Biomedical Engineering Reference
In-Depth Information
Cono-spherical tip
R
a
h
max
Sample
h
c
FIGURE 16.4
Schematic representation of the tip-
specimen contact at the maximum load
P
max
for a cono-spherical tip (
h
max
maximum
indentation depth and
h
c
contact depth).
relationship between the indentation loading rate,
P
, and strain rate,
ε
, can be derived by differentiat-
ing the mean contact pressure during indentation, i.e., Eq. (16.2), with respect to time to obtain
[33]
h
h
1
2
P
P
H
H
−
ε
(16.9)
Provided that the value of
H
is invariant with time,
ε
produced by the conventional triangular or
trapezoidal load function will decrease with an increase in
P
, since
P
is a constant. Therefore, in
order to obtain a constant
ε
,
P
/ has to be made a constant, say,
k
. In cases where there is no preload-
ing, integrating with respect to time yields
exp( )
Therefore,
P
has to be applied in the form of an exponential time function in order to obtain a
constant
ε
.
Bone, dentin, cementum, and enamel exhibit different degrees of viscoelasticity depending on
their relative amount of organic material. The most obvious effect of viscoelasticity on the nanoinden-
tation data is the presence of a “nose” in the unloading curve due to the superimposition of the down-
ward viscoelastic creep onto the upward elastic recovery upon unloading (
Figure 16.5A
).
The value of
S
would appear larger than if the material response is purely elastic and, in extreme
cases,
S
could become negative. One of the practical ways devised to overcome this problem is to
have a sufficient hold at the maximum load before unloading, coupled with a fast unloading rate
(
Figure 16.5B-D
). The viscoelastic effect in the load-displacement data could be further reduced by
modifying the contact stiffness term in the Oliver and Pharr method with the relation
[34]
P
kt
(16.10)
h
P
1
1
h
u
(16.11)
S
S
e
c
The three parameters required for eon (11) are shown schematically in
Figure 16.5B-D
.
For viscoelastic materials, the apparent stiffness is the result of both their elastic and viscous com-
ponents, i.e.,
P
h
P
u
u
S
(16.12)
c
h
h
