# Fatigue Crack Growth in Managing Steel Obtained by Selective Laser Melting
## 3 FCG Analysis Based on Plastic CTOD Range
### Identification of Material Parameters  
The high precision in the FE simulation results of the plastic CTOD depends on the accurate modeling of the material behavior. In this work, an elastic-plastic model was used: the isotropic elastic behavior is modeled by the generalized Hooke's law; the plastic behavior is described by the von Mises yield criterion coupled with a mixed isotropic-kinematic hardening law under an associated flow rule. The von Mises yield surface is described as follows:  
\[(\Sigma_{22}-\Sigma_{33})^{2}+(\Sigma_{33}-\Sigma_{11})^{2}+(\Sigma_{11}- \Sigma_{22})^{2}+3\Sigma_{23}^{2}+3\Sigma_{13}^{2}+3\Sigma_{12}^{2}\ =\ 2\gamma^{2} \tag{1}\]  
where \(\Sigma\) represents the effective stress tensor (\(\Sigma=\sigma^{\prime}\)- \(\mathbf{X}^{\prime}\), where \(\sigma^{\prime}\) and \(\mathbf{X}^{\prime}\) are the deviatoric components of the Cauchy stress tensor and back-stress tensor, respectively); \(Y\) is the yield stress, and its evolution during plastic deformation is modeled by the [PERSON] isotropic hardening law [17]:  
\[Y\ =\ Y_{0}+(Y_{\rm Sat}-Y_{0})[1-\exp(-C_{\gamma}\overline{z}^{\rm P})] \tag{2}\]  
Figure 3: da/dN-K\({}_{\rm max}\) curves (18 Ni300 steel; t = 6 mm).  
where \(Y_{0}\), \(Y_{\text{Sat}}\) and \(C_{\text{Y}}\) are material parameters and \(\vec{\epsilon}^{\text{P}}\) is the equivalent plastic strain. The non-linear kinematic hardening is modeled by the Armstrong-Frederick law [18], as follows:  
\[\dot{\mathbf{X}} = C_{\text{X}}\bigg{[}X_{\text{Sat}}\frac{\boldsymbol{\Sigma}}{ \sigma}-\mathbf{X}^{\prime}\bigg{]}\vec{\epsilon}^{\text{P}} \tag{3}\]  
where \(C_{\text{X}}\) and \(X_{\text{Sat}}\) are material parameters and \(\vec{\epsilon}^{\text{P}}\) is the equivalent plastic strain rate. An optimization procedure was carried out to obtain the set of material parameters that best model the cyclic plastic behavior of 18 Ni300, by minimization of the following least-squares function:  
\[F(\mathbf{A}) = \sum_{i=1}^{N}\bigg{(}\frac{\sigma^{\text{Fit}}(\mathbf{A})- \sigma^{\text{Exp}}}{\sigma^{\text{Exp}}}\bigg{)}_{i}^{2} \tag{4}\]  
where \(\sigma^{\text{Fit}}(\mathbf{A})\) and \(\sigma^{\text{Exp}}\) are, respectively, the analytically fitted and experimentally measured values of true stress at data point i (that corresponds to an equivalent plastic strain value); N is the number of experimental data points and \(\mathbf{A}\) is the set of Voce and Armstrong-Frederick parameters to be identified. The fitting procedure was carried out for 33 loading cycles (N = 6600), representing about 80% of total life, using a non-linear gradient-based optimization algorithm available in the Microsoft Excel SOLVER tool [19]. Table 3 shows the fitted material parameters that describe the elastic-plastic behavior of 18 Ni300. In this table, the Voce isotropic hardening parameters follow \(Y_{0}=Y_{\text{Sat}}\), and therefore the hardening is purely kinematic, which leads to a cyclically stable stress-strain fitting. This is acceptable in case of 18 Ni300, where mild cyclic softening occurs during the first 33 loading cycles. Accordingly, the fitted material parameters describe adequately the experimental results, as can be seen in Figure 4c. Table 3 also shows material parameters for other five metal alloys, which were characterized by the authors in previous works.  
Figure 4: (**a**) Low cycle fatigue specimens. (**b**) Stress-strain plot (\(\Delta\varepsilon=\pm\) 0.8%). (**c**) Experimental versus fitted curve (18 Ni300 steel).