\chapter{Material Properties}
\label{cha:material_char}

The two processes modeled in this thesis apply to stock of related but distinct materials:
The bars of the induction test rig are made from 50CrMo4 Steel which is widely available and well characterized in literature\autocite{vieweg2017phase, vieweg2017effects, eggbauer2018inductive, eggbauer2019situ, jaszfi2019influence}, while the crankshafts are made from a modified C38 Steel dubbed \emph{C38p}, the composition of which was given by the manufacturer.
Table~\ref{tab:steel_compositions} shows the compositions for both materials in comparison to each other.

Material data for 

\begin{table}[htbp]
    \centering
    \caption[Chemical compositions]{Chemical compositions of both steels examined in this chapter.  }
    \label{tab:steel_compositions}
    \begin{tabular}{cSS}\toprule
& \textbf{50CrMo4} & \textbf{C38p}\\
& \si{\wtpercent}& \si{\wtpercent}\\\midrule
C & 0.49 & \numrange{0.36}{0.40} \\
Mn & 0.71 & \numrange{1.30}{1.45} \\
Cr & 1.05 & \numrange{0.10}{0.20} \\
Mo & 0.18 & \le 0.050 \\
Si & 0.27 & \numrange{0.50}{0.65} \\
P & 0.016 & \le 0.025 \\
S & 0.01 & \numrange{0.050}{0.065} \\
N & & \numrange{0.013}{0.017} \\
Cu & & 0.25 \\
Al & & \numrange{0.010}{0.030} \\
Ni & & \le 0.15 \\
V & & \numrange{0.08}{0.12}\\\bottomrule
    \end{tabular}
\end{table}

% \begin{table}[htbp]
%     \centering
%     \caption{Chemical composition of the 50CrMo4 steel.}
%     \label{tab:crmo_composition}
%     \begin{tabular}{ccccccccc}\toprule
% Element & C & Mn & Cr & Mo & Si & P & S & Fe \\
% \si{\wtpercent} & 0.49 & 0.71 & 1.05 & 0.18 & 0.27 & 0.016 & \num{0.01} & \num{\sim 97.1} \\\bottomrule
%     \end{tabular}
% \end{table}
% \begin{table}[htbp]
%     \centering
%     \caption{Chemical composition of the C38p steel, as disclosed by the manufacturer.}
%     \label{tab:c38_composition}
%     \begin{tabular}{cSSSS}\toprule
% Element & C & Mn & Si & P \\
% \si{\wtpercent} & \numrange{0.36}{0.40} & \numrange{1.30}{1.45} & \numrange{0.50}{0.65} & \le 0.025 \\\midrule
% Element & S & Cr & N & Cu\\
% \si{\wtpercent} & \numrange{0.050}{0.065} & \numrange{0.10}{0.20} & \numrange{0.013}{0.017} & 0.25 \\\midrule
% Element & Mo & Al & Ni & V \\
% \si{\wtpercent} & \le 0.050 & \numrange{0.010}{0.030} & \le 0.15 & \numrange{0.08}{0.12}\\\bottomrule
%     \end{tabular}
% \end{table}



\section{Sample Preparation}
\label{sec:sample_preparation}

Crankshafts of various processing stages were supplied by the manufacturer, from which two pieces of forged blanks were chosen as sources for C38p samples.
These were untouched by subsequent heat treatments or surface machining, thus allowing for the characterization of a material microstructure to be used as a baseline for subsequent heat treatment and property evolution. 
Sample locations were strategically placed in the crankshafts to allow for necessary dimensions for the different testing equipment while staying as close as possible to bearing surfaces of interest, i.e. the section closest to the center rotation axis of the crank bearings.

