Analyzing the effect of residual stresses on the fatigue life of colddrawn steel wire specimens

Fatigue and residual stress are responsible for many mechanical failures. Although there are many areas where fatigue can be applied, such as civil engineering, Laterza et al. [1] or bioengineering, Teoh [2], the focus of this manuscript will be set on mechanical engineering. Most of the residual stress literature references are experimental, as in Tabatabaeian et al. [3]. In this manuscript, however, the residual stress approach will be developed in an analytical way. These two important failure causes, one widely known as fatigue and the other lesser known as residual stress, will be combined to form a compound analytical model. A step-by-step explanation will be given for the residual stress analytic approach, leaving the fatigue basics available in the Appendix. Residual stress can develop in numerous ways in manufacturing processes, such as welding or machining. Any procedure that partially yields a mechanical part cross-section can generate residual stress. Textbooks about the mechanics of solids, such as Crandall et al. [4], present, as a basic subject, the residual stress distribution in static beams submitted to one semi-cycle bending moment, for an elastic perfectly plastic material. Although analytical approaches can be used to gain insight into cross-sectional residual stress distribution, they usually handle only simple geometries, such as rectangular or circular cross-sections. Stok and Halilovic [5] show an elastoplastic analytical solution for bending beams with rectangular cross-sections. Rimovskis and Sabaliauskas[6] analyzed, for rectangular and circular beam cross-sections, the drift of the neutral axis relative to the centroidal axis of the beam as a function of dissimilar yielding parameters in tension and compression. Castro and Meggiolaro [7] and Jirasek and Bazant[8] presented improvements in residual stress analytical models. De Castro et al. [9] show different cases of analytical models' applications to estimate the residual stress distribution of partially yielded structures of rectangular cross sections. Experimental techniques were used to estimate residual stress on surfaces, as in Schajer [10]. Atienza and Elices [11] analyze the residual stress in cold-drawn wires experimentally and numerically, through the influence of residual stresses on the tensile test, where the residual stress was relieved based on mechanical and/or thermomechanical treatments. Yet, the availability of the combination of fatigue and residual stress, in technical literature, is quite uncommon. One exception is the interesting manuscript of McClung [12], that provides a broad literature survey of the current understanding of significant residual stress issues for fatigue lifetime for shot peening, cold expansion of holes, welding, and machining. Also, see Webster et al. [13], James et al. [14] and Vaara et al. [15] for additional references.
This article proposes an analytical model to describe the effect of residual stress on the fatigue life of partially yielded colddrawn steel wires, modeled as a bi-linear material, of bulged rectangular cross-section. This research is a development of former references, such as Zaidan et al. [16]. The results obtained in this manuscript can contribute to improving the knowledge about the effect of residual stress on fatigue performance of partially yielded cold-drawn steel wires, extensively used, for instance, in the offshore industry.

The compound analytical model, which intends to reproduce the experimental part of the manuscript, is divided into residual stress and fatigue approaches, which refer, respectively, to the experimental part of phases 1 and 2 . In phase 1, a monotonic displacement is imposed on the specimen, as shown in Fig.1.b. A circular forming tool acts transversally on the specimen, supported by a 3 -point bending gripper, yielding partially its cross-section. After spring-back, as shown in Fig.1.c, a residual stress distribution pattern is generated in its cross-section. Phase 1 is described by the residual stress approach of this analytical model, which is considered a relevant contribution, represented by Eqns. (1-11). In phase 2 a variable transverse load $\mathrm{P}(\mathrm{t})$$\mathrm{P}(\mathrm{t})$P(t)\mathrm{P}(\mathrm{t})$\mathrm{P}(\mathrm{t})$ is applied on the specimen that has already passed by phase 1 , turned upside down, as shown in Fig. 1.d. The variable transverse load is applied until the specimen completely fails. The phase 2 is described by the fatigue approach of this analytical model. The applied fatigue theory is the classical one, so it was placed in the Appendix section, as shown in Eqns. (1A-7A). The compound analytical model, as shown in Eqns. (12-22), merges two different phenomena, residual stress and fatigue, to produce a comprehensive analysis of the effect of residual stresses on the fatigue life of colddrawn steel wire specimens. Mathcad software is used to perform calculations.

