## 2 Waterjet Erosion Model of an Abrasive Particle
### Abrasive Particle Energy and Erosion Model  
The mass and velocity of the abrasive constitute the input energy of the abrasive particle (\(E_{p}\)) for material removal, which is expressed as follows:  
\[E_{p}=\frac{\pi d_{p}^{3}\rho_{p}v_{p}^{2}}{12}. \tag{22}\]  
The removal work is a function of a resistance force as a hardness [9]. Considering the energy transfer effect during impact due to the relative hardness of the particle and the target that absorbs the input energy of the abrasive particle, the effective erosion kinetic energy (\(E_{e}\)) can be expressed as [39; 40]  
\[E_{e}=E_{p}\bigg{(}\frac{1}{1+\sqrt{H_{t}/H_{p}}}\bigg{)}, \tag{23}\]  
where \(H_{t}\) and \(H_{p}\) represent the hardness of the target and particle, respectively. As the input energy of the abrasive particle is converted into target removal work based on energy conservation, the volume of the target removed (\(V_{r}\)) is associated with the energy of the particles that collide with the target (\(E_{e}\)) in the following manner:  
\[\frac{dV_{r}}{dt}\propto\frac{dE_{e}}{dt}. \tag{24}\]  
Therefore, the relationship between energy and work is given as follows:  
\[\frac{\pi}{2}\cdot b^{2}\cdot H_{t}\cdot d_{i}=E_{p}\cdot\Bigg{(}\frac{1}{1+ \sqrt{H_{t}/H_{p}}}\Bigg{)}=\frac{\pi d_{p}^{3}\rho_{p}v_{p}^{2}}{12\Big{(}1+ \sqrt{H_{t}/H_{p}}\Big{)}}. \tag{25}\]  
The deformation area can be expressed in terms of the erosion depth (\(d_{i}\)) and particle diameter (\(d_{p}\)) (Figure 1):  
\[b^{2}=\Bigg{(}\frac{d_{p}}{2}\Bigg{)}^{2}-\Bigg{(}\frac{d_{p}}{2}-d_{i}\Bigg{)} ^{2}. \tag{26}\]  
Substituting Equation (26) into Equation (25), a cubic equation for the erosion depth (\(d_{i}\)) can be obtained based on the abrasive particle energy (\(E_{p}\)) and hardness ratio (\(H_{t}/H_{p}\)), as follows:  
\[d_{i}^{3}-d_{i}^{2}d_{p}+\frac{2\cdot E_{p}}{\pi\cdot H_{t}}\Bigg{(}\frac{1}{ 1+\sqrt{H_{t}/H_{p}}}\Bigg{)}=d_{i}^{3}-d_{i}^{2}d_{p}+E_{c}=0,\ E_{c}=\frac{2 \cdot E_{p}}{\pi\cdot H_{t}}\Bigg{(}\frac{1}{1+\sqrt{H_{t}/H_{p}}}\Bigg{)}. \tag{27}\]  
In this case, the velocity of the abrasive is the exit velocity (Equation (18)), which is determined based on the properties of the abrasive particle. The cubic formula for erosion depth (\(d_{i}\), Equation (27)) can be expressed as  
\[d_{i}=\frac{d_{p}}{3}+\sqrt[3]{\frac{1}{54}\Big{(}\Big{(}2d_{p}^{3}-27E_{c} \Big{)}+\sqrt{27E_{c}\Big{(}27E_{c}-4d_{p}^{3}\Big{)}}\Big{)}}+\sqrt[3]{\frac {1}{54}\Big{(}\Big{(}2d_{p}^{3}-27E_{c}\Big{)}-\sqrt{27E_{c}\Big{(}27E_{c}-4d _{p}^{3}\Big{)}}\Big{)}}. \tag{28}\]  
By calculating the sphere cap volume, the erosion volume \(V_{r}\) can be obtained as follows:  
\[V_{r}=\frac{\pi d_{p}d_{i}^{2}}{2}-\frac{\pi d_{i}^{3}}{3}. \tag{29}\]The characteristics of the erosion volume \(V_{r}\) can be expressed as a function of the effective erosion kinetic energy and hardness of the target, as shown below:  
\[V_{r}\propto\cdot\frac{d_{p}^{3}\cdot v_{p}^{3}\cdot\rho_{p}^{3/2}}{\left(H_{t} \Big{(}1+\sqrt{H_{t}/H_{p}}\Big{)}\right)^{3/2}}. \tag{30}\]  
As it provides a simple expression for energy conversion into work during erosion, this model is called the simple energy-work erosion model. This model considers the effective erosion kinetic energy and relative hardness and proves that increasing the particle hardness increases the removal rate. This finding is similar to that of previous experimental and theoretical studies [41; 42].