Contact Mechanics · Residual stress
Residual stress from indentation: how stress shifts the curve
Squeeze or stretch the surface below and watch the load, the pile-up and the apparent hardness change.
How does residual stress change a nanoindentation curve?
A compressed surface pushes back harder, so the same depth needs more load; a stretched surface needs less.
Try it: move the stress from −1 to +1 times the yield: the load at 500 nm goes from 10.7 to 6.9 mN (stress-free 9.2 mN), and the pile-up turns into sink-in.
How do you estimate residual stress from indentation?
Compare an indent in the stressed part with one in a stress-free piece of the same material, at the same depth.
Try it: at A/A0 = 1.08 and H = 3 GPa the estimate is −532 MPa; set the reference uncertainty to 5% and the estimate spreads from −864 to −199 MPa.
What to take away
Compression adds load
At the same depth a compressed surface needs more load, a stretched one less.
True hardness barely moves
Tsui, Oliver and Pharr found the hardness stays about the same when the contact area is measured directly. The change comes from pile-up.
You need a reference
Every method compares with the same material free of stress. A few percent of error in the reference is hundreds of MPa.
Models disagree
The same data give different stresses in different models. Report which one you used.
More detail: the equations, assumptions and limits
The starting assumption
Suresh and Giannakopoulos (1998) assume the mean contact pressure (the true hardness H) does not change with an elastic residual stress. Tsui, Oliver and Pharr (1996) measured this in an aluminium alloy (8009) under applied elastic stress: with the contact area measured from images of the indents, the hardness stayed about constant. Bolshakov, Oliver and Pharr (1996) showed with finite element models that the pile-up changes with stress, and that the Oliver-Pharr method, which works out the area from the depth, does not see this change. So the stress shows up as a change in load at a fixed depth, and in the true contact area.
Suresh and Giannakopoulos
An equibiaxial stress σR in the surface is split into a hydrostatic part, which does not change plastic flow, and a part along the indentation axis. For tensile stress they take the full stress: A0/A = 1 + σR/H. For compressive stress they argue that a load of σR sin α × A goes into the hydrostatic part and does not add to the hardness, which brings in sin α: A0/A = 1 − |σR| sin α / H (the same equations are in their US patent 6,155,104). Here A is the true contact area in the stressed part and A0 in the stress-free reference, both at the same depth. α is the angle between the indenter face and the surface: 22° for Vickers (136° between opposite faces) and 24.7° for Berkovich (65.3° from the axis to a face). Since H does not change, P = H A, so the load ratio at the same depth equals the area ratio: P/P0 = A/A0.
What picture 1 computes
H = 3σy (Tabor's rule for a fully plastic contact in a metal). The stress-free load is P0 = H × 24.5 h2, with 24.5 h2 the ideal area of a Berkovich or Vickers tip. The stressed load is P0 × A/A0 from the equations above. The apparent hardness assumes Oliver-Pharr finds the stress-free area at that depth, so it changes by the same fraction as the load. The pile-up height is set so the true contact area fits the model; the shape of the surface outside the contact is only a sketch. Elastic sink-in, tip rounding and work hardening are left out.
Lee and Kwon
Lee and Kwon (2003, 2004) take the part of the stress that changes plastic flow: for equibiaxial stress its component along the indentation axis is −2σR/3. It adds to or takes away from the load at a fixed depth, P0 − P = (2/3) σR A, so σR = 3 (P0 − P) / 2A. With P = HA this is σR = 1.5 H (A0/A − 1) for both signs. For a stress that is not equibiaxial, with σy = p σx, the load change becomes (1 + p) σx A / 3, so one indent gives only the sum of the two stresses, not each one.
Limits
Both models assume the stress is uniform over a depth several times the contact size, stays below yield, and does not change the material's own hardness. Carlsson and Larsson (2001) found with finite elements that the pile-up depends on σR/σy and on work hardening, so a stress estimate also needs the yield strength and the hardening of the material. Stress near yield plastically relaxes under the tip. Near a surface of a thin film, the substrate also changes the curve. The error band in picture 2 only covers the reference; tip area errors and scatter between indents add to it.
