J. Cent. South Univ. (2019) 26: 2393-2403

DOI: https://doi.org/10.1007/s11771-019-4182-8

Distribution of contact loads in crushed zone between tunnel boring machine disc cutter and rock

SHI Yu-peng(史余鹏)^{1, 2}, XIA Yi-min(夏毅敏)^{1, 2}, TAN Qing(谭青)^{1, 2},ZHANG Yi-chao(张逸超)³, QIAO Shuo(乔硕)^{1, 2}

1. State Key Laboratory of High Performance Complex Manufacturing, Central South University,Changsha 410083, China;

2. College of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China;

3. Patent Examination Cooperation Sichuan Center of the Patent Office, SIPO, Chengdu 610000, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: The construction efficiency and quality of tunnel boring machines (TBMs) is largely determined by the service life of cutting tools, which is the result of contact loads in the crushed zone between cutter ring and rock. In this paper, a series of rock breaking tests were conducted with a 216 mm diameter disc cutter and concrete samples. Based on the superposition principle, the distribution of contact loads between disc cutter and rock were obtained by using the truncated singular value decomposition (TSVD). The results show that both the peak value and the whole numerical distribution of the radial strains on the cutter ring increase with the increase of the penetration. The distribution curves of the contact loads show an approximate parabola going downwards, which indicates contact loads are more concentrated. The front non-loading area with a ratio from 1.8% to 5.4% shows an increasing trend with the increase of penetration. However, the change of rear non-loading area is not obvious. It is believed that the conclusions have guidance for the study of rock breaking mechanism and manufacturing process of the disc cutter.

Key words: disc cutter; contact loads; superposition principle; non-loading area

Cite this article as: SHI Yu-peng, XIA Yi-min, TAN Qing, ZHANG Yi-chao, QIAO Shuo. Distribution of contact loads in crushed zone between TBM disc cutter and rock [J]. Journal of Central South University, 2019, 26(9): 2393-2403. DOI: https://doi.org/10.1007/s11771-019-4182-8.

1 Introduction

Tunnel boring machine (TBM) has been widely used in tunnel excavations, water diversion projects and other tunnel constructions with the advantages of large-scale, automation, high-speed and safety [1]. As the core component of TBM, disc cutter accomplishes the breaking of rock in tunnel face by the extrusion and rotating of cutter-head. With the development of underground space utilization, TBMs need to be applicable for extremely complex geological conditions, such as high confining stress, high hydrostatic pressure and high strength rock [2-3]. The abnormal wear and damage with the cutter ring in those cases, which are the result of contact loads in the crushed zone between cutter ring and rock, seriously affect the construction progress and cost [4]. Therefore, studying the distribution of loads applied to the disc cutter ring with a typical rock has great importance for the guidance of manufacturing process of disc cutter ring and the improvement of TBM construction quality.

Compared with the harsh field conditions during TBM construction, laboratory tests have the advantages in feasibility, repeatability and reliability. Based on the self-designed rotary and linear TBM cutting machines, a preliminary study of rock breaking mechanism by disc cutter has been done [5-7]. ROXBOROUGH et al [8] firstly proposed a formula to estimate the forces applied to a V-shape disc cutter, which included the parameters of disc diameter, tip edge angle and the UCS of rock sample. Based on a large amount of hard rock breaking data by linear cutting machine (LCM), HASSANPOUR et al [9] and ROSTAMI [10] developed the CSM model to estimate the cutting forces applied to the constant cross section disc cutters (CCS). The strain distribution curves of the cutter ring were also obtained in Ref. [11]. The contact load distribution while cutting hard rock was deduced by numerical fitting at the same time. WANG et al [12, 13] developed the disc cutter wear prediction model based on the mechanics and energy analysis of the contact area. The validity of the prediction model was confirmed by the project data obtained on site. Several influencing factors on the wear of the slurry shield cutting tools were also analyzed through Ruhr-University Bochum (RUB) tunneling device [14]. The influence of geological environment on the loads of the disc cutter was considered as well. CHO et al [15, 16] collected the forces applied to the disc cutter in the process of breaking rock with different combinations of penetration and cut spacing by LCM. To reduce the specific energy consumption, the optimum combination of the cut spacing and penetration was derived. ZARE et al [17], and TAN et al [18] looked into the effect of joints in the rock on the performance of TBM. The results show that there are several modes for the expansion and development of cracks according to the direction of joints in a rock. The smaller the join angle is, the easier the fragmentation of the rock is. TAN et al [19] studied the breaking properties of disc cutter whiling penetrating the rock samples on the saturated. The experimental results showed that the strength of saturated rock reduced and the loads decreased by 6%-27%. According to the fragmentation collected in the single-axis combination load tests, ZHU et al [20] found that impact load component was favorable to the initiation and propagation of internal cracks in rocks. XIA et al [21] proposed a multicriteria decision method for determining the size of disc cutter which matches the geological conditions. The result was applied in a certain TBM project.

