Shear-induced rigidity in athermal materials: A unified statistical framework

Recent studies of athermal systems such as dry grains and dense, non-Brownian suspensions have shown that shear can lead to solidification through the process of shear jamming in grains and discontinuous shear thickening in suspensions. The similarities observed between these two distinct phenomena suggest that the physical processes leading to shear-induced rigidity in athermal materials are universal. We present a nonequilibrium statistical mechanics model, which exhibits the phenomenology of these shear-driven transitions, shear jamming and discontinuous shear thickening, in different regions of the predicted phase diagram. Our analysis identifies the crucial physical processes underlying shear-driven rigidity transitions, and clarifies the distinct roles played by shearing forces and the packing fraction of grains.

Figure 1

Mapping to spin model: (a) A typical sheared packing undergoing the SJ transition [15], color coded according to the strength of the nonaffine strain ($D_{\min }^2$ [16]) at a grain. (b) Mapping to spin-1 Ising variables: grains with more than two contacts (green) are assumed to have $S=0$ (stress anisotropy below threshold), and grains with two contacts have either $S=1$ (red) or $S=-1$ (blue), depending on whether the contact is aligned along the compressive [yellow broken line in (a)] or dilational direction. (c) Enlargement of a small section of (a) illustrating the grain-spin mapping. (d) A schematic configuration of the spin model on a square lattice, color coded by the strength of $h_i$, which represents the nonaffine strain at site $i$. The external field $H$ is not shown.

Figure 2

Model behavior: (a) A typical forward shear trajectory from the model for $\alpha =4$ demonstrating the appearance of $M_{\mathrm{peak}}$ in the $M\left(H\right)$ (red, continuous line) plot, which is concomitant with saturation of $X\left(H\right)$ (blue, dashed line). (b) Mean-field phase diagram for $\alpha =4$: The color bar indicates $M_{\mathrm{peak}}$. The critical point $({\Delta }_c,R_c)$ (yellow circle) marks the end point of three transition lines (see text): $R_t\left({\Delta }_0\right)$ (black, continuous line), $R_{DST}\left({\Delta }_0\right)$ (light blue, dotted line), and $R_m\left({\Delta }_0\right)$ (white dashed line). Detailed descriptions of $R_t$, $R_m$, and $R_{DST}$ are presented in the Appendix. The region to the left of the pink dash-dotted line has $M_{\mathrm{peak}}\le {10}^{-10}$ and the region to the right of $R_m$ has $M_{\mathrm{peak}}\sim 1$. (c),(d) $M\left(X\right)$ at different values of $R$ and ${\Delta }_0$ can be used to characterize and distinguish between different shear-induced rigidity transitions. The different colors correspond to the values of $R$ indicated in the phase diagram. For ${\Delta }_0>{\Delta }_c$ (c), $M$ vs $X$ is nonmontonic, whereas for ${\Delta }_0<{\Delta }_c$ (d), the functional form changes from nonmonotonic to monotonic as $R$ is decreased. $R$ values beyond $R_t$ are marked by dashed lines.

Figure 3

Scaling of $M_{\mathrm{peak}}$: The system achieves peak anisotropy $M_{\mathrm{peak}}$ at the rigidity transition. In (a) we show the dependence of $M_{\mathrm{peak}}$ on $R$ for several ${\Delta }_0$ [legend in (b)]. For ${\Delta }_0>{\Delta }_c=\sqrt{2/\pi e}\approx 0.4839$, the peak value $M_{\mathrm{peak}}$ is continuous but for ${\Delta }_0<{\Delta }_c$, $M_{\mathrm{peak}}$ has a discontinuity at $R_m$. (b) $M_{\mathrm{peak}}$ has a scaling form as a function of $R/R_J$ with $R_J\left({\Delta }_0\right)={\Delta }_0/6$. For ${\Delta }_0<{\Delta }_c$, the scaling form is valid for $R\le R_m$, and the discontinuity at $R_m$ is evident in this scaling plot. The peak anisotropy at $R_J$ is $\approx 0$, which suggests that the system undergoes a rigidity transition without going through any anisotropic state, reminiscent of the approach to the isotropically jammed state [2].