\begin{figure}[htbp]
    \centering
    \begin{tabular}{c}
        \subfloat[Crankshaft 1\label{fig:probenlage_1}]{\includegraphics[width=0.95\textwidth]{Abbildungen/Probenlage_side.png}}\\
        \subfloat[Crankshaft 2 ]{\includegraphics[width=0.95\textwidth]{Abbildungen/Probenlage_side2.png}}
    \end{tabular}
    \caption[Crankshaft sample positions, side view]{Sample position within the two sacrificial crankshafts, side view.}
    \label{fig:sample-loc-1}
\end{figure}


Figures~\ref{fig:sample-loc-1} and~\ref{fig:sample-loc-2} show the detailed locations of the samples that were distributed as follows:
\begin{description}
    \item[Section A] 38 dilatometer samples D\qtyproduct{4 x  10}{\mm} (black)
    \item[Section B - C] 28 threaded tension test samples M\qtyproduct{8 x 58}{\mm} (red), 1 density sample D\qtyproduct{20 x 30}{\mm} (green)
    \item[Section D] 10 Gleeble samples M\qtyproduct{10 x 65}{\mm} (orange), 2x electrical resistivity samples D\qtyproduct{4 x 75}{\mm} (blue)
    \item[Section E] 7 dilatometer samples D\qtyproduct{4 x  10}{\mm} (black), 1 thermal conductivity sample D\qtyproduct{12.55 x 40}{\mm} (cyan) , 1 heat capacity sample D\qtyproduct{6 x 40}{\mm} (purple), 2 thermal expansion samples  D\qtyproduct{6 x 25}{\mm} (yellow)
    \item[Secfion F] 14 threaded tension test samples M\qtyproduct{8 x 58}{\mm} (red)
    \item[Secfion G] 10 Gleeble samples M\qtyproduct{10 x 65}{\mm} (orange), 3x magnetic hysteresis samples D\qtyproduct{3 x 3}{\mm} (blue)
\end{description} 


\begin{figure}[htbp]
    \centering
    \begin{tabular}{ccc}
         \subfloat[\label{fig:sample-loc-A}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_A.png}}&
         \subfloat[\label{fig:sample-loc-B}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_B.png}}&
         \subfloat[\label{fig:sample-loc-C}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_C.png}}\\
         \subfloat[\label{fig:sample-loc-D}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_D.png}}&
         \subfloat[\label{fig:sample-loc-E}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_E.png}}&
         \subfloat[\label{fig:sample-loc-F}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_F.png}}\\&
         \subfloat[\label{fig:sample-loc-G}]{\includegraphics[width=0.3\textwidth]{Abbildungen/Probenlage_G.png}}&\\
    \end{tabular}
    \caption[Crankshaft sample positions, axial view]{Sample position within the two sacrificial crankshafts, frontal view at slices as indicated in figure~\ref{fig:sample-loc-1}. The material state is as-forged without any further mechanical or thermal treatments. Sample positions were chosen to allow for the required dimensions to be reached while staying close to bearing surfaces for in}
    \label{fig:sample-loc-2}
\end{figure}


\section{Thermophysical Properties}
\label{sec:thermophysical_properties}

A wide regimen of experiments was applied to the materials to gather all properties needed for process simulation.
Table~\ref{tab:equipment_th} gives an overview of the equipment and norms used to generate direct data.

