Chemistry Reference
In-Depth Information
In general, for an experiment
in which stresses
σ
0
,
σ
1
,
σ
2
,
...
,
σ
n
were
applied at times
t
5
0,
u
1
,
u
2
,
...
,
u
n
,
X
n
i5
1
σ
n
D
εð
t
Þ 5σ
0
D
ð
t
Þ 1
ð
t
2
u
n
Þ
(4-55)
If the loaded specimen is allowed to elongate for some time and the stress is
then removed, creep recovery will be observed. An uncross-linked amorphous
polymer approximates a highly viscous fluid in such a mechanical test. Hence the
elongation-time curve of Fig. 1-3c is fitted by an equation of the form
εð
t
Þ 5σ
0
½
D
ð
t
Þ 2
D
ð
t
2
u
1
Þ
(4-56)
Here a stress
σ
0
is applied at
t
5
0 and removed at
t
5
u
1
. (This is equivalent
to the application of an additional stress equal to
2σ
0
.)
EXAMPLE 4-2
A particular grade of polypropylene has the following tensile creep compliance when mea-
sured at 35
C: D(t) 51.2 t
GPa
21
, where t is in seconds. The polymer is subjected to the
following time sequence of tensile stresses at 35
C.
σ5
0.1
0
t ,
0
1 MPa (10
23
GPa)
σ5
0
# t ,
2000 s
10
23
GPa)
σ5
1.5 MPa (1.5
3
1000
# t ,
2000 s
2000
Find the tensile strain at 1500 s and 2500 s using the Boltzmann superposition
principle.
At t 51500 s, the total tensile strain ε(1500) 5 ε0(1500) 1 ε1(1500) (
Eq. 4-54
).
Here,
σ5
0
t $
ε
0
(1500) 510
23
31.23 (1500)
0.1
ε
1
(1500) 51.5310
23
and
31.23 (1500
1000)
0.1
.
Therefore, ε(1500) 55.84310
23
.
At
t 52500 s,
the total
tensile strain
ε(2500) 5 ε
0
(2500) 1 ε
1
(2500) 2ε
2
(2500)
(
Eqs. 4-55 and 4-56
).
Here,
ε
0
(2500) 510
23
31.23 (2500)
0.1
,
ε
1
(2500) 510
23
31.23 (2500
1000)
0.1
and ε
2
(2500) 5 (111.5) 310
23
31.23 (25002000)
0.1
.
Therefore, ε(2500) 50.78310
23
.
4.7.2.2
Use of Mechanical Models
Equation (4-36)
summarized purely elastic response in tension. The analogous
expression for shear deformation (
Fig. 4.14
)is
τ 5
G
γ
(4-57)
This equation can be combined conceptually with the viscous behavior of
Eq. (4-39)
in either of two ways. If the stresses causing elastic extension and vis-
cous flow are considered to be additive, then
τ 5 τ
elastic
1τ
viscous
5
G
γ 1η
d
γ=
dt
(4-58)
