# The Effect of Stick-Slip Friction on the Stability of Nanostructuring Burnishing
## 3 Results  
With the dynamical model described in the previous section, we obtain classic stick-slip dynamics on individual tool passes. This by itself creates variations in the indentation depth, per the proposed coupling mechanism. This is illustrated in Figure 6. Note that the spikes of indentation depth correspond to the stick phases, as the static coefficient of friction is higher than the dynamic one and the effective hardness in the normal direction therefore lower (Equation (7)).  
Generally, these periodic indentations remain relatively independent from previous passes for reasonable values of the empirical parameters. Whether this irregularity results in recognizable, large-scale patterns depends very strongly on the damping constant \(\eta\). While high damping tends to stabilize the dynamics on a small scale, it has, ironically, the opposite effect on the large scale, because high damping tends to synchronize the phases of the stick-slip motion across passes very quickly, leading to the observed irregularities.  
Figure 7 shows a representative example of the resulting surface pattern: \(y\) is the coordinate of the primary tool pass (circumferential) and \(x\) is the coordinate along the shaft. The darker lines correspond to increased indentation due to stick. Note the nonuniform inclination of the tracks along the length of the workpiece. This is due to the variation of stiffness along the \(x\)-axis (cf. Figure A1). Figures 8-10 show relevant sections from the beginning, middle, and end section of the workpiece and the qualitative differences in behavior associated with them.  
Of particular note is the initial part of the track, where no stick-slip and therefore no surface patterning is observed due to the high stiffness close to the mount point. This is consistent with the observations of [PERSON] (1998) and generally with displacement-controlled operating conditions. At some point, there is a relatively abrupt beginning of stick-slip with the phase offset slowly synchronizing and stabilizing.  
In the middle section, we observe the most stable behavior. This is actually somewhat at odds with observation, but we are considering a simplified model here and are more interested in basic effects rather than quantitative prediction.  
The stability of phase synchronization decreases again as the other end of the bar is approached. Unlike the rigid mount on the left side, this side is permitted to rotate and therefore has a different stiffness profile, which in turn affects the surface pattern.  
Figure 6: Example of stick–slip behavior over one circumferential tool pass. (**Left**): lateral displacement showing typical staircase structure. (**Right**): resulting variations in indentation depth; the spikes are aligned with the stick phases.  
In addition, it was found that the produced instabilities and surface patterns are quite sensitive to the operating parameters, in particular the ratio of rotational period to the natural frequency of oscillations in the \(y\)-direction.  
Figure 8: Enlarged section of the indentation tracks from the beginning of the bar.  
Figure 7: Simulated indentation tracks. The indenter makes passes along the \(y\)-axis (circumferential, periodic boundary conditions apply), while slowly advancing in the \(x\)-direction (axial). The simulation parameters are \(v_{0}=1\) m/s, \(F_{n}=300\) N, \(R_{indenter}=2\) mm; axial feed rate: \(\
icefrac{{\
u}}{{2}}\) contact radius per pass.  
Figure 9: Enlarged section of the indentation tracks from the middle of the bar.