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Mechanism and numerical analysis of cutting rock and soil by TBM cutting tools
Tunnelling and Underground Space Technology 81 (2018) 428–437
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Mechanism and numerical analysis of cutting rock and soil by TBM cutting tools
T
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Guohui Lia, , Wenjin Wanga, Zhijuan Jinga, Leibin Zuoa, Fubin Wangb, Zhen Weia a b
China Petroleum Pipeline Engineering Corp, Langfang 065000, China Technology Center of China Petroleum Pipeline Bureau, Langfang 065000, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Cutter Disc cutter Rock and soil cutting Finite element Numerical analysis
The rock and soil cutting efficiency and reliability of cutting tools are critical factors to tunneling. However, due to its complexity, in-depth researches on some problems existing in rock fragmentation has not been conducted. This paper introduces the mechanism of cutting rock and soil by cutting tools. On the basis of this mechanism, it establishes finite element models using ABAQUS software to simulate the cutting processes when a cutter and disc cutter are cutting rock and soil masses. The results show that when the cutting force of a cutter is stronger than the shear strength of soil mass, breakage occurs at the interface between the cutting edge and the soil mass, leading to a successful cutting. When a disc cutter is pressed into a rock mass, with penetration depth increasing, the maximum Mises stress of the rock increases almost linearly, and plastic strain accumulates constantly, resulting in the stress growing as well. The plastic deformation zone in the rock mass is larger than the area compressed by the disc cutter. Along the working direction of the disc cutter, the rock mass is subject to continuous compression, producing plastic deformation until a fragment is broken off from its parent body. In this paper, a finite element model is established to simulate the process of cutting rock and soil by cutting tools, and a method for analyzing the interactions between cutting tools and the rock and soil is provided.
1. Introduction
driving process is too complicated to be satisfied by traditional calculation methods, and many problems are still unresolved. With the rapid development of numerical calculation methods, especially the development of finite element technique, this issue is being solved step by step (Qing, 2013a, 2013b). Modern numerical simulation methods are applied more and more widely and deeply in the field of tunneling analysis, which includes traditional analysis method (Laikuang, 2013), discrete element method and finite element method. This paper introduces the mechanism of cutting rock and soil by cutting tools, establishes a finite element model of the cutting processes on the basis of the finite element method, and studies the characteristics of the interactions between cutting tools and the rock and soil mass while tunneling.
Shield driving and pipe jacking technologies are widely applied in tunnel construction. A tunnel boring machine (herein after referred to as a TBM) may run into different layers with various characteristics while tunneling such as mucks, sands, clays, soft rocks and hard rocks. Cutters and disc cutters shown in Figs. 1 and 2 are a TBM’s major tools to break and strip rock and soil, which directly bear loadings and shocks from cutting activities. Their rock breaking abilities are strongly associated with TBM driving efficiency and liability (Piaoping, 2014). Therefore, the aspect of tunneling technique that focuses on the interaction between cutting tools and rock and soil mass is a key technique in the tunnel construction process (Marilena and Pierpaolo, 2012). Domestic and overseas scholars have carried out some researches on the rock fragmentation mechanism (Gunes Yilmaz et al., 2007, Hegadekatte et al., 2010, Moon et al., 2007). Because the form of the knife plate, the configuration and arrangement of the cutter are different, and the soil layer is complicated. There are many uncertainties in the fragmentation process, it is difficult to select an accurate and reliable model. Some studies have been carried out by using traditional and theoretical analysis methods (Zhiyong 2009), but the tunnel
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2. Failure mechanisms of cutting rock and soil 2.1. 2.1Rock and soil failure zones As shown in Fig. 3, disturbed zones of the soil mass at the working face induced by tunnel construction include: compresso-disturbed zone, shear-disturbed zone and two unloading-disturbed zones (Qinghe et al.,
Corresponding author. E-mail address: [email protected] (G. Li).
https://doi.org/10.1016/j.tust.2018.08.015 Received 6 June 2017; Received in revised form 6 June 2018; Accepted 16 August 2018 0886-7798/ © 2018 Published by Elsevier Ltd.
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Fig. 4. Cutting schematic diagram of the cutter.
similar to ③, except that the stress was a little stronger because this area is located below the pipe. Fig. 1. Cutter.
2.2. Cutting principle of the cutter As shown in Fig. 4, as the TBM moves on, the cutter produces a shearing force along tunneling direction and a cutting force in the tangential direction of rotating to the excavation face. With the joint action of these two forces, stress and deformation occurs inside the rock or soil mass at the excavation face (Yimin et al., 2012). When the stress is stronger than the yield strength of the rock and soil mass, a portion of the face is broken and released from the excavation face, obtaining rock and soil fragments.
2.3. 2.3Cutting principle of the disc cutter Fig. 2. Disc Cutter.
When a disc cutter cuts and breaks a rock face, the following phenomenona occur: At the interface between the rock and the disc cutter, there is a hard area called the dense core generated by plastic deformation of the rock where a large number of deformation energies gather (Bilgin et al., 2012). Hence, many cracks appear around the core. Under the pressure of the disc cutter, they grow and meet each other continuously. When lateral cracks completely mix with free surface or with adjacent cracks, broken blocks are made. To explain why the rock breaks around the dense core suddenly, scholars hold different opinions and put forward different rock fragmentation mechanisms. Most representative of them are the following three: (1) Compressional rock fragmentation mechanism: The disc cutter breaks the rock mass because it overcomes compressive strength. (2) Shear rock fragmentation mechanism: There are compressive deformation and shear deformation on the rock mass induced by the disc cutter. (3) Tension and shear rock fragmentation mechanism: in the fragmentation process, shear failure occurs accompanied by compressional deformation and tension deformation. The compressive failure mainly happens near to the dense core, and tension failure and shear failure generate cracks.
Fig. 3. Longitudinal disturbed zones induced by tunneling.
