Content deleted Content added
No edit summary |
|||
Line 53:
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 equation x with x lead to the definition of <math>G'</math> and <math>G''</math>
<ref name="Ferry">{{cite journal|last=Ferry|first=J.D.|author2 = Myers, Henry S |year=1961|title=Viscoelastic properties of polymers|journal=The Electrochemical Society |volume=108 }}</ref>
==Applications==
| |||