The indentation size effect, and its impostor
Push a diamond a hundred nanometres into a metal and it will look harder than the same metal indented a micron deep. This is real, it is reproducible, and it has a theory. It also has a near-perfect impostor: a tip that is slightly rounded, analysed as though it were perfectly sharp, produces the same rise from a material with no size effect at all.
The real effect comes from dislocations you are geometrically forced to store. To accommodate the shape of a rigid pyramid pressed into a flat surface, the crystal underneath has to contain a net density of dislocations of one sign: geometrically necessary dislocations. The volume they occupy scales with the indent, but the number you need scales with the indent's surface, so the density goes as 1/h. Shallow indents force a huge stored density into a tiny volume, and Taylor hardening does the rest.
Nix and Gao turned that argument into one line. Add the geometrically necessary density to whatever statistically stored density the material already had, put the total into the Taylor relation, and the depth dependence falls out:
H / H0 = √(1 + h*/h) ⇔ H² = H0² (1 + h*/h) H0 is the depth-independent hardness the material would have at infinite depth; h* is a characteristic length that sets where the size effect switches on. The second form is the useful one: plot H² against 1/h and a material obeying Nix–Gao gives a straight line, intercept H0², slope H0²h*.That straight line is the whole reason the model became standard. It is a one-parameter test you can run on a spreadsheet. It is also, unfortunately, a shape that a blunt tip reproduces almost exactly.
Why hardness rises: the dislocations the geometry forces you to store
h* is not a free parameter with a story attached: it is the depth at which the dislocations the indent geometry forces on you match the ones the material already had. Above it the size effect is negligible; below it, hardness climbs.
Where h* comes from
Because the whole model is Taylor hardening applied to a density you can count geometrically, h* is not fitted: it is predicted from things you can look up:
ρG = 3 tan²θ / (2 b h)h* = (81/2) · b · α² · tan²θ · (µ / H0)² b is the Burgers vector, α ≈ 0.5 the Taylor constant, µ the shear modulus, and θ the angle between the specimen surface and the indenter surface: 19.7° for the cone equivalent to a Berkovich, so tan θ = cot 70.3° = 0.358. The (µ/H0)² makes h* enormously sensitive to how soft the material is: annealed copper has h* of order a micron, a nitride coating of order tens of nanometres.
Two things follow immediately. Soft metals show a big size effect over the depth range everyone uses; hard ceramics barely show one, because h* is already smaller than the shallowest indent you can make. And because H0 appears squared in the denominator, the same material work-hardened will show a much weaker size effect than in the annealed state, which is exactly what Nix and Gao's copper data did.
Go deeper: why the impostor is so convincing, and what actually breaks the tie
A real Berkovich is not a point. Its end is roughly spherical, radius anywhere from 20 nm on a fresh tip to several hundred on a tired one. Below the tangent depth ht = R(1 − sin ψ) the contact is spherical, above it conical, and the projected area is bigger than the ideal 24.5hc² at every depth, approaching it only asymptotically. Analyse with the ideal function and you divide by an area that is too small, so hardness reads high, and it reads worst where the discrepancy is largest: shallow.
Expand the blunt-tip area at depths well above ht and you get Areal/Aideal ≈ (1 + 0.062 R/h)², so the apparent hardness is H0(1 + 0.062 R/h)² and the apparent H² is H0²(1 + 0.062 R/h)4. To first order that is H0²(1 + 0.25 R/h): a straight line on Nix–Gao axes with an apparent h* of about a quarter of the tip radius. A 300 nm tip end fakes h* ≈ 75 nm without any dislocation physics at all. The residual curvature is fourth-order and disappears under a few per cent of scatter.
What breaks the tie? Not the fit. Three things do. Calibrate the area function down to depths shallower than your shallowest data point and use it: this is the single most effective step and the one most often skipped. Change the tip: the artefact scales with R, the physics does not. Change the indenter geometry: Swadener, George and Pharr showed that with spheres the hardness depends on sphere radius rather than depth, which is a prediction the strain gradient argument makes and the tip artefact does not.
Pharr, Herbert and Gao's 2010 review is worth reading in full on this. Their conclusion is not that the size effect is an artefact: it is real, and it survives careful area calibration in single crystals. Their conclusion is that a substantial fraction of the published h* values were never separated from tip bluntness, surface preparation and oxide layers, and that the numbers should not be pooled.
Because the Nix–Gao intercept is H0² and the slope is H0²h*, a small error in the intercept propagates straight into h*. Fitting a short lever arm in 1/h (the usual case, because nobody indents four decades of depth) makes the intercept badly conditioned, and h* along with it. If your deepest indent is not well past h*, the fitted h* is mostly an extrapolation.