1999). ① is the compresso-disturbed zone, where the soil mass is a little far away from the excavation face and mainly undertakes compressional deformation caused by compressive stress. As the compressive stress increases with tunneling, the horizontal stress of the soil mass increases accordingly. The vibrating load and cutting force of the cutter head has little impact on this zone. ② is shear-disturbed zone in front of the TBM. Due to combined action of TBM thrust, shearing force from the cutter head and the vibrating load, the stress state here is very complicated. On the one hand, horizontal stress decreases because of stress relaxation induced by excavation. On the other hand, it increases because of the TBM thrust and the pressure from slurry or excavated material in the excavation chamber. ③ is one unloading-disturbed zone, where the soil mass is near to the excavation face. It affected by the compressive stress and shear stress from the compressor-disturbed zone. ④ is another unloading-disturbed zone under the pipe. The stress state in this zone is
This paper is intended to analyze the cutting performance of a single disc cutter, the rock fragmentation mechanism of which is shown in Fig. 5. The disc cutter squeezes the rock mass under it. As a result, cracks appear in the rock mass and form a central broken zone. Outside this zone, loads from the disc cutter have less impact on the rock mass. Thus, this part bears minor stress and is called the transition area. The rock under the disc cutter has to undertake differential pressures. That is to say, the area nearer to the disc cutter suffers a large pressure while the farther area bears minor load. That is why tensile stress are formed inside the rock, and then cracks are created (Alber, 2008). 429
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When ABAQUS is used for 3D cutting simulation for the D-P constitutive model of the soil mass, because of its peculiarity and complexity, the original stress-strain curve can't indicate the mechanical material property accurately at the moment the tools are working on the soil mass (Cho et al., 2010). Otherwise, it would lead to mesh-dependence in the computing process. Subjected to the imposed loadings, it is a continuous changing process from the soil mass being yielded to being destroyed. ABAQUS describes it as the process of yield stress softening and elasticity modulus degradation. Therefore, to simulate the birth and death condition of soil elements, ABAQUS introduces stress-strain curve based on damage evolution shown in Fig. 6. InFig.6, D is the damage variable. δy0 and ε0pl are the stress and equivalent plastic strain when the soil mass suffers damage at the initial stage. εfpl is the equivalent plastic strain when the soil is completely damaged. δ is the stress tensor neglecting material damage. The process of damage and failure includes four phases: OM is the elastic deformation phase; MN is the plastic deformation phase; NP is the damage evolution phase; PQ is the stress softening phase. N is the initial damage and breaking point where D = 0 and micro cracks will be produced inside the soil mass. Along with the loads’ continuous impact, those cracks lower the macro-mechanical properties during the damage evolution. Material stiffness is weakened until it totally loses shear resistance capacity. The NPQ segment describes the stiffness degradation and stress softening rule of the material. At this moment, the stress tensor can be written as:
Fig. 5. Failure zones compressed by the single disc cutter.
(1)
δ = (1−D)δ
where D = 1, that is point Q in Fig. 6, the soil element is in complete failure. The imaginary line displays the nondestructive hardening stage of the material. To analyze shear failure, there are mainly two methods to simulate the cutting process by finite element analysis. One is a physical separation criterion based on the density of stress-strain energy. The other is geometric separation criterion based on geometric dimensions. Usually, the physical separation criterion can better reflect actual cutting performance, and an element failure model including an element deletion function can effectively manage large deformations and interlaces of the material induced by loading. The damage and failure occur after the material with specific constitutive relation reaches its ultimate strength, and the stiffness of the material drops to zero as per a certain rule. At that time, the material element loses bearing capacity completely and exits the model calculation. The model of shear failure criterion in ABAQUS is the model of equivalent plastic damage based on element integration points. It defines the quantity of state ωs which describes the progressive increase of equivalent plastic strain εpl to make sure whether the material element is at failure or not. And:
Fig. 6. Stress-strain curve based on damage evolution. Table 1 Material Parameters of Cutting tools. Density
Young’s modulus
Poisson’s ratio
7800 kg/m3
210 GPa
0.3
pl
3. Establishing finite element models for cutting rock and soil
ωs =
∫ ε pl (dεθs, ε pl)
(2)
3.1. Constitutive models and material parameters of rock and soil mass
θs =
q + ks p τmax
(3)
This paper has adopted the finite element software ABAQUS to simulate and analyze rock and soil cutting performances of TBM cutting tools. ABAQUS has an extended classical Drucker-Prager (D-P) model, so the D-P model can be used to simulate materials containing frictional angles and cohesions. Moreover, the soil mass is a typical one. Therefore, when simulation calculations are carried out in ABAQUS, the linear D-P model is used to simulate the constitutive relation of the soil mass affected by the cutter-head and cutting tool (Bejari et al., 2011).
where: εpl—Equivalent plastic strain; θs—Stress ratio; ε pl —Rate of strain; ks—Material parameter, the value of 0.3 is assumed; p—Compressive stress;
Table 2 Material Parameters of Clay. Density (kg/m2)
Young’s modulus (MPa)
Poisson's ratio
Angle of internal friction (°)
Dilation angle
Flow stress ratio
Breaking strain
Damage displacement
1850
10.9
0.35
26
0.1
0.86
0.9
0.002
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Table 3 Material Parameters of Rock. Density (kg/m3)
Uniaxial compressive strength (MPa)
Tension strength (MPa)
Adhesive force (MPa)
Angle of internal friction (°)
Angle of external friction with steels
Young’s modulus (GPa)
Poisson’s ratio
2350
58–64
4–6
1.15
43.5
18
7.5
0.23
Plastic
Plastic
deformation ε0pl
deformation εspl
0.001
0.002
Fig. 7. Geometric dimensions of the cutter (mm).
Fig. 9. Contact property.
Fig. 8. 3D entity model of cutting soil mass by the cutter.
q—Mises Stress. where ωs = 1, the material element satisfies its shear failure criterion. That is to say, it arrives at its initial breaking point N. In the simulation process, effective displacement is introduced to define the damage regularity of the soil mass (Rojek, 2007). When the material is damaged and broken, the effective plastic displacement can be written as:
variable can be calculated as follow:
μ pl = L·ε pl
ΔD =
Fig. 10. Finite element model of cutting soil mass by the cutter.
(4)
In this formula:
μ pl Lε pl = pl μ fpl μf
(5)
In this formula:
μ pl —Effective plastic displacement L—Characteristic length
μ fpl —Effective displacement when the soil element is at failure
As the constitutive model is a linear D-P model, the damage variable D can be defined that has a linear relation with the characteristic length. When the soil element is in complete failure, the damage
When ABAQUS carries out the simulation calculation, the software monitors the changes of value D. At the moment of D = 1, complete failure occurs and the quantity of state changes to 1. This soil element is 431
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Fig. 11. Geometric parameters of a single disc cutter.
