Steel Quench Hardening: The Martensite Transformation Front

Steel & Quenchant

Carbon content 0.40 wt% C

Quenching medium

Bar radius 18 mm

Simulation speed 6×

Live Readout

850°C

martensite

Elapsed time

0.0 s

M_s start temp

357 °C

Surface temp

850 °C

Core temp

850 °C

Avg. martensite

0 %

Surface hardness

~10 HRC

How it works

The bar's cross-section is split into concentric radial rings and cooled by explicit finite-difference conduction — the axisymmetric heat equation with a convective surface boundary:

Interior:  ∂T/∂t = a·[ ∂²T/∂r² + (1/r)·∂T/∂r ]
Center:    ∂T/∂t = 4a·(T₁-T₀)/Δr²        (symmetry, r→0)
Surface:   ρc(Δr/2)·∂T/∂t = k(T_{N-2}-T_{N-1})/Δr − h(T_{N-1}-T∞)
a = k/(ρc) ≈ 1.17×10⁻⁵ m²/s  (plain-carbon steel)

Each ring tracks the coldest temperature it has reached, T_min. Once T_min drops below the martensite-start temperature M_s, the diffusionless austenite→martensite transformation follows the Koistinen-Marburger relation:

Ms(°C) ≈ 539 − 423·(%C)              (Andrews' relation, carbon term)
f_M(T) = 1 − exp[ −0.011·(Ms − T) ]   for T < Ms, else 0

Because M_s falls as carbon content rises, a higher-carbon steel needs to be chilled further before martensite starts forming — but it ends up harder. Hardness is estimated per ring by blending a soft, untransformed baseline with a carbon-dependent maximum:

HRC_max(%C) ≈ min(66, 20 + 58·%C)
HRC(ring)   ≈ 10·(1 − f_M) + HRC_max·f_M

Carbon content — sets M_s and the achievable hardness ceiling.
Quenching medium — sets the surface heat-transfer coefficient h: water quenches fast and hard but risks cracking; oil is gentler; still air barely hardens a thick bar at all.
Bar radius — a thicker bar has a longer conduction path, so its core cools far more slowly than its surface — the classic "mass effect" that limits how deep a quench can harden a real part.
The colour front sweeping inward is the actual computed transformation boundary, not a canned animation — slow the quenchant or thicken the bar and you can watch the core stay soft, unhardened austenite while the skin turns to hard martensite.

Hot austenite Martensite

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