Dynamic modulus

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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#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.^[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).

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.^[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.

Ratio between loss and storage modulus#edit

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.

References#edit

^ The Open University (UK), 2000. T838 Design and Manufacture with Polymers: Solid properties and design, page 30. Milton Keynes: The Open University.
^ "PerkinElmer "Mechanical Properties of Films and Coatings"" (PDF). Archived from the original (PDF) on 2008-09-16. Retrieved 2009-05-09.
^ a b c d e Meyers and Chawla (1999): "Mechanical Behavior of Materials," 98-103.
^ Rubinstein, Michael, 1956 December 20- (2003). Polymer physics. Colby, Ralph H. Oxford: Oxford University Press. p. 284. ISBN 019852059X. OCLC 50339757.

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Dynamic modulus

Viscoelastic stress–strain phase-lag#edit

Storage and loss modulus#edit

Ratio between loss and storage modulus#edit

See also#edit

References#edit