Experimental study of penetration of pre-cut free surface rock materials based on orthogonal tests and response surface testing methods

The parameters of the pre-cut kerf as the free surface on rock material have an important influence on the rock-breaking force of the disc cutter of TBM. The rock-breaking force of the disc cutter can be simplified as the normal penetration load of the metal indenter. In order to study the effect of the pre-cut free surface parameters on the penetration load, orthogonal experiments and response surface tests were carried out on sandstone and granite samples, and extreme deviation analysis and analysis of variance (ANOVA) were performed. The results showed that: (1) the order of priority of the factors affecting the penetration load of the disc cutter was Kerf depth > Kerf spacing > Kerf angle. (2) The fitting degree of the regression equation for the predicted values of the penetration loads of the sandstone and granite samples was R2 (Sandstone) = 0.9810 and R2 (Granite) = 0.9995, respectively, indicating that there was a good fit between the measured and predicted values of the penetration load of the two samples.


Introduction
The tunnel boring machine (TBM), which is now a crucial piece of equipment for tunnel construction, is capable of safe, economical, efficient, and environmentally friendly tunnel excavation [1].In recent years, based on the actual problems encountered in the project, water jetting [2] and other new technologies and methods for rapid excavation of rock have emerged continuously.Unlike the traditional disc cutters which break the rock by extrusion, water jets cut rock material by impact energy.On the surface of the rock, when the cutting is finished, there are macroscopic cuts of various sizes.At present, the development of high-pressure hydraulic coupling TBM [3] has verified the high feasibility and broad application prospect for rock-breaking with high-pressure water jet pre-cutting kerf to assist the TBM mechanical disc cutter.It is predicted that the TBM will operate in a synergistic manner for rock-breaking by using a mechanical disc cutter and pre-cutting kerf, as shown in Figure 1.In this case, the rock-breaking behavior of the mechanical disc cutter can also be significantly influenced by the dimensional parameters of the pre-cutting kerf in the rock mass.and (b) Simplified design scheme of coupling disc cutter and the nozzle of water jet.The force of the disc cutter for crushing the rock can be affected by the spacing, depth, and angle of the kerf when it is pre-cut in the rock mass by using a water jet and other methods.After creating a prefabricated kerf by using a high-pressure water jet, Zhang et al. [4] conducted a linear rock-breaking cutting test and found that the kerf depth significantly affects the force of the disc cutter for crushing the rock and specific energy.A series of rock-breaking disc cutter systems with water jet assistance were developed by Ciccu et al. [5].The water jet cut the rock mass at an incidence angle of 20° from the vertical line (Jet inclination from the vertical line: 20°).Lower kerf spacing can contribute to rock sample destruction on both sides of the disc cutter, according to the research by Cheng et al. [6].
The normal penetration force is the primary initiator of the internal crack system within the rock mass during TBM rock breaking [7].Laboratory experiments [8] have shown that the rock-breaking procedure using a disc cutter can be reduced to the issue of penetration beneath various -sized indenters.However, rock materials in penetration tests are mostly intact cubes, penetration experiments on rock specimens with pre-cut kerf are less well-studied, and the parameters of the pre-cut free surfaces are still insufficient for penetration loading and effects.
In this paper, the effect of kerf spacing, depth, and angle on the normal load of the indenter was explored by range analysis through the penetration tests on rock samples, and the normal load of the indenter was tested by the Box-Behnken design scheme of the response surface method.

Materials
China's Hubei and Sichuan provinces provided the granite and sandstone samples utilized in the penetration tests, respectively.To determine the physical and mechanical properties of the materials, three tests were conducted: uniaxial compression (Φ 50 mm × 100 mm), direct shear (100 mm × 100 mm × 100 mm), and Brazilian splitting (Φ 50 mm × 25 mm), as shown in Table 1 [9], the whole plate rock samples were measured 210 mm in length, 100 mm in breadth, and 40 mm in height.To pre-cut the plate sample, a 150 mm diameter diamond grinding wheel was employed, and the width of all the kerf was 2 mm, as shown in Figure 2.

