## 2 The Fatigue Theory and Assumptions for AM Materials  
The fundamental model of mechanical fatigue was initially developed by [PERSON] and [PERSON] based on the formation of surface extrusions and intrusions via dislocation dipole pileups [8]. The model has been further revised by [PERSON] [9; 10] using the true strain definition, and the fatigue crack nucleation life is given by:  
\[N_{c}=\frac{8(1-v)w_{s}}{\mu b}\Delta\gamma^{-2} \tag{1}\]  
\[N_{c}=\frac{2\mu w_{s}}{(1-v)b}(\Delta\tau-2k)^{-2} \tag{2}\]  
where \(w_{s}\) is the surface energy (J/m\({}^{2}\)), \(\mu\) is the shear modulus (N/m\({}^{2}\)), \(v\) is the Poisson's ratio, \(b\) is the Burgers vector (m), \(k\) is the lattice friction stress (N/m\({}^{2}\)), \(\Delta\gamma\) is the cyclic plastic shear strain range, and \(\Delta\tau\) is the cyclic shear stress range. Equation (1) is applicable to fatigue crack nucleation life with appreciable plastic strain, typically for LCF, which has been validated on a number of conventional metals and alloys [9]. Equation (2) is applicable to stress-controlled fatigue crack nucleation where plastic strain is negligible, typically in the HCF regime. Therefore, the model in the context of Equations (1) and (2) is called the Tanaka-Mura-Wu (TMW) model. An extensive application of the TMW model over the full range of fatigue life (LCF + HCF) for commonly used engineering alloys has been documented elsewhere [11].  
With regards to AM materials, it has been recognized that the effects of intrinsic AM defects, both surface and internal, on fatigue performance are the most critical properties of the AM metal components. The morphology of various internal defects such as LOF defects, keyhole or gas porosities, and un-melted particles have been characterized in the review articles [1; 2; 3; 4; 5], which will not be discussed in this paper. Considering the characteristics of AM materials with the microstructure and surface roughness profile, as schematically shown in Figures 1 and 2, two effect factors are introduced into Equations (1) and (2) such that:  
\[N_{c}=\frac{8(1-v)w_{s}R_{s}}{3\mu b}\frac{1}{\left(\text{M}\Delta\varepsilon _{p}\right)^{2}} \tag{3}\]  
\[N_{c}=\frac{6\mu w_{s}R_{s}}{(1-v)b}\frac{1}{\left(\Delta\sigma-\frac{2\sigma _{0}}{M}\right)^{2}} \tag{4}\]Here, the surface roughness factor, \(R_{s}\), and microstructure factor, \(M\), are introduced with the following assumptions:  
1. As the applied stress is uniform within the material coupon, in the continuum sense, the presence of LOF defects and pores raise the local stress concentration such that it effectively reduces the microstructural resistance to fatigue by a factor \(M\), in accordance with the Murakami-Endo model [12], as opposed to \(M=1\) for the baseline (wrought) microstructure. Note that \(\sigma_{0}\) is the fatigue limit for the baseline material.
2. The surface roughness may either facilitate the formation of extrusions and intrusions directly or cause local stress concentration at the surface, so the effect can be represented by a surface finish factor \(R_{s}\). Note that \(R_{s}=1\) for idealized smooth surface. The Taylor factor is used to convert stress/strain from shear to normal components as: \(\gamma=\sqrt{3}\epsilon\) and \(\sigma=\sqrt{3}\tau\). Equation (3) applies above the material's yield strength and Equation (4) applies below the material's yield strength.