The schedule is the experiment

Eight schedules, one instrument. Each one is answering a different question, and each one breaks in a way you can read off the curve, if you know what you are looking for. That last part is what this section is really about.

Parts 1 and 3 took the load–displacement curve apart. This one goes one step earlier, to the load–time curve, because that is the only thing you actually control. Everything downstream (the shape of the P–h loop, whether you see a pop-in or average over it, whether a cracked coating shows up as a burst or as nothing at all) is decided by the schedule.

Three questions, asked of every mode: what the schedule looks like, when to use it, and what failure looks like in it. The third column is the one that gets left out of instrument brochures, so it is drawn in copper below.

Eight loading schedules Pick a schedule; every panel and every slider follows it

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Peak load P_max

Static hold load

Coating cracks at h_c

Hold at peak load

Number of cycles

Partial unload to

Number of cycles N

Load ratio P_min/P_max

Cyclic damage rate

Harmonic amplitude h_os

Oscillation frequency

Relaxation time τ

Accumulated damage

Strain rate ḣ/h

Impact energy

Number of impacts

Reference coating toughness

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What you get

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Use it when

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Failure signature

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– –

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What the sandbox is actually computing

Every mode above is driven from the same contact model, the one from part 1: iterate the contact depth until it is self-consistent with the load and the stiffness it implies.

A = A(h_c) · P = H·A/10⁶ · S = 2β·E_r·√(A/π)·10⁻⁶ · h_c = h_max − 0.75·P/S h in nm, P in mN, A in nm², H and E_r in GPa, S in mN/nm. A(h_c) is a blunted Berkovich: a sphere of 100 nm radius tangent to the 70.32° equivalent cone. β = 1.034. Everything else in this section (bursts, staircases, fatigue traces, impact steps) is that one loop run on a schedule.

The film–substrate blend, the cracking event and the two damage laws are the parts that are models rather than mechanics, so they are worth stating plainly.

Go deeper: the damage law, and why it is a teaching model

Cracking. When the contact depth crosses the critical depth you set with the slider, the coating through-cracks. That is implemented as a jump in a damage variable d, which softens the contact: H is multiplied by (1 − 0.55d) and E_r by (1 − 0.75d). The size of the damage jump is chosen so the resulting depth excursion at constant load is the burst length quoted in the readouts. The point of doing it this way is that a single mechanism then produces all four of the signatures you see: the displacement burst on a ramp, the step down in CSM stiffness, the departure of the multi-cycle staircase, and the bulge on the unloading curve. They are not four separate cartoons.

Fatigue. Damage accumulates per cycle as

Δd = c·(ΔP/P_max)²·[ (1 − d/d_sat) + r ] , and Δd is multiplied by (1 + 6(d − d_c)/d_c) once d > d_c c is the cyclic damage rate from the slider, r is a non-saturating "ratchet" fraction, d_sat = 0.28 and d_c = 0.42. r = 0 gives shakedown, small r gives ratcheting, and large c crosses d_c inside the test and produces the knee. This is a damage-mechanics caricature with the right qualitative behaviour and no claim to quantitative validity: it is not calibrated against any material.

Impact. Each impact deposits a fixed energy, and the penetration is found by equating that energy to the work of indentation, ∫P dh, at the current damage level. Wear damage accumulates smoothly with each impact; discrete fracture events are drawn from a seeded random number generator with a probability that rises once the damage passes a threshold set by the coating's toughness. The random draw is seeded, so the trace is stable when you move other sliders. Real impact data is not that reproducible.

The film–substrate blend is a weighted average, w·(film) + (1 − w)·(substrate), with w = exp[−(h_c/0.25t)^1.4] for hardness and a longer range for modulus, because the elastic field reaches further than the plastic zone. This is a convenience, not a model; part 2 treats substrate effects properly.

The thing to remember

A single load–unload test on a cracked coating will hand you a hardness and a modulus that look entirely normal. The crack happened during loading and the analysis only looks at the unloading slope, so the event is averaged into the answer instead of being reported. If you need to know whether something broke, not just what the average stiffness was afterwards, the schedule has to sample continuously (CSM), or repeatedly (multi-cycle, fatigue, impact). No amount of post-processing recovers an event the schedule never looked at.

