Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
3.6. Viscoelasticity
3.6.1. The uniaxial tensile stress-strain behavior of a
viscoelastic material is idealized using the spring-damper systems illustrated
in the figure, as discussed in Section 3.6.3.
3.6.1.1.
Derive the
differential equations relating stress to strain for each system.
3.6.1.2.
Calculate
expressions for the relaxation modulus for the Maxwell material and the 3
parameter model.
3.6.1.3.
Calculate
expressions for the creep compliance of all three materials
3.6.1.4.
Calculate
expressions for the complex modulus for all three materials.
3.6.1.5.
Calculate
expressions for the complex compliance for all three materials.
3.6.2. The shear modulus of a viscoelastic material can be
approximated by a Prony series given by .
3.6.2.1.
Find the creep
shear compliance of the material
3.6.2.2.
Find the complex
shear modulus of the material
3.6.2.3.
Find the complex
shear compliance of the material
3.6.3. A uniaxial tensile specimen is made from a
viscoelastic material with time independent bulk modulus K, and has a shear modulus that can be approximated by the Prony
series . The specimen is subjected to step increase in
uniaxial stress, so that with all other stress components zero. Find an expression for the history of strain
the specimen. It is easiest to solve
this problem by first calculating the creep shear compliance for the material.
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3.6.4. A floor is covered with a pad with thickness h of viscoelastic material, as shown in
the figure. The pad is perfectly bonded
to the floor, so that . The pad can be idealized as a viscoelastic
solid with time independent bulk modulus K,
and has a shear modulus that can be approximated by the Prony series . The surface of the pad is subjected to a
history of displacement .
3.6.4.1.
Calculate the
history of stress induced in the pad by
3.6.4.2.
Calculate the
history of stress induced in the pad by
3.6.4.3.
Assume that the
pad is subjected to a displacement for long enough for the cycles of stress and
strain to settle to steady state. Calculate the total energy dissipated per
unit area of the pad during a cycle of loading.
27 0C
|
35 0C
|
35 0C
|
55 0C
|
65 0C
|
Time
(Sec)
|
Modulus
(GPa)
|
Time
(Sec)
|
Modulus
(GPa)
|
Time
(Sec)
|
Modulus
(GPa)
|
Time
(Sec)
|
Modulus
GPa
|
Time
(Sec)
|
Modulus
GPa
|
9
|
23.95
|
13
|
18.89
|
17
|
8.81
|
15
|
5.67
|
17
|
4.92
|
17
|
23.91
|
23
|
18.46
|
31
|
8.61
|
27
|
5.16
|
30
|
4.86
|
29
|
23.84
|
41
|
18.04
|
55
|
8.34
|
48
|
5.00
|
55
|
4.76
|
52
|
23.72
|
73
|
17.36
|
98
|
7.91
|
85
|
4.97
|
97
|
4.59
|
93
|
23.72
|
131
|
16.27
|
174
|
7.27
|
152
|
4.95
|
174
|
4.32
|
166
|
23.51
|
233
|
14.67
|
310
|
6.46
|
271
|
4.92
|
309
|
3.92
|
295
|
23.16
|
415
|
12.63
|
552
|
5.67
|
482
|
4.86
|
550
|
3.41
|
525
|
22.61
|
738
|
10.61
|
982
|
5.16
|
857
|
4.76
|
979
|
2.92
|
933
|
21.79
|
1312
|
9.29
|
1746
|
5.00
|
1524
|
4.59
|
1742
|
2.60
|
1660
|
20.73
|
2334
|
8.81
|
3106
|
4.97
|
2711
|
4.32
|
3097
|
2.50
|
2952
|
19.66
|
4151
|
8.61
|
5523
|
4.95
|
4822
|
3.92
|
5508
|
2.50
|
5250
|
18.89
|
7381
|
8.34
|
9822
|
4.92
|
8574
|
3.41
|
8000
|
2.50
|
9337
|
18.46
|
13126
|
7.91
|
17467
|
4.86
|
15248
|
2.92
|
17500
|
2.50
|
16600
|
18.04
|
|
|
|
|
|
|
|
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3.6.5.
The table above
lists measured relaxation (shear) modulus for a (fictitious) polymer at various
temperatures. The polymer has a glass
transition temperature of 30 0C.
3.6.5.1.
Plot a graph of
the modulus as a function of time for each temperature, using a log scale for
both axes
3.6.5.2.
Hence, show that
the data for various temperatures can be collapsed onto a single master-curve
by scaling the times in each experiment by a temperature dependent factor ,
as described in Section 3.6.1. Plot the
master-curve corresponding to relaxation at 27 0C.
3.6.5.3.
Plot a graph of as a function of temperature, and show that the
data can be fit by a function of the form . Determine the values of that best fit the data
3.6.5.4.
Hence, determine
the constants for the material, as discussed in Section
3.6.1
3.6.5.5.
Hence, scale the
times in the experimental data to plot the relaxation modulus at the glass
transition temperature .
3.6.5.6.
Find a Prony
series fit to . Use four terms in the series, together with
an appropriate value for
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3.6.6. An instrument with mass m=10kg is mounted a set of rubber pads with combined cross
sectional area A=5 and
height h=3cm as shown in the
figure. The pads are made from
polyisobutylene, with properties listed in Section 3.6.6. The base vibrates harmonically with amplitude
and (angular) frequency ,
causing the instrument to vibrate (also harmonically) with amplitude .
3.6.6.1.
Find an
expression relating to the harmonic modulus of the material and m, A, h and
. Assume that the pads are all subjected to a
uniaxial state of stress.
3.6.6.2.
Find an
expression for the harmonic modulus in terms of the material properties and .
3.6.6.3.
Hence, plot a
graph showing the variation of as a function of frequency, for temperatures