Contact Mechanics · Nanoindentation mapping
Nanoindentation mapping: spacing, depth and phase peaks
A hardness map is a grid of tiny indents, and how deep and how close you push them decides whether you see the phases or a blur.
How far apart should nanoindents be in a map?
Each indent leaves a plastic zone (permanently deformed material) a few contact radii wide around it.
Try it: in DP steel, raise the depth from 150 to 600 nm: mixed indents go from 32% to 89% and the martensite islands smear into the ferrite. Back at 150 nm, set the spacing to 1 µm: the zones turn coral.
How do you get phase hardness from a nanoindentation histogram?
Grid indentation places indents at random spots and fits one bell curve per phase to the hardness histogram.
Try it: in DP steel at 50 nm the fit gives 4.1 and 6.4 GPa at 65% and 35% (true: 70% and 30%). At 600 nm the coral mixed indents fill the gap and the fit gives 4.4 and 5.2 GPa at 43% and 57%.
What to take away
Space by depth
For a Berkovich tip, spacing of 10 times the depth was enough in about 50,000 tests (Phani and Oliver 2019).
Zones may overlap
At 10 times the depth the plastic zones already overlap. The same study found this did not change the result.
Shallow for small phases
An indent feels a zone a few contact radii wide. Keep the depth below about a tenth of the phase size.
Check the fit
A mixture fit returns as many peaks as you ask for. Mixed indents shift them, so compare with the microstructure.
More detail: the models behind the pictures and their limits
Contact and plastic zone
A Berkovich tip has a projected contact area A = 24.5 h², so the equivalent contact radius is a = (24.5/π)1/2 h = 2.79 h, and the triangle side is 7.52 h. The plastic zone radius c comes from Johnson's expanding cavity model: c/a = (E tanβ / 3Y)1/3 with β = 19.7° (the equivalent cone), and Y found from H = (2/3) Y [1 + ln(E tanβ / 3Y)]. This gives c/a of about 2.5 for ferrite, 2.0 for martensite, 1.7 for cobalt and 1.5 for WC. Soft metals with a large E/Y reach 3 to 5. The model assumes a uniform, isotropic solid; real zones near a phase edge are not circles.
What one indent measures
Here each indent reads a weighted average of the phase hardness inside its plastic zone, with a weight that falls linearly from the centre to c. An indent counts as mixed when no phase has 90% of that weight. This is a simple averaging model, not a finite element result. A hard particle in a soft matrix can read lower than this average, because the soft phase yields around it.
Phase values used
DP 780 steel: ferrite 4.0 GPa, martensite 6.7 GPa (Webber and Knezevic 2024, 1 µm spacing, about 50 nm deep). WC-Co: cobalt binder 10 GPa, WC 25 GPa, moduli 230 and 470 GPa (Roa et al. 2018). Cement paste: low density C-S-H 0.45 GPa, high density C-S-H 0.83 GPa, portlandite 1.31 GPa, with moduli 18, 29 and 40 GPa. These are typical values of the size reported in grid indentation of cement paste (Constantinides and Ulm 2007; Ulm et al. 2007); the moduli near 20 and 30 GPa for the two C-S-H types are well established, but treat the hardness values as illustrative. The microstructures are made up for the picture; the phase fractions shown are what the page counts in them.
The mixture fit
Widget 2 gives each indent a random position (grid spacing much larger than the phases), adds 8% random scatter, then fits K Gaussians with the expectation maximisation (EM) method: guess peaks, give each indent a share of each peak, refit the peaks, repeat 400 times. Real studies often fit hardness and modulus together, which separates peaks better (Besharatloo and Wheeler 2021). The older rule of 3 indent widths (about 20 h) is stricter than the 10 h found by Phani and Oliver.
On this site: Nanoindentation hardness: making the number · When nanoindentation lies · Indentation and grain orientation · Reading EBSD IPF maps · Beyond hardness
Questions people ask
What is the minimum spacing between nanoindents?
