## 4 Conclusions  
Research of the slide diamond burnishing process of butt joints made using the FSW method at a variable feed rate in the range of \(f=500\) mm/min to \(f=1500\) mm/min and burnishing force in the range of \(P=40\) N to \(P=60\) N, carried out in accordance with the methodology of the PS/DC 3\({}^{2}\) plan, showed that:  
* During friction stir welding, small tensile stresses (2 MPa) and grooves with an average depth of 9.77 um (max 31.5 um) are generated in the surface layer of the weld, which may contribute to a reduction in the fatigue strength of the joints.
* Burnishing enables the introduction of compressive stresses both in the transverse and longitudinal directions. Compressive stresses during exploitation add up to stresses from external forces, changing the symmetrical cycle into an asymmetric one with reduced, harmful tensile stresses, increasing the fatigue strength of the joints. The highest compressive stress value in the transverse direction was obtained with a burnishing  
Figure 13: Plot of the objective function of a multi-criteria optimization (a) u\({}_{i}=0.142\), (b) u\({}_{1,2}=0.25\), u\({}_{3-7}=0.1\).  
force of \(P=50\) N and a feed rate of \(f=500\) mm / min (\(\sigma_{x}~{}=~{}-130\) MPa), while in the longitudinal direction with a force of \(P=60\) N and a feed rate of \(f~{}=~{}500\) mm / min (\(\sigma_{x}=-242\) MPa).
* Plastic deformation of the surface layer as a result of the burnishing process increases the microhardness of the surface layer of the weld from 84.19-174.53% and reduces the surface roughness Sa from 73.68-94.84%. This ensures a microgeometry of the surface characterized by large radii of the recesses and small angles of inclination of the sides of the irregularities, thus, eliminating geometric micro-cuts.
* Multi-criteria optimization of the burnishing process allows us to obtain a compromise solution ensuring compressive stresses in the surface layer (\(\sigma_{x}=-123\) MPa and \(\sigma_{y}=-202\) MPa) and microhardness \(H_{V}=362.56\) mm / mm\({}^{2}\) with the roughness of the weld surface Sa = 0.28 \(\upmu\)m, Sku = 3.93 and Spc = 35.88 1/mm, while ensuring high efficiency of the burnishing process.
* The highest microhardness was obtained at the force \(P=50\) N. By further increasing the burnishing force in the area under the diamond burnishing tip, the contact area increases, which in turn results in a decrease in the value of the maximum stress (especially in the transverse direction \(\sigma_{x}\)). In the presence of lower stresses, fewer structural defects are created, which results in a decrease in the hardness of the surface layer of the joint.  
Conceptualization, M.B. and R.K.; methodology, M.B., R.K., K.A., A.D., W.B. and K.O.; software, M.B. and R.K.; validation, K.A., M.B. and R.K.; formal analysis, M.B., R.K., and A.D.; investigation, M.B., R.K., A.D, W.B. and K.O.; resources, M.B. and R.K.; data curation, R.K., A.D. and M.B.; writing--original draft preparation, M.B. and R.K. writing--review and editing, and M.B. and K.A.; visualization, R.K.; supervision, R.K. and M.B.; project administration, R.K. and M.B.; funding acquisition, K.A. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Not applicable. Not applicable. Not applicable. Not applicable. Not applicable.  
The authors declare no conflict of interest.