Biomedical Engineering Reference
In-Depth Information
from the harmonic average of two tip geometries (conical and spherical)
and in which the only two parameters are the effective tip radius
R
and
the included cone angle
. Of course, this expression is less useful at
extremely small indentation depths since it gives
A
c
(
h
c
)
0 at
h
c
= 0, but
it is very effective in estimating a tip effective radius. Any area function
A
c
at each contact depth
h
c
:
=
h
max
− 0.75
P
max
S
h
c
(5-15)
The calibration is performed by assuming the fixed value of
E
R
for
a material with known (and constant with depth) elastic modulus (and
thus reduced modulus). This calibration standard is typically the glass
fused silica (a.k.a. fused quartz) with an elastic modulus
E
= 72 GPa
and Poisson's ratio = 0.17 such that, with diamond (
E
= 1.02 TPa and
= 0.07) the reduced modulus
E
R
= 69.6 GPa.
In addition to the area function, there is an additional calibration-
derived quantity, the machine (frame) compliance. The measured
stiffness,
S
, is the series combination of the sample stiffness,
S
s
, and the
frame compliance,
S
f
, as:
1
S
=
1
S
s
+
1
S
f
(5-16)
The contact area (
A
c
) and the frame compliance (
C
f
=
S
f
-1
) are related,
and the calibration protocol used to obtain these values has been
discussed at length in other sources
1
and in the manuals for commercial
nanoindentation instruments.
Once the full calibration (frame compliance and tip area function) is
known, the analysis of elastic-plastic DSI data using the Oliver-Pharr
protocol is straightforward. Three parameters are obtained directly from
raw
P
-
h
data, the peak load (
P
max
), the peak displacement (
h
max
) and the
stiffness
S
is corrected by
Eq. 5-16
to obtain the sample stiffness and the
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