Chemistry Reference
In-Depth Information
where
G
is a constant,
γ
0
is the initial strain imposed on the polymer,
τ
0
is
τ
0
5γ
0
2
, and
the residue stress that is related to
γ
0
in the form of
λ
is the
relaxation time.
(a) What is the relaxation modulus of the polymer?
(b) What are the initial and final stresses?
(c) Given that
G5
0.15, calculate the percentage of the
original stress that has decayed when
t 5λ
0.8 MPa and
γ
0
5
.
(d) It has been found that the above model is a better model than the
Maxwell model to describe the stress of the polymer, especially at
long times. Why?
γ
and complex shear stress
τ
are given by the
4-18 The complex shear strain
following expressions:
5γ
0
e
iωt
5τ
0
e
iðωt1δÞ
γ
;
τ
Based upon the above expressions, show that the real and imaginary parts
of the complex shear compliance
J
are given by the following equations:
J
0
5
γ
0
v 5
γ
0
τ
0
cos
δ;
J
τ
0
sin
δ
τ
are 10%
and 10
5
Pa and the phase angle of the material is 45
, calculate the amount
of energy that is dissipated per full cycle of deformation (J/m
3
).
Also show that
J
00
/J
0
5
tan
δ
.
If the magnitudes of
γ
and
4-19 Consider a sinusoidal shear strain with angular frequency
ω
and strain
amplitude
).
(a) What is the corresponding time dependent shear stress for a perfectly
elastic material that has a shear modulus of
G
and is subjected to the
above sinusoidal shear strain?
(b) Show that the shear stress of a Newtonian liquid with a viscosity of
γ
0
(i.e.,
γðtÞ 5γ
0
sin
ðωtÞ
η
still oscillates with the same angular frequency but is out-of-phase
with the sinusoidal shear strain by
/2.
(c) What is the time-dependent shear stress of a viscoelastic material with
a stress amplitude
π
σ
0
and in which stress leads the strain by a phase
?
(d) What is the corresponding expression of the above described sinusoi-
dal shear strain written in the complex number form?
(e) If a sinusoidal shear strain is in the form of
angle
δ
5γ
0
e
iωt
, determine
the magnitudes of the shear strains and the corresponding shear
stresses for a viscoelastic material when
γ
(
t
)
ω
t
5
0,
π
/2,
π
,3
π
/2, and 2
π
.
Note that the time-dependent shear stress has a stress amplitude
σ
0
.
(f) The following expression shows the complex compliance of a visco-
elastic material which is subjected to a sinusoidal shear strain in the
and that its stress leads the strain by a phase angle
δ
