Spherical indentation: a stress–strain curve from one indent

A sharp tip is self-similar, so it imposes the same strain at every depth and gives you a single hardness number. A sphere is not. It starts in fully elastic contact and sweeps through elastic–plastic into fully plastic contact as it sinks in, so the load–displacement record of one spherical indent contains a whole stress–strain curve, provided you can find the point where the tip first touched.

The elastic part is Hertz, and Hertz is exact. For a sphere of radius R pressed into a flat elastic half-space by an elastic displacement h_e, the load, the contact radius and the contact stiffness are all fixed by one modulus:

P = (4/3)·E_r·√R·h_e^3/2 · a = √(R·h_e) · S = 2·E_r·a ⇒ P = (2/3)·S·h_e E_r is the reduced modulus of the tip–sample pair, a is the contact radius, S is the contact stiffness measured directly by a CSM oscillation. The last identity is the useful one: it links three measured quantities with no unknown material constant in it at all.

Kalidindi and Pathak turned that into a stress–strain definition. Divide the load by the contact area to get an indentation stress, and form an indentation strain from the ratio of depth to contact radius:

σ_ind = P / (π a²) · ε_ind = (4 / 3π) · (h_t / a) h_t is the total penetration below the original surface. In the elastic regime these two definitions collapse to σ_ind = E_r · ε_ind exactly, which means the elastic part of the curve must be a straight line through the origin with slope E_r. That is not a result. It is a check.

The effective zero point

Here is the problem. Surface detection on a nanoindenter is a threshold crossing (a small jump in load, stiffness or phase), and it is always slightly wrong. The recorded origin sits at some unknown offset (P₀, h₀) from the real first contact. On a sharp tip driven a few hundred nanometres deep, a couple of nanometres of error is a rounding difference. On a sphere, where the entire elastic segment may be only twenty or thirty nanometres long, it is the whole measurement.

The fix uses the Hertz identity above, rewritten for offset data:

P − P₀ = (2/3) · S · (h − h₀)

Rearranged, that is a straight line in two quantities you can build out of the raw channels:

y ≡ P − (2/3)·S·h = P₀ − (2/3)·h₀ · S ≡ intercept + slope · x, with x ≡ S Regress y against x over the initial elastic segment. The intercept is P₀; the slope is −(2/3)h₀. No modulus, no radius, no area function: just three signals the instrument already records. Outside the elastic segment the relation fails, which is also how you find where the elastic segment ends.

Spherical stress–strain workbench Move the injected offsets, then switch the correction on

Recorded P–h Elastic segment used Hertz through the fitted zero Recorded vs effective origin

Inset: the first few tens of nanometres, where the whole elastic segment lives.

Recovered σ_ind–ε_ind True law (what was simulated) Elastic fit, slope = E_r 0.2% offset yield

If the elastic part is not a straight line through the origin, the zero point is wrong.

Sphere radius R

True Young's modulus E

True indentation yield Y_ind

Hardening exponent n

Injected depth offset h₀

Injected load offset P₀

Noise level

Let it load elastically past yield and pop in

Recovered E_r

–

GPa · true – · –

Recovered Y_ind (0.2% offset)

–

GPa · true – · –

Fitted h₀

–

nm · injected –

Fitted P₀

–

mN · injected –

Elastic segment used

–

points · to h = – nm

Zero-point fit residual

–

µN rms about the line

– –

Set the depth offset to +6 nm and leave the raw origin selected. The P–h curve looks completely normal, and the stress–strain curve it produces has the wrong modulus and a yield point that is not there. Then switch to the corrected origin without touching anything else.

The thing to remember

The elastic segment of a spherical indent on a real metal is short. For a 10 µm sphere on a material with an indentation yield near 4 GPa it ends around 25 nm; on annealed aluminium with a 13 µm sphere it ends before 5 nm. That is the scale on which your surface-detection error has to be small, and no threshold-crossing algorithm is that good. The zero-point regression is not a refinement. It is the step that makes the measurement possible.

Go deeper: what the model behind the sandbox actually does

The synthetic experiment is built forward from a constitutive law in indentation stress–strain space: elastic with slope E_r up to Y_ind, then σ = Y_ind + K(ε − ε_y)ⁿ, with K set so the material hardens 35% by five per cent of plastic indentation strain. Contact kinematics are Hertzian with a sink-in term that relaxes as plasticity accumulates, so the contact radius grows from a = √(Rh_e) towards √(2Rh_t). Load, depth and stiffness are then generated from that state, offset by the injected (P₀, h₀) and given Gaussian noise on all three channels.

