Lecture 14 - Frequency and Temperature Dependence of Complex Elastic Modulus Notes | EduRev

: Lecture 14 - Frequency and Temperature Dependence of Complex Elastic Modulus Notes | EduRev

 Page 1


Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
The Lecture Contains:
Complex Elastic Modulus of VEM
Frequency and Temperature Dependence of VEM
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
The Lecture Contains:
Complex Elastic Modulus of VEM
Frequency and Temperature Dependence of VEM
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1a.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
There are about 23 VEMs frequently used for vibration damping. A few of these materials and their loss
factors are listed below
Table 14.1: Important VEM for damping
     Name of VEM   Approx,  Loss Factor
PVA 2.5
Thiokol M-5 1.22
Butyl Rubber 1.5
3M tap 466 1.17
M 169A Butyl Gum 1.04
Polyisobutylene 0.8
Dupont Viton A 0.8
Silicone 0.6
Neoperene 0.5
PU 0.3
Filled Silicon Rubber 0.13
Natural Rubber 0.1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
The Lecture Contains:
Complex Elastic Modulus of VEM
Frequency and Temperature Dependence of VEM
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1a.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
There are about 23 VEMs frequently used for vibration damping. A few of these materials and their loss
factors are listed below
Table 14.1: Important VEM for damping
     Name of VEM   Approx,  Loss Factor
PVA 2.5
Thiokol M-5 1.22
Butyl Rubber 1.5
3M tap 466 1.17
M 169A Butyl Gum 1.04
Polyisobutylene 0.8
Dupont Viton A 0.8
Silicone 0.6
Neoperene 0.5
PU 0.3
Filled Silicon Rubber 0.13
Natural Rubber 0.1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_2.htm[6/25/2012 12:33:37 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
Complex Modulus of Viscoelastic Materials
Consider the generalised stress-strain relationship as follows:
 
For harmonic excitation at steady state applying
    and     
 
We get
 
 
Therefore
 
 
This ratio could be denoted as complex Young's modulus  such that
 
 
where  is the storage modulus and  is the loss modulus. The loss factor  is expressed as
 
 
Hence, 
Similarly, the shear modulus of VEM
 
 
and the bulk modulus of VEM
 
 
The various modululli are interrelated as
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
The Lecture Contains:
Complex Elastic Modulus of VEM
Frequency and Temperature Dependence of VEM
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1a.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
There are about 23 VEMs frequently used for vibration damping. A few of these materials and their loss
factors are listed below
Table 14.1: Important VEM for damping
     Name of VEM   Approx,  Loss Factor
PVA 2.5
Thiokol M-5 1.22
Butyl Rubber 1.5
3M tap 466 1.17
M 169A Butyl Gum 1.04
Polyisobutylene 0.8
Dupont Viton A 0.8
Silicone 0.6
Neoperene 0.5
PU 0.3
Filled Silicon Rubber 0.13
Natural Rubber 0.1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_2.htm[6/25/2012 12:33:37 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
Complex Modulus of Viscoelastic Materials
Consider the generalised stress-strain relationship as follows:
 
For harmonic excitation at steady state applying
    and     
 
We get
 
 
Therefore
 
 
This ratio could be denoted as complex Young's modulus  such that
 
 
where  is the storage modulus and  is the loss modulus. The loss factor  is expressed as
 
 
Hence, 
Similarly, the shear modulus of VEM
 
 
and the bulk modulus of VEM
 
 
The various modululli are interrelated as
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_3.htm[6/25/2012 12:33:37 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
Variation of the storage modulus and the loss factor of VEM with frequency and temperature are shown
in Figs. 14.1a to 14.1d.
Figure 14.1: Variation in storage modulus and the loss factor with frequency
The figures depict the following facts:
a. The shear modulus is low at low frequency and increase sharply beyond a critical frequency ?
cr
b. The loss modulus also reaches its maxima at the same frequency
c. A reverse change is observed in the shear modulus with respect to temperature
d. Here also, the loss modulus increases sharply at a critical temperature
The critical frequency and temperature actually depict a phase change in the polymer. For frequency,
the change occurs from rubbery to glassy phase and the reverse for temperature.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 5


Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
The Lecture Contains:
Complex Elastic Modulus of VEM
Frequency and Temperature Dependence of VEM
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_1a.htm[6/25/2012 12:33:36 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
There are about 23 VEMs frequently used for vibration damping. A few of these materials and their loss
factors are listed below
Table 14.1: Important VEM for damping
     Name of VEM   Approx,  Loss Factor
PVA 2.5
Thiokol M-5 1.22
Butyl Rubber 1.5
3M tap 466 1.17
M 169A Butyl Gum 1.04
Polyisobutylene 0.8
Dupont Viton A 0.8
Silicone 0.6
Neoperene 0.5
PU 0.3
Filled Silicon Rubber 0.13
Natural Rubber 0.1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_2.htm[6/25/2012 12:33:37 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
Complex Modulus of Viscoelastic Materials
Consider the generalised stress-strain relationship as follows:
 
For harmonic excitation at steady state applying
    and     
 
We get
 
 
Therefore
 
 
This ratio could be denoted as complex Young's modulus  such that
 
 
where  is the storage modulus and  is the loss modulus. The loss factor  is expressed as
 
 
Hence, 
Similarly, the shear modulus of VEM
 
 
and the bulk modulus of VEM
 
 
The various modululli are interrelated as
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_3.htm[6/25/2012 12:33:37 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
Variation of the storage modulus and the loss factor of VEM with frequency and temperature are shown
in Figs. 14.1a to 14.1d.
Figure 14.1: Variation in storage modulus and the loss factor with frequency
The figures depict the following facts:
a. The shear modulus is low at low frequency and increase sharply beyond a critical frequency ?
cr
b. The loss modulus also reaches its maxima at the same frequency
c. A reverse change is observed in the shear modulus with respect to temperature
d. Here also, the loss modulus increases sharply at a critical temperature
The critical frequency and temperature actually depict a phase change in the polymer. For frequency,
the change occurs from rubbery to glassy phase and the reverse for temperature.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///D|/chitra/vibration_upload/lecture14/14_4.htm[6/25/2012 12:33:37 PM]
 Module 3: Dynamic Properties and Selection of Materials
 Lecture 14: Frequency and Temperature Dependence of Complex Elastic Modulus
 
The frequency-dependence of the complex modulus we have just discussed can be explained through a
linear viscoelastic model. For example, consider the simple, three-element model shown in the figure
below:
Figure 14.2: 3 Element model
The stress-strain relation for this model is given by the following equation
(14.1)
where  is a geometric parameter.
Assuming a harmonic loading of frequency , we substitute ( ) for the operator  in this
equation. Then, we get the complex modulus as
(14.2)
Taking the real and imaginary parts of this equation, we obtain
(14.3)
It can be seen from eqns. (14.3) that the loss modulus  has a maxima at ,
where  is the relaxation parameter of the viscous branch.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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