Load control or displacement control: the same event, two faces

A pop-in and a load drop can be the same physical event. Which one you see is decided by the feedback loop, not by the material. Under load control the instrument holds P and lets the depth run; under displacement control it holds h and lets the load fall.

Write the contact as P = f(h) with tangent stiffness k_c = df/dh. An instability is a sudden change of state: the material now needs an extra depth δ to carry the same load, so the whole curve shifts right by δ. The machine sits in series with the contact, with a feedback stiffness K_m that is small when it is chasing a load setpoint and large when it is chasing a displacement setpoint. Balancing the two after the jump gives one expression that contains both experiments:

Δh = δ · k_c / (k_c + K_m) ΔP = − K_m · δ · k_c / (k_c + K_m) K_m → 0 is load control: the full burst Δh = δ appears and the load never moves. K_m → ∞ is displacement control: no burst at all, and the load drops by k_cδ. Everything real is in between, which is why "displacement control" on a load-controlled instrument gives you a partial burst and a partial load drop.

One event, two feedback loops Sweep the feedback stiffness from one limit to the other

Before the event After the event The excursion Actual path at this K_m

Load control: P is the setpoint, so the depth is free to run.

Before the event After the event The excursion Actual path at this K_m

Displacement control: h is the setpoint, so the load has to give.

Event magnitude δ

Load at which it happens

Feedback stiffness K_m/k_c

Burst length Δh

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nm · load control gives –

Load drop ΔP

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mN · displacement control gives –

Contact tangent stiffness k_c

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mN/nm

Machine work, load control

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pJ = P·δ

Stored energy released, disp. control

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pJ ≈ ½k_cδ²

Energy ratio

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× more energy under load control

– –

Drive the feedback stiffness slider from one end to the other with everything else fixed. Nothing about the material changed. The burst became a load drop.

The energy readouts are the reason this matters. Under load control the actuator keeps pushing at P while the tip advances by δ, so it does work P·δ on the sample. Under displacement control the actuator does no work at all: the event runs on the elastic energy already stored, roughly ½k_cδ². Load control pumps energy into an instability that is trying to release it. That ratio, 2P/(k_cδ), is often in the tens, which is why a load-controlled machine flies across a softening event and lands somewhere on the far side rather than walking through it.

The practical division: load control is the default and is entirely fine for hardness and modulus. Displacement control is what you want for serrated flow in a metallic glass, for strain softening, for anything where the interesting physics is the instability itself and you need to stay on the curve rather than jump over it.

A trap worth naming

Most commercial nanoindenters are load-controlled machines: an electromagnetic coil or an electrostatic actuator applies a force, and displacement is the measured output. "Displacement control" on those instruments is a fast feedback loop adjusting the load to chase a displacement setpoint, with a bandwidth of order a kilohertz. A pop-in lasts tens of microseconds. The loop cannot possibly follow it, so what you record is the load-control answer with a slower recovery afterwards. Genuine displacement control needs a stiff, fast actuator in the loop, and you should check what your instrument really does before claiming you measured a load drop.

CSM up close: the oscillation, the phase, and the artifact

Continuous stiffness measurement is the most useful thing on a nanoindenter and the easiest to misread. It works by treating the whole instrument as a damped mass on springs with the contact in the middle, and every term in that model has to be right before the stiffness it reports means anything.

Superimpose a small sinusoidal force on the loading ramp, measure the displacement amplitude and the phase lag with a lock-in, and you have two numbers at every depth instead of one number at the peak. The dynamic model Oliver and Pharr wrote down in 1992 relates those two numbers to the contact:

| P_os / h_os | = √( [ (1/S + 1/K_f)⁻¹ + K_s − mω² ]² + (ωC)² ) tan φ = ωC / [ (1/S + 1/K_f)⁻¹ + K_s − mω² ] S is the contact stiffness you want, K_f the load-frame stiffness in series with it, K_s the stiffness of the springs holding the indenter column up, m the mass of that column, C the damping (air gap, gauge, and the contact's own), ω = 2πf. Everything is in SI here (N/m, kg, N·s/m) and S in mN/nm is S × 10⁶ N/m.

