## 3 Numerical Applications and Discussion
### Test Case 1: Validation and Applicability of the Mode I Fast Procedure  
The mode I _Fast_ procedure has been implemented in the ABAQUS FEM code [41] by using a Python script interface. A refined FE model [42] has been introduced in the delamination region, to accurately evaluate the local buckling of the delaminated region. Indeed, a mesh sensitivity analysis has been performed to choose the optimum discretization for the finite element model. The structural configuration adopted for this first test-case [38] has been schematized in Figure 5. The mechanical properties of the adopted AS4/3501-6 material system are reported in Table 1. The stacking sequences of the skin, stringer cap, and stringer web are respectively [+45\({}^{\circ}\)/0\({}^{\circ}\)/90\({}^{\circ}\)/\(-45^{\circ}\)]\({}_{3s}\), [+45\({}^{\circ}\)/\(-45^{\circ}\)/0\({}^{\circ}\)/0\({}^{\circ}\)/90\({}^{\circ}\)]\({}_{s}\), and [+45\({}^{\circ}\)/\(-45^{\circ}\)/0\({}^{\circ}\)/0\({}^{\circ}\)/90\({}^{\circ}\)]\({}_{2s}\). A 20 mm radius delamination between the third and fourth ply (90\({}^{\circ}\)/\(-45^{\circ}\) interface) has been considered. Figure 5: Test case 1 geometrical description (dimensions in mm) and boundary conditions. The FEM model has been discretized with eight-nodes SC8R layered elements. The stringer feet and the sub-laminates have been connected to the skin by means of TIE constraints, as shown in Figure 6. Indeed, a more refined mesh has been adopted in the local delamination region while a coarser discretization has been used for the rest of the panel. This choice allowed to obtain more accurate results while reducing the computational cost. As remarked in [38; 39], a number of linear analyses on the configurations with delamination size \(A\) and \(A+\Delta A\) are needed to evaluate the delamination growth initiation with the mode I fast procedure. First, the delamination buckling displacement and load for the structure with delamination size \(A\) are obtained by means of a linearized buckling (Figure 7). Then, the critical applied displacements \(u^{cr}\), the critical compressive load \(F^{cr}\), and the linear nodal opening \(\Delta w^{i}\) trend on the delamination front with a size \(A\) are evaluated. Hence, the node with maximum ERR is identified, as shown in Figure 8a. A further linear buckling analysis is performed by releasing the node with maximum ERR on the delamination front, thus considering the delamination size \(A+\Delta A\) shown in Figure 8b, to obtain the critical load and displacement \(F^{cr^{\prime}}\) and \(u^{cr^{\prime}}\). Then, the stiffness \(K^{0}\) of undamaged structure, the post-buckling stiffness \(K^{A}\) of the structure with delamination size \(A\), and the post-buckling stiffness \(K^{A+\Delta A}\) of the structure with delamination size \(A+\Delta A\) are evaluated by means of static linear analyses. In order to evaluate the stiffness \(K^{A}\) and \(K^{A+\Delta A}\), the thinner sub-laminate has been removed from the finite element model to take in account the stiffness lost related to the delamination buckling. Finally, the applied loads and displacements at growth initiation and the ERR on the delamination front are evaluated. Several analyses have been carried out by considering different mesh densities along the tangential and radial directions of the local delamination region, in order to assess the mesh-dependency for the proposed approach. In particular, different mesh densities have been considered in a circular area around the delamination front, where the nodal displacements are evaluated. Therefore, the number of elements in the radial direction has been progressively increased. In Figure 9, the configurations characterized by two, four, and eight elements are shown, and the influence of the mesh densities on the critical propagation strain is reported as well.