High-Frequency Induction Hardening Simulation

The Physics and Theory of Induction Hardening

The Skin Effect Decides Everything

Induction hardening heats only the surface of a workpiece rapidly, using the Joule heat of the eddy currents that the alternating magnetic field of a coil induces in the surface layer, then quenches immediately to form a martensitic case. What governs the spatial distribution of that heating is the skin effect: the current density decays exponentially from the surface over the penetration depth \( \delta \).

$$ \delta = \sqrt{\frac{2\rho}{\mu \omega}} = \sqrt{\frac{\rho}{\pi \mu f}} $$

For steel at room temperature, \( \delta \) is of the order of 0.1 mm at \( f = 10 \) kHz and a little under 1 mm at \( f = 1 \) kHz — the choice of frequency is directly the design variable for case depth. On top of this comes a twist peculiar to induction heating: as the temperature rises the resistivity \( \rho \) increases, and at the Curie point (about 770 °C in steel) ferromagnetism disappears and \( \mu \) collapses, so the penetration depth jumps several-fold at a stroke. In other words, the place being heated itself moves during heating — this strong nonlinearity is the core of what separates induction hardening simulation from ordinary thermal analysis.

Coupling of the Three Physics: Electromagnetic, Thermal, Metallurgical

Because six orders of magnitude separate the supply frequency from the heating time, the standard split is to solve the electromagnetic field as a frequency-domain (time-harmonic) analysis, obtain the cycle-averaged heat generation density, and feed it as the source term of the thermal analysis. The whole is assembled as a weakly coupled iteration in which the properties (ρ, μ) are updated and the electromagnetic field re-solved every time the temperature changes.

The Quenching Side — What Counts as "Hardened"

Hardening has two conditions. (1) Heating must exceed the austenitising temperature (Ac3, shifted upward by the heating rate); (2) the cooling rate must exceed the critical cooling rate so that martensitic transformation occurs (below Ms, following the Koistinen-Marburger law). The deliverable of the simulation is therefore not a temperature contour but the case pattern and hardness distribution given by "the region that austenitised ∩ the region that was quenched fast enough". Cooling is governed by boiling heat transfer from a spray or an immersion quench (a boiling curve with violent temperature dependence), and is strongly nonlinear in its own right.

Numerical Methods

Implementing the Electromagnetic-Thermal Coupling

1. Electromagnetic field (harmonic analysis) — impose the coil current (or the supply power) and compute the eddy currents and the distribution of heat generation density. The iron rule for the mesh is at least two to three elements within the skin depth (based on the thin δ before the Curie point)
2. Thermal analysis (transient) — advance the temperature with the heat generation density as the source
3. Property update and re-solve — update ρ and μ (the B-H curve) as the temperature rises and solve the electromagnetic field again. Control the update interval by the temperature change, above all through the Curie point
4. Cooling and transformation analysis — quench with boiling heat transfer and compute phase fractions and hardness with transformation models based on CCT and TTT data (JMAK-type for the diffusional products, the KM equation for martensite)

Where Convergence and Accuracy Are Won or Lost

• Passing the Curie point — the collapse of μ transforms the heating distribution. Unless the properties are updated finely through this interval (temperature increments of 10–20 °C), the heating history is missed
• B-H nonlinearity — the region near the surface operates in magnetic saturation. A linear μ approximation overestimates the heating. A measured B-H curve, temperature dependent, is the foundation of accuracy
• Motion (scan hardening) — where the coil or the workpiece moves, treat it either as harmonic analyses at each position assembled into a moving heat source, or as a steady-state problem in a moving frame
• Modelling the power supply — real equipment runs constant-power or constant-current control through a matching circuit. Where the assumption of constant coil current does not match the machine, bring the supply control into the loop

Calibrating the Quench Heat Transfer

The heat transfer coefficient of spray cooling depends on surface temperature (film boiling, then transition boiling, then nucleate boiling) and varies by more than an order of magnitude, and published values scatter widely — h on the cooling side is a parameter to be calibrated independently of the heating side. In practice the temperature-dependent h curve is inversely identified from cooling curves measured with thermocouples embedded in a test piece (the same idea as identifying h for reflow, in a more strongly nonlinear version).