Meanwhile, numerical simulation methods, such as finite element analysis (FEA) and discrete element analysis (DEA), were also used in this filed. ENTACHER et al [22] established the three- dimensional model of breaking rock by finite element software ABAQUS and got the deformation law of cutter ring. On this basis, the method of measuring three direction forces in real time was proposed and had been verified by experiments. WANG et al [23] simplified the rock cutting process as a two-dimensional contact problem between the cylinder and the elastic plane. Based on the contact mechanics, the load- approximate parabolic distribution within the contact area was solved, which was validated by numerical simulation. ZHAO et al [24] who used a hybrid modeling method of finite element and smoothed particle hydrodynamics (SPH) studied on pressure distribution in the crushed zone and proposed the pressure distribution was concentrated in the center of contact area. Based on the finite element software LS-DYNA, WU et al [25] established a three-dimensional model of breaking rock. The pressure distribution curve between cutter ring and rock was obtained and a trigonometric function was applied to fulfill a nonlinear fitting of the curve. The optimal rock-cutting parameters and energy consumption analysis were also done by discrete element method in Refs. [26-29].

The review of available literature shows that some headway has been made in the analysis of rock breaking features through numerical simulation and laboratory tests. However, the results based on the laboratory tests usually pay attention to the average load acting on the disc cutter. It is considered that the loads in the contact area are uniform or increase linearly [30]. Numerical simulation results also lack rigorous verification. In this paper, the relation between a single load and the radial strains of cutter ring was obtained through the self-designed TBM cutter performance test bench. And the distribution of contact loads between disc cutter and rock was studied based on the superposition principle. It is believed that the results can provide important reference to the study of wear mechanism and design of cutter ring.

2 Mathematical model

The process of breaking rock is actually a dynamic three-dimensional process, which includes vertical cutting and horizontal scrolling. Figure 1 shows the forces in three directions applied to the disc cutter. In the process of penetration of TBM cutter head, the micro-cracks inside the rock begin to grow and spread around under the continuing effect of loads, resulting in the formation of fragmentation. The rock under the disk cutter has plastically fractured. It cannot be simplified as concentrated load model or solved by theory of elastic mechanics any more. However, the strain distribution of cutter ring is the comprehensive effect of contact loads. According to the mechanics of materials, if dividing the contact area into finite elements, the strain in the element i produced by the load on element j can be calculated by the following formula:

                       (1)

where y(i, j) denotes the strain that generates in the element i produced by the load on element j, g_j(i, j) denotes the function representing the strain that generates in the element i produced by the unit load on element j, p(j) denotes the load on element j.

Figure 1 Forces applied to disc cutter

Furthermore, according to the superposition principle, the total strain of element i generates by load vector P in contact area can be expressed as:

              (2)

where y(j) denotes the total strain in element i, n denotes the number of elements.

