Heat Treatment of Steel: Quench Severity, the Jominy Curve and the 4% Volume Change That Quietly Decides Your Tolerance

Grossmann Di and quench severity H, Jominy end-quench curves, Ms and retained austenite, Hollomon-Jaffe tempering, carburising and nitriding kinetics, induction skin depth — with worked numbers and the distortion cost ledger.

·

Heat Treatment of Steel: Quench Severity, the Jominy Curve and the 4% Volume Change That Quietly Decides Your Tolerance

A drawing that says EN19, harden and temper to 28–32 HRC is not a heat treatment specification. It is three specifications wearing one line of text: the hardness at the surface, the hardness at the core, and how much the part will move while getting there. A mid-size Indian job shop will quote the line at ₹80–90/kg and return parts that pass a Rockwell check at three points — and still bend 0.2 mm over 300 mm, still grind 0.4 mm off a bore to clean up a scale pocket, still scrap 4% of the batch to quench cracks that only appear after finish grinding opens them.

None of that is mysterious. It is arithmetic. Heat treatment is the only manufacturing operation whose input and output are both a dimension — and the operation itself changes volume by about 4%.

This guide works the numbers end to end: what martensite actually is, how to read a Jominy curve the way a bearing manufacturer reads a datasheet, how Grossmann's ideal critical diameter turns chemistry into a through-hardening limit, why a 50 mm EN19 shaft hardens to a different number than a 25 mm one, how much of the quench is spent under a vapour blanket doing nothing, where retained austenite comes from and what it costs, why tempering is a time-temperature equivalence problem rather than a temperature table, and what all of it adds to the price of a part.

1. What martensite is, and why every heat-treating shop is really selling distortion control

Austenite is face-centred cubic — four atoms per unit cell, octahedral interstices that dissolve up to 2.11 wt% carbon, lattice parameter around 3.65 Å. Martensite is body-centred tetragonal: the same iron lattice sheared by a Bain strain, with the carbon trapped in one set of octahedral sites because there was no time to diffuse out. The trapped carbon inflates the c-axis, and the lattice has to fit more volume into less.

The numbers that matter, in densities:

A 3.9% volumetric expansion becomes a 1.3% linear expansion — (1.039)^{1/3} - 1 = 1.28\% — if it happens uniformly in all three directions. On a 50 mm diameter that is 0.64 mm on the diameter. Nobody measures 0.64 mm of growth on a hardened shaft, because the expansion does not happen uniformly and never happens freely: the surface transforms first (it cools first), while the core is still austenitic, hot and plastic. The core restrains the surface; the surface puts the core into tension; by the time the core transforms, part of the expansion has already been spent as plastic strain. The measured growth on a 50 mm through-hardened shaft is typically 0.10–0.25 mm on diameter — a third of the free-expansion figure — and the difference went into residual stress and shape change.

That is the central point of this entire subject: the expansion that did not become size became stress, and stress that exceeds yield at temperature becomes shape change. Size change is predictable and can be absorbed by a tolerance. Shape change is the enemy, because it is asymmetric and consumes grinding allowance unevenly.

One more number to anchor the next section. The carbon that martensite cannot hold also raises hardness. For a fully martensitic microstructure, carbon content sets as-quenched hardness almost independently of everything else. The practical relation in HRC terms, fitted across plain-carbon and low-alloy steels, is:

At 0.40%C that returns 61.6 HRC, and after accounting for retained austenite and incomplete transformation, real 4140 quenched from 850 °C reads 55–57 HRC. At 0.80%C — a carburised case — the same equation says 74.7 HRC, which does not exist; the maximum attainable hardness in a 0.8%C martensite is around 64–66 HRC because retained austenite takes the rest. Carbon buys hardness with brutal diminishing returns, and past about 0.6%C it mostly buys retained austenite instead.

2. Hardenability is not hardness — the Jominy curve is the real datasheet

Hardness is what you measure at a point. Hardenability is how deep the steel can be made hard, and it is a completely different property. A 0.4%C plain carbon steel and a 0.4%C chrome-moly steel have the same maximum hardness (about 57 HRC) and utterly different hardenability: the plain carbon grade will not through-harden a 20 mm bar in water, the chrome-moly will through-harden a 100 mm bar in oil.

