Research on Cutting Characteristics of Rock Plate with Two Sides Fixed and Two Sides Free

College of Transportation, Shandong University of Science and Technology, Qingdao 266590, China College of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qingdao 266590, China College of Information Science and Engineering, Shandong Normal University, Jinan 250358, China Advanced Manufacturing Technology Center, Shandong University of Science and Technology, Qingdao 266590, China


Introduction
Roadheader is the main engineering machinery for roadway excavation. As the main cutting tool, conical pick wear frequently occurs in the driving process of rock roadway, especially when encountering hard rock. e increase of pick wear leads to the increased cost of roadway excavation. Besides, the rock cutting efficiency is markedly reduced. e research of rock cutting has great significance for guiding production.
Considerable researches have been carried out on the interaction between rock and cutters. e most classical theory is proposed by Evans [1], who proved that tensile and compressive strength of rock and cutting depth of cutter are the major factors influencing cutting performance. Evans also pointed out that when the cone angle of conical pick decreased to zero, the cutting force was not equal to zero. erefore, his cutting theory had defects, and it is hoped that other scholars can solve the problem. Goktan [2] and Roxborough and Liu [3] optimized the theory of Evans [1], taking into account friction angle between cutter and rock. Results demonstrated that the new theory was closer to the experimental results. Goktan and Gunes [4] carried out the full-scale tests to establish the equations to predict the cutting force and the equations suitable to the uniaxial compressive strength of rock within 30∼170 MPa. Bilgin et al. [5] researched the relationship between cutting force and cutting parameters with 22 different property rocks and demonstrated that friction angle included in the cutting theory was more reasonable. Wang et al. [6] found that cone angle of conical pick had a significant influence on cutting force and specific energy and proposed the empirical prediction model. Triaxial testing apparatus was applied to research confining pressure influencing conical pick cutting performance, and it was found that the cuttability of rock decreased and then increased with the increasing confining pressure [7]. Huang et al. [8] carried out numerical method to research the unidirectional confining pressure influencing cutting force, which was verified by experimental method. e fracture mode of rock with different cutting depth was investigated at the lower cutting depth. Rock presented ductile failure in lower cutting depth and brittle failure in higher cutting depth [9]. Based on indentation tests and Evans' cutting theory, a specific energy model was proposed that the fracture toughness, elasticity modulus of rock, and cutting depth of conical pick had significant influence on specific energy [10]. Zhao et al. [11] employed ANSYS/LS-DYNA to establish the interaction between spiral drum and rock, giving the stress distribution of spiral drum, and demonstrated that the main reason for coal and rock collapse was the interface between the broken rock and coal. Rock fragments formation at different cutting speeds was investigated through LS-DYNA, with the results showing that the finite element method was an effective method to research fragments separation [12]. Zhou and Lin [13] demonstrated that, with the cutting depth increasing, the rock failure was a process from ductile to brittle. Park et al. [14] combined experimental and numerical simulation to research the cutting efficiency of conical pick, with the results showing that compressive strength, cutting depth, and cutting angle had a significant influence on cutting performance of conical pick. Wang et al. [15] established the constitutive model between rock and conical pick, giving the relationship between cutting depth and specific energy, and the feasibility was illustrated by comparison with experiment. About pick wear, Dewangan et al. [16] employed scanning electron microscopy to research the pick wear mechanisms and found that the plastic deformation was one of the main reasons causing pick wear. Liu et al. [17] pointed out that high impact and stress caused the pick wear and that higher cutting angle was an effective method to reduce the point attack pick wear. Yang et al. [18], by establishing rotary cutting model of point attack pick, demonstrated that lower rotary angle and higher cutting angle were beneficial for reducing pick wear. Lu et al. [19][20][21] proposed a multi-free-surface-rockcutting method under different constraints and carried out a numerical simulation study on cutting force in crushing process. Jeong et al. [22] used the smooth particle hydrodynamics (SPH) technique to build the rock and conical pick interaction mode and the Drucker-Prager model to simulate the brittle fracture. Su and Akkaş [23] researched the wear mechanism of two different types of conical picks and pointed out that metallurgical composition, pick position, and other environmental conditions had influence on wear mechanism. Liu [24] and Geng researched the influence of self-rotation on rock wear. Zhou et al. [25] combined theory and experiment to study dust generation in the process of rock breaking by pick, showing that the tip angle and attack angle had important influence on the dust generation. Zou et al. [26] used the discrete element method to study the influence of pick cutting parameters on rock fracture, showing that cutting depth and attack angle had significant influence on cutting force.
For the traditional rock cutting method, considerable researches have been carried out for rock cutting performance and pick wear with experimental and numerical simulation and theoretical methods. However, the new rock cutting method has seldom been related in past researches. In the paper, the diamond saw blade and conical pick combined rock cutting method is proposed. e research is based on rock plate formation. To predict the bending and preferential fracture position of rock plate, the theoretical research is proposed. e cutting process is studied by numerical simulation method. e experimental approach is carried out to research the fracture morphology and cutting force of rock plate.