Read that expression as a competition. At low frequency the stiffness terms dominate and the amplitude ratio is essentially the contact stiffness plus the support springs. At the frequency where mω² cancels the stiffness terms, the bracket goes to zero and only the damping is left: the amplitude ratio is then exactly ωC and the phase passes through 90°. On a real, well-damped head that is not a sharp dip; it is a shoulder. But the phase crossing is unmistakable, and any analysis that dropped the mass term is now reporting garbage that looks like a measurement.

The CSM dynamic model Push the depth down and the frequency up until the resonance walks into view

|P_os/h_os| Damping floor, ωC Contact stiffness S Test frequency

Both axes logarithmic. The curve leaves the flat S plateau where the mass term takes over; whether that shows as a dip depends on how damped the head is. Set C to zero to see the undamped antiresonance.

Phase φ 90°: resonance Test frequency

A few degrees on a stiff elastic contact. Tens of degrees on a polymer.

True S Full model inversion Neglecting m and K_s Neglecting the frame K_f

The full inversion sits exactly on the truth by construction: that is the point of having the model. The other two are what you get by leaving a term out.

Test frequency f

Depth h

Harmonic amplitude h_os

Damping C

Support stiffness K_s

True contact stiffness S

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mN/nm · –

|P_os/h_os| at f

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N/m

Phase φ at f

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degrees

Harmonic load P_os

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µN at h_os

Resonance f₀

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Hz · f/f₀ = –

Error if you drop m and K_s

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in the reported S

– –

Column mass m = 0.30 g and frame stiffness K_f = 5 × 10⁶ N/m are held fixed; the sample is fused silica. A few N·s/m of damping is a well-behaved head; tens of N·s/m is a lossy one or a lossy contact. Set the depth to 4 nm and the resonance drops into the audio band, where a 45 Hz test is no longer comfortably far from it.

The artifact: a hardness that rises because you oscillated

Now the uncomfortable part. The oscillation is not free. Each cycle does a little extra plastic work on the material under the tip, and when the harmonic amplitude is a significant fraction of the total depth, which it always is at the start of an indent, the measured hardness reads high at shallow depth for a reason that has nothing to do with the material.

Pharr, Strader and Oliver laid this out in 2009: the error grows as the ratio of harmonic to total displacement grows, it depends on the loading rate through the number of oscillations per nanometre, and it is worse in materials that are more strain-rate sensitive. The magnitude is instrument-dependent and the effect is still argued about. What is not in dispute is that it produces a hardness that rises towards the surface, which is exactly the shape of an indentation size effect.

Merle, Maier, Göken and Durst pushed the same signal in the opposite direction, using continuously measured unloading stiffness to pin down the effective indenter shape and the sink-in factor ε rather than assuming them. That is worth holding next to part 1: ε = 0.75 and the area function are not constants of nature, they are measurements, and a continuous stiffness signal is one of the few ways to check them on your own tip.

Two ways to make hardness rise at shallow depth Match the Nix–Gao curve to the artifact and see how well it fits

True H, depth-independent H with the CSM artifact Genuine Nix–Gao size effect Nix–Gao fitted to the artifact

Harmonic amplitude h_os

Artifact strength κ

Nix–Gao length h*

Apparent H at h = 25 nm

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GPa · true 6.00

Inflation at 25 nm

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from the oscillation alone

h* from a blind Nix–Gao fit

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nm, fitted to the artifact

H₀ from that fit

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GPa · fit R² –

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The artifact model here is H_app = H·(1 + κ·h_os/h), a caricature with one honest knob: κ absorbs everything the real effect depends on: material, loading rate, the number of oscillations per nanometre, the lock-in settings. Its size is instrument-dependent and still argued about. Note where a blind Nix–Gao fit to it lands: the shape matches beautifully and h* comes out at roughly 2κh_os, a small but entirely plausible-looking number.

