Exponential Stress Relaxation Driven by Elementary Plastic Events in Non-Ageing Liquid Foams

Abstract

Liquid foams are archetypal athermal amorphous solids whose elasticity arises from the jamming of densely packed bubbles. We investigate the stress relaxation of non-ageing liquid foams following flow cessation, using fast X-ray tomo-rheoscopy. Thanks to in situ, time-resolved measurements, we uncover robust linear affine relationships between shear stress, plastic activity, and coordination number throughout the relaxation toward a residual stress state below the yield value. In contrast to previous studies on amorphous solids, we observe an exponential relaxation governed by the duration of individual plastic events, rather than by cascades of correlated ones associated with much longer, shear-rate-dependent timescales or power-law relaxations. Our results are consistent with a recent theoretical framework proposed by Cuny et al., suggesting that residual stress originates from the orientation of the stress tensor.

Figures (16)

• Figure 1: Experimental setup. In the plate–plate geometry, the shear rate in the pre-shear phase varies with radius according to $\dot{\gamma}(r) = \Omega r / h$, where $\Omega$ is the relative rotation speed. Measurements are performed within toroidal regions at various radii and corresponding shear rates, with color coding from green ($r = 0.1375$ mm, $\dot{\gamma} = 0.0006$ s$^{-1}$, near the center) to red (2.6125 mm, 0.0109 s$^{-1}$, at the outer edge).
• Figure 2: (a) Shear stress $\sigma$ and (b) plastic activity $P$ relaxation curves as functions of the relaxation time $\Delta t$ for the reference experiment. The color scale, from green to red, represents the local shear rate prior to flow cessation. $\sigma_{ss}$ and $P_{ss}$ are indicated for the largest $\dot{\gamma}$. The yield stress $\sigma_Y$ is shown as a horizontal line, with the gray band representing its standard deviation, indicating that most stress curves relax below $\sigma_Y$.
• Figure 3: (a) Shear stress $\sigma$ as a function of plastic activity $P$ in the reference experiment (data of Fig. \ref{['fig:fig2']} excluding the innermost and outermost tori), showing affine relations for each shear rate. Solid lines are affine fits, whose intercepts define the $\sigma_{\infty}$ values, as used in Eq. \ref{['eq:stressplasticity']}. (b) Normalized $\sigma$ versus normalized $P$ for the same dataset. (c) $\sigma$ as a function of $P$ for different liquid fractions at a fixed shear rate, $\dot{\gamma} = 0.0098$ s$^{-1}$. Symbols associated with each liquid fraction are shown next to the lower color bar. (d) Normalized $\sigma$ versus normalized $P$ for the same dataset.
• Figure 4: (a) Mean coordination number $Z$ as a function of relaxation time $\Delta t$ for various shear rates in the reference experiment. (b) $Z$ as a function of the plastic activity $P$ for the same experiment, revealing affine relations for each shear rate. Solid lines are best fits, with intercepts defining the residual coordination number $Z_\infty$. Same color coding as in Fig. \ref{['fig:fig3']}.
• Figure 5: Normalized shear stress, $(\sigma - \sigma_\infty)/(\sigma_{ss} - \sigma_\infty)$, as a function of $\dot{\gamma} \Delta t$ (a) and $\Delta t/\Delta t_{T1}$ (b) for the reference experiment (main panels) and for different liquid fractions at a fixed shear rate $\dot{\gamma} = 0.0098$ s$^{-1}$ (inset of (b)). The black line is a guide to the eye, representing an exponential decay of equation $e^{-0.6 \Delta t/\Delta t_{T1}}$. Same color scale as in Fig. \ref{['fig:fig3']}.