\chapter{State of the Art}

This chapter compiles the state of the art for the fundamental topics necessary for the simulation of an inductive heat treatment process and its validation.

\section{Inductive Surface Hardening}
\label{sec:sota_induction}

In modern physics, Maxwell's Equations are the basis for understanding electromagnetic phenomena.
Their differential forms, names, and approximate meaning are as follows:

\begin{equation}
\div{\vect{D}} =  \rho \label{eq:gauss}
\end{equation}
\paragraph{Gauss's law} describes how the electric field $\vect{D}$ originates at electric charges.

\begin{equation}
\div{\vect{B}} = 0 \label{eq:gaussmag} 
\end{equation}
\paragraph{Gauss's law for magnetism} states that the magnetic field $\vect{H}$ has no points of origin and always forms closed loops.

\begin{equation}
\curl\vect{E} = -\pdv{\vect{B}}{t} \label{eq:faraday} 
\end{equation}
\paragraph{Faraday's law} bridges the electrical and magnetic fields by describing the curl of the former as the time derivation of the latter.
Put simply, a changing electric field \emph{induces} some form of change in the magnetic field.

\begin{equation}
\curl\vect{H}=  \vect{J}_e+\vect{J}_f+\pdv{\vect{D}}{t} \label{eq:ampere}
\end{equation}
\paragraph{Amp\`ere's circuital law} concludes the relation by defining the magnetic field's curl in relation to the temporal change of the the electric field. 
Moving electric charges (a current $\vect{J}$) are also taken into account here as well.

Faraday's and Ampere's laws in conjunction describe how electric and magnetic field are inextricably linked. 
A changing magnetic field will induce an electric field, and with it often a current will flow, while any current flowing will in turn induce a magnetic field. 

These interactions can be leveraged for a plethora of uses: 
Electromagnets can be magnetized instantly by a current flowing through a wire wrapping around an iron core.
The moving rotors of a generator induce alternating currents in it's stator coils, while a motor reverses the effect.
Transformers can use disparate amounts or wire turns to turn one alternating voltage into another.

In all of these, \emph{eddy currents} are an undesirable consequence of Maxwell's laws cutting both ways:
As the changing electric field of the transformer induces a magnetic flux in the yoke, it is changing with time and therefore induces an electric field.
This brings with it small current loops which in turn induce a magnetic field that is opposed to the one already being generated in the yoke, therefore weakening that one by superposition and reducing output power.
The eddy currents also heat the material through it's electrical resistance.

Where in most other applications great effort is applied to reduce eddy currents, in inductive heating they are the intended effect.
Here, AC current through induction coils is used to generate electromagnetic losses within a conductive work piece and so heat it up rapidly.

The process has many variables that can be adjusted to achieve desired heating results, but the most useful for controlling the exact volume of heated material within a part are the the current amplitude, its frequency, and the inductor geometry.

The current is in direct relation to the amount of energy dumped into the target volume.
The frequency of the current through the coil influences the rise time of the induced magnetic field, and through interactions of \ref{eq:faraday} and \ref{eq:ampere} the magnitude of the induced opposed field. 
This interaction repeats and produces an effect where the current and magnetic flux through a conductor get displaced towards the surface with increasing frequency.
The current flow drops exponentially towards the center with the \emph{skin depth} being defined as the depth where the flux has decreased to $\frac{1}{e}$ or \qty{\sim 37}{\percent} of the surface strength~\cite{skin depth??}.

The skin depth for conductive materials can be calculated using the frequency (or more commonly, the angular frequency $\omega = 2\pi f$) the resistivity $\rho$, and the magnetic permeability $\mu$~\cite{skin depth calc??}:

\begin{equation}
    \delta = \sqrt{\frac{2\rho}{\omega\mu}} \label{eq:skin}
\end{equation}

The geometry of the induction coil is the most difficult aspect of inductive hardening, where it's broad range of application leads to a multitudes of designs. 
Continuous heating of rods may require several conscutive soilenoids, where  gears and sprockets are briefly held in a single loop. 
In general only the most simple geometries allow for analytical solutions of the heated volume and so inductor design still greatly relies on the experience of process engineers.
Many iterations of experiments are needed to dial in the geometry if the hardened zone and arrive at a usable product.
Numerical simulation has been a boon in this regard as it allows initial experiments to be run digitally, freeing valuable production equipment and saving the cost of bespoke copper inductors in exotic geomtries.


\section{Finite Element Method}
\label{sec:sota_fem}

The \acrfull{fem} is one of several numerical approaches to subdividing a volume into smaller chunks in which a simplification of physics can be calculated numerically.
It can achieve solutions to differential equations for complex volumes, for which analytical solutions are all but impossible.
\acrshort{fem} in particular is mostly used for the continuum mechanics problems, such as stress/strain calculations, heat transfer problems, mass transport, and electromagnetic potential.

