Chemistry Reference
In-Depth Information
the Maxwell element can also be solved (
Eq. 4-59
) for creep deformation.
However, the resultant equations do not describe creep deformation well.
Therefore, they are seldom used in practice. On the other hand, the Voigt element
cannot be solved (
Eq. 4-58
) in a meaningful way for stress relaxation (an instanta-
neous strain is applied at
t
0) because the dashpot cannot be deformed instan-
taneously. In a stress relaxation experiment, a strain
5
γ
0
is imposed at
t
5
0 and
held constant thereafter (
d
is monitored as a function of
t
. Under
these conditions,
Eq. (4-59)
for a Maxwell body behavior becomes
γ
/dt
5
0) while
τ
d
dt
1
τ
1
G
0
5
(4-60)
η
This is a first-order homogeneous differential equation and its solution is
τ 5τ
0
exp
ð2
=ηÞ
Gt
(4-61)
where
τ
0
is the initial value of stress at
γ5γ
0
.
Another way of writing
Eq. (4-59)
is
dy
dt
5
d
dt
1
τ
1
G
(4-62)
ζ
G
where
ζ
(zeta) is a relaxation time defined as
ζ η=
G
(4-63)
An alternative form of
Eq. (4-61)
is then
τ 5 τ
0
exp
ð2
t
=ζÞ
(4-64)
The relaxation time is the time needed for the initial stress to decay to 1
/e
of
its initial value.
If a constant stress
τ
0
were applied to a Maxwell element, the strain would be
γ 5τ
0
=
G
1τ
0
t
=η
(4-65)
This equation is derived by integrating
Eq. (4-59)
with boundary condition
γ 5
0. Although the model has some elastic character the viscous
response dominates at all but short times. For this reason, the element is known
as a
Maxwell fluid
.
A simple creep experiment involves application of a stress
0,
τ 5τ
0
at
t
5
0
and measurement of the strain while the stress is held constant. The Voigt model
(
Eq. 4-58
) is then
τ
0
at time
t
5
τ
0
5
G
γ
1η
dy
=
dt
(4-66)
or
τ
0
η
5
G
γ
η
1
d
dt
5
γ
d
dt
ζ
1
(4-67)