\begin{sidewaystable}[p]
    \centering
    \caption[Thermophysical test equipment]{Summary of methods and equipment used to characterize the thermophysical and thermoelectric properties.}
    \label{tab:equipment_th}
    \begin{tabular}{ccccccrr} \toprule
         \multirow{2}{*}{\thead{Measurement\\Method}} & \multicolumn{3}{c}{\multirow{2}{*}{\thead{Physical Property}}} &  \multirow{2}{*}{\thead{Device}} &  \multirow{2}{*}{\thead{Norm}} & \multicolumn{2}{c}{\thead{Uncertainty in \\ Measurement ($\sigma \sim \qty{95}{\percent}$)}} \\
         &&&&&&at RT & at \qty{1000}{\celsius}\\ \midrule
         \makecell{Dynamic\\Differential\\Calorimetry} & Specific Heat Capacity &  $c_p$ & [\unit{\joule\per\gram\per\kelvin}] & Netzsch DSC 404 & EN 821-3 (2005) & \qty{\pm 3}{\percent} & \qty{\pm 4}{\percent} \\[20pt]
         Dilatomety & Thermal Strain & $\nicefrac{\Delta l}{l}$ & [\unit{\percent}] & Netzsch DIL 402 CD & DIN 51045-1 (2005) & \makecell{\qty{\pm 0.004}{\percent}\\(at \qty{100}{\celsius})} & \qty{\pm 0.015}{\percent} \\[20pt]
         \makecell{Laser Flash\\Method} & Thermal Diffusivity & $a$ & [\unit{\mm\squared\per\s}] & Netzsch LFA 427 &  EN 821-2 (1997) & \qty{\pm 5}{\percent} & \qty{\pm 5}{\percent} \\[20pt]
         Buoyancy Method & Density & $\rho$ & [\unit{\kg\per\m\cubed}] & Sartorius ED224S & DIN EN 993-1 &  \qty{\pm 0.1}{\percent} & ---\\[10pt]
         \makecell{Four-Terminal\\Sensing} & Electrical Resistivity & $\rho_{el}$ & [\unit{\micro\ohm\m}] & ÖGI in-house setup & DIN EN 993-1 &   \qty{\pm 2}{\percent} &   \qty{\pm 3}{\percent} \\
         \bottomrule
    \end{tabular}
    \label{tab:characterization_summary_c36p}
\end{sidewaystable}

From that data some more fundamental properties were derived for use in the simulations' material data sets.
Namely, Thermal conductivity $k$ was derived from thermal diffusivity $a$, specific heat $c_p$ and density $\rho$ by rearranging the diffusivity's standard formula for $k$:
$$ a = \frac{k}{c_p \rho} \Rightarrow k = a c_p \rho $$

Likewise, the thermal Expansion coefficient $\alpha$ was calculated from the thermal strain and the temperature differential to the previous data point:
$$a = \frac {1}{l} \frac{\Delta l}{\Delta T} $$

Data was collected for heating and cooling behaviors, but data from the heating cycle as preferred where the two disagreed due to transformation temperature varying.

Data was collected with higher resolution around the transformation temperatures \qtyrange{700}{800}{\celsius}. 
For use in the following simulations, all data was interpolated to a fixed set of temperature points, with $c_p$ requiring extrapolation at room temperature, and above \qty{1000}{\celsius}.
The resulting modeal data is listed in tables~\ref{tab:results_50CrMo4} and \ref{tab:results_c38p}, and compared in figure~\ref{fig:comp-char}. 
Both materials behave appreciably close to one another with the biggest discrepancy in their thermal conductivities below $A_{C1}$.


\begin{table}[htbp]
    \centering
    \begin{tabular}{S[table-format=4]SSSS[table-format=4]SS}\toprule
        {T} & {$c_p$} & {$\varepsilon^T$} & {$\alpha$} & {$\rho$} & {$a$} & {$k$}\\
        {[\unit{\celsius}]} & {[\unit{\joule}]} & {[\unit{\percent}]} & {[\unit{10^{-6}\per\kelvin}]} & {[\unit{\kg\per\meter\cubed}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\watt\per\meter\per\kelvin}]} \\ \midrule
        20 & \color{red}0.460 & 0.000 &{---}& 7827 &  12.47 & 44.9 \\
        100 & 0.488 & 0.097 & 12.085 & 7804 &  11.64 & 44.3 \\
        200 & 0.525 & 0.233 & 12.930 & 7773 &  10.53 & 43.0 \\
        300 & 0.561 & 0.380 & 13.559 & 7739 & 9.45 & 41.0 \\
        400 & 0.603 & 0.533 & 14.024 & 7703 & 8.39 & 39.0 \\
        500 & 0.655 & 0.690 & 14.377 & 7667 & 7.29 & 36.6 \\
        600 & 0.731 & 0.849 & 14.641 & 7631 & 6.11 & 34.1 \\
        700 & 0.884 & 1.010 & 14.855 & 7595 & 4.71 & 31.6 \\
        800 & 0.625 & 0.924 & 11.843 & 7614 & 5.56 & 26.5 \\
        900 & 0.632 & 1.189 & 13.507 & 7554 & 5.77 & 27.5 \\
        1000 & 0.639 & 1.419 & 14.480 & 7503 & 5.99 & 28.7 \\
        1100 & \color{red}0.645 & 1.649 & 15.264 & 7452 &6.23 & 30.0 \\
        1200 & \color{red}0.652 & 1.880 & 15.932 & 7402 & 6.45 & 31.1 \\
         \bottomrule
    \end{tabular}
    \caption[Thermophysical properties of 50CrMo4]{Thermophysical properties of 50CrMo4. The numbers highlighted in red denote where the specific heat capacity was extrapolated to room temperature, and to \qty{1000}{\celsius} and \qty{1200}{\celsius}.}
    \label{tab:results_50CrMo4}
\end{table}


