Sub-Yield Dynamics in Yield-Stress Materials

Abstract

The mechanical response of yield-stress materials below the yield point remains a subject of debate. Two of the most widely used constitutive models for these materials offer fundamentally conflicting views: one permits plastic flow at all stress levels, the other assumes entirely recoverable viscoelasticity below yield. Using parallel superposition rheometry, we test the sub-yield behaviour of a microgel and an emulsion. When residual slip effects are properly accounted for, both fluids exhibit bounded, periodic strain responses, offering compelling evidence that they do not flow in the studied regime. Our results indicate that the sub-yield regime is underpinned by nonlinear viscoelasticity and underscore the need for improved constitutive relations that capture such effects without treating yielding as a precursor for nonlinearity.

Figures (6)

• Figure 1: Comparison between the predicted strain responses of the SHB model (solid blue line), the numerical solution of the KDR model (solid red line), and the analytical approximation of the KDR model (dashed black line) under parallel superposition of steady and oscillatory stress. Constitutive parameters correspond to a 2 g/L Carbopol gel (see Table S1 in the Supplemental MaterialSM). The applied stress components are $\sigma_0 = \epsilon = 0.1\tau_Y$, and $\omega = 1$ rad/s.
• Figure 2: Linear strain drift in the parallel superposition measurements in a Carbopol gel. Main: Strain rate drift from experiments (green markers), SHB model (blue solid), KDR model (red solid); red dashed: analytical KDR approximation [Eq.\ref{['eq:linear_term_kdr']}]; purple dashed: slip-only term, $\dot{\gamma}_\mathrm{drift}=\dot{\gamma}_\mathrm{slip}=\sigma_0\beta$. At $\sigma_0=0.1\tau_Y$, four overlapping markers correspond to $\epsilon/\tau_Y \in \{0.03,0.1,0.3,0.5\}$; all other data and model curves use $\epsilon=0.1\tau_Y$. The orange square shows the drift measured with smooth plates at $\sigma_0=\epsilon=0.1\tau_Y$. The shaded region marks the fully unyielded limit in the parallel-disk geometry, $\sigma_0+\epsilon\le 2\tau_Y/3$. Inset: phenomenological slip parameter (red circles) versus gap between crosshatched plates at $\sigma_0=\epsilon=1 \, \mathrm{Pa}$ and $\omega=1~\mathrm{rad \, s^{-1}}$; error bars denote the standard error; blue solid: best linear fit.
• Figure 3: Comparison of parallel-superposition rheometry data (green circles) with full numerical predictions from the SHB (blue) and KDR (red) models over the first 20 cycles for Carbopol. The imposed stress is $\sigma(t)=\sigma_{0}+\epsilon\sin(\omega t)$, with $\sigma_{0}/\tau_Y\in \{0.03,\,0.1,\,0.3,\,0.5\}$ (top to bottom), $\epsilon=0.1 \tau_Y$ and $\omega=1~\mathrm{rad \, s^{-1}}$.
• Figure 4: (a, b) Comparison between experimental strain offsets (red circles) and predictions from the SHB model (blue line) in parallel superposition tests on Carbopol gel. Error bars indicate the standard error across independent repetitions. (a) fixed oscillatory amplitude $\epsilon = 0.1\tau_Y$; (b) fixed mean stress $\sigma_0 = 0.1\tau_Y$. (c) Storage (red) and loss (blue) compliances as functions of $\epsilon$ for Carbopol gel. Markers denote experimental values from parallel superposition rheometry; solid lines correspond to conventional oscillatory shear measurements; dashed lines show predictions from the SHB model. At $\epsilon=0.1 \tau_Y$ there are four overlapping markers for values of $\sigma_0 = [0.03, \, 0.1, \, 0.3, \, 0.5]\tau_Y$. The remaining points were measured for $\sigma_0 = 0.1\tau_Y$. For all measurements $\omega=1~\mathrm{rad \, s^{-1}}$.
• Figure 5: Comparison of parallel superposition rheometry data (green circles) with full numerical predictions from the SHB (blue) and KDR (red) models over the first 20 cycles for the body lotion. The imposed stress is $\sigma(t)=\sigma_{0}+\epsilon\sin(\omega t)$, with $\sigma_{0}/\tau_Y\in \{0.03,\,0.1,\,0.3,\,0.5\}$ (top to bottom), $\epsilon=0.1 \tau_Y$ and $\omega=1~\mathrm{rad \, s^{-1}}$.