New Rock Cutting Method.
e new cutting method is mainly used to cut hard rock and solve the problems of lower rock cutting performance and pick wear in the traditional rock cutting method. e main implementation processes of the rock cutting method are as follows: (a) diamond saw blade is employed to slice kerf on the rock; thus the rock plate is formed and the strength of rock decreases obviously; (b) conical pick is used to cut the rock plate, as is shown in Figure 1(a). After the saw blade slices the rock, the rock plate's formation is with three sides fixed and one side free. During the rock plate cutting process, it is likely to form the rock plate with two sides fixed and two sides free as is shown in Figure 1(b) [19]. e research content of the paper is mainly based on the rock plate with two sides fixed and two sides free. e rock plate is cut by single point attack pick, and the cutting performance and rock fracture morphology are researched.

eoretical Model.
According to the reciprocity theorem of works [27][28][29], within the elastic range of rock plate, the sum of the work done by the first group of force and moment on the displacement generated by the second group of force and moment is equal to the sum of the work done by the second group of force and moment on the displacement generated by the first group of force and moment. Two different force systems are involved: one is defined as basic force system and the other as actual force system. In the paper, the basic bending system is the rock plate bending with four sides simply supported under the concentrated cutting force 1, as is presented in Figure 2(a); thus the system has some particularities. e actual bending system is the rock plate bending with arbitrary boundary conditions under the concentrated cutting force P, as is presented in Figure 2(b); thus the system has universality. e solution of the actual system should be based on the basic system.
Based on the reciprocity theorem, the relationship between the basic force system and the actual force system is presented in the following equation: 2 Advances in Materials Science and Engineering where a and b are the dimensions of rock plate, P is cutting force, ω (ξ, η) is the rock plate bending of actual force system, ω 1 (ξ, η) is the rock plate bending of basic force system, (ξ, η) is cutting position of conical pick on the rock plate, M is the bending moment of each actual force system side, V is the shear force born by each side of basic force system, R is fulcrum resistance force of basic force system, and k is the fulcrum displacement of actual system. e bending equation of rock plate is a function about cutting force, bending moment, and shear force. For the actual force system with specific boundary condition, the bending moment and the corresponding deflection can be assumed with trigonometric series to obtain the solution of ω (ξ, η). rough (1), it can be seen that the solution of ω (ξ, η) must be based on the solution of ω 1 (ξ, η). e basic force system is a known system according to [27][28][29], and R, V, ω 1 (ξ, η), and ω 1 in the basic force system are the known parameters. e actual force system of the rock plate with two sides fixed and two sides free is presented in Figure 3(a). OA and OC are fixed sides, and AB and BC are free sides. e fixed boundary can be transformed to the combination of simple support and bending moment, as is presented in Figure 3(b).
According to (1), for the rock plate with two sides fixed and two sides free, the relationship between the basic force system shown in Figure 2(a) and the actual force system shown in Figure 3(b) is shown in the following equation: (2) e boundary condition of rock plate with two sides fixed and two sides free is presented as follows:

Advances in Materials Science and Engineering
In order to solve the bending problem of rock plate, the following assumptions are made for bending moment and deflection of the actual force system: In (4), there are five unknowns of A n , B m , C n , D m , and K B . By combining the five boundary conditions in (3), the unknowns can be solved.
According to actual working conditions, the conical pick generally acts on the free boundary of the rock plate. Taking a � b, h/a � 0.1, and v � 0.3, the cutting force P acting on midpoint of the free side with the coordinates is (a, b/2). Due to one end fixed and one end free, the free side presents asymmetric bending. e deflection of the free side x � a is shown in Figure 4(a), and the maximum reflection is 0.37(Pa 2 /D) obtained at the cutting position, where D is the bending strength. e bending moment of fixed side y � 0 is presented in Figure 4(b), and the maximum bending moment is 1.08 (−P), presented at the junction between free side and fixed side. erefore, when cutting force P acts on the midpoint of the free side, the cutting position and the junction between free side and fixed side are preferential fracture.

