Biomedical Engineering Reference
In-Depth Information
E
(1 − ν
P
= κ
2
R
)
h
(5-19)
2
where
,
R
/
t
f
) . In this instance, the contact dimension is fixed and
R
=
a
. The original work
21
required a look-up table to determine values
of
based on the numerical solution, but of course in the intervening
years the analysis for the film-substrate problem has become near
routine, at least in the instance in which the modulus mismatch between
the film and substrate (
E
f
/
E
s
) is not more than an order of magnitude. The
problem of larger modulus mismatch has also been examined within a
framework of analytical elasticity.
22
In practice, it is common to use
relatively simple empirical functions to extrapolate apparent elastic
modulus-depth data to estimate the modulus of a thin layer
23
or to keep
the indentation depth well below 10% of the total layer thickness.
1
κ = κ
(
ν
Figure 5-9. Schematic illustration of indentation of a layered structure, in which the
observed response depends on the ratio of the indentation length-scale (as given by either
the indentation depth
h
or contact radius
a
) relative to the layer thickness (
t
f
).
6. Time-Dependent Contact
A complication arises in the application of nanoindentation techniques
for examination of biological materials, in that substantial time-
dependent deformation can occur during the experimental time-frame.
As such, the assumption of elastic unloading in the Oliver-Pharr
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