Numerical Simulation of Rock Plate Cutting with Three Sides Fixed and One Side Free

In order to overcome conical pick wear in the traditional rock cutting method, a new cutting method was proposed on account of increasing free surface of the rock.-emechanical model of rock plate bending under concentrated force was established, and the first fracture position was given. -e comparison between experimental and numerical results indicated that the numerical method is effective. A computer code LS-DYNA (3D) was employed to study the cutting performance of a conical pick. To study the rock size influenced on the cutting performance, the numerical simulations with different thickness, width, and height of a rock plate was carried out. -e numerical simulation with the different cutting parameters of cutting speed, cutting angle, and cutting position influenced on cutting performance was also carried out. -e numerical results indicated that the peak force increased with the increasing thickness of rock plate. With the increasing width and height of the rock plate, the peak force decreased and then became stable. Besides, the peak force decreased with the increasing of cutting position lxp/lx. Moreover, the peak force increased and then decreasedwith the increasing of cutting angle.-e cutting speed has nonsignificant influence on the peak force.-e strong exponential relationship was obtained between the peak force and cutting position, thickness, height, and width of the rock plate at a confidence level of 0.95. A binomial relationship was observed between the peak force and cutting angel. -e cutting force comparison between traditional rock cutting and rock plate cutting indicated that the new cutting method can effectively reduce peak cutting force.


Introduction
Rock cutting is frequently encountered in some industry, for example, tunnel excavation, coal and rock mining, oil exploitation, and metal mining.As a typical excavation machine for rock cutting, a roadheader is widely used in underground tunneling engineering.For a given rock formation, the cutting method and the peak cutting forces during rock cutting process are very important for researchers to design the roadheaders and cutting tools.
Many scholars have made several attempts to research the rock fracture mechanism and predict cutting forces using theoretical, experimental, and numerical approaches.e pioneering work on conical pick cutting theories was performed by Evans [1][2][3].Evans proposed that the interaction between conical pick and coal was essentially threedimensional and concluded that the tensile and compressive strength were the primary factors influenced on mean peak cutting force in the rock cutting progress.Goktan [4] and Roxborough et al. [5] put forward revisions on Evans' cutting theory.ese theories are more accurate and scientific to predict the cutting forces.With the high tensile strength rock, shear failure is the dominant property during the cutting progress [6].Bilgin et al. [7] and Kel et al. [8] carried out the experimental researches with the linear cutting machine, in order to study the dominant rock properties that influenced on conical pick performance.Liu et al. [9] proposed a new rock cutting method and carried out the experiment to study the cutting performance.e numerical simulations are more detailed and faster than the experimental research.Huang et al. [10], Li et al. [11,12], and Jiang et al. [13] used the finite element method (3D) in three dimensions to model rock cutting progress.ey pointed out that there are close correlation between experimental and theoretical researches.Huang et al. [14] and Lei and Kaitkay [15] used PFC (2D) to simulate the rock cutting process in two dimensions, and the failure mode and hydrostatic pressure were created to research the rock failure performance.
Although considerable researches have been carried out on the classical rock cutting method, there was seldom consideration for a new cutting method.e conical picks are damaged rapidly when encounter hard rock due to the inability of conical pick and the stiffness of the cutting head.In this paper, a new rock cutting method was proposed, and new type equipment was shown.
e availability of the numerical simulation was verified by theoretical analysis and experimental research.In the simulations, a damage material and the erosion criteria were used in the code to dominant the rock failure.e influence of height, thickness, and width of rock plates on crack extends, and peak force was discussed.e cutting parameters of cutting speed, cutting angle, and cutting position of the conical pick influenced on cutting performance were also discussed.

