## 3 Analytical Description of Temporary Temperature Distribution  
In the solution allowing for the calculation of the temperature field, a heat source based on a two-distribution model was adopted. From the total amount of heat of the electric arc, a part of the heat needed to melt the electrode was separated and transferred by molten material to the weld pool. The remaining heat part emitted by the electric arc was considered to be directly affecting the welded object:  
\[\theta(x,y,z,\tau)-\theta_{i}=\theta_{a}(x,y,z,\tau)+\theta_{w}(x,y,z,\tau), \tag{1}\]  
where: \(\theta_{a}(x,y,z,\tau)\)--the rise of temperature caused by heat input of the electric arc; \(\theta_{w}(x,y,z,\tau)\)--the rise of temperature as a result of the heat transferred from the liquid electrode metal to the weld pool; \(\theta_{i}\)--initial temperature of the overlaid object; and \(\tau\)--the time of the process. The scheme of the applied padding segment with its characteristic dimensions is shown in Figure 2. In the considered case of flat surface welding overlaying, \(d_{p}\) is equal to 0. The volume of the deposited padding material is determined by the weld height (\(h_{w}\)) and its width (\(w_{w}\)), and \(vd\tau\) results from the padding volume equal to the volume of molten electrode material determined by its diameter and wire feed speed. Equations describing the temporary temperature field, including the molten metal heat transferred from the electrode to the overlaid weld, are deduced in [42]. The heat source model with Gaussian power distribution is shown in Figure 3.  
The solution of the differential heat conduction equation for the volume heat source determined by such a distribution is described in [43]. In this discussion, this model was adopted to represent the impact of the electric arc. Figure 1: Metallographic specimen in the middle cross section. Figure 2: Padding weld geometry. The temperature (1) during single-pass arc overlaying welding (Figure 4) in time \(\tau\leq\tau_{t}\), where \(\tau_{t}\) is defined by:  
\[\tau_{t}=l/v, \tag{2}\]  
is determined by the equation:  
\[ \begin{array}{l}\theta(x,y,z,z_{0},\tau)-\theta_{0}=\\ =A_{w}\int\limits_{0}^{\tau_{t}}[\iota_{H}\{\tau^{\prime\prime}\}[-\Phi_{1}[y, z]+\Phi_{2}[y,z]+\Phi_{3}[y,z]-\Phi_{4}[y,z]]]d\tau^{\prime\prime}+A_{H}\int \limits_{0}^{\tau}\Phi_{H}(\tau^{\prime\prime})d\tau^{\prime\prime}\end{array}  \tag{3}\]  
on the other hand, after the bead ends (for time \(\tau>\tau_{t}\)), the equation takes the form:  
\[ \begin{array}{l}\theta(x,y,z,z_{0},\tau)-\theta_{0}=\\ =A_{w}\int\limits_{0}^{\tau_{t}}[\iota_{C}\{\tau\}[-\Gamma_{1}(y,z)+\Gamma_{2} [y,z]+\Gamma_{3}[y,z]-\Gamma_{4}(y,z)]]+A_{C}\int\limits_{0}^{\tau_{t}}\Phi_{C }(\tau)d\tau,\end{array}  \tag{4}\]  
where \(A_{w}\), \(A_{C}\), \(H_{h}\), \(H_{c}\) are defined by the following equations:  
\[A_{w}=\dot{q}_{v}/8C_{p}\rho(\pi a)^{1.5}, \tag{5}\]  
\[A_{H}=\frac{3\dot{q}}{16C_{p}\rho(\pi a)^{1.5}z_{0}}exp\Bigg{(}-\frac{\xi v}{2 a}-\frac{v^{2}\tau_{0}}{4a}\Bigg{)}, \tag{6}\]  
\[A_{C}=3\dot{q}/\Big{(}16C_{p}\rho\pi a_{2}\Big{)}, \tag{7}\]  
\[\begin{array}{l}F_{H}(\tau^{\prime\prime})=\frac{1}{\tau^{\prime\prime}+ \tau_{0}}exp\Bigg{(}-\frac{\xi^{2}}{4a(\tau^{\prime\prime}+\tau_{0})}-\frac{v^ {2}\tau^{\prime\prime}}{4a}\Bigg{)}\Bigg{(}\Big{(}1-\frac{z^{2}  
\[F_{\rm C}(\tau\
u)=\frac{1}{\tau+\tau_{0}-\tau}exp\bigg{(}-\frac{(x- \
u\tau-x_{0})^{2}+(y-y_{0})^{2}}{4a(\tau+\tau_{0}-\tau)}\bigg{)}\] \[\bigg{\{}\bigg{(}1-\frac{z^{2}+2a(\tau-\tau)}{z_{0}^{2}}\bigg{)} \bigg{(}er\bigg{(}\frac{z+z_{0}}{\sqrt{a_{t}(\tau-\tau)}}\bigg{)}-\Omega(z)er \bigg{(}\Omega(z)\frac{z-z_{0}}{\sqrt{(\tau-\tau)}}\bigg{)}\bigg{)}+ \tag{11}\] \[\frac{4a(\tau-\tau)}{z_{0}^{2}}\bigg{(}\frac{z+z_{0}}{2(\pi(\tau- \tau))^{0.5}}exp\bigg{(}-\frac{(z-z_{0})^{2}}{4a(\tau-\tau)^{2}}-\frac{z-z_{0} }{2(\pi(\tau-\tau))^{0.5}}exp\bigg{(}-\frac{(z+z_{0})^{2}}{4a(\tau-\tau)} \bigg{)}\bigg{)}\bigg{)}\] \[\xi=x-v(\tau+\tau_{0})-x,\] (12) \[\Omega(z)=\bigg{\{}