# Discrete/Finite Element Modelling of Rock Cutting with a TBM Disc Cutter

- 9k Downloads
- 8 Citations

## Abstract

This paper presents advanced computer simulation of rock cutting process typical for excavation works in civil engineering. Theoretical formulation of the hybrid discrete/finite element model has been presented. The discrete and finite element methods have been used in different subdomains of a rock sample according to expected material behaviour, the part which is fractured and damaged during cutting is discretized with the discrete elements while the other part is treated as a continuous body and it is modelled using the finite element method. In this way, an optimum model is created, enabling a proper representation of the physical phenomena during cutting and efficient numerical computation. The model has been applied to simulation of the laboratory test of rock cutting with a single TBM (tunnel boring machine) disc cutter. The micromechanical parameters have been determined using the dimensionless relationships between micro- and macroscopic parameters. A number of numerical simulations of the LCM test in the unrelieved and relieved cutting modes have been performed. Numerical results have been compared with available data from in-situ measurements in a real TBM as well as with the theoretical predictions showing quite a good agreement. The numerical model has provided a new insight into the cutting mechanism enabling us to investigate the stress and pressure distribution at the tool–rock interaction. Sensitivity analysis of rock cutting performed for different parameters including disc geometry, cutting velocity, disc penetration and spacing has shown that the presented numerical model is a suitable tool for the design and optimization of rock cutting process.

## Keywords

Rock cutting Disc cutters TBM Numerical model Discrete/finite element method Simulation## 1 Introduction

TBMs are widely used in various tunnelling projects in civil engineering (road and railway tunnels), mining industry (tunnels for access to underground excavations, conveyance of ore and waste, drainage, exploration, water supply and diversion) and other geotechnical engineering applications. The use of TBMs is continuously growing mainly due to their efficiency. Nevertheless, there is still a need of improvement of TBM performance depending mainly on rock properties, operational parameters (machine trust, penetration and rate of advance) and design of the cutter head design, including design of the disc cutters (Roby et al. 2008) and design of their layout (Huo et al. 2010).

Historically, design of rock cutting tools for the excavation machinery has been based on a combination of the experience of engineers and real size laboratory tests, resulting many times in an inefficient process, and involving high costs for the excavation companies. Different empirical models have been developed for the estimation of the principal parameters involved (Nilsen and Ozdemir 1993; Rostami and Ozdemir 1996). These models are useful in certain cases; nevertheless, their use is restricted by the availability of historical data and range of rock material properties (Ramezanzadeh et al. 2004).

Application of numerical analysis improves the design methodology and allows to obtain more efficiently optimized designs of rock cutting tools and machines. Numerical methods can be used to optimize the TBM cutter layout (Sun et al. 2015) or simulate interaction between TBM components and rock mass (Zhao et al. 2014). The present work is focused on simulation of rock cutting with a single TBM disc cutter. Such an analysis can help the designer to understand better rock cutting mechanisms, to detect the reasons of the cutting tool wear and failure, and finally to improve the cutter design and determine optimum operational parameters. The aim of the analysis is to substitute or at least reduce number of laboratory tests used in the design process of rock cutting tools (Nilsen and Ozdemir 1993; Rostami and Ozdemir 1993), and finally speed-up the design process and reduce its costs.

Simulation of excavation processes, in general, and of rock cutting process, in particular, is not an easy task. Continuum-based simulation techniques, such as the finite element method (FEM), encounter serious difficulties in modelling of fracture and fragmentation of a rock material occurring in an excavation process (Jonak and Podgórski 2001; Yu and Khair 2007; Shenghua 2004; Loui and Karanam 2012). The discrete element method (DEM) employing a discrete material model offers a more realistic way to simulate discontinuous phenomena. The DEM has been successfully applied to simulation of different rock cutting processes by Stavropoulou (2006), Rojek et al. (2011), Su and Akcin (2011), Huang et al. (2013), Labra et al. (2008), Wyk et al. (2014).

