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.

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).

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.

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.

• Dynamic mechanical analysis
• Elastic modulus
• Palierne equation