Mechanical Properties Test and Rock Plate Cutting Test.
Four different rock specimens are gathered to obtain the mechanical properties. To study rock property on rock cutting performance and to establish the numerical model, uniaxial compressive strength (UCS), elastic modulus (E), density (D), and tensile strength (T) are tested by electrohydraulic servo compression machine. e cylindrical specimens with dimensions of φ50 mm in diameter and 100 mm in height are used to conduct the compression test, and others with dimensions of φ50 mm in diameter and 50 mm in height are used to conduct the Brazilian splitting test. e compressive test is shown in Figure 5, and sample fracture result is shown in Figure 6. e strain-stress curve obtained from the compressive test is presented in Figure 7. UCS is the maximum stress of strain-stress curve, and E is obtained at the half of the maximum stress. e quality of the samples is measured; meanwhile, the volume of the sample can be calculated.
us, D can be measured from quality divided by volume. In order to avoid the contingency of the results, tests are conducted three times for each mechanical property. e main properties of rock are shown in Table 1.
e rock plate cutting test bed consists of hydraulic system, rock cutting component, and signal acquisition system, as is shown in Figure 8. e working principles of rock plate cutting bed are the following: (a) e cylinder telescopes with control of hydraulic system control unit, with power provided by pumping station. (b) e rock plate is fixed on the rock plate fixture and moves on the sliding guide under the action of cylinder. (c) us, the moving rock plate is cut by the stationary conical pick. (d) e static torque sensor generates deformation under the action of rock cutting force and outputs the voltage signal in mV unit. (e) e voltage signal is amplified to V unit through transmitter. (f ) e enlarged voltage signal is acquired by signal acquisition device and displayed on the computer. In the rock cutting process, the sampling frequency of signal acquisition device is 10 kHz. (g) e schematic diagram of signal acquisition system is shown in Figure 9.      Test one Test two Test three Figure 7: e stress-strain curve obtained from compression test. Advances in Materials Science and Engineering degrees. In other words, the direction of pick velocity is the same as positive direction of y-axis. l xp /l x is defined as the cutting position, set as 0.5 in the paper. e hourglass energy and element penetration significantly affect the cutting force. In order to avoid the above phenomenon, the full integration algorithm is adopted in the numerical simulation. e impact angle and the tip angle of conical pick are 57°and 80°, respectively. Besides, the friction angle between the cutting pick and rock plate is assigned as 8.5° [5,31]. e parameters of granite are shown in Table 2.

Rock Cutting Process.
e cutting process of rock plate under the numerical simulation condition is shown in Figure 11. With the contact between conical pick and rock plate, rock elements failure around the cutting position under the pressure of conical pick is shown in Figure 11(a), with the main feature being initial crushing. is is mainly because the contact area between the cutting teeth and the rock is too small, and the rock can bear less force. As a result, the rock is compressed and destroyed at the contact position. With the increase of pressure between conical pick and rock plate, the rock plate breaks and generates crack along with the middle line of rock plate starting from the cutting position; meanwhile, crack also occurs at the junction of the left fixed side and free side, as is presented in Figure 11(b). e initial fracture position of rock plate is consistent with the theoretical result. After the initial crushing of the rock plate, the contact area between the pick and the rock plate increases, and the interaction force between the pick and the rock increases. e resulting cutting force makes the stress generated at the contact position greater, resulting in the rock plate cracking from the middle, and it also makes the bending moment at the junction of base rock and rock plate larger, resulting in tensile failure at this point. Afterwards, with the movement of the pick, the main crack is formed, dividing the rock plate into two pieces, and then the rock plate breaks at bottom and the junction between bottom fixed side and right free side is shown in Figure 11(c). is is mainly due to the deformation of the rock plate caused by the movement of the conical pick, and the tensile stress is formed in the middle of the rock plate, which makes the rock plate continue to fracture along the original crack. After the fracture reaches the bottom of the rock plate, a large bending moment is formed near the crack, and then cracks are generated along the boundary between the rock plate and the base rock. From now on, the rock plate breaks wholly. e existing cracks continue to expand under the action of pick until they are connected to each other inside and outside of rock plate, as is shown in Figure 11(d) and Figure 11(e). Subsequently, with rock plate separated from base rock, the cutting process of rock plate finishes.

