IN29 · Separate contact from machine flex

Nanoindenter

Why does indentation depth alone not determine hardness and modulus? Hardness compares the supported load with the contact area. Elastic modulus comes from how rapidly the load falls as the contact recovers during unloading. Two tests can reach the same depth yet have different contact areas and recovery slopes. The tip shape and machine flexibility must be calibrated to separate them.

Press and unload a calibrated tip. Keep the force, travel and time samples, then separate frame compliance, contact stiffness and area.

Press, unload and compare recovery

Reduced modulus from unloading $100\,\mathrm{GPa}$

Conditional indentation hardness $2.00\,\mathrm{GPa}$

Contact stiffness $79.8\,\mathrm{\mu N/nm}$

Work returned on unloading $16.8\%$

Contact modelPeak load

P_{\max}/\mathrm{\mu N}

1000

Material hardness input

H_{\mathrm{in}}/\mathrm{GPa}

2

Reduced modulus input

E_{r,\mathrm{in}}/\mathrm{GPa}

100

Unload duration

T_{\mathrm{unload}}/\mathrm s

8

Drag the gold peak-load slider to record a fresh loading cycle.

A calibrated plastic indentation and its unloading are ready.

Machine, protocol and assumptions

Frame compliance

C_f/(\mathrm{nm/\mu N})

Contact-datum offset

h_0/\mathrm{nm}

Quadratic area coefficient

C_2

Tip-rounding area coefficient

C_1/\mathrm{nm}

Loading duration

T_{\mathrm{load}}/\mathrm s

Peak hold duration

T_{\mathrm{hold}}/\mathrm s

Unsupported hold-motion challenge

v_{\mathrm{hold}}/(\mathrm{nm/s})

Displacement noise estimate

\sigma_h/\mathrm{nm}

Force noise estimate

\sigma_P/\mathrm{\mu N}

Assumed specimen Poisson ratio

\nu

Sensor seed

s

Specimen interpretation

A changed frame or tip acquires new reference curves. Earlier cycles retain their own references and applied calibration.

Model assumptions

A synthetic axisymmetric rounded-cone specimen follows a declared passive elastoplastic loading/unloading law. Effective-paraboloid unloading uses a 0.75 contact-depth factor and axisymmetric geometry factor one. It is not a Berkovich reconstruction or finite-element plasticity solution. A separately acquired known-radius Hertz sphere first calibrates frame compliance and the common coplanar contact datum; four known-modulus reference curves then fit a two-term tip area function. Separate sphere and sharp-tip checks are withheld. Reference properties are declared synthetic benchmarks, not certified materials. Force and capacitive machine-travel conversion are known. Their raw noise, quantization, timing and clipping flags are retained. The inverse uses only raw unloading and measured references, with contact-datum, released-depth, coverage and passivity guards. Hertz mode reports elastic mean pressure, not plastic hardness. Reduced modulus combines specimen and diamond-tip compliance; converting it to specimen modulus requires the stated Poisson ratio. Pile-up, substrate effects, adhesion, anisotropy and viscoelasticity are not solved. Time-dependent hold motion and unvalidated specimen domains withhold contact-property conclusions. Enlarged cutaway surfaces are geometric models, not acquired topography. Mechanical supports are schematic; full axisymmetric contact remains in the calculation.

Simulated force and capacitive travel readings

Unloading carries a different piece of information

The cutaway surface above is an enlarged geometric model. Acquired observations are the force, machine-travel and time samples below.

Inspect cycle

Measured travel and corrected contact displacement

$P/\mathrm{\mu N}$

$h/\mathrm{nm}$

Machine travelCorrected contact travelUnloading fit

The retained loading protocol

$P/\mathrm{\mu N}$

$(t-t_0)/\mathrm s$

Calibration and every observed sample

Fitted frame compliance

C_f=0.00300\,\mathrm{nm/\mu N}

Fitted projected tip area

A(h_c)=24.5h_c^2+1000\,\mathrm{nm}\,h_c

Independent reference errors

(0.0\%,0.9\%)

Recorded force, machine travel and clock time
Time $t/\mathrm s$	Phase	$P/\mathrm{\mu N}$	$h_{\mathrm{machine}}/\mathrm{nm}$

The acquisition clock starts at instrument time zero; these are synchronous sample instants, not wall-clock dates. CSV retains all eight sample/reference curves, their full row counts, protocol durations, sensor units, seeds, flags and known-reference definitions. Applied frame and area calibrations are archived; replay recomputes them from raw references.

A separate generating reference

These inputs describe the synthetic specimen and enlarged contact model. They are excluded from the inverse and raw CSV/JSON.

E_{r,\mathrm{in}}=100\,\mathrm{GPa}

Equations, scope and primary sources

h=h_{\mathrm{machine}}-h_0-C_fP,\quad h_c=h_{\max}-0.75\frac{P_{\max}}{S}

E_r=\frac{\sqrt{\pi}}{2\beta}\frac{S}{\sqrt{A(h_c)}},\quad\beta=1,\quad \bar p=\frac{P_{\max}}{A(h_c)}

The equivalent rigid axisymmetric tip acts on a half-space with the reduced contact modulus; explicit diamond deformation is not drawn. The rounded tip uses an effective-paraboloid unloading correction. The half-space is homogeneous and isotropic, with a coplanar transferred contact datum and unchanged linear frame compliance across mounted reference tips. The sphere radius and reference elastic properties are known independently. The sharp area function is calibrated over its observed depth range. Unloading is a declared phenomenological power law, not a finite-element plasticity calculation.