The measurement is the Jominy end-quench test, standardised in ASTM A255 and ISO 642. A 25 mm diameter × 100 mm bar, normalised, austenitised at the grade's temperature (845–870 °C for most alloy steels), transferred in under five seconds to a fixture, and quenched by a 24 °C water jet through a 12.5 mm orifice against one end face only. The bar then cools along a controlled gradient: over 300 °C/s at the quenched end, under 5 °C/s at the far end. Two flats are ground 0.4 mm deep, 180° apart, and hardness is surveyed from 1.5 mm onward — every 1.6 mm for the first 12.7 mm, then every 3.2 mm.

The result is a curve, and the curve is the steel's fingerprint. The cooling rate along the bar follows, to a good engineering approximation:

At x = 1.5 mm that gives 300 °C/s; at 10 mm, 25 °C/s; at 20 mm, 8 °C/s. Those three numbers, and the hardness at each, are what every subsequent calculation in this article hangs on.

For EN19 / 42CrMo4 / AISI 4140 — 0.38–0.45%C, 0.60–0.90%Mn, 0.90–1.20%Cr, 0.15–0.30%Mo — a typical published Jominy band reads:

Distance from quenched end · Cooling rate · Hardness band

J 1.5 mm · ~300 °C/s · 54–58 HRC

J 5 mm · ~75 °C/s · 52–56 HRC

J 10 mm · ~25 °C/s · 47–53 HRC

J 15 mm · ~15 °C/s · 42–49 HRC

J 20 mm · ~8 °C/s · 38–45 HRC

J 40 mm · ~2 °C/s · 30–38 HRC

Now read the same table as a purchasing engineer. Two suppliers both ship "42CrMo4". One holds the top of the band at J20 (45 HRC), the other the bottom (38 HRC). At J10 the gap is 6 HRC, which is roughly 200 MPa of tensile strength and a factor of two in fatigue life at the same stress. The grade name does not contain that information. Only the Jominy band does, and it is not in the EN 10083 mill certificate unless you ask for it.

3. Grossmann's ideal critical diameter: turning chemistry into a size limit

Jominy curves are measured, not designed. To design, Grossmann (1942) collapsed the curve into a single number — the ideal critical diameter D_I: the diameter of an infinitely long bar that hardens to 50% martensite at its centre under an infinitely severe quench (H = \infty).

D_{I,C} is the base diameter from carbon content and ASTM austenite grain size. At 0.40%C and grain size 7, D_{I,C} \approx 0.21 in (5.3 mm). The multipliers are read off charts; to first order they behave like f_x \approx 1 + k_x w_x, with chromium around k \approx 2.2 per wt% and molybdenum around k \approx 3.

Worked for EN19 at 0.40C / 0.80Mn / 0.25Si / 1.05Cr / 0.20Mo:

Compare the same arithmetic on EN8 (0.40C, 0.80Mn, 0.25Si, no Cr, no Mo): D_I \approx 0.21 \times 2.9 \times 1.2 \approx 0.73 in ≈ 18 mm. Same carbon, same maximum hardness, five times less hardenability. This is why EN8 hardens in water and EN19 hardens in oil, and why specifying EN8 for a 60 mm section and then asking for 30 HRC at the core is a drawing error, not a supplier error.

The bridge from D_I to a real part is the quench severity factor H, defined as H = h/k where h is the surface heat-transfer coefficient and k the steel's thermal conductivity. H has units of inverse length and is the single number that describes your quench tank:

Medium · Agitation · H (in⁻¹)

Air · still · 0.02

Air · forced · 0.05

Oil · still · 0.25–0.30

Oil · moderate · 0.35–0.50

Oil · strong · 0.50–0.80

Water · still · 0.90–1.00

Water · moderate · 1.00–1.50

Water · violent · 1.50–2.00

Brine 10% NaCl · moderate · 2.0–2.5

Brine · violent · 2.5–5.0

D_I and H together give the actual critical diameter D_c — the bar that hardens 50% martensite at the centre in your tank. Grossmann's nomograph is the classical way to read it; the shape of the relation is that D_c saturates as H \to \infty and collapses as H \to 0. For D_I \approx 95 mm the chart gives roughly:

Which answers the question the drawing never asked: EN19 in still oil will not through-harden a 50 mm shaft. The centre sees less than half martensite. The surface reads 55 HRC, the core reads 38–42 HRC, and the part is not wrong — it is exactly what the physics promised. If the requirement was 45 HRC at the core, the fix is agitation (raising oil to H \approx 0.5 moves D_c to 52 mm), a deeper-hardening grade (EN24/4340 with D_I \approx 150 mm), or a smaller section. Changing the tempering temperature will not fix it, and neither will complaining to the heat treater.

