Scratch hardness is not indentation hardness
Both tests press a diamond into a surface and divide the load by an area. They divide by different areas, on purpose. An indenter is pushed straight down and the whole contact carries load; a stylus is dragged, and only the leading half of the contact is touching anything that pushes back.
That is the entire content of the ASTM G171 definition. Take the residual groove, measure its width w, treat the contact as a circle of that diameter, and use half of it:
Aload = ½ · (π w² / 4) = π w² / 8 HS = P / Aload = 8 P / (π w²) H = P / (π w² / 4) = 4 P / (π w²) HS[GPa] = 10³ · P[N] / A[µm²]P is the normal load and w the groove width. The half is not an empirical correction: it is the statement that the trailing half of the contact has nothing beneath it, because the stylus has already moved on and the material behind is either recovering elastically or gone. Compare with indentation hardness over the same circular contact, H = P/A = 4P/(π w²), which is exactly half of HS.So a factor of two is definitional and should not surprise anyone. What is more interesting is that measured scratch hardness usually exceeds independently measured indentation hardness by more than that factor, and for three separate reasons. The contact is being sheared as well as compressed, so the stress state in front of the stylus is more severe than the pure-normal Prandtl field under an indenter. The strain rate is higher: a stylus at 10 µm/s through a contact 40 µm across strains the material at of order 0.25 s⁻¹, against 0.05 s⁻¹ for a slow indent, and hardness is rate-sensitive in almost everything. And the groove you measure is the recovered groove, narrower than the loaded contact, which pushes HS up again.
Vertical scale exaggerated.
Plan view, to scale. Travel is left to right.
How was the width measured?
Why the width convention is the whole ball game
HS depends on w−2. Differentiate and you get the sensitivity that makes this section necessary:
δHS / HS = −2 · δw / w A 5% error in width is a 10% error in hardness. A 10% error is 21% (the exact figure is 1/0.9² − 1 = 23% if you under-measure and 1 − 1/1.1² = 17% if you over-measure; the error is not symmetric, which is its own small trap). Nothing else in the measurement chain is this leveraged.The four conventions in the workbench are all things people actually do, and all four are defensible in isolation:
- Optical, groove floor only. Under a light microscope the groove reads as a dark band. The eye picks the boundary of the visibly disturbed floor, which sits a little below the true edge at the original surface, so the width comes out narrow and the hardness high. Diffraction blurs the boundary by a few tenths of a micron on top of that.
- Optical, shoulder to shoulder. The operator measures to the outer visible edge of the disturbed band, which includes the pile-up ridges. Widest of the four, so the lowest hardness. This is the most common way to get a scratch hardness that disagrees with everyone else's.
- Profilometry at the original surface level. Level the trace on the untouched surface far from the groove and take the width where the profile crosses zero. This is the ASTM G171 intent and the one to report.
- Profilometry at the deepest point. Take the maximum depth of the trace and convert it to a width through the known tip radius, w = 2√(2Rh − h²). Clean in principle, but a profilometer levelled on the pile-up crests reports h + hp instead of h, and the inferred width comes out too large.
Go deeper: what the model behind the four numbers actually assumes
The groove is a circular arc of the stylus radius R, so its half-width at the original surface is a = √(2Rh − h²) and the exact cross-sectional area of the groove is R²·arcsin(a/R) − a(R − h). The pile-up is modelled as one bump per side, rising from the groove lip to a crest of height hp and returning to the original surface a distance L further out, with shape 4u(1 − u) in the normalised lateral coordinate u. L grows with the pile-up height, because bigger ridges spread further.
None of that is a theory of pile-up. It is a shape with the right qualitative behaviour, chosen so the four width conventions separate the way they separate in real data. The fraction of displaced material sitting in the shoulders is then a derived number, printed in the readouts, and it is well below 100%, because in a scratch a good part of the displaced volume goes forward into the ridge ahead of the stylus rather than sideways. That is section 2's subject.
