# Effect of Process Parameters on Distortions Based on the Quantitative Model in the SLM Process
## 3 Results  
### Mechanism of Distortion  
Observing the stress field on the twenty-layer thin wall (as shown in Figure 3), the longitudinal residual stress is significantly reduced, and distortion is formed after removing the substrate. This means that residual stresses influence the distortion before the substrate is removed. Therefore, an analytical model is established to investigate the distribution and evolution of residual stresses in the thin wall to study the distortion mechanism. A thin wall has n layers with a thickness of h, width of w, and length of L. During the SLM process, the newly added layer is deposited on the previous layer or substrate. The newly added layer shrinks due to the cooling effect, which drives the deposited layers and substrate to shrink (as shown in Figure 4). It is assumed that the newly added layer has the same displacement as the structure below. The plastic strain in the substrate is neglected since the plastic area is relatively small in the substrate. Since the distortion is mainly affected by the residual stress in the scanning direction [14], we only discuss stresses and strains in the longitudinal direction. The manufacturing process is divided into the following steps (as shown in Figure 4) to calculate the residual stress on each layer. Figure 3: The longitudinal residual stress distribution before and after removing the substrate. Figure 2: The schematic diagram of scanning strategies: Case (**a**) is the unidirectional scanning; Case (**b**) has 90\({}^{\circ}\) rotation angles; Case (**c**) is the 45\({}^{\circ}\) inclined scanning with 90\({}^{\circ}\) rotation angles; Case (**d**) has 45\({}^{\circ}\) rotation angles; Case (**e**) is the out–in scanning; Case (**f**) is the in–out scanning. (1) Step 1. When the first layer is deposited and cooled down, the residual stress in the substrate \(\sigma_{\text{s}}^{(1)}\) can be expressed as:  
\[\sigma_{\text{s}}^{(1)}=E_{s}\cdot\varepsilon_{\text{s\_r}}^{(1)}=E_{s}\cdot \left(\frac{u^{(1)}}{L}-\varepsilon_{\text{s\_th}}^{(1)}\right) \tag{3}\]  
where \(E_{s}\) is the elastic modulus of the substrate, \(\varepsilon_{\text{s\_r}}^{(1)}\) is the elastic strain of the substrate, \(u^{(1)}\) is the displacement of the component, and \(\varepsilon_{\text{s\_th}}^{(1)}\) is the thermal strain of the substrate. The residual stress in the first layer \(\sigma_{f1}^{(1)}\) is defined as:  
\[\sigma_{f1}^{(1)}=E_{f}\cdot\varepsilon_{f\_01}^{(1)}=E_{f}\cdot\left(\frac{u ^{(1)}}{L}-\varepsilon_{f\_01}^{(1)}-\varepsilon_{f\_01}^{(1)}\right) \tag{4}\]  
where \(E_{f}\) is the elastic modulus of the deposited material, and \(\varepsilon_{f\_01}^{(1)}\), \(\varepsilon_{f\_01}^{(1)}\), and \(\varepsilon_{f\_01}^{(1)}\) are the elastic strain, the plastic strain, and the thermal strain of the deposited material, respectively. The force balance in the longitudinal direction can be expressed as:  
\[\sigma_{\text{s}}^{(1)}\cdot A_{s}+\sigma_{f1}^{(1)}\cdot A_{f1}=0 \tag{5}\]  
where \(A_{s}\) is the equivalent sectional area of the substrate, and \(A_{f1}\) is the sectional area of the first layer. Therefore, the displacement of the first layer \(u^{(1)}\) can be expressed as:  
\[u^{(1)}=\frac{E_{s}\cdot A_{s}\cdot\varepsilon_{\text{s\_th}}^{(1)}+E_{f} \cdot A_{f1}\cdot\left(\varepsilon_{f\_01}^{(1)}+\varepsilon_{f\_01}^{(1)} \right)}{E_{s}\cdot A_{s}+E_{f}\cdot A_{f1}} \tag{6}\]  
(2) Step 2. When the second layer is added, residual stresses in the substrate, the first layer, and the second layer can be expressed as:  
\[\sigma_{\text{s}}^{(2)}=E_{s}\cdot\varepsilon_{\text{s\_r}}^{(2)}=E_{s}\cdot \left(\frac{u^{(1)}+u^{(2)}}{L}-\varepsilon_{\text{s\_th}}^{(2)}\right) \tag{7}\]  
Figure 4: The schematic diagram of the SLM process.