1
The six assumptions
Oliver-Pharr is a chain: measure the unloading stiffness S, get the contact depth from it, turn that into a contact area through a calibrated area function, and divide. Each link carries an assumption, and the whole chain is only as good as the weakest one.
- The unloading is elastic. S is read at the top of the unloading curve and treated as pure elastic recovery. Anything time-dependent still happening (creep, viscoelasticity) is not elastic and inflates S.
- The area function is right. A(hc) is a calibration, not a geometric fact. A real tip is blunt at the scale of a shallow indent.
- The contact area is the one the model predicts. The material is assumed to sink in. If it piles up, the true contact is larger than anything the curve can tell you.
- The surface is flat and the zero point is real. Depth is measured from first contact with a plane, not from a peak or a valley on a rough surface.
- The specimen is a half-space. No substrate, no interface, no free edge inside the stressed volume.
- The instrument is corrected. Frame compliance and thermal drift have been measured and removed, not assumed away.
Assumptions 1 and 6 break in the curve: the shape changes, and you can see it if you look. Assumptions 2, 3, 4 and 5 break outside the curve: the load-displacement data stay perfectly well behaved while the number they produce is wrong. Those are the failures that survive peer review.
2
The failure map
One synthetic experiment, ten ways of spoiling it. The true material is fixed (H = 5 GPa, E = 100 GPa, Berkovich, 10 mN), so every error below is measured against a known answer: the page runs the same Oliver-Pharr analysis you would run, on the data each failure would have given you.
How the synthetic experiment is built
The true material is H = 5 GPa and E = 100 GPa with a perfect Berkovich (A = 24.5 hc²) at Pmax = 10 mN. That fixes the contact area (2.0 × 106 nm²), the contact depth (286 nm), the reduced modulus (100 GPa against a diamond tip) and therefore the true stiffness S = 2βEr√A/√π = 0.165 mN/nm. Loading is P = C h²; unloading is the usual power law P = α(h − hf)m with m = 1.5, and α and hf follow from Pmax and S, which puts hf/hmax at 0.73.
Every failure then acts on that experiment the way it acts on a real one, and the page analyses the result exactly as you would: fit the unloading branch over the window, differentiate at hmax for S, take hc = hmax − 0.75 P/S, read A(hc), and divide. Nothing is fudged toward a known answer, which is why the healthy case does not come out exactly 5.00 GPa: the power-law fit itself carries a fraction of a per cent.
Blunt tip, pile-up, substrate and roughness do not change the shape of the load-displacement curve at all. Switch between them above and the recorded curve sits exactly on the true one while the hardness moves by tens of per cent. No amount of staring at the curve, and no error bar computed from repeat indents, will find them. They are found by calibrating, by imaging the imprint, by knowing the film thickness and by measuring the roughness.
3
How flat does the surface have to be?
The scatter comes almost entirely from one thing: the instrument decides where zero is by touching the surface, and on a rough surface that first touch is a peak or a valley, not the mean plane. That scatter is predictable. The bias that comes with it is not, because two effects pull against each other, and this is where the usual rule of thumb comes from.
The zero-point error is the surface height where the tip lands, so its spread is the roughness Rq itself. Depth enters the hardness through the area, and the area goes as hc², so a depth error of δ costs about 2δ/hc in hardness. Setting that to 10% gives h > 20 Rq, which is where the usual rule of thumb comes from, and setting it to 3% gives h > 65 Rq. Nothing here is fitted: it falls out of the geometry.
Averaging fixes the scatter and not the bias, and the bias can go either way
Scatter falls as 1/√N, so twenty-five indents give a mean whose standard error is five times smaller than one indent. That is real, and it is why reporting a mean feels safe. What does not fall with N is the bias, and on a rough surface there are two of them pulling opposite ways.
The first is arithmetic. Hardness goes as 1/hc², which is a convex function, so scattering the depth symmetrically about the right value still lifts the average hardness: the shallow indents gain more than the deep ones lose. Switch the asperity term off above and that is what is left, a few per cent high at 200 nm on a 15 nm surface.
The second is contact. At a given depth below the local surface the neighbouring material sits lower, so less of the pyramid carries load than the area function assumes, and the hardness reads low. The page models that with a single-parameter form; the strength of it depends on the shape of the surface, not just its Rq, which is exactly why the net bias is not something a rule of thumb can give you. Report the roughness beside the depth and let the reader judge.
