# 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.

The stress relaxation modulus ${\displaystyle G\left(t\right)}$  is the ratio of the stress remaining at time ${\displaystyle t}$  after a step strain ${\displaystyle \varepsilon }$  was applied at time ${\displaystyle t=0}$ : ${\displaystyle G\left(t\right)={\frac {\sigma \left(t\right)}{\varepsilon }}}$ ,

which is the time-dependent generalization of Hooke's law. For visco-elastic solids, ${\displaystyle G\left(t\right)}$  converges to the equilibrium shear modulus[4]${\displaystyle G}$ :

${\displaystyle G=\lim _{t\to \infty }G(t)}$ .

The fourier transform of the shear relaxation modulus ${\displaystyle G(t)}$  is ${\displaystyle {\hat {G}}(\omega )={\hat {G}}'(\omega )+i{\hat {G}}''(\omega )}$  (see below).

### 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.

### Ratio between loss and storage modulus

The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the ${\displaystyle \tan \delta }$ , (cf. loss tangent), which provides a measure of damping in the material. ${\displaystyle \tan \delta }$  can also be visualized as the tangent of the phase angle (${\displaystyle \delta }$ ) between the storage and loss modulus.

Tensile: ${\displaystyle \tan \delta ={\frac {E''}{E'}}}$

Shear: ${\displaystyle \tan \delta ={\frac {G''}{G'}}}$

For a material with a ${\displaystyle \tan \delta }$  greater than 1, the energy-dissipating, viscous component of the complex modulus prevails.