The progressive-load scratch test
Put a rounded diamond on a coated surface, start the stage moving, and ramp the normal load from nothing to a hundred newtons over ten millimetres. The coating does not fail all at once. It fails in a sequence, and the loads at which each new kind of damage first appears are the critical loads.
Three of them are worth naming, and the standards do name them. Lc1 is cohesive: the coating cracks but stays where it is. Fine through-thickness cracks appear in and around the track, usually curving backwards like the wake of a boat, because the stress that opens them is the tensile field dragged along behind the contact. Lc2 is adhesive: the coating starts coming off, in flakes at the track edge or in patches inside it. That is the number people mean when they say "the scratch adhesion was 30 newtons". Lc3 is total removal: continuous exposure of the substrate along the track. It is the least interesting of the three physically and the easiest of the three to see.
The model below builds all three out of one sliding spherical contact. The contact starts elastic and Hertzian; when the mean pressure reaches the coating's hardness it stops rising and the contact goes fully plastic. That single pressure drives everything else: the penetration depth, the tensile stress behind the tip, and the elastic energy stored in the film.
Drag anywhere on the track, or use the stylus-position slider, to read the trace at a point
The chain of reasoning
Everything starts with how hard the diamond is pressing. For an elastic sphere on a flat, Hertz gives the contact radius and therefore the mean pressure. That pressure cannot exceed the material's hardness, so once it gets there it stops and the contact spreads instead:
aH = (3PR / 4E*)1/3 · pH = P / π aH² · pm = min(pH, Hc) · a = √(P / π pm) · d = a² / 2R E* is the reduced modulus of coating and diamond together (Ei = 1141 GPa, νi = 0.07). The last relation is the spherical cap: a contact of radius a on a sphere of radius R has sunk in by a²/2R, accurate while d ≪ R, which for a 200 µm stylus and a few microns of penetration is very comfortable.Now the two stress criteria. A sliding sphere does not just press; it drags. Hamilton and Goodman solved the full stress field under a sliding Hertzian contact, and the piece that matters here is the maximum tensile stress, which sits at the trailing edge of the contact, directly behind the tip. It has a pressing term and a friction term, and the friction term wins quickly:
σt = pm [ (1 − 2ν)/3 + π(4 + ν) µf / 8 ] · σ = σt + σres ν is the coating's Poisson ratio and µf the friction coefficient. σres is the coating's own intrinsic stress, which is added straight on: compressive (negative) residual stress subtracts from the driving tension and pushes the critical loads up, which is one of several reasons hard PVD coatings are deposited with a gigapascal or two of compression in them. At ν = 0.25 the friction term overtakes the pressing term at µf ≈ 0.10.Cracking starts when that stress reaches the coating's fracture strength. Delamination is an energy argument instead: the strained coating is a spring, and if releasing it liberates more energy per unit area than the interface costs to break, the interface breaks.
Lc1 : σ ≥ σf · Lc2 : G = σ² t / 2Ec ≥ Γi · Lc3 : d ≥ t σf is the coating fracture strength, tied here to hardness as σf = Hc/5 so there is one fewer slider; that ratio is the right order for a hard ceramic film and nothing more than that. G is the elastic strain energy release rate per unit interface area, Γi the interfacial toughness in J/m². Only tensile σ contributes: this criterion has no buckling mode in it, which is a real gap and is discussed below.Go deeper: where Lc3 comes from, in closed form
On the fully plastic branch the mean pressure is pinned at the coating hardness, and that makes the penetration depth a linear function of load with no fractional powers left in it. The rest is substitution:
pm = Hc ⇒ a = √( P / π Hc ) ⇒ d = a² / 2R = P / (2 π Hc R) ⇒ d = t at Lc3 = 2 π Hc R t In SI that is exactly it: Hc in Pa, R and t in m, load in N. In the practical units used on this page, Lc3[N] = 2π × 10⁻³ × Hc[GPa] × R[µm] × t[µm].Two things fall out of that expression that are worth carrying around. The load needed to punch through a coating is linear in both thickness and tip radius, so a report of Lc3 without both of those numbers beside it is uninterpretable. And it does not contain the modulus, the interface, or the residual stress at all. Lc3 is a geometry measurement wearing a mechanics costume.