Fig. 14. t = 0.4 s Equivalent stress contour of the soil mass (Unit: Pa).
material constitutive model and then to reasonably simplify that model on the basis of meeting project actual requirements while insuring calculation accuracy. The purpose of this simulation is to study the stress and plastic deformation status of the rock and soil mass, the changing rule of the cutting force, the process of cutting the soil mass and its flow behavior. Therefore, to cut down computer operation time, the tools were set as rigid bodies during the simulation process. That is to say, deformation of the tools themselves was not considered, and value assignment of material properties was not required. Because forces always come in pairs, the change of forces on the cutting tools could be reflected indirectly by analyzing the change of forces on the rock and soil mass. Besides, nonlinear dynamic analysis was adopted, which made the simulation calculations more convergent. Meanwhile, it improved calculation efficiency significantly. Cutting rock and soil mass is a nonlinear and dynamic process. It is assumed that the cutting tools were rigid bodies, and would not be deformed in the simulation process. In that way, setting parameters such as Young’s modulus, Poisson’s ratio and density was enough. For the parameters of cutting tools, please refer to Table 1. When analyzing the cutter cutting the soil mass, it is necessary to set the frictional angle, dilation angel and stress ratio in the D-P model. Yield stress and absolute plastic strain were defined in D-P hardening. Fracture strain and damage displacement were defined in shear damage. In addition, Young’s modulus, Poisson’s ratio and density were
Fig. 12. Coupling constraint of the disc cutter and the reference node.
Fig. 13. t = 0.2 s Equivalent stress contour of the soil mass (Unit: Pa).
removed from the calculation model immediately and the next element is labeled. In this way, the software calculates in order and obtains the calculation results. The first step for numerical simulation analysis is to select a proper 432
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Fig. 15. Mises stress contour of the rock (h = 1 mm).
Fig. 16. Mises stress contour of the rock (h = 2 mm).
Fig. 17. Mises stress contour of the rock (h = 3 mm).
Fig. 18. Mises stress contour of the rock (h = 4 mm).
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applied. In addition, the top, right and left surfaces are set as free surfaces. The amount and quality of the meshes directly impact the accuracy and scale of the finite element calculation. In the simulation model, the soil mass deforms greatly, so more concentrated meshing was required. Since the depth of cutis fixed in the whole process, it is necessary to increase the density of the mesh at the cutting area. Also, the ALE Adaptive Mesh Technique is used so that the deformation of those soil meshes can be more coherent in the cutting process. For the cutter, as it is a rigid body, only mesh elements matching the soil mass need to be created. The finite element model obtained from meshing is shown in Fig. 10. Fig. 19. Rock maximum Mises stresses under different penetration depths.
3.3. The model of cutting rocks by a single disc cutter
defined in the linear elastic model. The specific parameters are listed below (see Table 2): When analyzing the single disc cutter cutting the rock mass, marble was selected as the material that the TBM was cutting, and the D-P model was selected as the material model as well. Its physical and mechanical parameters are shown in Table 3. In this model, the size of the rock mass is 200 mm × 100 mm × 20 mm.
Fig. 11 shows the geometric dimensions of the selected single disc cutter. There are two processes to simulate the cutting performance of the disc cutter: one is the process of pressing in at right angle to the rock mass; the other is forward cutting rotation. Accordingly, the finite element simulation contains two analysis steps. As shown in Fig. 12, in order to control the movement of the disc cutter in the following analysis steps, a reference node is selected in the middle and coupling is established between it and the internal surface of the disc cutter. Similarly to analyzing the cutting performance of the cutter, surface-tosurface contact between the disc cutter and the rock is created. In the penetration process, the downward speed is 2 mm/s. The depth of cut is 1 mm, 2 mm, 3 mm and 4 mm, so in the first analysis step, the duration is set as 0.5 s, 1 s, 1.5 s and 2 s accordingly. The speed is specified only in the perpendicular direction to the rock surface, but the disc cutter’s degree of freedom is constrained in other directions. In the second analysis step, the disc cutter cut and rolls at a certain penetration depth. To realize its translation and rotation round its axis, the reference node’s translational degree of freedom is constrained along the X-axis and Y-axis, and the rotational degree of freedom round the Z-axis and Y-axis. At the same time, a translational velocity is applied to the reference node along the Z-axis and an applied rotational velocity around the X-axis. The translational velocity is set as 6.65r/min. Also, it is assumed that there is no relative sliding between the disc cutter and the rock. Therefore, v = 332.5 mm/s and w = 1.539/s is set, which means the disc cutter is subjected to pure rolling on the rock surface.
3.2. The model of cutting soil mass by the cutter Fig. 7 shows the geometric dimensions of the selected cutter. Its width is 120 mm. The size of the soil mass model is 200 mm × 160 mm × 100 mm. The 3D model established by the ABAQUS/CAE pre-processing program is depicted in Fig. 8. Usually, material failures always occur on cutting target and object in the cutting process. At the same time, the contact surface changes constantly with the material failing at each cross section. However, the analysis not only refers to the contact surface of the soil mass but also its inside. Therefore, it is necessary to establish coupling between the cutter and the soil mass. In this cutting model, a rigid body constraint is created for a reference node on the cutter and in the analysis steps, anode constraint was created for the cutting layer of the soil mass. Surface-to-surface contact is used to describe the contact between the cutter and the soil mass. As shown in Fig. 9, in the simulation process, the cutter moves with the movement of the reference node. During the analysis, the cutter is set as rigid body and one point on it is selected as the reference node. And then, a certain speed along X-axis is applied to it. At the same time, a displacement constraints are applied to the cutter along the Yaxis and Z-axis and rotation constraints along the X-axis, Y-axis and Zaxis. For the model of the soil mass, fixed constraints to the basal surface and symmetry constraints to the front and rear surfaces are
4. Finite element analysis results 4.1. Numerical simulation of cutting clay The equivalent stress contours at different time of cutting of the clay
Fig. 20. Plastic deformation contour of the rock mass (t = 0.1 s, h = 1 mm). 434
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Fig. 21. Plastic deformation contour of the rock mass (t = 0.2 s, h = 1 mm).
Fig. 22. Plastic deformation contour of the rock mass (t = 0.3 s, h = 1 mm).
Fig. 23. Plastic deformation contour of the rock mass (t = 0.4 s, h = 1 mm).
by the cutter are shown from Figs. 13 and 14. These figures show that:
(2) The forward motion of the cutter leads to the changes of redistribution stress for the soil mass. The deformation is accumulated piecemeal. When the cutter cuts the soil mass, it creates a shear zone inside. With the increase of shear displacement, the angle of internal friction and cohesive force are gradually mobilized. (3) Developing Flow Cutting
(1) The zone bearing maximum force moves forward with the movement of the cutter, and the maximum force changes also. The maximum stress is mainly distributed over the interface between the cutter and the soil mass and front end of the soil mass which is about to be cut. With fixed depth of cut, the axial force on the cutter is mainly to bear the soil mass’s resistance and remains unchanged. Therefore, additional stress induced by compression is unchanged as well. The closer the area is to the cutter, the stronger is the redistribution of stress. Along with the cutter heading, the stress of the soil ahead adjusts accordingly. When the maximum Mises stress exceeds the cohesive force of clay, shear failure occurs.
