Biomedical Engineering Reference
In-Depth Information
Prior to indentation on the material of interest, the same cantilever tip has to indent on two refer-
ence samples each with a known elastic modulus, in order to solve for the cantilever sensitivity,
A
, and
the cantilever-tip constant,
α
[39]
. If the sample's deformation is viscoelastic instead of purely elastic,
Eq. (16.14) would not be applicable, but it was shown that in this case,
E
r
can be measured by a load
schedule involving a rate jump, such as the connection between loading and unloading in
Figure 16.6
[40]
.
E
r
would still be given by an equation of the same form as Eq. (16.14), albeit that
K
is now the
ratio between the change in the sample's velocity across the jump, and the change in the rate of the
photodiode signal across the jump. It is noteworthy that a critical requirement for AFM indentation is
that the deformation at the tip-sample contact is significant compared with the deflection of the AFM
cantilever. The ratio
α/E
r
in Eq. (16.14) in fact measures the tip-sample deformation compared to the
cantilever's deflection, and this provides a useful guideline for the choice of the cantilever-tip prop-
erty,
α
[39]
. In addition, a further requirement is that
E
sample
E
tip
so that an appreciable amount of
the deformation at the tip-sample contact is due to the material of interest, rather than due to the tip.
For commercially available silicon tips, their elastic modulus is ~169 GPa
[41]
which may not satisfy
the above requirement if they are to indent enamel which may have
E
as high as 115 GPa
[42]
. In this
case, other available options include tips made of Si
3
N
4
which has an elastic modulus
200 GPa
[43]
.
Enamel, dentin, cementum, and bone are anisotropic materials. The loading direction with respect
to the microstructures in the tissues, therefore, has an effect on the measured properties. For most
cases, the orientation of the microstructure with respect to the indentation surface is already deter-
mined during the initial sectioning process. Full understanding of the deformation behavior of the
oral tissues requires independent characterization of the elastic properties along the different axes.
However, as the stress field under a Berkovich tip is multiaxial, the deformation response elicited
would represent some kind of unknown average over different directions. A two-step technique could
be used for characterizing the mechanical properties of anisotropic materials using nanoindenters.
The elastic constants,
c
ijkl
, of the biological tissue in question are first characterized by ultrasonic
means. An effective indentation modulus can then be calculated from the nanoindentation-derived
data using the
c
ijkl
[44]
. A relatively easier, and more direct, method for investigating anisotropic
effects in mineralized tissues involves making and testing specimen volumes of small (e.g., micron)
dimensions and of known orientations. One method for producing such samples is focused ion beam
(FIB) milling. Using this technique, micron-sized cantilevers which can be orientated in any desired
direction can be produced as shown schematically in
Figure 16.7
[23]
.
The free ends of these micro-cantilevers are loaded by the tip of a nanoindenter until the cantile-
vers fail. From the load-displacement data,
E
can be calculated as
3
SL
I
E
(16.15)
3
where
L
is the length between the loading point and the fixed end of the cantilever,
I
wd
3
/36 is the
second moment of area of the cantilever's triangular cross section,
w
is the width of the cantilever,
d
is the height of the cantilever's triangular cross section, and
S
is the slope of load-deflection curve.
This technique also enables the localized flexural strength,
σ
cant
, to be evaluated as
σ
cant
PLd
I
(16.16)
3
where
P
is the load at fracture.
