Theory 7
Agilent Nano Indenter G200 User’s Guide
7-6
Determining the Contact Stiffness and Contact Area
and
, it is clear that in order to calculate the
hardness and elastic modulus from indentation load-displacement data,
one must have an accurate measurement of the elastic contact stiffness
(
S
) and the projected contact area under load (
A
).
One of the primary distinctions between IIT and traditional hardness
testing is the manner in which the contact area is derived. Rather than by
imaging, the area is established from an analysis of the indentation
load-displacement data.
The Power-Law Relation
The most widely used method for calculating the contact area was
developed by Oliver and Pharr. The Oliver-Pharr data-analysis
procedure begins by fitting the load-displacement data acquired during
unload to the power-law relation:
(8)
P
is the load applied to the test surface,
h
is the resulting penetration,
B
and
m
are empirically determined fitting parameters, and h
f
is the final
displacement after complete unloading (also determined by curve
fitting). The
contact stiffness
,
S
, is then established by analytically
differentiating
and evaluating at the maximum depth of
penetration, h = h
max
, or
(9)
Experience has shown that
does not always provide an
adequate description of the entire unloading curve, especially for films
on substrates. It is thus prudent practice to determine the contact
stiffness by fitting only the upper portion of the unloading data; fitting
the upper 25 % to 50 % of the data is usually sufficient.
P
B h
h
f
–
m
=
S
Bm h
h
f
–
m
1
–
h
h
max
=
=