The analysis path is the one you would run on real data. First the growing-window regression of y = P − (2/3)Sh against S, extended point by point until the residual leaves the noise band: that gives the offsets and the end of the elastic segment. Then E_r from a through-the-origin Hertz fit on the corrected elastic segment, then a = S/(2E_r) at every point, then σ_ind and ε_ind. Yield is read with a 0.2% offset construction against the fitted elastic line, so even a perfect measurement reads a few per cent above the true Y_ind when hardening is steep. That bias is a property of the construction, not of the instrument, and it is worth knowing which one you are looking at.

Turn on the pop-in. The material then loads elastically well past Y_ind, up to an indentation stress of 2.2 Y_ind here, and bursts at constant load onto the flow curve. In the P–h record that is a displacement excursion at fixed load. In stress–strain space it is a horizontal excursion, because the CSM stiffness signal cannot follow a burst that lasts a few milliseconds and so the contact radius used in the analysis does not update until afterwards. Note what the offset construction then reports as "yield": the pop-in stress, which is a nucleation event, not a flow stress. That is the subject of the next section.

Pop-in statistics

The first burst of plasticity under a spherical tip is a nucleation event. Nucleation is probabilistic, so the pop-in load is a distribution, not a number, and a paper that reports one pop-in load from one indent has reported a sample from a distribution it did not measure.

The reason this measurement is interesting at all is what the stress reaches before the burst. Under a Hertzian sphere the maximum shear stress sits a little below the surface, on the axis, and its magnitude follows from the load and the geometry alone:

τ_max = 0.31 · ( 6 · P · E_r² / (π³ · R²) )^1/3 the numerical factor is the peak of the Hertzian shear field for ν ≈ 0.3, at a depth of about 0.48a below the contact centre. Everything on the right is measured, so τ_max at pop-in is about as close to a direct measurement of a nucleation stress as contact mechanics gets.

Compare that to the shear modulus. The theoretical shear strength of a perfect lattice, the stress to shear one plane over another with no dislocation to help, is somewhere between G/10 and G/30 depending on the interatomic potential you trust. Pop-ins in well-annealed, well-polished single crystals land in that band. That is the evidence that no pre-existing dislocation was involved: the material was loaded to the strength of a defect-free crystal because, in that particular few hundred cubic nanometres, it was one.

Put dislocations in the volume and the story changes completely. The stressed zone under the tip is small (a contact radius of 200 nm probes a region of order 10⁻¹³ m² in cross-section), so whether it contains a dislocation is a Poisson question. At 10¹² m⁻² the answer is almost always no; at 10¹⁴ m⁻² it is always yes. In between, some indents nucleate homogeneously and some find a source, which is why heterogeneous pop-in populations are not just lower but dramatically more scattered.

Pop-in population Drag N down to 8 and watch the confidence interval

Measured staircase Fitted Weibull Model distribution Median & 95% CI

Cumulative probability plotted at median ranks, (i − 0.3)/(N + 0.4).

τ_max at pop-in Theoretical strength band, G/30 – G/10 Median

Inside the band means homogeneous nucleation. Below it means you found a source.

Tip radius R

Dislocation density ρ

Number of indents N

Temperature

Median pop-in load

–

τ_max at the median

–

GPa

τ_max / G

–

Fitted Weibull modulus k

–

model –

95% CI on the median

–

Dislocations in the zone

–

expected, Poisson

– –

Leave everything else alone and sweep the dislocation density from 10¹¹ to 10¹⁴ m⁻². The median load falls by more than an order of magnitude and the Weibull modulus collapses: the population stops being a nucleation measurement and becomes a survey of whatever sources happened to be underneath.

Go deeper: why Weibull, and how the crossover is modelled

Thermally activated nucleation gives a survival probability that decays with the integral of an attempt rate over the loading history, and for a stress that rises smoothly with load the result is extremely steep in load. A two-parameter Weibull captures that shape with the fewest assumptions:

F(P) = 1 − exp( −(P/λ)^k ) λ is the scale (the 63rd percentile load) and k the Weibull modulus. Large k means a narrow, deterministic-looking population; k of order 2 means the pop-in load tells you almost nothing about the next indent. Coefficient of variation is roughly 1.2/k.

