en>Nealmcb: clarify odd reference to real numbers

2012-12-28T18:08:14Z

clarify odd reference to real numbers

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'''Dynamic modulus''' 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>[http://las.perkinelmer.com/content/ApplicationNotes/APP_FilmsandCoatings.pdf PerkinElmer "Mechanical Properties of Films and Coatings"]</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(t\omega)</math>
*Stress: <math> \sigma = \sigma_0 \sin(t\omega + \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.

=== Storage and loss modulus ===

The storage and loss modulus in viscoelastic solids 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]].

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

==References==
<references/>

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

Complex modulus - Revision history

en>Nealmcb: clarify odd reference to real numbers