Figure 4

SJ regime (${\Delta }_0=0.9$): (a) $X\left(H\right)$ and (b) $M\left(H\right)$ for $R=$ 0.3 (blue, continuous line), 0.4 (green, long-dashed line), 0.57 (orange, dash-dotted line), 0.84 (red, short-dashed line), and 1.31 (brown, dotted line). The disorder values are chosen in such a way that $R/R_J-1$ increases logarithmically between 1 and 10. (c) Plots (see text) of $g_X(H/H_{\mathrm{peak}}\left(R\right))$ (main figure) and $g_M(H/H_{\mathrm{peak}}\left(R\right))$ (inset): $H_{\mathrm{peak}}\sim {(R/R_J-1)}^{1.2}$. (d) The scaling functions for $f_{\mathrm{iso}}$ (main) and stress anisotropy (inset) obtained from the SJ experiments [15] with packing fractions much below ${\varphi }_J[0.02\le (1-\varphi /{\varphi }_J)\le 0.09]$. The experimental data show exactly the same scaling behavior as the model [cf. (c)] if $R/R_J-1$ is mapped to $1-\varphi /{\varphi }_J$ and $H$ is mapped to $\gamma$.

Figure 5

Hysteresis: For ${\Delta }_0>{\Delta }_c$, the mean-field solution does not show hysteresis. The simulation of the model [Eq. (4)] in a 2D lattice for ${\Delta }_0=2>{\Delta }_c^{\mathrm{sim}}\approx 1.4$, however, shows hysteresis, which we show in (a) for $R=2$ (blue, continuous line), 3.5 (green, dashed line), 5 (red, dotted line), and 8 (brown, dash-dotted line). $H$ is varied cyclically between $-0.5$ and $0.5$. (b) The area of the hysteresis loops exhibits a nonmonotonic behavior and decreases with increasing disorder value beyond a peak. (c) For ${\Delta }_0<{\Delta }_c$, the mean-field solution exhibits hysteresis for $R_m\le R\le R_{DST}$. A few representative hysteresis curves are shown for $R=0.4\sim R_{DST}$ (red, dotted line), 0.3 (green, dashed line), 0.2 (light blue, dash-dotted line), and 0.1 $\sim R_m$ (blue, continuous line). The hysteresis loop first appears at $R_{DST}$, and increases in size as $R_m$ is approached from above. Below $R_m$, no hysteresis loops exist. (d) The size of the hysteresis loop [measured as $H_{+}-H_{-}$, where $H_{+}\left(H_{-}\right)$ is the maximum (minimum) value of $H$, where a loop exists] increases as $R$ is decreased from $R_{DST}$. The size increases as a power law with exponent 3/2 [9].

Figure 6

Phase diagram based on the value of $M_{\mathrm{peak}}$ from MF analysis (left) compared to the phase diagram obtained from simulations (right). The color bar shows the value of $M_{\mathrm{peak}}$. The simulations were performed on ${64}^2$ spins, and averaged over 20 different configurations for each $\left(R,{\Delta }_0\right)$. In both figures, the region to the left of the pink dash-dotted line has $M_{\mathrm{peak}}\le {10}^{-10}$ and region to the right of $R_m$ (white, dashed line) has $M_{\mathrm{peak}}\sim 1$. The blue dotted line shows $R_{DST}\left({\Delta }_0\right)$.

Figure 7

Hysteresis of $M$ and $X$ illustrating the behavior of the model for different disorders ($\alpha =4\text{and}{\Delta }_0=0.2$). $R_m\sim 0.097$, $R_{DST}\sim 0.4$, and $R_c\sim 0.8$ are the special disorders for ${\Delta }_0=0.2$ (see section 3 in Appendix). At $R_m$, the peak value of $M$, $M_{\mathrm{peak}}$, changes discontinuously [Fig. 3]. (a) $R=0.09$ (just below $R_m$), (b) $R=0.1$ (just above $R_m$), (c) $R=0.3$ (below $R_{DST}$), (d) $R=0.4$ ($R_{DST}$), (e) $R=0.6$ ($R_{DST}<R<R_t$), and (f) $R=0.9$ ($R>R_t$).

Figure 8

Comparison of trajectories with different $\alpha$, ${\Delta }_0>{\Delta }_c$. The asymptotic ($H\gg H_{\mathrm{peak}}$) dynamics is governed by $\alpha$. $M$ monotonically increases to 1 while $X$ decreases to zero for $\alpha <1$ trajectories (a). The exact opposite trend is observed for $\alpha >1$ trajectories (c). For $\alpha =1$, both $M$ and $X$ increase monotonically and saturate to a value less than 1 (b). The saturation value depends on the disorder.

Figure 9

$\alpha =1$: $X$ (a) and $M$ (b) as functions of the field $H$ for $\alpha =1$ (${\Delta }_0=0.9$) trajectories for a few typical disorder strengths, obtained from mean-field calculations. Both order parameters increase monotonically and saturate to a value less than 1. The saturation value depends on $R$. For $M$, the saturation value increases with $R$ while for $X$ it decreases.

Figure 10

Numerically obtained asymptotic $(H\gg H_{\mathrm{peak}})$ spin configuration for $\alpha =1$ trajectories shows microphase separation between spins 1 and 0. ${\Delta }_0=2>{\Delta }_c$ and $R=2$.