Where radius ${}_1$${}_1$_(1)_{1}${}_1$ is the forming tool radius. Radius ${}_2$${}_2$_(2){ }_{2}${}_2$ is the radius of curvature of the specimen after spring back. Radius ${}_3$${}_3$_(3){ }_{3}${}_3$ is the radius of the specimen's lateral cross-section. $L$$L$LL$L$ is the length of the specimen between the 3-point bending gripper external rollers. $\delta$$\delta$delta\delta$\delta$ is transversal displacement applied towards the specimen. $\mathrm{P}(\mathrm{t})$$\mathrm{P}(\mathrm{t})$P(t)\mathrm{P}(\mathrm{t})$\mathrm{P}(\mathrm{t})$ is the fatigue loading. ${\mathrm{b}}_1$${\mathrm{b}}_1$b_(1)\mathrm{b}_{1}${\mathrm{b}}_1$ and ${\mathrm{b}}_2$${\mathrm{b}}_2$b_(2)\mathrm{b}_{2}${\mathrm{b}}_2$ are, respectively, the minor and the major width of the specimen's cross-section, $y$$y$yy$y$ is a vertical distance and $dy$$dy$dyd y$dy$ is a differential of vertical distance. And $c$$c$cc$c$ is the semi-height and $b$$b$bb$b$ is the height of the specimen's cross-section. ${\mathrm{y}}_{\mathrm{y}}$${\mathrm{y}}_{\mathrm{y}}$y_(y)\mathrm{y}_{\mathrm{y}}${\mathrm{y}}_{\mathrm{y}}$ is the elastic-plastic border.

Every step of the residual stress part of the analytical model is explicitly presented and commented on. The value of the bending moment $\mathrm{M}(\varrho \mathrm{i})$$\mathrm{M}(\varrho \mathrm{i})$M(ϱi)\mathrm{M}(\varrho \mathrm{i})$\mathrm{M}(\varrho \mathrm{i})$ imposed by the forming tool is inside the range ${\mathrm{M}}_{\mathrm{y}}<\mathrm{M}(\varrho \mathrm{i})<{\mathrm{M}}_{\mathrm{p}}$${\mathrm{M}}_{\mathrm{y}}<\mathrm{M}(\varrho \mathrm{i})<{\mathrm{M}}_{\mathrm{p}}$M_(y) < M(ϱi) < M_(p)\mathrm{M}_{\mathrm{y}}<\mathrm{M}(\varrho \mathrm{i})<\mathrm{M}_{\mathrm{p}}${\mathrm{M}}_{\mathrm{y}}<\mathrm{M}(\varrho \mathrm{i})<{\mathrm{M}}_{\mathrm{p}}$. The maximum elastic moment ${\mathrm{M}}_{\mathrm{y}}$${\mathrm{M}}_{\mathrm{y}}$M_(y)\mathrm{M}_{\mathrm{y}}${\mathrm{M}}_{\mathrm{y}}$ and the plastic moment ${\mathrm{M}}_{\mathrm{p}}$${\mathrm{M}}_{\mathrm{p}}$M_(p)\mathrm{M}_{\mathrm{p}}${\mathrm{M}}_{\mathrm{p}}$ equations are presented.

