*3.1. Numerical Framework*

When a crack grows from the initiation region, due to the reduction of cross-sectional area in the plane where the crack propagates, it can be expected that the propagation speeds will be higher than in the reference lug, at least when the crack propagates through the region where the capillary is located. Hence, the objective of this section is to quantify the repercussions of the acceleration on the crack growth life. This quantification is done in function of capillary shape and size. To that purpose, fatigue crack growth simulations have been run on lugs equipped with different configurations of the eSHM, namely those of Figure 5a,c. The results are then compared to computations run on the reference lug (in which there is no capillary). Two types of initial defect are considered: a part-through crack of initial length *a*<sup>0</sup> = 1 mm (to be consistent with the study of Schijve [29]), and a quarter-elliptical crack of initial lengths *a*<sup>0</sup> = 1 mm and *b*<sup>0</sup> = 1 mm, both located at the initiation region (top of the lug hole surface). Part-trough cracks and quarter-elliptical cracks are actually the most fore-coming type of defects in aeronautical lugs [30].

The objective here is to make a comparison between a reference lug and lugs equipped with the eSHM, so that to be able to infer up to what extend the capillaries do affect the crack growth life. The crack propagation model used may thus remain simple, and therefore, the Paris law has been referred to [31,32], considering a Paris exponent n = 3.58 and a Paris coefficient of *C* = 1.3651*e* − 13 *mm cycle* · (*MPa*√*mm*)−*n*. The fracture toughness of aluminum 2024-T3 is *Kc* <sup>=</sup>1010 MPa√*mm*, so that the lugs are considered to have failed when the cracks reach a length of *a <sup>f</sup>* =12 mm (the maximum stress intensity factors (mode I) on the crack front reach *Kc* for that crack length).

All the fatigue crack growth simulations have been performed with Morfeo, developed by Cenaero [33,34]. In the software, the propagation is driven by a user defined crack propagation step Δ*a*. The software then computes the corresponding Δ*N*, number of cycles required to propagate the crack by Δ*a*. One should note that Δ*a* has to be set to a value ensuring that the propagation path is properly computed, and that the time integration yielding the Δ*N* is correctly evaluated. Since the software is based on the XFEM method, the crack position in the mesh is spotted by level sets, which are at each step updated based on propagation length and direction [33–35]. All the meshes used in this section have been generated using the open source pre- and post-processor "Gmsh", developed by Geuzaine and Remacle [36]. They were obtained after several refinement steps, ensuring the use of a converged mesh. The mesh was highly refined in the crack region, and particularly in the initial defect region (see Figure 7), to ensure for converged stress intensity factors on the complete crack path. Depending on the configuration, the mesh totalizes between 850 k and 930 k linear tetrahedron elements.

**Figure 7.** Mesh used in the fatigue crack propagation simulations. (**a**): overview. (**b**): highly refined region around the crack.
