## 4 Discussion  
To analyse the variations of stress reduction with the cyclic loading, a dimensionless parameter (\(\Delta\sigma/\sigma\)) is defined as the reduction of the axial stress (\(\Delta\sigma\)) divided by the axial residual stress itself (\(\sigma\)). This parameter is evaluated at the wire surface, \(\Gamma\) (maximum stress in the tensile zone, cf. Figure 2b), and in the wire core (maximum stress in the compressive zone, cf. Figure 2b). In addition, it was also applied for the average stress as a measure of the uniformity of the stress distribution. Taking this into account, the values of this parameter are represented in Figure 8 against the value of the maximum in-service stress (\(\sigma_{\max}\)) for the first study (uniform cyclic loading) and against the overload ratio (\(\upmu\)) for the second one (cyclic loading with overload). As observed in Figure 8, a linear growing trend is exhibited for the stress reduction at the wire surface. Figure 8 reveals different trends for the three analysed stress reductions. Thus, at the wire surface, a highly linear growing trend is observed with the maximum cyclic loading. The correlation coefficient is as high as 0.999. Consequently, the fitting equation (\(\Delta\sigma/\sigma=-77.234+0.104~{}\sigma_{\max}\)) could be useful for estimation of the stress reductions expected when another maximum stress loading level is applied. The stress reduction can be as high as 50% for stress within the elastic regime (Loading I and II in Table 1), and as high as 90% for the highest loading considered (Loading V in Table 1) within the plastic regime. On the other hand, in the wire core, the stress reduction grows linearly with the maximum stress loading, but in two different ways, depending on if the maximum stress is higher or lower than the material yield strength. Thus, the slope (variation of the stress reduction) of the first stage (\(\sigma_{\max}<\sigma_{\rm Y}\)) is lower (due to the negligible effect of local plasticity in the wire core) than for the second stage (\(\sigma_{\max}>\sigma_{\rm Y}\)), where full plasticity appears. This way, the stress reduction for the lowest loading (Loading I in Table 1, 1000 MPa) is only 20%, and can be as high as 30% for the case in which the maximum stress reaches the material yield strength (Loading III in Table 1, \(\sigma_{\max}=\sigma_{\rm Y}\)). However, in those cases which generate plastic strains at the wire core, the stress reduction reaches the same values obtained at the wire surface. Finally, the variations of the stress reduction for the average stress obviously exhibit the same trend observed in the case of the wire core: soft reductions (low slope in Figure 8) as the maximum cyclic loading is increased for the cases of local plastic strains near the wire surface, and high reductions (high slope in Figure 8) when the maximum cyclic loading overcomes the material yield strength, so a higher stress reduction can be achieved applying lower increments of the maximum cyclic stress. In the case of the results obtained for cyclic loading with overloads, the stress reduction achieved for the two base cyclic loading (0.75 \(\sigma_{\rm Y}\) and 0.90 \(\sigma_{\rm Y}\)) are shown in Figure 9 for the wire surface and  
Figure 8: Variation of the relative stress reduction at the wire surface with the maximum stress factor for the two base cyclic loading considered. in Figure 10a,b for the wire core and average stress, respectively. As in the case of uniform constant maximum stress loading, a highly linear growing trend is achieved for the stress reduction at the wire surface, reaching reductions as high as 90% for the highest overload (Loading Id and Loading IIc in Table 2). It is worth highlighting that the same stress reductions can be obtained by using a higher overload stress and lower maximum in-service loading (Loadings Ia-d in Table 2) or by applying a lower overload stress and a higher maximum in-service loading (Loadings IIa-c in Table 2).