the following empirical relation to evaluate the strength of jointed rock masses under triaxial stress:

(8)

where A_j and B_j are constant coefficients.

4.2 Application in ICJRM

The values of the elastic modulus and failure strength of ICJRM under various confining pressures are predicted by combining the two joint factor methods and the above empirical equations. According to Eqs. (7) and (8), the uniaxial compression strength is the basic parameter required to predict the deformation and strength of the ICJRM under triaxial stress. The empirical relationships for the uniaxial compression strength determined by QUE et al [28] based on the TJFM and IJFM are given as:

(9)

(10)

The prediction effect of the IJFM on the uniaxial compression test results of the ICJRM samples is greater than that of the TJFM. This is because the improved joint factor was presented on the basis of the unique joint distribution of CJRM.

Table 2 exhibits the calculated coefficients J_n, J_sn, n, J_f and J_mf for ICJRMs with different inclination angles. In this test, the friction angle φ_j of the joint plane is 32°; thus, the roughness parameter r is equal to 0.625. When the inclination angle is 0°, the frequency parameters J_sn and J_n along the vertical load direction are both 0; thus, these parameters are not listed.

For the deformation of the ICJRM samples under triaxial stress, Eq. (7) is fitted with the triaxial test results obtained in this study. The empirical equation of deformation is

(11)

Equations (9) and (10) are substituted into the empirical relationship for deformation (Eq. (11)), and the elastic moduli of the ICJRM with different inclination angles under different confining pressures are predicted. The test E_j,exp and predicted E_j,pre values of the elastic modulus are plotted in Figure 8. The empirical relationships for deformation based on the two joint factor methods perform well in general, but the prediction effects are poor for the test data of the ICJRM with inclination angles of 30° and 45°. A total of 71.429% of the data points in Figure 8(a) are within the upper/lower limit of 85%, and 89.286% of the data points shown in Figure 8(b) are within this range. This indicates that the prediction quality of the empirical relationship for deformation based on the improved method is higher than that of the relationship based on the traditional method.

Figure 8全屏查看

Comparison between E_j,exp and E_j,pre: (a) Traditional joint factor method; (b) Improved joint factor method

For the strength parameters of the ICJRM under triaxial stress, Eq. (8) is fitted with the test results obtained in this paper. The empirical equation has poor fitting performance for the overall data, but the fitting effect on the test data corresponding to each inclination angle is good. According to the method used by RAMAMURTHY et al [31], the relationships between the two constant terms (A_j and 1/B_j) and are established:

(12)

(13)

Both constants show a significant exponential change with . Equations (12) and (13) are substituted into Eq. (8) to obtain an empirical expression for strength. Subsequently, Eqs. (9) and (10) are substituted into the determined strength empirical equation to predict the failure strength of ICJRM. The experimental σ_1,exp and predicted σ_1,pre values of the failure strength are plotted in Figure 9. Both empirical relationships for strength based on the two joint factor methods perform well, and the prediction effects are better than those of the empirical relationships for deformation. A total of 92.857% of the data points corresponding to TJFM are within the upper/lower limit of 85%, while all the data points corresponding to the IJFM are within this range. This means that the relationship for strength based on the IJFM is more efficient than the relation based on the traditional method.

Figure 9全屏查看

Comparison between σ_exp and σ_1,pre: (a) Traditional joint factor method; (b) Improved joint factor method

The above results show that the empirical relationships based on the IJFM are superior to the relationships based on the TJFM in estimating deformation and strength under triaxial stress. This means that the IJFM is more suitable than the TJFM for the ICJRM.