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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**

Consider the generalised stress-strain relationship as follows:

For harmonic excitation at steady state applying

We get

Therefore

This ratio could be denoted as complex Young's modulus E^{*} such that

where E' is the storage modulus and E" is the loss modulus. The loss factor n is expressed as

Hence,

Similarly, the shear modulus of VEM

and the bulk modulus of VEM

The various modululli are interrelated as

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 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:

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 w, we substitute (jw) 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 E_{iw} has a maxima at where λ_{1} is the relaxation parameter of the viscous branch.

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