For a Berkovich tip, Phani and Oliver (2019) found that 10 times the indentation depth is enough for accurate hardness and modulus. The older rule of 3 times the lateral size is about 20 times the depth. Closer indents land in material that the neighbour has already work hardened or cracked.
How deep should I indent to measure one phase?
Shallow enough that the plastic zone stays inside the phase. Grid indentation work uses a depth below about one tenth of the phase size. Too shallow brings surface roughness and size effect problems, so there is a window.
What is grid indentation?
A method where many indents are placed on a grid with spacing larger than the phases, so each indent lands on a random spot. The histogram of results then holds one peak per phase. Constantinides, Ulm and co-workers set it up for cement, bone and shale.
What is statistical deconvolution in nanoindentation?
Fitting the histogram of hardness or modulus with a sum of bell curves, one per phase. The centre of each curve estimates the phase property, and its area estimates the phase fraction. It works best when the phases differ well and the indents are small compared with the phases.
How many indents does a nanoindentation map need?
Enough that each phase peak has a few hundred indents. With a few dozen, the fitted peaks move a lot from one sample to the next. High speed mapping, with each indent taking under a second, makes thousands of indents routine (Phani and Oliver 2019; Hintsala et al. 2018).
References
Show the 9 references
- G. Constantinides, K. S. Ravi Chandran, F.-J. Ulm, K. J. Van Vliet, Grid indentation analysis of composite microstructure and mechanics: principles and validation, Materials Science and Engineering A 430, 189 to 202 (2006). doi:10.1016/j.msea.2006.05.125
- F.-J. Ulm, M. Vandamme, C. Bobko, J. A. Ortega, K. Tai, C. Ortiz, Statistical indentation techniques for hydrated nanocomposites: concrete, bone, and shale, Journal of the American Ceramic Society 90, 2677 to 2692 (2007). doi:10.1111/j.1551-2916.2007.02012.x
- G. Constantinides, F.-J. Ulm, The nanogranular nature of C-S-H, Journal of the Mechanics and Physics of Solids 55, 64 to 90 (2007). doi:10.1016/j.jmps.2006.06.003
- P. Sudharshan Phani, W. C. Oliver, A critical assessment of the effect of indentation spacing on the measurement of hardness and modulus using instrumented indentation testing, Materials & Design 164, 107563 (2019). doi:10.1016/j.matdes.2018.107563
- E. D. Hintsala, U. Hangen, D. D. Stauffer, High-throughput nanoindentation for statistical and spatial property determination, JOM 70, 494 to 503 (2018). doi:10.1007/s11837-018-2752-0
- H. Besharatloo, J. M. Wheeler, Influence of indentation size and spacing on statistical phase analysis via high-speed nanoindentation mapping of metal alloys, Journal of Materials Research 36, 2198 to 2212 (2021). doi:10.1557/s43578-021-00214-5
- J. J. Roa, P. Sudharshan Phani, W. C. Oliver, L. Llanes, Mapping of mechanical properties at microstructural length scale in WC-Co cemented carbides: assessment of hardness and elastic modulus by means of high speed massive nanoindentation and statistical analysis, International Journal of Refractory Metals and Hard Materials 75, 211 to 217 (2018). doi:10.1016/j.ijrmhm.2018.04.019
- E. Webber, M. Knezevic, Assessing strength of ferrite and martensite in five dual phase and two martensitic steels via high throughput nanoindentation to elucidate origins of strength, Journal of Materials Research and Technology 33, 3635 to 3648 (2024).
- K. L. Johnson, Contact Mechanics, Cambridge University Press (1985), chapter 6 (the expanding cavity model of indentation).
BibTeX
@misc{tripathy2026nanoindentationmapping,
author = {Tripathy, Manisha},
title = {Nanoindentation Mapping Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/indentation/nanoindentation-mapping}},
note = {Interactive web tool}
}