Proposal of the New Rock Cutting Method
Increasing the cutting power is an effective method for improving the cutting ability of the tunneling machine for cutting hard rock [16].But on the contrary, the cutting tool wear would aggravate with the increase of cutting power as shown in Figure 1(a).Furthermore, the pick head have a great possibility to be stripped away from the conical pick as shown in Figure 1(b).
erefore, it is necessary to propose a new cutting method for cutting the hard rock.
In order to reduce the cutting force and solve the problem of conical pick wear, the diamond saw-conical pick sequential cutting method was proposed.Diamond is the hardest substance in nature, and diamond saw blades are widely used in stone exploitation and processing industry.Especially in hard rock cutting, it can obtain a good cutting performance on the basis of negligible wastage.e diamond saw was employed to slot the base rock and obtain the saw kerf as shown in Figure 2(a).e freed surfaces of the base rock increased with increasing saw kerfs and the rock plates formed.On this basis, conical picks were applied to cut the rock plates as shown in Figure 2 2 Advances in Materials Science and Engineering new rock cutting method, the saw-pick combined cutting machinery was produced as shown in Figure 3. e interaction between conical pick and rock has a considerable value to research.e rock plate was approximated as rectangular plates because it was narrow and long so that the radian can be ignored.e constraint conditions of rectangular rock plates depend on the base rock constraint conditions, and the three sides xed rock plate formed as shown in Figure 4.

Theoretical, Experimental, and Numerical Studies
3.1.eoretical Studies.Based on the reciprocal theorem [17,18], the work of the rst force group on the displacement corresponding to the second force group equals to the work of second force group on the displacement corresponding to the rst force group.e rock plate with four edges simply supported is a basic mechanical system as shown in Figure 5(a), and the rock with the arbitrary boundary condition as the actual mechanical system is shown in Figure 5(b).e basic mechanical system and actual mechanical system have the relationship as shown in the following equation:

Advances in Materials Science and Engineering
where ω 1 (ξ, η) is the de ection of the basic system, ω 2 (ξ, η) is the de ection of the actual system, V is the boundary shear force, R is the angular corner force, k is the corner replacement, M is the bending moment, and P is the concentrated force.What needs to be stressed is that all the parameters in the basic system can be expressed by trigonometric series and they are known [17,18].e parameters are related to v (Poisson's ratio), E (modulus of elasticity), h (thickness of the rock plate), and a and b. e rock plate with three sides xed and one side free is shown in Figure 6(a).In order to solve the de ection of the rock plate, the xed boundary was decomposed into the combination of simply support and bending moment as shown in Figure 6(b).Reciprocal theorem was applied between Figure 6(b) and Figure 5(a).
where M x 0 , M x a , M y 0 , and ω y b can be represented by trigonometric series: where αm mπ/a, βn nπ/b, and A n , B n , C m , and D m are unknown quantities in (3).e boundary condition should satisfy the following equations: In order to give the solution of the rock plate bending, a b, h/a 0.1, and v 0.3 were taken.e concentrated force was acted on (a/2, b). e de ection of free side is shown in Figure 7(a), where D is the bending strength.e bending moments of M x 0 and M y 0 are shown in Figures 7(b) and 7(c).
e maximum de ection of the free side was 4.6 (Pa 2 /D × 10 −3 ), obtained from the middle of the free surface.Advances in Materials Science and Engineering e maximum bending moment on the xed side was 1.02 (−P × 10 −1 ), obtained from the border of free side and xed side.With the side y 0, the maximum bending moment of 0.67 (−P × 10 −1 ) was obtained.e de ection and bending moment can cause the stress of the rock plate, and the bigger the bending moment, the bigger the tensile stress [19].erefore, the rock plate rst breaks at the junctional point of free side and xed side, the middle of the free side, and the middle of xed side y 0.

Rock Plate Cutting
Test.Mechanical properties of the rock are essential for rock plate cutting and numerical simulation.Uniaxial compression tests for uniaxial compressive strength (UCS) were carried out on a standard core sample: the cylinder which has a diameter of 50 mm and length of 100 mm.Axial loading rate was limited to 0.5∼1.0MPa/s.Axial displacement velocity is 0.1 mm/min.e Brazilian tensile strength (BTS) test was conducted on the core sample with a diameter of 50 mm and length of 50 mm.Elasticity modulus (E) was measured at 1/2 of the rock ultimate uniaxial compressive strength.In order to obtain accurate measurements, three group experiments were carried out.e density of the rock was obtained from the ratio of sample mass to sample volume.e crushed sample and the stress-strain curve are shown in Figures 8(a) and 8(b).e measurements of the granite are shown in Table 1.
e rock cutting test bed schematic in the experiments is shown in Figure 9. e static torque sensor can monitor the torque ranging from −3,000 N•m to 3,000 N•m. e transmitter output voltage ranges from −2.5 V to 2.5 V. e signal acquisition device includes ten independent input channels 6 Advances in Materials Science and Engineering and monitors the voltage data.Data sampling frequency is 10,000 HZ.A computer is used to control the acquisition device and record data.e conical pick is stable and the xture moved linearly on the slide.In order to agree with the real rock plate cutting, three sides of the rock plate were xed by xture.

