Kramers–Kronig relations for nonlinear rheology. Part I: General expression and implications
Sachin Shanbhag, Yogesh M. Joshi
Abstract
The principle of causality leads to linear Kramers–Kronig relations (KKR) that relate the real and imaginary parts of the complex modulus G∗ through integral transforms. Using the multiple integral generalization of the Boltzmann superposition principle for nonlinear rheology, and the principle of causality, we derived nonlinear KKR, which relate the real and imaginary parts of the nth order complex modulus Gn∗. For n=3, we obtained nonlinear KKR for medium amplitude parallel superposition (MAPS) rheology. A special case of MAPS is medium amplitude oscillatory shear (MAOS); we obtained MAOS KKR for the third-harmonic MAOS modulus G33∗; however, no such KKR exists for the first harmonic MAOS modulus G31∗. We verified MAPS and MAOS KKR for the single mode Giesekus model. We also probed the sensitivity of MAOS KKR when the domain of integration is truncated to a finite frequency window. We found that (i) inferring G33′′ from G33′ is more reliable than vice versa, (ii) predictions over a particular frequency range require approximately an excess of one decade of data beyond the frequency range of prediction, and (iii) G33′ is particularly susceptible to errors at large frequencies.