Where ${\mathrm{S}}_{\mathrm{y}0}$${\mathrm{S}}_{\mathrm{y}0}$S_(y0)\mathrm{S}_{\mathrm{y} 0}${\mathrm{S}}_{\mathrm{y}0}$ is the material yielding resistance and ${\mathrm{S}}_{\mathrm{ut}0}$${\mathrm{S}}_{\mathrm{ut}0}$S_(ut0)\mathrm{S}_{\mathrm{ut} 0}${\mathrm{S}}_{\mathrm{ut}0}$ is the material's ultimate resistance, both, as received, ${\mathrm{E}}_{\mathrm{t}}$${\mathrm{E}}_{\mathrm{t}}$E_(t)\mathrm{E}_{\mathrm{t}}${\mathrm{E}}_{\mathrm{t}}$ is the plastic tangent modulus, ${\rho }_i,{\rho }_y$${\rho }_i,{\rho }_y$rho_(i),rho_(y)\rho_{i}, \rho_{y}${\rho }_i,{\rho }_y$, and ${\rho }_p$${\rho }_p$rho_(p)\rho_{p}${\rho }_p$ are the radius of curvature of the specimen, respectively, at loading, at the beginning of yielding, and the full cross-section yielding. The ${\epsilon }_{\mathrm{a}}$${\epsilon }_{\mathrm{a}}$epsi_(a)\varepsilon_{\mathrm{a}}${\epsilon }_{\mathrm{a}}$ is the strain at point $a$$a$aa$a$ of Fig 1.a, ${\epsilon }_{\mathrm{y}}$${\epsilon }_{\mathrm{y}}$epsi_(y)\varepsilon_{\mathrm{y}}${\epsilon }_{\mathrm{y}}$ is the yielding strain, and ${\epsilon }_{\mathrm{u}}$${\epsilon }_{\mathrm{u}}$epsi_(u)\varepsilon_{\mathrm{u}}${\epsilon }_{\mathrm{u}}$ is the ultimate strain. The elastic-plastic border $y_y$$y_y$y_(y)y_{y}$y_y$ equation is presented:

The equation of the bending moment $M\left({\rho }_i\right)$$M\left({\rho }_i\right)$M(rho_(i))M\left(\rho_{i}\right)$M\left({\rho }_i\right)$ in function of the imposed radius of curvature of the specimen:

The stress $\sigma$$\sigma$sigma\sigma$\sigma$ and strain $\mathcal{E}$$\mathcal{E}$E\mathcal{E}$\mathcal{E}$ evolution of the specimen point $a$$a$aa$a$ (shown in Fig. 1.a) is presented in Fig. 2.a, whose values are shown in Eqn. (06), Castro and Meggiolaro [6].

where, $\alpha =\mathrm{M}\left({\varrho }_{\mathrm{i}}\right)/{\mathrm{M}}_{\mathrm{y}}$$\alpha =\mathrm{M}\left({\varrho }_{\mathrm{i}}\right)/{\mathrm{M}}_{\mathrm{y}}$alpha=M(ϱ_(i))//M_(y)\alpha=\mathrm{M}\left(\varrho_{\mathrm{i}}\right) / \mathrm{M}_{\mathrm{y}}$\alpha =\mathrm{M}\left({\varrho }_{\mathrm{i}}\right)/{\mathrm{M}}_{\mathrm{y}}$ is a multiplicative factor. ${\sigma }_{sb}(-c)$${\sigma }_{sb}(-c)$sigma_(sb)(-c)\sigma_{s b}(-c)${\sigma }_{sb}(-c)$ is the springback stress at specimen point $a$$a$aa$a$, of Fig 1.a, estimated by Eqn. (10). The Eqns. (07) and (08) represent, respectively, in Fig. 2.a, the material behavior (light gray dash lines) and the loading/unloading equations (black lines for point $a$$a$aa$a$, and gray lines for point $b$$b$bb$b$ ), where points $a$$a$aa$a$ and $b$$b$bb$b$ are shown in Fig 1.a.

In Fig. 2.a, in the final state of point $a$$a$aa$a$, labeled with the number 3, there is a change in the stress signal from its former state, labeled with the number 2, while the strain signal remains the same signal. Also, in Fig. 2.a the abscissa axis was maintained as strain (not divided by ${\epsilon }_{\mathrm{y}0}$${\epsilon }_{\mathrm{y}0}$epsi_(y0)\varepsilon_{\mathrm{y} 0}${\epsilon }_{\mathrm{y}0}$ ) to show that at the end of phase 1 , point $a$$a$aa$a$ has a residual strain of 0.06 (or $6\mathrm{\%}$$6\mathrm{\%}$6%6 \%$6\mathrm{\%}$ ). The Eqns. (0911) were used to build the various cross-section stress distributions in Fig. 2.b.