Pile-up and sink-in
Oliver–Pharr assumes the surface around the indenter is pulled down. In soft metals that do not work-harden, it is pushed up instead, and the material rides up the flanks of the tip. The contact is then larger than any part of the load–displacement record can tell you, so you divide by an area that is too small and every number comes out too big.
The sink-in correction, hc = hmax − ε Pmax/S, is derived from elastic contact. It is a purely elastic statement about how a surface deflects outside a punch, and it can only ever move hc below hmax. Pile-up is plastic: material displaced from under the tip has to go somewhere, and if there is no work hardening to spread the deformation outward, it goes straight up around the contact. The Oliver–Pharr equation has no term that can produce that, so it does not.
Bolshakov and Pharr ran the finite-element experiment that settled what governs it. Two parameters do almost all the work: how much of the depth is permanent, hf/hmax, and the work-hardening exponent n. Pile-up matters when hf/hmax is above about 0.7 and hardening is weak. Below 0.7 there is enough elastic recovery that the surface sinks in whatever the material does, and at n ≥ 0.3 the hardening pushes the plastic flow outwards and downwards instead of upwards.
| Material | hf/hmax | n | Atrue/AOP | Error in H | Error in E | Regime |
|---|
What it costs you
Only one thing is wrong (the area), but it enters the two answers with different powers:
Hreported / Htrue = Atrue / AOPEreported / Etrue = √( Atrue / AOP ) because H = P/A while Er = (√π/2)·S/(β√A). A 60% area error is therefore a 60% hardness error and a 26% modulus error, in the same direction, on the same indent. They are not two independent measurements that happen to agree.
The area ratio used in the sandbox above is an explicit phenomenological fit to Bolshakov and Pharr's finite-element results: it is a curve drawn through their published trends, not a derivation:
Atrue/AOP = 1 + 0.60 · max(0, (hf/hmax − 0.7)/0.3)1.2 · max(0, 1 − n/0.3) It is built to reproduce the two limits they identified: 1.00 at hf/hmax = 0.7 or at n ≥ 0.3, where Oliver–Pharr is sound, and 1.60 for a fully plastic, non-hardening contact, which is about the worst case reported for annealed soft metals. Use it to reason about direction and order of magnitude. Do not use it to correct data.Go deeper: how to actually get the area, and the one number that is free
Image the impression. An AFM scan, or an SEM image of a larger indent, gives you the contact perimeter directly, and pile-up lobes on a Berkovich are unmistakable: three of them, on the faces rather than the corners. This is the only method that measures rather than infers, and it is what Bolshakov and Pharr used to anchor their simulations. The cost is that it is slow and it is post hoc, so it does not scale to a mapping experiment.
Sidestep the area entirely. Joslin and Oliver pointed out that the combination P/S² does not contain the contact area:
P / S² = (π / 4β²) · H / Er² so if you are prepared to report the ratio H/Er² rather than H alone, or if you know E independently, pile-up drops out. Continuous stiffness measurement gives you P and S at every depth, so this can be evaluated as a continuous function of depth rather than at one point.And the diagnostic is free. hf/hmax comes out of the unloading fit you already did: hf is the extrapolated final depth, hmax is on the axis. There is no reason not to report it. If it is above 0.7 in a soft metal, say so, and say what you did about it. A paper that reports H, E and hf/hmax is much easier to trust than one that reports H and E to four significant figures.
Two caveats on the criterion itself. It was established for sharp pyramidal indenters on elastic–plastic solids with power-law hardening; it does not transfer cleanly to spherical indentation, to films on substrates, or to materials that densify rather than flow, such as silica and many glasses. And residual stress shifts it: a compressive surface stress suppresses pile-up, a tensile one encourages it, which is one of the reasons indentation is used to measure residual stress at all.
Pile-up does not make your data noisy. It makes it precisely wrong. Ten indents in annealed aluminium will agree with each other to a few per cent and all be forty per cent high together, and no statistic computed from those ten indents can tell you so.
How thin is too thin: the 10% rule
Everyone who has measured a coating has been told to keep the indent below a tenth of the film thickness. It is a good habit and it is not a law. Whether a tenth is safe depends on how big the plastic zone gets, and that depends on the contrast between the film and what is underneath it.
The rule goes back to Bückle, who was working with microhardness indents in the 1970s and needed something a technician could apply without a calculation. The physics behind it is simply that the disturbed volume under an indent is much larger than the indent, so the surface you are probing has to be much thicker than the mark you leave. How much larger is the part the rule hides.
