Self-Consistent Fourier-Tschebyshev Representations of the First Normal Stress Difference in Large Amplitude Oscillatory Shear

TL;DR

This paper introduces a Fourier-Tschebyshev framework to represent the first normal stress difference $N_1$ in Large Amplitude Oscillatory Shear (LAOS), complementary to the conventional shear-stress analysis. By expressing $N_1$ as a harmonics-based expansion with time- and strain-dependent coefficients, the authors establish self-consistency relations with linear viscoelastic moduli in the quasilinear limit and extend the framework to higher harmonics for nonlinear materials. They validate the approach through analytical results for the UCM and Giesekus models, apply Gaborheometry strain sweeps to extract $N_1$ material functions from PDMS and TPU data, and demonstrate that higher harmonics and Pipkin diagrams provide rich, material-specific fingerprints. The work enables rapid, time-local extraction of $N_1$ fingerprints and opens avenues for improved constitutive modeling and machine-learning-based rheology, including potential digital fluid twins for nonlinear viscoelastic fluids.

Abstract

Large Amplitude Oscillatory Shear (LAOS) is a key technique for characterizing nonlinear viscoelasticity in a wide range of materials. Most research to date has focused on the shear stress response to an oscillatory strain input. However, for highly elastic materials such as polymer melts, the time-varying first normal stress difference $N_1(t;ω,γ_0)$ can become much larger than the shear stress at sufficiently large strains, serving as a sensitive probe of the material's nonlinear characteristics. We present a Fourier-Tschebyshev framework for decomposing the higher-order spectral content of the $N_1$ material functions generated in LAOS. This new decomposition is first illustrated through analysis of the second-order and fourth-order responses of the quasilinear Upper Convected Maxwell model and the fully nonlinear Giesekus model. We then use this new framework to analyze experimental data on a viscoelastic silicone polymer and a thermoplastic polyurethane melt. Furthermore, we couple this decomposition with the recently developed Gaborheometry strain sweep technique to enable rapid and quantitative determination of the $N_1$ material function from experimental normal force data obtained in a single sweep from small to large strain amplitudes. We verify that asymptotic connections between the oscillatory shear stress and $N_1$ in the quasilinear limit are satisfied for the experimental data, ensuring self-consistency. This framework for analyzing the first normal stress difference is complementary to the established framework for analyzing the shear stresses in LAOS, and augments the content of material-specific data sets, hence more fully quantifying the important nonlinear viscoelastic properties of a wide range of soft materials.

Figures (14)

• Figure 1: The evolution of $N_1 (t)$ calculated using the Upper Convected Maxwell model (here $G = 1$ Pa, $\tau = 1$ s, $\omega = 1$ rad/s) for different strain amplitudes, presented as (a) time series data, (b) Lissajous figures of the first normal stress difference vs. strain, and (c) single-valued curves with an abscissa $z = \cos(\omega t - \Phi_2/2)$ that incorporates the phase angle $\Phi_2$. In this specific 2D projection of the (naturally 3D) Lissajous orbits, curves of $N_1(\omega,\gamma_0)$ overlap and enclose no area.
• Figure 2: (a) The evolution of $N_1$ calculated using the Upper Convected Maxwell model ($G = 1$ Pa, $\tau = 1$ s, $\gamma_0 = 10$) for different angular frequencies, presented as stress-strain Lissajous curves. (b) and (c) Analytical solutions for the zeroth and second harmonics of the UCM model as well as asymptotic values given in the text. Note the change in sign for $\tan \Phi_2$ at a critical Deborah number $De_c = \omega_c \tau = \sqrt{1/2}$. The asymptotic expressions for $n_1^d$, $|n_{1,2}^*|$ and $\Phi_2$ at small and large $De$ (equations \ref{['UCMsmallDe_n1d']} to \ref{['UCMasymptotic_end']}) are marked as dashed lines.
• Figure 3: The input to the Giesekus model ($G = 1$ Pa, $\tau = 1$ s, $\omega = 1$ rad/s, $\alpha = 0.4$) is (a) an amplitude-modulated strain sweep that is linearly increasing in strain amplitude, producing (b) an oscillatory normal stress difference signal that oscillates about a non-zero mean at a frequency $2\omega$. (c) The resultant Lissajous curves of $N_1$ grow in size with time. These signals can be analyzed using the windowed Gabor transform to obtain material functions for $N_1$ in LAOS at each strain amplitude applied. Three representative discrete single cycles at $t_i = 9.425, 84.82, 153.9$ s (corresponding to strain amplitudes $\gamma_0 (t_i) = 2.226, 4.035, 5.694$) are highlighted in green and blue to show the progression in the size and shape of the curves with time.
• Figure 4: Comparison between the discrete strain amplitude simulations (black symbols) and the continuous strain sweep ($\gamma_0(t) = 2 + 0.024t$) simulations of the Giesekus model ($G = 1$ Pa, $\tau = 1$ s, $\omega = 1$ rad/s, with $\alpha = 0.1, 0.4$). The maximum mutation number was $Mu_{\text{max}} = 0.075$ at $t = 0$. The contributions to the zeroth, second, and fourth harmonics of the $N_1$ signal are presented. The predictions of the zeroth and second harmonics from the UCM model (i.e. $\alpha = 0$ in the Giesekus model) are shown as black dashed lines in (a), (b) and (c). Simulations for $\alpha = 0.1$ are shown by the black triangles (discrete imposed strain amplitude) and closely spaced blue dots (GaborSS), while the case when $\alpha = 0.4$ is shown by the black circles (discrete strains) and closely spaced red dots (GaborSS).
• Figure 5: The asymptotic low $\gamma_0$ limit for the zeroth and second harmonics of $N_1$ signal predicted by the Giesekus model ($G = 1$ Pa, $\tau = 1$ s, $\omega = 1$ rad/s, $\alpha = 0.4$) match well with the appropriate combinations of the first harmonic measurements of the complex modulus. In each figure, the black line is computed from the ramped oscillatory $N_1$ data while the red line is obtained from the oscillatory shear stress data. Dotted lines in (c) indicate negative values of the coefficient due to a change in sign of $n_{1,2}'(\omega)$ at a critical strain amplitude.