Analysis of recorded waveforms using Chebyshev polynomial analysis

As the rheometer is operated in strain-controlled mode, the oscillatory strain imposed on the sample can be described as γ = γ_osin(ωt). The response of the material is assumed to be sinusoidal and can be written as σ = σ_osin(ωt + δ). Where, γ_o is the strain amplitude, σ_o is the magnitude of stress, ω is the oscillatory frequency of rheometer, δ is the phase angle between the input (strain) and output (stress) waveforms and t denotes time. Using the equations described in Ferry et al.⁸⁹ and assuming that stress waveforms are sinusoidal (for various strain amplitudes), one can decompose the total stress into elastic and viscous components and calculate the elastic (G') and viscous modulus (G″) of the material.

However, at large strain amplitudes the material enters the nonlinear regime and the stress waveform is not a simple sinusoid anymore. As a result, higher order harmonics need to be considered in order capture the true meaning of the waveform. The non-sinusoidal stress waveform can therefore be written in terms of Fourier expansion as:

where n represents the higher order harmonics. Only odd harmonics are considered because stress is assumed to bear odd symmetry with respect to shear strain or strain rate⁴⁹. Ewoldt et al.⁵⁰ proposed the use of orthogonal Chebyshev polynomials of first kind to approximate the nonlinear waveforms, as the higher order Chebyshev coefficients have physical meanings. Therefore Eq. (1) can be rewritten in terms of Chebyshev polynomials as:

where, e_n, ν_n are the n-th order elastic and viscous Chebyshev coefficents, T_n denotes the Chebyshev polynomial of first kind of n-th order, x = γ∕γ_o = sin(ωt) and y=γ˙∕γo=cos(ωt). Furthermore, by using the recursion identities of Chebyshev polynomials T_ncos(ωt) = cos(nωt) and sin(ωt) = cos(π∕2 − ωt), which yields Tnsin(ωt)=(−1)n−12sin(nωt), one can directly express the Chebyshev coefficients in terms of the n-th order moduli:

For n = 1, one can recover e1=G1′ and ν1=G1″ω. Traditionally, G1′~G′ is known as the elastic modulus, G1″=ν1ω~G″ is known as the viscous modulus and ν₁ is the viscosity. The higher order coefficients can be related their respective moduli and in particular, the third order elastic (e₃) and viscous (ν₃) Chebyshev coefficients represent a physical meaning. A positive value, i.e. e₃ > 0 and ν₃ > 0 represents intracyle strain hardening and shear thickening, respectively; whereas negative values e₃ < 0 and ν₃ < 0 represents strain softening and shear thinning. These measures represent the a quantitative way of describing the elastic and viscous nonlinearities occurring in the material. One can also calculate the dimensionless strain stiffening index (S) and shear thickening index (T), which can be defined in terms of the higher order Chebyshev coefficients as follows:

A more detailed description of the derivations of the above-mentioned equations and the physical interpretation can be found in the articles by Ewoldt et al.^50,90,91.