Editing Dynamic mechanical analysis (section)

==Theory==

===Viscoelastic properties of materials===
[[Image:Dynamic+Tests+Setup+Chem+538.jpg|thumb|325px|Figure 1. A typical DMA tester with grips to hold the sample and an environmental chamber to provide different temperature conditions. A sample is mounted on the grips and the environmental chamber can slide over to enclose the sample.]]
Polymers composed of long molecular chains have unique viscoelastic properties, which combine the characteristics of [[Elasticity (physics)|elastic solid]]s and [[Newtonian fluid]]s. The classical theory of elasticity describes the mechanical properties of elastic solids where stress is proportional to strain in small deformations. Such response to stress is independent of [[strain rate]].  The classical theory of hydrodynamics describes the properties of viscous fluid, for which stress response depends on strain rate.<ref name="Ferry1980">{{cite book|last=Ferry|first=J.D.|title=Viscoelastic properties of polymers|publisher=Wiley|year=1980|edition=3}}</ref> This solidlike and liquidlike behaviour of polymers can be modelled mechanically with combinations of springs and dashpots, making for both elastic and viscous behaviour of viscoelastic materials such as bitumen.<ref name="Ferry1991">{{cite journal|last=Ferry|first=J.D|year=1991|title=Some reflections on the early development of polymer dynamics: Viscoelasticity, dielectric dispersion and self-diffusion|doi=10.1021/ma00019a001|journal=Macromolecules|volume=24|issue=19|pages=5237–5245|bibcode = 1991MaMol..24.5237F }}</ref>

===Dynamic moduli of polymers===
The viscoelastic property of a polymer is studied by dynamic mechanical analysis where a sinusoidal force (stress σ) is applied to a material and the resulting displacement (strain) is measured. For a perfectly elastic solid, the resulting strain and the stress will be perfectly in phase. For a purely viscous fluid, there will be a 90 degree phase lag of strain with respect to stress.<ref name="Meyers1999">{{cite book|last=Meyers|first=M.A.|author2=Chawla K.K.|title=Mechanical Behavior of Materials|publisher=Prentice-Hall|year=1999}}</ref> Viscoelastic polymers have the characteristics in between where some [[phase lag]] will occur during DMA tests.<ref name=Meyers1999/> When the strain is applied and the stress lags behind, the following equations hold:<ref name="Meyers1999"/>

*Stress: <math> \sigma = \sigma_0 \sin(t\omega + \delta) \,</math><ref name=Meyers1999/>
*Strain: <math> \varepsilon = \varepsilon_0 \sin(t\omega)</math>

where
:<math> \omega </math> is the frequency of strain oscillation,
:<math>t</math> is time,
:<math> \delta </math> is phase lag between stress and strain.

Consider the purely elastic case, where stress is proportional to strain given by [[Young's modulus]] <math>E</math> .  We have <br>
<math> 
\sigma(t) = E \epsilon(t)
\implies  \sigma_0 \sin{(\omega t + \delta)} = E  \epsilon_0 \sin{\omega t}
\implies \delta = 0
</math>

Now for the purely viscous case, where stress is proportional to strain ''rate''.<br>
<math>
 \sigma(t) = K \frac{d\epsilon}{dt}
\implies  \sigma_0 \sin{(\omega t + \delta)} = K  \epsilon_0 \omega \cos{\omega t}
\implies \delta = \frac{\pi}{2}
</math>

The storage modulus measures the stored energy, representing the elastic portion, and the loss modulus measures the energy dissipated as heat, representing the viscous portion.<ref name=Meyers1999/> The tensile storage and loss moduli are defined as follows:

*Storage modulus:  <math> E' = \frac {\sigma_0} {\varepsilon_0} \cos \delta </math>
*Loss modulus: <math> E'' =  \frac {\sigma_0} {\varepsilon_0} \sin \delta </math>
*Phase angle: <math> \delta = \arctan\frac {E''}{E'} </math>

Similarly, in the shearing instead of tension case, we also define [[Shear modulus|shear storage]] and 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'' = \frac {\sigma_0} {\varepsilon_0} e^{i \delta} \,</math>
:<math>G^* = G' + iG'' \,</math>
where
:<math>{i}^2 = -1 \,</math>

====Derivation of dynamic moduli====

Shear stress <math>\sigma(t)=\int_{-\infty}^t G(t-t') \dot{\gamma}(t')dt'</math> of a finite element in one direction can be expressed with relaxation modulus <math>G(t-t')</math> and strain rate, integrated over all past times <math>t'</math> up to the current time <math>t</math>. With strain rate <math> \dot{\gamma(t)}=\omega \cdot  \gamma_0 \cdot \cos(\omega t)</math>and substitution <math>\xi(t')=t-t'=s </math> one obtains  <math>\sigma(t)=\int_{\xi(-\infty)=t-(-\infty)}^{\xi(t)=t-t} G(s) \omega \gamma_0 \cdot \cos(\omega(t-s))(-ds)=\gamma_0\int_0^{\infty} \omega G(s)\cos(\omega(t-s))ds</math>. Application of the trigonometric addition theorem <math>\cos(x \pm y)=\cos(x)\cos(y) \mp \sin(x)\sin(y)</math> lead to the expression 
:<math>
\frac{\sigma(t)}{\gamma(t)}=\underbrace{[\omega\int_o^{\infty}G(s)\sin(\omega s) ds]}_{\text{shear storage modulus }G'} \sin(\omega t)+\underbrace{[\omega\int_o^{\infty}G(s)\cos(\omega s) ds]}_{\text{shear loss modulus }G''} \cos(\omega t).
\,</math> 
with converging integrals, if <math>G(s) \rightarrow 0</math> for <math>s  \rightarrow \infty </math>, which depend on frequency but not of time. Extension of <math>\sigma(t)=\sigma_0 \cdot \sin (\omega \cdot t + \Delta \varphi) </math> with trigonometric identity <math> \sin(x \pm y)=\sin(x)\cdot \cos(y) \pm \cos(x)\cdot \sin(y)</math> lead to 
:<math> \frac{\sigma(t)}{\gamma(t)}=\underbrace{\frac{\sigma_0}{\gamma_0} \cdot \cos(\Delta \varphi)}_{G'}\cdot \sin (\omega \cdot t)+ \underbrace{\frac{\sigma_0}{\gamma_0} \cdot \sin(\Delta \varphi)}_{G''} \cdot \cos (\omega \cdot t)
\,</math>. 
Comparison of the two <math>\frac{\sigma(t)}{\gamma(t)}</math> equations lead to the definition of <math>G'</math> and <math>G''</math>.<ref name="Ferry">{{cite book|last=Ferry|first=J.D.|author2 = Myers, Henry S |year=1961|title=Viscoelastic properties of polymers|publisher=The Electrochemical Society |volume=108 }}</ref>