Guidelines for Practical Application

The Standard Workflow

1. Assemble the material data — temperature-dependent ρ, B-H curves and thermal properties, the CCT diagram, and, if distortion is in scope, high-temperature mechanical properties. This is the largest preparation effort of all
2. Calibrate the heating — heat a test piece at production conditions and fit the electromagnetic-thermal model to the temperature history from a pyrometer or thermocouples (identifying the effective current and the efficiency)
3. Calibrate the cooling — identify the boiling h curve from cooling curves
4. Predict and validate the case pattern — compare against cross-sectional macrographs and hardness distributions (a Vickers traverse). Case depth within roughly ±10–15% is what a calibrated model delivers
5. Explore the conditions — study frequency, power, heating time, coil geometry and quench conditions to narrow down the settings that give the target case with minimum distortion

Typical Situations Where Simulation Pays

• Coil design — for shapes such as stepped shafts and gears, working out the coil geometry and coupling distance that at once avoid overheating at edges (corners concentrate flux and risk quench cracking) and unhardened roots. The saving in prototype coils is large
• Setting up a new product — narrowing down initial conditions for a geometry change where conditions cannot simply be carried over from a similar part
• Countermeasures for distortion and cracking — predicting distortion and residual stress from transformation expansion plus thermal stress, and locating the sites at risk of quench cracking (tensile residual stress × rapid cooling)

How to Compare Against Measurement

The strongest reference data for validation is a cross-sectional map of the hardness distribution. It carries more information than point measurements of temperature history, and it lets you check the validity of both the heating (the austenitised region) and the cooling (the region that reached the critical cooling rate) at the same time. Check not only the case depth but the shape of the pattern — the wrap-around at edges, the breaks at the roots: if the depth matches but the pattern does not, that points to an error on the electromagnetic side (the coil model or the coupling distance).

Tool Coverage

Tool Comparison

How to Approach Selection

Decide on (1) the accuracy demanded on the electromagnetic side (optimising coils calls for B-H handling of the dedicated-electromagnetics class), (2) how far the chain must reach through microstructure, hardness and distortion (do you need material models of the DANTE class?), (3) who will use it (a dedicated analyst, or the coil design shop floor). One caveat applies throughout: the material databases built into these tools frequently do not match your own steel grades, and preparing at least your own B-H curves and CCT data is a precondition for accuracy.

Frontiers

Automatic Optimisation of Coil Geometry

Research and implementations that optimise coil geometry, number of turns and coupling distance — parametrically or topologically — against a target case pattern are advancing. Now that a single electromagnetic-thermal coupled case fits within minutes, optimisation over a practical number of iterations has become possible. Combined with 3D-printed manufacture of complex coils, "build exactly the coil the computation designed" is becoming reality.

Integration with Digital Twins and Process Control

Control research is under way that assimilates measured supply waveforms and infrared temperature measurements into the model in real time to compensate for variability (material lot, coil wear, quenchant degradation). The physical model is reduced to a ROM or a surrogate for millisecond response, and the industrial driver behind it is the vision of full quality assurance in which the estimated case depth of every workpiece is logged.

Deeper Prediction of Microstructure and Properties

At the research front are kinetic models for the non-equilibrium austenitisation peculiar to rapid short-time heating (the heating-rate dependence of Ac3, undissolved carbides), the relationship between prior austenite grain size and hardness or fatigue strength, and coupling all the way through to fatigue life prediction including residual stress. The goal of prediction is moving one level deeper, from "case depth" to "component performance (fatigue limit)".

Troubleshooting

Symptoms, Causes, and Fixes

When It Misses, Which Physics to Suspect First

Related: index of industrial thermal process articles, arc welding simulation (a close relative, sharing the moving heat source and phase transformation), reflow oven simulation.