# Smoothed Particle Hydrodynamics Simulation of Orthogonal Cutting with Enhanced Thermal Modeling
## 2 Governing Equations  
Before presenting the numerical framework for modeling of a cutting process, we provide an overview of the underlying physics in this section. The drawings in Figure 1 show a graphical illustration of the key thermal aspects of metal cutting. In these figures, the main heat sources associated with each deformation zone are shown, as well as a representation of the thermal boundary conditions. Figure 1: Illustration of heat sources and thermal boundary conditions in orthogonal cutting. The continuum mechanical equations in a Lagrangian formalism are expressed by the field equations:  
\[\frac{d\rho}{dt} =-\rho\
abla\cdot\underline{v} \tag{1}\] \[\frac{dv}{dt} =\frac{1}{\rho}\
abla\cdot\underline{v}+\frac{1}{\rho}\underline{b}\] (2) \[\frac{d\underline{r}}{dt} =\underline{v} \tag{3}\]  
where \(\rho\) is the density, \(\underline{r}\) the position vector, \(\underline{v}\) the velocity, \(\underline{\sigma}\) the Cauchy stress tensor, and \(\underline{b}\) the volumetric body forces. It is worth mentioning that all numerical simulations in this work are based on single-resolution SPH formulations; thus, a uniform constant mass is allocated to each computational node and the momentum equation (i.e., Equation (2)) does not require the utilization of mass scaling. An elaboration of this subject, as well as further details on the numerical stabilization procedure, can be found in [19; 20]. This mechanical system is coupled with thermal effects by solving the heat conduction equation:  
\[\frac{dT}{dt}=\frac{k}{\rho\,c_{p}}\
abla^{2}T+\frac{1}{\rho\,c_{p}}(Q^{\text{ plastic}}+Q^{\text{fric}}) \tag{4}\]  
in which \(k\) is the heat conductivity assuming an isotropic heat conduction, \(c_{p}\) the specific isobaric heat capacity, including the source terms \(Q^{\text{plast}}\) and \(Q^{\text{fric}}\) due to the conversion of the plastic and frictional work. According to the works in [17; 23], these heat sources are computed from  
\[Q^{\text{plast}} =\chi\left(\sigma_{y}\ \hat{e}^{pt}\right) \tag{5}\] \[Q^{\text{fric}} =\eta\left(\frac{\rho\,|f^{\text{fric}}|\cdot|\underline{v}|^{ \text{rel}}|}{m}\right) \tag{6}\]  
if \(\chi\) and \(\eta\) are dimensionless parameters specifying the fraction of plastic and frictional work converted into heat (see in [23; 24]). According to the illustration in Figure 1, the boundary conditions of Equation (4) are given by  
\[T_{c} =T_{\infty} \longrightarrow\text{on fixed boundaries} \tag{7}\] \[q_{l} =-\Big{[}\underbrace{h_{c}\left(T_{s}-T_{\infty}\right)}_{\text {convection}}+\underbrace{e\sigma\left(T_{s}^{4}-T_{\infty}^{4}\right)}_{\text {radiation}}\Big{]} \longrightarrow\text{on open boundaries} \tag{8}\]  
with \(h_{c}\) denoting the heat convection coefficient, \(T_{s}\) and \(T_{\infty}\) the surface and background temperatures, respectively; \(\epsilon\) the emissivity factor; and \(\sigma\) the Stefan-Boltzmann constant. For material modeling, the Johnson-Cook (JC) flow stress according to [25] is used to define the plastic response of the workpiece material. This model calculates the flow stress of the material, \(\sigma_{y}\), and incorporates three effects: strain hardening, strain rate hardening, and thermal softening. The JC flow stress is computed from  
\[\sigma_{y}^{\text{JC}}=\left[A+B\left(\bar{e}^{pt}\right)^{n}\right]\left[1+ C\ \ln\left(\frac{\bar{e}^{pt}}{\bar{e}_{0}^{pt}}\right)\right]\left[1-\left(\frac{T -T_{r}}{T_{m}-T_{r}}\right)^{m}\right] \tag{9}\]  
where \(\bar{e}^{pt}\) is the equivalent plastic strain and \(T_{r}\) and \(T_{m}\) are the reference temperature and melting point, respectively. In Section 5, the values of five JC parameters (\(A,B,C,m\), and \(n\)) will be provided for the titanium alloy under consideration. For friction modeling, Coulomb's law with a constant coefficient of \(\mu=0.35\) is considered as a typical choice in LS-DYNA [13] and other SPH cutting models like [17], unless otherwise mentioned.