4. The quench itself: three stages, and the one that does nothing useful

Heat extraction from a hot part into a liquid is not a constant coefficient. It runs through three regimes, and the part spends a disturbing fraction of its cooling time in the first one.

Stage A — vapour blanket (film boiling). The part enters above the Leidenfrost temperature; the liquid instantly vaporises and the part is wrapped in a stable gas film. Heat transfer is poor, h \approx 100–500 W/m²K, and cooling is slow and — critically — uneven, because the film is not of uniform thickness.

Stage B — nucleate boiling. Once the surface drops below the Leidenfrost point, the film collapses and the liquid contacts metal directly. Bubbles nucleate, grow and depart at high frequency; this is where h peaks, typically 3,000–8,000 W/m²K for oil and 10,000–30,000 W/m²K for water. This is the stage that decides whether the steel misses the pearlite nose.

Stage C — convective cooling. Below the boiling point of the quenchant, heat leaves by convection. h drops to a few hundred W/m²K. The part is already martensitic or already lost; the only question here is whether the cooling through the M_s–M_f range is gentle enough to avoid cracking.

What this means in the tank: the vapour-blanket stage is a delay you pay for in seconds and distortion, and its length depends on the quenchant's vapour pressure, viscosity and — above all — agitation. Stopping the pump "to reduce cracking" lengthens Stage A, makes cooling less uniform, and typically makes cracking worse while making hardness worse at the same time. Agitation does two jobs: it strips the vapour film (raising effective H) and it equalises the surface temperature around the part (reducing distortion).

Two counter-intuitive facts that separate a good heat treater from an expensive one:

  1. Cold oil is slower than warm oil. For conventional cold quench oils, raising the bath from 20 °C to about 50–65 °C lowers viscosity, shortens the vapour-blanket stage, and increases the cooling rate through the critical range. Most oils are specified for use at 50–65 °C; running them at ambient in a cold workshop is a performance loss, not a safety margin. Above ~65 °C the convection stage slows and the fire risk rises, so the window is real and narrow.
  2. Agitation matters more than the medium's name. Still oil (H \approx 0.3) and well-agitated oil (H \approx 0.6) differ by more than still oil and still water do at the low end. A shop that "uses oil because water cracks" and then never moves the oil is running at the bottom of the oil band and blaming the steel for shallow hardening.

The re-wetting front — the boundary between the still-blanketed and the nucleate-boiling regions — sweeps across the part as it cools, and wherever it stalls (a shoulder, a keyway, a fixture contact point) there is a soft spot and a stress concentration at the same time. This is why the same steel and the same tank produce good parts on a 50 mm shaft and bent parts on a 50 mm flange: geometry, not grade.

5. Ms, Mf and the austenite you cannot see

Martensite does not start at the quenchant temperature. It starts at M_s, set by composition. Andrews' 1965 regression, still the standard reference, is:

Calibration range 0.03–0.64%C, up to 9.4%Cr, 4.3%Mo; standard deviation ±17 °C. For EN19 (0.40C, 0.80Mn, 1.05Cr, 0.20Mo, 0.25Si) that gives M_s \approx 325 °C. For a 0.8%C carburised case it gives 175 °C. Carbon is the dominant depressant at −423 °C per wt% — a brutal coefficient that explains almost everything about high-carbon tool steels.