The cutter applied penetrative force and cutting force to the contact soil mass are generally called the cutting force. When it is greater than the shear strength of the soil mass, mesh elements start to accumulate damage until an element fails and is removed. That is, breakage to the soil mass occurs at the contact surface and cutting is successful. As shown in Fig. 14, instead of disappearing, the soil stripped from the 435
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parent body flows at the front surface with the forward motion of the cutter. Therefore, this model can better reflect the cutting and flow processes when the cutter is cutting the soil mass.
not have to undergo a large equivalent plastic strain; or could have been damaged and stripped from the parent body like rocks in the other zones along the trajectory. As cutting time increases, the disc cutter’s trajectory advances. More and more parts of the rock and soil mass are damaged. Because the rock mass is brittle, its damage does not present plastic flow characteristics, but develops into tension cracks due to the compression by the disc cutter. Under the joint action from tensile stress and shear stress, the rock mass is broken into crushed pieces in different sizes. After that, the contact area between the rock and the disc cutter is adjusted. The maximum equivalent plastic strain is undulatory at different times.
4.2. Numerical simulation of cutting rock by a single disc cutter 4.2.1. Simulating the penetration process of the disc cutter In the penetration process, Mises stress contours of the rock mass are shown from Figs. 15–18. Comparing the stress distributions of rock breaking simulations under different penetration depths, it turns out that: on the action line of the disc cutter, lattices deform relatively due to big pressures on the inner rock particles, and show different color1 zones. That means, the closer it is to the action area, the bigger the stress is. On the contrary, the further it is from the acting force, the smaller the stress is. Besides, stresses at the two side faces of the cutting edge are strongest, but they are getting smaller gradually to the middle. The stress distribution of the rock mass is basically symmetric which explains the diffusion phenomenon of the stresses. At the initial stage of penetration, the stresses are concentrated at the contact surface, and a failure zone appears on the rock surface. Because the stress was very high, some parts in the failure zone are crushed or show obvious plastic deformation. The above results explain the stress distribution condition in the penetration process. Fig. 19 depicts the maximum Mises stresses in the case of different penetration depths. When the penetration depth is 1 mm, 2 mm, 3 mm and 4 mm, the Mises stress is 0.8531 MPa, 0.9005 MPa, 1.029 MPa and 1.401 MPa respectively. This figure proves that the maximum Mises stress grows linearly with the increase of penetration depth. The rock mass accumulates plastic deformation gradually, and its stress increases accordingly.
5. Conclusions This paper has introduced the mechanism of cutting rock and soil by cutting tools. On the basis of this mechanism, by using the finite element method, finite element models have been established to simulate and analyze the cutting processes. The following conclusions are drawn from the simulations: (1) In this paper, a finite element model is established to simulate the process of cutting rock and soil by cutting tools, and a method for analyzing the interactions between cutting tools and the rock and soil is provided. (2) When the cutting force of a cutter is stronger than the shear strength of the soil mass, breakage occurs at the interface between the cutting edge and the soil mass, which means, cutting is successful. In the cutting process, the zone bearing maximum force moves forward with the movement of the cutter, and the maximum force changes also. In addition, the cutter’s forward movements also lead to the redistribution of the stress in the soil mass. (3) In the penetration process of the disc cutter, the closer the location is to the action area, the bigger the stress is. On the contrary, the further it is from the acting force, the smaller the stress is. In addition, the stress at the two side faces of the cutting edge is strongest, but it is getting smaller gradually towards the middle. At the initial stage of penetration, stresses concentrate at the contact surface, and the failure zone appears on the rock surface. Because the stress is very high, some parts of the rock mass crush or obvious plastic deformation is created in the failure zone. With penetration depth increasing, the maximum Mises stress of the rock increases almost linearly, and plastic strain accumulates constantly, resulting in the stress growing as well. (4) When the disc cutter cuts the rock mass, the plastic deformation zone in the rock is larger than the area that is compressed by the disc cutter. Along the working direction of the disc cutter, the rock mass is compressed harder and harder, accumulating plastic deformation until it is broken and a fragment is stripped from its parent body. The maximum equivalent plastic strain appears at the leading edge of the disc cutter’s trajectory. The rock damage does not present plastic flow characteristics, but develops into tension cracks due to the compression by the disc cutter. Under the joint action from tensile stress and shear stress, the rock mass could be broken into crushed pieces in different sizes. After that, the contact area between the rock mass and the disc cutter is adjusted. The maximum equivalent plastic strain is undulatory at different times.
4.2.2. Simulation of roll-cutting process in the rock mass by the disc cutter Figs. 20–23 are plastic deformation contours of the rock mass when the penetration depth is 1 mm. The values of equivalent plastic strain are given in the figures. First, to analyze the characteristics of equivalent plastic strain distribution induced by cutting the rock mass, those figures only exhibit the equivalent plastic strain of rock but not rolling activities so that the equivalent plastic strain distribution could be better revealed. The cutting direction of the disc cutter is from right to left. The blue color represents the part of the rock far away from the disc cutter which is not affected by cutting. Plastic deformation cannot be found in this zone. The black rectangle is the trajectory of the disc cutter. The deeper color outside the rectangle is the region where the plastic deformation occurs due to the compression and cutting by the disc cutter. The red zone at the leading edge of the rectangle is the region with the biggest equivalent plastic strain. It can be obtained from the figures that the plastic deformation zone is larger than the area compressed by the disc cutter, the size of which can be affected by many factors including the disc cutter’s parameters, motion parameters and properties of the rock and soil mass. The color gradient change rule represents the changing size of the equivalent plastic strain. There is one line at the leading edge of the disc cutter’s trajectory which is in deep red. Along the cutting direction the color changes from red to yellow and finally to blue-green, the same as the color surrounding it. The reason why the maximum equivalent plastic strain appears at the leading edge of the disc cutter’s trajectory is that: the rock undergoes continuous compression along the cutting direction of the disc cutter, and then experiences plastic deformation until it is broken and stripped from its parent body. Therefore, the maximum plastic strain occurs in the area bearing compression and damage. Other parts, for example the blue-green zone outside the disc cutter’s trajectory, might
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1 For interpretation of color in Figs. 15-18, 20-23, the reader is referred to the web version of this article.
