Power fluctuations in sheared amorphous materials: A minimal model

The importance of mesoscale fluctuations in flowing amorphous materials is widely accepted, without a clear understanding of their role. We propose a mean-field elastoplastic model that admits both stress and strain-rate fluctuations, and investigate the character of its power distribution under steady shear flow. The model predicts the suppression of negative power fluctuations near the liquid-solid transition; the existence of a fluctuation relation in limiting regimes but its replacement in general by stretched-exponential power-distribution tails; and a crossover between two distinct mechanisms for negative power fluctuations in the liquid and the yielding solid phases. We connect these predictions with recent results from particle-based, numerical microrheological experiments.

Figure 1

(a) Fully resolved, lattice-based $M\times N$ system of elemental stresses ${\sigma }_{i,j}$ and strain rates ${\dot{\gamma }}_{i,j}$, with external shear applied at boundaries. (b) The zero-dimensional HL model discards any notion of space at the mean-field level [23]. (c) Our extended model homogenizes strain rate $\dot{\gamma }$ along, but allows stochastic variation between, streamlines [44]. Each streamline $j\in \{1,\cdots ,M\}$ carries a set of $N$ stress elements ${\sigma }_k$ with no further spatial structure, creating an effectively 1D model with translational symmetry along the flow direction.

Figure 2

(a) The log ratio of the power distribution is plotted for HL ($N\to \infty )$, and for NHL at various finite $N$. The straight line for HL follows from the fluctuation relation (7), which breaks down at finite $N$. Common parameters for the lines at finite $N$ are $\alpha =0.8$, $\mathrm{\Sigma }=0.2$, and $\eta =0.1$. The $N=\infty$ curve is produced with $\alpha =0.8$, and $\dot{\gamma }=0.07$, which is the common mean shear rate for the NHL models. (b)–(e) Power distribution on either side of the yielding transition. In the fluid regime [(b), (c) $\alpha =0.70]$, the decays could be mistaken for a two-sided exponential. In the yielding solid regime [(d), (e) $\alpha =0.37]$, there is a clear departure from the two-sided exponential. The right panels are fit-free plots of $\mathcal{P}$ against $p^{2/3}$ to test (8) which predicts linear behavior as a function of $p^{2/3}$. The remaining parameters are $\eta =0.1$, $\mathrm{\Sigma }=0.2$, $N=2^{10}$. (f) The probability $\mathcal{P}(p<0)$ of negative injected power is nonmonotonic in $\alpha$. The inset decomposes this into $({\sigma }^{+},{\dot{\gamma }}^{-})$ (green dots and dashes) and $({\sigma }^{-},{\dot{\gamma }}^{+})$ (red dashed line) for $\mathrm{\Sigma }=0.2$, showing the crossover on moving from afluid to solid phase. Same parameters as in (b)–(e). Lines are from numerical solutions of (4) and (5) and circles are stochastic simulations of the full NHL dynamics. As $N$ increases, the minimum deepens and moves to the right, and it vanishes for $N\to \infty$.