A framework for determining the first nonlinear relaxation modulus of a viscoelastic fluid from a medium-amplitude oscillatory shear (MAOS) deformation is constructed. Knowledge of this “MAOS relaxation modulus” allows one to predict the weakly nonlinear stress response of a material under an arbitrary transient deformation via a memory integral expansion. Our framework is demonstrated by explicitly determining the MAOS relaxation modulus for a dilute suspension of Brownian spheroids subject to a dual-frequency oscillatory shear flow. Specifically, we first calculate the second normal stress difference for such a deformation from a corotational memory integral expansion. Second, the microstructural stress response of the model system of Brownian spheroids is determined via a regular perturbation expansion of the orientation distribution function at small dimensionless strain-rate amplitude, or Weissenberg number. An analytical expression for the MAOS relaxation modulus is resolved by comparing the second normal stress difference results of the memory integral expansion and microstructural stress calculation. Finally, using the MAOS relaxation modulus, we reconstruct the stress response of the model system for the start-up and cessation of simple shear and uniaxial extension. In summary, our work offers an approach to utilizing medium (and large) amplitude oscillatory shear results to predict stress dynamics of viscoelastic fluids in other transient, nonlinear flows.
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