The simulation does not use the closed form. It solves d(P) = t numerically, because at the loads and radii a Rockwell C stylus actually runs, the contact is often still on the elastic branch where pm < Hc. A softer contact is a deeper one at the same load, so the numerical answer is always at or below the closed form, and the closed form is best read as an upper bound:
Critical loads are not material constants
This is the single most abused result in coating characterisation. A critical load is the answer to "at what load did this particular tip, on this particular ramp, on this particular coating thickness, over this particular substrate, first produce damage I could see". Change any one of those and the number changes, even though the coating and its adhesion are untouched.
The table below is live. The right-hand columns recompute every critical load for a stylus of twice the radius with everything else held fixed: same coating, same interface, same residual stress, same friction.
| Critical load | Criterion | Scaling with the test | Lc now | Lc at 2R | Change |
|---|
Doubling the tip radius multiplies Lc1 by very nearly four, because holding the contact pressure constant while the radius grows requires the load to grow as R². Lc3 only grows as R1/2, so at the default settings a blunt enough stylus reaches the substrate before the interface fails and the ordering of the modes inverts, after which the friction jump against the bare substrate drags Lc2 in early and its ratio comes out well below four. Either way the numbers move by factors, not per cent, with the coating untouched. Critical loads are comparable between two coatings tested identically on the same instrument on the same day. They are not comparable across laboratories, across tip conditions, or across coating thicknesses, and a worn stylus quietly drifts a lab's numbers upward over months.
What the damage looks like
ASTM C1624 exists largely to give these patterns names, because a critical load with no failure mode attached to it is not a measurement. The sketches below are plan views of the track. The ones the current settings predict are outlined in copper.
The highlighting is a rule-of-thumb reading of Bull's failure-mode maps (thickness, residual stress sign, elastic recovery and which criterion fired first), not a calculation. Real mode selection also depends on the substrate, which this model does not have.
The three critical loads answer three unrelated questions. Lc1 is about the coating's own strength, Lc2 is about the interface, Lc3 is about geometry. Only Lc2 has anything to do with adhesion, and even that only through an energy balance you cannot invert without knowing the stress state, which is why Lc2 is a ranking tool, not an adhesion measurement.
The three-scan method
A scratch instrument makes three passes down the same line. A light pre-scan of the original surface, the scratch itself under full load, and a light post-scan of whatever is left. Subtracting them is what separates the surface's roughness from the depth the tip reached and from the damage that stayed.
The pre-scan matters more than it sounds. A profilometer measuring a groove one micron deep on a surface that is already half a micron rough is not measuring the groove; it is measuring the sum, and the tilt of the sample on top of that. Running the same tip down the same line at a load small enough to do nothing gives you the baseline to subtract, in the same coordinates, with the same tip, so the tip's own shape cancels too.
The difference between the on-load and post-scan profiles is elastic recovery, and for hard coatings it is not small:
recovery = (don − dres) / don · dres / don = 1 − k · H / Er (k ≈ 5) A Cheng–Cheng-type scaling: the fraction of the penetration that is recovered on unloading is set by the ratio of hardness to reduced modulus, not by either one alone. Clipped to [0, 1]. It is a fit to the elastic–plastic contact problem, not a derivation, and k is the fitted number.Go deeper: what the subtraction still does not fix
Pile-up is not roughness and does not subtract out. A ploughing contact pushes material sideways into ridges along the track. Those ridges are above the original surface in the post-scan, so a naive "mean depth" underestimates the damage, and a cross-section area measurement has to decide whether to count them. Standards differ on this. ISO 20502 and ASTM G171 both end up leaning on the groove width rather than the depth, partly for this reason.
The post-scan tip rides on the shoulders. A 200 µm sphere cannot reach the bottom of a groove narrower than about 2√(2Rd): for a one-micron-deep groove that is roughly 40 µm wide, which is fine, but for the fine cracks inside the track it is hopeless. Everything you learn about cracking comes from the microscope, not the profile.
Recovery is time-dependent. The post-scan happens seconds to minutes after the scratch, and viscoelastic and anelastic recovery continue over that interval. On polymers the "permanent" depth measured immediately after the scratch and the one measured an hour later can differ by tens of per cent, which makes the recovery number a function of how fast the instrument runs its scans.
The one thing subtraction does fix perfectly is tilt and long-wavelength form, because both scans see the same sample in the same position. That alone is worth the extra two passes.
