Dynamic modulus

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Dynamic modulus 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[edit]

Viscoelasticity is studied using dynamic mechanical analysis where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.[1]

  • 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.[2]

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

  • Strain:  \varepsilon = \varepsilon_0 \sin(t\omega)
  • Stress:  \sigma = \sigma_0 \sin(t\omega + \delta) \, [2]


 \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[edit]

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.[2] The tensile storage and loss moduli are defined as follows:

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

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'' \, [2]

where i is the imaginary unit.

See also[edit]


  1. ^ PerkinElmer "Mechanical Properties of Films and Coatings"
  2. ^ a b c d e Meyers and Chawla (1999): "Mechanical Behavior of Materials," 98-103.