# Fatigue Crack Growth from Notches:
## 3 Numerical Model  
The numerical simulations were carried out with the in-house static implicit finite element code DD3 IMP. FCG was studied in single-edge notch tension specimens, Figure 2a, in which the notch depth was kept constant and equal to 8 mm. A crack was placed 96 \(\upmu\)m ahead of the notch, leading to an initial crack length, \(a_{0}\), equal to 8.096 mm. Different notch radii, \(r\), equal to 8, 4, 2, and 1 mm were modeled, as shown in Figure 2c-f, respectively. As Figure 2 indicates, only \(\frac{1}{4}\) of the specimen was modeled considering adequate boundary conditions. A small thickness of 0.1 mm was considered in the simulations, in order to reduce the numerical effort. Both plane stress and plane strain states were  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \(E\) & \(\mathbf{\
u}\) & \(\mathbf{Y_{0}}\) & \(\mathbf{Y_{Sat}}\) & \(\mathbf{C_{Y}}\) & \(\mathbf{X_{Sat}}\) & \(\mathbf{C_{X}}\) \\  & **(GPa)** & **(\(\mathbf{\
u}\))** & **(MPa)** & **(MPa)** & **(-)** & **(MPa)** & **(\(\mathbf{\
u}\))** \\ \hline AA6082-T6 [24] & 70 & 0.29 & 238.15 & 487.52 & 0.01 & 83.18 & 244.44 \\ AA7050-T6 kinematic [25] & 71.7 & 0.33 & 420.50 & 420.50 & 0 & 198.35 & 228.91 \\ AA7050-T6 isotropic & 71.7 & 0.33 & 420.50 & 420.50 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Set of elastic-plastic properties.  
Figure 1: Schematic view of the study.  
studied, applying proper boundary conditions, as indicated in Figure 2g,h, respectively. In the case of plane strain, an additional boundary condition is necessary to eliminate out-of-plane deformation.  
Figure 2: Overview of the geometry, finite element mesh, and boundary conditions of the notched specimens. (**a**) Loading, boundary conditions, and main dimensions, in mm. (**b**) Detail of refined mesh. (**c**) \(\rho\) = 8 mm. (**d**) \(\rho\) = 4 mm. (**e**) \(\rho\) = 2 mm. (**f**) \(\rho\) = 1 mm. (**g**) Plane stress boundary. (**h**) Plane strain boundary conditions.  
A cyclic load was applied remotely and normal to the crack growth, as can be seen in Figure 2a, resulting in mode I loading. The maximum and minimum applied loads were \(F_{\mathrm{max}}\) = 400 N and \(F_{\mathrm{min}}\) = 4 N, leading to a stress ratio \(R\) = 0.01 and to a maximum remote stress of 80 MPa.  
A refined mesh was placed at the crack tip region (Figure 2b), with \(8\times 8\) um\({}^{2}\) elements, to accurately quantify stress and strain gradients. Out of the crack tip region, a coarser mesh was introduced to reduce the computational cost. The finite element mesh comprised 7175 3D linear isoparametric elements and 7359 nodes. Two load cycles were applied between crack propagations, each one conducting to a crack increment of 8 um that took place at minimum load. The simulations stopped when the total crack propagation reached 1272 um, i.e., after 159 crack increments. The contact of the crack flanks was modeled considering a rigid flat surface aligned with the crack symmetry plane. Some simulations were conducted with the possibility to overlap the crack flanks, i.e., for these simulations the flat surface was removed enabling the interference of the crack flanks and disabling crack closure.  
To study the effect of the notch on FCG, a comparison was made with an unnotched-cracked specimen. This unnotched model was created by removing the notch in the model presented in Figure 2a, maintaining the crack with a\({}_{0}\) equal to 8.096 mm.  
The radii considered for the notch in the present work (1-8 mm) are within the values proposed in the literature. [PERSON] et al. [32] considered radii of 0.4, 1 and 2 mm, while [PERSON] et al. [33] studied strain ratcheting at notch roots with radius of 1.5, 4.5, and 7.5 mm. [PERSON] et al. [4] considered radius of 3 mm in AA2024-T351 and of 1 mm in AA7749-T7951. [PERSON] et al. [12] considered disk-shaped compact specimens with radii of 0.10, 0.8, and 3.18 mm. The onset of fatigue crack growth is usually defined for a crack length of 0.2-0.25 mm.