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.

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 (${\displaystyle \pi /2}${\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: ${\displaystyle \varepsilon =\varepsilon _{0}\sin(\omega t)}${\displaystyle \varepsilon =\varepsilon _{0}\sin(\omega t)}
• Stress: ${\displaystyle \sigma =\sigma _{0}\sin(\omega t+\delta ),円}${\displaystyle \sigma =\sigma _{0}\sin(\omega t+\delta ),円} ^[3]

where

${\displaystyle \omega =2\pi f}${\displaystyle \omega =2\pi f} where ${\displaystyle f}${\displaystyle f} is frequency of strain oscillation,

${\displaystyle t}${\displaystyle t} is time,

${\displaystyle \delta }${\displaystyle \delta } is phase lag between stress and strain.

The stress relaxation modulus ${\displaystyle G\left(t\right)}${\displaystyle G\left(t\right)} is the ratio of the stress remaining at time ${\displaystyle t}${\displaystyle t} after a step strain ${\displaystyle \varepsilon }${\displaystyle \varepsilon } was applied at time ${\displaystyle t=0}${\displaystyle t=0}: ${\displaystyle G\left(t\right)={\frac {\sigma \left(t\right)}{\varepsilon }}}${\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, ${\displaystyle G\left(t\right)}${\displaystyle G\left(t\right)} converges to the equilibrium shear modulus^[4] ${\displaystyle G}${\displaystyle G}:

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

The Fourier transform of the shear relaxation modulus ${\displaystyle G(t)}${\displaystyle G(t)} is ${\displaystyle {\hat {G}}(\omega )={\hat {G}}'(\omega )+i{\hat {G}}''(\omega )}${\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: ${\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }${\displaystyle E'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }
• Loss: ${\displaystyle E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta }${\displaystyle E''={\frac {\sigma _{0}}{\varepsilon _{0}}}\sin \delta } ^[3]

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

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

${\displaystyle E^{*}=E'+iE'',円}${\displaystyle E^{*}=E'+iE'',円}

${\displaystyle G^{*}=G'+iG'',円}${\displaystyle G^{*}=G'+iG'',円} ^[3]

where ${\displaystyle i}${\displaystyle i} is the imaginary unit.

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

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

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

For a material with a ${\displaystyle \tan \delta }${\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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