The scatter, on the other hand, is robust: it comes from the height distribution alone, which is why the rule is written against Rq rather than Ra. For a Gaussian surface Rq ≈ 1.25 Ra, so the same rule quoted in Ra is about 25% looser.
4
The nose: unloading that is not elastic
Oliver-Pharr reads the stiffness at the very top of the unloading curve, where the material is assumed to be recovering elastically and nothing else. If the specimen is still creeping when unloading begins, the depth keeps increasing while the load comes down, and the curve bends the wrong way. Holding longer does two separate things, and they are worth keeping apart: it lowers the hardness, which is real, and it fixes the stiffness, which was the artifact.
Feng and Ngan wrote the correction that makes this quantitative: 1/Smeasured = 1/Selastic + ḥhold/|dP/dt|, where ḥhold is the creep rate at the end of the hold and dP/dt is the unloading rate. The two knobs that fix a nose are in that expression: hold longer, so the creep rate falls, or unload faster, so the elastic term dominates. The page computes the curve rather than the correction, so the fitted S carries the error the correction is meant to remove.
Polymers, biological tissue, soft metals at room temperature (indium, lead, tin), amorphous alloys near the glass transition, and anything at elevated temperature. In these, a nose is normal at ordinary hold times and the reported modulus can be tens of per cent high, or even negative in extreme cases, which is the one failure that does announce itself.
5
Is this result usable?
Eight checks. None of them is exotic and all of them are things you either did or did not do, so this is a record of the experiment rather than a judgement of the material. Tick what is true.
A failed check is not a failed experiment. It is a number you should quote with the caveat attached, or a measurement worth repeating with one thing changed. The checks that most often turn a published number around are the first two: an area function calibrated on a standard, and a drift rate you actually measured rather than assumed.
6
What this page does not cover
The synthetic experiment is a single Berkovich indent on an isotropic, elastic-plastic solid, analysed by the standard Oliver-Pharr chain. It carries the failures as parametric distortions of that experiment, which is enough to show direction and rough size but is not a simulation of contact: for the real models, the linked pages do the work. Four things are deliberately outside it. Anisotropy, which makes the modulus a direction-dependent quantity that a single indent cannot resolve. Residual stress, which shifts hardness without touching any assumption above. Sharp-tip cracking in brittle materials, where the unloading curve is no longer describing one contact. And the continuous stiffness method, whose own artifacts at shallow depth are a separate discussion, in part 4.
Report the diagnostics beside the number. hf/hmax, the measured drift rate, the depth as a fraction of the film thickness, the roughness against the depth, and when the area function was last calibrated cost one line in a methods section, and they are what lets a reader (or you, a year later) tell a hardness from a number.
Sources
Show the eight references
- 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 method, and the assumptions it states for itself.
- 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 review that revisits pile-up, the area function and where the method goes wrong.
- G. Feng, A. H. W. Ngan, “Effects of creep and thermal drift on modulus measurement using depth-sensing indentation,” J. Mater. Res. 17, 660 (2002): the correction 1/Smeasured = 1/Selastic + ḥ/|dP/dt| used in module 4.
- T. Chudoba, F. Richter, “Investigation of creep behaviour under load during indentation experiments and its influence on hardness and modulus results,” Surf. Coat. Technol. 148, 191 (2001): hold times, the nose, and how long is long enough.
- A. C. Fischer-Cripps, Nanoindentation, 3rd ed., Springer (2011): chapters on the sources of error, surface roughness and the frame compliance calibration.
- Y.-T. Cheng, C.-M. Cheng, “Scaling, dimensional analysis, and indentation measurements,” Mater. Sci. Eng. R 44, 91 (2004): what hf/hmax does and does not tell you.
- ISO 14577-1 (2015), Metallic materials: instrumented indentation test for hardness and materials parameters: the surface, drift and calibration requirements a test has to meet to be called a standard test.
- M. F. Doerner, W. D. Nix, “A method for interpreting the data from depth-sensing indentation instruments,” J. Mater. Res. 1, 601 (1986): the earlier analysis whose linear unloading assumption Oliver-Pharr replaced, and a useful reminder that the fit function is a choice.