Experimental design
The orthogonal experimental design can consider three factors such as kerf depth, spacing, and angle simultaneously to make a scientific and reasonable experimental arrangement.To examine the impact of these three variables on the penetration load, three-factor four-level orthogonal tests were conducted with the kerf depth, spacing, and angle as the orthogonal test factors.Without taking into account the interactions between the components, a total of nine groups of trials were designed by using the L9(3 4 ) experimental table.
The process situated in the middle of the test space's edge is incorporated into the Box -Behnken response surface design, which arranges parallel experiments to minimize experimental error.For this experiment, a three-factor Box-Behnken experimental design technique was adopted.Five of the seventeen groups of trials that were created were parallel experiments.The corresponding coding values for the three independent variables -kerf depth, spacing, and angle-were X1, X2, and X3 in order to investigate the impact of prefabricated kerf on penetration load.Then, the corresponding levels were -1, 0, 1.The specific design method is shown in Table 2.

Experiment procedures
The tests were carried out on the RMT-150C electro-hydraulic servo rigidity tester, as shown in Figure 3.A self-designed basic restraint device was employed to impose lateral limitations on both sides of the sample during the test.The indenter's simplified cross-sectional dimension was 40 mm × 13 mm, and it was manufactured of high-strength alloy steel, as illustrated in Figure 3(c).The loading rate of the RMT was 0.005 mm/s, and the control method was vertical piston stroke control.The indenter was continually punctured throughout the loading process until the sample was destroyed and the sample's typical penetration load was determined.Throughout the loading procedure, the RMT testing apparatus automatically monitored and recorded the penetration load value.

Results of orthogonal tests and range analysis
The results of nine groups of orthogonal tests of sandstone and granite rock samples are shown in Table 3, which shows the normal penetration load of the two kinds of rock samples from loading to failure.The test results were analyzed by range analysis, as shown in Table 4. the sum of the penetration loads in column j for level i; kij indicates the average of the penetration loads in column j for level i, and the extreme difference R=kijmax -kijmin; i=1,2,3, was the level number.
The mean range Rj reflected the variation range of the penetration load when the level of the factor in column j changed.The factor's impact on the penetration load increased with increasing Rj.The range study indicated that the following sequence of parameters affected the disc cutter's penetration load: Kerf depth > Kerf spacing > Kerf angle.

Response surface test and variance analysis
The results of the penetration tests for the sandstone and granite samples are presented in Table 5, and then the multiple regression and variance analysis are carried out.According to the reference [10], the connection between the response value and the independent factors in the response surface test can be characterized according to Equation (1) as follows: ) where Y represents the predicted value of the response, i.e. the penetration load; is the intercept; is the linear coefficient; is the squared coefficient; is the interaction coefficient; and are the values of the influencing factors.A f t e r m u l t i p l e r e g r e s s i o n s o f t h e p e n e t r a t i o n l o a d o f t w o k i n d s o f s a m p l e s , t h e q u a d r a t i c polynomial regression equation was obtained.In Equations ( 2) and (3), YS represents the predicted value of the penetration load response of sandstone samples, and YG represents the predicted value of the penetration load response of the granite samples.And the Equation is as follows: YS=103.38-18.81A+6.79B-3.45C+1.85AB-0.44AC+0.57BC+9.49A 2 -0.94B 2 +0.94C 2  (2) YG=280.96-37.03A+5.69B-3.46C+2.11AB-0.13AC-0.30BC-5.91A 2-0.40B 2 +1.07C 2  (3) where A, B, and C represent the independent variables Kerf depth, Kerf spacing, and Kerf angle, respectively.
After RSM analysis, the ANOVA table can be obtained, as shown in Tables 6 and 7.Each element's substantial impact on the experimental outcomes was indicated by the p-value; if p < 0.05, the factor was found to have a significant influence.The factor significantly affected the experimental outcomes if p < 0.0001.In the model, p < 0.0001 indicated that the regression effect was very significant.Notice: R 2 = 0.9995 The fitness R 2 test can be used to evaluate the reliability of the model.The discrepancy between the response value and the actual value can be shown by the R 2 indicator [11], and the value range of R 2 is [0, 1].The closer R 2 was to 1, the smaller the difference between them was, and when R 2 = 1, they were completely coincident.
In Equation ( 4), SR is the sum of regression squares value; ST is the sum of total deviation squares; * , , are the measured values, predicted values of the "i th " group of experimental results, and the mean values of all experimental results; n represents the sample size, i.e. the number of experiments.
As shown in Tables 6 and 7, the fitness of the regression equation for the predicted values of the penetration loads of the sandstone and granite samples were R 2 (Sandstone) = 0.9810 and R 2 (Granite) = 0.9995, respectively, showing that the observed and expected values of the penetration loads of the two samples fit each other well.For each group, scatter plots representing the measured and estimated penetration loads of the samples were created.The scatter points were distributed around the straight line Y = X, indicating that the predicted re sults of the m odel we re approxim ately equal to the experimental sample data, which proved the excellent fitness of the model, as shown in Figure 4.

