Biomedical Engineering Reference
In-Depth Information
c
ð
t
Þ¼
Z
t
g
ðÞ
ds
:
ð
37
Þ
0
The stress may be written:
T
t
c
ð
t
Þ¼
r
P
ð
t
Þ;
De
T
De
k
0\t\T
r
ðÞ¼
ð
38
Þ
De
T
k D
ðÞ
c
ð
t
Þ
c
ð
t
T
Þ
ð
Þ ¼
r
H
ð
t
Þ;
t [ T
where r
P
(t) and r
H
(t) are the predictions of the time variation of stress over the
ramp-loading and hold-relaxation phases, respectively.
Here we use the experimental stress during the hold phase to calibrate the
parameters of the reduced relaxation function, g(t). Parameters of the reduced
relaxation functions are those that minimize the following integral:
2
Z
þ1
I
¼
X
n
r
H
ðÞ
H
ðÞ
r
H
ðÞ
dt
:
ð
39
Þ
H
ð
T
Þ
T
Usually, if g(t) is known parametrically c(t) can be determined parametrically
as well. Once the g(t) function and the associated c(t) function are determined, the
nonlinear function k(e) may be calculated algebraically using the experimentally
recorded stress during the ramp-loading phase:
T
t
¼
T
De
r
P
ð
t
Þ
c
ð
t
Þ
k
:
ð
40
Þ
De
3.3.2 Fung QLV Model
Substituting the strain from Eq. (
35
) into the Fung QLV model given by Eq. (
2
)
yields the following prediction for stress:
<
Þ
dr
ð
e
Þ
D
T
s
de
Z
t
De
T
gt
s
ð
ds
¼
r
P
ð
t
Þ;
0\t\T
0
r
ðÞ¼
ð
41
Þ
Þ
dr
ð
e
Þ
D
T
s
de
:
Z
T
De
T
gt
s
ð
ds
¼
r
H
ð
t
Þ;
t [ T
0
Unlike the Adaptive QLV model, no simple procedure exists for calibration the
Fung QLV model to a ramp-and-hold test, and calibration requires far more
computation. The challenge is that the r
(e)
(e) function, an unknown function,
appears inside the convolution integral in both the ramp stress and the hold stress.
g(t) and r
(e)
(e) may be calibrated by trial and error and by testing different choices
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