G.M. Pharr et al.: Understanding nanoindentation unloading curves
14. W.D. Nix and H. Gao, J. Mech. Phys. Solids 46, 411 (1998).
15. A. Bolshakov, W.C. Oliver, and G.M. Pharr, J. Mater. Res. 11,
760 (1996).
16. A. Bolshakov and G.M. Pharr, J. Mater. Res. 13, 1049 (1998).
17. T.Y. Tsui, W.C. Oliver, and G.M. Pharr, J. Mater. Res. 11, 752
(1996).
18. J.C. Hay, A. Bolshakov, and G.M. Pharr, J. Mater. Res. 14, 2296
(1999).
19. K.L. Johnson, Contact Mechanics (Cambridge University Press,
Cambridge, U.K., 1985).
20. K.L. Johnson, in Engineering Plasticity, edited by Heyman and
Leckie (Cambridge University Press, Cambridge, U.K., 1968).
21. W. Hirst and M.G.J.W. Howse, Proc. R. Soc. A 311, 429 (1969).
formation of the permanent hardness impression. The sur-
face distortions can be accounted for by defining an
effective indenter with a shape such that when pressed
into a flat elastic half space, the resulting normal surface
displacements are the same as those produced by a coni-
cal indenter pressed into the plastically deformed surface
of the hardness impression. Simple arguments based on
the assumption of a constant pressure distribution under the
indenter show that the shape of the effective indenter is
well-described by the relation in Eq. (11) (see Sect. V),
where p is the pressure, amax is the contact radius at peak
load, E is Young’s modulus, and is Poisson’s ratio. The
exponent describing the unloading curve consistent with
this shape is m ס
1.38. Slightly larger values of m are
expected if the pressure under the indenter decreases with
distance from the center of the contact, as is the case for
soft metals. The same arguments show that indentation
unloading curves (P ס
load; h ס
displacement) can be
quantitatively approximated by the relation in Eq. (14)
(see Sect. V), where Pmax is the maximum load, H is the
hardness, and hf is the final depth after full unloading.
The analysis also reveals that the parameter ⑀ used in the
Oliver–Pharr method for analyzing nanoindentation data
to obtain hardness and elastic modulus should have a
value close to 0.76.
APPENDIX: EFFECTIVE INDENTER SHAPE
FOR A LINEARLY DECREASING
PRESSURE DISTRIBUTION
The effective indenter shape for a pressure distribu-
tion, p(r), that decreases linearly with distance from the
center of the contact, r, can be derived using a method
outlined by Johnson.19 The pressure distribution is as-
sumed to be of the form
r
p r͒ = p − ⌬p
,
(A1)
͑
0
a
where p0 is the maximum pressure at the center of the
contact, ⌬p is the reduction in pressure at the contact
edge relative to the peak value, and a is the radius of the
contact circle. Following Johnson,19 the position of each
point in the contact circle can be defined by a set of
coordinates (s, ), for which the linear distribution of
pressure can be expressed as
ACKNOWLEDGMENT
Research through the Oak Ridge National Laboratory
SHaRE User Program was sponsored by the Division of
Materials Sciences and Engineering, United States De-
partment of Energy, under Contract DE-AC05-
00OR22725 with UT-Battelle, LLC.
2
2
͌
p s, ͒ = p − ⌬p
r + s + 2rs cos
.
(A2)
͑
0
The vertical displacements of the surface are then given by
REFERENCES
1 − v2
E
s
w r͒ =
2p s, ͒ ds d
s
1
,
(A3)
͑
͑
͐ ͐
1. J.B. Pethica, R. Hutchings, and W.C. Oliver, Philos. Mag. A 48,
593 (1983).
2. J.L. Loubet, J.M. Georges, O. Marchesini, and G. Meille, J. Tribol.
106, 43 (1984).
3. M.F. Doerner and W.D. Nix, J. Mater. Res. 1, 601 (1986).
4. W.C. Oliver and G.M. Pharr, J. Mater. Res. 7, 1564 (1992).
5. G.M. Pharr and W.C. Oliver, MRS Bull. 17, 28 (1992).
6. G.M. Pharr, Mater. Sci. Eng. A 253, 151 (1998).
7. J.L. Hay and G.M. Pharr, in ASM Handbook Volume 8: Mechani-
cal Testing and Evaluation, 10th ed., edited by H. Kuhn and
D. Medlin (ASM International, Materials Park, OH, 2000),
pp. 232–243.
0
where the limits of integration are
1 2
ր
͖
s1,2 = −r cos ͒ ע
r2 cos2 + a2 − r2͒
.
͑
͑
͕
(A4)
Substitution of (A1) and (A2) into (A3) and evaluating
the displacement at the center of contact circle yields for
the effective indenter shape
4 1 − v2͒p a
1 ⌬p
͑
o max
8. I.N. Sneddon, Int. J. Eng. Sci. 3, 47 (1965).
9. G.M. Pharr, W.C. Oliver, and F.R. Brotzen, J. Mater. Res. 7,
613–617 (1992).
z r͒ =
͑
1 −
− E r
ր
amax
͒
͑
ͭ
ͩ ͪ ͩ
ͪ
E
2 po
2
r2 ⌬p
2
ր
10. S.V. Hainsworth, H.W. Chandler, and T.F. Page, J. Mater. Res.
+
sin2
11, 1987 (1996).
͐
2
0
po
4amax
11. K.W. McElhaney, J.J. Vlassak, and W.D. Nix, J. Mater. Res. 13,
1300 (1998).
12. N.A. Stelmashenko, M.G. Walls, L.M. Brown, and Y.V. Milman,
Acta Metall. 41, 2855 (1993).
13. Q. Ma and D.R. Clark, J. Mater. Res. 10, 853 (1995).
2
͌
͌
1 +
1 − r
ր
amax͒ sin2
͑
ln
d
.
ͫ
ͬ
ͮ
2
1 −
1 − r
ր
amax͒ sin2
(A5)
͑
J. Mater. Res., Vol. 17, No. 10, Oct 2002
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