# Design and Assessment of an Austenitic Stainless Alloy for Laser Powder Bed Additive Manufacturing
## 2 Materials and Methods
### Criteria for Alloy Design
#### 2.1.4 Physical and Mechanical Properties  
* Maximization of the solid solution hardening effect For crystalline materials, yield stress can be increased by mechanisms such as the grain size effect (Hall-Petch), precipitation hardening, strain hardening or solid solution hardening. In this particular study, we have ignored the effect of precipitation because we focus on the design of single-phase alloys. The influence of grain size was also omitted from the optimization criteria, as it is assumed to be mostly process dependent. Strain hardening is also ignored as we primarily focus on AM--i.e., non-deformed alloys. A remaining possible level of action to increase alloy strength is to maximize solid solution hardening. [PERSON] has suggested that the solid solution hardening can be represented as follows [56]: \[\Delta\sigma_{SSH}=\sum_{i}B_{i}X_{i}^{\frac{2}{3}}\] (4) where \(\text{X}_{i}\) is the atomic fraction of component i and \(\text{B}_{i}\) is a constant defined by the following [12; 57]:
* \(\text{B}_{i}=3\mu\epsilon_{i}{}^{(4/3)}\text{Z}\), with \(\mu\) the shear modulus, \(\epsilon_{i}\) the mismatch parameter and \(Z\) a fitting constant;
* \(\epsilon_{i}=(\text{n}_{i}{}^{2}+\alpha^{2}.\text{s}_{i}{}^{2})^{0.5}\), with \(\text{n}_{i}\) the term of the elastic misfit. Using [PERSON]'s law [58], for Fe-based alloys, one can estimate the atomic size misfit \(\delta_{i}\) by the following: \[\delta_{i}=\frac{dr}{rdX_{i}}=\frac{r_{i}-r_{Fe}}{(1-X_{i})r_{Fe}+X_{i}\cdot r _{i}}\] (5) with \(r_{i}\) representing the atomic radius of the solute and \(r_{Fe}\) the atomic radius of the iron solvent. As for the term of the elastic misfit, in general, the variation of shear modulus with composition is assumed to be linear [59] and so the parameter \(\text{n}_{i}\) can be estimated as follows:
\[\eta_{i}=2\frac{\mu_{i}-\mu_{Fe}}{\mu_{i}+\mu_{Fe}} \tag{6}\]  
where \(\mu_{Fe}\) and \(\mu_{i}\) are the shear moduli of solvent Fe and solute i, respectively. Finally, by considering a value of \(\alpha=16\) which is typically used for FCC alloys to account for the interaction of solute atoms with edge dislocations [60], is it possible to calculate a solid solution strengthening index according to Equations (4)-(6). For alloy design, the solid solution index \(\Delta\sigma_{SSH}\) is an objective to be maximized.  
* Minimization of residual stress and distortions  
The addition and melting of layers in LPBF directly affect residual stresses and distortions in the built component. These issues in AM components result from the spatial temperature gradient due to localized heating and cooling, leading to uneven strain distribution [61; 62]. Assessing the risk of distortion and residual stresses is a very complex task but, for a given thermal history (as in thermal shock [63]), the extent of thermal strains scales, overall, with the coefficient of thermal expansion, \(\alpha_{\text{CTE}}\). The stresses are then proportional to a product of strains and elastic constants. For different alloy elements \(i\), with element atomic fraction \(X_{i}\), \(\alpha_{CTE}\) can be estimated using a simple rule of mixtures [64], where \(\alpha_{CTEi}\) represents the coefficient of thermal expansion for pure element \(i\), as follows:  
\[\alpha_{CTE}=\sum_{i}X_{i}\alpha_{CTEi} \tag{7}\]  
Elastic coefficients usually exhibit a limited variation, especially for austenitic stainless alloys (e.g., steels). Therefore, for alloy design it is here proposed to minimize \(\alpha_{CTE}\) and to ignore any possible variation in elastic properties.