Energy Fluctuations in Slowly Sheared Granular Materials

Here we show the first experimental measurement of the particle-scale energy fluctuations $\mathrm{\Delta }E$ in a slowly sheared layer of photoelastic disks. Starting from an isotropically jammed state, applying shear causes the shear-induced stochastic strengthening and weakening of particle-scale energies, whose statistics and dynamics govern the evolution of the macroscopic stress-strain curve. We find that the $\mathrm{\Delta }E$ behave as a temperaturelike noise field, showing a novel, Boltzmann-type, double-exponential distribution at any given shear strain $\gamma$. Following the framework of the soft glassy rheology theory, we extract an effective temperature $\chi$ from the statistics of the energy fluctuations to interpret the slow startup shear (shear starts from an isotropically jammed state) of granular materials as an “aging” process: Starting below one, $\chi$ gradually approaches one as $\gamma$ increases, similar to those of spin glasses, thermal glasses, and bulk metallic glasses.

Figure 1

(a) The schematic of the experimental setup. (b) A typical stress-strain curve. An inset shows the enlarged curve for $\gamma <2.8\%$. A dashed line denotes the position of yield strain ${\gamma }_y$. A blue arrow labels the $\gamma$, where (c)–(e) are referring to. (c) An experimentally recorded image of force chains. (d) A computer-constructed image (upper panel) using the measured contact forces. The examples of measured contact forces are drawn using four red arrows on a yellow-circled disk in the lower panel. (e) The map of the measured particle-scale energies. The lower panels in (c)–(e) plot a magnified small portion of the force chain image, the corresponding computed constructed images of force chains, and the particle-scale energies, which correspond to the same group of particles labeled in those three red squares in the upper panels in (c)–(e).

Figure 2

The spatial maps of the energy fluctuations $\mathrm{\Delta }E$ in (a) and the coarse-grained local diagonal strains ${\epsilon }_n\equiv {\epsilon }_{xx}-{\epsilon }_{yy}$ in (b). Here, at a given axial strain $\gamma$, $\mathrm{\Delta }E(\gamma )\equiv E(\gamma +\delta \gamma )-E(\gamma )$. In the first row, ${\gamma }_1=0$, and in the second row, ${\gamma }_2=1.37{\gamma }_y$. The spatial features of local strain are nearly unobservable in the distributions of $\mathrm{\Delta }E$. The $\mathrm{\Delta }E$ are in units of $E^{*}\equiv P_y\times D^2$, where $P_y$ is the confining pressure and $D$ is the average particle diameter.

Figure 3

The relaxations of energy fluctuations decay much more rapidly compared to those of strain in (a). In (a), the correlation functions $C(\mathrm{\Delta }\gamma )$ of the energy fluctuations $\mathrm{\Delta }E$ are nearly $\delta$ correlated, in a sharp contrast to that of the strain ${\epsilon }_n$, which can be fit well using a stretched exponential function (the dotted line). Here, $C(\mathrm{\Delta }\gamma )\equiv ⟨A(\gamma ,\overrightarrow{r})A(\gamma +\mathrm{\Delta }\gamma ,\overrightarrow{r})⟩$ for a variable $A$, in which the $⟨⟩$ stands for the spatial average over different positions $\overrightarrow{r}$. The probability distribution functions (PDFs) of the energy fluctuations $\mathrm{\Delta }E$ in (b) show a Boltzmann-type, double-exponential distribution for both $\mathrm{\Delta }E>0$ and $\mathrm{\Delta }E<0$, with $\mathrm{PDF}(\mathrm{\Delta }E<0)\sim \exp (-|\mathrm{\Delta }E|/{\alpha }_{-})$ and $\mathrm{PDF}(\mathrm{\Delta }E>0)\sim \exp (-|\mathrm{\Delta }E|/{\alpha }_{+})$. Results are shown for two different axial strains of ${\gamma }_1=0$ and ${\gamma }_2=1.37{\gamma }_y$.

Figure 4

The “aging time” ${\gamma }_a=2.24\%$ is defined when the ratio $\chi \equiv {\alpha }_{-}/{\alpha }_{+}$, derived from the two energy scales ${\alpha }_{\pm }$, first approaches $\chi =1$, as shown in (a). Beyond ${\gamma }_a$, the values of ${\alpha }_{\pm }$ become comparable, and thus the $\chi$ is pinned around $\chi \approx 1.0$. In SGR theories [25, 26, 28], when $\chi <1$, the system is in the glassy state where the aging takes place. The ${\alpha }_{\pm }$ are obtained from the PDF of energy fluctuations $\mathrm{\Delta }E$ as shown in Fig. 3. (b) shows the evolution of the ensemble-averaged effective temperature $\overline{\chi }$ (main panel) and the ensemble-averaged shear stress $\overline{\tau }$ (inset). The ensemble average is over ten different experimental runs under the same conditions.