The Oliver–Pharr workbench
Push a diamond into a surface, then pull it back out. On the way in, the material bends and flows together. On the way out, only the springback is elastic. And elastic contact is a problem we can solve exactly. That is the whole trick: the unloading curve is the only part of the experiment that obeys a theory clean enough to invert.
The chain of inference runs like this. The slope of the unloading curve at maximum load, called the contact stiffness S, tells you the size of the contact, because for any axisymmetric punch, stiffness scales with the square root of contact area. Knowing S lets you work backwards to the depth over which the diamond was actually touching, hc, which is smaller than the depth you measured because the surface around the indent sinks in. Feed hc into a calibration curve for the tip's shape and you get a projected contact area. Divide the load by that area and call it hardness. Take the stiffness and that same area and call it modulus.
Four inferences, each one leaning on the last. The sandbox below lets you run that chain yourself on a synthetic experiment where, unusually, the true answer is known, so you can see exactly how far off you land.
Drag the copper band, or use the slider, to change how much of the unloading curve is fitted
What each step is doing
The fit itself is a power law through the unloading data, with the final depth left free:
P = α (h − hf)m α is a scale factor with no physical meaning on its own; hf is the depth the indent settles to after full unload; m is an exponent that comes out of the fit, typically 1.2–1.6 for a Berkovich tip. A perfect cone would give m = 2, a flat punch m = 1, a paraboloid m = 1.5.Differentiate at maximum load and you have the contact stiffness. Then the two equations that do the real work:
hc = hmax − ε Pmax / S (ε = 0.75) H = Pmax / A(hc) Er = (√π / 2) · S / (β √A) (β = 1.034 for Berkovich) ε accounts for elastic sink-in of the surface around the indenter; 0.75 is the paraboloid value used almost universally. β corrects the axisymmetric stiffness relation for a three-sided pyramid. A(hc) is the tip area function, which is a calibration, not a measurement.Then the sample modulus is separated from the diamond's, which is not infinitely stiff:
1 / Er = (1 − ν²) / E + (1 − νi²) / Ei for diamond, Ei = 1141 GPa and νi = 0.07. On a stiff ceramic this correction is worth tens of per cent; on a polymer it is negligible. Note that you need to assume a Poisson's ratio for your sample to get E at all.Go deeper: why the fit window changes the answer
The unloading curve is not actually a power law. It is close to one over the top part, and drifts away lower down, partly because the contact is not self-similar and partly because time-dependent deformation keeps contributing on the way out. So the exponent you fit depends on how much of the curve you feed the fitter, and the stiffness at maximum load is the derivative of whatever curve you fitted.
ISO 14577 and common practice fit somewhere between the top 25% and top 50% of the unloading curve. Fitting too little makes the fit noise-sensitive and R² becomes meaningless; fitting too much pulls the exponent down and systematically underestimates S. On a creeping material such as PMMA, using more of the curve is actively harmful. Watch R² in the workbench: it stays high across a wide range of windows even while S moves by several per cent, which is exactly why R² is a bad guide to whether you fitted the right thing.
An alternative that sidesteps the whole question is continuous stiffness measurement, where a small oscillation superimposed on the loading segment gives S at every depth without any unloading fit at all. It brings its own artefacts, which is a story for part 3.
Hardness is a load divided by an area that was never observed. Every number downstream of A(hc) inherits whatever is wrong with the tip calibration; and the same area appears under a square root in the modulus, so a 10% area error is a 10% hardness error and a 5% modulus error. They are not independent measurements.
The error budget nobody plots
Three systematic errors dominate instrumented indentation, and all three are invisible in the raw curve. Below, each one is switched on alone, with the others corrected, so you can see what it is worth by itself, and then all together, where they partly cancel.
Frame compliance is the machine bending while it pushes. The frame and the sample are springs in series, so what you record is always the sum. Because compliances add rather than stiffnesses, the error grows with contact stiffness: on a polymer it is nothing, on tungsten at a micron deep it can dominate everything else. Thermal drift is the instrument slowly changing length while the experiment runs; a tenth of a nanometre per second sounds harmless until you notice a hold segment lasts thirty seconds. The area function is the tip's real shape, which is never the ideal pyramid and gets blunter with every indent; and its error is worst at exactly the shallow depths people use for thin films.
