Biomedical Engineering Reference
In-Depth Information
Eq. (17.18), the time-dependent processes end at about 5
τ
1
. For long processes, creep compliance
function usually contains more exponential terms,
C
j
[1
exp(
t
/
τ
j
)].
K
and
m
in Eq. (17.17) are
constants, depending on the geometry of loading. For pointed indenters,
m
2 and
K
π
/(2 tan
α
);
α
is the semiangle of the equivalent cone (70.3° for Berkovich and Vickers). For a spherical indenter,
m
3/2 and
K
3/(4
R
), where
R
is the tip radius. Pointed indenters nearly always cause perma-
nent deformations. As the deformations of teeth in biting are mostly reversible, the corresponding vis-
coelastic properties are better studied using spherical indenter and low contact pressure (sufficiently
lower than the hardness obtained by a pointed indenter).
Indentation tests for obtaining viscoelastic parameters usually proceed as follows. The indenter
is loaded quickly to the nominal load
P
, which is then held constant. After the time
t
, the indenter is
unloaded—to zero in the simplest case, or to a very low load
P
low
, which is held constant some time and
unloaded to zero. The material constants are determined from the creep during the dwell under nominal
load, while the back-creep during the low-load dwell can serve for a check of the number of elements in
the model. A universal function for fitting the indenter penetration under constant load is
[34,35]
m
h
( )
t
PK B
[
c t
Σ
D
exp
(
t
/
τ
)]
(17.19)
0
v
j
j
This function characterizes the instantaneous elastic as well as plastic deformations (the term
B
0
),
irreversible viscous flow (
c
v
t
) and delayed reversible deforming,
D
j
exp(
t
/
τ
j
); the number of
D
j
terms depends on the duration of creep processes. If no irreversible viscous flow occurs, the term
c
v
t
is omitted. The determination of parameters proceeds in three steps
[34,35]
:
Step 1.
Calculation of regression constants
B
0
,
c
v
,
D
j
and retardation times
τ
j
in the proposed model
by fitting the
h
m
(
t
) data from the creep period (for
t
t
R
) by Eq. (17.19). A curve fitter or a solver
combined with the least-squares method is necessary.
Step 2.
Calculation of the constants
C
j
from
D
j
as
C
j
D
j
/ρ
j
; ρ
j
are the ramp correction factors
[33]
,
considering the fact that the load increase to
P
was not instantaneous, but lasted
t
R
:
ρ
(
τ
/
t
)[exp(
t
/
τ
)
1
]
(17.20)
j
j R
R
j
Step 3.
Determination of the constant
C
0
as:
C
B
c t
/
Σ
C
(17.21)
0
0
v R
j
If no permanent deformations occurred,
C
0
is related to the reduced modulus as
C
0
1/
E
r
.
Figure 17.6
shows penetration of a Berkovich indenter into enamel. Equation (17.19) with
j
=
1,2,3,
fitted the experimental data very well
[34]
. After unloading, the deformations diminished,
but small permanent imprint remained
[34,35]
.
Besides these measurements, common load-unload tests can also be done in order to obtain the
“elastic” modulus and hardness, with the corrections from Section 16.3.4, if necessary. The time-
dependent component of deformation is important in some cases, while sometimes it may be neglected.
Simple, but very illustrative information about the significance of viscoelastic processes is the ratio of
the time-dependent component of penetration to the total penetration. Useful also is the ratio of the
residual deformation (after unloading) to the total deformation under load. For more see Ref.
[36]
.
