# Dynamic modulus

Dynamic modulus (sometimes complex modulus[1]) is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free or forced vibration tests, in shear, compression, or elongation). It is a property of viscoelastic materials.

## Viscoelastic stress–strain phase-lag

Viscoelasticity is studied using dynamic mechanical analysis where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.[2]

• In purely elastic materials the stress and strain occur in phase, so that the response of one occurs simultaneously with the other.
• In purely viscous materials, there is a phase difference between stress and strain, where strain lags stress by a 90 degree ($\pi/2$ radian) phase lag.
• Viscoelastic materials exhibit behavior somewhere in between that of purely viscous and purely elastic materials, exhibiting some phase lag in strain.[3]

Stress and strain in a viscoelastic material can be represented using the following expressions:

• Strain: $\varepsilon = \varepsilon_0 \sin(\omega t)$
• Stress: $\sigma = \sigma_0 \sin(\omega t+ \delta) \,$ [3]

where

$\omega =2 \pi f$ where $f$ is frequency of strain oscillation,
$t$ is time,
$\delta$ is phase lag between stress and strain.

### Storage and loss modulus

The storage and loss modulus in viscoelastic materials measure the stored energy, representing the elastic portion, and the energy dissipated as heat, representing the viscous portion.[3] The tensile storage and loss moduli are defined as follows:

• Storage: $E' = \frac {\sigma_0} {\varepsilon_0} \cos \delta$

• Loss: $E'' = \frac {\sigma_0} {\varepsilon_0} \sin \delta$ [3]

Similarly we also define shear storage and shear loss moduli, $G'$ and $G''$.

Complex variables can be used to express the moduli $E^*$ and $G^*$ as follows:

$E^* = E' + iE'' \,$
$G^* = G' + iG'' \,$ [3]

where $i$ is the imaginary unit.