The crossover here is driven by one number: the expected count of dislocation lines threading the stressed zone, n_d ≈ ρ·πa², with a the contact radius at the homogeneous pop-in load. The probability that the zone is not defect-free is w = 1 − exp(−n_d), and the model interpolates the scale logarithmically between the homogeneous value and 5% of it, and the modulus logarithmically between a temperature-dependent k of order 10–20 and k ≈ 2. The interpolation is continuous because the physical transition is: at intermediate densities a real dataset is a mixture, some indents nucleating homogeneously and some not, and a mixture of two narrow distributions at different loads reads as one broad one.

Temperature enters twice. The athermal shear strength is reduced by thermal assistance (τ(T) = τ₀·(1 − (T/T_a)^2/3) here, a Kocks-type form), and the scatter grows, because a thermally activated process samples more of its own statistics. Cooling to 77 K both raises the pop-in load and tightens the distribution, which is exactly what low-temperature pop-in experiments show.

On sample size. The width of the confidence interval on the median scales roughly as 1/(k√N). With k = 14 and N = 10 you know the median to about ±8%; with N = 100, to about ±2.5%. With k = 2 (a dislocated sample), N = 10 buys you almost nothing. The readout above computes the interval by bootstrap on the actual sample, so it responds to both.

A trap worth naming

A pop-in load is only a nucleation measurement if the tip radius is known. τ_max goes as R^−2/3, so a tip you believe is 1 µm but is actually 1.4 µm gives a τ_max 20% too high, enough to move a result from "G/14, homogeneous" to "G/11, suspiciously close to theoretical". Calibrate the sphere on a known elastic material by fitting Hertz, every time, and report the radius you fitted rather than the one on the box.

Strain-rate jumps, creep, and the drift that pretends to be creep

If hardness depends on how fast you push, that dependence is a measurement in its own right: it tells you about the rate-limiting obstacle to dislocation motion. The catch is that the two experiments used to get at it, rate jumps and constant-load holds, are both corrupted by artefacts that look exactly like the signal.

The indentation strain rate is defined from the displacement signal, and the rate sensitivity from how hardness responds to it:

ε̇ = (1/h)·(dh/dt) · m = ∂ln H / ∂ln ε̇ · V* = 3√3 · k_BT / (m · H) m is the strain-rate sensitivity, typically 0.005 in a coarse-grained bcc metal, 0.01–0.03 in nanocrystalline fcc, and above 0.1 where grain-boundary processes dominate. V* is the apparent activation volume, reported in units of b³ with b the Burgers vector: around 100–1000 b³ for forest-dislocation cutting, 1–20 b³ where grain boundaries or lattice friction control the rate.

A rate-jump test gets m from a single indent. Hold a constant strain rate until hardness plateaus, jump the rate, wait for the new plateau, jump again. Because the same material volume is tested throughout, the comparison is internal and the usual scatter between indents cancels. The whole method lives or dies on whether each plateau has actually settled before you average it.

Strain-rate jump test Shrink the averaging window until the transient stops biasing m

Measured hardness Settled plateau value Averaging window Segment excluded from fit

Rates run 0.05 → 0.005 → 0.05 → 0.5 s⁻¹, 45 s each.

Segment averages Fitted slope = m True slope

Both axes logarithmic; the slope of this line is the whole result.

True rate sensitivity m

Transient time constant

Noise on H

Averaging window

Hardness level H

Test temperature

Include segment 1 Include segment 2 Include segment 3 Include segment 4

Each segment's imposed rate, its settled value, and the average actually fed to the fit. The gap between the last two columns is the transient bias.
Segment	ε̇ (s⁻¹)	Settled H (GPa)	Averaged H (GPa)	Points averaged	In fit

Fitted m

–

true – · –

Apparent V*

–

b³ (b = 0.25 nm)

V* from the true m

–

b³

Fit R²

–

– –

Set the time constant to 20 s and the averaging window to 100%. The fitted m can be half the true value, and V*, which goes as 1/m, is then twice what it should be. Nothing about the trace looks wrong.

The creep hold, and the drift that imitates it

The other rate experiment is a constant-load hold: park the tip and watch the depth grow. Real indentation creep follows a power law in time, h ∝ t^p, with p small. The problem is that thermal drift also moves the depth signal, linearly in time, and over a sixty second hold a tenth of a nanometre per second is six nanometres of depth that has nothing to do with the material.