Numerical Model.
e FEM software LS-DYNA (3D) was employed for all simulations.A conical pick and a rock were considered for modeling as shown in Figure 10(a).
e material type MAT_111_JOHNSON_HOLMQUIST_ CONCRETE (JHC) was given to the rock [20,21].e parameters of granite rock are given in Table 2. e rock consisted of two parts: base rock and rock plate.e impact angle and the tip angle of conical pick were 57 °and 80 °, respectively.α and β in Figure 10(b) are the tip angle and impact angle, respectively.Furthermore, the friction angle between the cutting pick and rock plates was assigned as 8.5 ° [7,22].In order to obtain crack propagation and fragment separation in the numerical simulation, a damage material model and erosion criteria were used for the base rock.
e ERODING_SURFACE_TO_ SURFACE and AUTOMATIC_GENERAL contact type was applied between the moved cutting pick and the stationary base rock.Hourglass and element distortion frequently occurred in the rock plate cutting progress.ese situations in uenced the cutting force signi cantly.erefore, the fully integrated calculation method was adopted.In Figure 10(a), l xp was used to express the position of the conical pick, and l z , l y , and l x were height, thickness, and width of the rock plates, respectively.e cutting angle was represented by θ, and it is the angle between cutting speed and zoy plane.Advances in Materials Science and Engineering z-direction.In order to avoid the in uence of the stress wave, nonre ecting boundary was applied to the entire surface.

Performance Check of Numerical Model.
e experimental simulation and numerical simulation were carried out with the same cutting parameters for granite.e cutting angle is 0 °, and cutting position l xp /l x 1/2.e thickness of granite was 20 mm. e experimental result and numerical result with l x of 400 mm and l y of 120 mm for granite are shown in Figure 11.
e crack propagation and cutting results are similar between experimental and numerical results.Fracture appeared at junctional position of free side and clamped side and at the middle of the free surface.It was consistent with the theoretical results.e variation of the cutting force is shown in Figure 12.It is clear that the regulation of curves is similar between experimental and numerical results.e peak cutting force obtained from numerical simulation and experiment are very similar; therefore, the numerical method is available.

Numerical Results and Discussion
e cutting forces were obtained from numerical and experimental methods, but there is no comparison with the cutting force obtained from traditional rock cutting.e advantage of the rock plate cutting has not been shown.e experiment of the traditional rock cutting had been carried out by a linear cutting machine [8].e mechanical properties of the rock in the experiment are shown in Table 3.With these parameters, the numerical simulations were carried out.e other numerical parameters were the same as given in Table 2.

e Rock Plate Cutting Process and Variation of Cutting
Force.
e numerical results of rock cutting process for a cutting speed of 3 m/s, cutting angle of 0 °, and cutting position l xp /l x of 1/2 for sandstone 1 are shown in Figure 13.With the movement of the conical pick, a cutting force was exerted on the rock plate, and then the fragment separated from the rock plate.As shown in Figure 13(a), the rock plate was crushed only at the position around the conical pick.e variation in cutting force is shown in Figure 13(f), stage I.With the increasing cutting distance of conical pick, the cutting force increased.With the element failure and deleted from the rock plate, the cutting force decreased rapidly.As shown in Figure 13(b), the rock plates fracture not only around the cutting pick, but also at the place of the border between rock plate and base rock.With the increasing   is can be explained by the factor that the middle of the rock plate bore greater tensile stress and was easier to failure.In this cutting process, the cutting force reached to the maximum value, as shown in Figure 13(f).Subsequently, some tiny cracks propagated from the main crack as shown in Figures 13(d); with the movement of the pick, the cracks propagated rapidly and irregularly on the surface and the inside of the rock plate.e rock plate border generated more crush regions, and the crack extended to the base rock.e cutting force in this process showed an increasing trend as shown in Figure 13(f), stage III.As shown in Figure 13(e), with increasing cutting distance, the cracks connected to each other in the surface and inside of the rock plate, and then the fragment with random size and shape was separated from the rock plate.e cutting force decreased in this process as shown in stage IV. Figure 13(e) also shows that considerable cracks remain on the rock plate as the results of rock cutting.ese results provided the foundation for reducing cutting force when the rock plate is cut again.