According to the results of laboratory and field tests, side force is so small that can be ignored compared with normal force and rolling force [2, 31]. Supposing the contact loads distribute uniformly along the width direction of cutter ring, the process of breaking rock can be simplified into a two-dimensional process, as shown in Figure 2. Divide the theoretical contact area whose range is from A to B in Figure 2 into a number of elements. Using F_t as the unit radial load, so the unknown contact loads can be divided into a number of components, such as x₁F_t, x₂F_t, …, x_nF_t. The strain at a certain element on the cutter ring is the cumulative effect caused by each individual load. Based on the superposition principle, the relationship between the contact loads and the strain distribution in the crushed zone is:

S=CL                                   (3)

where S denotes the measured strain vector, L denotes the load vector, C denotes the coefficient matrix, which can be obtained from the calibration tests. The element C(i, j) in the matrix represents the strain that can be generated in the element i produced by the unit load on element j. It has the same meaning as g_j(i, j) in Eq. (1). The coefficient matrix C has a similar form as follows after normalization:

            (4)

Figure 2 Schematic diagram of contact loads between disc cutter and rock (ω-Angular velocity of disc cutter; h-Penetration; θ-Corresponding angle of theoretical contact area)

Obviously, the order of matrix C is the number of element. The following formula can be used to obtain the contact load vector L.

L=C^-1S                                 (5)

where C^-1 denotes the inverse matrix of C.

3 Experimental details

3.1 Test equipment

The experiments were performed on the multifunctional TBM cutter performance test bench at the State Key Laboratory of High Performance Complex Manufacturing (Central South University), China. It is mainly composed of vertical load system (VLS), rolling cut system (RCS) and data collection system. The VLS consists of a vertical hydraulic cylinder and a vertical-active crossbeam whose mechanical locking nuts are to guarantee a constant penetration while breaking rock. The RCS is a platform that can be moved freely on sliding guide rails in the horizontal by driving hydraulic cylinders. Its basic idea is to allow the disc cutter cutting rock at preset cut spacing. In order to minimize the influence of boundary effect on the results, the rock samples to be cut are placed in a pre-prepared steel box and fixed with high strength concrete. The average cutting velocity of the cutter in the experiments is 10 mm/s. In accordance with the design requirements, the maximum loading capacity of normal force is 120 kN. A three directional (3-D) force sensor is installed between vertical-active crossbeam and cutter mount. Besides, a computer-based data acquisition system including data acquisition card, voltage-stabilized source and the National Instruments LabVIEW (2012) is used to record the outputs from the sensors. Figures 3 and 4 are the structural and physical maps of multifunctional TBM cutter performance test bench.

3.2 Disc cutter and rock sample

In order to guarantee the continuity of the cutter ring strain curve and simplify the operation, the disc cutter used in the experiments had a diameter of 216 mm, which was the reduction scale of 1:2 of the 17 inches constant cross-section disc cutter. Strain rosettes were pasted with an interval of 60° on the cutter ring which were 8 mm away from tip after rough grinding and polishing (No.1 to No. 3, and No.3 was a spare strain rosette), as shown in Figure 2.

Figure 3 A schematic view of multifunctional TBM cutter performance test bench

Figure 4 A physical map of multifunctional TBM cutter performance test bench

The position where strain rosettes were pasted should be corresponding to the screw on the surface of the cutter. In this way, the right position of the strain rosettes could be easily confirmed which facilitates the interception of signals in the contact area. It was necessary to make a few alterations to the disc cutter considering that fragmentation may scratch its surface. The restructuring process of disc cutter is shown in Figure 5. To minimize the influence of the initial damage and mechanical performance defects of the rock samples on the experimental results, concrete with size of 900 mm×380 mm×260 mm was used as the cutting sample. The standard of concrete production and conservation can refer to Ref. [32]. The physical parameters gotten through the mechanical tests of the cutting samples are shown in Table 1.

3.3 Experimental procedures

According to penetrations, the rock sample is divided into several data layers. Each layer includes a few of cuts with the same penetration and different spacing, whose spacing is also called the cut spacing, as shown in Figure 6. It should be noted that the cutting paths are farther than one cut spacing to any block boundaries to minimize the adverse effects of the boundary effect on the strain signals. Table 2 shows the test matrix in the rock breaking experiments. Since this paper concentrated on the effect of penetration on the contact loads, the cut spacing was set to 100 mm to avoid the interaction between adjacent cutting paths. The disc cutter was under break cutting model in this situation [33].

