Protocol Dependence and State Variables in the Force-Moment Ensemble

Stress-based ensembles incorporating temperaturelike variables have been proposed as a route to an equation of state for granular materials. To test the efficacy of this approach, we perform experiments on a two-dimensional photoelastic granular system under three loading conditions: uniaxial compression, biaxial compression, and simple shear. From the interparticle forces, we find that the distributions of the normal component of the coarse-grained force-moment tensor are exponential tailed, while the deviatoric component is Gaussian distributed. This implies that the correct stress-based statistical mechanics conserves both the force-moment tensor and the Maxwell-Cremona force-tiling area. As such, two variables of state arise: the tensorial angoricity ($\widehat{\alpha }$) and a new temperaturelike quantity associated with the force-tile area which we name keramicity ($\kappa$). Each quantity is observed to be inversely proportional to the global confining pressure; however, only $\kappa$ exhibits the protocol independence expected of a state variable, while $\widehat{\alpha }$ behaves as a variable of process.

Figure 1

(a) Example photoelastic images at $\mathrm{\Gamma }=2.1\times {10}^{-3}\mathrm{N}\mathrm{m}$ for each of three loading schemes—biaxial, uniaxial, and ${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^A$—shown schematically according to which walls move. ${\mathrm{S}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^B$ (dashed lines) is the opposite of ${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^A$ (solid lines). Bright particles are under larger force; large-scale light gradients have been removed by image processing. (b) Joint probability distributions $\mathcal{P}({\tilde{v}}^f,\mathrm{\Gamma })$ for dimensionless mean local free volume ${\tilde{v}}^f$ (averaged over clusters of size $m=8$) and global confining pressure $\mathrm{\Gamma }$ for each protocol. Dashed lines are linear fits using a weighted average. (c) A force-balanced particle within a jammed packing has an associated Maxwell-Cremona tile, formed by a closed loop of all vector forces acting on a particle.

Figure 2

(a) Histograms of local normal force-moment component $p$ on clusters of $m=8$ particles, at fixed ${\mathrm{\Gamma }}_i$, for all three loading protocols (${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^A$ and ${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^B$ combined). (b) Associated histogram log-ratios $\mathcal{R}$ taken between probability distributions observed at confining pressure ${\mathrm{\Gamma }}_i$ and ${\mathrm{\Gamma }}_j=\frac{1}{2}\max ({\mathrm{\Gamma }}_i)$. Each $\mathcal{R}(p)$ is fit to a straight line with slope ${\alpha }_j^p-{\alpha }_i^p$ [Eq. (4), with ${\kappa }_j-{\kappa }_i\approx 0$].

Figure 3

(a) Histograms of local deviatoric force-moment component $\tau$ on clusters of $m=8$ particles, at fixed ${\mathrm{\Gamma }}_i$, for all four loading protocols (${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^A$ and ${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^B$ separated). (b) Associated histogram log-ratios $\mathcal{R}$ taken between probability distributions observed at confining pressure ${\mathrm{\Gamma }}_i$ and ${\mathrm{\Gamma }}_j=\frac{1}{2}\max ({\mathrm{\Gamma }}_i)$. Each $\mathcal{R}(\tau )$ is fit to a parabola with coefficients ${\alpha }_j^{\tau }-{\alpha }_i^{\tau }$ and ${\kappa }_j-{\kappa }_i$ [Eq. (5)].

Figure 4

Temperaturelike variables: (a) Normal inverse angoricity, $1/{\alpha }^p$ against confining pressure $\mathrm{\Gamma }$, as computed for three loading conditions [$({\alpha }^p{\mathrm{\Gamma })}^{-1}=0.29\pm 0.01$ (biaxial), $0.34\pm 0.01$ (uniaxial), and $0.43\pm 0.01$ (shear)]. (b) Deviatoric inverse angoricity, $1/{\alpha }^{\tau }$ plotted against the same, indicating opposite dependencies on confining pressure based on the loading protocol [$({\alpha }^{\tau }{\mathrm{\Gamma })}^{-1}=-1.5\pm 0.1$ (uniaxial), $1.6\pm 0.1$ (biaxial), $0.59\pm 0.03$ (${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^A$), and $-1.0\pm 0.3$ (${\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}^B$)]. (c) The inverse keramicity ($1/\kappa$) evolves linearly with confining pressure and does so with equal proportionality under all protocols [$(\kappa {\mathrm{\Gamma })}^{-1}=(3.0\pm 0.1)\times {10}^{-3}$]. Dashed lines represent linear fits.