Dynamic modulus (sometimescomplex modulus^[1]) is the ratio of stress to strain undervibratory conditions (calculated from data obtained from either free or forced vibration tests, in shear, compression, or elongation). It is a property ofviscoelastic materials.

Viscoelasticity is studied usingdynamic mechanical analysis where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.^[2]

• In purelyelastic materials the stress and strain occur inphase, so that the response of one occurs simultaneously with the other.
• In purelyviscous materials, there is aphase 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:${\displaystyle \epsilon ={\epsilon }_0\sin (\omega t)}$
• 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${\displaystyle \epsilon }$ was applied at time${\displaystyle t=0}$:${\displaystyle G\left(t\right)=\frac{\sigma \left(t\right)}{\epsilon }}$,

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

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

Thefourier transform of the shear relaxation modulus${\displaystyle G(t)}$ is${\displaystyle \hat{G}(\omega )={\hat{G}}^{\prime }(\omega )+i{\hat{G}}^{″}(\omega )}$ (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:${\displaystyle E^{\prime }=\frac{{\sigma }_0}{{\epsilon }_0}\cos \delta }$
• Loss:${\displaystyle E^{″}=\frac{{\sigma }_0}{{\epsilon }_0}\sin \delta }$^[3]

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

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

${\displaystyle E^{\ast }=E^{\prime }+i{E}^{″}}$

${\displaystyle G^{\ast }=G^{\prime }+i{G}^{″}}$^[3]

where${\displaystyle i}$ is theimaginary 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:${\displaystyle \tan \delta =\frac{E^{″}}{E^{\prime }}}$

Shear:${\displaystyle \tan \delta =\frac{G^{″}}{G^{\prime }}}$

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

• Physical quantities
• Solid mechanics
• Non-Newtonian fluids
• Viscoelasticity

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