Figure 5 Restructuring process of the disc cutter:

Table 1 Physical properties of concrete sample

There are two parts in the experiments, which are calibration experiment and rock breaking experiment. According to the direction of rotation of the disc cutter, No. 3 spare strain rosette was located at the lowest point of the cutter ring before the experiment, as shown in Figure 2. In the calibration experiment, a piece of steel plate was placed under the disc cutter, and the vertical hydraulic cylinder was under jog operation constantly until the preset load was reached through the data collection system. And then the front and back hydraulic cylinders were driven to collect the strain signals of the cutter ring. The outlet and inlet of oil of the vertical hydraulic cylinder were closed to ensure the constant vertical load during the rolling calibration experiments. In the rock breaking experiments, the concrete samples were cut at the pre-set cut spacing and penetration through VLS and RCS. To increase the reliability of the results, the strain signals were obtained from at least four cutting paths under the same penetration.

Figure 6 Cutting parameters in test

Table 2 Test matrix in rock breaking experiments

4 Results

4.1 Calibration experimental results analysis

Considering the high stiffness of the steel plate, it can be considered as a point contact while disc cutter rolls on it. The contact point is the lowest point of the cutter ring. The strain curves of No. 1 and No. 2 strain rosettes with 1.9 kN contact load are shown in Figure 7. It can be seen that the symbols of strain at 0°(tangential) and 45°have changed as time goes by. But the strain at 90°(radial) is always positive or zero, and the maximum value is greater than that in the other two directions. Therefore, the radial strain curve is used as the calibration results to solve the contact load distribution.

Figure 7 Strain curves obtained in calibration experiments when normal force is 1.9 kN:

The radial strain curves of the calibration experiments under different loads are clipped and stacked, as shown in Figure 8. The abscissa in the drawing is the angle that the point has rotated about the axis of the disc cutter from the contact point. Thus, zero point represents the contact point. The angle value is positive when turning anticlockwise. The radial strain curves are basically symmetrical about 0° and drastically decrease with the increase of angle, as shown in Figure 8. When the angle value exceeds 20°, the radial strains tend to zero. Under the same calibration load, the strain curves are basically coincident with each other, and the radial strain value at the same angle has a linear relationship with the point contact load.

4.2 Solving contact loads in process of breaking rock

Based on the radial strain values and the pre-set contact load, the elements in matrix C, which have the same meaning as that in the function g_j(i, j), can be obtained according to Eq. (1). Taking matrix C of 10 orders as an example, Figure 9 shows the changing trend of elements in the first row of matrix C after normalization. The value gradually decreases from 1 to 0.

According to Eq. (5), it can be inferred that the accuracy of solutions is largely determined by the characteristics of C. However, the condition number of the matrix C of 10 orders reaches 535, which exceeds the critical value and indicates that it belongs to the ill-conditioned matrix [34]. The noise and vibration are inevitably mixed in the testing process, which leads to the difference between the calculated results and the accurate values. Truncated singular value decomposition (TSVD) is an effective solution to the problem of ill- conditioned equations. The least square solution of Eq. (5) using TSVD can be obtained as follows [35]:

                   (6)

whereanddenote the transposed vectors of singular value decomposition matrix of matrix C; S denotes the true value of strains; δ denotes the interference signals; σ_i denotes the ith singular value of matrix C; k denotes the reserved number of singular values.

Figure 8 Calibrated strain curves after clipping and stacking

Figure 9 Numerical distribution of elements in first row of matrix C with 10 orders

Obviously, when k=N:

L_TSVD=L                                (7)

where N denotes the order of matrix C; L denotes the solution of Eq. (5).

The basic idea of the truncated singular value decomposition is to eliminate the corresponding small σ_i term in Eq. (6) and keep the corresponding part of larger σ_i to modify the solution of the equation. The influence of interference signals is reduced and the solutions of the equations can reflect the main features more accurately. L-curve method is one of the effective ways to select truncation parameters [36]. The L-curve method is to draw the curve between the residual norm ||CL-S|| and the regularized solution norm ||L|| for different regularization parameters. In general, the curve presents L-shape, and the inflection point of the curve is the optimal value of the regularization parameters. Taking the matrix C with 10 orders as an example, the truncation coefficient k of Eq. (6) is finally determined to be 3.