The known disadvantage of the DEM is that it usually requires the use of a large number of elements, which leads to long computation times. This paper put forwards the idea allowing to reduce the computation cost of the DEM simulation of rock cutting by coupling the DEM with the FEM and using them in different subdomains of the cut material according to expected material behaviour. In rock cutting problems, a sufficiently large specimen must be taken in order to avoid artificial boundary effects. In such specimens, a large part of the rock material is not damaged and can be treated as a continuous material. A continuous material deformation is usually modelled efficiently using the FEM. Therefore, an optimum model of rock cutting can be obtained combining the DEM with the FEM in such a way that discrete elements are used only in a portion of the analysed domain where material fracture occurs, while outside the DEM subdomain finite elements are used (Oñate and Rojek 2004; Labra et al. 2008). The numerical method proposed in this paper is based on the formulation presented by Rojek and Oñate (2007). Preliminary results for a 2D rock cutting problems obtained with the DEM/FEM coupled method have been presented by Labra et al. (2008). In the present paper, this algorithm is applied to 3D simulation of the rock cutting problem. This work is original with respect to other research. As far as we know, DEM/FEM coupling has not been used for rock cutting simulations by other authors. Other studies of rock cutting, cf. Stavropoulou (2006), Rojek et al. (2011), Su and Akcin (2011), Huang et al. (2013), Wyk et al. (2014), have employed pure DEM models.

Numerical simulations presented here have been performed for the linear cutting machine (LCM) test. The LCM test is one of the most representative testing procedures for prediction of TBM disc cutters (Nilsen and Ozdemir 1993; Rostami and Ozdemir 1993), and it is used extensively in the design of rock cutting tools. Both unrelieved and relieved cutting modes have been studied. The simulation results have been compared with available experimental data and the predictions of the empirical model developed by Rostami (1997). The use of numerical models and simulation has allowed us to study some aspects of rock cutting such as the stress field in rock and disc or pressure in the contact zone which are difficult to study using experimental techniques. Direct measurements of the load distribution in the contact between disc cutter and rock performed by Rostami (2013) have proved that the problem of the load and stress concentration in the contact zone is far from being fully understood and requires further studies. The numerical model presented in this work can be a useful tool to verify theoretical models for rock cutting with disc cutters. Rock fragmentation induced by TBM disc cutters has also been studied by other authors using DEM models (Gong et al. 2006; Moon et al. 2006), but in those works the authors employed 2D models so the simulations consisted in analysis of indentation under plane stress conditions. The present work is the first one employing a full 3D model reproducing real cutting conditions for a TBM disc cutter.

## 2 Linear Cutting Test

The LCM test is a suitable test to validate a numerical model of rock cutting. With the simulations of the LCM test for a single disc cutter, it is possible to perform an analysis of the effect of different parameters such as velocity, penetration rate or tool geometry, on the resultant forces, as well as their influence on the rock fracture.

## 3 Theoretical Predictions of Rock Cutting Performance

### 3.1 Cutting Forces Prediction Models

Cutting forces estimation is based on the correlation of different parameters, such as the disc cutter geometry, spacing, penetration rate, disc rolling velocity and rock material properties. Early prediction models for single V-shape disc cutters were proposed by Roxborough and Phillips (1975), Sanio (1985) and Sato and Itakura (1991). Colorado School of Mines developed a model for the CCS-shape cutters (Rostami and Ozdemir 1993; Rostami 1997). This model will be used here for the comparison with the simulation results for the LCM test.

The first version of the Colorado School of Mines model was developed by Ozdemir (1977), and later was updated by Rostami and Ozdemir (1993), Rostami (1997). The CSM model estimates the cutting forces considering a given penetration, rock mass properties, cutter geometry and cutting conditions. The model is based on a large database of full-scale LCM tests and does not consider rock mass conditions such as fractures or joints.

*P*in the crushed zone as

*C*is a dimensionless constant (usually \(C=2.12\)), and

*s*the spacing between cutters.

*T*is the cutter tip width and

*R*the cutter radius.

*CC*(also called rolling coefficient) is used, which is the ratio of both forces defined by the angle \(\beta \) as

This model has been used for the estimation of the TBM cutterhead performance in many tunnelling projects with a high degree of success (Rostami and Ozdemir 1996; Rostami 2008; Exadaktylos et al. 2008).

### 3.2 Specific Energy

*SE*is defined as the amount of energy required to excavate a unit volume of rock.

*SE*for a single disc cutter is defined as

*L*the cutting distance, and

*V*the cutting volume. Assuming optimum cutting performance, the cutting volume

*V*can be expressed in terms of the penetration depth

*p*and spacing between disc cutters

*s*as \(V=p \, L \, s\), which allows us to rewrite Eq. (9) as follows

*s*/

*p*) is a suitable variable to investigate cutting efficiency with TBMs (Sato and Itakura 1991; Rostami 1997, 2008).