\begin{table}[htbp]
    \centering
    \begin{tabular}{S[table-format=4]SSSS[table-format=4]SSS}\toprule
        {T} & {$c_p$} & {$\varepsilon^T$} & {$\alpha$} & {$\rho$} & {$a$} & {$k$} & {$\rho_{el}$}\\
        {[\unit{\celsius}]} & {[\unit{\joule}]} & {[\unit{\percent}]} & {[\unit{10^{-6}\per\kelvin}]} & {[\unit{\kg\per\meter\cubed}]} & {[\unit{\mm\squared\per\second}]} & {[\unit{\watt\per\meter\per\kelvin}]} & {[\unit{\micro\ohm\m}]}  \\ \midrule
        20 &  \color{red}0.470 & 0 &  & 7797 & 10.27 & 37.6 & 0.283 \\
        100 & 0.495 & 0.096 & 11.997 & 7775 & 10 & 38.5 & 0.332 \\
        200 & 0.528 & 0.233 & 12.926 & 7743 & 9.38 & 38.4 & 0.402 \\
        300 & 0.563 & 0.380 & 13.572 & 7709 & 8.67 & 37.7 & 0.482 \\
        400 & 0.604 & 0.534 & 14.043 & 7674 & 7.85 & 36.4 & 0.573 \\
        500 & 0.657 & 0.691 & 14.391 & 7638 & 6.91 & 34.7 & 0.674 \\
        600 & 0.733 & 0.850 & 14.660 & 7601 & 5.84 & 32.5 & 0.781 \\
        700 & 0.878 & 1.010 & 14.853 & 7565 & 4.51 & 29.9 & 0.905 \\
        800 & 0.621 & 0.920 & 11.796 & 7586 & 5.55 & 26.1 & 1.068 \\
        900 & 0.634 & 1.142 & 12.979 & 7536 & 5.81 & 27.8 & 1.143 \\
        1000 & 0.644 & 1.370 & 13.984 & 7485 &  6.02 & 29.0 & 1.175 \\
        1100 &  \color{red}0.660 & 1.593 & 14.753 & 7436 &6.24 & 30.6 & 1.212 \\
        % 1000 & 0.644 & 1.359 &  & 7488 & 6.09 & 29.4 &  \\
        % 900 & 0.634 & 1.127 &  & 7539 & 5.87 & 28.1 &  \\
        % 800 & 0.621 & 0.899 &  & 7591 &  5.63 & 26.5 &  \\
        % 700 & 0.878 & 0.696 &  & 7636 & 5.09 &  {\color{red}transf.} &  \\
        % 600 & 0.733 & 0.792 &  & 7615 & 5.88 & 32.8 &  \\
        % 500 & 0.657 & 0.628 &  & 7652 &  6.96 & 35.0 &  \\
        % 400 & 0.604 & 0.470 &  & 7688 & 7.87 & 36.5 &  \\
        % 300 & 0.563 & 0.317 &  & 7723 & 8.70 & 37.9 &  \\
        % 200 & 0.528 & 0.174 &  & 7757 & 9.45 & 38.7 &  \\
        % 100 & 0.495 & 0.039 &  & 7788 & 10.02 & 38.6 &  \\
        % 20 &  \color{red}0.470 & -0.076 &  & 7815 & 10.20 & 37.4 &  \\
         \bottomrule
    \end{tabular}
    \caption[Thermophysical properties of C38p]{Thermophysical properties of C38p. Red highlights again denote where the specific heat capacity was extrapolated to room temperature, and to \qty{1100}{\celsius}.}
    \label{tab:results_c38p}
\end{table}