Hertz mode assumes small elastic spherical contact and reports mean pressure. In the plastic model this mean load over projected peak area is conditional indentation hardness. A specimen Young modulus additionally needs its Poisson ratio and the assumed diamond constants of 1141 gigapascals and 0.07. Pile-up, substrates, adhesion and viscoelasticity need another model. The hold-motion challenge is an additive time-dependent penetration example; its inverse is withheld. Noise and calibration systematics are not propagated into certified uncertainty intervals.

Start with the essentials

Focus question: Why does indentation depth alone not determine hardness and modulus?
One-sentence intuition: Hardness needs peak load and projected contact area. Reduced modulus needs contact stiffness and that area, after frame correction.

Core mathematical model

Subtract machine flexibility and the transferred contact datum

h=h_{\mathrm{machine}}-h_0-C_fP

Compression and penetration are positive. Force is recorded in micronewtons and displacement in nanometres; frame compliance uses the corresponding displacement-per-force unit.

Fit the nonlinear unloading branch

P=B(h-h_f)^m,\qquad S=\left.\frac{\mathrm dP}{\mathrm dh}\right|_{P_{\max}}

The fit reads the observed unloading points. The forward benchmark uses exponent 1.5; the inverse estimates it and screens incompatible branches. An effective-paraboloid contact correction of 0.75 is adopted.

Contact area separates pressure and elastic response

h_c=h_{\max}-0.75\frac{P_{\max}}S,\quad H=\frac{P_{\max}}{A(h_c)},\quad E_r=\frac{\sqrt\pi}{2\beta}\frac S{\sqrt{A(h_c)}}

The geometry factor is one for this axisymmetric equivalent tip. The projected area is calibrated over measured depths. These symbolic relations use coherent units. In Hertz mode the force-over-area quantity is elastic mean pressure, not plastic hardness.

Both contact bodies contribute elastic compliance

\frac1{E_r}=\frac{1-\nu^2}{E}+\frac{1-\nu_i^2}{E_i}

Reduced modulus is measured conditionally on the contact model. A specimen Young modulus additionally needs its Poisson ratio and the diamond constants of 1141 gigapascals and 0.07.

Common difficulties

Contact stiffness depends on contact size

Typical misconceptionA steeper unloading curve always means a larger material modulus.

Better mental modelA larger contact also stiffens the measurement. The calibrated projected area is needed to turn the measured stiffness into reduced modulus.

Frame compliance and tip area need independent information

Typical misconceptionOne unknown curve can calibrate any frame flexibility and tip shape at once.

Better mental modelHere a known-radius Hertz reference first identifies the frame and zero. Four known-modulus curves then identify the two area coefficients; separate references check both steps.

The enlarged surface is an equivalent contact model

Typical misconceptionThe apparatus viewport shows a measured surface map or explicit deformation of a diamond tip.

Better mental modelThe scene draws a rigid equivalent tip on a reduced-modulus half-space. Full axisymmetric contact remains in the calculation despite the visual cutaway. The acquired data are force, machine travel and time.

Run the experiment

01
Follow one force-controlled cycle
Start with A lasting impression. Watch the replay, then compare measured travel and corrected contact displacement below.
What to observe: The blue curve includes the contact datum and frame flexibility. The green curve removes the separately calibrated machine contribution.
02
Change the real force control
Choose Sensors, then drag the gold peak-load slider while keeping the pointer down.
What to observe: The physical slider moves and a fresh cycle is acquired. The transducer and capacitive plate follow the same series travel used by the record.
03
Reach the same depth with more recovery
Compare A lasting impression and Same depth, more recovery.
What to observe: Their maximum corrected contact displacements nearly coincide. The softer model returns more work on unloading, even though its conditional hardness is the same.
04
Try an elastic sphere
Choose The elastic sphere returns and inspect the pressure label and recovered-work value.
What to observe: The sphere returns to the contact datum. Its mean elastic pressure is nonzero during contact, although this model has no plastic hardness.
05
Keep an earlier measurement
Keep this cycle, change a setting and select the earlier cycle. Export that selection as CSV or JSON.
What to observe: Its original sensor rows, eight sample/reference curves and applied calibration stay fixed. Replay refits the retained reference observations.
06
Check when the interpretation fails
Try Motion during the hold and A substrate needs another model.
What to observe: The observed curves remain available, but quasistatic bulk contact properties are withheld. The unsupported hold-motion example is not a solved viscoelastic constitutive law.

Nanoindenter

Unloading carries a different piece of information

Separate load support from elastic recovery

Core mathematical model

Subtract machine flexibility and the transferred contact datum

Fit the nonlinear unloading branch

Contact area separates pressure and elastic response

Both contact bodies contribute elastic compliance

Common difficulties

Contact stiffness depends on contact size

Frame compliance and tip area need independent information

The enlarged surface is an equivalent contact model

Run the experiment

Follow one force-controlled cycle

Change the real force control

Reach the same depth with more recovery

Try an elastic sphere

Keep an earlier measurement

Check when the interpretation fails

Related experiments and articles

How physics instruments measure