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Mechanism and numerical analysis of cutting rock and soil by TBM cutting tools
T
⁎
Guohui Lia, , Wenjin Wanga, Zhijuan Jinga, Leibin Zuoa, Fubin Wangb, Zhen Weia a b
China Petroleum Pipeline Engineering Corp, Langfang 065000, China Technology Center of China Petroleum Pipeline Bureau, Langfang 065000, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Cutter Disc cutter Rock and soil cutting Finite element Numerical analysis
The rock and soil cutting efficiency and reliability of cutting tools are critical factors to tunneling. However, due to its complexity, in-depth researches on some problems existing in rock fragmentation has not been conducted. This paper introduces the mechanism of cutting rock and soil by cutting tools. On the basis of this mechanism, it establishes finite element models using ABAQUS software to simulate the cutting processes when a cutter and disc cutter are cutting rock and soil masses. The results show that when the cutting force of a cutter is stronger than the shear strength of soil mass, breakage occurs at the interface between the cutting edge and the soil mass, leading to a successful cutting. When a disc cutter is pressed into a rock mass, with penetration depth increasing, the maximum Mises stress of the rock increases almost linearly, and plastic strain accumulates constantly, resulting in the stress growing as well. The plastic deformation zone in the rock mass is larger than the area compressed by the disc cutter. Along the working direction of the disc cutter, the rock mass is subject to continuous compression, producing plastic deformation until a fragment is broken off from its parent body. In this paper, a finite element model is established to simulate the process of cutting rock and soil by cutting tools, and a method for analyzing the interactions between cutting tools and the rock and soil is provided.
1. Introduction
driving process is too complicated to be satisfied by traditional calculation methods, and many problems are still unresolved. With the rapid development of numerical calculation methods, especially the development of finite element technique, this issue is being solved step by step (Qing, 2013a, 2013b). Modern numerical simulation methods are applied more and more widely and deeply in the field of tunneling analysis, which includes traditional analysis method (Laikuang, 2013), discrete element method and finite element method. This paper introduces the mechanism of cutting rock and soil by cutting tools, establishes a finite element model of the cutting processes on the basis of the finite element method, and studies the characteristics of the interactions between cutting tools and the rock and soil mass while tunneling.
Shield driving and pipe jacking technologies are widely applied in tunnel construction. A tunnel boring machine (herein after referred to as a TBM) may run into different layers with various characteristics while tunneling such as mucks, sands, clays, soft rocks and hard rocks. Cutters and disc cutters shown in Figs. 1 and 2 are a TBM’s major tools to break and strip rock and soil, which directly bear loadings and shocks from cutting activities. Their rock breaking abilities are strongly associated with TBM driving efficiency and liability (Piaoping, 2014). Therefore, the aspect of tunneling technique that focuses on the interaction between cutting tools and rock and soil mass is a key technique in the tunnel construction process (Marilena and Pierpaolo, 2012). Domestic and overseas scholars have carried out some researches on the rock fragmentation mechanism (Gunes Yilmaz et al., 2007, Hegadekatte et al., 2010, Moon et al., 2007). Because the form of the knife plate, the configuration and arrangement of the cutter are different, and the soil layer is complicated. There are many uncertainties in the fragmentation process, it is difficult to select an accurate and reliable model. Some studies have been carried out by using traditional and theoretical analysis methods (Zhiyong 2009), but the tunnel
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2. Failure mechanisms of cutting rock and soil 2.1. 2.1Rock and soil failure zones As shown in Fig. 3, disturbed zones of the soil mass at the working face induced by tunnel construction include: compresso-disturbed zone, shear-disturbed zone and two unloading-disturbed zones (Qinghe et al.,
Corresponding author. E-mail address: [email protected] (G. Li).
https://doi.org/10.1016/j.tust.2018.08.015 Received 6 June 2017; Received in revised form 6 June 2018; Accepted 16 August 2018 0886-7798/ © 2018 Published by Elsevier Ltd.
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Fig. 4. Cutting schematic diagram of the cutter.
similar to ③, except that the stress was a little stronger because this area is located below the pipe. Fig. 1. Cutter.
2.2. Cutting principle of the cutter As shown in Fig. 4, as the TBM moves on, the cutter produces a shearing force along tunneling direction and a cutting force in the tangential direction of rotating to the excavation face. With the joint action of these two forces, stress and deformation occurs inside the rock or soil mass at the excavation face (Yimin et al., 2012). When the stress is stronger than the yield strength of the rock and soil mass, a portion of the face is broken and released from the excavation face, obtaining rock and soil fragments.
2.3. 2.3Cutting principle of the disc cutter Fig. 2. Disc Cutter.
When a disc cutter cuts and breaks a rock face, the following phenomenona occur: At the interface between the rock and the disc cutter, there is a hard area called the dense core generated by plastic deformation of the rock where a large number of deformation energies gather (Bilgin et al., 2012). Hence, many cracks appear around the core. Under the pressure of the disc cutter, they grow and meet each other continuously. When lateral cracks completely mix with free surface or with adjacent cracks, broken blocks are made. To explain why the rock breaks around the dense core suddenly, scholars hold different opinions and put forward different rock fragmentation mechanisms. Most representative of them are the following three: (1) Compressional rock fragmentation mechanism: The disc cutter breaks the rock mass because it overcomes compressive strength. (2) Shear rock fragmentation mechanism: There are compressive deformation and shear deformation on the rock mass induced by the disc cutter. (3) Tension and shear rock fragmentation mechanism: in the fragmentation process, shear failure occurs accompanied by compressional deformation and tension deformation. The compressive failure mainly happens near to the dense core, and tension failure and shear failure generate cracks.
Fig. 3. Longitudinal disturbed zones induced by tunneling.
