Mathematical Problems in Engineering

Volume 2019, Article ID 5432470, 13 pages

https://doi.org/10.1155/2019/5432470

## The Effects of Confining Stress on Rock Fragmentation by TBM Disc Cutters

^{1}College of Civil Engineering and Architecture, Henan University of Technology, Zhengzhou 450000, China^{2}Department of Civil Engineering, Chongqing University, Chongqing 400045, China

Correspondence should be addressed to S. F. Zhai; moc.621@uqcgnafuhsiahz

Received 13 September 2018; Accepted 11 October 2018; Published 9 January 2019

Academic Editor: Roberto G. Citarella

Copyright © 2019 S. F. Zhai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, General Particle Dynamics (GPD3D) is developed to simulate rock fragmentation by TBM disc cutters under different confining stress. The processes of rock fragmentation without confining pressure by one disc cutter and two disc cutters are investigated using GPD3D. The crushed zone, initiation and propagation of cracks, and the chipping of rocks obtained from the proposed method are in good agreement with those obtained from the previous experimental and numerical results. The effects of different confining pressure on rock fragmentation are investigated using GPD3D. It is found that the crack initiation forces significantly increase as the confining stress increases, while the maximum angle of cracks decreases as the confining stress increases. The numerical results obtained from the proposed method agree well with those in previous indentation tests. Moreover, the effects of equivalent confining stress on rock fragmentation are studied using GPD3D, and it is found that rock fragmentation becomes easier when the equivalent confining stress is equal to 15MPa.

#### 1. Introduction

In recent years, tunnel boring machine (TBM) has been extensively used in the underground engineering because of high advance rates and safety performance. During the tunneling process, the TBM machine interacts with rock masses, so the rock mass structure, properties, and geological environment (such as the jointing growth condition, confining stress, and groundwater) have significant impact on the performance of TBM machine. Up to now, many researches have investigated the effects of confining stress on rock fragmentation by TBM disc cutters, and there are two main approaches to study the issue, which are, respectively, laboratory indentation test and numerical simulation.

Experimentally, Gehring [1] found that the cutter consumption under high confining stress is greater than that under lower overburden in the field of TBM tunneling. Huang et al. [2] studied the influence of the lateral confining stress on the development of damaged rock and the initiation of tensile fractures and found that the position of the maximum tensile stress is dramatically deviated from the indentation axis with a small increment in the confining stress. Innaurato et al. [3] found that a slight effect of the lateral confinement on the load may cause the rock breakage, if the cutter spacing is efficient. Yin et al. [4] found from the indentation test that the crack initiation stress and crushed zone size increase, and cracks tend to propagate towards the free surface when the confining stress increases.

In the numerical simulation, Gong et al. [5] introduced discrete element method (DEM) to investigate the rock fragmentation by disc cutters. The results obtained from these numerical simulations are in good agreement with the experimental ones. Based on a micromechanics-based numerical code, Ma et al. [6] studied the rock fragmentation processes under different confinement, and the confining conditions for TBM tunneling are divided into three general classes through an index of the confining ratio defined as the ratio of confining stress to uniaxial compressive strength. On the basis of theoretical investigation and numerical simulation of PFC, Liu et al. [7] found that cutting efficiency increases as the confining stress increases in a certain degree, while the efficiency is restrained by the increasing confining stress as the confining stress exceeds a critical value.

From the above studies, in the past years, several numerical approaches were used to solve the rock fragmentation problems, such as finite element method (FEM) and discrete element method (DEM). However, FEM is limited because the finite element meshes coincide with cracks. This limitation makes the task of generating the mesh very difficult. When cracks propagate during rock fragmentation, remeshing is inevitable, which makes it nearly impossible to model rock fragmentation problems with FEM [8].

DEM serves an important approach to investigate the rock chipping mechanism induced by the TBM cutters. Unfortunately, as mentioned in works by Donze et al. [9], fracture is closely related to the size of elements and that is the so-called size effect, cross effect exists because of the difference between the size and shape of elements with real grains, and in order to establish the relationship between the local and macroscopic constitutive laws, data obtained from classical geomechanical tests which may be impractical are used.

Besides, most numerical studies were conducted on two-dimensional models, which may not be identical to the true physics of rock fragmentation by TBM cutters. In this study, a 3D numerical method known as General Particle Dynamics (GPD3D) [10, 11] is introduced to investigate the rock fragmentation by disc cutter under different confining stress. This mesh-free numerical method overcomes the shortcomings of Smooth Particle Hydrodynamics (SPH) and can determinate the paths of crack growth and fragmentation under loads. In GPD3D, it is assumed that one particle is killed when its stresses satisfy the critical value, and the killed particle has no effect on its neighbors during the deformation process. This is reasonable as the kernel usually distributes larger weightage just around the particle of interest, and the quantity of these immediate particles is enough. During the deformation process, one particle’s life is determined based on the accumulated stress and a fully damaged particle is considered as the initiation of crack. In this paper, an elastic brittle damage model based on 3D Hoek-Brown criterion [12] is adopted to simulate the process of rock fragmentation by TBM cutters.

The paper is organized as follows. In Section 2, the strength criterion and damage model for GPD3D is outlined. In Section 3, the processes of rock fragmentation without confining stress by one disc cutter and two disc cutters are investigated using GPD3D. In Section 4, the effectiveness of GPD3D on rock fragmentation by one TBM disc cutter under different confining stress is validated. In Section 5, the effects of equivalent confining stress on rock fragmentation by one TBM disc cutter are investigated using GPD3D. Conclusions are drawn in Section 6.

#### 2. The Strength Criterion and Damage Model for General Particle Dynamics (GPD3D)

In this paper, only the strength criterion and the damage model implemented in the GPD are described in detail because the general formulations of the methods such as constitutive model, governing equations, and correction for consistency have been stated in the previous works [10, 11].

In most cases, rock-like materials fail in a brittle manner. All the particles have the same parameters as those of the parent material (i.e., uniaxial compressive strength, Young’s modulus (), and Poisson ratio ()); therefore, the damage initiation and growth in the particles are determined by the damage model of materials. The algorithm is described as follows.

In this paper, the 3D Hoek-Brown criterion is applied to determine damage initiation. Damage is believed to be initiated from one particle when stresses in the particle satisfy the 3D Hoek-Brown criterion. In order to enhance applicability of the 3D Hoek-Brown criterion, there are several correct versions and the latest version is proposed for rocks as follows [13]:where is a uniaxial compressive strength of an intact rock material, is the second invariants of the stress deviator, is Lode’s angle of stress varying from 0 to , and and are the material parameters in the Hoek-Brown criterion, which are defined by Hoek et al. and take the following form [14, 15]:where * i*s a disturbance coefficient which varies from 0.0 for the undisturbed in situ rock masses to 1.0 for very disturbed rock masses. GSI is short for Geological Strength Index which reflects the fragmentation degree of rock mass and it is equal to 100 for the intact rock, is the value of m for intact rock and can be obtained from experiments, and the parameter varies from 4 for very fine weak rock-like claystone to 33 for coarse igneous light-colored rock-like granite.

Now we introduce a parameter , coined as the “interaction factor” which defines the level of interaction between the -th and the -th particles. The interaction factor is determined based on the damage in particles. Initially, for undamaged particles, , which implies “full interaction”. With progress of damage, finally becomes zero for fully damaged particle. In order to model the damage growth, a linear elastic brittle law, as shown in Figure 1, is used.where is the damage factor and is the interaction factor.