Some singular values are dropped when choosing the truncation parameter, which reduces the resolution of the results. Therefore, the amplitude of the contact load distribution curve needs to be modified. The actual contact loads are set to the following expression:

L=λL_N                                  (8)

where L_N denotes the fitting curve of contact loads when matrix C is N order, λ denotes the amplitude amplification factor.

Based on the sum of the absolute values of the strain difference between the calculated and measured strains under different amplitude amplification factors, which can be calculated through Eq. (3), the best amplification factor is 2.6. The fitting degree between the calculated and the experimental measured strain curve with the penetration of 4 mm is 0.95, as shown in Figure 10, which verifies the correctness and accuracy of the model. For the convenience of illustration, the abscissas 0 and 1 in Figure 10 indicate the beginning and the end of the contact area, which is A and B respectively in Figure 2.

Figure 10 Comparison of calculated and measured strains

5 Discussion

5.1 Strain analysis of disc cutter ring

According to the test matrix, four typical radial strain curves of disc cutter ring under different penetrations are edited and stacked together, as shown in Figure 11. Under the same penetration, the measured strain curves appear several fluctuations in some parts. This is probably due to the overbreak of the upper layers in the process of flatting the rock surface, resulting in the reduction of the loads applied to a disc cutter in the cutting paths of the lower layers. This is difficult to be avoided owing to the random distribution of cracks in rock. It is necessary to repeat the experiments for a number of times to obtain the average strains to reduce the random error. Fortunately, the overall trend of measured results is acceptable to study on the contact loads between cutter ring and rock.

Figure 11 Radial strain curves at different penetrations:

The abscissa is the angle that the point has rotated from the start of theoretical contact area. And the angle value is positive when turning anticlockwise. The central angle θ corresponding to the theoretical contact area can be calculated by the following formula:

                           (9)

where R denotes the diameter of disc cutter; h denotes the penetration.

Figure 11 shows that the range of strains on the cutter ring is larger than that of the theoretical contact area at all the penetrations. But the amplitude of strain outside the contact area is rather small, which can be inferred from Eq. (1). The strains of the cutter ring within the contact area experience a process of increase-peak-decrease. Figure 11(d) shows the trend of average strain curves with different penetrations. Taking into account the central angle corresponding to the theoretical contact area under different penetrations is different, which can be calculated from Eq. (9), the abscissa is normalized. It has the same meaning as that in Figure 10. It is obvious to see that the change rules of the strain curves at different degrees of penetration are the same. The overall average value and peak of the strains increase with increasing penetration. The peak value of the strains is more than 500 με at the penetration of 6 mm. It can be inferred that there is a positive correlation between the contact loads and penetration.

5.2 Analysis of contact loads between cutter ring and rock

According to the 4th chapter, the contact loads between disc cutter and rock under different penetrations are solved. The varying curves of the contact loads in theoretical contact area are obtained as shown in Figure 12. Generally, the distribution of loads appears to be a parabola going downwards, and its maximum value is near the middle of the theoretical contact area. The contact loads are overall small, and the amplitudes increase slightly while raising the level of penetration, which has a direct relationship with the low hardness of the cutting samples. When the penetration increases by 2 mm, the peak value of contact loads will increase by about 0.5 kN. It shows that the contact loads and penetration are positively related.

