Defining nonlinear rheological material functions for oscillatory shear

Material functions underlie our understanding of rheology. They form the descriptive language of rheologists and require clear definitions. Here, it is shown that the definitions of oscillatory material functions depend on how the oscillating input is mathematically referenced, as a sine or cosine. Depending on this seemingly arbitrary trigonometric reference choice, the (3rd, 7th, 11th, etc.) Fourier coefficients of a nonlinear shear response change sign. Additionally, the even harmonic coefficients of a shear normal stress response are transposed. This impacts large-amplitude oscillatory shear (LAOS) characterization in both shear strain-control (LAOStrain) and shear stress-control (LAOStress) modes. It is important to resolve this issue, because it involves the leading-order nonlinearities and the signs of these higher harmonics convey important information. This paper provides a resolution, in two parts. First, it is shown that the deformation-domain Chebyshev coefficients are immune to the arbitrary trigonometric reference in the time domain, and therefore the Chebyshev-coefficient material functions can be used and interpreted without risk of inconsistency. Second, this paper proposes the convention of referencing to a sine input for strain-control tests (currently the typical convention) and using a cosine input for stress-control (where there is not currently a convention). Finally, clarity is brought to the practical issue of data processing a digital signal, which is required for numerical simulations and every instrument that performs oscillatory characterization.