Biomedical Engineering Reference
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equation subject to the case when the strain history is given by
E
12
¼
E
o
h
(
t
) and no
other loading is applied to the model. Determine the stress response to this strain
history loading.
Solution
: The general solution to the first-order ordinary differential equation
d
y
d
t
þ
f
ð
t
Þ
y
g
ð
t
Þ
is (see Sect. A.17)
Z
t
Z
t
e
F
e
F
d
s
y
ð
t
Þ¼
c
þ
g
ð
s
Þ
;
where
F
ð
t
Þ¼
f
ð
s
Þ
d
s
;
t
o
t
o
and where
c
is a constant of integration. In the differential equation of interest
y
(
t
)
¼
T
12
(
t
),
F
(
t
)
¼
(
t
/
t
r
),
g
(
s
)
¼
(
d
/d
t
)
E
12
¼
(
d
/d
t
)(
E
o
h
(
s
)) and, due to the initial
condition
c
¼
0, thus
e
t=
t
r
Z
t
0
e
t=
t
r
Z
t
0
G
o
E
o
d
h
ð
s
Þ
e
s=
t
r
d
s
e
s=
t
r
d
s
T
12
ð
y
Þ¼
¼
G
o
E
o
dð
s
Þ
d
s
e
t=
t
r
¼
G
o
E
o
h
ð
t
Þ
:
The special form of the relaxation function for the Maxwell model may then be
identified as
e
t=
t
G
dev
ð
t
Þ¼ð
1
=
E
o
Þ
T
12
¼
G
o
h
ð
t
Þ
:
r
Problems
6.5.1 Derive (
6.50
) and (
6.51
) from (5.36V).
6.5.2 Derive (
6.52
) and (
6.53
) from (
6.41
).
6.5.3 Determine the strain response to this stress loading for an isotropic viscoelas-
tic material subjected to a step loading in shear stress
T
12
. The magnitude of
the step loading is
T
o
. This problem is formally similar to the Example 6.5.1,
the only change being that the step loading is now in shear stress rather than
shear strain. The step loading is represented by
T
12
¼
T
o
h
(
t
). Show that the
creep function is just the value of the resulting strain divided by
T
o
. Note that
typical creep functions are of the form of increasing functions of time like
that shown in Fig.
6.11
.
6.5.4 The Voigt model is a lumped viscoelastic model that is a combination of a
spring and a dashpot in parallel (Fig. 1.10b). When a force applied to a Voigt
model is changed from zero to a finite value at an instant of time and then held
constant thereafter, extension occurs only after the dashpot begins to move.
Thus the Voigt model is initially rigid; then it begins to creep asymptotically
under the constant applied load to a rest value. In parallel models like the
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