Conclusions 1)
The rock-breaking technology with the combination of prefabricated kerf and mechanical cutters can effectively use the water jet, laser, and other technologies to pre-cut kerf to create a free surface in the rock mass.The rock-breaking efficiency of the mechanical cutters can be effective after prefabricated kerf.The three primary preset kerf parameters -kerf depth, kerf spacing, and kerf anglehad a significant impact on the disc cutter's ability to break through rocks.
2) The L9 (3 4 ) orthogonal test and response surface test were carried out for sandstone and granite samples.The test results were analyzed by range analysis and variance analysis, respectively.The range analysis indicated that kerf depth, kerf spacing, and kerf angle were the elements that affected the disc cutter's penetration load in that order.
3) The fitness of the regression equation of penetration load prediction value of sandstone and granite samples was R 2 (Sandstone) = 0.9810 and R 2 (Granite) = 0.9995, respectively, demonstrating that the observed and expected penetration load values for the granite and sandstone samples matched together well.

Figure 1 .
Figure 1.Combining mechanical disc cutters with water jets to crush rocks, (a) TBM Cutter Head [3],and (b) Simplified design scheme of coupling disc cutter and the nozzle of water jet.The force of the disc cutter for crushing the rock can be affected by the spacing, depth, and angle of the kerf when it is pre-cut in the rock mass by using a water jet and other methods.After creating a prefabricated kerf by using a high-pressure water jet, Zhang et al.[4] conducted a linear rock-breaking cutting test and found that the kerf depth significantly affects the force of the disc cutter for crushing the rock and specific energy.A series of rock-breaking disc cutter systems with water jet assistance were developed by Ciccu et al.[5].The water jet cut the rock mass at an incidence angle of 20° from the vertical line (Jet inclination from the vertical line: 20°).Lower kerf spacing can contribute to rock sample destruction on both sides of the disc cutter, according to the research by Cheng et al.[6].The normal penetration force is the primary initiator of the internal crack system within the rock mass during TBM rock breaking[7].Laboratory experiments[8] have shown that the rock-breaking procedure using a disc cutter can be reduced to the issue of penetration beneath various -sized indenters.However, rock materials in penetration tests are mostly intact cubes, penetration experiments on rock specimens with pre-cut kerf are less well-studied, and the parameters of the pre-cut free surfaces are still insufficient for penetration loading and effects.In this paper, the effect of kerf spacing, depth, and angle on the normal load of the indenter was explored by range analysis through the penetration tests on rock samples, and the normal load of the indenter was tested by the Box-Behnken design scheme of the response surface method.

Figure 2 .
Figure 2. Schematic diagram of rock sample with various prefabricated kerf at granite and sandstone.

Figure 3 .
Figure 3. Testing system, (a) testing apparatus; (b) lateral limiting mechanism; and (c) solid and dimensional indenter designs.The loading rate of the RMT was 0.005 mm/s, and the control method was vertical piston stroke control.The indenter was continually punctured throughout the loading process until the sample was destroyed and the sample's typical penetration load was determined.Throughout the loading procedure, the RMT testing apparatus automatically monitored and recorded the penetration load value.

Figure 4 .
Figure 4. Comparison between measured value and predicted value of crack initiation strength, (a) sandstone and (b) granite.

Table 1 .
. Characteristics of rock samples in terms of mechanics and physicality.
a The meanings of the numbers in brackets are kerf depth/mm, kerf spacing/mm, and kerf angle/° respectively.

Table 4 .
Range analysis of orthogonal experiment.
a The meanings of the numbers in brackets are kerf depth/mm, kerf spacing/mm, and kerf angle/°, respectively.

Table 6 .
Analysis of the variance table of sandstone.

Table 7 .
Analysis of the variance table of granite.