# Fatigue Crack Growth in Managing Steel Obtained by Selective Laser Melting
## 3 FCG Analysis Based on Plastic CTOD Range
### Numerical Determination of \(\delta_{p}\)  
The C(T) specimen (Figure 1) presents geometric, material and loading symmetries; therefore, 1/4 of the specimen was modeled numerically considering adequate boundary conditions, as indicated in Figure 5a,b. A pure plane stress state was simulated by assuming a small thickness equal to 0.1 mm. Several initial crack lengths a\({}_{0}\) were considered, equal to 7 mm, 10 mm, 13 mm, 16 mm, 19 mm, 22 mm and 24 mm, to replicate a wide range of experimental crack lengths. The simulations were done under load control, similar to the experimental tests. The maximum and minimum values of the remote load were defined considering the experimental loads listed in Table 2.  
The finite element mesh (Figure 5a,c) comprised 7142 linear isoparametric elements and 14,606 nodes, with two main regions: (i) An ultra-refined mesh near the crack tip, composed of elements with 8\(\times\)8 \(\upmu\)m side; and (ii) a coarser mesh in the remaining specimen, to reduce the computational overhead. Only one layer of elements' through-thickness was used. The crack propagation occurs at the minimum load, by successive debonding of both crack front nodes over the thickness. A total of 159 crack propagations were modeled, each propagation corresponding to the size of one finite element (= 8 \(\upmu\)m), with a total crack advance (\(\Delta\)a) of 1272 \(\upmu\)m (i.e., \(\Delta\)a = \((160-1)\times 8\)\(\upmu\)m). Five load cycles were applied between each crack increment.  
The numerical simulations were performed using the DD3 IMP (Deep-Drawing 3D IMPlicit) in-house code, originally developed to model deep-drawing processes [24]. The evolution of the deformation is modeled by an updated Lagrangian scheme, assuming a hypoelastic-plastic material model. The material plastic behavior was modeled considering the set of elastic-plastic parameters  
Figure 5: Model of the C(T) specimen. (**a**) Load and boundary conditions. (**b**) Boundary conditions for plane stress state. (**c**) Detail of finite element mesh.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{3}{c}{**Hooke\({}^{\prime}\)s Law**} & \multicolumn{3}{c}{**Isotropic Hardening (Voee)**} & \multicolumn{3}{c}{**Kinematic Hardening**} \\  & \multicolumn{3}{c}{**Parameters**} & & & & **(Armstrong-Frederick)** \\ \cline{2-7}  & \(E\) [GPa] & \(N\) [-] & _Y_\({}_{0}\) [MPa] & _Y_\({}_{\text{Sat}}\) [MPa] & _C_\({}_{\text{Y}}\) [-] & _C_\({}_{\text{X}}\) [-] & _X_\({}_{\text{Sat}}\) [MPa] \\ \hline
18 Ni300 & 160 & 0.30 & 683.62 & 683.62 & 0 & 728.34 & 402.06 \\ AA7050-T6 [15] & 71.7 & 0.33 & 420.50 & 420.50 & 0 & 228.91 & 198.35 \\ AA2050-T8 [20] & 77.4 & 0.30 & 383.85 & 383.85 & 0 & 97.38 & 265.41 \\ AA6028-T6 [21] & 70 & 0.29 & 238.15 & 487.52 & 0.01 & 244.44 & 83.18 \\ AA6016-T4 [22] & 70 & 0.29 & 124.00 & 415.00 & 9.5 & 146.50 & 34.90 \\
304 LS SS [23] & 196 & 0.3 & 117 & 87 & 9 & 300 & 176 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Elastic-plastic parameters (AA—Aluminum Alloy; SS—Stainless Steel).  
shown in Table 3. The contact between crack flanks is modeled considering a rigid plane surface aligned with the crack symmetry plane. A master-slave algorithm is used; an augmented Lagrangian approach is used for the contact problem treatment.