A cutter penetrates rock when the shield tunneling machine is working (depending on the tunneling surface of the ground). As Fig. (1) shows, F_N is the normal force vertical to the surface F_s is the side force induced by the revolution of the cutters, and F_R is the rolling force which is parallel to the working face and points to the movement direction of the cutters. The cutter head is thrusted into the tunneling surface mainly by the normal force. The rolling force is mainly provided by the torque of the cutter head. The side force is mainly derived from the extrusion pressure during the rotary operation of the cutters. Under the action of the cutter head, the cutters rotate around their axis and revolute around the center of the cutter head. The main consideration of this paper is the horizontal scroll of the cutters, and therefore the side force is further ignored in this 2D problem. In this paper, the test employed an approximate constant-cross-section cutter whose dimensional parameters are shown in Table 1. As the cutter ring is made of hardened steel, this article considered the cutter as a rigid body which is shown in Fig. (2).

Fig. (1). Cutter forces of the cutter.

Fig. (2). Diagram of the cutter.

In particle flow method, the rock is divided into many rigid elements, and uses the explicit central difference method to solve the elements’ motion equations. These equations all satisfy the Newton's second law. The overall state of motion is given as follows:

(1)

Based on the law of force displacement, the contact force can be superposed by the amount of overlapping displacement of two contact particles (balls). The contact force of the two balls, and that of the ball and the wall all occurs on the corresponding contact point. By a certain predetermined contact rule, the amount of contacted force in the given models (ball-ball model see Fig. (3), ball-wall model see Fig. (4)) can be calculated. In Fig. (3A and B) are the two particles in contact. In Fig. (4), W is the wall which can be used to simulate non-particle rigid objects, such as cutter.

Fig. (3). Ball-ball contact model.

Fig. (4). Ball-wall contact model.

Particle contact constitutive model is expressed by the action of the contact force and the relative displacement relationship between the two entities (Fig. 5). The total contact force with respect to the displacement is determined by the normal stiffness (Fig. 6) and tangential stiffness (Fig. 7). The overall contact force constituted of the normal component and tangential component is located within the contact surface. Its expression is:

Fig. (5). Contact-stiffness model.

Fig. (6). Normal direction.

(2)

Assuming that the normal force between the blocks is proportional to the normal “overlap” U ⁿ, like:

(3)

Actually, the “overlap” is an supposed parameter and is multiplied by kⁿ as a measure of the normal force. For example, one can increase the value of kⁿ or decrease Uⁿ to get the same normal force. Due to the fact that the shear force is relevant with the movement of the block and load history, the expression should be defined as a increment ∆Fⁿ. Assume that relative displacement amount between the two bodies is ∆Uⁿ, so:

(4)

The shear force of one time can be calculated by the sum of tangential increment and the previous shear force:

(5)

Fig. (7). Shear direction.

Due to the experimental conditions, soft rock is selected as the cutting object. As it is difficult to detect the original defects of natural rock, concrete samples are used to replace rock samples to increase the repeatability of the experiment in this paper. The aggregate of concrete is coarse sand. The macroscopic mechanical parameters of the concrete samples are tested on the INSTRON-1342 electro hydraulic servo control material testing machine (Fig. 8). The results are shown in Table 2. The concrete grade is equivalent to C20 whose compressive strength and other properties are close to soft rock such as natural mudstone and sandstone. In order to establish the relationship between micro-parameters of particle flow model and macroscopic mechanism of the materials, the basic sizes of particles and other parameters need to be confirmed (Table 3). The micro-parameters were calibrated by uniaxial compression tests. In Brazil split and direct shear tests (Fig. 9), the left and right walls in the uniaxial compression experiments are the servo loading plates (Fig. 9a) and then the compressive strength of the materials can be obtained. In the vertical direction, two mobile walls in the Brazilian split tests are servo loading plates (Fig. 9b). The tensile strength of the material can be obtained. In the shear test (Fig. 9c), two side walls of the upper are fixed. Both the upper polymer particle wall and the bottom wall are servo loading plates. Carrying out the direct shear test under different vertical stresses, the relationship between shear stress and shear displacement can be obtained. By extracting the peak values of shear stress, cohesion and internal friction angles can be obtained. Generally, by using the trial-and-error method, we can adjust the micro-parameters repeatedly until the curve of the output of the simulation is close to the curve of the physical calibration. The calibration parameters of the particle are shown in Table 4.