e Failure Mode of the Rock.
Failure mode of the rock plate is very important for researching the cutting force and fracture process.As shown in Figure 14, ten failure elements in di erent places were selected to research the failure mode in the saw blade-conical pick cutting method.
ree failure modes may occur in the rock cutting process: tensile, compressive, and shear failure.4. It can be concluded that the main failure mode of rock is tensile failure, and a few elements experienced compressive failure at the position around the conical pick.
e ultimate tensile strength is the main dominant property for the rock failure during rock plate cutting process.

e Height of the Rock Plate In uenced on the Cutting
Performance.
e cutting results for the simulation that were carried out at a cutting speed of 3 m/s, l x of 500 mm, l y of 16 mm, and l z of 20 mm, 30 mm, 60 mm, and 120 mm with sandstone 1 are shown in Figure 15.During the cutting process, the fragments separated from rock plate with random morphology and size and cracks propagated irregularly.It was easier to generate fragments near the pickrock interaction region.e rock fracture from the bottom border with the lower l z is shown in Figure 15(a).At the end of the cutting process, there were no cracks remained on the rock plate.With the increasing l z , more cracks generated on the rock plate as shown in Figures 15(c  120 mm, all of the cracks were generated on the edge of the rock plate.erefore, the larger l z was more favorable for next step rock cutting because there were more cracks.is can reduce cutting force e ectively.e peak cutting forces for l z of 20 mm, 30 mm, 60 mm, and 120 mm were 3.1 kN, 3.0 kN, 2.2 kN, and 2.2 kN as shown in Figures 15(b), 15(d), 15(f), and 15(h), respectively.It is clear that the peak cutting force decreased with the increasing l z and nally tended to be stable.
e corresponding cutting distances to the peak cutting forces were 0.0012 m, 0.0013 m, 0.0017 m, and    0.0017 m.It increased with the increasing l z and nally became stationary.It also can be seen that the variation of cutting force was similar before peak force generated when compared with Figures 15(b), 15(d), 15(f), and 15(h), while after peak force, the variation of cutting force was signicantly di erent.e cutting force decreased rapidly with the higher l z and decreased slowly with the lower l z .e relationships between the peak cutting force and height of rock plate for di erent rock properties are shown in Figure 16.e peak force decreased with increasing height of the plate.However, when the height increased to a certain value, the peak cutting force varied nonsigni cantly.It also indicated that when both the height and the thickness of rock plate increased to a certain value, the peak force is stable for the certain thickness of plate.e minimum value of the peak force for a certain thickness of plate with various height and width was named as "stable peak force."With the thickness of 16 mm, the minimum height and width corresponded to the stable peak force were 60 mm and 200 mm, as shown in Table 5.It means if the height is more than 60 mm and width is more than 200 mm, the peak cutting force should be stable peak force.

