# The dynamic mechanical behavior of an isotropic polymer matrix material may be characterized by two independent complex

## Question:

The dynamic mechanical behavior of an isotropic polymer matrix material may be characterized by two independent complex moduli such as the complex extensional modulus, \(E^{*}(\omega)\) and the complex shear modulus, \(G^{*}(\omega)\). Based on experimental evidence, however, the imaginary parts of \(E^{*}(\omega)\) and \(G^{*}(\omega)\) are not independent, because the material can be assumed to be viscoelastic in shear but elastic in dilatation (i.e., the shear modulus, \(G^{*}(\omega)\) is complex and frequency dependent, but the bulk modulus, \(k\) is real and frequency independent). Use this simplifying assumption to develop an expression for the shear loss factor, \(\eta_{\mathrm{G}}(\omega)\) in terms of the extensional loss factor, \(\eta_{\mathrm{E}}(\omega)\), the extensional storage modulus, \(E^{\prime}(\omega)\), and the bulk modulus, \(k\). Assume all loss factors \(\ll 1\).

## Step by Step Answer:

**Related Book For**

## Principles Of Composite Material Mechanics

**ISBN:** 9781498720694

4th Edition

**Authors:** Ronald F. Gibson