# Recent Progress in Hybrid Additive Manufacturing of Metallic Materials
## 5 Effect of Heat Treatments in AM Processed Maraging Steels
### Strengthening Mechanisms' Models
#### 5.1.3 Precipitation Strengthening
Some discrepancies can be observed for this model as it shows that the peak strength at 560 \({}^{\circ}\)C is lower (475 HV) than the experimental value (520 HV) at a total aging time of about 50 h. Conversely, the predicted hardness is higher than the actual value at 520 \({}^{\circ}\)C. The model failed to predict the complete behavior, potentially due to errors from the acquired SANS data. phases, Reference [106] used APT to calculate the spacings between the precipitates and to determine the contribution of the \(\beta\) and \(\eta\) strengthening phases in Fe-12 Ni. In this study, two models were used. Model 1 included the contribution of the grain size, dislocation density, and solid solution strengthening. This was later used to validate the results from Model 2, which incorporates the mean free radius and distance and the surface-to-surface distance of precipitates as measured from the APT data. It was found that the precipitation strengthening of the combined \(\beta\)and \(\eta\) phases was close to 1000 MPa at an aging temperature of 510 \({}^{\circ}\)C for 16 h and that the relative strengthening effect of each phase was sensitive to the Ti and Al concentration. Low titanium concentrations (approximately 1 wt.%) and high Al concentrations (approximately 1.6 wt.%) in the \(\beta\) phase contributed more to the overall strength compared to high Ti concentrations (approximately 1.5 wt.%) and Al concentrations (0.9 wt.%). In the case of more complex microstructures, such as those observed in LPBF-Corrax [107], precipitation strengthening was found to be the main contributor to strength via the shearing mechanism. Three factors can be considered to contribute to the overall strength according to Equation (16) [108]. \[\sigma_{p}=\frac{1}{b}\frac{2S}{\sqrt{\pi}w_{q}}\Big{(}\frac{\gamma\ w_{r}r_{ p}}{S}\Big{)}\frac{\sqrt[3]{f}}{r_{p}} \tag{6}\]2  
where \(\Delta\sigma_{o}\) is the order strengthening, \(\Delta\sigma_{c}\) is the coherency strengthening, and \(\Delta\sigma_{m}\) is the modulus mismatch strengthening. Order strengthening arises from the antiphase boundaries during dislocation shear through the ordered \(\beta\)-NiAl, and it is expressed as:  
\[\sigma_{p}=\frac{1}{b}\frac{2S}{\sqrt{\pi}w_{q}}\Big{(}\frac{\gamma\ w_{r}r_{ p}}{S}\Big{)}\frac{\sqrt[3]{f}}{r_{p}} \tag{6}\]3  
where \(\Upsilon_{\text{APB}}\) is the antiphase boundary energy of the precipitates. \(M\) is the Taylor factor, and f is the volume fraction of precipitates. Coherency strengthening, which arises from the mismatch strain between the coherent precipitates and the matrix, is given by:  
\[\sigma_{p}=\frac{1}{b}\frac{2S}{\sqrt{\pi}w_{q}}\Big{(}\frac{\gamma\ w_{r}r_{ p}}{S}\Big{)}\frac{\sqrt[3]{f}}{r_{p}} \tag{6}\]4  
where \(r\) is the precipitate mean radius and \(\epsilon\) is the lattice mismatch parameter. Lastly, the modulus mismatch strengthening arises from the variation in the dislocation energy as dislocations approach a precipitate with a different shear modulus to the matrix:  
\[\sigma_{p}=\frac{1}{b}\frac{2S}{\sqrt{\pi}w_{q}}\Big{(}\frac{\gamma\ w_{r}r_{ p}}{S}\Big{)}\frac{\sqrt[3]{f}}{r_{p}} \tag{6}\]5  
where \(\Delta G\) is the shear modulus mismatch between the precipitate and matrix (\(G\) and \(\text{Gp}\), respectively) and m is a constant (0.85). The estimation of the final yield strength was estimated to be about 1530 MPa as a contribution of the hierarchical structure of the microstructure. However, based on [101; 103] previously discussed, the volume and size fractions need to be considered over the aging times for a more-accurate representation, in addition to the contribution of the high dislocation densities in AM parts. Maraging steels are highly valued for their combination of strength, toughness, ductility, and weldability. Heat-treatment procedures can be used to optimize their properties by controlling the precipitation of intermetallic phases within the low-carbon Fe-Ni/Mn martensitic matrix.