1999). ① is the compresso-disturbed zone, where the soil mass is a little far away from the excavation face and mainly undertakes compressional deformation caused by compressive stress. As the compressive stress increases with tunneling, the horizontal stress of the soil mass increases accordingly. The vibrating load and cutting force of the cutter head has little impact on this zone. ② is shear-disturbed zone in front of the TBM. Due to combined action of TBM thrust, shearing force from the cutter head and the vibrating load, the stress state here is very complicated. On the one hand, horizontal stress decreases because of stress relaxation induced by excavation. On the other hand, it increases because of the TBM thrust and the pressure from slurry or excavated material in the excavation chamber. ③ is one unloading-disturbed zone, where the soil mass is near to the excavation face. It affected by the compressive stress and shear stress from the compressor-disturbed zone. ④ is another unloading-disturbed zone under the pipe. The stress state in this zone is
This paper is intended to analyze the cutting performance of a single disc cutter, the rock fragmentation mechanism of which is shown in Fig. 5. The disc cutter squeezes the rock mass under it. As a result, cracks appear in the rock mass and form a central broken zone. Outside this zone, loads from the disc cutter have less impact on the rock mass. Thus, this part bears minor stress and is called the transition area. The rock under the disc cutter has to undertake differential pressures. That is to say, the area nearer to the disc cutter suffers a large pressure while the farther area bears minor load. That is why tensile stress are formed inside the rock, and then cracks are created (Alber, 2008). 429
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When ABAQUS is used for 3D cutting simulation for the D-P constitutive model of the soil mass, because of its peculiarity and complexity, the original stress-strain curve can't indicate the mechanical material property accurately at the moment the tools are working on the soil mass (Cho et al., 2010). Otherwise, it would lead to mesh-dependence in the computing process. Subjected to the imposed loadings, it is a continuous changing process from the soil mass being yielded to being destroyed. ABAQUS describes it as the process of yield stress softening and elasticity modulus degradation. Therefore, to simulate the birth and death condition of soil elements, ABAQUS introduces stress-strain curve based on damage evolution shown in Fig. 6. InFig.6, D is the damage variable. δy0 and ε0pl are the stress and equivalent plastic strain when the soil mass suffers damage at the initial stage. εfpl is the equivalent plastic strain when the soil is completely damaged. δ is the stress tensor neglecting material damage. The process of damage and failure includes four phases: OM is the elastic deformation phase; MN is the plastic deformation phase; NP is the damage evolution phase; PQ is the stress softening phase. N is the initial damage and breaking point where D = 0 and micro cracks will be produced inside the soil mass. Along with the loads’ continuous impact, those cracks lower the macro-mechanical properties during the damage evolution. Material stiffness is weakened until it totally loses shear resistance capacity. The NPQ segment describes the stiffness degradation and stress softening rule of the material. At this moment, the stress tensor can be written as:
Fig. 5. Failure zones compressed by the single disc cutter.
(1)
δ = (1−D)δ
where D = 1, that is point Q in Fig. 6, the soil element is in complete failure. The imaginary line displays the nondestructive hardening stage of the material. To analyze shear failure, there are mainly two methods to simulate the cutting process by finite element analysis. One is a physical separation criterion based on the density of stress-strain energy. The other is geometric separation criterion based on geometric dimensions. Usually, the physical separation criterion can better reflect actual cutting performance, and an element failure model including an element deletion function can effectively manage large deformations and interlaces of the material induced by loading. The damage and failure occur after the material with specific constitutive relation reaches its ultimate strength, and the stiffness of the material drops to zero as per a certain rule. At that time, the material element loses bearing capacity completely and exits the model calculation. The model of shear failure criterion in ABAQUS is the model of equivalent plastic damage based on element integration points. It defines the quantity of state ωs which describes the progressive increase of equivalent plastic strain εpl to make sure whether the material element is at failure or not. And:
Fig. 6. Stress-strain curve based on damage evolution. Table 1 Material Parameters of Cutting tools. Density
Young’s modulus
Poisson’s ratio
7800 kg/m3
210 GPa
0.3
pl
3. Establishing finite element models for cutting rock and soil
ωs =
∫ ε pl (dεθs, ε pl)
(2)
3.1. Constitutive models and material parameters of rock and soil mass
θs =
q + ks p τmax
(3)
This paper has adopted the finite element software ABAQUS to simulate and analyze rock and soil cutting performances of TBM cutting tools. ABAQUS has an extended classical Drucker-Prager (D-P) model, so the D-P model can be used to simulate materials containing frictional angles and cohesions. Moreover, the soil mass is a typical one. Therefore, when simulation calculations are carried out in ABAQUS, the linear D-P model is used to simulate the constitutive relation of the soil mass affected by the cutter-head and cutting tool (Bejari et al., 2011).
where: εpl—Equivalent plastic strain; θs—Stress ratio; ε pl —Rate of strain; ks—Material parameter, the value of 0.3 is assumed; p—Compressive stress;
Table 2 Material Parameters of Clay. Density (kg/m2)
Young’s modulus (MPa)
Poisson's ratio
Angle of internal friction (°)
Dilation angle
Flow stress ratio
Breaking strain
Damage displacement
1850
10.9
0.35
26
0.1
0.86
0.9
0.002
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Table 3 Material Parameters of Rock. Density (kg/m3)
Uniaxial compressive strength (MPa)
Tension strength (MPa)
Adhesive force (MPa)
Angle of internal friction (°)
Angle of external friction with steels
Young’s modulus (GPa)
Poisson’s ratio
2350
58–64
4–6
1.15
43.5
18
7.5
0.23
Plastic
Plastic
deformation ε0pl
deformation εspl
0.001
0.002
Fig. 7. Geometric dimensions of the cutter (mm).
Fig. 9. Contact property.
Fig. 8. 3D entity model of cutting soil mass by the cutter.
q—Mises Stress. where ωs = 1, the material element satisfies its shear failure criterion. That is to say, it arrives at its initial breaking point N. In the simulation process, effective displacement is introduced to define the damage regularity of the soil mass (Rojek, 2007). When the material is damaged and broken, the effective plastic displacement can be written as:
variable can be calculated as follow:
μ pl = L·ε pl
ΔD =
Fig. 10. Finite element model of cutting soil mass by the cutter.
(4)
In this formula:
μ pl Lε pl = pl μ fpl μf
(5)
In this formula:
μ pl —Effective plastic displacement L—Characteristic length
μ fpl —Effective displacement when the soil element is at failure
As the constitutive model is a linear D-P model, the damage variable D can be defined that has a linear relation with the characteristic length. When the soil element is in complete failure, the damage
When ABAQUS carries out the simulation calculation, the software monitors the changes of value D. At the moment of D = 1, complete failure occurs and the quantity of state changes to 1. This soil element is 431
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Fig. 11. Geometric parameters of a single disc cutter.
