Closed-form solution for horizontal and vertical shiftings of viscoelastic material functions in frequency domain

The closed-form shifting (CFS) algorithm is a simple mathematical methodology which determines the unique solution in the process of constructing master curves at selected reference temperature and pressure conditions. In a previous paper, the CFS algorithm has been fully described for monotonically increasing or monotonically decreasing functions only. This paper presents detailed steps of the generalized CFS methodology for non-monotonic functions, like the loss tangent. Performing shifting on the loss tangent, which does not require vertical shifting, is particularly important for materials which require vertical adjustment of dynamic viscoelastic functions, i.e., loss and storage moduli. Thus, based on horizontal shifting of the loss tangent, the CFS-based procedure of consecutive horizontal-vertical superposition for the storage modulus is proposed and analyzed. The analysis is done on the example of two synthetically generated non-monotonic tan delta segments and corresponding storage modulus segments in respect to different experimental parameters. It has been shown that the error brought by the shifting method into non-monotonic loss tangent and storage modulus master curves is twice smaller than the corresponding experimental noise level.