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The principle of causality leads to linear Kramers-Kronig relations (KKR)that relate the real and imaginary parts of the complex modulus $G^{*}$ throughintegral transforms. Using the multiple integral generalization of theBoltzmann superposition principle for nonlinear rheology, and the principle ofcausality, we derived nonlinear KKR, which relate the real and imaginary partsof the $n^\text{th}$ order complex modulus $G_{n}^{*}$. For $n$=3, we obtainednonlinear KKR for medium amplitude parallel superposition (MAPS) rheology. Aspecial case of MAPS is medium amplitude oscillatory shear (MAOS); we obtainedMAOS KKR for the third-harmonic MAOS modulus $G_{33}^{*}$; however, no such KKRexists for the first harmonic MAOS modulus $G_{31}^{*}$. We verified MAPS andMAOS KKR for the single mode Giesekus model. We also probed the sensitivity ofMAOS KKR when the domain of integration is truncated to a finite frequencywindow. We found that that (i) inferring $G_{33}^{\prime\prime}$ from$G_{33}^{\prime}$ is more reliable than vice-versa, (ii) predictions over aparticular frequency range require approximately an excess of one decade ofdata beyond the frequency range of prediction, and (iii) $G_{33}^{\prime}$ isparticularly susceptible to errors at large frequencies.

Kramers-Kronig Relations for Nonlinear Rheology: 1. General Expression and Implications