The subsequent analysis of the contact area is shocking. As shown in Figure 12, there is a non-loading area at the beginning of theoretical contact area as well as a non-loading area directly underneath the disc cutters. It shows that the loads are more concentrated compared with the initially assumption that there is a linear increasing or uniform distribution of load in the field of cutter ring. This result is a good evidence that the previous assumptions about the range of the contact area and the amplitudes of the loads look so idealized. As shown in Figure 13, the ratio of peak load to average load at the penetration of 2, 4 and 6 mm is 1.91, 2 and 1.63, respectively. It can be understood that the cutter ring, which is designed according to an average load applied to it, has a short service time. Table 3 lists the summary of calculated distribution of both loading area and non-loading area for the concrete samples under different penetrations. The front non-loading area is between the start of the contact area to the actual loading area, and the rear non-loading area is between the loading area and the vertical line. As can be seen, there are not obvious front and rear non-loading areas compared to the results in hard rocks [11]. It is probably due to the low strength of test samples, which mainly occurs plastic failure without lots of brittle cracks. With the increase of penetration, the proportion of non-loading area increases. The ratio of front non-loading areas increases slightly from 1.8% to 4.2% with penetration increased. The extent of the front non-loading area is a little larger at the penetration of 4 mm. It is probably due to a certain degree of local extrusion in the process of breaking rock, resulting in the presence of compacting rock cores, which increases the degree of stress concentration. However, the change of rear non- loading area is not obvious. The overall amplitude is maintained at 1.3%-3.1%.

These results allow us to have a new understanding of the process of rock breaking. Figure 14 shows the schematic view of distribution of contact load in the crushed area. The front non- loading area could simply be interpreted as the generation of the fragmentation owing to the production of high pressure zone and the propagation of stress waves. In the middle of contact area, the compacting rock cores generate a stress bubble accomplishing the actual crushing. The contact loads increase to their peaks. With the rotation of the disc cutter, the fragmentation created on the side moves backwards, and the pressure relieves as well as the loads. Therefore, there is a non-loading area directly underneath the disc cutter.

Figure 12 Curves of contact loads between cutter ring and rock

Figure 13 Comparison of peak load and average load

Table 3 Test data of contact loads distribution between cutter ring and rock

Figure 14 Schematic view of distribution of contact load in crushed area (-Theoretical contact area; - Actual contact area; -Front non loading area; - Rear non loading area)

6 Conclusions

In this paper, the radial strain distribution of cutter ring under different penetrations is obtained by the multifunctional TBM cutter performance test bench. The contact loads between disc cutter and rock are solved by TSVD. The following conclusions could be drawn:

1) The range of the cutter ring strains which experiences a process of increase-peak-decrease is larger than the theoretical contact area. The peak value of the strains is more than 500 με at the penetration of 6 mm. It can be inferred that there is a positive correlation between the contact loads and penetration.

2) The distribution of contact loads appears to be a parabola going downwards, and its maximum value occurs near the middle of the theoretical contact area. An increasing trend in contact loads with the increase of penetration can be observed. The peak value can be twice higher than the average load in the process of breaking rock.

3) The actual contact area is smaller than the theoretical contact area, which indicates that contact loads are more concentrated. The front non-loading area with a ratio from 1.8% to 5.4% increases slightly with the increase of penetration. However, rear non-loading area is not sensitive to it.

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(Edited by FANG Jing-hua)

中文导读

隧道硬岩掘进机盘形滚刀破岩过程中刀岩接触载荷

摘要：隧道硬岩掘进机(TBM)破岩刀具的使用寿命是刀岩接触载荷作用的结果，这与工程的施工效率和质量息息相关。本文采用直径为216 mm的盘形滚刀进行了一系列混凝土切削试验，基于叠加原理，利用截断奇异值法(TSVD)得到了盘形滚刀与岩石接触区接触载荷的分布规律。研究结果表明：随着贯入度的增加，刀圈径向应变峰值和整体数值均增加；刀岩接触载荷分布曲线近似为开口向下的抛物线，接触载荷更加集中；随着贯入度的增加，前卸荷区占比从1.8%增加至5.4%，而后卸荷区无明显变化；研究结果对盘形滚刀破岩机理和制造工艺的研究有一定的指导意义。

关键词：盘形滚刀；接触载荷；叠加原理；卸荷区

Foundation item: Project(51475478) supported by the National Natural Science Foundation of China; Project(2013CB035401) supported by the National Basic Research Program of China

Received date: 2018-06-29; Accepted date: 2018-10-24

Corresponding author: XIA Yi-min, PhD, Professor; Tel: +86-731-88876926; E-mail: xiaymj@csu.edu.cn; ORCID: 0000-0001-6174- 0377