Biomedical Engineering Reference
In-Depth Information
c
is a constant obtained by fitting the measured
X
(
ξ
) values. Very good results were received
[21]
with
a more complex weight function Φ(
ξ
) proposed by Gao et al.
[24]
. An example of the determination
of elastic modulus of a metallic coating on a glass substrate is shown in
Figure 17.5
.
When determining hardness, two cases can be distinguished: soft coating on a hard substrate and a
hard coating on a more compliant substrate. While the plastic deformations are limited to the coating
in the former case, a hard coating on a soft substrate deforms elastically and plunges into plastically
deformed substrate, and, eventually, breaks. The simplest empirical approximations are Eq. (17.15)
for hard films on softer substrates, and
2
Φ( )
ξ
exp
[
(
c
ξ
) ]
(17.16)
for soft films on hard substrates;
c
is a constant,
ξ
h/t
;
X
c
,
X
s
correspond to
H
c
and
H
s
. More
approximations can be found in Refs
[21,22,25-28]
.
17.4.2
Multiphase Microstructure
Several kinds of values can be obtained by nanoindentation, depending on the imprint size. If this
is relatively large compared to the size of individual phases, only average properties are measured.
If the device enables very low loads and good positioning of the indenter, it is possible to study sin-
gle components. (One must be aware that even the pointed indenters have a rounded tip, with radius
several tens of nm.) It is also possible to make a series of indents for various forces and see, how the
apparent
X
(
h
) values gradually change due to increasing influence of the surrounding material. In this
way, properties of tooth enamel were studied
[3,6]
. At very low loads, the indenter touched only one
(relatively stiff) enamel rod. Under increasing load, more rods appeared beneath the indenter, but also
more of softer interrod material, and the apparent hardness decreased. Such effects can contribute to
the dependence of measured values on the imprint size (so-called indentation size effect).
Information can also be obtained by statistical analysis of many indentations, arranged in a grid
over some area. Due to random influences, the measured values vary from a test to test. In a homoge-
neous material, their histogram is a bell-like curve (e.g., Gauss, log-normal, or Weibull distribution).
The presence of two “hills” indicates a two-phase structure, e.g., enamel rods and interrod material.
Using statistical tools, one can decompose the original group of values into two distributions
[29]
and determine the typical values of each. The situation is easier, if the volume fractions of individual
components are known. A role is also played by their mutual arrangement. With more phases, the
decomposition becomes more difficult or impossible.
17.5
CHARACTERIZATION OF TIME-DEPENDENT LOAD RESPONSE
Dentin and other biological tissues, as well as enamel or resins used for dental restorations, belong
to viscoelastic materials. At the instant of loading, they deform elastically, but under load the defor-
mations continue growing to a small extent
(Figure 17.6)
. After unloading they gradually diminish.
At a first approximation these materials can be considered as elastic and characterized by Young's
modulus
E
. In a more detailed analysis or in the nanoindentation determination of properties, their
viscoelastic behavior must be taken into account. In common load-unload tests, the unloading part
of the
P-h
curve is more distorted than with elastic materials (dotted curve in
Figure 17.1
). As a