16
Advances in Materials Science and Engineering area no longer increases.At the cutting distance of 0.0008 m, the cutting force reached to the maximum value of 3.1 kN as shown in Figure 17(b) and larger than 3.0 kN shown in Figure 15(c).So, the cutting force decreased with the increasing l x .e cutting distance corresponding to the peak cutting force was increased with the increasing l x .e peak force varied signi cantly with the lower width of plate as shown in Figure 18.When compared with Figure 16, the peak force had a sharper decrease with decreasing width.Speci cally, width had a greater in uence on peak force than height.It can be attributed to constraint condition of the rock plate.In detail, the constrained left side and right side lead to the rock plate harder to deform in the width direction.In the height direction, only the bottom was constrained, and therefore, it is easy to deform and crush.
e minimum height and width corresponding to stable peak force were obtained from Figure 18 and were the same as given in Table 5.

e ickness of the Rock Plate In uenced on the Cutting
Performance.
e cutting results with di erent thickness of 10 mm, 16 mm, 20 mm, and 30 mm as shown in Figures 19(a) and 19(c) and 13(e) were compared.ere were little fragments generated in the cutting process with the thickness of 10 mm as shown in Figure 19(a).As for crack, only the main crack generated.When the thickness was increased to 16 mm, both middle and border obtained cracks as shown in Figure 19(b).When the thickness is of about 20 mm and 30 mm, cracks appeared on the rock plate randomly and the quantity increased rapidly.Accordingly, the quantity and size of cracks and fragments increased with increasing thickness.e peak force obtained in thickness of 20 mm was 3.5 kN, while that in the thickness of 30 mm was 7.4 kN. e higher the thickness, the higher the cutting force.
To study the thickness in uenced on the peak cutting force, 8 group numerical results were extracted for analyzing the regularities.e thickness of the rock plate had a significant in uence on peak cutting force as shown in Figure 20.All of the minimum peak forces were obtained in the thickness 10 mm, while all of the maximum peak forces were obtained in the thickness of 30 mm. e peak force increased with increasing thickness of rock plate signi cantly.It was because the bending strength of the plate increased with the thickness of the rock, and the rock plate with large bending strength would bring out greater resistance.

4.6.
e Cutting Angle In uenced on the Cutting Performance.By comparing the cutting results shown in Figures 19(b), 21(a), and 21(c) with θ of 0 °, 5 °, and 10 °, the cutting results with θ of 0 °obtained the fewest cracks.In contrary, θ of 10 °obtained the most cracks.It is indicated that the quantity of the cracks had a strong association with θ. e quantity of the cracks and fragments increased with increasing θ.However, θ of 30 °obtained fewer cracks when compared with θ of 5 °and 10 °.It is interesting that cracks obtained only at one side of the rock plate as shown in Figure 21(e) with θ of 30 °. So, it can be concluded that too higher values of θ reduced the crush region of the rock plate.e peak force obtained in θ of 0 °was 2.2 kN.It is lower than 2.9 kN and 3.9 kN that were obtained in θ of 5 °and 10 °as shown in Figures 21(b) and 21(d).However, the fewest peak force with 1.9 kN was obtained in θ of 30 °as shown in Figure 21(f).e wave form is also signi cantly di erent from Figures 21(b) and 21(d).ere are three peak forces in the cutting process for θ of 30 °, and the frequency of the cutting force increased.Meanwhile, the cutting distance obviously increased with the increasing cutting angel.
e variation in peak force with cutting angle is shown in Figure 22. e in uence of cutting angle on peak force is signi cant.e curve shows that peak force increased rst and then decreased with the increasing cutting angle.All of the maximum peak force obtained at cutting angle of 10 °, while all of the minimum peak force obtained at cutting angle of 30 °.

4.7.
e Cutting Position In uenced on the Cutting Performance.Cutting position (l xp /l x ) is a key factor inuence on peak cutting force, and l xp is as shown in Figure 10.e previous studies were all carried out at l xp /l x of 1/2.To study l xp /l x in uence on the peak force, l xp /l x of 3/8, 1/4, and 1/8 were also carried out for simulation.Crack and fragment vary nonsigni cantly with the decrease of l xp /l x as shown in Figures 19(b As shown in Figure 24, the peak force decreased signi cantly with l xp /l x of 1/8 increase to 1/4.However, the cutting force was stable with l xp /l x increase from 1/4 to 1/2.As a conclusion, peak force decreased with increasing l xp /l x and then became stable.e cutting position near the border between rock plate and base rock should be avoided during the cutting process.e Cutting Speed In uenced on the Cutting Performance.It is indicated that cutting speed has nonsigni cant in uence on the cutting performance by comparing cutting results with cutting speed of 3 m/s as shown in Figure 19(b), 0.2 m/s as shown in Figure 25(a), and 1 m/s as shown in Figure 25(b).Cracks obtained in the three cutting speeds were very similar to each other, although there was a little di erence (Figure 25).e cutting speed had no obvious in uence on cutting force as shown in Figure 26.
is result is same to the traditional rock cutting that, in the lower cutting speed, the cutting force varies nonsigni cantly [23].erefore, it was meaningless to research the cutting speed in uence on cutting performance and cutting peak force.

