## 1 Introduction  
### Modeling of Selected Phenomena in the Diamond Sliding Burnishing Process  
Diamond sliding burnishing is a quite well-known technological process. The main advantages include, among others, improved quality of the surface layer, improved fatigue strength or improved corrosion resistance. Numerous papers describe different advantages of using diamond burnishing as an alternative method in relation to the turning process. Below, several papers are shown as an example of the huge popularity of diamond sliding burnishing. The Authors generally research the influence of technological parameters or shape of the tool on the state of the surface layer after research. However, all the papers have the same common features. The Authors will mention these after a brief analysis of the literature. In paper [1], the Authors present the geometric changes taking place on the surface of an object made of 16 HG steel, hardened and burnished with a sliding diamond. They conducted real and simulation tests in the ABAQUS program on the basis of a two-dimensional section of the surface of a real object. In works [2; 3], it was found that the multiplicity of the load (the number of passes) during burnishing should not be too high, as it may cause excessive crushing with all its harmful consequences. Increasing the number of cycles above several is not advisable for economic reasons either, because as a result of the phenomenon of metal hardening, each cycle brings worse and worse effects [2; 3]. The model of material deformation using a non-deformable rigid ball, known as the Hertz model, is quite often and with good results, adopted for an analysis of deformation processes in the contact zone of two elements, e.g., in an analysis of strengthened burnishing [4; 5]. However, this model is not useful in the case of burnishing and hardening, or even in the case of burnishing itself. The formulas obtained on its basis in comparison with the values of forces used in experimental studies [6; 7] indicate significant differences in the results. The model of material deformation by means of a ball rolling on the machined surface is a better approximation of the burnishing processes, and due to its high similarity to sliding burnishing, it can be used to calculate the tool clamping force. However, attempts to calculate the tool clamping force during smooth burnishing provide inflated results; the values obtained differ significantly from the experimental data [8; 9; 10]. The model of material deformation by means of a ball sliding on the surface of the workpiece was considered by many authors, among which the studies described in [11] belong to the classics. The effects of burnishing, as practice shows, depend not only on the elastic-plastic properties of the workpiece and the geometry of the contact between the tool and the workpiece, but also on the initial surface roughness, and this factor is not taken into account by classical models, treating output surfaces as perfectly smooth. This is confirmed by the experimental formulas presented in the literature [12; 13] that are used to select the burnishing parameters, but do not take into account the effect of the initial roughness of the surface machined. Therefore, it is necessary to take this factor into account in theoretical considerations. For the reasons mentioned above, in an analysis of burnishing, it is necessary to adopt a different model than the one found in the literature. Such a model should take into account the stereometric state of the surface and, in particular, the height of the initial surface roughness, as the roughness of the surface after burnishing depends to a large extent on it [14; 15; 16; 17; 18; 19; 20]. Work hardening of the surface layer increases the resistance to the development of fatigue processes, especially in materials without very high hardness, such as aluminum alloys [21]. The hardening depth should be up to about 0.25 of the cross-sectional dimension of the part; at greater depths, no improvement in fatigue strength is observed [22].