Its method of subdivision are the eponymous finite elements, that (in a 3D case) can take the shape of tetrahedrons, pyramids, wedges and cuboids which are in turn defined through their vertex points.



\subsection{Electromagnetic Analysis}
\label{sub:sota-fem-em}

% \begin{align}
% \div{\vect{D}} & =  \rho \label{eq:gauss} \\
% \div{\vect{B}} & = 0 \label{eq:gaussmag} \\
% \curl\vect{E}& = -\pdv{\vect{B}}{t} \label{eq:faraday} \\
% \curl\vect{H}& =  \vect{J}_e+\vect{J}_f+\pdv{\vect{D}}{t} \label{eq:ampere}
% \end{align}

\begin{align}
\vect{D} &= \tens{\varepsilon}\:\vect{E} \label{eq:de}\\
\vect{B} &= \tens{\mu}\:\vect{H} \label{eq:bh}\\
\vect{J}_e &= \tens{\sigma}\:\vect{E} \label{eq:je}
\end{align}

\begin{equation}
    \vect{B} = \curl{\vect{A}} \label{eq:def_a}
\end{equation}

\begin{equation}
\vect{E} = - \pdv{\vect{A}}{t} - \grad{\varphi}  \label{eq:def_phi}
\end{equation}

\begin{equation}
\curl(\tens{\mu}^{-1}\:\curl\vect{A}) + \tens{\sigma} \pdv{\vect{A}}{t} -\vect{J}_f = \vect{0} \label{eq:strong_form}
\end{equation}

\begin{align}
    \vect{A} = \vect{A}^0 \exp(i\omega t)\\
    \vect{J} = \vect{J}^0 \exp(i\omega t)
\end{align}

\begin{equation}
\curl(\mu^{-1}\tens{I}\:\curl\vect{A}^0) +
\tens{\sigma} i\omega\vect{A}^0  = \vect{J}_f  \label{eq:timeharmonic}
\end{equation}

\label{sub:em_fem}
\subsubsection{Non-Linear Materials}

Labridis!




\subsection{Thermal Analysis}

\subsubsection{Latent Heat}

\subsection{Phase Transformation Models}

\subsubsection{Austenite - Johnson-Mehl-Avrami-Kolmogorow}

JMAK-Equation \autocite{johnson1939reaction}

conversion from isothermal TTT to continual TTT using Scheil sum\autocite{geijselaers2003numerical}

\begin{equation}
    \tau_1 = \dv{T_1}{c}
\end{equation}

\begin{equation}
    \int_{0}^{t_1}\frac{1}{\tau_1(T)} \mathrm{d}t = 1
\end{equation}

\begin{equation}
    f_{phase} = 1- \exp \left[-bt^n\right]
\end{equation}

\begin{equation}
    b \propto \exp \left[ -\frac{\frac{const}{(T - A_s)^2}+Q_s}{R T} \right]
\end{equation}

\begin{equation}
    b(T,c_c, \rho) = D(\rho, c_c) \exp{\left[ -\frac{\frac{E(c_c)}{\Delta T^2}+F(c_c)}{R T} \right]}
\end{equation}

50CrMo4 using parameters $n_{Aust}$, $D$, $E$, $F$ \autocite{nusskern2013simulation}

Scheil-Approach\autocite{scheil1935anlaufzeit}:
\begin{equation}
    t_i^* = \left[ \frac{-\ln{\left(-f_{i-1}\right)}}{b_i} \right]^{1/n_i}
\end{equation}

\begin{equation}
    f_i^* = 1- \exp \left[b_i(t_i^*+\Delta t)^n\right]
\end{equation}

\begin{equation}
    f = f_i^*\left( f_{i-1}^\gamma + f_{i-1}\right) f_{max}
\end{equation}



\begin{equation}
    t_i^* = \left[ \frac{-\ln{\left(-f_{i-1}\right)}}{k} \right]^{1/n_i}
\end{equation}\

\begin{equation}
    k = k_max \exp \left[ - {\left(\frac{T-T_{nose}}{P_1}  \right)}^6 \right]
\end{equation}

C38p using $n_{Aust}$, $k_max$, $T_{nose}$, $P_1$ \autocite{garcia1998modelling}



\subsubsection{Martensite - Koistinen-Marburger}
\autocite{nusskern2013simulation, koistinen1959general}

\begin{equation}
    f_M = f_M^{max} \left(1-\exp{\left[ -k(M_s-T) \right]}\right)
\end{equation}