Go deeper: how these are actually calibrated
Frame compliance and area function are calibrated together, usually on fused silica, because they are entangled: both are unknowns in the same two equations. The classic procedure indents a reference material of known modulus over a wide load range and exploits the fact that at large contacts the measured compliance approaches a constant, the frame's, while the area function is whatever makes the modulus come out right at every depth. The output is a polynomial,
A(hc) = C₀hc² + C₁hc + C₂hc1/2 + C₃hc1/4 + … C₀ = 24.5 for an ideal Berkovich. The later terms are pure curve-fitting to describe a blunt tip end; they have no geometric meaning and should never be extrapolated below the shallowest depth used in the calibration.Two consequences worth carrying around. First, if you calibrate on fused silica and then test something with a very different modulus, some of the calibration error transfers: the reference material's assumed modulus is baked into your tip shape. Second, area functions age. A tip that has done a thousand indents into a nitride coating is not the tip that was calibrated last year, and the drift always goes the same way: blunter, so shallow hardness reads high.
Thermal drift is measured separately, from a low-load hold at the end of the unloading segment where the contact should be doing nothing. If that segment is not flat, the correction is not reliable and neither is the test.
A blunt tip analysed with an ideal area function produces hardness that rises as depth falls. That looks exactly like the indentation size effect, which is a real physical phenomenon with a real theory behind it. Distinguishing the two is the subject of part 2. And the honest answer is that a good fraction of the published size-effect literature never ruled out the boring explanation.
Which tip, and why it matters
Choosing an indenter is choosing a strain. A sharp pyramid imposes a fixed, large strain on the material no matter how deep you go; a sphere starts nearly elastic and gets more severe as it sinks in. That single difference is why a Berkovich gives you one hardness number and a sphere gives you a whole stress–strain curve.
| Indenter | Face / semi-angle | Equivalent cone | A(hc) | A at this depth | Mean pressure | Rep. strain |
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Go deeper: representative strain, and what "sharp" means
Tabor's insight was that a conical or pyramidal indenter imposes a strain set purely by its angle, independent of depth: the geometry is self-similar, so a deep indent is just a scaled copy of a shallow one. The usual estimate is
εr ≈ 0.2 cot ψ ψ is the semi-angle of the equivalent cone, measured from the axis. Berkovich: ψ = 70.3°, εr ≈ 7%. Cube corner: ψ = 42.3°, εr ≈ 22%. For a sphere the strain is not fixed: εr ≈ 0.2 a/R, so it grows as the contact widens.This is why sharp indenters are blunt instruments, in the useful sense: a Berkovich drives the material so far past yield that the answer is insensitive to where yield actually was. You get one number, robustly. A sphere sweeps through elastic, elastic–plastic and fully plastic contact as you push, so the load–displacement record contains the yield point and the early hardening, at the cost of needing a much more careful analysis to extract it.
The other reason tip choice matters is cracking. Whether an indent produces radial cracks in a brittle solid depends on the wedging action of the tip, which scales with how sharp it is. Cube corner indents crack ceramics at loads a Berkovich never would, which is the basis (a contested one) of indentation fracture toughness measurement.
Sources & further reading
- W. C. Oliver, G. M. Pharr, “An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,” J. Mater. Res. 7, 1564 (1992): the original method.
- W. C. Oliver, G. M. Pharr, “Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology,” J. Mater. Res. 19, 3 (2004): the twelve-years-later reassessment, including where the method breaks.
- A. C. Fischer-Cripps, Nanoindentation, 3rd ed., Springer (2011): the standard textbook treatment of the corrections in section 2.
- ISO 14577-1, Metallic materials: Instrumented indentation test for hardness and materials parameters: defines the fit window conventions used above.
- R. B. King, “Elastic analysis of some punch problems for a layered medium,” Int. J. Solids Struct. 23, 1657 (1987): the origin of the β correction factor.
- D. Tabor, The Hardness of Metals, Oxford (1951): representative strain, and the idea that a hardness test is a constrained yield test.