Generalization of Kramers-Kronig transforms and some approximations of relations between viscoelastic quantities

Abstract

On the basis of some very plausible assumptions about the response of physical systems to stimuli, such as Boltzmann's superposition principle and the causality principle, Spence showed that the following characteristics obtain for the modulus and compliance functions: (i) They are analytic in the lower half of the complex frequency plane, (ii) they are limited if the frequency tends to infinity, and (iii) the real and imaginary parts are even and odd functions, respectively, of the frequencyω. It can generally be demonstrated that the real and imaginary parts of every function satisfying these three requirements and (iv) without singularities on the real frequency axis, are interrelated by Kramers-Kronig transforms. Similar relations hold between the logarithm of the modulus and the argument of the function.

Under certain conditions the Kramers-Kronig relations may be approximated by rather simple equations. For linear viscoelastic materials, for instance, the following approximate relations were obtained for the components of the complex dynamic shear modulus,G ^* (iω) = G′(ω) + iG″(ω) = G _d (ω) expiδ(ω):

$$\begin{gathered} G'' (\omega ) \simeq \frac{\pi }{2}\left( {\frac{{dG'(u)}}{{d In u}}} \right)_{u = \omega } , \hfill \\ G' (\omega ) - G'(o) \simeq - \frac{{\omega \pi }}{2}\left( {\frac{{d[G''(u)/u]}}{{d In u}}} \right)_{u = \omega } , \hfill \\ \delta (\omega ) \simeq \frac{\pi }{2}\left( {\frac{{d In G_d (u)}}{{d In u}}} \right)_{u = \omega } . \hfill \\ \end{gathered} $$

The first of these relations was published long ago by Staverman and Schwarzl and is useful over broad frequency ranges, as is the second relation. The last equation is the most general one, and also is better supported by experiment.