# 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
\[\Delta\sigma_{Oro}=\frac{0.81M_{T}Gbln\frac{2r_{s}}{b}}{4\pi r_{s}\left[\left( \frac{\pi}{4b}\right)^{0.5}-1\right]\left(1-v\right)^{0.5}} \tag{10}\]  
\[\Delta\sigma_{\beta}=min\left(\Omega_{\beta,\text{\tiny{Coo}}}\mid\kappa_{Partio }\Omega_{\beta,\text{sh}}\right) \tag{11}\]  
\[\Delta_{\beta,\text{\tiny{Coo}}}=\left(\Phi\right)^{0.5}\frac{0.81M_{T}Gbln \frac{2r_{s}}{b}}{4\pi r_{s}\left[\left(\frac{\pi}{4b}\right)^{0.5}-1\right] \left(1-v\right)^{0.5}}\Delta\sigma_{\beta,\text{sh}}=\left(\Phi\right)^{0.5} \frac{2M_{T}S}{br\left(\pi r\omega_{q}\right)^{0.5}}\left(\frac{\pi r\omega_{ r}Y}{S}\right)^{0.5} \tag{12}\]  
where \(r_{s}=\left(\frac{2}{3}\right)^{0.5}r,M_{T}\) is the Taylor factor, \(G\) is the matrix shear modulus, \(b\) is the magnitude of the dislocations of [PERSON]'s vector, \(v\) is Poisson's ratio, \(\omega\), \(q\), and \(r\) are constants derived from particle statistics, \(S\) is the dislocation line tension given by \(S=Gb^{2}/2\), and \(r\) and \(\Phi\) are the particle radius and volume fraction, respectively. This was used to determine the strength contribution of NiAl precipitates in a 9922 alloy (maraging steel) undergoing shearing and Orowan looping. The precipitate sizes were normalized with a standard deviation of one under the assumption that the overall strength is a combination of the two mechanisms. In addition to precipitate strengthening, the increase in the strength of maraging steels involves contributions from the lath structure (\(\sigma_{l}\)), the intrinsic strength of iron (\(\sigma_{Fe}\)), and Solid Solution strengthening (SS). As a limitation of most models, the contribution of the SS is not well defined. Various researchers have identified different relationships between the composition (\(\chi\)) of solute elements and the resulting strengthening in maraging steels. These relationships often exhibit varying values of the composition exponent, with commonly used values being 1 or 1.5. Additionally, different proportionality constants are used [104; 105]. The strengthening varies in a complex way with changes in the elemental concentrations. The contribution of these factors is represented in Equations (13) and (14). The hardness of the lathe structure and Fe matrix is also included. \[\Delta\sigma_{ss}(HV)=14.5x_{Al}^{\prime}+14.5x_{Mo}^{\prime}-4.5x_{Ni}^{\prime }+14.5x_{w}^{\prime} \tag{13}\]  
\[\sigma_{p}=\frac{1}{b}\frac{2S}{\sqrt{\pi}w_{q}}\Big{(}\frac{\gamma\ w_{r}r_{ p}}{S}\Big{)}\frac{\sqrt[3]{f}}{r_{p}} \tag{6}\]0  
For the 992 alloy, the size and distribution of precipitates was determined using Atomic Probe Topography (APT) and Small-Angle Neutron Scattering (SANS), and the overall strength can be represented as shown in Equation (15). \[\sigma_{p}=\frac{1}{b}\frac{2S}{\sqrt{\pi}w_{q}}\Big{(}\frac{\gamma\ w_{r}r_{ p}}{S}\Big{)}\frac{\sqrt[3]{f}}{r_{p}} \tag{6}\]1  
\(\kappa\mathrm{L}\), \(\kappa\beta\), and \(\kappa\beta\)ratio are fitting variables. The results of this model (Figure 14) showed that the strengthening via the Orowan model decreases with extended aging time because of precipitate coarsening. A noticeable jump in the NiAl precipitate was observed based on the SANS measurements. The estimated strength contribution of the Laves phase was difficult to determine as evidenced by the small-sized precipitates reporting a similar hardness with the precipitates with a large size fraction. Solid solution strengthening decreased with the aging time because of the diffusion of strengthening elements towards the Laves phases. This suggests that the growth of the Laves phases is faster at higher temperatures, and hence, the contribution of the solid solution to the overall hardness is expected to diminish at higher temperatures. In contrast, at low aging times, the NiAl precipitates, along with other intrinsic strengthening contributions such as solid solution, lath structure, and matrix effects, play a dominant role. However, at extended aging times exceeding 50 h, the dominance shifts to the formation of Laves phases.