Fig. (8). Electro-hydraulic and servo-controlled material testing machine.

Fig. (9). Meso-parameter determination in PFC2D.

Based on the above physical and mechanical parameters of the concrete samples, the related microscopic parameters of the particle flow code are determined by using PFC2D. Based on the above work, the rock-breaking model with joint by a cutter (Fig. 10) is established. x and y are the moving directions of the cutter, S is the joint spacing, α is the joint inclination. The sample is placed at a square groove which is surrounded by three fixed walls. The size of the model is 400mm ×200mm. Joint characteristics of the joint model are set by the JSET command. Since the stiffness of the cutter is much greater than soft rock, the cutter will be treated as a rigid body with the property of wall type. The bearings in the cutter are replaced by a hinge. The cutter rolls on rock surface under the drive of the friction generated by the contacting surface. We can obtain useful information such as stress distribution of the cutter ring by implying the FISH statements. The contact friction coefficients between the cutter and rock is set as 0.3. In the simulation, rock failure can be considered as the key connection between particles during the loading process under the action of faults. There are two forms of key joints, namely contact and parallel keys. By setting the size/strength of the key and particle, different constitutive models are expressed. When the load exceeds the preset bond strength, it will make the particles broke away from contact and then make them destroyed. Macro cracks and statistical micro-cracks which are generated underneath the rock mass can be observed in the simulation.

Fig. (10). A numerical rock-breaking model with jointed rock by a cutter.

Joint characteristics have great influence on the cutting mechanism of the cutter. The joint distribution characteristics in nature rock is generally complex, and the relative position of the cutter and joints is changing constantly. As for this, some necessary simplification has been done in this paper. It assumes that there only exists one group of equally distributed joints in this numerical model (S=40mm). The rock strength is indirectly designed by the micro-parameters of particles (friction coefficient, normal intension, tangential intension) of the joint surface. By changing the micro-parameters of particles of the joint surface, different strength of joint rock with different cohesion and angle of internal friction, specific values of joint strength can be obtained by direct shear experiment.

It can be found that when μ is constant (i.e. set μ to be 0.3), with the normal and shear strength increases, the intensity of jointed rock mass increases. In this paper, we use three joint surface strength (5kPa/50kPa/500kPa) to describe various intensity in the joint (low strength of the joint/ medium strength of the joint/ high strength of the joint).

With different joint characteristics, the rock-breaking state diagrams of the cutter are shown in Figs. (11-13) (J=250mm, p=8mm,v=0.1m/s). In Fig. (11), connection strength of joint surface is 5kPa, so its intensity is low. From the Fig. (11a-d), it can be observed that the main crack rarely extends inward. There are few main cracks generated in the internal parts of rock mass, as low-strength joint surface is easy to be penetrated and it will constrain the cutter force from transmitting to the deeper side. From these subgraphs, we can draw the conclusion that there are a large number of fragments collapsed from the rock surface; most of crack initiation direction is perpendicular to the joint plane. In Fig. (12), connection strength of joint plane is 50kPa. Its intensity is moderate. When the joint angle is 0° or 45°, it can be observed that the main crack tends to perpendicularly propagate to the joint surface. Some main cracks can span much more widely and deeply. It means that comparing with the barrier effect of high joint surface on crack propagation, low joint surface intensity have a more effect.

Fig. (11). Connect strength of the joint surface is 5kPa. (PFC2D).

Fig. (12). Connect strength of the joint surface is 50kPa. (PFC2D).

Fig. (13). Connect strength of the joint surface is 500kPa. (PFC2D).

When the joint angle is 90° or 135°, it shows that the direction of the main crack tends to be parallel to the joint surface. In Fig. (13), connection strength of joint plane is 500kPa. Its intensity is comparatively high. The joint has little effect on the depth of crack propagation. Because of the influence of joint angles, propagation angles of the main crack still have certain regularity.

As shown in Fig. (14), the angle between the joint plane and the horizontal plane is defined as a joint angle α. The angle between the main crack and the horizontal plane angle is defined as the β. The angle between the main crack and the joint plane is defined as crack angle γ. Obviously, γ = α + β. As Fig. (15) shows, when the strength of joint surface is 5kPa, γ is always within 90° ± 30°. This conclusion is consistent with the research of H. S. Ma et al. [17]. When the strength of joint surface is 50kPa, γ increases with the increasing of α. When the strength of joint surface is 500kPa, γ is similar to the condition in which the connection strength of joint plane is 50kPa. When the connection strength of joint plane reaches a certain level, the deflection of γ tends to be stable.

