# 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  
The presence of precipitates hinders dislocation gliding. They act as obstacles, which reduce their mobility, hence reducing slip and grain boundary sliding. Based on the size of the precipitate and its coherency inside, the matrix dislocations have two main ways of passing these obstacles, viz. bowing (Orowan-Ashby model) [98] and shearing (Friedel model) [99] the precipitates, Equations (5) and (6), respectively. \[\sigma_{p}=0.26\frac{\mu b}{r_{p}}f^{\frac{1}{2}ln}\Big{(}\frac{r_{p}}{b}\Big{)} \tag{5}\]  
\[\sigma_{p}=\frac{1}{b}\frac{2S}{\sqrt{\pi}w_{q}}\Big{(}\frac{\gamma\ w_{r}r_{ p}}{S}\Big{)}\frac{\sqrt[3]{f}}{r_{p}} \tag{6}\]  
where \(r_{p}\) is the particle radius (this model assumes that particles are spherical), \(S\) is the dislocation line tension (\(\mu b^{2}/2\)), \(w_{r}\) and \(w_{q}\) are particle statistical constants [100], \(\gamma\) is the antiphase boundary energy associated with the sheared precipitates, and \(f\) is the fractional volume of precipitates. In the case of maraging steels, the pile-up model by [PERSON] and [PERSON] assumes that yielding will occur after a critical number of dislocations' pile up exceeding the maximum stress [101] (Equation (7)). The Orowan-Ashby model and the modified Orowan-Ashby model [102] (Equation (8)) are considered to be the most-relevant, according to [PERSON] et al. [95]. In contrast to [PERSON] et al. [97], the Orowan and the Friedel  
Figure 13: Sub-structure sketch of a typical martensite grain. Reprinted with permission from [92]. Copyright 2016 Acta Materialia Inc.  
models had good fitting results with experimental results based on a PH 13-8 Mo stainless steel under aging and overaging conditions. \[\sigma_{p}=2\frac{\mu t}{4C}\left(\frac{f^{\frac{1}{3}}}{0.82-f^{\frac{1}{3}}}\right) \tag{7}\]  
\[\sigma_{p}=\frac{2\mu b\phi ln\left(\frac{d-2r_{p}}{2b}\right)}{4\pi\left(d-r_{ p}\right)} \tag{8}\]  
where \(\mu^{\prime}\) is the shear modulus of the precipitates, \(C\) is a constant, \(d\) is the inter-particle spacing, and \(\Phi\) is related to the Poisson ratio (\(\
u\)) by the following relationship: \(\Phi=[1+1/(1-\
u)]/2\). The previously mentioned scenarios assume that only one form of precipitate exists, which is not the case as purported by [PERSON] et al. [92]. Ni-Ti-rich clusters were associated with Mo-rich phases. According to [PERSON] [102], the total precipitate strength is the summation of each particle's contribution, represented by Equation (9). \[\sigma_{p}=\left(\sum_{j}\sigma_{j}^{2}\right)^{\frac{1}{2}} \tag{9}\]  
where \(\sigma_{j}\) is the strength increase due to precipitate \(j\) with \(f\) and \(r\), following Equations (5) to (8). There are limitations in using these precipitation models (Equations (2)-(9)) [103], as their exact contribution of precipitation to strength is difficult to determine because their exact interplanar spacing is not accounted for from the volume and fraction sizes and the lattice misfits between the matrix and precipitates. For example, it was shown that the Orowan model outputs different magnitudes with a factor of 10 in some instances based on the type of distributions used [95]. In the case where several mechanisms are expected to proceed with different precipitates sizes, [PERSON] et al [101] suggested that no single model can account for the peak hardness. Hence, based on the precipitate sizes, strength can be estimated by the shearing model below a critical size value and the Orowan model above this value. The previously mentioned Ansell and Lenel model can be used to describe the contribution of fine precipitates; however, the model is appropriate for a constant volume fraction. To accommodate varying precipitate sizes during different aging times, a new formulation (Equations (10)-(12)) is better suited.