The optical offsets are deliberately crude: convention (a) is evaluated at a depth of (0.18 + 0.22·hp/h)·h below the surface, convention (b) out to 90% of the shoulder's lateral extent, and both carry a few tenths of a micron of edge blur. Change those constants and the spread changes; the point that survives any choice of constants is the sign of each bias and the factor-of-two leverage of w−2.
Set P = 5 N and w = 40 µm with the profilometry-at-surface convention. The load-bearing area is π(40)²/8 = 628.3 µm². Hardness is 5 N / 628.3 µm² = 7.96 × 10⁻³ N/µm², and since 1 N/µm² = 10⁶ MPa = 10³ GPa, that is 7.96 GPa, about what you would expect from a hardened tool steel. The card marked c should read 7.96.
A scratch hardness without a stated width definition is not a measurement, it is a number. Say which instrument measured the width, at what height the width was taken, and whether pile-up was included, and if you are comparing two materials, make sure the pile-up behaviour is similar in both, because a material that piles up heavily will look softer than it is under convention (b) and harder than it is under convention (a).
Ploughing, wedging, cutting
Whether a hard asperity pushes material out of the way or removes it is a geometry question, and it has a surprisingly sharp answer. The same question decides whether a surface wears mildly for years or is machined away in an afternoon: a single-asperity scratch is the unit event of abrasive wear, and everything two-body abrasion does is this event repeated a few million times.
The controlling variable is the degree of penetration, the groove depth divided by the contact half-width:
Dp = h / a, a = √(2Rh − h²), θ = arcsin(a / R) h is the groove depth, a the contact half-width, R the tip radius and θ the effective attack angle: the slope of the stylus surface where it meets the original surface. Substitute and the two collapse into one exact identity: θ = 2·arctan(Dp), with no R in it. The tip radius sets how big the scratch is; Dp alone sets what kind of scratch it is.The second variable is how well the interface sticks. Write the shear strength of the interface as a fraction of the shear yield strength of the material being scratched, f = τ/k, running from 0 for a perfectly slippery contact to 1 for a fully seized one. Hokkirigawa and Kato mapped the three regimes in this (f, Dp) plane in 1988, and the map below is that map.
Drag the marker, or use the sliders: they are the same two numbers
Vertical scale exaggerated.
Ploughing friction is pure geometry. Adhesion is the part a lubricant changes.
What the boundaries are, and what they are not
Hokkirigawa and Kato's diagram is experimental, backed by a slip-line analysis that gets the trend but not the numbers. The version drawn above is a two-point linear digitisation of the published fields:
ploughing → wedge : Dp = 0.040 + 0.100 · f wedge → cutting : Dp = 0.155 + 0.125 · fAnchored on the published values Dp ≈ 0.06 and ≈ 0.18 at f = 0.2, and on boundaries that rise roughly linearly to f = 1. These are digitised approximations of a figure, not a theory and not a fit to raw data. Use them to reason about which way a change pushes you, not to predict a transition to two significant figures.The physical story is straightforward. At shallow penetration the material can flow around the stylus and close up behind it: nothing is removed, everything is displaced, and the surface simply acquires a groove with ridges. As the attack angle steepens the flow can no longer get around, and material starts accumulating as a stable wedge, a prow, riding in front of the tip, shedding some of itself as loose debris. Steeper still and the flow field switches to the one a lathe tool produces: a continuous chip on a shear plane, and most of the groove volume leaves.
A sticky interface delays all of this. High f means the material cannot slide along the stylus face, so it is dragged around with the tip rather than up it, which is why the boundaries rise with f, and why a lubricant that lowers f can move you into the cutting regime rather than out of it. That is not intuitive, and it is one of the more useful things this map tells you.