\begin{figure}[htbp]
    \centering
    \begin{tabular}{cc}
         \subfloat[Specific Heat Capacity]{\includegraphics[width=5cm]{Abbildungen/mat_char_cp.png}}&\quad
         \subfloat[Density]{\includegraphics[width=5cm]{Abbildungen/mat_char_rho.png}}\\
         \subfloat[Thermal Expansion]{\includegraphics[width=5cm]{Abbildungen/mat_char_expan.png}}&\quad
         \subfloat[Thermal Expansion Coefficient]{\includegraphics[width=5cm]{Abbildungen/mat_char_epsilon_T.png}}\\
         \subfloat[Thermal Diffusivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_th_diff.png}}&\quad
         \subfloat[Thermal Conductivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_htc.png}}\\
         \subfloat[Electrical Resistivity]{\includegraphics[width=5cm]{Abbildungen/mat_char_rhoe.png}}
    \end{tabular}
    \caption[Thermophysical material data]{Graphed thermophysical data. The local discontinuities at \qtyrange{700}{800}{\celsius} are caused by $\alpha$-$\gamma$ transformation happening in that range.}
    \label{fig:comp-char}
\end{figure}

\section{Mechanical Properties}
\label{sec:mechanical_properties}

A series of tensile tests was conducted for both materials at various temperatures.
Since phase transformations are being simulated, subsets of tensile test samples were heat treated to obtain austenitic and martensitic microstructures in addition to the ferritic-pearlitic initial states, labelled "base microstructure".

Tensile tests were conducted at overlapping temperature ranges for each material and their phases as far as transition times would allow:

\begin{description}
    \item[Base Microstructure:] 25, 300, 400, 500, 600, 700
    \item[Martensite:] 25, 100, 200, 300, 350
    \item[Austenite:] 600, 650, 700, 800, 900, 1000, 1100 
\end{description}

Samples were tested to EN ISO 6892-2, with sample geometries of B4x20. 
They were prestressed at \qty{5}{\mega\pascal} strained at $\dot{\varepsilon} = \qty{0.001}{\per\s}$.
Martensitic Samples were tested to failure, the others until they started necking.


\begin{table}[htbp]
    \centering
    \caption{Stress-stress parameters of base microstructure.}
    \label{tab:tensile_test_base}
    \begin{tabular}{S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S}\toprule
           {$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$}\\
        {[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]}  \\ \midrule
        25 & 204 & 601 & 616 & 892 & 7,8 \\
        300 &  191 & 511 & 551 & 856 & 9,4 \\
        400 &189 & 469 & 506 & 710 & 6,2 \\
        500 &165 & 401 & 430 & 537 & 3,3 \\
        600 & 141 & 306 & 330 & 374 & 2,2 \\
        700 & 102 & 156 & 166 & 174 & 1,6 \\
        \bottomrule
    \end{tabular}
\end{table}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_BASE.png}
    \caption{Stress-strain curves for base microstructure.}
    \label{fig:stress_base}
\end{figure}