where, ${\sigma }_{\text{curv}}(y),{\sigma }_{sb}(y)$${\sigma }_{\text{curv }}(y),{\sigma }_{sb}(y)$sigma_("curv ")(y),sigma_(sb)(y)\sigma_{\text {curv }}(y), \sigma_{s b}(y)${\sigma }_{\text{curv }}(y),{\sigma }_{sb}(y)$ and ${\sigma }_{\text{res\_c}}^{\ast }(y)$${\sigma }_{\text{res\_c }}^{\ast }(y)$sigma_("res_c ")^(**)(y)\sigma_{\text {res_c }}^{*}(y)${\sigma }_{\text{res\_c }}^{\ast }(y)$ are, respectively, loading stress distribution, the spring-back stress distribution, and the residual stress distribution, all within phase $1.I$$1.I$1.I1 . I$1.I$ is the area moment of inertia for a straight specimen.

Fig. 2.a shows, the $\sigma (\epsilon )/{\mathrm{S}}_{\mathrm{y}0}$$\sigma (\epsilon )/{\mathrm{S}}_{\mathrm{y}0}$sigma(epsi)//S_(y0)\sigma(\varepsilon) / \mathrm{S}_{\mathrm{y} 0}$\sigma (\epsilon )/{\mathrm{S}}_{\mathrm{y}0}$ vs $\epsilon$$\epsilon$epsi\varepsilon$\epsilon$ evolution diagram of specimen points $a$$a$aa$a$ and $b$$b$bb$b$ during phase 1, and Fig. 2.b shows the specimen cross-section parameterized stress distribution in phase $1-$$1-$1-1-$1-$ loading $({\sigma }_{\text{curv}}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0})$$\left({\sigma }_{\text{curv }}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0}\right)$(sigma_("curv ")(y)//S_(y0))\left(\sigma_{\text {curv }}(\mathrm{y}) / \mathrm{S}_{\mathrm{y} 0}\right)$({\sigma }_{\text{curv }}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0})$, spring back $({\sigma }_{\mathrm{sb}}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0})$$\left({\sigma }_{\mathrm{sb}}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0}\right)$(sigma_(sb)(y)//S_(y0))\left(\sigma_{\mathrm{sb}}(\mathrm{y}) / \mathrm{S}_{\mathrm{y} 0}\right)$({\sigma }_{\mathrm{sb}}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0})$, and residual $({\sigma }_{\text{res\_c}}^{\ast }(y)/{\mathrm{S}}_{\mathrm{y}0})$$\left({\sigma }_{\text{res\_c }}^{\ast }(y)/{\mathrm{S}}_{\mathrm{y}0}\right)$(sigma_("res_c ")^(**)(y)//S_(y0))\left(\sigma_{\text {res_c }}^{*}(y) / \mathrm{S}_{\mathrm{y} 0}\right)$({\sigma }_{\text{res\_c }}^{\ast }(y)/{\mathrm{S}}_{\mathrm{y}0})$.

Note that Fig.2.b axes are dimensionless to improve conceptual understanding. For instance, when ${\sigma }_{\text{curv}}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0}=1$${\sigma }_{\text{curv }}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0}=1$sigma_("curv ")(y)//S_(y0)=1\sigma_{\text {curv }}(\mathrm{y}) / \mathrm{S}_{\mathrm{y} 0}=1${\sigma }_{\text{curv }}(\mathrm{y})/{\mathrm{S}}_{\mathrm{y}0}=1$, the material begins to yield. Also, an interesting point for mechanical parts severely yielded is that the residual stresses at the elastic-plastic border ( $\mathrm{y}/\mathrm{c}=\pm {\mathrm{y}}_{\mathrm{y}}/\mathrm{c}$$\mathrm{y}/\mathrm{c}=\pm {\mathrm{y}}_{\mathrm{y}}/\mathrm{c}$y//c=+-y_(y)//c\mathrm{y} / \mathrm{c}= \pm \mathrm{y}_{\mathrm{y}} / \mathrm{c}$\mathrm{y}/\mathrm{c}=\pm {\mathrm{y}}_{\mathrm{y}}/\mathrm{c}$ ) are bigger than the residual stresses at the external surfaces ( $\mathrm{y}/\mathrm{c}=\pm 1$$\mathrm{y}/\mathrm{c}=\pm 1$y//c=+-1\mathrm{y} / \mathrm{c}= \pm 1$\mathrm{y}/\mathrm{c}=\pm 1$ ).

