## 3 Numerical Framework  
### Cutting Simulations  
In this study, Johnson-Cook (JC) material damage model is applied for numerical cutting simulation of AA2024. The constitutive equation describing the von Mises stress due to chipped off material is described as [3]:  
\[\overline{\sigma}=\underbrace{\left(\sigma_{y}+B\overline{\varepsilon}^{n} \right)}_{\text{Plastic term}}\underbrace{\left[1+C\ln\left(\frac{\frac{ \bullet}{\overline{\varepsilon}}}{\overline{\varepsilon}_{0}}\right)\right]}_{ \text{Viscosity term}}\underbrace{\left[1-\left(\frac{T-T_{room}}{T_{melt}-T_{ room}}\right)^{m}\right]}_{\text{Softening term}}, \tag{1}\]  
where \(\overline{\sigma}\) represents the equivalent stress, \(A\) is the material yield strength (MPa); \(B\) is the hardening modulus; \(n\) shows the work hardening exponent; \(C\) is the material constant for strain hardening rate; \(m\) is the thermal softening coefficient; \(T_{room}\) is the reference ambient temperature.  
The damage evolution of an element can be expressed by using the classical damage law:  
\[D=\sum\frac{\Delta\overline{\varepsilon}}{\overline{\varepsilon}_{f}}, \tag{2}\]  
where \(\Delta\overline{\varepsilon}\) is the increment of equivalent plastic strain and \(\overline{\varepsilon_{f}}\) is the equivalent strain to fracture. Fracture occurs when \(D\) is equal to 1. The plastic strain at damage initiation \(\dot{\overline{\varepsilon}_{0}}\) is expressed as [3]:\[\overline{\epsilon}_{0i}=\left[D_{1}+D_{2}\exp\left(D_{3}\frac{P}{\overline{\sigma}} \right)\right]X\left[1+D_{4}\ln\left(\frac{\frac{\bullet}{\overline{\epsilon}} }{\overline{\epsilon}_{0i}}\right)\right]\left[1+D_{5}\left(\frac{T-T_{room} }{T_{melt}-T_{room}}\right)\right], \tag{3}\]  
where \(D_{1}\) represents the initial failure strain; \(D_{2}\) gives the exponential factor; \(D_{3}\) deals with triaxiality factor; \(D_{4}\) is the strain rate factor; \(D_{5}\) is the temperature factor; \(P\) is the average normal stress; and \(\overline{\sigma}\) is the von Mises equivalent stress.  
Generally, \(D_{1}\) to \(D_{5}\) values are determined through experiments [43]. The equivalent plastic strain with a scalar damage parameter \(\omega\) can be expressed by [44; 45]:  
\[\omega=\frac{\sum\Delta\overline{\epsilon}}{\overline{\epsilon}_{0i}}, \tag{4}\]  
[PERSON] et al. [46] proposed the fracture energy (\(G_{f}\)) per unit area of the crack [6]:  
\[G_{f}=\int_{\overline{\epsilon}_{0i}}^{\overline{\epsilon}_{f}}L\overline{ \sigma}_{y}d\overline{\epsilon}=\int_{0}^{\overline{\epsilon}_{f}}\overline{ \sigma}_{y}d\overline{u}, \tag{5}\]  
where \(L\) is the characteristic length of an element and \(\overline{U_{f}}\) is the equivalent plastic displacement at failure. \(\overline{U_{f}}\) is computed by the following equation [47]:  
\[\overline{u}_{f}=\frac{2G_{f}}{\sigma_{f}}, \tag{6}\]  
Under the applied force, damage evolution laws can be expressed as [42]:  
\[D=\frac{L\overline{\epsilon}}{\overline{u}_{f}}=\frac{\overline{u}}{\overline {u}_{f}}, \tag{7}\]  
\[D=1-\exp\left(-\int_{0}^{\overline{u}}\frac{\overline{\sigma}}{G_{f}}d \overline{u}\right), \tag{8}\]  
In dry turning operation, heat is generated due to the plastic deformation, which can be expressed as:  
\[\dot{q}_{p}=\eta_{p}\overline{\sigma}\dot{\overline{\epsilon}}, \tag{9}\]  
where \(q_{p}\) is the heat generation rate and \(\eta_{p}\) is the inelastic heat fraction.  
Heat generated due to the contact friction is expressed as:  
\[q_{f}=\eta_{f}I\tau_{f}\gamma, \tag{10}\]  
where \(\tau_{f}\) is the shear stress given by Coulomb friction law and \(\gamma\) is the slip strain rate.