## 3 Influence of the Maximum Cyclic Loading  
To reveal the effects of the cyclic loading on the residual stress distribution appearing in a heavily drawn wire, the radial distributions of axial stress (obtained after applying a uniform cyclic loading according to the different cases of study depicted in Table 1) are compared in Figure 5 with the residual stress distribution obtained after wire drawing (Figure 2c). For the sake of clarity, a new variable--the effective axial stress \(\sigma_{\mathrm{z}}\)*--is defined as the difference between the axial stress and the in-service applied stress, as shown in the following equation:  
\[\sigma_{\mathrm{z}}\mathrm{{}^{\star}}=\sigma_{\mathrm{z}}-\sigma_{\mathrm{ app}} \tag{1}\]  
As a consequence of the cyclic loading, the distribution of the residual axial stress undergoes significant changes at both the tensile zone (near the wire surface, Figure 2c) and the compressive zone (close to the wire core, Figure 2c). The variations consist of a redistribution of the residual stress profile with stress reductions for all points. The intensity of the stress reduction increases with the maximum cyclic loading applied to the wire. Thus, for loadings under the material yield strength, the stress reduction at the wire surface is as high as 28% for the lowest loading (Loading I, 1000 MPa) and 48% for Loading II (1200 MPa). In the case of the cyclic loading with maximum stress equal to the material yield strength (Loading III), the stress reduction is slightly lower than that corresponding to Loading II. Consequently, a similar stress relief can be obtained by using in-service loading within the elastic regime without generating a generalized plasticity in the wire. Nevertheless, the highest reductions are obtained when the maximum cyclic loading overcomes the material yield strength, as seen in the distributions for Loadings IV and V (Figure 5), where a huge reduction is achieved in the wire surface surroundings and the wire core, resulting in an almost plain (uniform) distribution of residual stress in the radial direction of the wire. Considering these results, plastic strains caused during the cyclic loading are key for understanding the observed residual stress redistribution. In this way, for the loading cases below the material yield strength, the plasticity locally appears in the tensile zone as a result of the addition of the residual stress (of tensile nature) and the in-service stress caused by the cyclic loading. Consequently, large and non-recoverable strains are generated in the near-surface surroundings. Out of this zone, the stresses remain within the elastic regime, thus being low and completely recoverable when loading is removed. Consequently, in these cases, the main changes appear in the wire surface vicinity with soft reductions in the wire core. However, when the loading is equal to or higher than the material yield strength, the whole wire undergoes plastic strains, and consequently, a huge reduction appears through the whole wire radius. According to the obtained results, a cyclic loading of just 15% higher than the material yield strength (Loading IV) causes a reduction of the stress at the wire surface as high as 80%. Figure 5: Radial distribution of the effective axial stress after applying diverse uniform cyclic loading to a heavily drawn wire. On the other hand, for the case considering an in-service overload, the first analysis consists of finding out the influence of the instant when the overload is applied. To achieve this goal, three cases were considered: initial overload, intermediate overload, and final overload (Figure 1b-d). The radial distributions of the axial stress for each case are plotted in Figure 6.  
As can be observed, no significant changes appear in the stress distribution at the end of the final cycle. Thus, the influence of the overload sequence can be considered as a second-order effect, and thereby negligible within the framework of this work.