## 2 Materials and Methods
### Mathematical and Finite Element Models  
During the process of burnishing treatment, heat is produced due to plastic deformation of the material surface layer and friction between the tool and the workpiece surface. Since the strain and stress fields depend on the temperature field, and the change in stresses and plastic strains, in turn, affects the temperature field, the problem must be solved within the framework of coupled problems of thermomechanics [14; 15]. In this case, simultaneous analysis of the temperature field due to heat generation and conduction as well as stress and strain fields is carried out. Local instantaneous temperatures can be rather high, so all material parameters (elastic moduli, heat capacity,  
Figure 1: The tool with a variable angle of cylindrical indenter inclination to the treated surface and with burnishing force adjustment. thermal conductivity coefficient, linear thermal expansion coefficient) are supposed to depend on temperature. For a more detailed description of stress and strain distributions in the contact zone, we adopt a two-dimensional formulation of the problem in plane strain conditions because three-dimensional calculations with comparable accuracy are computationally expensive. The central issue determining the level of stresses and deformations in the material under treatment is the choice of the constitutive relations and parameters of the selected model. To describe the deformation of steel, we adopt an isotropic elastoplastic model of the material. To define plastic strain rates, the \(J_{2}\) plastic flow theory is adopted, which demands determining the flow stress. One of the most common models of metal material mechanical processing is the Johnson-Cook model [16]. This model allows us to take into account not only strain hardening but also the dependence on the strain rate and thermal softening at a temperature increase. In the Johnson-Cook model, the equation for the yield stress has the following form:  
\[\sigma_{y}=(A+B\varepsilon^{n})\Big{(}1+C\ln\hat{\varepsilon}^{*}\Big{)}(1-T^{*m }), \tag{1}\]  
where \(\varepsilon\) is the equivalent plastic deformation; \(\hat{\varepsilon}^{*}=\hat{\varepsilon}/\hat{\varepsilon}_{0}\) is the normalized plastic deformation rate; \(\hat{\varepsilon}_{0}\) is the specified rate of deformation at which the model parameters are determined (in the original article by [PERSON] and [PERSON] and many other works, it is assumed that \(\hat{\varepsilon}_{0}=1\) s\({}^{-1}\)); \(T^{*}=(T-T_{r})/(T_{m}-T_{r})\) is homological temperature; \(T_{r}\) is the specified temperature at which the model parameters are determined (usually a room temperature); \(T_{m}\) is the melting point. Thus, five material constants \(A\), \(B\), \(n\), \(C\), \(m\), and some more parameters \(T_{r}\) and \(T_{m}\) are introduced. The values of the Johnson-Cook model parameters for AISI 52100 steel are presented in Table 1. There are parameter values from other authors' articles that are accepted in this study. The parameters of the model differ significantly from different authors, as well as the stress-strain diagrams of AISI 52100 steel, which can be seen, for example, in Figure 6 in the article [17]. Our parameters were determined by approximating our experimental data at different temperatures, which will be shown further on in the description of the experimental results. The \(T_{m}\) parameter was considered not as the melting point, but as a model parameter that allows us to more accurately describe experimental data using Formula (1). The temperature value \(T\) to be used in Equation (1) is calculated from the heat balance equation (thermal conductivity). Two factors can be taken into account as heat sources. First, during deformation, a part \(\eta\) of the work of stresses on plastic deformations is converted into heat (it is usually assumed that the parameter \(\eta\) = 0.9).