*SE*(in kWh/\(\hbox {m}^3\)), the net production rate

*NPR*(in \(\hbox {m}^3\)/h) of the excavating machine can be calculated from the following equation:

*k*is the energy transfer ratio from the cutting head to the tunnel face, it is usually taken as equal to 0.8 for TBM, cf. Bilgin et al. (2014), and

*P*(in kW) is the power used to excavate rock expressed in terms of the TBM torque

*T*(in kNm) and the rotational velocity of the cutterhead

*N*(in revolutions/sec) as follows:

## 4 Formulation of the Model

### 4.1 Basic Assumptions

A numerical model allowing us to simulate the LCM test has been developed. A system consisting of a tool and rock sample is considered in the model. The tool is considered as a rigid body. The rock is modelled using a hybrid discrete-finite element method approach in which a part of the rock near the tool is modelled using the discrete element method (DEM) and the other part is considered using the finite element method (FEM). The DEM and FEM subdomains are coupled using special kinematic constraints. The tool–rock interaction is modelled assuming the Coulomb friction model. Rock fracture during cutting is assumed to be localized in the DEM subdomain, and the FEM subdomain is assumed to be continuous and linearly elastic.

### 4.2 Discrete Element Method Formulation

The discrete element model assumes that material can be represented by an assembly of distinct particles or bodies interacting among themselves. Generally, discrete elements can have arbitrary shape. In this work, the formulation employing spherical rigid particles is used. Basic formulation of the particle-based discrete element method was first proposed by Cundall and Strack (1979). A similar formulation of the DEM has been implemented in the explicit dynamic finite element method programme by Rojek and Oñate (2004), Oñate and Rojek (2004), starting the development of the DEM/FEM code DEMpack (CIMNE 2010).

*i*-th element, we have

*i*-th element due to external load, \(\mathbf{F}^{\text {ext}}\) and \(\mathbf{T}^{\text {ext}}\), contact interactions with neighbouring spheres and other obstacles \(\mathbf{F}^{\text {cont}}\) and \(\mathbf{T}^{\text {cont}}\), as well as forces and moments resulting from external damping in the system, \(\mathbf{F}^{\text {damp}}\) and \(\mathbf{T}^{\text {damp}}\), respectively, which can be written as

*i*-th element with the contact point with the

*j*-th element.

### 4.3 Constitutive Contact Models

The overall behaviour of the system is determined by the contact laws assumed for the particle interaction. Models of contact in the discrete element method can include force and moment interaction between particles. In the present work, however, contact moments are not considered.

*l*(also called branch length) and their radii \(r_i\) and \(r_j\)

Two approaches to evaluation of contact model parameters, \(K_t\), \(K_n\), \(R_t\) and \(R_n\), can be distinguished. In the first approach, the stiffness and strength parameters of the contact model are assumed to depend on the size of contacting particles and are evaluated locally as certain functions of contacting pair radii (Potyondy and Cundall 2004). In the second approach, uniform microscopic properties are assumed in the whole discrete element assembly (Kruyt and Rothenburg 2004). The latter approach is adopted in the present work. Numerical studies performed by Rojek et al. (2012) have shown that uniform global parameters lead to a more brittle behaviour of discrete element models than local size-dependent parameters.

### 4.4 Discrete/Finite Element Method Coupling

It is assumed that the DEM and FEM can be applied in different subdomains of the same body. The DEM and FEM subdomains can overlap each other. The common part of the subdomains is the part where both discretization types are used with gradually varying contribution of each modelling method. This allows us to avoid or minimize unrealistic wave reflections at the interface between the DEM and FEM subdomains. The idea of such coupling follows that used by Xiao and Belytschko (2004) for bridging molecular dynamics with a continuous model.

### 4.5 Determination of DEM Parameters

The discrete element model can be regarded as a micromechanical material model, with the contact model parameters being micromechanical parameters. Assuming adequate micromechanical parameters, we obtain required macroscopic rock properties. The most important macroscopic rock properties include the Young’s modulus *E*, Poisson’s coefficient \(\nu \), compressive strength \(\sigma _c\) and tensile strength \(\sigma _t\), which will be used for the model calibration in this work. The contact stiffness moduli, \(K_n\) and \(K_t\), and bond strengths, \(R_n\) and \(R_t\), as well as the Coulomb friction coefficient \(\mu \) will be taken as the most significant micromechanical parameters influencing the macroscopic elastic and strength properties.

*L*and

*A*are characteristic lengths and areas of the particle system, and \(\Phi \) is a function representing the assembly characterization parameters influence, as the porosity of the particle assembly

*e*, or the average particle radius \(\bar{r}\).