Go deeper: the friction split and the wear-rate scaling
Total friction in a scratch splits into a term from making and breaking the interface and a term from pushing material out of the way. The second is pure geometry: for a cone or a sphere with effective attack angle θ, the ratio of the horizontal projected area to the vertical projected area gives
μ = μadhesion + μploughing, μploughing = (2/π)·tan θ With θ = 2 arctan Dp this is (2/π)·2Dp/(1 − Dp²), which for shallow grooves is just (4/π)·Dp. A blunt stylus at Dp = 0.05 contributes 0.06 to μ; the same stylus at Dp = 0.3 contributes 0.42.The adhesion term used here is the textbook one, τ over the mean contact pressure, with the mean pressure taken as H and H ≈ 3Y ≈ 6k for a fully plastic contact. That gives μadhesion ≈ f/6, so even total seizure buys only about 0.17, famously too small, which is the observation that motivated junction-growth theory. Treat it as a floor, not a prediction.
The wear-rate scaling drops out of the same geometry. If a fraction β of the groove leaves as a chip, the volume removed per unit sliding distance is β times the groove cross-section, and the normal load is the hardness times the load-bearing half of the contact:
V / s = β · Ag ≈ β · (4/3)·a²·Dp, P ≈ H · (π a² / 2) k = V / (P·s) = (8 / 3π) · β · Dp / HWith H in GPa the result is already in mm³/(N·m), the standard unit for specific wear rate, because 1/(1 GPa) = 1 mm³/(N·m) exactly. The contact radius cancels in this idealised derivation: wear rate depends on the shape of the asperity, not its size (real wear can win back a size dependence through plasticity, fracture, debris and scale effects). Everything severe about severe wear is in β and Dp.The β(Dp, f) curve in the readouts is a smooth interpolation anchored on the values Hokkirigawa and Kato report for their three fields: near zero in ploughing, a few tenths in wedge forming, approaching one in cutting. It is a shape, not a law, and it is the least trustworthy thing on this page.
Within this single-asperity abrasive model, mild-looking and severe-looking wear are not two different mechanisms competing: they are the same mechanism at two different attack angles. (The broader mild-to-severe transition in sliding wear brings oxidation, adhesion, delamination and heat into it.) The design levers are therefore geometric: blunt the asperities (raise R at fixed load, lowering Dp), harden the surface (lower h at fixed load), or change f. Making the surface harder works partly because it is harder and partly because it is now being scratched more shallowly, and those two effects are not separable from a wear-rate measurement alone.
H/E and H³/E²: why the hardest coating is not the most wear-resistant
Everything above says wear is set by how deep the asperity gets, which is set by how the contact deforms, which is set by hardness and stiffness together. Hardness alone is the wrong single number, and it has been the wrong single number since 1951.
Two combinations do most of the work. H/E is an index of elastic strain tolerance (roughly the strain at which a contact stops responding elastically), and it scales how much of a contact springs back when the load comes off. A high-H/E surface accommodates an asperity elastically and lets it go; a low-H/E surface takes a permanent dent from the same event. H³/E² scales the resistance to the onset of plastic deformation: the load needed to first yield a Hertzian contact goes as H³/Er², so it is the natural figure of merit for "how hard is it to start damaging this surface at all".
H / E : elastic-strain-tolerance index, sets the recovered fraction of a contact H³ / E² : load to first yield in a Hertzian contact, in GPaBoth are ratios, so both are dimensionless up to the GPa carried by H³/E². Leyland and Matthews (2000) showed that for coatings the Archard wear coefficient correlates far better with H³/E² than with H, and Oberle had already noticed the H/E part in 1951. The lesson took fifty years to be acted on, and is still not universal in coating datasheets.| Rank | Material | H (GPa) | E (GPa) | H/E | H³/E² (GPa) | 5H/Er |
|---|
The DLC argument
Hydrogenated diamond-like carbon is about 20 GPa hard, against 45 GPa for tetrahedral amorphous carbon and 28 GPa for sapphire. It is not the hardest thing in the list and it is nowhere near. But its modulus is around 150 GPa (low for a carbon, because the amorphous hydrogenated network is compliant), and that puts H/E at 0.13 and H³/E² at 0.36 GPa, both the highest here.