UCS Influencing Cutting Performance.
In the experiment, two sides of rock plate are fixed and two sides are free. e cutting results of rock plate with 200 mm height, 500 mm width, 20 mm thickness, and 20 mm cutting depth are shown in Figure 12. Cutting depth is defined as the distance from cutting position to the top side of rock plate. It can be seen that, with the difference of UCS, the fracture morphology of rock plate is significantly different. With the increasing UCS, the dimension of fragment obtained from rock plate cutting process increases, but the quantity of fragment decreases. All of the rock plate breaks at the bottom fixed side, and the fracture position extends to the right free side.  e cutting force for different rock properties with time is presented in Figure 13. It can be seen that the variation of cutting force is the same. However, for the rock with different properties, the peak cutting force (PCF) is obviously different. e PCF obtained in the experiment is illustrated in Table 3 and the variation in PCF with UCS is shown in Figure 14. e minimum PCF is 0.596 kN for marble with the least UCS, while the maximum PCF is 1.975 kN for sandstone-2 with the greatest UCS. PCF increases significantly with UCS. According to linear fitting, it can be seen that linear relationship exists between PCF and UCS. R 2 equal to 0.97356 indicates that the relevance is strong.

ickness of Rock Plate Influencing Cutting Performance.
e thickness of rock plate is an important factor that influences cutting force. e cutting force obtained at 500 mm width, 200 mm height, and thickness of 16 mm, 20 mm, 26 mm, and 30 mm is shown in Figure 15. With the different thickness of rock plate, the cutting force and PCF are    Figure 16. PCF increases with the increasing thickness. e fitting results show that exponential relationship exists between peak cutting and thickness of rock plate. R 2 of 0.99158 shows that the exponential relationship is strong and credible.   Advances in Materials Science and Engineering rock plate fractures along the bottom side until the free side, and, on the left part of rock, two main random cracks are generated. e cutting result obtained at width of 300 mm is shown in Figure 17(a), and that of 400 mm is shown in Figure 17(b), basically with the same width of 500 mm shown in Figure 12(c). Especially for widths of 400 mm and 500 mm, the cutting results are the same. erefore, the width of rock plate has little effect on rock plate fracture morphology on the condition of two sides fixed and two sides free. e cutting force obtained at thickness of 20 mm, height of 200 mm, and width of 200 mm, 300 mm, 400 mm, and 500 mm is presented in Figure 18. e boundaries between each curve are confusing, significantly different from Figures 13 and 15. e reason is that the change regulation and PFC are similar in each cutting force. PCF with different width of rock plate and the relationship between PCF and width are shown in Table 5 and Figure 19. PCF decreases with the increasing width of rock plate, and when the width decreases to 300 mm, the peak cutting force becomes stable,   Advances in Materials Science and Engineering no longer decreasing with the increasing width. At the widths of 200 mm and 500 mm, the peak cutting forces are 2.001 kN and 1.726 kN, respectively. When the width reduces by 300 mm, the corresponding cutting force decreases by 0.275 kN. It indicates that although the cutting force has a tendency of decrease, the amplitude is very small. erefore, the width of rock plate has little influence on cutting force. From the fitting result in Figure 19, exponential relationship exists between width and PCF with R 2 of 0.98847.

Height of Rock Plate Influencing Cutting Performance.
e cutting result of granite with width of 500 mm, thickness of 20 mm, and height of 80 mm and 120 mm is illustrated in Figure 20. At the height of 80 mm, the cutting result significantly differs from Figure 12(c) with 200 mm in height. e fracture area is mainly near the conical pick and relatively smaller when compared with Figure 12(c). Besides, the rock plates on the left and right sides are retained on the base rock. At height of 120 mm, the right part of rock plate breaks, with the decreased ratio between width and height and increased fracture area. erefore, rock plate with higher ratio of width to height corresponds to smaller regional fracture. e cutting force obtained at heights of 80 mm, 120 mm, 160 mm, and 200 mm is illustrated in Figure 21. At height of 80 mm, the value of cutting force is obviously bigger than that obtained at other heights. At heights of 120 mm, 160 mm, and 200 mm, little difference exists between the cutting forces. e statistics of PCF are given in Table 6 and the relationship between PCF and height of rock plate is shown in Figure 22.
e PCF of 2.559 kN obtained at height of 80 mm is still significantly greater than others. It is because lower height results in the lower distance between conical pick and bottom of rock plate. In order to obtain the same bending moment at bottom fixed side, lower distance corresponds to higher cutting force. At the height within 120 mm to 200 mm, although the cutting fore has the tendency to decrease, the value change is very small and basically stable. It is because when the height of rock plate increases, the cutting force depends on the distance between the cutting position and the left fixed boundary. erefore, PCF decreases with the increasing height of rock plate and it remains stable until the height increases to a certain value. Exponential relationship exists between PCF and height of rock plate with R 2 of 0.9595 and P value of 0.02.

