Biomedical Engineering Reference
In-Depth Information
β
is the correction factor for indenter shape (
β
1 for circular contact and 1.05 for Berkovich
indenter).
E
r
is related to the elastic (Young) modulus
E
and Poisson's ratio
ν
of the sample (no sub-
script) and indenter (subscript i) as:
2
2
E v E v E
r i i
Equations (17.1-17.5) are valid for elastic as well as elastic-plastic indentation and any indenter
shape. For elastic contact with a spherical indenter, also Hertz' formulae
[5]
may be used:
1
/
(
1
)/
(
1
)/
(17.5)
2
9
16
P
RE
h
,
h
h
/
2
(17.6)
3
c
2
r
2
1 16
9
PE
R
r
p
3
(17.7)
m
2
π
R
is the indenter radius and
p
m
is the mean contact pressure. Sometimes,
p
m
is denoted as hardness,
but the term mean contact pressure is more general, suitable for elastic as well as plastic defor-
mations, while hardness is more often used to characterize the material resistance to permanent
deformations.
The values of enamel hardness and elastic modulus, measured by nanoindentation
[2,3,6-8]
, vary
between 3-4 GPa for
H
and 70-120 GPa for
E
; a role is played by the age, position on the tooth, and
other factors. Hardness obtained under high loads is often significantly lower, also the modulus and
hardness of dentin or other parts of a tooth.
17.2.2
Harmonic Contact Stiffness
Nanoindentation tests mostly use monotonic loading and unloading (
Figure 17.1
), and the contact
stiffness
S
(
d
P
/d
h
) is determined from the regression function fitted to the upper part of the unload-
ing curve
[4]
. Some devices also enable continuous stiffness measurement (CSM or DMA mode). In
this mode, a small harmonic signal (amplitude several nm or a fraction of a millinewton) is added
[4]
to the monotonously increasing basic load, and the harmonic contact stiffness
S
f
is defined
[9]
as the
ratio of the load and displacement amplitudes of these oscillations,
f
∆ /
the subscript f denotes the exciting frequency.
S
f
is measured continuously from zero to the nomi-
nal load (
Figure 17.2
). This significantly simplifies the measurement in cases, where the properties
and load response change with indenter depth, for example in enamel. In these measurements, the
“monotonic” contact stiffness d
P
/d
h
must be obtained from unloading curve in addition to
S
f
, as it
is necessary for the determination of contact depth
[9]
. The CSM mode is also suitable in the study
of materials with time-dependent response (e.g., polymers)
[10]
. For this purpose,
dynamic elastic
modulus E
r,f
can be determined from Eq. (17.4) with
S
replaced by
S
f
. Another useful quantity is the
shift between the load and displacement amplitudes (
phase angle ψ
), which informs about the inter-
nal friction in the material.
S
F h
(17.8)
