Editing Dynamic mechanical analysis (section)

====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>