# 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 (${\displaystyle \pi /2}$ 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: ${\displaystyle \varepsilon =\varepsilon _{0}\sin(\omega t)}$
• Stress: ${\displaystyle \sigma =\sigma _{0}\sin(\omega t+\delta )\,}$ [3]

where

${\displaystyle \omega =2\pi f}$ where ${\displaystyle f}$ is frequency of strain oscillation,
${\displaystyle t}$ is time,
${\displaystyle \delta }$ 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: ${\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }$

• Loss: ${\displaystyle E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta }$ [3]

Similarly we also define shear storage and shear loss moduli, ${\displaystyle G'}$ and ${\displaystyle G''}$.

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

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

where ${\displaystyle i}$ is the imaginary unit.