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\title{Non-linear Regression Model for $Ac_3$, $Ac_1$ Transformation Temperatures}
\author{Gemini analysis by H. K. D. H. Bhadeshia}
\date{February 9, 2026}

\begin{document}

\maketitle

\section{$Ac_3$ Introduction}
This document presents a non-linear mathematical relationship between the chemical composition, heating rate, and the $Ac_3$ transformation temperature of various steel alloys. The model was developed using a second-order polynomial regression analysis based on a dataset of 824 experimental observations.

\section{Methodology}
The features used for the model include the weight percentages $w_i$  of carbon, silicon, manganese, nickel, chromium, and molybdenum, alongside the heating rate ($\dot{T}$). To capture non-linearities and synergistic effects between elements, the model incorporates:
\begin{itemize}
    \item Linear terms for primary alloying elements.
    \item Quadratic terms to account for diminishing or increasing returns of specific elements.
    \item Interaction terms to model how the presence of one element alters the influence of another.
\end{itemize}

\subsection{Derived Equation}
The resulting regression equation for $Ac_3$ (in $^\circ$C) as a function of the wt\%\ of solute and the heating rate in K\,s$^{-1}$ is given by:

\begin{equation}
\begin{split}
Ac_3 \approx \ & 918.88 - 529.00 w_{\rm C} + 97.78 w_{\rm Si} - 15.08 w_{\rm Mn} - 30.95 w_{\rm Ni} + 29.25 w_{\rm Mo} \\
& + 430.98 w_{\rm C}^2 + 0.99 w_{\rm Ni}^2 + 6.99 w_{\rm Mo}^2 - 84.13 w_{\rm C} \times w_{\rm Si} \\
& + 50.44 w_{\rm C} \times w_{\rm Mn} + 37.52 w_{\rm C} \times w_{\rm Ni} + 7.95 w_{\rm C} \times w_{\rm Cr} \\
& - 58.25 w_{\rm C} \times w_{\rm Mo} - 29.28 w_{\rm Si} \times w_{\rm Mn} + 18.13 w_{\rm Si} \times w_{\rm Mo} \\
& + 2.53 w_{\rm C} \times \dot{T} - 2.38 w_{\rm Si} \times \dot{T} + 0.75 w_{\rm Ni} \times \dot{T} \\
& - 0.14 w_{\rm Cr} \times \dot{T} + 1.75 w_{\rm Mo} \times \dot{T}
\end{split}
\end{equation}

\subsection{Statistical Summary}
The model demonstrates a strong fit to the experimental data, as summarised in the table below:

\begin{table}[h]
\centering
\begin{tabular}{ll}
\toprule
\textbf{Metric} & \textbf{Value} \\
\midrule
Number of Observations & 824 \\
R-squared ($R^2$) & 0.806 \\
Adjusted $R^2$ & 0.801 \\
F-statistic & 166.7 \\
\bottomrule
\end{tabular}
\caption{Model performance metrics.}
\end{table}

\subsection{Discussion}
The high coefficient for $w_{\rm C}^2$ indicates a strong curvature in the $Ac_3$ response to Carbon content. Furthermore, the significance of the interaction terms between alloying elements and the heating rate ($\dot{T}$) confirms that the kinetic response of the transformation is highly dependent on the specific chemical system of the steel.\\

\hrule

\section{$Ac_1$ Introduction}
This report provides a non-linear mathematical model relating the $Ac_1$ transformation temperature to the chemical composition and heating rate of various steel alloys. The model was derived from 824 experimental observations using polynomial regression.

\subsection{Methodology}
The regression model incorporates weight percentages $w_i$ of alloying elements (C, Si, Mn, Ni, Cr, Mo) and the heating rate ($\dot{T}$). The model considers linear, quadratic, and interaction terms, with only statistically significant parameters ($p < 0.05$) retained to ensure predictive reliability.

\subsection{Derived Equation}
The equation for $Ac_1$ (in $^\circ$C) is:

\begin{equation}
\begin{split}
Ac_1 \approx \ & 836.00 - 225.36 w_{\rm C} - 135.23 w_{\rm Mn} - 27.44 w_{\rm Ni} + 11.06 w_{\rm Cr} + 2.16 \dot{T} \\
& + 131.13 w_{\rm C}^2 + 33.65 w_{\rm Mn}^2 - 0.81 w_{\rm Cr}^2 - 0.0004 \dot{T}^2 \\
& + 25.64 w_{\rm C} \times w_{\rm Si} + 99.39 w_{\rm C} \times w_{\rm Mn} + 23.61 w_{\rm C} \times w_{\rm Ni} \\
& + 5.60 w_{\rm Si} \times w_{\rm Cr} + 5.84 w_{\rm Mn} \times w_{\rm Cr} - 1.15 w_{\rm Ni} \times w_{\rm Cr} \\
& + 1.95 w_{\rm Cr} \times w_{\rm Mo} - 3.61 w_{\rm Si} \times \dot{T} - 0.73 w_{\rm Mn} \times \dot{T} \\
& + 0.85 w_{\rm Ni} \times \dot{T}
\end{split}
\end{equation}
\subsection{Statistical Summary}
The performance of the $Ac_1$ model is summarized below:

\begin{table}[h]
\centering
\begin{tabular}{ll}
\toprule
\textbf{Metric} & \textbf{Value} \\
\midrule
Number of Observations & 824 \\
R-squared ($R^2$) & 0.812 \\
Adjusted $R^2$ & 0.808 \\
F-statistic & 183.3 \\
\bottomrule
\end{tabular}
\caption{Model performance for $Ac_1$.}
\end{table}

\subsection{Observations}
\begin{itemize}
    \item \textbf{Manganese}: Mn exhibits a strong depressive effect on $Ac_1$, but the positive $w_{\rm Mn}^2$ term indicates this effect diminishes as concentration increases.
    \item \textbf{Interactions}: Significant interactions between chromium and molybdenum ($w_{\rm Cr} \times w_{\rm Mo}$) and nickel and chromium ($w_{\rm Ni}\times w_{\rm Cr}$) highlight the complex synergy of these elements in determining the onset of austenite formation.
    \item \textbf{Heating kinetics}: The $Ac_1$ temperature shows a non-linear sensitivity to the heating rate, influenced significantly by the presence of silicon and nickel.
\end{itemize}


\end{document}