At the end of phase 1, the specimen cross-section stress distribution ${\sigma }_{\text{res}\mathrm{\_}c}^{\ast }(y)$${\sigma }_{\text{res }\mathrm{\_}c}^{\ast }(y)$sigma_("res "_c)^(**)(y)\sigma_{\text {res } \_c}^{*}(y)${\sigma }_{\text{res }\mathrm{\_}c}^{\ast }(y)$ is shown in Fig. 2.b. Before beginning phase 2 the specimen is turned upside down, obeying Eqn. (12) to convert ${\sigma }_{\text{res}\mathrm{\_}c}^{\ast }(y)$${\sigma }_{\text{res }\mathrm{\_}c}^{\ast }(y)$sigma_("res "_c)^(**)(y)\sigma_{\text {res } \_c}^{*}(y)${\sigma }_{\text{res }\mathrm{\_}c}^{\ast }(y)$ of Fig. 2.b into ${\sigma }_{\text{res}\mathrm{\_}}(y)$${\sigma }_{\text{res }\mathrm{\_}}(y)$sigma_("res "_)(y)\sigma_{\text {res } \_}(y)${\sigma }_{\text{res }\mathrm{\_}}(y)$ of Fig. 3.

Combining the residual stress and the fatigue stress loading distributions, using Eqns. (12) and Eqns. (4A.a) and (5A.a), results in Eqns. (13) and (14), as shown, graphically, in Fig. 3.

Fig. 3 shows the specimen cross-section stress distribution resulting from Eqns. (4А.a, 5A.a, 12, 13, 14). Both, ${\sigma }_{\text{fad\_ max}}(y)/S_{y0}$${\sigma }_{\text{fad\_ max }}(y)/S_{y0}$sigma_("fad_ max ")(y)//S_(y0)\sigma_{\text {fad_ max }}(y) / S_{y 0}${\sigma }_{\text{fad\_ max }}(y)/S_{y0}$ and ${\sigma }_{\text{fad\_ min}}(y)/S_{y0}$${\sigma }_{\text{fad\_ min }}(y)/S_{y0}$sigma_("fad_ min ")(y)//S_(y0)\sigma_{\text {fad_ min }}(y) / S_{y 0}${\sigma }_{\text{fad\_ min }}(y)/S_{y0}$ of Fig. 3.a and 3.b, respectively, have elastic stress distribution, but for the ${\sigma }_{f_{-}max}(y)/S_{y0}$${\sigma }_{f_{-}max}(y)/S_{y0}$sigma_(f_(-)max)(y)//S_(y0)\sigma_{f_{-} \max }(y) / S_{y 0}${\sigma }_{f_{-}max}(y)/S_{y0}$ distribution, shown in Fig.3.a, it seems to have some yielding at external specimen surfaces. The main reason for applying such severe combined loading is to impose short lives on the specimens, intending to finish the experimental part of this research in a reasonable period.

