## 2 Background  
### Residual Strength of Panels with MSD  
A variety of analytical and computational approaches can theoretically be used for estimating the residual strength of panels with MSD cracks. These approaches include analytical, semi-analytical (empirically corrected), computational and data-driven, and the accuracy of these approaches varies significantly [27]. Figure 1 illustrates the configuration of a typical panel having a lead crack along with adjacent MSD cracks, which is being considered here. Linear Elastic Fracture Mechanics (LEFM) is one of the most fundamental theories in fracture mechanics, and it can be used for evaluating residual strength for different types of crack configurations [43]. However, the LEFM is more applicable to brittle materials since it assumes a linear-elastic behavior for the material at the crack tip. According to LEFM, failure (or unstable crack extension) will occur when the value of the crack-tip SIF reaches a critical value. This limiting value of the SIF is called the fracture toughness (K\({}_{\text{C}}\)). The fracture toughness is a material property, but for thin sheets, it is also slightly dependent on thickness, grain orientation and crack length. For a panel with a major crack and collinear adjacent MSD cracks subjected to tension, as illustrated in Figure 1, the Mode-I stress intensity factor (K\({}_{\text{I}}\)) for the lead crack is:  
\[\text{K}_{\text{I}}\ =\ \sigma\sqrt{\pi a}\ \beta_{\text{W}}\ \beta_{a/\ell} \tag{1}\]  
where  
\(\sigma\): The remote applied stress. \(a\): Lead crack half-length. \(\beta_{\text{W}}\): Finite width correction factor. \(\beta_{a/\ell}\): Cracks interaction correction factor for the effect of MSD cracks on the lead crack. Thus, the residual strength (\(\sigma_{\text{C}}\)) of the panel can be found as:  
\[\sigma_{\text{c}}\ =\ \frac{\text{K}_{\text{C}}}{\sqrt{\pi a}\ \beta_{\text{W}}\ \beta_{a/\ell}} \tag{2}\]  
Experimental investigations have shown that the LEFM is not able to accurately predict the residual strength of panels with MSD neither for brittle nor ductile materials where it consistently over predicts the strength [29]. [PERSON] [28] introduced an analytical model that is specially formulated for the prediction of the residual strength of panels having MSD cracks. This model is called the \"Linkup\" model, and it is based on the concept that the ligament between the lead crack and the adjacent MSD crack will fail when the remote stress reaches a level that causes the lead crack-tip plastic zone and the adjacent MSD crack-tip plastic zone to touch each other (i.e., merge together). As the linkup occurs, the entire panel will fail (assuming MSD cracks exist on all subsequent holes and no crack arresting structures are used); therefore the stress causing linkup of the lead and MSD cracks is practically equal to the residual strength of the panel. Therefore, the residual strength of panels with MSD is also sometimes referred to as linkup stress. The Linkup model estimates the plastic zone size using the Irwin plane-stress plastic zone model. The Irwin model is based on the crack-tip SIF (K\({}_{\text{I}}\)), and it gives the size (diameter) of plastic zone as:  
\[\text{r}_{\text{P}}\ =\ \frac{1}{2\pi}\bigg{(}\frac{\text{K}_{\text{I}}}{ \sigma_{\text{y}}}\bigg{)}^{2} \tag{3}\]where \(\sigma_{y}\) is the yield strength of the material. Therefore, according to the linkup model, the remote stress that causes failure of the ligament (i.e., touch of the plastic zones), which is called \"linkup\" stress (\(\sigma_{\rm LU}\)), is found as:  
\[\sigma_{\rm c}\ =\ \sigma_{\rm LU}\ =\ \sigma_{y}\sqrt{\frac{2{\rm L}}{a\ \beta_{a} ^{2}+\ell\ \beta_{\ell}{}^{2}}} \tag{4}\]  
where  
L: Length of the ligament between the lead crack and MSD crack. \(\ell\): MSD crack half-length. \(\beta_{a}\): The overall SIF correction factor for the lead crack tip. \(\beta_{\ell}\): The overall SIF correction factor for the adjacent MSD crack tip.