# Numerical Simulation of Rock-Breaking Mechanism by Spherical Tooth Impact in Granite Formation
## 2 Model and Methodology
### Simulation Settings  
#### 2.2.1 Simulation Model  
Recently, the FDEM (Finite-Discrete Element Method) model has been widely used for simulating material fractures. This method combines the advantages of continuum and discontinuum approaches. It employs zero-thickness cohesive elements to simulate the rock fracture process. The constitutive response of the zero-thickness cohesive elements can be directly defined using a traction-separation law to evaluate fracture behavior. In the model, a bilinear traction-separation law is adopted to describe the response of the zero-thickness cohesive elements [25].  
Before damage initiation, the normal traction \(t_{n}\) increases with the increase in normal displacement \(\delta_{n}\). When the normal traction \(t_{n}\) reaches its peak value \(t_{n0}\), a linear damage evolution law is adopted to describe the degradation of the normal stiffness \(k_{n}\) of the cohesive elements until the element fails, as expressed below [31]:  
\[t_{n}=\begin{cases}k_{n}\delta_{n},&\delta_{n}\leq\delta_{n}^{0}\\ (1-D)k_{n}\delta_{n},&\delta_{n}^{0}\leq\delta_{n}\leq\delta_{n}^{f}\\ 0,&\delta_{n}^{f}\leq\delta_{n}\end{cases} \tag{12}\]  
Similarly, there is a linear relationship between the tangential traction force \(t_{s}\) and the relative shear displacement \(s\). After reaching the tangential peak \(t_{s0}\), the linear damage evolution law is activated to describe the degradation of the tangential stiffness \(k_{s}\), which is expressed as follows [32]:  
\[t_{s}=\begin{cases}k_{s}\delta_{s},&\delta_{s}\leq\delta_{s}^{0}\\ (1-D_{s})k_{s}\delta_{s},&\delta_{s}^{0}\leq\delta_{s}\leq\delta_{s}^{f}\\ 0,&\delta_{s}^{f}\leq\delta_{s}\end{cases} \tag{13}\]  
where \(\delta_{s}^{0}\) is the relative shear displacement at the beginning of the damage and \(D_{s}\) is the damage variable.  
The damage initiation is relative to a maximum nominal stress criterion. The damage is assumed to initiate if the maximum nominal stress ratio is to a value of one:  
\[\text{max}\bigg{\{}\frac{\langle t_{n}\rangle}{t_{n}^{0}},\frac{t_{s}}{t_{s}^{ 0}}\bigg{\}}=1 \tag{14}\]  
where \(\langle\rangle\) is the Macaulay bracket, which is adapted to distinguish whether the cohesive element is under compression or tension:  
\[\langle t_{n}\rangle=\begin{cases}t_{n},&t_{n}\geq 0\text{ (tension)}\\ 0,&t_{n}<0\text{ (compression)}\end{cases} \tag{15}\]  
The damage is described by a scalar damage variable \(D\), which represents the unrecoverable overall damage of the cohesive element from 0 to 1. In this model, the damage variable \(D\) adopts a linear softening law in independent mode [31], which can be expressed as:  
\[D_{n}=\frac{\delta_{n}^{f}(\delta_{n}^{\text{max}}-\delta_{n}^{0})}{\delta_{n} ^{\text{max}}\bigg{(}\delta_{n}^{f}-\delta_{n}^{0}\bigg{)}},\;\delta_{n}^{ \text{max}}\geq 0 \tag{16}\]  
\[D_{s}=\frac{\delta_{s}^{f}\big{(}\delta_{s}^{\text{max}}-\delta_{s}^{0}\big{)} }{\delta_{s}^{\text{max}}\bigg{(}\delta_{s}^{f}-\delta_{s}^{0}\bigg{)}} \tag{17}\]  
\[\delta_{n}^{f}=\frac{2\sigma_{I}^{C}}{\delta_{n}^{0}} \tag{18}\]  
\[\delta_{s}^{f}=\frac{2\sigma_{II}^{C}}{\delta_{s}^{0}} \tag{19}\]  
where \(\delta_{n}^{\text{max}}\) and \(\delta_{s}^{\text{max}}\) are the maxima of the relative displacements obtained in the loading history of the normal and shear direction, respectively. \(D_{n}\) and \(D_{s}\) are corresponding to the damage variables in the normal and shear direction. \(G_{I}^{C}\) and \(G_{II}^{C}\) respectively denote the critical fracture energy in the normal and shear direction.  
Based on the damage of the cohesive element in the normal and shear direction, an overall damage variable U that synthetically reflects the damage in all directions is used to estimate whether the integration points of the cohesive element are fully damaged or not:  
\[D=\sqrt{D_{n}^{2}+D_{s}^{2}} \tag{20}\]  
If the overall damage variable U reaches 1 at all of its integration points, the cohesive element is treated as completely failed and will be deleted from the model.