Cutting Depth Influencing Cutting Performance.
e cutting results of rock plate with width of 500 mm, height of 200 mm, thickness of 20 mm, and cutting depth of 4 mm and 36 mm are shown in Figure 23. At the cutting depth of 4 mm, the rock plate fracture morphology is similar to cutting depth of 20 mm. At cutting depth of 36 mm, the cutting results have a significant difference with two others, with only half of rock plate breaking and the other half remaining on the base rock. e cutting force at different cutting depths is shown in Figure 24, with significant difference of cutting force for different cutting depths. At the higher cutting depth, the cutting force fluctuates during the rising process, for each part of rock plate does not break at the same time. PCF with different cutting depths is illustrated in Table 7. At cutting depth of 4 mm, the minimum PCF is obtained with the value of 1.853 kN. Relatively, at cutting depth of 45 mm, the maximum PCF is obtained with the value of 4.314 kN. e relationship between PCF and cutting depth for rock plate with two sides fixed and two sides free is shown in Figure 25. It is obvious that PCF increases significantly with the increasing cutting depth. From variation of numerical    value, cutting depth influences PCF greater than width and height of rock plate. e fitting results show that positive exponential relationship exists between PCF and cutting depth with R 2 of 0.94776. e fitting results between PCF and cutting parameters are shown in Table 8, with all the P values less than 0.05, indicating that the fitting results are all credible.

Advantages of the Method.
e existing roadway excavation technology has been relatively mature, but the hard rock roadway driving speed is slow and pick wear is serious, leading to cost increase. Reducing the cutting force is an effective means to reduce pick wear. By presplitting the rock and increasing the free surface, rock strength can be effectively reduced and the brittle fracture failure of the rock can be generated. According to the data in this paper, the influence of rock plate thickness on cutting force is the most obvious. e maximum thickness of granite studied in this paper is 30 mm, and the corresponding peak cutting force is 5.646 kn. In [5], 22 kinds of rocks are provided, in which the compressive strength of limestone is consistent with that of granite in this paper, which is 121 MPa. e tensile strength of granite in this paper is 12.1 MPa, harder than the limestone (7.8 MPa) in [5]. In the traditional rock breaking mode of continuous coal and rock, the cutting force required for breaking limestone is 29.4 kN when the cutting depth is 9 mm. According to [32], the ratio of peak cutting force to mean cutting force is 2.45 kN. e peak cutting force for limestone cutting should be 72.03 kN, which is far greater          than that of 5.646 kN when the granite thickness is 30 mm. erefore, the method proposed in this paper can effectively reduce the cutting force and thus reduce the pick wear.

Conclusion
To increase the cutting performance and reduce the production cost, based on increasing free surface, a new rock cutting method is proposed. e cutting force and fracture morphology of rock plate are researched according to theoretical, numerical, and experimental methods: (1) e foremost fracture position is deduced by theoretical method. e calculation results show that, at the cutting position and junction between free sides and fixed sides, rock plate breaks preferentially due to bore maximal bending moment and generated maximal deflection. (2) rough numerical simulation of rock plate cutting, it can be seen that the rock plate had preferential fracture at the cutting position, with crack generated at the middle of rock plate, and the results are consistent with the theoretical value. Subsequently, the rock plate breaks at junction between free sides and fixed sides. After cracks connect with each other, rock plate separates from base rock. e numerical method employed in the paper is an effective method for rock plate cutting process. (3) Under the constraint condition of rock plate with two sides fixed and two sides free, PCF increases linearly with the increasing UCS of rock plate, increases exponentially with the increasing thickness of rock plate and cutting depth of conical pick, and decreases exponentially with the increasing width and height of rock plate. e cutting depth of conical pick and thickness of rock plate influence PCF greater than width and height of rock plate. Besides, the peak cutting force produced by this method is obviously less than that of continuous coal and rock cutting. (4) e size of rock fragments separated from base rock increases with the increasing UCS of rock. Higher width and height and lower cutting depth are more beneficial for rock cutting, due to generation of lower cutting force and occurrence of higher fracture area. erefore, in the sawing process, rock plate with higher free surface area is more expected to be obtained.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.