## 2 Materials and Methods
### Mathematical and Finite Element Models
The heat flux per unit volume is computed using the equation:  
\[\dot{Q}_{pl}=\eta\varepsilon^{pl}, \tag{2}\]  
where \(\sigma\) is the stress, \(\dot{\varepsilon}^{pl}\) is the rate of plastic strain \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Authors, Paper** & **\(A\), MPa** & **\(B\), MPa** & \(n\) & \(C\) & \(m\) & \(T_{r}\), \({}^{\circ}\)C & \(T_{m}\), \({}^{\circ}\)C \\ \hline [PERSON] et al. [18] & 11.032 & 4783 & 0.0946 & 0 & 1 & 0 & 775 \\ [PERSON] et al. [19] and [PERSON] et al. [20] & 688.17 & 150.82 & 0.3362 & 0.04279 & 2.7786 & & 1370 \\ [PERSON] et al. [21] & 635.926 & 101.703 & 0.649 & & 2.259 & 635.926 & \\ [PERSON] et al. [22] and [PERSON] et al. [23] & 2482.24 & 1498.5 & 0.19 & 0.027 & 0.66 & & \\ This paper & 322 & 994 & 0.34 & 0.043 & 0.597 & 20 & 6510 \\ \hline \end{tabular}
\end{table} 
Table 1: Johnson–Cook model parameters. The second source of heat is the work of friction (due to frictional sliding). The rate of frictional energy dissipation is given by  
\[\dot{Q}_{fr}=\tau\dot{\gamma}, \tag{3}\]  
where \(\tau\) is the frictional stress and \(\dot{\gamma}\) is the slip rate. Some part of this energy releases as heat fluxes into the surfaces being in contact. Since the simulation was proposed to be performed in the formulation of a plane strain field, there arose a problem of choosing the values of the corresponding burnishing forces. It is necessary to establish the relationship between the forces that are applied to the instrument in a real three-dimensional formulation and the specified forces in two-dimensional calculations of the plane strain field. We will carry out calculations in the SI unit system. This means that the width of the sample and the indenter in the contact zone will be equal to 1 m.  
The basis for determining the correspondence between real forces and calculated ones is the condition that the compression stresses under the indenter in the two-dimensional case and in the real three-dimensional case are the same, \(\sigma_{\rm 2D}=\sigma_{\rm 3D}\). Then the relationship between the forces in the two-dimensional case \(P_{\rm 2D}\) and in the three-dimensional one \(P_{\rm 3D}\) will be as follows  
\[P_{\rm 2D}=P_{\rm 3D}\frac{S_{\rm 2D}}{S_{\rm 3D}}, \tag{4}\]  
where \(S_{\rm 2D}\) and \(S_{\rm 3D}\) are the areas of the contact zones in the two-dimensional and three-dimensional cases, respectively. To derive a further formula, it is important to determine which three-dimensional case we are considering. Firstly, our indenter is cylindrical. Secondly, we will assume that we exert pressure with this indenter on the plane surface of the sample. Here are expressions for the areas of the contact zones in the two-dimensional and three-dimensional cases:  
\[S_{\rm 2D}=l_{c}\cdot 1\,{\rm m}^{2},\;S_{\rm 3D}=\frac{k}{2}l_{c}\cdot L_{c}, \tag{5}\]  
where \(k>1\), because the shape of the contact zone is of teardrop shape (parabolic), and not triangular, as assumed in Formula (5);  
\(L_{c}\) is the width of the contact zone in meters in the three-dimensional case, and the corresponding width of the indenter contact (perpendicular to the plane of simulation) with the workpiece in the case of two-dimensional simulation;  
\(l_{c}\) is the length of the contact zone in meters, and the corresponding contact length of the indenter with the contacting bodies' surface plane in the case of two-dimensional simulation in the formulation of a plane strain condition. The diagram of the contact interaction of an inclined cylindrical indenter at an angle \(\alpha\) with the workpiece is shown in Figure 2.  
Figure 2: Contact interaction diagram of a cylindrical indenter inclined to a plane surface during nanostructuring burnishing.