\begin{table}[htbp]
    \centering
    \caption{Stress-stress parameters of martensite.}
    \label{tab:tensile_test_mart}
    \begin{tabular}{S[table-format=3]S[table-format=3]S[table-format=4]S[table-format=4]S[table-format=4]SS}\toprule
           {$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$} & {$A$}\\
        {[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]}  & {[\unit{\percent}]}  \\ \midrule
        25 & 209 & 1041 & 1217 & 1994 & 3,5 & 3,5 \\
        100 & 199 & 1155 & 1362 & 2101 & 2,5 & 2,5 \\
        200 & 185 & 1141 & 1342 & 2011 & 5,0 & 6,0 \\
        300 & 180 & 926 & 1094 & 1540 & 3,0 & 5,5 \\
        350 & 181 & 852 & 986 & 1284 & 2,5 & 4,5 \\
        \bottomrule
    \end{tabular}
\end{table}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_MART.png}
    \caption{Stress-strain curves for martensite.}
    \label{fig:stress_base}
\end{figure}

\begin{table}[htbp]
    \centering
    \caption{Stress-stress parameters of austenite.}
    \label{tab:tensile_test_aust}
    \begin{tabular}{S[table-format=4]S[table-format=3]S[table-format=3]S[table-format=3]S[table-format=3]S}\toprule
           {$T$} & {$mE$} & {$R_{p0,1}$} & {$R_{p0,2}$} & {$R_m$} & {$A_g$}\\
        {[\unit{\celsius}]} & {[\unit{\giga\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\mega\pascal}]} & {[\unit{\percent}]}  \\ \midrule
        600 & 132 & 202 & 217 & 262 & 3,0 \\
        650 & 127 & 160 & 173 & 189 & 2,0 \\
        700 & 108 & 92 & 96 & 135 & 9,9 \\
        800 & 105 & 69 & 72 & 100 & 11,7 \\
        900 & 70 & 49 & 51 & 64 & 6,9 \\
        1000 & 57 & 30 & 31 & 37 & 9,5 \\
        1100 & 56 & 19 & 19 & 22 & 9,4 \\
        \bottomrule
    \end{tabular}
\end{table}

\begin{figure}[htbp]
    \centering
    \includegraphics[width=7cm]{Abbildungen/C38_Zugdaten_AUST.png}
    \caption{Stress-strain curves for austenite.}
    \label{fig:stress_base}
\end{figure}


\acrfull{rve}\autocite{fischer2000new}

\section{Phase Transformation Behavior}
\label{sec:phase_transformation}

Phase transformation data was gathered by examination of the dilatometer experiments\autocite{antretter2002thermo}


\subsection{Martensite}

\

\autocite{schemmel2014modelling}

\section{Electromagnetic Properties}
\label{sec:electromagnetic_properties}

A measurement setup for magnetic hysteresis behavior at high temperatures was designed in-house\autocite{jaszfi2022indirect}, but was not ready in time to deliver the material data for this project.
As a recourse, magnetic model data was taken from M. Schwenk's thesis on induction heating\autocite[111]{schwenk2012numerische}.
His subject material of a 42CrMo4 steel was deemed close enough in its characteristics to stand in for the materials of this work.

A recent proposal\autocite{baldan2020improving} for analytic descriptions of magnetization curves incorporates the quadratic region of the curve at the origin, but the numeric capabilities of common solvers do not support the inflection point that this increased adherence to measured data brings.
This thesis will therefore use the analytic formula described by Trutt et al.\autocite{trutt1968representation} using the arc tangent, as Schwenk did:

$$B(H,T) = \mu_0 H + \frac{2 B_{sat}}{\pi}\cdot \mathrm{atan} \left(  \frac{(\mu_{r0} - 1)\mu_0 \pi}{2 B_{sat}} H \right)\cdot {e}^\frac{T-T_C}{C}$$

With a saturation flux $B_s=\qty{1.4}{\tesla}$, initial relative permeability $\mu_{r0}=2500$, Curie temperature $T_C=\qty{785}{\celsius}$, and a temperature scaling constant $C=\qty{60}{\celsius}$.

