Determination of Relaxation and Retardation Spectrum from Modulus of Complex Frequency-Domain Material functions

摘要

The paper is devoted to improving and simplifying determination of the relaxation and retardation spectrum (RRS). A concept is postulated that determination of RRS from some specially selected material responses differing from the explicitly defined material functions, such as the real or imaginary parts of complex compliance and complex modulus, may improve the recovery performance at the price of better measurability of these specific material responses. As one of possible implementations of the postulated concept, we propose to recover RRS from the modulus (absolute value) of a complex frequency-domain (dynamic) material function, which, compared to the real or imaginary part, can be more accurately and easy acquired by measuring the amplitudes of harmonic responses of a material. It is demonstrated that RRS recovery problem from the modulus of a complex frequency-domain material function may be interpreted as a filtering task with a diffuse magnitude response bounded by the responses of the Mellin deconvolution filters corresponding to the minimum (zero) and maximum imaginary parts according to the Kramers-Kronig relation. A discrete RRS recovery filter operating with geometrically sampled data is constructed for recovering RRS from the modulus and the simulation results are presented. A measurement system is proposed implementing RRS recovery through the modulus of a complex frequency-domain material function, where a material under test is subjected to multi-harmonic excitation at geometrically spaced frequencies with subsequent measuring the amplitudes of multi-harmonic responses and processing them by a discrete RRS recovery filter.