Dynamic mechanical analysis: Difference between revisions

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====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).