Iterative calculation of relaxation spectrum from free vibration data

Abstract

AbstractThe complex elastic modulus G*(ω) = iG′(ω) + iG′(ω) is shown to be a function not only of the frequency (ω) but also of the damping factor a if the strain is of the form x = x0 exp {− at} sin ωt. The general equations for G*(ω,a) = G′(ω,a) + iG″ (ω,a) = are derived in terms of the relaxation spectrum. The use of a spring to aid the specimen and reduce the ratio a/ω is discussed, and its desirability demonstrated. If data are thus available in terms of either G″(ω) or G″(ω,a) the iterated second approximation of Ninomiya and Ferry provides a rapid and powerful method of finding the relaxation spectrum. To the accuracy to which the time–temperature reduction factor aT is known or can be predicted by means such, for example, as the WLF equation, the function G″(ω,a) over a temperature range at nearly constant frequency can be translated into terms of G″(ω,a) at constant temperature and varying frequency. In such cases, the relatively simple torsional pendulum, or some analogue of it, can economically provide a characterization of the viscoelastic behavior of the material over an extended time or frequency range.