Fig. 14. t = 0.4 s Equivalent stress contour of the soil mass (Unit: Pa).
material constitutive model and then to reasonably simplify that model on the basis of meeting project actual requirements while insuring calculation accuracy. The purpose of this simulation is to study the stress and plastic deformation status of the rock and soil mass, the changing rule of the cutting force, the process of cutting the soil mass and its flow behavior. Therefore, to cut down computer operation time, the tools were set as rigid bodies during the simulation process. That is to say, deformation of the tools themselves was not considered, and value assignment of material properties was not required. Because forces always come in pairs, the change of forces on the cutting tools could be reflected indirectly by analyzing the change of forces on the rock and soil mass. Besides, nonlinear dynamic analysis was adopted, which made the simulation calculations more convergent. Meanwhile, it improved calculation efficiency significantly. Cutting rock and soil mass is a nonlinear and dynamic process. It is assumed that the cutting tools were rigid bodies, and would not be deformed in the simulation process. In that way, setting parameters such as Young’s modulus, Poisson’s ratio and density was enough. For the parameters of cutting tools, please refer to Table 1. When analyzing the cutter cutting the soil mass, it is necessary to set the frictional angle, dilation angel and stress ratio in the D-P model. Yield stress and absolute plastic strain were defined in D-P hardening. Fracture strain and damage displacement were defined in shear damage. In addition, Young’s modulus, Poisson’s ratio and density were
Fig. 12. Coupling constraint of the disc cutter and the reference node.
Fig. 13. t = 0.2 s Equivalent stress contour of the soil mass (Unit: Pa).
removed from the calculation model immediately and the next element is labeled. In this way, the software calculates in order and obtains the calculation results. The first step for numerical simulation analysis is to select a proper 432
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Fig. 15. Mises stress contour of the rock (h = 1 mm).
Fig. 16. Mises stress contour of the rock (h = 2 mm).
Fig. 17. Mises stress contour of the rock (h = 3 mm).
Fig. 18. Mises stress contour of the rock (h = 4 mm).
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applied. In addition, the top, right and left surfaces are set as free surfaces. The amount and quality of the meshes directly impact the accuracy and scale of the finite element calculation. In the simulation model, the soil mass deforms greatly, so more concentrated meshing was required. Since the depth of cutis fixed in the whole process, it is necessary to increase the density of the mesh at the cutting area. Also, the ALE Adaptive Mesh Technique is used so that the deformation of those soil meshes can be more coherent in the cutting process. For the cutter, as it is a rigid body, only mesh elements matching the soil mass need to be created. The finite element model obtained from meshing is shown in Fig. 10. Fig. 19. Rock maximum Mises stresses under different penetration depths.
3.3. The model of cutting rocks by a single disc cutter
defined in the linear elastic model. The specific parameters are listed below (see Table 2): When analyzing the single disc cutter cutting the rock mass, marble was selected as the material that the TBM was cutting, and the D-P model was selected as the material model as well. Its physical and mechanical parameters are shown in Table 3. In this model, the size of the rock mass is 200 mm × 100 mm × 20 mm.
Fig. 11 shows the geometric dimensions of the selected single disc cutter. There are two processes to simulate the cutting performance of the disc cutter: one is the process of pressing in at right angle to the rock mass; the other is forward cutting rotation. Accordingly, the finite element simulation contains two analysis steps. As shown in Fig. 12, in order to control the movement of the disc cutter in the following analysis steps, a reference node is selected in the middle and coupling is established between it and the internal surface of the disc cutter. Similarly to analyzing the cutting performance of the cutter, surface-tosurface contact between the disc cutter and the rock is created. In the penetration process, the downward speed is 2 mm/s. The depth of cut is 1 mm, 2 mm, 3 mm and 4 mm, so in the first analysis step, the duration is set as 0.5 s, 1 s, 1.5 s and 2 s accordingly. The speed is specified only in the perpendicular direction to the rock surface, but the disc cutter’s degree of freedom is constrained in other directions. In the second analysis step, the disc cutter cut and rolls at a certain penetration depth. To realize its translation and rotation round its axis, the reference node’s translational degree of freedom is constrained along the X-axis and Y-axis, and the rotational degree of freedom round the Z-axis and Y-axis. At the same time, a translational velocity is applied to the reference node along the Z-axis and an applied rotational velocity around the X-axis. The translational velocity is set as 6.65r/min. Also, it is assumed that there is no relative sliding between the disc cutter and the rock. Therefore, v = 332.5 mm/s and w = 1.539/s is set, which means the disc cutter is subjected to pure rolling on the rock surface.
3.2. The model of cutting soil mass by the cutter Fig. 7 shows the geometric dimensions of the selected cutter. Its width is 120 mm. The size of the soil mass model is 200 mm × 160 mm × 100 mm. The 3D model established by the ABAQUS/CAE pre-processing program is depicted in Fig. 8. Usually, material failures always occur on cutting target and object in the cutting process. At the same time, the contact surface changes constantly with the material failing at each cross section. However, the analysis not only refers to the contact surface of the soil mass but also its inside. Therefore, it is necessary to establish coupling between the cutter and the soil mass. In this cutting model, a rigid body constraint is created for a reference node on the cutter and in the analysis steps, anode constraint was created for the cutting layer of the soil mass. Surface-to-surface contact is used to describe the contact between the cutter and the soil mass. As shown in Fig. 9, in the simulation process, the cutter moves with the movement of the reference node. During the analysis, the cutter is set as rigid body and one point on it is selected as the reference node. And then, a certain speed along X-axis is applied to it. At the same time, a displacement constraints are applied to the cutter along the Yaxis and Z-axis and rotation constraints along the X-axis, Y-axis and Zaxis. For the model of the soil mass, fixed constraints to the basal surface and symmetry constraints to the front and rear surfaces are
4. Finite element analysis results 4.1. Numerical simulation of cutting clay The equivalent stress contours at different time of cutting of the clay
Fig. 20. Plastic deformation contour of the rock mass (t = 0.1 s, h = 1 mm). 434
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Fig. 21. Plastic deformation contour of the rock mass (t = 0.2 s, h = 1 mm).
Fig. 22. Plastic deformation contour of the rock mass (t = 0.3 s, h = 1 mm).
Fig. 23. Plastic deformation contour of the rock mass (t = 0.4 s, h = 1 mm).
by the cutter are shown from Figs. 13 and 14. These figures show that:
(2) The forward motion of the cutter leads to the changes of redistribution stress for the soil mass. The deformation is accumulated piecemeal. When the cutter cuts the soil mass, it creates a shear zone inside. With the increase of shear displacement, the angle of internal friction and cohesive force are gradually mobilized. (3) Developing Flow Cutting
(1) The zone bearing maximum force moves forward with the movement of the cutter, and the maximum force changes also. The maximum stress is mainly distributed over the interface between the cutter and the soil mass and front end of the soil mass which is about to be cut. With fixed depth of cut, the axial force on the cutter is mainly to bear the soil mass’s resistance and remains unchanged. Therefore, additional stress induced by compression is unchanged as well. The closer the area is to the cutter, the stronger is the redistribution of stress. Along with the cutter heading, the stress of the soil ahead adjusts accordingly. When the maximum Mises stress exceeds the cohesive force of clay, shear failure occurs.