Johnson's expanding cavity model gives an estimate. Treat the material immediately under the indenter as an incompressible core being inflated inside an elastic–plastic half-space, and the radius of the plastic zone that develops around it follows from the elastic mismatch:
(c/a)³ = [ (E/σy) · tan β + 4(1 − 2ν) ] / [ 6(1 − ν) ] c is the plastic-zone radius, a the contact radius, β the angle between the indenter surface and the specimen surface, so tan β = 0.358 for a Berkovich. σy ≈ H/3 is Tabor's constraint relation. ν is taken as 0.25 throughout the sandbox below. Because a = √(24.5/π)·hc = 2.79 hc for a Berkovich, a c/a of 3 already means the plastic zone reaches more than eight times the contact depth below the surface.That factor of eight is what the 10% rule is really trying to be a proxy for; and it is not a constant. A stiff film over a soft substrate has an enormous E/σy, so the zone runs deep and the substrate starts carrying load long before you reach a tenth of the thickness. A compliant film over a hard substrate has the opposite problem, and the rule wastes a perfectly usable depth range.
Weighting by the plastic zone
To turn the plastic-zone size into a hardness you need some rule for how the film and substrate share the load. The sandbox uses the simplest one that has the right limits: volume weighting, and it is a teaching model, not a method you should fit data with:
d = max(0, t − hc)fsub = (c − d)² (2c + d) / (2c³) for 0 < d < c
Hmeasured = Hfilm + (Hsubstrate − Hfilm) · fsub The plastic zone is treated as a hemisphere of radius c hanging beneath the indenter tip, and d is the film still left under the tip. fsub is then the volume fraction of that hemisphere lying deeper than the interface: the spherical cap of height (c − d). It is 1 when the tip has gone through the film and 0 while the zone is still contained. Real composite-hardness models (Jönsson and Hogmark's area-weighted model, Korsunsky's energy-based one, Bull's work) do considerably better than this, and none of them agrees with the others.
Go deeper: the argument about how deep is too deep
Bückle's rule was empirical and it was written for a different experiment: optical microhardness on electroplated layers, where "the indent" meant the diagonal you could see. Carrying it over to instrumented indentation on a 200 nm PVD coating is an act of faith, and the literature has been arguing about it ever since.
Chen and Bull measured plastic-zone sizes directly and concluded that the expanding cavity estimate is too small for hard coatings on soft substrates, because the substrate yields under the elastic field well outside the film's own plastic zone. Their revision of Bückle's rule pushes the safe depth down: for a superhard coating on a soft metal, substrate influence is detectable at a few per cent of the thickness, not ten. In the sandbox above you can see the mechanism by comparing TiN on steel with TiN on aluminium: the only thing that changed is σy of the substrate, and the zone doubles in reach.
The converse is just as real and much less discussed. For a compliant film on a stiff substrate the plastic zone is small and shallow, and the valid depth window is often two or three times wider than the rule allows. If you are indenting a 2 µm polymer coating on glass and dutifully stopping at 200 nm, you are throwing away signal and fighting surface roughness for no reason.
Two practical notes. First, the rule as usually quoted is written for hardness. Modulus is worse: the elastic field is longer-ranged than the plastic one, so substrate influence on E shows up at shallower depths than it does on H, often below 5% of the thickness. Second, none of this helps if the film is rough. At depths of tens of nanometres the dominant error is usually the surface finish, not the substrate, and the honest answer is sometimes that the film is too thin to measure this way and you should be doing a cross-sectional or a bulge test instead.
All three failures here share a signature: the data look fine. The size-effect impostor produces a straight line, pile-up produces tight repeatability, substrate influence produces a smooth trend. None of them shows up as scatter, so none of them shows up in an error bar. The only defences are calibration you can point to, a diagnostic ratio you report rather than hide, and a second, independent measurement.
Sources & further reading
- W. D. Nix, H. Gao, “Indentation size effects in crystalline materials: A law for strain gradient plasticity,” J. Mech. Phys. Solids 46, 411 (1998): the model and the H² versus 1/h plot used in section 1.
- G. M. Pharr, E. G. Herbert, Y. Gao, “The indentation size effect: A critical examination of experimental observations and mechanistic interpretations,” Annu. Rev. Mater. Res. 40, 271 (2010): the review that separates the effect from its artefacts.
- A. Bolshakov, G. M. Pharr, “Influences of pileup on the measurement of mechanical properties by load and depth sensing indentation techniques,” J. Mater. Res. 13, 1049 (1998): the finite-element study behind the hf/hmax and n criterion.
- K. L. Johnson, “The correlation of indentation experiments,” J. Mech. Phys. Solids 18, 115 (1970): the expanding cavity model for the plastic-zone radius.
- J. Chen, S. J. Bull, “On the relationship between plastic zone radius and maximum depth during nanoindentation,” Surf. Coat. Technol. 201, 4289 (2006): measured zone sizes and the case for revising the depth rule.
- H. Bückle, in The Science of Hardness Testing and Its Research Applications, ASM (1973): the origin of the depth rule that became "10%".