A trap worth naming

A blind Nix–Gao fit to the artifact curve above returns a perfectly respectable h* and a perfectly respectable R², from which you could compute a geometrically-necessary dislocation density and publish it. Part 2 covers the other impostor: a blunt tip analysed with an ideal area function, which produces the same rising hardness. Three different causes, one curve shape. The only way to separate them is to change the thing you suspect: run the same material at two harmonic amplitudes, or with CSM off and single unloads instead, and see whether the size effect moves.

nanoDMA, and why the damping term has to be done properly

On a polymer the phase angle is no longer a small correction. The contact itself dissipates, and if you resolve the response into a real and an imaginary part you get storage and loss moduli directly:

E′ = (S·√π) / (2β·√A) E″ = (ω·C_s·√π) / (2β·√A) tan δ = E″/E′ = ω·C_s/S C_s is the damping of the contact alone: the measured damping minus the instrument's own, which has to be characterised out of contact at every frequency you intend to use. β and A are the same correction factor and contact area as in the static analysis. Note the last identity: tan δ needs no area function at all, which makes it the most robust number nanoDMA produces.

Two consequences. First, the instrument's damping is not a constant: the air gap in a capacitance gauge, the leaf springs and the electronics all contribute frequency-dependent terms, and subtracting a single number measured at one frequency will put a slope into your tan δ that looks like a relaxation. Second, on a soft sample the contact damping can be comparable to the instrument's, so the subtraction is a difference of two similar numbers and its uncertainty is much larger than either. Herbert, Oliver and Pharr worked through the whole dynamic characterisation for viscoelastic solids in 2008, and it is the reference to read before reporting a tan δ from a nanoindenter.

Choosing

The honest summary of the previous three sections is a table. Read the last column first: if you cannot say what failure would look like in the mode you picked, you have not designed the experiment yet.

ISO 14577-1 defines the load-application schedules for rows 1–3; the rest are established practice rather than standardised methods. "Where it fails" means where the mode gives you a wrong or empty answer, not where the sample breaks.
Mode	What it measures	Best for	Where it fails	Failure signature
Direct ramp monotonic to peak, no unload	Events and critical loads. No H, no E: there is no unloading slope to fit.	Finding pop-in, coating breakthrough, cracking onset. The cheapest reconnaissance you can run.	Anything that needs a property. Also anything where the event is slow, because you have no repeat.	Displacement burst at constant load; a staircase of bursts as successive layers give way.
Load–unload the standard quasi-static test	H and E at one depth by Oliver–Pharr, plus the energy dissipated in the loop.	Routine hardness and modulus on a homogeneous sample. Everything in part 1.	Depth profiles, creeping materials (the unload slope is corrupted), anything that broke on the way in.	A kink or bulge on the unloading curve; h_f/h_max near 1 warning of pile-up.
Progressive multi-cycle load–partial-unload, CMC	S, and therefore H and E, at many depths from one indent location.	Depth profiling when CSM is unavailable; rough or precious samples where you cannot afford many sites.	Fine depth resolution: you get one point per cycle, not one per nanometre. Reloading disturbs the contact.	The depth jumps out of the smooth staircase and the extracted stiffness drops; that cycle's peak load is a critical load.
Cyclic at constant peak nanofatigue	Depth at peak load against cycle number, and the loop area per cycle.	Coating durability, cyclic softening or hardening, contact fatigue at small scale.	Absolute fatigue life. The stress state under an indenter is nothing like a component's.	The knee in depth-versus-N where the per-cycle increment accelerates; loop area rising instead of settling.
CSM continuous stiffness, ~45 Hz	S, H and E continuously against depth, from one indent, with no unloading fit.	Depth profiles, thin films, size-effect studies, soft and creeping materials.	Very shallow depths, where the plasticity artifact inflates H; and near the column resonance.	A step down in the continuously measured stiffness the instant a crack forms: fracture a single unload would average away.
nanoDMA frequency sweep	Storage modulus E′, loss modulus E″ and tan δ against frequency.	Polymers, gels, tissue, adhesives; anything viscoelastic.	Stiff elastic solids, where the measured loss is mostly the instrument's own damping.	Not fracture: a tan δ peak marking a relaxation or the glass transition, and a rising loss modulus as damage accumulates.
Constant strain rate Ṗ/P = 2ḣ/h, load rises exponentially	H at a fixed strain rate across depth; rate sensitivity when combined with rate jumps.	Rate-sensitive materials, and any comparison of hardness across depths that should be at matched strain rate.	Very shallow starts, where h is small and the commanded ḣ becomes unachievably slow.	Serrated flow: in metallic glasses the size and spacing of the serrations depend on the imposed rate, and they vanish above a critical rate.
Nano-impact repetitive impact at fixed energy	Depth against impact number. A ranking, not a property.	Accelerated coating durability and fracture-resistance comparison between candidates.	Anything you want in units. There is no toughness number at the end of this, only an order.	A step-like depth-versus-impact trace; each abrupt jump is a fracture event, and the impact number of the first big jump is the ranking.