The cutter applied penetrative force and cutting force to the contact soil mass are generally called the cutting force. When it is greater than the shear strength of the soil mass, mesh elements start to accumulate damage until an element fails and is removed. That is, breakage to the soil mass occurs at the contact surface and cutting is successful. As shown in Fig. 14, instead of disappearing, the soil stripped from the 435
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parent body flows at the front surface with the forward motion of the cutter. Therefore, this model can better reflect the cutting and flow processes when the cutter is cutting the soil mass.
not have to undergo a large equivalent plastic strain; or could have been damaged and stripped from the parent body like rocks in the other zones along the trajectory. As cutting time increases, the disc cutter’s trajectory advances. More and more parts of the rock and soil mass are damaged. Because the rock mass is brittle, its damage does not present plastic flow characteristics, but develops into tension cracks due to the compression by the disc cutter. Under the joint action from tensile stress and shear stress, the rock mass is broken into crushed pieces in different sizes. After that, the contact area between the rock and the disc cutter is adjusted. The maximum equivalent plastic strain is undulatory at different times.
4.2. Numerical simulation of cutting rock by a single disc cutter 4.2.1. Simulating the penetration process of the disc cutter In the penetration process, Mises stress contours of the rock mass are shown from Figs. 15–18. Comparing the stress distributions of rock breaking simulations under different penetration depths, it turns out that: on the action line of the disc cutter, lattices deform relatively due to big pressures on the inner rock particles, and show different color1 zones. That means, the closer it is to the action area, the bigger the stress is. On the contrary, the further it is from the acting force, the smaller the stress is. Besides, stresses at the two side faces of the cutting edge are strongest, but they are getting smaller gradually to the middle. The stress distribution of the rock mass is basically symmetric which explains the diffusion phenomenon of the stresses. At the initial stage of penetration, the stresses are concentrated at the contact surface, and a failure zone appears on the rock surface. Because the stress was very high, some parts in the failure zone are crushed or show obvious plastic deformation. The above results explain the stress distribution condition in the penetration process. Fig. 19 depicts the maximum Mises stresses in the case of different penetration depths. When the penetration depth is 1 mm, 2 mm, 3 mm and 4 mm, the Mises stress is 0.8531 MPa, 0.9005 MPa, 1.029 MPa and 1.401 MPa respectively. This figure proves that the maximum Mises stress grows linearly with the increase of penetration depth. The rock mass accumulates plastic deformation gradually, and its stress increases accordingly.
5. Conclusions This paper has introduced the mechanism of cutting rock and soil by cutting tools. On the basis of this mechanism, by using the finite element method, finite element models have been established to simulate and analyze the cutting processes. The following conclusions are drawn from the simulations: (1) In this paper, a finite element model is established to simulate the process of cutting rock and soil by cutting tools, and a method for analyzing the interactions between cutting tools and the rock and soil is provided. (2) When the cutting force of a cutter is stronger than the shear strength of the soil mass, breakage occurs at the interface between the cutting edge and the soil mass, which means, cutting is successful. In the cutting process, the zone bearing maximum force moves forward with the movement of the cutter, and the maximum force changes also. In addition, the cutter’s forward movements also lead to the redistribution of the stress in the soil mass. (3) In the penetration process of the disc cutter, the closer the location is to the action area, the bigger the stress is. On the contrary, the further it is from the acting force, the smaller the stress is. In addition, the stress at the two side faces of the cutting edge is strongest, but it is getting smaller gradually towards the middle. At the initial stage of penetration, stresses concentrate at the contact surface, and the failure zone appears on the rock surface. Because the stress is very high, some parts of the rock mass crush or obvious plastic deformation is created in the failure zone. With penetration depth increasing, the maximum Mises stress of the rock increases almost linearly, and plastic strain accumulates constantly, resulting in the stress growing as well. (4) When the disc cutter cuts the rock mass, the plastic deformation zone in the rock is larger than the area that is compressed by the disc cutter. Along the working direction of the disc cutter, the rock mass is compressed harder and harder, accumulating plastic deformation until it is broken and a fragment is stripped from its parent body. The maximum equivalent plastic strain appears at the leading edge of the disc cutter’s trajectory. The rock damage does not present plastic flow characteristics, but develops into tension cracks due to the compression by the disc cutter. Under the joint action from tensile stress and shear stress, the rock mass could be broken into crushed pieces in different sizes. After that, the contact area between the rock mass and the disc cutter is adjusted. The maximum equivalent plastic strain is undulatory at different times.
4.2.2. Simulation of roll-cutting process in the rock mass by the disc cutter Figs. 20–23 are plastic deformation contours of the rock mass when the penetration depth is 1 mm. The values of equivalent plastic strain are given in the figures. First, to analyze the characteristics of equivalent plastic strain distribution induced by cutting the rock mass, those figures only exhibit the equivalent plastic strain of rock but not rolling activities so that the equivalent plastic strain distribution could be better revealed. The cutting direction of the disc cutter is from right to left. The blue color represents the part of the rock far away from the disc cutter which is not affected by cutting. Plastic deformation cannot be found in this zone. The black rectangle is the trajectory of the disc cutter. The deeper color outside the rectangle is the region where the plastic deformation occurs due to the compression and cutting by the disc cutter. The red zone at the leading edge of the rectangle is the region with the biggest equivalent plastic strain. It can be obtained from the figures that the plastic deformation zone is larger than the area compressed by the disc cutter, the size of which can be affected by many factors including the disc cutter’s parameters, motion parameters and properties of the rock and soil mass. The color gradient change rule represents the changing size of the equivalent plastic strain. There is one line at the leading edge of the disc cutter’s trajectory which is in deep red. Along the cutting direction the color changes from red to yellow and finally to blue-green, the same as the color surrounding it. The reason why the maximum equivalent plastic strain appears at the leading edge of the disc cutter’s trajectory is that: the rock undergoes continuous compression along the cutting direction of the disc cutter, and then experiences plastic deformation until it is broken and stripped from its parent body. Therefore, the maximum plastic strain occurs in the area bearing compression and damage. Other parts, for example the blue-green zone outside the disc cutter’s trajectory, might
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1 For interpretation of color in Figs. 15-18, 20-23, the reader is referred to the web version of this article.
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