Why a reference is needed
Every formula uses the ratio A0/A or P0/P. The reference must be the same material, in the same condition (grain size, work hardening, phase), with no stress: for example an annealed piece, or a piece cut free so the stress is released. A handbook hardness is not good enough: grain size or cold work alone can change the hardness by more than the stress does.
On this site: Nanoindentation hardness: making the number · When the number lies (pile-up) · When Oliver-Pharr fails · Scratch testing: critical loads · Hardness scales and conversions
Questions people ask
Does residual stress change hardness?
It changes the apparent hardness from Oliver-Pharr analysis: compressive stress raises it and tensile stress lowers it. Tsui, Oliver and Pharr (1996) found that the hardness from the directly measured contact area stayed about the same. The apparent change comes from more or less pile-up, which the depth-based area does not see.
Can nanoindentation measure residual stress?
Yes, within limits. You compare indents in the stressed part with indents in a stress-free piece of the same material, at the same depth. Models such as Suresh and Giannakopoulos then turn the change in contact area or load into a stress. The result assumes an equibiaxial stress that is uniform over the indent depth.
Why is there a sin α only for compressive stress?
It is an assumption of the Suresh and Giannakopoulos model. For tension they use the full stress. For compression they argue that the stress acts through the sloped indenter faces, so only sin α of it changes the load, where α is the angle between a face and the surface. Finite element studies such as Carlsson and Larsson (2001) show the real response also depends on work hardening.
What if I have no stress-free reference?
Then the stress cannot be found from one test with these models. Options are annealing a piece to remove the stress, cutting a piece free, or using a method that estimates the stress-free curve from other data, such as the contact shape. X-ray diffraction or hole drilling can give an independent check.
Does this work for a stress that is not the same in both directions?
A sharp, symmetric tip feels only the sum of the two in-plane stresses. Lee and Kwon's extension gives that sum, not the two values. Knoop tips or directional pile-up can split them.
References
Show the 9 references
- S. Suresh and A. E. Giannakopoulos, A new method for estimating residual stresses by instrumented sharp indentation, Acta Materialia 46, 5755 to 5767 (1998). doi:10.1016/S1359-6454(98)00226-2
- T. Y. Tsui, W. C. Oliver and G. M. Pharr, Influences of stress on the measurement of mechanical properties using nanoindentation: Part I. Experimental studies in an aluminum alloy, Journal of Materials Research 11, 752 to 759 (1996). doi:10.1557/JMR.1996.0091
- A. Bolshakov, W. C. Oliver and G. M. Pharr, Influences of stress on the measurement of mechanical properties using nanoindentation: Part II. Finite element simulations, Journal of Materials Research 11, 760 to 768 (1996). doi:10.1557/JMR.1996.0092
- Y.-H. Lee and D. Kwon, Measurement of residual-stress effect by nanoindentation on elastically strained (100) W, Scripta Materialia 49, 459 to 465 (2003). doi:10.1016/S1359-6462(03)00290-9
- Y.-H. Lee and D. Kwon, Estimation of biaxial surface stress by instrumented indentation with sharp indenters, Acta Materialia 52, 1555 to 1563 (2004). doi:10.1016/j.actamat.2003.12.006
- S. Suresh and A. E. Giannakopoulos, Method and apparatus for determining preexisting stresses based on indentation or other mechanical probing of a material, US Patent 6,155,104 (5 December 2000).
- S. Carlsson and P.-L. Larsson, On the determination of residual stress and strain fields by sharp indentation testing. Part I: theoretical and numerical analysis, Acta Materialia 49, 2179 to 2191 (2001). doi:10.1016/S1359-6454(01)00122-7
- W. C. Oliver and G. M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, Journal of Materials Research 7, 1564 to 1583 (1992). doi:10.1557/JMR.1992.1564
- D. Tabor, The Hardness of Metals, Clarendon Press, Oxford (1951): H ≈ 3 times the flow stress for a fully plastic indent.
BibTeX
@misc{tripathy2026residualstressindentation,
author = {Tripathy, Manisha},
title = {Residual Stress from Indentation Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/indentation/residual-stress-from-indentation}},
note = {Interactive web tool}
}