Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev

Civil Engineering (CE) : Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev

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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
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev

Consider the generalised stress-strain relationship as follows:
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
For harmonic excitation at steady state applying
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
We get
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
Therefore
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
This ratio could be denoted as complex Young's modulus E* such that
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
where E' is the storage modulus and E" is the loss modulus. The loss factor n is expressed as
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev

Hence,Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
Similarly, the shear modulus of VEM
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
and the bulk modulus of VEM
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
The various modululli are interrelated as
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev

Variation of the storage modulus and the loss factor of VEM with frequency and temperature are shown in Figs. 14.1a to 14.1d.
Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
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 ωcr
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.

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:

Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev
Figure 14.2: 3 Element model

The stress-strain relation for this model is given by the following equation

Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev                  (14.1)

where  Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev is a geometric parameter.

Assuming a harmonic loading of frequency w, we substitute (jw) for the operator  Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev in this equation. Then, we get the complex modulus as

Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev                                             (14.2)

Taking the real and imaginary parts of this equation, we obtain

Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev                                         (14.3)

It can be seen from eqns. (14.3) that the loss modulus Eiw has a maxima at  Frequency and Temperature Dependence of Complex Elastic Modulus Civil Engineering (CE) Notes | EduRev where λ1 is the relaxation parameter of the viscous branch.

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