## 2 Theoretical Background
With this additional assumption, the global buckling phenomenon involves a displacement field, in the delaminated zone, compatible with fracture mode II which causes a mixed mode I and mode II  
Figure 2: Stiffened composite panel stiffness: \(K^{0}\): Before delamination buckling; \(K^{A}\): After the local delamination buckling; and \(K^{A+A}\): After the propagated delamination buckling. delamination growth. Therefore, it is necessary to evaluate the global buckling relating it to local instability to evaluate the mode II contribution to delamination growth initiation. Assuming the critical global displacement of the skin \(u_{\text{glo}}\), two cases can be considered. _Case 1: Delamination growth initiation occurring before the global buckling (\(u^{*}<u_{\text{glo}}\))._ This case is representative of mode I dominant propagation mode. Therefore, the theory developed for pure mode I is still effective, as shown in Figure 3. In addition, a coefficient \(\alpha\), which correlates the contribution of \(G_{IC}\) and \(G_{IIC}\) to the total ERR, is introduced, which ranges between zero (pure mode II) and one (pure mode I). Hence, the total amount of energy released during the propagation can be evaluated as:  
\[\frac{\Delta E(u*)}{\Delta A^{m}}\cdot\frac{(\Delta w^{m})^{2}}{\sum\limits_{i }\left((\Delta w^{i})^{2}\right)}\Bigg{[}\frac{\alpha}{G_{IC}}+\frac{(1-\alpha )}{G_{IIC}}\Bigg{]}=1. \tag{4}\]  
In this work, the coefficient \(\alpha\) has been tuned by means of a non-linear VCCT analysis through Equation (5):  
\[\alpha=\frac{G_{I}^{\text{max}}}{\Delta E(u*)}, \tag{5}\]  
where \(G_{I}^{max}\) is the maximum value of \(G_{I}\) found by non-linear VCCT analyses along the delamination front. Indeed, with this approach an anticipation in delamination growth initiation is expected with respect to the pure mode I approach. This anticipation in growth initiation is due to the presence of the mode II contribution (neglected in the frame of pure mode I approach). _Case 2: Delamination growth initiation occurring after the global buckling (\(u^{*}>u_{\text{glo}}\))._ In this case, the structure experiences the global buckling of the skin before delamination growth initiation takes place. Indeed, after the global buckling of the skin, the structure resulting stiffness can be assumed equal to the stiffness of the stringers, since the skin loses all the capabilities to further carry compressive load. Hence, the mode II released energy is related to both local and global critical load and the unknown  
Figure 3: Schematic representation of energy release rate evaluation considering mode I and mode II interaction: \(u^{*}<u_{\text{glo}}\). propagation displacement \(u^{*}\). Therefore, according to Figure 4, the additional energy related to mode II can be evaluated as:  
\[F^{A}=K^{0}u^{\sigma}+K^{A}\big{(}u_{glob}-u^{\sigma}\big{)}, \tag{6}\]  
\[\frac{G^{i}(u\star)}{G^{k}(u\star)}=\left(\frac{\Delta w^{i}}{\Delta w^{k}} \right)^{2}i=1,\ldots,N;\ k=1,\ldots,N, \tag{1}\]0  
\[\frac{G^{i}(u\star)}{G^{k}(u\star)}=\left(\frac{\Delta w^{i}}{\Delta w^{k}} \right)^{2}i=1,\ldots,N;\ k=1,\ldots,N, \tag{1}\]1  
\[\frac{G^{i}(u\star)}{G^{k}(u\star)}=\left(\frac{\Delta w^{i}}{\Delta w^{k}} \right)^{2}i=1,\ldots,N;\ k=1,\ldots,N, \tag{1}\]2  
Once the global energy release rate has been related to the applied displacements, in order to determine the delamination growth initiation load, the maximum energy release rate along the delamination front should be equated to critical energy release rate in mixed mode configuration.