Accuracy of local Kramers–Kronig relations between material damping and dynamic elastic properties

Abstract

The local Kramers–Kronig (KK) relations are the differential form approximations of the general KK integral equations linking the damping properties (loss modulus or loss factor) and dynamic modulus of elasticity (shear, bulk, etc.) of linear solid viscoelastic materials. The local KK relations are not exact; therefore, their accuracy is known to depend on the rate of frequency variations of material dynamic properties. The accuracy of the local KK relations is investigated in this paper under the assumption that the frequency dependence of the loss modulus obeys a simple power law. It is shown by analytic calculations that the accuracy of prediction of the local KK relations is better than 10% if the exponent in the loss modulus-frequency function is smaller than 0.35. This conclusion supports the result of an earlier numerical study. Some experimental data verifying the theoretical results will be presented. The conclusions drawn in the paper can easily be extended to acoustic wave propagation, namely to the accuracy of local KK relations between attenuation and dispersion of phase velocity.