The experimental part of this research was done in LAMAT Laboratory (https://dippg.cefet-rj.br/ppemm/index.php/pt/laboratorios/57-estrutura/296-laboratorio-de-materiais-lamat) of the Federal Center for Technological Education Celso Suckow da Fonseca - CEFET/RJ (https://www.cefet-rj.br/), in Rio de Janeiro, Brazil. It was used a servo-hydraulic material testing machine INSTRON 8801 of 100 kN of capacity, with a three-point bending grip setup with a phase 2 specimen positioned, as shown in Fig.4.a. The frequency of the fatigue test was limited to 10 Hz , due to the difficulties in adjusting the PID feedback control for such flexible specimens. The fatigue test was made in load control, and an excessive displacement interlock was set to stop the test immediately after the specimen had failed. Fig. 4.b shows an example of the experimental graph output displacement vs load of a specimen in phase 2. Fig. 4.c shows the two specimens' groups ( $\mathit{a}$$\mathit{a}$a\boldsymbol{a}$\mathit{a}$ and $\mathit{b}$$\mathit{b}$b\boldsymbol{b}$\mathit{b}$ ) after phase 1 , where the curvatures were imposed, before being used in phase 2 of the fatigue test.
Each specimen has 25 cm length, and cross-section nominal values of $14\mathrm{mm}\times 6\mathrm{mm}$$14\mathrm{mm}\times 6\mathrm{mm}$14mmxx6mm14 \mathrm{~mm} \times 6 \mathrm{~mm}$14\mathrm{mm}\times 6\mathrm{mm}$, which were taken from the manufacturer's wire spools of cold-drawn steel wires. Figs. 4.d and 4.e show examples of specimens' final fractures, from an external and an internal point of view.
In Fig. 4.b the abscissa axis shows the displacement of the piston rod, and the ordinate axis shows the load cell measurements during the fatigue test. The test began with the cycles positioned at the right part of the graphic. As the test heads towards the end, the distance between cycles increases until the specimen final failure, when the excessive displacement activates the interlock, ending the test. In the macrography of Fig. 4.e, it is possible to see a fatigue crack propagation region (the smooth region at the cross-section superior area) and a final failure region (the rough surface at the cross-section central area).

I$n$$n$nn$n$ the experimental part of this research, an unexpected behavior of the tested specimens occurred. The specimens submitted to phases 1 and 2 had surprisingly long lives. The experimental results are summarized from the experimental approach of the original research developed in Zaidan [17], which is not entirely available because of a confidentiality agreement. Nevertheless, the results presented in the text are sufficient to support the development of the proposed analytical model.
It was initially assumed that the specimens taken from the manufacturer's wire spools of cold-drawn steel wires, did not have appreciable residual stress. As the specimens were taken from the manufacturer's wire spools, they were marked on the upper side with a yellow marker. The specimens were divided into groups $\mathit{a}$$\mathit{a}$a\boldsymbol{a}$\mathit{a}$, and $\mathit{b}$$\mathit{b}$b\boldsymbol{b}$\mathit{b}$. These groups were related to how the specimens were positioned on the 3 -point bending gripper, with the wire's original yellow marking above or below. The objective was to verify if the specimen side, with the yellow marking top side or the bottom side, affects the specimen's
fatigue performance. Fig. 5 shows the theoretical SN curve ( ${\mathrm{SN}}_{\text{teo}}0\mathrm{\%}$${\mathrm{SN}}_{\text{teo }}0\mathrm{\%}$SN_("teo ")0%\mathrm{SN}_{\text {teo }} 0 \%${\mathrm{SN}}_{\text{teo }}0\mathrm{\%}$ ), with experimental results of groups $\mathit{a}$$\mathit{a}$a\boldsymbol{a}$\mathit{a}$, and $\mathit{b}$$\mathit{b}$b\boldsymbol{b}$\mathit{b}$, each group with three specimens, and the notable point $OF$$OF$OFO F$OF$.