Below M_s, the fraction transformed follows Koistinen–Marburger:

Quenched to 20 °C, EN19 retains:

Three and a half percent retained austenite is harmless. A 0.8%C case is not: M_s = 175 °C gives f_\gamma = \exp(-0.011 \times 155) = 18\% — and in real carburised cases with 1%Cr and 0.3%Mn the measured figure is routinely 20–35%, because austenite stabilises as it is constrained by the surrounding martensite and because the nickel-bearing grades stabilise it further. ISO 6336 and DIN 3990 cap retained austenite at 30% for material quality MQ; AGMA 2001 allows 30% for grades 2 and 3, and sub-zero treatment is the standard route to bring it below 20%.

Why it matters, in order of commercial pain:

The practical fixes are all cycle changes, not grade changes: lower the carbon potential so surface carbon lands at 0.75–0.85% instead of 1.0%; quench to below 50 °C rather than pulling the basket at 120 °C; sub-zero treat at −70 to −80 °C (or −196 °C for tool steels) immediately after the quench, before the austenite stabilises; and temper immediately after the sub-zero soak, because fresh martensite is brittle and full of stress.

6. Tempering: two variables, one parameter, and two embrittlement windows

Quenched martensite at 57 HRC is not a usable engineering material — it has almost no toughness and residual stresses near yield. Tempering is the operation that converts hardness into a specification, and it is governed by a time-temperature equivalence that most shops treat as folklore.

Hollomon and Jaffe (1945) showed that hardness after tempering is a function of a single parameter:

with C \approx 20 for plain-carbon and low-alloy steels, 14–15 for tool steels, and 15–19.5 depending on the source's convention. The consequence is that time and temperature are interchangeable in a very specific ratio: at 600 °C, one extra decade of time (\times 10) is worth about 30 °C. Which means a 10 °C furnace error at 600 °C is equivalent to a three-hour time error on a one-hour temper — the furnace's control thermocouple is a bigger quality variable than the soak clock.

Worked: EN19 quenched to 57 HRC, target 300 HBW (≈ 32 HRC).

The two are within 2% of each other in parameter space, which is the point: nobody sinters a 72-hour temper, but the equivalence explains why a well-controlled continuous furnace at 560 °C for 90 minutes and a batch furnace at 540 °C overnight can both hit the same hardness — and why the overnight one is more forgiving of a thermocouple that reads 8 °C high.

Working hardness ranges for the two grades that carry most of Indian job-shop work:

Grade · Austenitise · Quench · Temper · Result

EN8 / C45 · 820–850 °C · water or oil · 550–660 °C · 22–30 HRC

EN19 / 42CrMo4 · 840–870 °C · oil or polymer · 540–680 °C · 240–310 HBW (≈ 24–33 HRC)

EN19, higher strength · 840–870 °C · oil · 400–500 °C · 36–45 HRC

EN31 / 100Cr6 · 830–860 °C · oil · 150–200 °C · 58–65 HRC

20MnCr5, carburised · 880–980 °C (case) · oil · 150–200 °C · 58–62 HRC case

Two windows to avoid, and they are not the same window:

The other tempering output nobody quotes is dimensional: tempering shrinks the part slightly (the trapped carbon precipitates as carbides, relieving the tetragonal distortion) and relieves residual stress, which can move the part again. A shaft straightened after quenching and then tempered at 600 °C will partially spring back; the correct sequence is temper first, then straighten (if you must), then stress-relieve below the tempering temperature.

7. Case hardening: three different processes with three different economics

When the requirement is 58–62 HRC on the surface and 30–40 HRC at the core, through-hardening cannot deliver it — a 0.4%C steel cannot reach 60 HRC at any depth. The surface has to be enriched or selectively heated, and the three routes have entirely different cycle times, distortion signatures and costs.

7.1 Carburising: diffusion in austenite, and the square root that sets your furnace bill

Carburising holds the part in a carbon-bearing atmosphere at 900–980 °C, where austenite dissolves up to 2.11%C. Carbon diffuses in, and the profile follows Fick's second law for a semi-infinite solid at constant surface concentration:

The engineering form of that solution is the Harris equation, and it is the only formula a production planner needs:

with d the case depth in mm, t in hours, and K \approx 0.6–0.65 mm/√h at 925 °C for total case depth. The square root is the whole story: doubling case depth takes four times the furnace time. At 925 °C, 1 mm takes about 2.5–3 h and 8 h yields 1.7–1.9 mm; a 2.5 mm case needs roughly 15 h, and a 4 mm case needs 40 h at a temperature where grain growth starts to matter.