*l*over all the bonded contacts of the assembly, as

## 5 Simulation of Unrelieved Rock Cutting

### 5.1 Numerical Model

The cutter disc has been discretized with 8880 triangular elements, considering a refinement in the cutter tip in order to reproduce its curvature. It has been treated as a rigid body with all the discretizing nodes slaved to its centre. The disc has a prescribed translational motion and can rotate about its axis of symmetry under the action of the rolling force. A real value of the disc moment of inertia has been used. In this way, a real kinematics and dynamics of the disc cutters during cutting have been reproduced.

The full-scale LCM tests are performed using 1.0 m \(\times \) 0.7 m \(\times \) 0.7 m block rock samples. In order to obtain results at a reasonable computational cost, a smaller sample with dimensions 0.4 m \(\times \) 0.15 m \(\times \) 0.4 m has been taken in the numerical model. Numerical tests have shown that it is sufficiently large to avoid boundary effects. The unrelieved rock sample with the adopted boundary conditions is shown in Fig. 8. The displacement of the bottom surface of the sample has been completely restricted, while lateral surfaces have had the out-of-surface displacements restricted.

Characterization of the particle assembly for the LCM test

Parameter | Value |
---|---|

Number of particles, \(N_p\) | 35,604 |

Characteristic radius, \(\widetilde{r}\) (mm) | 2.7971 |

Coordination number, \(n_c\) | 11.449 |

Porosity, | 24.912 |

Mechanical rock properties

Parameter | Value |
---|---|

Uniaxial compressive strength, \(\sigma _c\) (MPa) | 147.3 |

Brazilian tensile strength, \(\sigma _t\) (MPa) | 10.2 |

Young modulus, | 40.0 |

Poisson’s ratio, \(\nu \) | 0.23 |

Density, \(\rho \) (kg/\(\hbox {m}^3\)) | 2650 |

DEM model parameters for the LCM test

Parameter | Value |
---|---|

Normal stiffness, \(k_{n}\) (MN/m) | 160.79 |

Tangential stiffness, \(k_{t}\) (MN/m) | 16.325 |

Normal bond strength, \(R_{n}\) (kN) | 0.8482 |

Tangential bond strength, \(R_{t}\) (kN) | 4.1759 |

Density, \(\rho \) (kg/\(\hbox {m}^3\)) | 3085 |

The density of the particles is calculated considering the solid fraction of the assembly, in order to preserve the equivalent mass.

Cutting process parameters

Parameter | Value |
---|---|

Penetration rate, | 3.9 |

Cutting velocity, | 2.37 |

The amount of penetration per revolution has been taken as the penetration depth in the LCM model. The penetration depth in all the simulations has been defined as the indentation depth of the cutter tip below the free undamaged surface of the rock sample. It was kept constant under the prescribed cutting tool trajectory. In the LCM test, one of the most important parameters is the spacing between the disc cutters (Rostami and Ozdemir 1993). This parameter cannot be taken into account in the unrelieved cutting. It will be included in simulation of the relieved cutting later on.

### 5.2 Numerical Results

Summary of the results for the LCM test with unrelieved specimen

Simulation | Experiment | |
---|---|---|

\(F_n\) (kN) | 191.1 | 231.8 |

\(F_r\) (kN) | 19.6 | – |

\(F_s\) (kN) | 0.9 | – |

| 0.102 | – |

The experimental normal force has been estimated from in-situ measurements of a real TBM thrust (Labra et al. 2008b). The in situ value of the normal force for one disc has been calculated by dividing the total thrust force by the total number of disc cutters. In principle, the thrust force obtained from a TBM represents the relieved cutting mode because of interaction between cutting grooves in a TBM excavation. However, in this case the cutter spacing was quite large so the interaction between disc cutters was small and the conditions were close to the unrelieved cutting mode.

The average normal force obtained in simulation, 191.08 kN, agrees quite well with the estimated experimental value, 231.8 kN. No comparison with the force predicted by the CSM model has been done because this model considers a spacing between disc cutters, which requires the use of the relieved rock sample.

Using the values of the average normal and rolling forces obtained in simulation, the cutting coefficient *CC* is evaluated. The value of the cutting coefficient given in Table 5, 0.102, is in the range of the *CC* values expected for a hard rock (Tarkoy 1983). Obviously, correctness of prediction of the normal forces and cutting coefficients verifies correctness of the predicted values of the rolling forces.

The information on stress distribution and evolution during cutting cannot be obtained directly in laboratory tests. Stress analysis is important for investigation of the mechanism of rock cutting and verification of the assumptions made in theoretical models of rock cutting.