Or answer two questions

Most people arrive at this table already knowing what they want to find out and what they are holding. Those two facts are usually enough.

Mode chooser Pick one from each row

What do you want to know?

What is your material?

Use this schedule

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Runner-up: – –

Thirty combinations, all written out by hand. There is no scoring function behind this: a lookup table you can read is more honest than a formula you cannot.

Where the honest answer is "this measurement is not worth much on that material", the card says so rather than picking a mode anyway.

The thing to remember

The four parts of this series have circled one point. Indentation gives you a number for anything you ask about, and the number always looks reasonable. Part 1 showed that the number depends on arithmetic you chose; part 2, that it depends on artefacts you did not; part 3, that the interesting quantities need a different analysis entirely. This part adds the earliest decision of all: the schedule decides which physics is even visible. Everything after that is bookkeeping.

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 3

Beyond hardness

Spherical indentation stress–strain curves, pop-in statistics, strain-rate jumps and activation volume, and the fracture-toughness equations that disagree with each other.

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 dynamic model used in section 3 appears here.
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): refinements, and CSM as it is actually practised.
X. Li, B. Bhushan, “A review of nanoindentation continuous stiffness measurement technique and its applications,” Mater. Charact. 48, 11 (2002).
G. M. Pharr, J. H. Strader, W. C. Oliver, “Critical issues in making small-depth mechanical property measurements by nanoindentation with continuous stiffness measurement,” J. Mater. Res. 24, 653 (2009): the CSM plasticity artifact in section 3.
B. Merle, V. Maier, M. Göken, K. Durst, “Experimental determination of the effective indenter shape and ε-factor for nanoindentation by continuously measuring the unloading stiffness,” J. Mater. Res. 27, 214 (2012): what continuously measured unloading stiffness does to the standard constants.
E. G. Herbert, W. C. Oliver, G. M. Pharr, “Nanoindentation and the dynamic characterization of viscoelastic solids,” J. Phys. D: Appl. Phys. 41, 074021 (2008): the damping bookkeeping behind E′, E″ and tan δ.
C. A. Schuh, A. C. Lund, T. G. Nieh, “New regime of homogeneous flow in the deformation map of metallic glasses: elevated temperature nanoindentation experiments and mechanistic modeling,” Acta Mater. 52, 5879 (2004): serrated flow and its rate dependence, used in mode 7.
B. D. Beake, J. F. Smith, “High-temperature nanoindentation testing of fused silica and other materials,” Philos. Mag. A 82, 2179 (2002); and B. D. Beake, S. R. Goodes, J. F. Smith, “Nanoscale repetitive impact testing of polymer films,” Surf. Eng. 17, 187 (2001): the repetitive-impact method in mode 8.
ISO 14577-1, Metallic materials: Instrumented indentation test for hardness and materials parameters: Part 1, Test method: defines the load-application schedules.
A. C. Fischer-Cripps, Nanoindentation, 3rd ed., Springer (2011): chapter on dynamic indentation and the instrument as a mass–spring–dashpot.