VERIFICATION OF LOCAL KRAMERS–KRONIG RELATIONS FOR COMPLEX MODULUS BY MEANS OF FRACTIONAL DERIVATIVE MODEL

Abstract

The local Kramers–Kronig (K–K) relations, which link the damping properties of solid materials at one frequency to the rate of frequency variation of dynamic modulus, are not exact. The validity and accuracy of the local K–K relations is theoretically investigated in this paper by means of material models, especially the fractional Zener model. It is shown that the local K–K relations qualitatively always properly predict the relation between the damping and the frequency dependence of dynamic modulus for any type of deformation and any linear mechanism of energy loss determining the frequency variations. Nevertheless, the accuracy depends on the rate of frequency variation of dynamic properties, mainly of the loss modulus and loss factor, and the weaker the frequency dependence, the better the accuracy. The accuracy is better than 10% if the slope of frequency increase or decrease of loss functions plotted in a log–log system is smaller than 0·35. The application of the local K–K relations to some experimental data is presented.