imported>Koidkiok: corrected capitalization of fourier

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corrected capitalization of fourier

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{{Short description|Ratio used in material engineering}}
'''Dynamic modulus''' (sometimes '''complex modulus'''<ref name=TOU>The Open University (UK), 2000. ''T838 Design and Manufacture with Polymers: Solid properties and design'', page 30. Milton Keynes: The Open University.</ref>) 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 ==

[[Viscoelasticity]] is studied using [[dynamic mechanical analysis]] where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.<ref>{{Cite web |url=http://las.perkinelmer.com/content/ApplicationNotes/APP_FilmsandCoatings.pdf |title=PerkinElmer "Mechanical Properties of Films and Coatings" |access-date=2009-05-09 |archive-url=https://web.archive.org/web/20080916080329/http://las.perkinelmer.com/content/ApplicationNotes/APP_FilmsandCoatings.pdf |archive-date=2008-09-16 |url-status=dead }}</ref>
*In purely [[Elasticity (physics)|elastic]] materials the stress and strain occur in [[Phase (waves)|phase]], so that the response of one occurs simultaneously with the other.
*In purely [[viscosity|viscous]] materials, there is a [[Phase (waves)#Phase difference|phase difference]] between stress and strain, where strain lags stress by a 90 degree (<math>\pi/2</math> [[radian]]) phase lag.
*Viscoelastic materials exhibit behavior somewhere in between that of purely viscous and purely elastic materials, exhibiting some phase lag in strain.<ref name=Meyers>Meyers and Chawla (1999): "Mechanical Behavior of Materials," 98-103.</ref>

Stress and strain in a viscoelastic material can be represented using the following expressions:
*Strain: <math> \varepsilon = \varepsilon_0 \sin(\omega t)</math>
*Stress: <math> \sigma = \sigma_0 \sin(\omega t+ \delta) \,</math> <ref name=Meyers/>
where
:<math> \omega =2 \pi f </math> where <math>f</math> is frequency of strain oscillation,
:<math>t</math> is time,
:<math> \delta </math> is phase lag between stress and strain.

The stress relaxation modulus <math>G\left(t\right)</math> is the ratio of the stress remaining at time <math>t</math> after a step strain <math>\varepsilon</math> was applied at time <math>t=0</math>:
<math>G\left(t\right) = \frac{\sigma\left(t\right)}{\varepsilon}</math>,

which is the time-dependent generalization of [[Hooke's law]].
For visco-elastic solids, <math>G\left(t\right)</math> converges to the equilibrium shear modulus<ref>{{Cite book|title=Polymer physics|last=Rubinstein, Michael, 1956 December 20-|date=2003|publisher=Oxford University Press|others=Colby, Ralph H.|isbn=019852059X|location=Oxford|oclc=50339757|page=284}}</ref><math>G</math>:
:<math>G=\lim_{t\to \infty} G(t)</math>.
The [[Fourier transform]] of the shear relaxation modulus <math>G(t)</math> is <math>\hat{G}(\omega)=\hat{G}'(\omega) +i\hat{G}''(\omega)</math> (see below).
=== Storage and loss modulus ===

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.<ref name=Meyers/> The tensile storage and loss moduli are defined as follows:
*Storage: <math> E' = \frac {\sigma_0} {\varepsilon_0} \cos \delta </math>
*Loss: <math> E'' = \frac {\sigma_0} {\varepsilon_0} \sin \delta </math> <ref name=Meyers/>
Similarly we also define shear storage and shear loss moduli, <math>G'</math> and <math>G''</math>.

Complex variables can be used to express the moduli <math>E^*</math> and <math>G^*</math> as follows:
:<math>E^* = E' + iE'' \,</math>
:<math>G^* = G' + iG'' \,</math> <ref name=Meyers/>
where <math>i</math> is the [[imaginary unit]].

=== Ratio between loss and storage modulus ===

The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the <math> \tan \delta </math>, (cf. [[loss tangent]]), which provides a measure of damping in the material. <math> \tan \delta </math> can also be visualized as the tangent of the phase angle (<math> \delta </math>) between the storage and loss modulus.

Tensile: <math> \tan \delta = \frac {E''} {E'} </math>

Shear: <math> \tan \delta = \frac {G''} {G'} </math>

For a material with a <math> \tan \delta </math> greater than 1, the energy-dissipating, viscous component of the complex modulus prevails.

==See also==
*[[Dynamic mechanical analysis]]
* [[Elastic modulus]]
* [[Palierne equation]]

==References==
{{reflist}}

[[Category:Physical quantities]]
[[Category:Solid mechanics]]
[[Category:Non-Newtonian fluids]]
[[Category:Viscoelasticity]]

Dynamic modulus - Revision history

imported>Koidkiok: corrected capitalization of fourier