## 2 Materials and Methods
### CEL Simulation of the SLD Process  
For the simulation of the drilling process, the CEL method was used to calculate the problem. This simulation method has already been successfully used for machining simulations [8; 11], mostly an orthogonal cutting process, and offers many advantages compared to an explicit Lagrangian simulation when dealing with problems with large deformations. The material 42 CrMo4+QT (AISI 4140) was used for the investigations. This is a quenched and tempered alloyed stainless steel with high strength and high toughness. The material is mostly used for highly stressed components with high toughness requirements. The temperature-dependent thermal properties describing the workpiece material behavior at different temperatures are listed in Table 2. These are of particular importance in deep drilling processes since the heat in a borehole propagates into the material in all directions. The inelastic heat fraction can be used in thermomechanical problems to determine which fraction of the inelastic energy is converted into heat. By default, a value of 0.9 is assumed. For the plastic material behavior, the so called Johnson-Cook (JC) model was used, which is widely applied for machining simulations. With the parameters \(A\), \(B\), \(C\), \(m\), \(n\) and the material parameter \(\dot{\epsilon}_{0}\) for the strain rate and the room temperature, as well as the melting temperature, the following equation is obtained:  
\[\sigma=[A+B\cdot\epsilon^{n}][1+C\cdot\frac{\dot{\epsilon}}{\dot{\epsilon}_{0} }][1-(\frac{T-T_{room}}{T_{melt}-T_{room}})^{m}], \tag{3}\]  
The values for the calculation of the plastic material behavior are presented in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Density \(\rho\)/ kg/m\({}^{3}\)** & **Young’s Modulus \(E\)/ MPa** & **Poisson’s’ratio \(\mu\)/ MPa** & **Theral Expansion \(\xi\)/ \(\mu\)/m/K** & **Conductivity \(k\)/ W/m/K** & **Specific Heat \(c_{p}\)/ J/kgK** \\ \hline
15,700 & 524,000 & 0.23 & 6.3 & 82.24 & 579.45 \\ \hline \end{tabular}
\end{table} 
Table 1: Material parameters of tungsten carbide adapted from [10]. \begin{table}
\begin{tabular}{c c c c c} \hline
**Young’s Modulus \(E\)/ MPa** & **Theral Expansion \(\xi\)/ \(\mu\)/m/K** & **Conductivity \(k\)/ W/m/K** & **Specific Heat \(c_{c}\)/ J/kgK** & **Temperature \(T\)/ K** \\ \hline
217,000 & 10.8 & - & 291.24 & 173 \\
213,000 & 11.7 & - & 354.03 & 273 \\
212,000 & 11.9 & 41.7 & 361.89 & 293 \\
207,000 & 12.5 & 43.4 & 389.36 & 373 \\
199,000 & 13.0 & 43.2 & 418.41 & 473 \\
192,000 & 13.6 & 41.4 & 445.88 & 573 \\
184,000 & 14.1 & 39.1 & 479.64 & 673 \\
175,000 & 14.5 & 36.7 & 531.45 & 773 \\
164,000 & 14.9 & 34.1 & 610.73 & 873 \\
69,000 & 14.9 & 34.1 & 610.73 & 1773 \\ \hline \end{tabular}
\end{table} 
Table 2: Material parameters of 42 CrMo4 adapted from [10]. To consider the damage of the material, the JC damage model was implemented, which can be described with the values in Table 4.  
For an optimization of the large computation times for three-dimensional problems, the density of the materials was increased by a manual mass scaling (MS) to such an extent that an acceptable stable time increment results. The maximum time increment \(dt\) depends on the characteristic element length \(L\) and the acoustic wave speed \(c_{d}\). Here the density \(\delta\) occurs in the denominator and provides a small propagation velocity, which results in larger time increments [14]  
\[dt=\frac{L}{c_{d}};\;c_{d}=\sqrt{\frac{E}{\rho}};\;\rho=\frac{m}{V}, \tag{4}\]  
Due to the relation of the density with the specific heat capacity, even the specific heat capacity has to be adjusted. With the CEL approach, increasing the density changes the right-hand term of equation:  
\[k\
abla^{2}T=\xi\lambda T_{0}tr(\dot{E})+\rho c_{p}\hat{T}, \tag{5}\]  
the thermal time constant. Here, \(\lambda\) and \(\mu\) represent the Lame coefficients, and \(T_{0}\) the reference temperature.