## 3 Results and Discussion
### Prediction of the Stress-Strain State of the Produced Metal 3D-Objects
The boundary conditions for the system of Equation (1) are as follows:  
* movements or pinching (in at least one point on the body's surface): \[u|_{S_{u}}=0\] (3) where \(S_{u}\) is the surface (or surface point), on which movement is assigned;
* symmetry \[n\times u|_{S_{u}}=0\] (4) where \(n=n_{i}\) is the vector of the normal to the surface; \(S_{su}\) is the surface of symmetry of the body;
* from the action of an external force \[\sigma\times n|_{S_{p}}=p\] (5) where \(p=p_{i}\) is the vector of external force, acting on surface \(S_{p}\);
* from the action of an external pressure \[(\sigma\times n)\times n|_{S_{p}}=p\] (6) where \(p\) is the external pressure assigned on the surface \(S_{p}\), Pa [30]. The stress-strain state in the deposited products depends on the thermal deformation processes proceeding during deposition. Analysis of the thermal processes of arc deposition was performed using a volumetric heat source according to [PERSON] with a normal (Gaussian) distribution of the specific thermal power along all the coordinate axes within the body, having the form of a volumetric semi-ellipsoid (Figure 6) [31]. A feature of the model is the independent assignment of the distribution of specific thermal power \(q_{v}\) in the front (index \(f\)) and tail (index \(r\)) parts of the ellipsoid:  
\[q_{vf}=f_{f}\frac{6\times\sqrt{3\times q}}{a_{f}\times b\times c\times\pi^{ \frac{3}{2}}}\exp\left\{-3\left[\left(\frac{x+v\times(t-\tau)}{a_{f}}\right)^{ 2}+\left(\frac{y}{b}\right)^{2}+\left(\frac{z}{c}\right)^{2}\right]\right\} \tag{7}\]  
\[q_{vr}=f_{r}\frac{6\times\sqrt{3\times q}}{a_{r}\times b\times c\times\pi^{ \frac{3}{2}}}\exp\left\{-3\times\left[\left(\frac{x+v\times(t-\tau)}{a_{r}} \right)^{2}+\left(\frac{y}{b}\right)^{2}+\left(\frac{z}{c}\right)^{2}\right]\right\} \tag{8}\]  
\[q=\eta\times\mathrm{I}\times\mathrm{U} \tag{9}\]  
where \(q\) is the effective thermal power of the heat source; \(\eta\) is the surfacing arc efficiency; \(I\) is the deposition current; \(U\) is the deposition voltage; \(\tau\) is the dwell time calculated from the start of the source action; \(t\) is the current time; \(v\) is the speed of the source movement (deposition rate); \(x\), \(y\), and \(z\) are the ellipsoid semi-axes in the direction of coordinate axes OX, OY, and OZ;\(f_{f}\) and \(f_{r}\) are the coefficients determining the ratio of heat applied to the frontal and the tail part of the ellipsoid; and \(a_{f}\), \(a_{r}\), \(b\), and \(\mathrm{c}\) are the respective radii of normal distribution. Coefficients \(f_{f}\) and \(f_{r}\) are related as follows:  
\[f_{f}=\frac{2\times a_{f}}{a_{f}+a_{r}}\ ;\ f_{r}=\frac{2\times a_{r}}{a_{f}+a_{r}} \ ;\ f_{f}+f_{r}=2 \tag{10}\]  
In our case, it was assumed that \(f_{f}\) = 0.4; and \(f_{r}\) = 1.6. During the calibration of the [PERSON] model, the linear dimensions of the real weld pool were taken into account to select the respective dimensions of \(a_{f}\) = 3 mm, \(a_{r}\) = 5 mm, \(b\) = 1.7 mm, and \(c\) = 0.8 mm parameters of the ellipsoid (Figure 6), as required for calculation in Equations (7) and (8). Calculations performed via the method of finite element modeling were performed, taking into account the model of the source of a 3D stress-strain state (1-2) with boundary conditions (3-6) and the thermal powder distribution (7-9) for the selected coefficients (10) [32]. Computations performed via the finite element method used an irregular grid of prismatic 3D elements, the size of which was 0.3 \(\times\) 0.3 \(\times\) 0.25, 0.6 \(\times\) 0.6 \(\times\) 0.5, and 1.25 \(\times\) 1.25 \(\times\) 1.0 mm. Outside the highly heated region, the size of finite elements was increased to 2.5 \(\times\) 2.5 \(\times\) 1.0 and 5.0 \(\times\) 5.0 \(\times\) 1.0 mm to shorten the computation time. To ensure convergence of the finite element model grid, the transition between zones modeled by elements with different base sizes was provided by finite elements with rectangular-trapezoidal bases of appropriate sizes.