Numerical calculation of storage and loss modulus from stress relaxation data for linear viscoelastic materials

Abstract

Numerical formulae are given for calculation of storage and loss modulus from the known course of the stress relaxation modulus for linear viscoelastic materials. These formulae involve values of the relaxation modulus at times which are equally spaced on a logarithmic time scale. The ratio between succeeding times corresponds to a factor of two. Bounds for the relative error of those formulae were derived. These bounds depend on the value of the damping, tanδ, at the angular frequency,ω. The lower the damping values, the easier is the calculation of the storage modulus. This calculation involves the value of the relaxation modulus at timet 0=1/ω, and that of its derivative with respect to the logarithm of time in a rather narrow region aroundt 0. By contrast, the calculation of the loss modulus is difficult. This calculation involves the value of the derivative of the relaxation modulus with respect to the logarithm of time in a broad interval aroundt 0. Especially the behaviour of the relaxation modulus at timest<t 0 affects the calculation of the loss modulus significantly. The consequences of this “short time truncation problem” for the calculation of the loss modulus are discussed. If the results of dynamic measurements are available in the short time region in addition to those of the stress relaxation measurement in the long time region, the calculation of the loss modulus from the stress relaxation measurement is considerably simplified. The short time truncation problem can then be solved by using information aboutG″ as obtained from the dynamic measurements at short times (high frequencies).