The fatigue results for both specimen groups $\mathit{a}$$\mathit{a}$a\boldsymbol{a}$\mathit{a}$ and $\mathit{b}$$\mathit{b}$b\boldsymbol{b}$\mathit{b}$ eventually show that they were quite similar. So, it was taken the average of the number of cycles generated by the specimens of both groups, $N_{\text{exp}}$$N_{\text{exp }}$N_("exp ")N_{\text {exp }}$N_{\text{exp }}$, shown in Fig. 5 . Note that the number of cycles that the specimens failed $N_{\text{exp}}$$N_{\text{exp }}$N_("exp ")N_{\text {exp }}$N_{\text{exp }}$ exceeded the number of cycles estimated by the proposed fatigue model $N_{\text{teo}}$$N_{\text{teo }}$N_("teo ")N_{\text {teo }}$N_{\text{teo }}$. With the aid of Eqns. (7A.a) and (7A.b), the equivalent alternate stress ${\sigma }_{a_{-}eq}$${\sigma }_{a_{-}eq}$sigma_(a_(-)eq)\sigma_{a_{-} e q}${\sigma }_{a_{-}eq}$ and the number of cycles until fatigue failure, Nteo are estimated, which correspond to point OF of Fig. 5. In effect, the point of coordinates ( $N_{\text{exp}},{\sigma }_{a_{-}eq}/S_{y0}$$N_{\text{exp }},{\sigma }_{a_{-}eq}/S_{y0}$N_("exp "),sigma_(a_(-)eq)//S_(y0)N_{\text {exp }}, \sigma_{a_{-} e q} / S_{y 0}$N_{\text{exp }},{\sigma }_{a_{-}eq}/S_{y0}$ ) is not on the theoretical SN curve ( ${\mathrm{SN}}_{\text{teo}}0\mathrm{\%}$${\mathrm{SN}}_{\text{teo }}0\mathrm{\%}$SN_("teo ")0%\mathrm{SN}_{\text {teo }} 0 \%${\mathrm{SN}}_{\text{teo }}0\mathrm{\%}$ ), described by Eqn. (2A). An explanation for this behavior is proposed next.

It was tentatively solved by adjusting the ${\mathrm{S}}_{\mathrm{ut}0}$${\mathrm{S}}_{\mathrm{ut}0}$S_(ut0)\mathrm{S}_{\mathrm{ut} 0}${\mathrm{S}}_{\mathrm{ut}0}$ for a higher value, considering $|6\mathrm{\%}|$$|6\mathrm{\%}|$|6%||6 \%|$|6\mathrm{\%}|$ residual strain present in specimen surfaces after phase 1, as shown in point $a$$a$aa$a$ of Fig. 2.a. This ${\mathrm{S}}_{\mathrm{ut}}$${\mathrm{S}}_{\mathrm{ut}}$S_(ut)\mathrm{S}_{\mathrm{ut}}${\mathrm{S}}_{\mathrm{ut}}$ correction generated a new theoretical SN curve (SN ${\mathrm{SN}}_{\text{teo}}6\mathrm{\%}$${\mathrm{SN}}_{\text{teo }}6\mathrm{\%}$SN_("teo ")6%\mathrm{SN}_{\text {teo }} 6 \%${\mathrm{SN}}_{\text{teo }}6\mathrm{\%}$ ) of Eqn. (2A) approximating to the experimental results, but not sufficiently. Moving the SN curve more in the "right" direction, a new SN curve was proposed, named the hypothetical SN curve ( ${\mathrm{SN}}_{\text{hyp}}6\mathrm{\%}$${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$SN_("hyp ")6%\mathrm{SN}_{\text {hyp }} 6 \%${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$ ), which passes through the fatigue experimental points. To describe the ${\mathrm{SN}}_{\text{hyp}}6\mathrm{\%}$${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$SN_("hyp ")6%\mathrm{SN}_{\text {hyp }} 6 \%${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$, shown in Fig. 6, Eqn. (15) is proposed