Regression Analysis between Peak Force and Variables.
e strong correlation was obtained between peak force and the cutting variables as shown in Figure 23.
e exponential relationships exist between peak cutting force and height, width, and thickness of rock plate as shown in Figures 27(a) and 27(c).e correlation coe cients of 0.8620, 0.8148, 0.9848, 0.9758, 0.9997, and 0.9768 showed strong agreement between the cutting variables and peak force.A binomial relationship exists between peak force and cutting angle, and an exponential relationship exists between peak force and cutting position of l xp /l x as shown in Figures 27(d e con dence level is very important to evaluate the accuracy of regression analysis results.0.95 is most frequently used con dence level [23].At the con dence of 0.95, the regression results to predict the peak cutting force are shown in Table 6.e maximum and minimum P values are 4.8e −2 and 6.0e 9 , lower than 0.05, certifying that the regression analysis results are valid and reliable.

e Comparison between Rock Plate Cutting and Linear
Rock Cutting.Since the new cutting method aim to decrease the cutting force, the comparison between rock plate cutting and classic linear rock cutting is necessary.e stable peak cutting force for sandstone 1, sandstone 2, sandstone 3, and limestone with thickness of 30 mm were 7.5 kN, 6.1 kN, 4.9 kN, and 5.6 kN, respectively.e peak cutting force that was obtained from linear rock cutting experiment was 48.7 kN.28.1 kN, 15.9 kN, and 29.4 kN for sandstone 1, sandstone 2, sandstone 3, and limestone, respectively [7].e comparison between rock plate cutting and linear rock cutting is shown in Figure 28.It was obvious that rock plate cutting force was signi cantly smaller than linear rock cutting.

Conclusion
In this research, a new rock cutting method was proposed.To predict the cutting performance and peak force of the   conical pick, numerical simulations were carried out.e feasibility of numerical method was veri ed by the experimental results.In the simulation, the compressive damage constitutive model was combined with erosion criteria for controlling the rock failure.e peak cutting force and cutting performance were studied with varying thickness, height, and width of the rock plate.e cutting angle, cutting speed, and cutting position of the conical pick in uence on peak force were also researched.
(1) e numerical result agreed with the experiment results.It indicated that the numerical method is an e ective method to study rock plate cutting.(2) e cutting process of rock plate in the simulation was in good correspondence with the theoretical.Rock plate fractures at the intersection of free side and xed sides, and the middle of the free side.(3) e cracks obtained during the cutting process varied signi cantly with varying thickness, height, and width of rock plate.However, the cracks varied nonsigni cantly with cutting speed and cutting position of the conical pick.With increasing cutting angle, cracks increased and then decreased.(4) e peak force also varied signi cantly with varying thickness, height, and width of rock plate.With increasing width, height of rock plate, and the cutting positing of l xp /l x , the peak cutting force decreased and remained stable.Moreover, the peak cutting force increased with increasing thickness and increased and decreased with increasing cutting angle.And nally, the peak force varied nonsigni cantly with varying cutting speed.(5) e correlation coe cients obtained between peak force and cutting parameters were ideal, indicating a strong relationship between peak force and cutting parameters of rock plate and conical pick.e P values obtained at the con dence level of 0.95 were all lower than those of other con dence levels, indicating that the regression results are reliable.

Figure 2 :
Figure 2: Diamond saw-conical pick cutting method: (a) the formation of kerf and (b) rock plates cutting by conical picks.

Figure 5 :Figure 6 :
Figure 5: Solving system of bending rock plate: (a) the basic system and (b) the actual system.

