*2.1. Materials*

The materials used in this work were rolled sheets of austenitic stainless steels: AISI 304, provided by SARITAS Celik Sanayi ve Ticaret A.S. (Istanbul, Turkey) and AISI 316L, provided by Acerinox Europa SAU (Los Barrios, Spain), both with 4 mm thickness. The chemical compositions of these austenitic steels and their basic mechanical properties are given in Table 1.

**Table 1.** Chemical compositions and mechanical properties of austenitic stainless steels AISI 304 and AISI 316L.


*2.2. Obtaining RR of the IV-th Type by BB Process, Implemented on CNC Milling Machine* 2.2.1. Calculating the Toolpath Trajectory of the Ball Tool

Because RR were formed onto planar surfaces in the present work, the needed kinematics for the BB process could be borrowed from the classical vibratory BB process [34], but adapted for implementation with a contemporary CNC milling machine. This way the needed complex toolpath of the deforming tool (shown in Figure 1a,b), which is essential for the formation of RR of the IV-th type (see Figure 1c), could be achieved much more efficiently, and with a greater accuracy.

**Figure 1.** (**a**) Ball burnishing (BB) toolpath trajectory basic parameters; (**b**) BB toolpath distribution within the burnished surface boundaries; (**c**) resulting regular reliefs (RR) of the IV-th type with rectangular cells.

If we take into account the principle of the CNC equipment programming (i.e., ISO code) that the tool moves from its current position to the coordinates of the next position according to the NC code, the complex toolpath needed in the BB process can be divided into a relatively large number of short rectilinear segments that interpolate it with

a sufficient accuracy. The end point X and Y coordinates of each segment were calculated using the following system of functions [34] (see Figure 1a):

$$\begin{cases} \chi\_{\boldsymbol{m},\boldsymbol{n}} = \left(\boldsymbol{c}\cdot\cos\left(\frac{2\cdot\boldsymbol{m}\cdot\boldsymbol{n}}{p}\right)\_{\boldsymbol{m}} + \frac{1}{2}\cdot\sqrt{D\_{0}^{2}+4\cdot\boldsymbol{c}^{2}\cdot\cos\left(\frac{2\cdot\boldsymbol{m}\cdot\boldsymbol{n}}{p}\right)\_{\boldsymbol{m}}^{2}}\right)\cdot\sin\left(\frac{2\cdot\boldsymbol{m}\cdot\boldsymbol{n}}{p}+i\_{\boldsymbol{p}\cdot}\boldsymbol{m}\right) + d\_{f\boldsymbol{n}}\cdot\boldsymbol{m} \\\ \quad\quad\quad\quad\quad Y\_{\mathcal{H},\boldsymbol{\mathcal{H}}} = \left(\boldsymbol{c}\cdot\cos\left(\frac{2\cdot\boldsymbol{m}\cdot\boldsymbol{\mathcal{H}}}{p}\right)\_{\boldsymbol{m}} + \frac{1}{2}\cdot\sqrt{D\_{0}^{2}+4\cdot\boldsymbol{c}^{2}\cdot\cos\left(\frac{2\cdot\boldsymbol{m}\cdot\boldsymbol{\mathcal{H}}}{p}\right)\_{\boldsymbol{m}}^{2}}\right)\cdot\cos\left(\frac{2\cdot\boldsymbol{m}\cdot\boldsymbol{n}}{p}+i\_{\boldsymbol{p}\cdot}\boldsymbol{m}\right) \end{cases} \tag{1}$$

where *p* is the number of the toolpath points; *n* is the index of the current point from the toolpath (*n* = 0, 1, 2, ... *p*); *m* is the index of the current segment of the toolpath (*m* = 0, 1, 2, ... *q*); *q* = *L*/*df n* is the number of all toolpath segments; *D*0, mm is the toolpaths' segment diameters; *e*, mm is half of the amplitude of the sinewaves; *dfn*, mm is the linear distance between the toolpath segments; *ip* is the fractional part of the ratio *i* = π·*D*0/*λ*.

The parameter *ip* determines the phase shift between sinewaves of the successive toolpath segments and can have values between 0 and 0.5. When *ip* ≈ 0.15, the RR have cells that resemble a hexagonal shape, and when *ip* ≈ 0.45 the cells are close to having a rectangular shape. The integer part of the parameter *i* sets the number of the sinewaves within each of the toolpath segments, thus determining their period *λ*, mm. It has an impact on the resulting RR cells' size along the Y axis (see Figure 1a).

The parameters *e* and *dfn* from Equation (1) have a significant impact on the RR cells' size along the X axis. One of the important requirements to be met is that *dfn* must be equal or less than *e* (i.e., *dfn* ≤ *e*) in order to guarantee obtaining RR of the IV-th type. Otherwise, if *dfn* > *e* there is a possibility of obtaining RR of types I-st, II-nd or III-th, which can be formed onto burnished surfaces, which contain "isles" with initial roughness obtained by the previous operation. This is undesirable because it can lead to non-uniformity of the physical and mechanical properties in the burnished surface layer. When the values of these parameters are set in the Equation (1), the results for the imprint diameter also must be taken into account.

Another important condition is that the toolpath points must be generated only within the burnished area boundaries, because there is no reason for the deforming tool to process the space outside the material. In [34] an algorithm is presented which is based on additional logical conditions to prevent generation of points outside the material boundaries. It also connects the sinewave segments with each other (see Figure 1a) and this way ensures the overall length of the toolpath is as short as possible. The outcome of the algorithm is a single polyline, defined by the points calculated by Equation (1), with an optimal length that depends on the shape and size of the area processed by the BB operation. The polyline created in this way can then be exported as a two-dimensional drawing (in DXF or DWG format), and be used in suitable CAM for further modeling of the BB operation.