Positive drift inflates the apparent creep. Negative drift can cancel it exactly. Both produce a clean-looking power-law fit with a wrong exponent.

Creep hold vs thermal drift Push the drift negative until the creep disappears

True creep, h ∝ t^p Thermal drift, linear What the instrument records

Depth measured from the start of the hold.

True, slope = p Fitted to the record Recorded points

A straight line here is not evidence that you measured creep.

True creep exponent p

Creep depth over the hold

Thermal drift rate

Hold duration

Apparent p

–

true – · –

Drift you would infer

–

nm/s · true –

Depth gained in the hold

–

nm · creep alone –

Drift share of the signal

of total depth change

– –

The "drift you would infer" is measured the standard way, from a low-load hold at the end of the unloading segment. It is contaminated by the same creep you are trying to correct for, which is why that number is never quite the drift.

A trap worth naming

Reporting an activation volume in b³ requires care with units. V* = 3√3 k_BT/(mH) with k_B = 1.380649 × 10⁻²³ J/K and H in pascals (so a hardness of 5.5 GPa is 5.5 × 10⁹, not 5.5) gives V* in m³. Divide by b³ = (0.25 × 10⁻⁹)³ = 1.5625 × 10⁻²⁹ m³. Getting this wrong by 10⁹ is a published-literature-grade mistake, and the giveaway is an activation volume that comes out in the millions.

Indentation fracture toughness: three equations, three answers

Press a sharp pyramid into a brittle solid hard enough and it cracks. Measure the cracks, put the numbers into a published equation, and out comes a fracture toughness. The method needs one small polished sample and five minutes. It is also the least reliable toughness measurement in routine use. The published equations disagree with each other by tens of per cent on the same measurement, and by more than a factor of two once you leave the narrow conditions each of them was calibrated on.

The geometry that matters is simple. Call a the half-diagonal of the residual impression, c the distance from the centre of the impression to the crack tip, and l = c − a the crack length visible on the surface outside the impression. Two crack systems are possible. Radial-median (half-penny) cracks form a continuous semicircular front under the indent and are conventionally assumed when c/a ≥ 2.5. Palmqvist cracks are shallow surface flaws hanging off the impression corners, not connected underneath, and are assumed when c/a < 2.5. They obey different equations, and you cannot tell which you have without sectioning or serial polishing, which nobody does.

Anstis, radial-median: K = 0.016 · (E/H)^1/2 · P / c^3/2 Laugier, Palmqvist: K = 0.015 · (a/l)^1/2 · (E/H)^2/3 · P / c^3/2 Niihara, Palmqvist: K = 0.0089 · (E/H)^2/5 · P / (a · l^1/2) Unit convention, and it matters: P in newtons, a, c and l in metres, E and H in pascals. K then comes out in Pa·m^1/2; divide by 10⁶ to display MPa·m^1/2. The prefactors are empirical calibrations against beam-method toughness on a handful of ceramics and glasses, which is the root of the disagreement: each was fitted to a different set.

Indentation crack measurement Drag a crack tip to change c

Residual impression, half-diagonal a Radial cracks, tip at c Dimensions

Drag a crack tip, or use the slider, to change the crack length

Half-penny (radial-median) Palmqvist Plastic zone under the indent

The current c/a picks one of these. You cannot see which is true from above.

Indentation load P

Impression half-diagonal a

Crack length c from centre

Young's modulus E

Hardness H

Anstis 1981

–

MPa·m^1/2 · radial-median · –

Laugier 1987

–

MPa·m^1/2 · Palmqvist · –

Niihara 1982

–

MPa·m^1/2 · Palmqvist · –

c / a

–

Spread between the three

–

of the mean value

Surface crack length l = c − a

–

µm

E / H

–

appears under a root in all three

– –

Park c/a just below 2.5 and read the three cards. They are answering the same question about the same crack and they do not agree; and the equation the regime flag tells you to use is not the one most papers on this material will have used.

Why this is not a standard method

Every one of those prefactors was obtained by indenting materials whose toughness had already been measured on a proper fracture-mechanics specimen, and fitting. The fits used different material sets, so they encode different residual-stress assumptions, different crack systems and different amounts of subcritical crack growth. On top of that, the crack length enters as c^−3/2, so a 10% error in reading a crack tip under an optical microscope (routine, because the tip is not sharp and the contrast is poor) is a 15% error in K. Crack lengths also keep growing for minutes to hours after unloading in humid air, so when you measured matters.

