Kramers–Kronig relations for nonlinear rheology. Part II: Validation of medium amplitude oscillatory shear (MAOS) measurements

Abstract

The frequency dependence of third-harmonic medium amplitude oscillatory shear (MAOS) modulus G33∗(ω) provides insight into material behavior and microstructure in the asymptotically nonlinear regime. Motivated by the difficulty in the measurement of MAOS moduli, we propose a test for data validation based on nonlinear Kramers–Kronig relations. We extend the approach used to assess the consistency of linear viscoelastic data by expressing the real and imaginary parts of G33∗(ω) as a linear combination of Maxwell elements: the functional form for the MAOS kernels is inspired by time-strain separability (TSS). We propose a statistical test based on fitting a sum of Maxwell elements using LASSO (least absolute shrinkage and selection operator) regression, and call it the SMEL test. It works well on a broad range of materials and models including those that do not obey TSS. It successfully copes with experimental data that are noisy or confined to a limited frequency range. When Maxwell modes obtained from the SMEL test are used to predict the first-harmonic MAOS modulus G31∗, it is possible to identify the range of time scales over which a material exhibits TSS.