2.1. Test equipment

2.1.1. Cutters

Disc cutters with different diameters are used in practice. Two cutters with relatively large diameter differences from one another were used to study the influence of the diameter. The choice of 14-inch and 20-inch cutters was justified based on their representation of commonly used sizes in practice. A small portion of the disc that mimics the real diameter of the disc cutter was made of MO40 steel, and a Rockwell hardness of 55 was considered. The disc cutters with 14- and 20-inch diameters and 8-mm edge radii are shown in Figures 1 and 2.

Figure 1. Fabricated cutters.

Figure 2. Drawings of the designed cutters with diameters: of (a) 20 in and (b) 14 in.

2.1.2. Confining box

The hydraulic jack and frame comprise the confining-pressure device. SVS150 alloy steel with a tensile strength of 1300 MPa was used to construct the metal frame. The two plates can be length-adjusted and fastened together with four steel rods and bolts. A five-ton manual hydraulic jack that provides variable confining stresses during loading by using a consistent load application approach. The sample was positioned within a metal frame and compressed from one side by a piston. Two 100 $ \times $ 100 mm steel plates were positioned on either side of the samples to provide confining stress on the entire sample face, as depicted in Figure 3. These initiatives reflect a well-structured approach to simulating real-world conditions by replicating the stresses encountered in subterranean environments.

Figure 3. Hydraulic jack and metal frame for confining pressure applications.

2.1.3. Loading system

A 200-ton indentation test device was used for testing in this study. The device includes two steel loading plates; the upper plate is fixed. The sample is loaded by moving the bottom plate, which stops automatically when the sample breaks. The force exerted during the test is displayed on the digital force measurement system’s display screen, as shown in Figure 4. The confining device is positioned with the sample on the lower plate, and the disc cutters created are fitted in place of the upper plate.

Figure 4. Test apparatus.

2.2. Sample preparation

Based on ISRM classification, three different material types with varying strengths were selected for this study, each with dimensions of 100 mm $ \times $ 100 mm $ \times $ 100 mm. Figure 5 illustrates Isfahan black granite as a typical high-strength rock, Isfahan Lashotor marble as a medium-strength rock, and low-strength concrete samples. Firuzkoh sand (type 131) was used to make the concrete samples, which were cured in water for 3 days to achieve lower strength, using a water-to-cement ratio of 0.5 and a sand-to-cement ratio of 2.5. The mechanical and physical properties of the materials are listed in Table 1. The unconfined uniaxial strength test was conducted using the recommended ISRM methodology. The tests were carried out on three types of materials. At least 5 samples were tested for each rock type, and the average value was taken as the unconfined compressive strength of the rock type.

Figure 5. The cube samples from right to left: granite, marble, and concrete.

Table 1. Physical and mechanical properties of the materials

2.2.1. Dry and saturated conditions

The samples used for the dry tests were stored in a dry environment for 2 weeks. For saturated tests, the marble and granite samples were placed in a container of de-aired water once they had been in the oven and reached a consistent weight. The weight of each wet sample was subsequently measured every hour, after which the wet sample was returned to the water. The weight increase was used to calculate the water content of the sample at various times as calculated by Eq. (1):

(1) $$ \omega =\frac{m_s-{m}_d}{m_d}\times 100\% $$

where $ {m}_s $ (g) and $ {m}_d $ (g) are the wet and dry masses of the sample, respectively, and ω ( $ \% $ ) is the water content. Because the weights of the samples were fixed, we determined that the sample was saturated, and the test was performed immediately. The samples were stored for 21 days in a container filled with water. In the marble samples, the average water absorbed was 0.14%, whereas that in the granite samples was 0.03%. The concrete samples were subjected to saturation tests after the curing period had passed.

3.1. Test design

Under three different confining stresses of 0, 2.5, and 5 MPa, all dry and saturated granite, marble, and concrete samples were broken with 14-inch and 20-inch disc cutters. The sample was placed inside a confining box, and steel plates were placed on both sides. The confining stress was applied using a hydraulic jack, and the sample was broken by a disc cutter by applying a vertical force at the center of the sample, as depicted in Figure 6.

Figure 6. Test system: (a) Details of testing setup (b) Indentation test schematic.

Each material type was tested with a minimum of three samples, and the failure load was determined by averaging the results.

3.2. Test procedure

There are six steps to perform the test, and the testing procedure is as follows:

1. (i) Finding the sample midpoint.

2. (ii) The device nuts were tightened after inserting the sample into the confining box and covering it with metal plates on both sides.

3. (iii) Place the confining box inside the loading system.

4. (iv) The designated disc cutters were placed at the center of the sample.

5. (v) The confining stress was applied using a hydraulic jack.

6. (vi) The machine was switched to loading mode, and force was applied until the sample failed. The failure load was recorded at that point.

7. (vii) After the test was completed and the machine was unloaded, the hydraulic jack was removed, and the specimen was removed from the machine.

A dry granite sample fractured using a 14-inch disc cutter at a confining pressure of 5 MPa is depicted in Figure 7.

Figure 7. Broken granite sample.

4.1. Effect of confining stress on disc cutter failure load

The results of the 14- and 20-inch disc cutter fracture tests conducted under confined conditions on dry and saturated granite are presented in Table 2. In these experiments, the lowest breaking load was 121.3 kN, and the highest was 180.5 kN. In studies of fractures in both dry and saturated granites, the CVs ranged from 0.09% to 2.8%. The relationships between the granite breaking load (kN) and confining stress (MPa) at failure when a 14-inch (Eq. 2) or 20-inch (Eq. 3) disc is used are depicted in Figure 8.

(2) $$ {F}_n=8.193\sigma +118.41 $$

(3) $$ {F}_n=7.72\sigma +138.58 $$

Figure 8. Relationship between the confining stress and failure load of granite.

