# Thermal Strain-Based Simplified Prediction of Thermal Deformation Caused by Flame Bending
## 2 Prediction of Thermal Deformation
### Phase Transformation during Heating and Cooling Cycle  
During the flame bending process, the heated plate is subjected to rapid cooling so that the phase transformation of the metal crystal structure occurs [11]. In general, a hull plate of the mild A-grade steel formed by flame bending is converted into austenite by a heat source during the flame heating. The deformation caused by line heating occurs only on the surface of the austenite transformation, which begins with multiple phases due to the high temperature as shown in Figure 2. Thus, the austenite parts are all under the  
Figure 1: Stress–strain curve of heat affected zone (HAZ) [10].  
influence of water cooling. In such cases, a certain depth from the representation will result in a martensite transformation [9].  
The \(Ac1\) is the lower-phase transformation temperature at which austenite begins to form during heating of the steel as shown in Equation (3). It can be calculated by the function of the chemical composition of steel as described in Table 1. The \(M_{S}\) temperature is the temperature at which martensite begins to form upon cooling [12]. When the steel is heated, and temperature reached \(Ac1\), the ferrite is converted to austenite. As soon as the temperature passes the \(M_{S}\), the remaining austenite starts being transformed into martensite. The temperature \(M_{S}\) is a function of chemical composition in steel as shown in Equation (4) [13].  
\[\mathrm{Ac1}=723-10.7\mathrm{Mn}-16.9\mathrm{Ni}+29.1\mathrm{Si}+16.9\mathrm{ Cr}+290\mathrm{As}+6.38\mathrm{W} \tag{3}\]  
\[M\mathrm{s}=512-453C-16.9 Ni+15 Cr-9.5 Mo+217(C)^{2}\] \[-71.5(C)(Mn)-67.6(C)(Cr)-11.1 Si \tag{4}\]  
Using the phase-starting temperature of each phase, the portion of austenite (\(\gamma\)) and martensite (\(M\)), depending on the temperature, can be calculated during the cooling cycle [11].  
In this paper, it is assumed that the heated area above the critical temperature is converted to austenite during the heating phase [14]. The phase transformation not only changes the thermal expansion coefficient, elastic modulus, and fraction of microstructure, but also changes the magnitude of the inherent strain. The temperature \(M_{f}\) at which the martensitic transformation finishes is lower than room temperature. When the temperature falls to room temperature, the mild steel has various crystal structures, such as residual austenite, martensite, ferrite, and pearlite. If the austenite is almost transformed into martensite, the volume fraction of martensite at temperature \(T\) can be represented by Koistinen and Marburger [14]. Therefore, it is also assumed that all austenite transforms  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Grade A** & **C** & **Si** & **Mn** & **P** & **S** \\ \hline Wt\% & 0.18 & 0.45 & 0.02 & 0.02 & 0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Chemical composition of mild A-grade steel.  
Figure 2: Phase transformation during the heating and cooling process.  
into martensite during the cooling process because the temperature \(M_{f}\), at which martensite transformation stops, is lower than room temperature. During the cooling phase, the portion of austenite (\(X_{\gamma}\)) and martensite (\(X_{M}\)) according to each temperature can be expressed as Equations (5) and (6) [14].  
\[X_{\gamma}(T)=e^{k\cdot max[(M_{\mathrm{c}}-T),0]} \tag{5}\]  
\[X_{M}(T)=1-X_{\gamma}(T)=1-e^{k\cdot max[(M_{\mathrm{c}}-T),0]} \tag{6}\]  
where \(k\) is a phase transformation velocity constant depending on the material. If the austenite is transformed into martensite as a carbon steel (0.2 wt.%C), the value of \(k\) is \(-0.026\)[15]. Once the portion of retained austenite is measured using X-ray diffraction, \(k\) can be updated using the experimental data.