## 2 Numerical Approaches
### Modeling of Rock  
Various strength models have been used to simulate the rock cutting process in previous studies. The function of the yield stress of a material was determined by the strength model in AUTODYN. The Drucker-Prager (DP) strength model (Figure 6) is suitable for elasto-plastic materials such as rock [32]. Moreover, the applicability of the DP strength model in rock cutting simulation has been shown in previous studies [6; 9; 11; 33; 34; 35]. Because it is a linear function, the DP model was converted to the non-linear Stassi-hardening function [36] to describe the brittle behavior that is significantly weaker in tension than in compression during fragmentation of rock cutting [9]. In this study, the yield strength under tensile stress was assumed to be 90% of the Brazilian tensile strength of rock, and yield strength under compressive stress was assumed to be 90% of the uniaxial compressive strengths of rock. Tension cut-off level is determined on the basis of the tensile strength of rock. In Figure 6, on the left hand side of the intersection between the Rankine and Stassi-hardening functions, tensile yielding is dominant; on the right hand side, shear yielding is dominant [9]. Figure 5: Computational cycle for SPH code [16]. Figure 6: Description of the Drucker–Prager (DP) model with Stassi-hardening function (from [9]). The AUTODYN provides several failure models on the basis of the strength and equation of state (EOS) of the material. The damage-based model is used to calculate the damage to a material on the basis of the plastic strain in the plastic strain region (i.e., where the stress level is above the yield stress of the material) according to the defined function. The damage factor is computed on the basis of the plastic strain, and the model reduces the elastic modulus and yield strength on the basis of the damage. The level of damage is also used to define the failure of the material. Cumulative damage (CD) was proposed by [PERSON] [37] to describe the macroscopic inelastic behavior of materials, such as ceramics and concrete, in which the strength of the material can be degraded significantly by crushing. Therefore, the CD model is also a reasonable model for simulating the rock cutting process because the area between adjacent cutters, where chipping occurs, is damaged by each preceding cutting action. The CD model computes a damage factor (\(D\)) to model the progressive crushing and subsequent weakening of materials. Equation (5) and Figure 7 show that the damage factor is zero for cases of plastic deformation in which the effective plastic strain (\(EPS\)) is less than a certain value, i.e., \(EPS1\). When the strain reaches the value of \(EPS1\), the damage factor increases linearly with the strain up to a maximum value of \(D_{MAX}\) (\(<\)1), which is the value of the effective plastic strain, i.e., \(EPS2\)[16]. \[D=D_{MAX}\bigg{(}\frac{EPS-EPS1}{EPS2-EPS1}\bigg{)} \tag{5}\]  
The current value of the damage factor (\(D\)) modifies the bulk modulus (K), shear modulus (G), and yield strength (\(Y\)) of the material. \(Y\) is reduced to \(Y_{dam}\) by the damage factor. If the hydrostatic pressure is positive (compressive), the reduced yield strength is defined as in Equation (6):  
\[Y_{dam}=Y(1-D) \tag{6}\]  
If the hydrostatic pressure is negative (tension), the reduced yield strength is defined as in Equation (7), and these reduced functions are presented in Figure 8a. In the same manner as the yield strength, the bulk and shear modulus are also reduced to \(K_{dam}\) and \(G_{dam}\), respectively, on the basis of the level of damage factor. They are unaffected by compression, but during tension they are progressively reduced to zero when the damage is maximized. Thus, during tension, they are both reduced by the damage factor (Figure 8b). The cutting process of rock involves the combined behavior of compressive and tensile stresses.