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).

See also

• Dynamic mechanical analysis
• Elastic modulus
• Palierne equation

References