The surface carbon is set by the atmosphere's carbon potential, and typical practice runs it at 0.8–0.9%C — deliberately not higher. Above about 1.0%C you get grain-boundary carbide networks, retained austenite above 30%, and a case that spalls rather than wears. Boost-diffuse cycles (high potential for 2–4 h, then drop the potential and hold) shorten the total cycle by 10–20% for the same effective case depth and flatten the carbon profile, which is why every modern atmosphere furnace runs them.

Two definitions that cause endless arguments at goods-inward inspection:

The consequences downstream: carburising at 925 °C distorts the part before the quench even starts (creep under its own weight, plus the thermal cycle), the quench then adds transformation distortion, and the case itself grows 0.09–0.12 mm on a 100 mm diameter because a 0.8%C martensite expands more than a 0.4%C core. Every carburised gear drawing should carry a grinding allowance and a "grind after hardening" note; the ones that do not are the ones that come back.

7.2 Nitriding: no quench, no transformation, no distortion — and a very expensive millimetre

Nitriding diffuses nitrogen into ferrite at 500–580 °C, below the A₁ temperature, so no austenite forms and nothing transforms on cooling. There is no quench, no martensite, no 4% volume change. Dimensional change is tens of micrometres, not tenths of a millimetre. That is the entire commercial case for nitriding: it can be the last operation on a finished part.

Hardening comes from fine alloy-nitride precipitates — CrN, MoN, AlN, VN — so the response depends on the steel, not the process. Plain carbon steel nitrides to 250–300 HV; 4140-type grades reach 550–700 HV; aluminium-bearing nitriding steels (EN41B / 34CrAlNi7, Nitralloy 135M) reach 900–1,200 HV. The grade choice is the design decision; the furnace only executes it.

Kinetics are brutal in the same square-root way, with a smaller constant:

So 0.4 mm needs about 45 h and 0.6 mm needs about 100 h. Where carburising charges by the millimetre per hour, nitriding charges by the day. Typical cycles: 0.15–0.20 mm in 15–24 h at 510 °C, 0.40–0.50 mm in 50–72 h.

The compound layer — the "white layer" — is 5–25 µm of γ′-Fe₄N and ε-Fe₂₋₃N. It is either the product or the defect depending on the application: for a cylinder liner or a sliding surface, an ε-rich layer is excellent for scuffing and corrosion; for a gear tooth root or a die-casting die, a thick mixed layer is a crack path and is specified ≤ 8–13 µm or removed by lapping. Control is through nitriding potential K_N = p_{NH_3}/p_{H_2}^{1.5} — the two-stage Floe process runs high K_N briefly to charge the surface, then low K_N to diffuse inward while keeping the white layer thin.

7.3 Induction hardening: skin depth is the case depth

Induction hardening heats a surface layer by eddy currents and then quenches it — usually by an integrated spray — leaving the core untouched and the distortion low because only a thin shell transforms. The depth is set by frequency through the electromagnetic skin depth:

For 42CrMo4 at 850 °C, resistivity is roughly 1.0\times10^{-6}\ \Omega\cdot\text{m} and \mu_r \approx 1 (above the Curie point the steel is paramagnetic), giving \delta \approx 16 mm at 1 kHz, 5 mm at 10 kHz, 1.6 mm at 100 kHz. Practical case depth runs 0.3–0.5 × the calculated skin depth because the power density falls off exponentially and the austenitising isotherm sits well inside. The rule of thumb every induction shop uses:

which gives 10 kHz for a 5 mm case, 80 kHz for 1.8 mm, 400 kHz for 0.8 mm. For a 45 mm shaft needing a 2–4 mm case, 10 kHz is the standard answer; for a gear tooth flank at 0.8–1.2 mm, 100–200 kHz.

Induction's distortion advantage is real but not zero: the thin transformed shell still expands 1.3% linearly, and because it is constrained by an untransformed core it mostly goes into compressive residual stress at the surface — which is exactly what you want for fatigue, and why induction-hardened shafts commonly carry 400–700 MPa of compressive residual stress in the case. The failure modes are different from furnace hardening: decarburised or scaled surface from a bad atmosphere, soft spots where the coil coupling was poor, and cracking at the tooth tips of gears where the case wraps around a sharp corner.

