## 2 Numerical Methods
### Erosion Scheme  
Element deletion or erosion schemes are mainly used to get rid of highly distorted elements in explicit analyses, to reach numerical convergence and subsequently proceed with the simulation. In addition, deletion of elements is also used to visualize fracture propagation. Highly distorted or damaged elements, including the nodes becoming orphans, can either be removed or left in the mesh as ghost elements that cannot bear any loadings. The former approach is more involved, as it requires restructuring and an assembly of the related data structures and computational entities, such as mass and stiffness matrices. In the present case, the latter option is chosen, and the elements marked to be deleted behave like a vacuum, i.e., the R.H.S. of the equation of motion is zero. However, the mass matrix is kept untouched, to secure continuation of the analysis (zeros on the diagonal would result in a halt). An erosion criterion indicating the elements to be marked as deleted is required. Many options exist in the literature, such as strain, stress, and damage-based criteria [35]. A damage-based criterion is selected for the present purposes in the following form:  
\[\omega_{t}\ \geq\omega_{\text{tresh}}\ \&\ \omega_{\text{c}}\ \geq\omega_{\text{tresh}} \tag{28}\]  
where \(\omega_{\text{tresh}}\) is a threshold value above which an element is regarded as destroyed. This criterion is a conservative one, requiring that both damage variables reach the threshold. In the present case of leaving the deleted elements in the mesh, the erosion scheme thus involves removing the contact nodes for which all neighboring elements are marked as deleted. Such a node, being left as an orphan in the space, cannot bear any loading and should thus be removed from the list of contact nodes. This can be achieved with the pseudocode in Algorithm 1. ``` FOR \(n=1\) to \(N_{\text{cents}}\) DO:
1. Find neighboring elements of \(cn_{n}\rightarrow\mathbf{NCeles}_{n}=[e_{1},\ldots,e_{k}]\)
2. FOR \(m=1\) to \(k\) DO:

IF \(\omega_{\text{t}}\geq\omega_{\text{tresh}}\) & \(\omega_{\text{c}}\geq\omega_{\text{tresh}}\) for \(e_{m}\in\mathbf{NCeles}_{n}\)

THEN \(\mathbf{Eroded}_{n}=[\mathbf{Eroded}_{n}\ \mathbf{\omega}_{m}]\)

END DO

3. IF \(\mathbf{Eroded}_{n}\equiv\mathbf{NCeles}_{n}\) THEN

**RemList** = [**RemList**\(cn_{n}\)]

END DO

Remove the contact nodes in **RemList** from \(\mathbf{G}\) and \(\mathbf{b}\)

In the algorithm, \(cn_{n}\)_is_ a contact node \(n\) and \(N_{\text{cents}}\) is the number of contact nodes. Moreover, a list of all eroded elements, in addition to the one in the algorithm, is generated, so that there is no need to calculate strains and stresses for an element if it is found in the list. In addition to updates in the algorithm, \(\mathbf{G}\) and \(\mathbf{b}\) are also updated based on the value of the gap function, i.e., only the rows for which the gap function \(\mathbf{G}\widetilde{\mathbf{u}}_{l+\Delta l}-\mathbf{b}\) is positive are used in Equations (22) and (27) at each time step. This fulfils the unilateral condition that contacts can carry only compressive loading. ## 3 Numerical Simulations

In this section, numerical simulations demonstrating the method are carried out. The validation cases are the single PDC cutter experiments on granite from [4]. First, the material properties and model parameters, along with the finite element mesh, are specified. Then, the validation simulations are carried out. Finally, the effect of confining pressures is tested. As the variation of cutting speed has a negligible effect on the cutting force at low cutting speeds [11], it is not tested here. ### Material Properties and Model Parameters

The material properties and model parameters are given in Table 1.

The cohesion and friction angle in Table 1 give a uniaxial compressive strength of 165 MPa.