On the interconversion integral equation for relaxation and creep

The linear viscoelasticity interconversion equation allows estimates of the relaxation modulus to be derived computationally from experimentally derived estimates of the creep compliance (retardation modulus), and vice versa. It is popular as it allows more efficient utilization of resources in a rheological laboratory. However, the interconversion from the creep compliance to the relaxation is known to exhibit a greater level of instability than the converse. Although various algorithms have been proposed for performing the interconversion computationally, no adequate theoretical explanation of the mentioned difference in stability has been given. The question remains open as to whether the observed difference is an essential feature of the theoretical structure of the interconversion approaches or purely of the numerical algorithms that have been implemented to-date. This article gives a theoretical analysis for the situation where the relaxation and creep compliance functions are modelled as sums of exponentials. For the single exponential models, bounds are derived which established that the interconversion from relaxation to creep is always stable, whereas that from creep to relaxation can, under appropriate circumstances, exhibit instability. In this way, it is established that the difference is an essential feature of the theoretical structure of the interconversion equations