Biomedical Engineering Reference
In-Depth Information
Fig. 6.6
A plot of stress
increments applied at specific
time increments
X
n
i¼
1
J
11
ð
E
1
ð
t
i
ÞD T
1
ð
t
Þ¼
t
t
i
Þ:
(6.43)
In the passage to the limit the series of times
t
i
,
i
¼
1,
...
,
n
will be replaced by
by
d
T
1
D T
1
ð
the continuous parameter
s
, and the step in the stress by
t
i
Þ
d
s
d
s
, thus
Z
t
0
J
11
ð
d
T
1
d
s
d
s
E
1
ð
t
Þ¼
t
t
i
Þ
:
(6.44)
Finally, going back to the beginning of this development there was nothing
special about applying the loading at
t
0; it could have been started at any time,
thus we set it back to the beginning of time,
¼
Z
s¼t
d
T
1
d
s
E
1
ð
J
11
ð
t
Þ¼
t
s
Þ
d
s
:
(6.45)
s¼1
It is now clear that (
6.45
) is a special case of (
6.41
). The physical significance of
the creep function
J
11
ð
Þ
is the following: it is the uniaxial normal strain
E
1
ð
Þ
vs.
time response of a specimen subjected to a step increase in the normal stress in the
same direction,
T
1
¼
s
t
.
The physical significance of the relaxation function
G
11
ð
T
o
h
ð
t
Þ
can be developed in a
similar manner by reversing the roles of stress and strain used in the case of the
creep function. A uniaxial tension specimen of homogeneous material is subjected
to a step increase in strain,
E
1
¼
s
Þ
. The strain loading function
E
1
¼
E
o
h
ð
t
Þ
E
o
h
ð
t
Þ
is
G
11
ð
plotted in the top panel of Fig.
6.7
. The relaxation function
is the uniaxial
stress vs. time response of the uniaxially strained specimen to the step increase in
strain,
E
1
¼
s
Þ
. The function
G
11
ð
is illustrated in the lower panel of Fig.
6.7
.
The stress response is given by
T
1
¼ G
11
ð
E
o
h
ð
t
Þ
s
Þ
Þ E
o
. The relaxation function defined and
measured in this way may be used to predict the relaxation response to a more
complicated strain history as described above for the creep function. Following
completely analogous steps one finds that
t
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