## 2 Theoretical Background  
Buckling is one of the most dangerous phenomena which a composite structure may be subjected to. Indeed, in delaminated stiffened panels, the buckling can interact with the delamination and, in particular conditions, the following scenarios may occur:  
* The buckling only occurs on the thinner sub-laminate, as reported in Figure 1a. * Buckling occurs on both thicker and thinner sub-laminates in the delamination region, as reported in Figure 1b. Generally, composite stiffened panels with thin bay delaminations subjected to compressive loads firstly experience _Type-a_ buckling scenarios. Then, the _Type-a_ buckling scenarios may evolve into the more critical _Type-b_ buckling scenarios, characterized by more dangerous forms of buckling and/or failure phenomena. Indeed, in the framework of damage tolerant design, the prediction of the delamination evolution, which may result in _Type-a_ scenarios evolving into _Type-b_ ones, becomes crucial. The proposed approach can be seen as an improvement of the previously developed mode I _Fast_ method presented in [38; 39]. According to the mode I _Fast_ approach, four linear analyses are needed to simulate the delamination growth, as shown in Figure 2. In particular, a first eigenvalue buckling analysis is used to evaluate the delamination buckling load and shape (location \(A\) in Figure 2, characterized by a critical load \(F^{cr}\) and displacement \(u^{cr}\)). Then, the buckling load and shape due to a propagation of the initial delamination are obtained by means of a second eigenvalue buckling analysis (location \(A\)' in Figure 2, characterized by a critical load \(F^{cr}\)' and displacement \(u^{cr}\)'). Once the local buckling has occurred, the residual contribution to the global stiffness of the buckled thinner sub-laminate can be neglected. Hence, considering an unknown propagation displacement \(u^{*}\) beyond the critical displacements \(u^{cr}\) and \(u^{cr}\)', the amount of energy \(E(u^{*})\) (area \(A\)'\(\Lambda\)\  
It can be assumed that the ERR distribution along the delamination front follows the same distribution of the square of the first buckling out-of-plane displacements:  
\[\frac{G^{i}(u\star)}{G^{k}(u\star)}=\left(\frac{\Delta w^{i}}{\Delta w^{k}} \right)^{2}i=1,\ldots,N;\ k=1,\ldots,N, \tag{1}\]  
where \(N\) is the number of the nodes belonging to the delamination front. Moreover, the location \(m\), characterized by the maximum ERR, can be evaluated as:  
\[\frac{G^{m}(u\star)}{G^{k}(u\star)}=\max_{i}\left\{\frac{G^{i}(u\star)}{G^{k}( u\star)}\right\}=\max_{i}\left\{\left(\frac{\Delta w^{i}}{\Delta w^{k}} \right)^{2}\right\}i=1,\ldots,N;\ k=1,\ldots,N. \tag{2}\]  
Hence, the delamination will propagate from the location \(m\), as soon as the local ERR \(G^{m}(u\star)\) equates the critical value \(G_{IC}\). \[\frac{G^{m}(u\star)}{G_{IC}}=\frac{\Delta E(u\star)}{\Delta A^{m}}\cdot\frac{ \left(\Delta w^{m}\right)^{2}}{\sum\limits_{i}\left(\left(\Delta w^{i} \right)^{2}\right)}\cdot\frac{1}{G_{IC}}=1. \tag{3}\]  
One of the assumptions of the mode I _Fast_ procedure is the predominance of mode I contribution on the total ERR. This assumption considerably reduces the field of applicability of the procedure to delamination very close to composite surface and characterized by very thin sub-laminates. Moreover, the instability of sub-laminates is mandatory for the application of the mode I based procedure. Hence, in order to extend the field of applicability of the fast procedure, in this paper, a _Fast_ mode II based approach is introduced, which is aimed to improve the described _Fast_ method by considering, in addition, the mode II contribution in the evaluation of the energy absorbed in the propagation phase of delamination. Indeed, in case of global buckling of the skin taking place far after the local buckling of the delaminated region (new assumption), the mode II contribution can be introduced into the evaluation of the released energy leading to the mixed modes (mode I and mode II) propagation.