Biomedical Engineering Reference
In-Depth Information
1.98
1.96
1.94
1.92
1.90
1.88
1.86
1.84
1.82
0
200
400
600
800
1000
t
(s)
FIGURE 17.6
Penetration of Berkovich indenter into human tooth enamel under
constant load 250 mN (measurements done by Dr. Li Hong He).
h
—depth,
t
—time. The “instantaneous” penetration during the initial
load increase period 20 s was about 1.82
μ
m.
consequence, the apparent contact stiffness
S
is higher than the actual value. This could lead to an
error in the determination of contact depth and area, as well as of the elastic modulus and hardness.
The influence of viscoelastic effects on the unload curve can be reduced in various ways. Often, a
dwell is inserted between the loading and unloading, so long that the indenter velocity becomes neg-
ligible. Also a quick unloading is recommended
[30]
and the use of effective contact stiffness
[31]
,
explained in detail in Section 16.3.4. A disadvantage is that the indenter depth
h
at the beginning
of unloading is larger than at the end of load increase. This results in larger contact area and lower
apparent hardness. Moreover, it is generally insufficient to characterize materials, which gradually
flow under load, only by a single value of hardness or elastic modulus. The time-dependent properties
are better described by universal rheologic (spring-and-dashpot) models, which can also be used in
computer codes for the finite element analysis and modeling.
The parameters in these models can be obtained by fitting the time course of indenter penetration
by a suitable creep function. Generally, the relationship between the load
P
and deformation
h
can be
expressed as
[5,32-35]
,
m
(17.17)
ψ
where
ψ
(
P
,
J
,
t
) is a response function depending on the load, material, and time.
J
is the creep com-
pliance function, which expresses the material response to the step-load of unit magnitude. For the
simplest case of instant elastic deformation accompanied by reversible delayed deforming,
h
( )
t
K P J t
(
,
,
)
J t
( )
C
C
[
1
exp
(
t
/
τ
)]
(17.18)
0
1
1
with the constants
C
0
,
C
1
, and
τ
1
.
C
0
is the elastic compliance (reciprocal of the elastic modulus),
C
1
expresses the extent of delayed deforming, and the retardation time
τ
1
characterizes its duration. In