where $S_f^{byp}$$S_f^{byp}$S_(f)^(byp)S_{f}^{b y p}$S_f^{byp}$ is the hypothetical fatigue resistance, and ${\mathrm{a}}_{\text{hyp}}$${\mathrm{a}}_{\text{hyp }}$a_("hyp ")\mathrm{a}_{\text {hyp }}${\mathrm{a}}_{\text{hyp }}$ and ${\mathrm{b}}_{\text{hyp}}$${\mathrm{b}}_{\text{hyp }}$b_("hyp ")\mathrm{b}_{\text {hyp }}${\mathrm{b}}_{\text{hyp }}$ are hypothetical constants. Note that equivalent alternate stress ${\sigma }_{a_{-}eq\mathrm{\_}6}$${\sigma }_{a_{-}eq\mathrm{\_}6}$sigma_(a_(-)eq_6)\sigma_{a_{-} e q \_6}${\sigma }_{a_{-}eq\mathrm{\_}6}$ of Eqn. (15.b) uses the same Eqn. (7A.a), but with two differences: a) the material has a $6\mathrm{\%}$$6\mathrm{\%}$6%6 \%$6\mathrm{\%}$ residual strain, instead of $0\mathrm{\%}$$0\mathrm{\%}$0%0 \%$0\mathrm{\%}$, and b ) the loading combines residual and fatigue stresses, instead of only fatigue stresses.
Considering that the average experimental number of cycles ${\mathrm{N}}_{\text{exp}}$${\mathrm{N}}_{\text{exp }}$N_("exp ")\mathrm{N}_{\text {exp }}${\mathrm{N}}_{\text{exp }}$ is the only undeniable experimental result, two different approaches were proposed to explain the difference between ${\mathrm{N}}_{\text{teo}}$${\mathrm{N}}_{\text{teo }}$N_("teo ")\mathrm{N}_{\text {teo }}${\mathrm{N}}_{\text{teo }}$ and ${\mathrm{N}}_{\text{exp}}$${\mathrm{N}}_{\text{exp }}$N_("exp ")\mathrm{N}_{\text {exp }}${\mathrm{N}}_{\text{exp }}$. The two notable points related to the two different proposed approaches are shown in Fig. 6.
The first approach requires the existence of the new SN curve, called SN ${\mathrm{SNy}}_{\text{hyp}}6\mathrm{\%}$${\mathrm{SNy}}_{\text{hyp }}6\mathrm{\%}$SNy_("hyp ")6%\mathrm{SNy}_{\text {hyp }} 6 \%${\mathrm{SNy}}_{\text{hyp }}6\mathrm{\%}$, which is described by Eqn. (15). This approach receives the name of "resistance increase". One explanation for this proposition can be reached through the modification of the miscellaneous-effects factor ${\mathrm{k}}_{\mathrm{f}}$${\mathrm{k}}_{\mathrm{f}}$k_(f)\mathrm{k}_{\mathrm{f}}${\mathrm{k}}_{\mathrm{f}}$ of Eqn. (1A), substituting the value 1 by a value greater than one, rising the ${\mathrm{S}}_{\mathrm{e}}$${\mathrm{S}}_{\mathrm{e}}$S_(e)\mathrm{S}_{\mathrm{e}}${\mathrm{S}}_{\mathrm{e}}$ value, and therefore generating the $a_{\text{hyp}}$$a_{\text{hyp }}$a_("hyp ")a_{\text {hyp }}$a_{\text{hyp }}$ and $b_{\text{hyp}}$$b_{\text{hyp }}$b_("hyp ")b_{\text {hyp }}$b_{\text{hyp }}$ values of ${\mathrm{SN}}_{\text{hyp}}6\mathrm{\%}$${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$SN_("hyp ")6%\mathrm{SN}_{\text {hyp }} 6 \%${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$ curve. The point ( $N_{\text{exp}},{\sigma }_{a_{-}\text{eq}}^{RI}/S_{y0}$$N_{\text{exp }},{\sigma }_{a_{-}\text{eq }}^{RI}/S_{y0}$N_("exp "),sigma_(a_(-)"eq ")^(RI)//S_(y0)N_{\text {exp }}, \sigma_{a_{-} \text {eq }}^{R I} / S_{y 0}$N_{\text{exp }},{\sigma }_{a_{-}\text{eq }}^{RI}/S_{y0}$ ) is represented by a black triangle RI on the ${\mathrm{SN}}_{\text{hyp}}6\mathrm{\%}$${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$SN_("hyp ")6%\mathrm{SN}_{\text {hyp }} 6 \%${\mathrm{SN}}_{\text{hyp }}6\mathrm{\%}$ curve in Fig. 6: