## 2 Trans-Scale Formulation for Fatigue Crack Growth of Steel Wire and Cable  
### Macro/Micro Dual Scale Crack Model  
In this study, the cracking process is depicted using the macro/micro dual scale crack model, which is based on the concept of restraining stress zone that reflects the material damage. Assuming that a region with the size of \(a\) is cut at the fatigue source point, a macro/micro dual scale crack model can be established as shown in Figure 1[22], where \(r\) is a distance measured from the crack tip, and related to the crack propagation segment size at each time. The front of the zone is a V-shape notch which is simplified from the intergranular and transgranular defects with different grain boundary conditions. The restraining stresses would prevail on the cut surfaces denoted by \(\sigma_{0}\). The damage degree can be expressed by the ratio of restraining stress to applied stress \(\sigma_{\infty}\). Initially, the stress ratio \(\sigma_{0}/\sigma_{\infty}=1\) when the value of crack size \(a\) is very small. As the damage development and the crack size \(a\) increase with the cycle number of local load, the stress ratio gradually drops from 1 to 0, indicating the development of a fatigue crack from micro-scale to macro-scale [18]. Hence, the macro/micro dual crack model can describe the total fatigue process from micro to macro scales in a consistent way instead of dividing a continuous fatigue process into two different stages [23].  
As wires and cables only bear tension, the tension crack (mode I) is the most common pattern, and, in this study, the stress along the wire (one dimensional) is analyzed for simplicity [22; 23; 24; 25; 26; 27], though the stress distribution is much more complex with three-dimensional stresses. Furthermore, at the micro scale, the material is anisotropy, while in the analysis at the macro scale, the isotropic elasticity assumption is often used [24]. The material properties may change as the damage develops from micro to macro scale [17; 28]. Therefore, different values for the Poisson's ratio and shear modulus are used at micro and macro scale, respectively. In the macro/micro dual scale crack model, the expression of the macro/micro dual scale strain energy density factor for the tension mode crack can be obtained usingEquation (1) [24], where the superscript \"macro\" and subscript \"micro\" indicate that \(\Delta S^{macro}_{micro}\) is related to both microscopic and macroscopic factors.  
\[\Delta S^{macro}_{micro}=a\frac{2\left(1-2\
u_{micro}\right)\left(1-\
u_{macro} \right)^{2}\sigma_{\Delta}\sigma_{m}}{G_{macro}}G^{*}\left(1-\sigma^{*}\right) ^{2}\sqrt{\frac{d}{r}}, \tag{1}\]  
\[G^{*}=\frac{G_{micro}}{G_{macro}},\,\sigma^{*}=\frac{\sigma_{0}}{\sigma_{ \infty}},\,d=d^{*}\cdot d_{0}, \tag{2}\]  
\[\sigma_{\Delta}=\sigma_{\max}-\sigma_{\min},\,\sigma_{m}=\frac{\sigma_{\max}+ \sigma_{\min}}{2}. \tag{3}\]  
\(v_{micro}\) and \(v_{macro}\) are the microscopic and macroscopic [PERSON]'s ratios, while \(G_{micro}\) and \(G_{macro}\) are the microscopic and macroscopic shear module, respectively. \(d^{*}\) is the micro/macro characteristic size ratio, and \(d\) is the characteristic size of the local region, which can distinguish the regions of microscopic and macroscopic effects. \(d_{0}\) is the grain characteristic size of the material, for steel, it can be taken as \(10^{-3}\) mm [25]. \(\sigma_{\Delta}\) and \(\sigma_{m}\) are the stress range and the mean stress caused by cycle loading, respectively, which can be obtained according to \(\sigma_{\max}\) and \(\sigma_{\min}\) of the stress time-history along the axis of the cable.