Efficient test to evaluate the consistency of elastic and viscous moduli with Kramers–Kronig relations

Abstract

The principle of causality constrains the real and imaginary parts of the complex modulus $G^{*} = G^{\prime } + i G^{\prime \prime }$ via Kramers–Kronig relations (KKR). Thus, the consistency of observed elastic or storage ($G^{\prime }$) and viscous or loss ($G^{\prime \prime }$) moduli can be ascertained by checking whether they obey KKR. This is important when master curves of the complex modulus are constructed by transforming a number of individual datasets; for example, during time-temperature superposition. We adapt a recently developed statistical technique called the ‘Sum of Maxwell Elements using Lasso’ or SMEL test to assess the KKR compliance of linear viscoelastic data. We validate this test by successfully using it on real and synthetic datasets that follow and violate KKR. The SMEL test is found to be both accurate and efficient. As a byproduct, the parameters inferred during the SMEL test provide a noisy estimate of the discrete relaxation spectrum. Strategies to improve the quality and interpretability of the extracted discrete spectrum are explored by appealing to the principle of parsimony to first reduce the number of parameters, and then to nonlinear regression to fine tune the spectrum. Comparisons with spectra obtained from the open-source program pyReSpect suggest possible tradeoffs between speed and accuracy.