When a 14-inch cutter was used, the coefficient of determination for failure was 0.85; when a 20-inch cutter was used, it was 0.95. Using a larger-diameter disc cutter reduces the influence of the confining stress on the breaking load when cutting granite, as shown in Eqs. (2) and (3). In novel ways, the confining stress effect on the rock-breaking load can be minimized, and drilling operations can be conducted more effectively by increasing the cutter disc diameter.

Table 3 presents the results of the marble saturation and dry breakage tests. Additionally, these tests were run under 0, 2.5, and 5 MPa stresses using 14 and 20-inch discs. Owing to the differences in the physical properties of granite and marble, the overall failure load was lower than that of granite. The failure load ranged from 55.71 kN at minimum to 94.35 kN at maximum. The CV ranged from 0.2% to 4.2%. The relationships between the dry and saturated marble fracture loads (kN) and the confined stress (MPa) when a 14-inch disc (Eq. (4)) or a 20-inch disc (Eq. (5)) is used are depicted in Figure 9.

(4) $$ {F}_n=4.832\;\sigma +58.952 $$

(5) $$ {F}_n=6.103\;\sigma +60.249 $$

Figure 9. Relationship between the confining stress and failure load of marble.

For the 14-inch and 20-inch cutters, the coefficients of determination for the breaking load were 0.89 and 0.88, respectively. A comparison of Eqs. (4) and (5) clearly reveals that the smaller disc cutter minimizes the impact of confinement stress during marble rock fracture.

The results of the fracture tests on the confining concrete samples cut with 14- and 20-inch cutters are shown in Table 4. The samples were subjected to both saturated and dry conditions. When a 14-inch cutter was used under saturated conditions, the minimum breaking load for the concrete samples was 22.95 kN, whereas when a 20-inch cutter was used under dry conditions, the maximum breaking load was 38.07 kN. Additionally, the CV of the samples increased from 1.1% to 6.3%. The relationship between the confined stress (MPa) and failure load (kN) of the concrete sample is shown in Figure 10, where a 14-inch disc (Eq. (6)) and a 20-inch disc (Eq. (7)) are used.

(6) $$ {F}_n=1.966\;\sigma +25.005 $$

(7) $$ {F}_n=1.804\;\sigma +28.673 $$

Figure 10. Relationship between the confining stress and failure load of concrete.

For the 14-inch cutter, the coefficient of determination was 0.77, and that for the 20-inch cutter was 0.84. Compared with those in granite and marble, the relationships between the confining stress and breaking load in concrete are more linear. These relationships demonstrate that employing a larger cutter diameter in concrete samples can decrease the impact of confining stress. While the linear equations provide a useful approximation within the tested stress range, it is important to acknowledge that the relationship between confining stress and failure load may become non-linear at higher stress levels. This could be due to the onset of non-linear elastic behavior or the development of micro-cracking within the rock samples. Future studies should investigate the behavior at higher confining stresses to determine the limits of the linear approximation.

4.2. Effect of confining stress on the failure load of different rocks

The graphs in Figures 8– 10 clearly shows the impact of restricting pressure. For all three materials, the failure loads increased as the confining pressure increased from 0 to 5 MPa, suggesting that greater confinement resulted in greater load-bearing capacities. This conclusion is consistent with the results of the indentation tests provided by Chen and Labuz (Reference Chen and Labuz2006) and Yin et al. (Reference Yin2014). By comparing Eqs (2)–(7) for the breaking tests using 14-inch and 20-inch cutters, the relationship between the breaking load and confining stress for all three material samples with varying compressive strengths is shown. The effect of confining stress on the failure load increased with increasing sample compressive strength; in other words, samples with greater compressive strength (e.g., granite) experienced a greater increase in the failure load due to the confining stress with different disc cutters. Additionally, according to the presented data, the confining stress effect on the failure of dry samples was greater than that on the failure of saturated samples. For example, in the fracture of a marble sample with a 14-inch disk cutter, when the confining stress changes from 0 to 2.5 MPa, the failure load changes by 32.4%, whereas in the saturated state with the same amount of increase in confining stress, the failure load changes by 31.2%. This suggests that confining stress has a more pronounced impact on the failure of high strength, dry rock materials than low strength or saturated rock samples.

4.3. Regression analysis

Numerous prediction models for TBM performance have been generated over time. Depending on the basis on which each model was built, each model was developed with some assumptions and restrictions. The experimental data were used to calculate the regression equation. The form of a multivariate linear regression model can be a useful starting point for understanding the relationship between failure load, confining stress, disc diameter, and compressive strength. A multivariate linear regression model was initially chosen for its simplicity and ease of interpretation. However, it is recognized that non-linear models may provide a better fit to the data, particularly if there are complex interactions between the variables. The coefficients of the equation were determined by calculating the best fit of the data with three independent variables: the confining stress ( $ \sigma \Big) $ , cutter disc diameter (d), and rock compressive strength ( $ {\sigma}_C $ ). The failure load ( $ {F}_n $ ) is the dependent variable.

(8) $$ {F}_n=-5.63+5.1\;\sigma +1.4\;d+0.79\;{\sigma}_C $$

Eq. (8) expresses $ \sigma $ in terms of MPa, d in terms of inch, and $ {\sigma}_C $ in terms of MPa. For the linear regression relationship, 0.96 is the coefficient of determination. In rock cutting, the coefficients of the confining stress, disc cutter diameter, and compressive strength of the rock indicate that when various types of rock are broken, the confining stress plays a more significant and useful role than other factors in the multivariate linear regression relationship.

These findings underscore the importance of confining stress, cutter dimensions, and compressive strength for optimizing the cutting performance of geological materials.