*CC*) obtained in the simulation are compared with the respective values predicted by the CSM model in Table 6. The theoretical values of the angle \(\beta \) according to the CSM model have been calculated as proposed by Gertsch et al. (2007), Rostami (2008) It can be noted that the values of the angle \(\beta \) and cutting coefficient (

*CC*) obtained in the simulation are very close to those estimated by the CSM model.

Comparison of angle \(\beta \) and cutting coefficient (*CC*) in the unrelieved LCM test

CSM model | Simulation | |
---|---|---|

\(\beta \) (deg) | 5.158 | 5.859 |

| 0.09027 | 0.10262 |

### 5.3 Sensitivity Analysis of Unrelieved Cutting

Influence of various parameters on the performance of the rock cutting process has been analysed. Sensitivity of the results to changes of process and design parameters is very important for an optimization of the process. Two different categories of the parameters involved in the rock cutting process can be distinguished. The first category is related to the mechanical properties of the rock specimen. Most of the studies reveal that the most relevant mechanical properties of the rock are the compressive and tensile strengths. The elastic constants of the rock are not considered important for the estimation of the cutting forces and are not analysed in this section. The second category involves the geometric settings of the cutting process. Here, we can include the profile of the disc cutter, the spacing between discs and penetration. Sensitivity analysis of unrelieved cutting has been performed for the uniaxial compressive strength, cutting velocity and penetration depth. The effect of disc spacing will be analysed later for relieved cutting.

#### 5.3.1 Effect of the Uniaxial Compressive Strength

Figure 15 shows the effect of the compressive strength on the value of the cutting coefficient. In contradiction to the CSM and other theoretical models in the literature, which relate the cutting coefficient with geometric parameters, such as the penetration and disc cutter radius, Fig. 15 shows a linear relationship between the compressive strength and the cutting coefficient.

#### 5.3.2 Effect of the Penetration Depth

The penetration depth, or penetration rate, is one of the most important geometric parameters, together with the disc spacing, that affect the performance of the TBM, as it is directly related with the advance rate of the TBM cutterhead. All the theoretical models employ this parameter in evaluation of cutting forces as well as in estimation of the volume of rock material excavated and amount of energy required for the excavation process.

When the penetration depth increases, the normal and rolling forces also increase. The influence of the penetration depth can be clearly seen in the value of the rolling force, that is directly related to the energy required in the cutting process.

## 6 Simulation of Relieved Rock Cutting

### 6.1 Numerical Model

*l*in Fig. 19 changes depending on the spacing between the disc cutter and the previous pass in order to maintain a sufficient distance between the disc cutter and the DEM/FEM coupling interface. Two different specimens, for penetration of 4 and 8 mm, have been generated. The DEM subdomains have been discretized with 51668 and 50714 particles, respectively. The main parameters characterizing the particles assemblies are summarized in Table 7. The DEM parameters for the new rock specimens are summarized in Table 8.

Characterization of the particles assemblies for relieved rock specimens

Parameter | Penetration | ||
---|---|---|---|

4 mm | 8 mm | ||

\(N_p\) | Number of particles | 51668 | 50714 |

\(\widetilde{r}\) | Characteristic radius (mm) | 3.0039 | 2.9992 |

\(n_c\) | Coordination number | 9.0635 | 9.1028 |

| Porosity (%) | 22.647 | 22.821 |

DEM model parameters for LCM test with relieved rock specimens

Parameter | Penetration | |
---|---|---|

4 mm | 8 mm | |

Normal stiffness, \(k_{n}\) (MN/m) | 160.79 | 160.79 |

Tangential stiffness, \(k_{t}\) (MN/m) | 16.325 | 16.325 |

Normal bond strength, \(R_{n}\) (kN) | 0.4642 | 0.4640 |

Tangential bond strength, \(R_{t}\) (kN) | 4.1779 | 4.1761 |

Density, \(\rho \) (kg/\(\hbox {m}^3\)) | 3085 | 3085 |

### 6.2 Numerical Results

Results of the numerical simulations have been compared with the predictions of the CSM model. The normal forces obtained in the simulation are slightly higher than those evaluated using the CSM model for both cases of penetration. So are the numerical rolling forces higher than the theoretical predictions although in this case the difference is smaller.

Comparison of the angle \(\beta \) and cutting coefficient in the relieved LCM test for spacing of 80 mm

CSM model | Simulation | |||
---|---|---|---|---|

4 mm | 8 mm | 4 mm | 8 mm | |

\(\beta \) (deg) | 5.224 | 7.397 | 5.240 | 6.826 |

| 0.0914 | 0.1298 | 0.0917 | 0.1197 |

*p*and disc cutters spacing

*s*, the specific energy

*SE*can be calculated from Eq. (10). The specific energy as a function of the spacing/penetration ratio is plotted in Fig. 23.

The specific energy computed in simulation of relieved cutting is compared in this figure with the specific energy estimated by the CSM model for different values of penetration and spacing. A good correlation between the CSM model and the simulation results has been obtained.

The specific energy estimated in simulations agrees quite well with field data reported in the literature. Bilgin et al. (2014) report the field specific energy for the Kadikoy–Kartal Metro where the mean compressive strength of excavated rocks was 50 MPa. The field specific energy varied in the range 7–18 kWh/\(\hbox {m}^3\) depending on the penetration rate. These values are slightly below the values given in Fig. 23, but this is understandable since the compressive strength in our case was higher (147.3 MPa). The field specific energy reported also by Bilgin et al. (2014) for the Beykoz tunnel excavated in rocks with higher compressive strength (100 MPa) varies from 5 to 27 kWh/\(\hbox {m}^3\). The field data show that the numerical simulations give reasonable predictions of the specific energy. This is an important result of our simulations since confirms a good performance of the model and its potential utility in real applications.

## 7 Conclusions

Numerical tests have demonstrated a good performance of the coupled discrete/finite element model of rock cutting. A good agreement of numerical results with experimental measurements and theoretical predictions has been found. Main parameters characterizing rock cutting with a TBM disc cutters such as cutting and rolling forces, cutting coefficient and specific energy have been estimated correctly in numerical simulations.

The numerical model is capable to represent properly complexity of rock cutting with TBM disc cutters. The numerical simulation can provide valuable information about the cutting phenomenon such as stress distribution in the rock and contact pressure distribution at the tool–rock interaction area. Numerical results have confirmed a non-uniform contact pressure distribution revealed in experimental investigations and shown that a uniform pressure distribution in theoretical models is a simplified assumption. The numerical model has provided a new insight into the cutting process enabling us to understand better rock cutting mechanism.

Sensitivity analysis of rock cutting performed for different parameters including disc geometry, cutting velocity, disc penetration and spacing has shown that the presented numerical model is a suitable tool for the design and optimization of rock cutting process.

## References

- Balci C, Bilgin N (2007) Correlative study of linear small and full-scale rock cutting tests to select mechanized excavation machines. Int J Rock Mech Min Sci 44:468–476CrossRefGoogle Scholar
- Bilgin N, Balci C, Acaroglu O, Tuncdemir H, Eskikaya S, Akgul M, Algan M (1999) The performance prediction of a TBM in Tuzla-Dragos Sewerage Tunnel. In: Proceedings of the 1999 World Tunnel Congress, pp 817–822Google Scholar
- Bilgin N, Copur H, Balci C (2014) Mechanicsal excavation in mining and civil industries. CRC Press, Taylor & Francis GroupGoogle Scholar
- Cho JW, Jeon S, Yu SH, Chang SH (2010) Optimum spacing of TBM disc cutters: a numerical simulation using the three-dimensional dynamic fracturing method. Tunn Undergr Space Technol 25(3):230–244CrossRefGoogle Scholar
- Choi S (1992) Application of the distinct element method for rock mechanics problems. Eng Comput 9:225–233CrossRefGoogle Scholar
- CIMNE (2010) DEMpack, explicit nonlinear dynamic analysis by the finite and discrete element method. Web: www.cimne.upc.edu/dempack
- Cundall P, Strack O (1979) A discrete numerical method for granular assemblies. Geotechnique 29:47–65CrossRefGoogle Scholar
- EMI (2016) Excavation Engineering and Earth Mechanics Institute. Colorado School of Mines. http://insideminesedu/EMI-homeGoogle Scholar
- Exadaktylos G, Stavropoulou M, Xiroudakis G, de Broissia M, Schwarz H (2008) A spatial estimation model for continuous rock mass characterization from the specific energy of a TBM. Rock Mech Rock Eng 41:797–834CrossRefGoogle Scholar
- Fakhimi A, Villegas T (2007) Application of dimensional analysis in calibration of a discrete element model for rock deformation and fracture. Rock Mech Rock Eng 40(2):193–211CrossRefGoogle Scholar
- Farrokh E, Rostami J, Laughton C (2012) Study of various models for estimation of penetration rate of hard rock TBMs. Tunn Undergr Space Technol 30:110–123CrossRefGoogle Scholar
- Gertsch R, Gertsch L, Rostami J (2007) Disc cutting tests in Colorado Red Granite: implications for TBM performance prediction. Int J Rock Mech Min Sci 44(2):238–246CrossRefGoogle Scholar
- Gong QM, Zhao J, Hefny A (2006) Numerical simulation of rock fragmentation process induced by two TBM cutters and cutter spacing optimization. Tunn Undergr Space Technol 21(3–4):1–8Google Scholar
- Huang H (1999) Discrete element modeling of tool–rock interaction. Ph.D. thesis, University of MinnesotaGoogle Scholar
- Huang H, Detournay E (2008) Intrinsic length scales in tool–rock interaction. Int J Geomech 8(1):39–44CrossRefGoogle Scholar
- Huang H, Lecampion B, Detournay E (2013) Discrete element modeling of tool–rock interaction I: rock cutting. Int J Numer Anal Methods Geomech 37(13):1913–1929CrossRefGoogle Scholar
- Huo J, Sun W, Su P, Deng L (2010) Optimal disc cutters plane layout design of the full-face rock tunnel boring machine (TBM) based on a multi-objective genetic algorithm. J Mech Sci Technol 24(2):521–528CrossRefGoogle Scholar
- Inc ICG (2006) \(\text{PFC}^3D\) Particle Flow Code in 3D, Theory and Background Manual. Minneapolis, MN, USAGoogle Scholar
- Jonak J, Podgórski J (2001) Mathematical model and results of rock cutting modelling. J Min Sci 37:615–618CrossRefGoogle Scholar
- Kruyt NP, Rothenburg L (2002) Micromechanical bounds for the effective elastic moduli of granular materials. Int J Solids Struct 39(2):311–324CrossRefGoogle Scholar
- Kruyt N, Rothenburg L (2004) Kinematic and static assumptions for homogenization in micromechanics of granular materials. Mech Mater 36(12):1157–1173CrossRefGoogle Scholar
- Labra C (2012) Advances in the development of the discrete element method for excavation processes. Ph.D. thesis, Technical University of CatalonyaGoogle Scholar
- Labra C, Rojek J, Oñate E, Zarate F (2008a) Advances in discrete element modelling of underground excavations. Acta Geotech 3(4):317–322CrossRefGoogle Scholar
- Labra C, Rojek J, Oñate E, Köppl F (2008b) Tunconstruct D2.1.3.3: report of numerical modeling of disc cutter wear. Tech. rep., CIMNEGoogle Scholar
- Loui J, Karanam UR (2012) Numerical studies on chip formation in drag-pick cutting of rocks. Geotech Geol Eng 30:145–161CrossRefGoogle Scholar
- Moon T, Nakagawa M, Berger J (2006) DEM analysis for optimizing TBM performance. In: ARMA Conference Paper 06–1125 (1–5)Google Scholar
- Nilsen B, Ozdemir L (1993) Hard rock tunnel boring prediction and field performance. In: Rapid excavation and tunneling conference RETCGoogle Scholar
- Oñate E, Rojek J (2004) Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems. Comput Methods Appl Mech Eng 193:3087–3128CrossRefGoogle Scholar
- Oñate E, Rojek J (2004) Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems. Comput Methods Appl Mech Eng 193(27–29):3087–3128CrossRefGoogle Scholar
- Ozdemir L (1977) Development of theoretical equations for predicting tunnel borability. Ph.D. thesis, Colorado School of Mines, Golden, Colorado, USAGoogle Scholar
- Potyondy D, Cundall P (2004) A bonded-particle model for rock. Int J Rock Mech Min Sci 41(8):1329–1364 rock Mechanics Results from the Underground Research Laboratory, CanadaGoogle Scholar
- Ramezanzadeh A, Rostami J, Kastner R (2004) Performance prediction models for hard rock tunnel boring machines. In: Proceedings of sixth Iranian tunneling conference, Tehran, IranGoogle Scholar
- Roby J, Sandell T, Kocab J, Lindbergh L (2008) The current state of disc cutter design and development directions. In: Roach M, Kritzer M, Ofiara D, Townsed B (eds) North American Tunneling 2008 proceedings, Society for Mining, Metallurgy and Exploration, Inc. (SME), pp 36–45Google Scholar
- Rojek J, Oñate E, Labra C, Kargl H (2011) Discrete element simulation of rock cutting. Int J Rock Mech Min Sci 48(6):996–1010CrossRefGoogle Scholar
- Rojek J, Labra C, Su O, Oñate E (2012) Comparative study of different discrete element models and evaluation of equivalent micromechanical parameters. Int J Solids Struct 49:1497–1517CrossRefGoogle Scholar
- Rojek J, Karlis G, Malinowski L, Beer G (2013) Setting up virgin stress conditions in discrete element models. Comput Geotech 48:228–248CrossRefGoogle Scholar
- Rojek J, Oñate E (2004) Unified DEM/FEM approach to geomechanics problems. In: Proceedings of computational mechanics WCCM VI in conjunction with APCOM04. Beijing, China, pp 5–10Google Scholar
- Rojek J, Oñate E (2007) Multiscale analysis using a coupled discrete/finite element model. Interact Multiscale Mech 1:1–31CrossRefGoogle Scholar
- Rostami J (1997) Development of a force estimation model for rock fragmentation with disc cutters through theoretical modelling and physical measurement of crushed zone pressure. Ph.D. thesis, Colorado School of MinesGoogle Scholar
- Rostami J (2008) Hard rock TBM cutterhead modeling for design and performance prediction. Geomech und Tunnelbau 1:18–28CrossRefGoogle Scholar
- Rostami J (2013) Study of pressure distribution within the crushed zone in the contact area between rock and disc cutters. Int J Rock Mech Min Sci 57:172–186Google Scholar
- Rostami J, Ozdemir L (1993) A new model for performance prediction of hard rock TBMs. In: Rapid excavation and tunneling conference RETC, Boston, USAGoogle Scholar
- Rostami J, Ozdemir L, Nilsen B (1996) Comparison between CMS and NTH hard rock TBM performance prediction models. In: Annual Technical Meeting of the Institute of Shaft Drilling and Technology (ISDT), pp 1–11Google Scholar
- Roxborough F, Phillips H (1975) Rock excavation by disc cutter. Int J Rock Mech Min Sci Geomech Abstr 12:361–366CrossRefGoogle Scholar
- Sanio H (1985) Prediction of the performance of disc cutters in anisotropic rock. Int J Rock Mech Min Sci Geomech 22(3):153–161CrossRefGoogle Scholar
- Sato K, F G, Itakura K (1991) Prediction of disc cutter performance using a circular rock cutting ring. In: Proceedings 1st international mine mechanization and automation symposium, Golden Colorado, USAGoogle Scholar
- Shenghua Y (2004) Simulation of rock cutting by the finite element method. In: International ANSYS conference proceedings, pp 61–71Google Scholar
- Stavropoulou M (2006) Modeling of small-diameter rotary drilling tests on marbles. Int J Rock Mech Mining Sci 43:1034–1051CrossRefGoogle Scholar
- Su O, Akcin NA (2011) Numerical simulation of rock cutting using the discrete element method. Int J Rock Mech Mining Sci 48:434–442CrossRefGoogle Scholar
- Sun B, Guo W, Zhu D, Song L (2015) Multi-objective optimization design of tunnel boring machines edge cutter layout based on well-balanced wear. In: The 14th IFToMM World Congress, Taipei, Taiwan, Oct 25–30. doi: 10.6567/IFToMM.14TH.WC.FA.024
- Tarkoy P (1983) Selecting used tunnel boring machines: the pros and cons. Tunn Tunn pp 20–25Google Scholar
- van Wyk G, Els D, Akdogan G, Bradshaw SM, Sacks N (2014) Discrete element simulation of tribological interactions in rock cutting. Int J Rock Mech Min Sci 65:8–19Google Scholar
- Wang Y, Tonon F (2010) Calibration of a discrete element model for intact rock up to its peak strength. Int J Numer Anal Methods Geomech 34:447–469CrossRefGoogle Scholar
- Xiao S, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193:1645–1669CrossRefGoogle Scholar
- Yagiz S, Rostami J, Kim T, Ozdemir L, Merguerian C (2010) Factors influencing performance of hard rock tunnel boring machines. In: Rock Engineering in difficult ground conditions-soft rocks and Karst. Taylor & Francis GroupGoogle Scholar
- Yang B, Jiao Y, Lei S (2006) A study on the effects of microparameters on macroproperties for specimens created by bonded particles. Eng Comput 23(6):607–631CrossRefGoogle Scholar
- Yu B, Khair A (2007) Numerical modeling of rock ridge breakage in rotary cutting. In: ARMA General Meeting, Vancouver, British Columbia, CanadaGoogle Scholar
- Zhao K, Janutolo M, Barla G, Chen G (2014) 3D simulation of TBM excavation in brittle rock associated with fault zones: The Brenner Exploratory Tunnel case. Eng Geol 181:93–111CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.