What that buys is the ability to deform elastically through an event that would crack something stiffer. Push an asperity into sapphire and the contact is stiff, the stress rises fast with penetration, and the load to reach the critical tensile stress at the contact edge arrives quickly. Push the same asperity into DLC and the surface deflects, spreading the contact and keeping the peak stress down, then springs back. Roughly two thirds of the depth recovers. That is why DLC survives contacts that spall a much harder ceramic coating, and it is the whole basis of the nanocomposite coating design philosophy Leyland and Matthews argued for: do not maximise hardness, maximise H³/E², which usually means accepting a slightly softer coating with a much lower modulus.
The metric has limits, and the table shows one of them plainly. Fused silica ranks third, above TiN, because it is moderately hard and remarkably compliant for a ceramic, but silica is brittle in a way that H³/E² knows nothing about, and it would not last five seconds as a wear coating. H³/E² is a resistance-to-first-yield scaling, not a toughness, not a fatigue life and not an adhesion strength. It is a better single number than hardness. It is still a single number.
Go deeper: where the H³/E² grouping comes from
Take a Hertzian sphere-on-flat contact. The mean contact pressure at first yield is a fixed multiple of the yield stress, pm ≈ 1.1 Y, and the mean pressure at a given load and radius goes as (P Er² / R²)1/3. Set the two equal and solve for the load at which yield begins:
Py ∝ R² · Y³ / Er² ∝ R² · H³ / Er² using H ≈ 3Y for a fully plastic contact. The R² is fixed by the counterface, so the material's whole contribution to "how much load can I put on before anything permanent happens" is the group H³/Er². This is a genuine derivation, unlike the β curve in section 2: the approximation is only in H ≈ 3Y and in using H measured at full plasticity as a proxy for Y.The elastic recovery estimate in the readouts is the Lawn–Howes scaling: the residual depth of a contact after unloading is hf/hmax ≈ 1 − 5H/Er, so the recovered fraction is 5H/Er. Both are printed above because the two are easy to confuse and only one of them goes up when a material gets springier. Er is the reduced modulus against diamond, 1/Er = (1 − ν²)/E + (1 − 0.07²)/1141, which matters for the stiff ceramics and is negligible for the polymers.
The other half of this story is the wear side, and it is not on this page. Archard's equation and what its wear coefficient actually means, the units problem in specific wear rate, how surface roughness parameters change with the filter you chose, and the Stribeck curve as a live model; those live in the tribology series, linked in the rail below.
Sources & further reading
- ASTM G171, Standard Test Method for Scratch Hardness of Materials Using a Diamond Stylus: defines HS = 8P/(πw²) and the 200 µm radius stylus used as the default above.
- K. Hokkirigawa, K. Kato, “An experimental and theoretical investigation of ploughing, cutting and wedge formation during abrasive wear,” Tribol. Int. 21, 51 (1988): the wear-mode map in section 2, and the degree-of-wear parameter β.
- J. A. Williams, “Analytical models of scratch hardness,” Tribol. Int. 29, 675 (1996): why scratch hardness and indentation hardness differ by more than the factor of two, and the slip-line fields behind it.
- A. Leyland, A. Matthews, “On the significance of the H/E ratio in wear control: a nanocomposite coating approach to optimised tribological behaviour,” Wear 246, 1 (2000): the paper that made H/E and H³/E² standard coating design targets.
- T. L. Oberle, “Properties influencing wear of metals,” J. Met. 3, 438 (1951): the original observation that elastic strain capacity (H/E), not hardness, tracks wear.
- J. Musil, “Hard nanocomposite coatings: Thermal stability, oxidation resistance and toughness,” Surf. Coat. Technol. 207, 50 (2012): H³/E² as a design parameter, with the resistance-to-plastic-deformation argument set out at length.