## 4 The Concept of the Phase Share Calculation  
In quantification, volumetric share changes of individual phases in steel during thermal welding cycle are defined by the JMAK (Johnson-Mehl-Avrami-Kolmogorov) law [46,47] in description of diffusive transformations and KM (Koistinen-Marburger) principle is used in description of non-diffusive (martensitic) transformation kinetics [48]. If, during the temperature rise in the thermal cycle, its value exceeds the critical temperature \(A_{1}\), the transformation of the initial structure into austenite begins. The austenitic transformation ends at the temperature \(A_{3}\), if the temperature peak in the thermal cycle exceeds \(A_{3}\) (full transformation). Otherwise (\(\theta_{max}<A_{3}\)) the transformation ends after reaching the maximum temperature (partial transformation). The volume fraction of austenite \(\phi_{\gamma}\) created during heating depends on the current material temperature in relation to the temperature  
Figure 4: Scheme of single-pass arc overlaying. range determined by \(A_{3}\) and \(A_{1}\). The share of new austenitic structure is calculated according with the JMAK law [46] by equation:  
\[\varphi_{\gamma}(\theta)=\sum_{i}\varphi_{j}^{0}\Big{(}1-exp\Big{(}-b_{i}(\theta )\tau^{n_{i}(\theta)}\Big{)}\Big{)}, \tag{16}\]  
where \(\varphi_{i}^{0}\) denote initial volume shares of bainite (\(i\cong B\)), ferrite (\(i\cong F\)) and pearlite (\(i\cong P\)). The values of \(b_{i}\) and \(n_{i}\) are specified depending on the values of critical temperatures \(A_{1}\) and \(A_{3}\):  
\[n_{i}=\ln(\ln(0.99))/\ln(A_{1}/A_{3}), \tag{17}\]  
\[b_{i}=0.01n_{i}/A_{1}, \tag{18}\]  
The volumetric share of an individual phase during cooling depends on the temperature and the cooling rate, and on the austenite share. In an incomplete transformation zone, initial austenite share \(\varphi_{\gamma}\) before cooling has a value 0 \(<\varphi_{\gamma}<\)1. The created volumetric share \(\varphi_{i}\) of a structural component can be estimated analogically to the JMAK formula [47] by the following equation:  
\[\varphi_{i}(\theta,\tau)=\varphi_{A}\varphi_{i}^{\max}\Big{\{}1-exp\Big{[}b_{ i}[\theta]\tau^{n_{i}[\theta]}\Big{]}\Big{\}}+\varphi_{i}^{0}, \tag{19}\]  
where \(\varphi_{i}^{0}\)--volumetric share of \(i\)-th phase component, which remained after the heating stage of the thermal cycle (not undergone a transformation); and \(\varphi_{i}^{max}\)--a maximum volume share of component \(i\) estimated for the calculated cooling rate. The sum of volumetric shares equals:  
\[\sum_{i=1}^{k}\varphi_{i}=1, \tag{20}\]  
In order to estimate the volume shares of individual structural components arising from austenite during cooling, it is indispensable to know the value of the start temperature of the phase transformation \(T_{s}\) and its end \(T_{f}\). The values of these temperatures depend on the type of steel and the cooling time \(t_{85}\), which denotes the residence time of the material between temperatures 800-500 \({}^{\circ}\)C. The use of cooling speed \(v_{85}\) is often more convenient to determine the temperature limits of transformation. The quantity \(v_{85}\) is defined as cooling rate of the material in the above temperature range, i.e., \(v_{85}\) = (800 - 500)/(\(t_{85}\)). Limiting temperatures of cooling changes are determined on the basis of TTT (time-temperature-transformation) welding diagrams (Figure 5) developed for particular steel grades. These graphs allow one to determine not only the beginning of the transformation, but also the progress of the process based on the temperature of the process, as well as the type of the structure being created. Limit values of cooling phase transitions (also \(t_{85}\) values) can be determined using analytical formulas dependent on the chemical composition of steel [48].