Standards bodies do not accept it. ASTM C1421 specifies the single-edge-precracked-beam (SEPB), surface-crack-in-flexure (SCF) and chevron-notch (CN) methods for advanced ceramics, and indentation fracture is not among them; ISO 15732 is similarly SEPB. Quinn and Bradt's 2007 review went through the derivations and the round-robin data and concluded that the technique has no sound fracture-mechanics basis and should be abandoned as a toughness measurement.

What survives is narrower and genuinely useful: comparative ranking within one material system, measured at one load, with one tip, by one operator, on one day. Indentation cracking is a fine way to say that this heat treatment cracks more readily than that one. It is not a way to publish a number with units of MPa·m^1/2 and compare it to someone else's beam test.

Go deeper: where the c^−3/2 comes from

All three equations descend from the same idea: after unloading, the plastic zone under the impression is a wedge of material that does not fit back into the hole it made, and the residual field it imposes acts on the radial cracks like a point force at the contact centre. For a half-penny crack of radius c loaded by a centre force P_r, the stress intensity is K = χ P_r/c^3/2, which is where the exponent comes from. At equilibrium K = K_c, so measuring c gives you K_c if you know χ.

χ is where the trouble starts. It scales with (E/H)^1/2 because the residual force scales with the elastic mismatch the plastic zone has to accommodate, but the exponent on E/H is 1/2 in the Anstis calibration, 2/3 in Laugier's and 2/5 in Niihara's, and E/H spans a factor of thirty across the ceramics people apply these to. Move the E/H slider with everything else fixed and watch the three curves fan apart: that divergence is not measurement noise, it is three different empirical fits extrapolated outside the sets they were fitted to.

The Palmqvist equations additionally carry l = c − a, which makes them very sensitive at small crack lengths: as l → 0, Laugier's (a/l)^1/2 and Niihara's l^−1/2 both diverge. Near the cracking threshold, where the interesting materials sit, the Palmqvist forms are at their least stable. That is not a coincidence: near threshold there is no well-defined crack system at all.

Part 1

Making the number

Where hardness and modulus actually come from: the Oliver–Pharr unloading fit, the contact area you never measure, and the error budget for drift, frame compliance and tip shape.

Part 2

When it lies

Indentation size effect and the Nix–Gao plot, pile-up and sink-in wrecking the contact area, and how thin a film can be before you are measuring the substrate.

Sources & further reading

S. R. Kalidindi, S. Pathak, “Determination of the effective zero-point and the extraction of spherical nanoindentation stress–strain curves,” Acta Mater. 56, 3523 (2008): the zero-point regression used in section 1.
S. Pathak, S. R. Kalidindi, “Spherical nanoindentation stress–strain curves,” Mater. Sci. Eng. R 91, 1 (2015): the full review, including pop-ins in the stress–strain representation.
A. Gouldstone, N. Chollacoop, M. Dao, J. Li, A. M. Minor, Y.-L. Shen, “Indentation across size scales and disciplines: Recent developments in experimentation and modeling,” Acta Mater. 55, 4015 (2007): incipient plasticity and pop-in, with D. Lorenz and co-workers' thermally activated nucleation work.
C. A. Schuh, “Nanoindentation studies of materials,” Mater. Today 9, 32 (2006): thermally activated pop-in statistics and how to treat them.
V. Maier, K. Durst, J. Mueller, B. Backes, H. W. Höppel, M. Göken, “Nanoindentation strain-rate jump tests for determining the local strain-rate sensitivity in nanocrystalline Ni and ultrafine-grained Al,” J. Mater. Res. 26, 1421 (2011): the jump-test protocol in section 3.
G. R. Anstis, P. Chantikul, B. R. Lawn, D. B. Marshall, “A critical evaluation of indentation techniques for measuring fracture toughness: I, Direct crack measurements,” J. Am. Ceram. Soc. 64, 533 (1981).
M. T. Laugier, “New formula for indentation toughness in ceramics,” J. Mater. Sci. Lett. 6, 355 (1987).
G. D. Quinn, R. C. Bradt, “On the Vickers indentation fracture toughness test,” J. Am. Ceram. Soc. 90, 673 (2007): the case against, and the one to read before you publish an indentation toughness.
ASTM C1421, Standard Test Methods for Determination of Fracture Toughness of Advanced Ceramics at Ambient Temperature: SEPB, SCF and CN, and no indentation.