When Ra creeps up towards the residual depth, the difference between the pre-scan and the post-scan stops being damage and starts being the same roughness measured twice in slightly different places. There is no filter that recovers a groove buried in noise of its own size: the fix is a better-polished sample or a higher load, and the second one changes the experiment.
Multi-pass: failure that needs more than one try
A coating that survives a single pass at five newtons can be gone after fifty passes at five newtons. Constant-load repeated scratching is not an adhesion test, it is a fatigue test, and it is much closer to what a coating experiences in service than a single ramp is.
The mechanism is unremarkable: each pass drives the same sub-critical tensile stress through the same material, each pass extends existing flaws a little, and eventually one of them reaches the interface. The bookkeeping is the standard damage-accumulation form, with a Basquin-like exponent doing all the work:
Dn+1 = Dn + C (σ / σf)k, failure at D = 1 ⇒ Nf = 1 / [ C (σ / σf)k ] = (σf / σ)k C is set to 1 so that σ = σf fails on the very first pass, which makes the single-pass criterion of section 1 exactly the n = 1 limit of this one, rather than a separate model. k is the fatigue exponent, 6–10 for hard ceramic coatings. Nothing here is derived; it is the shape that fatigue data has, fitted.The consequence is severe because k is large. Dropping the stress to 80% of the fracture strength buys you a factor of 1/0.8k, which at k = 9 is about seven passes. Dropping it to half buys you five hundred. Between those two lies almost every coating that "passed" a single-pass qualification and then failed in the field.
Depth growth before failure is phenomenological: the fatigue model predicts when, not how deep.
An S–N curve for a coating. The steepness is the whole story.
Go deeper: why the exponent is the only parameter that matters
Take logs of Nf = (σf/σ)k and you get a straight line on log–log axes with slope −1/k. A 10% error in the stress you infer becomes a factor of 1.1k ≈ 2.4 error in lifetime at k = 9. Everything in section 1 that made the stress uncertain (the friction coefficient, the residual stress, whether the contact was elastic or plastic) comes back here amplified to the ninth power.
That cuts both ways, and the useful direction is this: a multi-pass test measures k far more reliably than a single-pass test measures anything. Run five loads, count passes to failure at each, and the slope of the resulting line is a genuine material parameter with a mechanism behind it: subcritical crack growth in the coating, or interfacial fatigue, depending on where the failure ends up. The intercept is contaminated by all the same test-dependence that ruins critical loads. The slope mostly is not.
The step in the depth curve is where the model stops being quantitative. Once the coating spalls locally, the tip contacts a mixture of coating shoulders and exposed substrate, the contact mechanics changes completely, and depth becomes a wear measurement governed by the substrate rather than a contact measurement governed by the film. It is drawn as a clean step to the interface because that is the honest cartoon; the real trace is jagged and the depth depends on the substrate.
Sources & further reading
- ASTM C1624, Standard Test Method for Adhesion Strength and Mechanical Failure Modes of Ceramic Coatings by Quantitative Single Point Scratch Testing: the failure-mode nomenclature used in the gallery above, with reference micrographs.
- ASTM G171, Standard Test Method for Scratch Hardness of Materials Using a Diamond Stylus: the groove-width route to a hardness number, which is part 2.
- ISO 20502, Fine ceramics (advanced ceramics, advanced technical ceramics): Determination of adhesion of ceramic coatings by scratch testing: the European counterpart, and the source of the three-scan procedure as normally implemented.
- G. M. Hamilton, “Explicit equations for the stresses beneath a sliding spherical contact,” Proc. Inst. Mech. Eng. C 197, 53 (1983); G. M. Hamilton, L. E. Goodman, “The stress field created by a circular sliding contact,” J. Appl. Mech. 33, 371 (1966): the trailing-edge tensile stress used for Lc1 and Lc2.
- S. J. Bull, “Failure mode maps in the thin film scratch adhesion test,” Tribol. Int. 30, 491 (1997): which mode you get, as a function of coating thickness, hardness and substrate.
- S. J. Bull, E. G. Berasetegui, “An overview of the potential of quantitative coating adhesion measurement by scratch testing,” Tribol. Int. 39, 99 (2006): the honest assessment of what a critical load can and cannot be turned into.
- P. J. Burnett, D. S. Rickerby, “The relationship between hardness and scratch adhesion,” Thin Solid Films 154, 403 (1987): the energy-based interfacial criterion behind Lc2, including its thickness scaling.