Fig. (14). Diagram of crack angle.

Fig. (15). Variation of crack angles with different joint characteristics.

The association rules of the number of micro cracks and the joint characteristics are shown in Fig. (16). When the strength of joint surface is high, the effect of the number of micro cracks on α decreases, and when the strength of joint surface is low, the number of micro cracks have greater fluctuation with the changing value of joint angle α. When the strength of joint surfaces is 5kPa and the joint angle is 90°, the most favorable condition for the development of micro cracks is obtained and the number of average tensile micro cracks is more than the number of shear micro cracks. Due to the limitation of the 2D simulation, the number of micro cracks cannot directly reflect the speed and tunneling efficiency.

Fig. (16). Variation of micro-crack with different joint characteristics.

The strength of the joint surface and α has significant effect on the depth of the main crack. Low strength joint surfaces are good for crack initiation [17] while it will hinder the main crack from expanding. α could change the propagation direction of the main crack, leading to the change of the crack depth.

The main crack cannot effectively contribute to the exfoliation of rock, since it consumes a lot of work done by the cutting forces. Therefore, a large crack depth may not have a direct benefit for improving the rock-breaking efficiency. The changing discipline of the main crack’s average depth shows that Fig. (17): for high-strength joint surface, it is easy for the main crack to extend to the deeper side of the rock mass. The lower the strength of joint surface, the smaller depth the main crack can reach. It implies that the low strength joint surface can improve the rock-breaking efficiency.

Fig. (17). Variation of the depth of main crack with different joint characteristics.

As for the given penetration depth, a larger broken area means a faster tunneling rate. By the auto measurement in CAD, the change discipline of rock-breaking area can be obtained (Fig. 18): the influence of joint orientation on the broken area is greater. As for the same intensity of joint surface, the broken area with 90° and 0° joint orientation is larger than other cases. The broken area will take the minimum when the strength of joint surface is 50kPa and α is 135°. It implies that this case is not conducive to improving the advance rate.

Fig. (18). Variation of breaking area with different joint characteristics.

The specific energy (SE) is often used to index the rock-breaking efficiency. It is defined as the energy consumed by the cutter cutting the unit volume rock. The lower the specific energy, the higher is the rock-breaking efficiency.

(6)

Generally, the work done by the normal force just account for 5% of the work done by the rolling force. Therefore, the normal force can be ignored in the actual calculation. Due to the limitation of 2D simulation, the crushing block cannot be discharged (Fig. 11d). The graph shows that the rock collapsed under the action of the squeeze by the cutter and therefore the rolling force increased. If bulk rock caving is observed in the simulation, the rolling force needs to be corrected by being multiplied by the coefficient g (g=0~1), which will make the simulation results more reasonable for the construction site.

Finally, Table 5 can be obtained by statistical analysis. According to the Table 5, the specific energy trend chart can be drawn (Fig. 19). As it can be seen in the figure, with the low strength joint surface, α is obviously related with specific energy. However, with the high strength joint surface, α has little impact on specific energy. Generally, the weaker the strength of joint surface, the lower is the specific energy. It shows that the low strength joint surface often prevents the main crack from propagating in the rock mass, reduces the reactive power dissipation of the cutter and improves the rock-breaking efficiency. As for the high strength joint surface, it is not easy to hinder the propagation of the main crack in the rock mass, and the rock-breaking efficiency is low. In general, the most favorable value of α is 0 and then the rolling-cutting efficiency is improved. With the low strength joint surface, the specific energy will get the minimum value when α is 135°. In this case, it is easy to form a triangle broken area (see Fig. 11d). With the high strength joint surface, the specific energy will be higher when α=135°. In this situation, the joint surface is not easy to be broken. When the strength of joint surface is 500kPa, the rock-breaking efficiency will be higher when α is 0° or 90°. When the strength of joint surface is 50kPa, the rock-breaking efficiency will be higher when α is 0° or 45°. When the strength of joint surface is 5kPa, the rock-breaking efficiency will be higher when α is 0° or 135°.

Fig. (19). Variation of specific energy with joint characteristics.