\subsubsection{Bainite - Mahnken}

\begin{equation}
    \Delta G_r = - \frac{4}{3}\pi r^3 \Delta G_V + 4\pi r^2 S_{\gamma b}
\end{equation}

\begin{equation}
    \dot{z} = A_5 \exp \left[ \frac{\theta^*[e_v]-\theta}{B_1}\right]
    \left[ \frac{G^*}{R\theta} \right]
    {\left\langle \frac{r-r^*}{r} \right\rangle}^n
    {\left( 1-z \right)}^\gamma
\end{equation}

\begin{equation}
    \dot{r} = \alpha_{2l} \exp \left[ \frac{\theta-\theta^*[e_v]}{B_l} \right] \exp [-\beta_{2l}e_v]
\end{equation}

\begin{equation}
    r^* = \frac{A_1\theta_0\Delta\theta^{0.5}}{\theta\Delta\theta Q^* + A_2\theta_0e_v\sigma_v}
\end{equation}

\begin{equation}
    G^* = \frac{A_3\theta_0^2\Delta\theta^{1.5}}{\left(\Delta\theta Q^* + A_4\theta_0e_v\sigma_v\right)^2}
\end{equation}


\autocite{mahnken2012multiphase, garrett2004model}


\subsection{Stress Analysis}

Calculating deflections and stresses resulting from various forces upon an object is one of the oldest use of \acrshort{fem}.
It uses Hooke's law (\eqref{eq:hooke}

\begin{equation}
    \label{eq:hooke}
    \tens{\sigma} = \tens{C}\tens{\varepsilon}
\end{equation}


\subsubsection{Plasticity}

\subsubsection{TRIP strain}


\subsubsection{Metallurgical Strain}

\begin{equation}
    \varepsilon^{met} = 
\end{equation}

\subsection{Multiphysical Simulation}


\section{Residual Stress}
\label{sec:sota_residual_stress}

Surface heat treatment enhances the longevity of hardenable steel parts by affecting the treated volume in multiple ways:

On the one hand, the material becomes harder and thus experiences less wear and surface damage during operation.
The hardness is quite easy to measure, in essence requiring only a polished surface and a hardness testing machine.
Surface measurements leave only microscopic indents and count as non-destructive testing, while cutting samples for in-depth testing does not alter the hardness when done carefully.
Thus, it is a preferred measure for quality control in a great many heat treatment facilities.

On the other hand, if the treatment is applied correctly, a residual stress field is imparted on the workpiece that compresses the surface regions and thus inhibits crack initiation and growth, thereby increasing the fatigue resistance of the part.

\subsection{Measuring Residual Stresses}


The residual stress distribution of heat treated parts is of great interest to manufacturers, since an unfavorable result may lead to a superposition of internal and external stresses during service, which greatly reduces the life span without any visible faults at production.
Unfortunately, an internal balanced force is impossible to measure directly and only its effects may be observed as residual elastic strain, as was observed by Hattori \autocite{hattori1929cause} in the deflection of one-sidedly treated rods.
The earliest methods of quantifying the residual elastic strains within work pieces strategically remove material and observe the resulting deformation that occurs as the internal forces re-balance themselves \autocite{sachs1927detection, mathar1934determination}.
Today's implementations of this principle rely on the change in electrical resistance in strain gauge rosettes for high precision measurements around drilled holes\nocite{schajer2013practical}\autocite{schajer2013hole, nobre2018plasticity, peng2021residual}.

A more direct but technologically more involved approach is the measurement of local elastic strain within a crystalline material by \gls{xrd} and subsequent evaluation of residual stresses\autocite{murray2013applied}.
The crystal lattice will scatter an incoming X-ray beam according to Bragg's law, where the only material parameter is the distance between the diffracting lattice planes, which is in turn directly influences by elastic strains.
In the case of \gls{hexrd} of poly-crystalline material performed in transmission mode, Debye-Scherrer rings are usually recorded by a two dimensional (2D) detector and represent diffraction on favorably oriented crystallites of various diffraction vector orientations.
Due to the presence of residual stresses, the rings may change their diameter and/or transform into ellipses, whereby the ratio of the principal axes holds information about the two dimensional residual elastic strain state normal to the beam  \autocite{withers2013applied}.
By scanning over the area of a thin slice of a sample, a two dimensional impression of the internal stress gradient can be imaged\autocite{todt2018gradient,bodner2021correlative}.

Thicker parts with more material not only weaken the exiting beam significantly, but also bear the problem of three-dimensional stress states, one axis of which will always be lost due to being in line with the beam.
Another issue with thicker samples is the difference of the surface stress state being two dimensional as opposed to within the volume.
Since the electron beam passes through both of these stress states and the gradient in between, the resulting Debye-Scherrer Rings will be an average of all stresses through the thickness of the sample and likely blurred and imprecise.
Methods such as neutron diffraction can take measurements from within the volume of 3D parts\autocite{withers2004depth1}, but with lower spacial resolution as a trade-off. 
Often, it is simpler to analyze  thin samples cut from a 3D part, which give clear measurements but come with their own challenges. 

When freed from the surrounding volume, a sample's residual strains are able to relax and greatly reduce the internal stresses that were present in the untouched part, although the resulting deformation of the cut plane can itself be used to glean information about the stress state before the cut.
This is achieved using \gls{fem} simulation to calculate the stresses it would take to reshape a deformed cut surface back to its original geometry\autocite{prime2013contour}.
 