Figure 7 :
Figure 7: e solution of rock plate bending: (a) the de ection of free side, (b) bending moment as the xed side x 0, and (c) bending moment as the xed side y 0.

Figure 8 :
Figure 8: e experiment of granite: (a) the compression test and (b) stress-strain curve.

Figure 11 :
Figure 11: Rock cutting results: (a) experimental results and (b) numerical results.

Figure 12 :
Figure 12: e comparison between experimental cutting force and numerical cutting force: (a) the variation of cutting force and (b) peak cutting force.

Figure 14 :
Figure 14: e variation in the stress with time: (a) test point 1 and (b) test point 2.
e tensile stress value reaching to 11.6 MPa indicates tensile failure, while the damage value of the element reaching to 1 indicates the compressive failure, otherwise the element shear failure.e three failure modes do not occur simultaneously.e positive value of the pressure indicates the rock bore compressive stress.e negative value of the pressure indicates the rock bear tensile stress.As shown in Figure 14(a), the test point 1 bore the minimum pressure of −8.9 MPa, less than limited tensile strength of −11.6 MPa, while the damage value of test point 1 reached to 1.It indicated that test point 1 experienced compressive damage failure.As shown in Figure 14(b), the damage value of test point 2 was zero, so the element failure was not caused by compressive stress.e minimum pressure of −11.5 MPa indicated that the failure mode of test point 2 was tensile failure.e failure pressure was slightly greater than −11.6 MPa due to the data sampling frequency.Among the ten test points, nine of them experienced tensile failure and one experienced compressive failure as shown in Table ) and 15(e) besides the fragment separation.e cracks basically were close to the arc.It agreed with experimental results.When l z increased to

ForceFigure 15
Figure 15: e cutting results with l x of 500 mm, l y of 16 mm, and cutting speed of 3 m/s for sandstone 1: (a), (c), (e), and (g) are l z of 20 mm, 30 mm, 60 mm, and 120 mm, respectively, and (b), (d), (f), and (h) are the corresponding cutting forces.

Figure 16 :
Figure 16: e variation in peak cutting force with height of rock plate for di erent rock properties: (a) sandstone 1, (b) sandstone 2, (c) sandstone 3, and (d) limestone.

Figure 17 :Figure 18 :
Figure 17: e cutting results with l x of 120 mm, l y of 16 mm, l z of 30 mm, and cutting speed of 3 m/s for sandstone 1: (a) fracture results and (b) the variation of cutting force with cutting distance.

Figure 19 : 1 Figure 20 :
Figure 19: e cutting results of the rock with a cutting speed of 3 m/s, l x of 500 mm, and l z of 200 mm: (a) l y of 10 mm, (b) l y of 16 mm, (c) l y of 20 mm, and (d) the variation of cutting force.

Figure 21 :Figure 22 :
Figure 21: e cutting results with l x of 500 mm, l y of 16 mm, and l z of 200 mm for sandstone 1: (a), (c), and (e) are cutting results for θ of 5 °,

Figure 23 :
Figure 23: e cutting results with l x of 500 mm, l y of 16 mm, and l z of 200 mm for sandstone 1: (a, c) the cutting results for l xp /l x of 1/8 and 1/4, and (b, d) the variation in cutting force with cutting distance for l xp /l x of 1/8 and 1/4.

1 Thickness = 16 mm
Figure 24: e variation in the peak force with cutting position of l xp /l x .
) and 27(e) e coe cients of 0.669, 0.9153, 0.9786, and 0.9763 indicate that the analysis results are satis ed.

Thickness = 16 mmFigure 25
Figure 26: e variation in the peak force with cutting speed.

y = 2 .ForceForceForceForce
Figure 27: e relationship between peak cutting force and cutting variables: (a) height of rock plate, (b) width of rock plate, (c) thickness of rock plate, (d) cutting angle, and (e) cutting position of l xp /l x .

Table 1 :
Mechanical properties of the granite.

Table 2 :
Parameters of JHC for granite.

Table 4 :
e stress and damage values of the test points.

Table 5 :
e minimum height and width corresponding to stable peak force for thickness of 16 mm.

Table 6 :
Regression results to predict peak force.