8. The distortion ledger — where heat treatment shows up in the price

Distortion is two different phenomena that shops quote as one:

The physics of shape change has three drivers:

  1. Thermal gradient. The surface cools first and contracts; the hot core restrains it; the surface yields in tension. When the core finally contracts, the surface is in compression and the part bows. The magnitude scales with \alpha \Delta T — for steel, \alpha = 12\times10^{-6}/K, so a 500 K surface-to-core difference in a thin section is 0.6% strain, well past yield at temperature.
  2. Transformation strain timing. The surface transforms first (it is colder), expands, and yields the still-austenitic core in compression; when the core transforms later, it expands against an already-rigid martensitic shell and the residual stress pattern inverts. Whether the part ends up bowed toward or away from a machined flat depends on which of those two events dominates — which is why distortion direction is genuinely hard to predict and why FEA heat-treatment simulation (SYSWELD, DANTE, DEFORM-HT) exists as a commercial product.
  3. Transformation plasticity (TRIP). During transformation, the weaker austenite deforms plastically at stresses well below its yield strength, in the direction of the applied stress. A finite-element study of an asymmetrically cut 4340 cylinder found that including TRIP was necessary to match the measured distortion at all, and that TRIP dominated over the plain volume expansion. In practical terms: a part with a milled flat, an off-centre bore or a keyway will distort in a way that a symmetric part of the same mass does not.

The numbers that make this a commercial problem rather than an academic one, on a 50 mm × 500 mm EN19 shaft (7.7 kg):

Item · Rate / value · Per part

Hardening and tempering · ₹85/kg · ₹654

Grinding, nominal 0.2 mm stock · 0.2 mm at 3 cm³/min, ₹600/h · ₹110

Additional bow cleanup (0.15 mm bow raises eccentric diameter stock from 0.2 to 0.3 mm) · +0.1 mm removal + extra dressing · +₹70

Straightening (press, 0.15 mm bow) · ₹40/piece, adds residual stress · ₹40

Scrap from quench cracking / unstraightenable bow, 3% at ₹3,800 material+prior ops · 0.03 × ₹3,800 · ₹114

Effective heat-treatment cost · · ≈ ₹990

Against a part price of ₹4,200, that is 23% — and only ₹654 of it appears on the heat treater's invoice. The rest is grinding time, straightening labour and scrap, and none of it was quoted. This is the real content of the sentence "heat treatment is expensive": the process fee is a third of the cost.

The levers, in order of leverage:

9. What to put on the drawing

A heat treatment note that can be executed without a phone call has six elements, and most drawings have two:

  1. Grade with a hardenability requirement — "EN19 per EN 10083-3, Jominy band to be declared; J20 ≥ 42 HRC."
  2. Process and target — "Through harden and temper to 285–320 HBW, measured at mid-radius on a sectioned sample" — not just a surface hardness number.
  3. Core requirement — "Minimum 80% martensite at the centre of the largest section" is the honest way to state it, because it forces the D_I conversation.
  4. Case depth with its definition and its measurement standard — "CHD 0.6–0.9 mm, defined at 550 HV1 per ISO 18203" beats "case harden 0.8 mm" by an entire quality department.
  5. Distortion allowance and the cleanup condition — "Grinding stock 0.25 mm per side after hardening; maximum bow 0.05 mm per 300 mm after grinding" — because the allowance and the straightness limit are the same conversation.
  6. Verification — hardness at three points, case depth on a sacrificial coupon from the same load, and a hardness traverse on the first-off. The coupon is cheap; the batch is not.

The reason to write all six is that the alternative is a purchase order that says "harden 55 HRC" and a supplier who delivers exactly that on the surface of a part whose core is 35 HRC, whose bore is oval by 0.06 mm, and whose Jominy band was the bottom of the mill's range. The heat treater did what was asked. The drawing did not.

The short version

Heat treatment is the operation where a machined part stops being geometry and becomes a material. Everything after it — grinding, honing, inspection, assembly — is correcting or confirming what the quench decided.

More FabFlow blog posts