**5. Discussion**

The proposed probabilistic model can be used for decision making. For example, in a lot of applications, a relief with certain parameters is needed, i.e., length and amplitude of the sinewave (B and C factors) are chosen due to technological requirements. So, the probabilistic question could look like: "Which force-feed rate (A–D) combination will give higher probability of reaching fatigue life gain of more than mean value (15 dB)?" To answer the question, the posterior distribution for the data (*y*) is formed by reversing the scaled data, fixing the regressors at a certain point, and incorporating the uncertainties of regression coefficients and data. For a relief with finer sells or a high degree of imbrication (B = 1, C = 1) the posteriors are shown in Figure 14. The probabilities *p*(*y* > *15*) calculated from the posteriors, for all combinations of relief parameters, are given in Table 9.

**Figure 14.** Posterior distributions of the fatigue life gain for relief with high degree of imbrication.



As an amplitude of the sinewave (factor C) gives higher effect to the relief's degree of imbrication (see Table 9), for reliefs with high C value the feed rate should be kept low. For these kinds of reliefs, the optimal burnishing parameters are high force and low feed rate (A, D = 1, −1). For reliefs with low C value, the optimal burnishing regime is strongly dependent on sinewave length (factor B). If relief with a high B is needed, the feed rate (factor D) should be kept to a low value, unlike the relief with low B value, where high force and feed rate (A, D = 1, 1) are needed. Of course, all kinds of relief, using A, D = 1, −1 combination, result in the probability of reaching more than 15dB fatigue life gain, more than 50%.

The results from the proposed model support in general the published experimental results for conventional ball burnishing. Travieso-Rodriuez et al. investigated the fatigue life of burnished carbon steel specimens [50]. Their experimental results show that increasing the burnishing force and number of passes benefits fatigue life of the specimens. Rich experimental data for the influence of the burnishing regime on the fatigue strength is presented by Swirad [6]. He used a diamond composite burnishing element on low-alloyed carbon steel 40HM. The results showed that by increasing the burnishing force fatigue, the strength increased only to a certain threshold value. Beyond the threshold value, the fatigue strength decreased rapidly. Maximov et al. reported similar results for the influence of the burnishing force [51]. They used a diamond tool on aluminum alloy specimens and registered decrease of the fatigue life for the higher values of the burnishing force.

Other results of Swirad relate to the influence of feed rate and velocity. For burnishing tools with higher diameters, increasing the feed rate slightly decreases the fatigue strength after a certain value and the change of the velocity seems irrelevant.

The microstructural analysis of the burnished AISI 304 specimens given in [52] reports the phase composition in the surface layer. After the fatigue testing, strain-induced

martensite is developed. In specimens with a large content of the martensite phase, a shortened fatigue life has been registered. Since for austenitic steels the martensite phase is strain-induced, the higher martensite content in some of the specimens can be a result of the local fluctuation of the material properties or the burnishing force (burnishing is done on rolled sheets without annealing). In [53] the effect of strain-induced martensite on fatigue behavior is investigated. Martensitic transformation is registered during the fatigue tests. The more pronounced transformation is for prestrained specimens. In [54], a strong influence of martensitic content on fatigue limit is emphasized and optimum martensite content for a predeformed specimen of 26% is reported. These phenomena look like a reason to adopt a two-side posterior distribution (p (ACcoeff < 0) = 16%) of AC regression coefficient in the first robust regression model (see Figure 3). Obviously, the chosen high value for burnishing force, combined with a high value of relief amplitude, is just below the threshold level and small random fluctuations can shorten the fatigue life.

In the above cited reference [52], a comparison of microhardness profiles of two burnished specimens is given, and the only difference in the relief is the sinewave length (factor B). There is no difference in the hardened layer depth. The only difference is the higher microhardness value, registered just below the surface in the specimen, burnished with a high B value regime, resulting in a slightly higher fatigue life. This phenomenon is captured by BD regression coefficient, whose posterior distribution lies on the negative side. This means that a low feed rate (factor D) in combination with a high sinewave length value (factor B) gives an additional improvement to the fatigue life.
