Force mobilization and generalized isostaticity in jammed packings of frictional grains

We show that in slowly generated two-dimensional packings of frictional spheres, a significant fraction of the friction forces lie at the Coulomb threshold—for small pressure $p$ and friction coefficient $\mu$, about half of the contacts. Interpreting these contacts as constrained leads to a generalized concept of isostaticity, which relates the maximal fraction of fully mobilized contacts and contact number. For $p\to 0$, our frictional packings approximately satisfy this relation over the full range of $\mu$. This is in agreement with a previous conjecture that gently built packings should be marginal solids at jamming. In addition, the contact numbers and packing densities scale with both $p$ and $\mu$.

Figure 1

Relation between the number of fully mobilized forces per particle, $n_{\mathrm{m}}$, and contact number $z$. The solid line indicates the maximum of $n_{\mathrm{m}}$, and such packings are marginal, while below this line one finds hyperstatic stable packings. The data points refer to numerically obtained values of $n_{\mathrm{m}}$ in two dimensions, for $p\sim 2\times {10}^{-4}$ (+), $p\sim 5\times {10}^{-5}$ (◻), $p\sim 2\times {10}^{-5}$ (×), and $p\sim 5\times {10}^{-6}$ (엯). $n_{\mathrm{m}}$ and $z$ behave smoothly as function of $p^{1∕3}$, and by extrapolation we obtain our $p=0$ estimate indicated by the solid squares (see inset and main text).

Figure 2

Mobilization at $p=2\times {10}^{-5}$. (a) Scatter plots of $f_{\mathrm{t}}∕\mu$ versus $f_{\mathrm{n}}$ for three packings at $\mu =0.001$, 0.32, and 1. The probability density of normalized tangential $(f_t∕\mu )$ and normal $\left(f_n\right)$ forces exhibits a singularity on the Coulomb cone for small $\mu$, which rapidly diminishes for larger $\mu$ (all forces are normalized so that $⟨f_n⟩=1$). (b) The cumulative distribution of the mobilization $C\left(m\right)\equiv {\int }^{m}d{m}^{\prime }P\left(m^{\prime }\right)$ exhibits a clear jump near $m=1$. Data shown here are for $\mu ={10}^0,{10}^{-0.5},{10}^{-1},\dots ,{10}^{-3}$.

Figure 3

Variation of contact numbers $z$ and packing density $\varphi$ as function of pressure $p$ and friction coefficient $\mu$. Error bars are smaller than the symbol size. (a)–(c) The variation of the contact number $z$, the packing density including rattlers ${\varphi }_{+\mathrm{R}}$, and the packing density excluding rattlers ${\varphi }_{-\mathrm{R}}$ as a function of $\mu$. Symbols indicate data at pressures $p\sim 4\times {10}^{-3}$ (▿), $5\times {10}^{-4}$ (◇), $2\times {10}^{-4}$ (+), $5\times {10}^{-5}$ (◻), $2\times {10}^{-5}$ (×), and $5\times {10}^{-6}$ (엯). Based on the extrapolation illustrated in panels (d)–(f), we also show the estimated values at $p=0$ (∎). Even though ${\varphi }_{+\mathrm{R}}$ and ${\varphi }_{-\mathrm{R}}$ differ substantially, their variation with $\mu$ appears very similar. (d)–(f) $z$ scales as $p^{1∕3}$ and ${\varphi }_{+\mathrm{R}}$ as $p^{2∕3}$, which allows us to extrapolate to zero pressure. Surprisingly, the packing density ${\varphi }_{-\mathrm{R}}$ does not scale convincingly with $p^{2∕3}$, but rather as $p^{1∕3}$. Symbols are as in panels (a)–(c).

Figure 4

Scaling of the zero pressure, extrapolated, contact numbers, and packing densities with the friction coefficient $\mu$. The extrapolated values at zero (infinite) friction are labeled as 0,0 $(0,\infty )$. (a)–(c) When $\mu \to 0$ and $p\to 0$, $z$ approaches $z^{0,0}\approx 3.975$ [20], while ${\varphi }_{+\mathrm{R}}$ approaches ${\varphi }_{+\mathrm{R}}^{0,0}\approx 0.8395$ [5]. For finite but small $\mu$, $z$ and ${\varphi }_{+\mathrm{R}}$ appear to scale as (a) $(z^{0,0}-z)\sim {\mu }^{0.70\left[10\right]}$ and (b) $({\varphi }_{+\mathrm{R}}^{0,0}-{\varphi }_{+\mathrm{R}})\sim {\mu }^{0.77\left[10\right]}$. (c) The contact number and packing deviate similarly from this scaling when $\mu$ approaches 1, and so $(z^{0,0}-z)\sim {({\varphi }_{+\mathrm{R}}^{0,0}-{\varphi }_{+\mathrm{R}})}^{0.91\left[10\right]}$. (d) In the limit of infinite friction and zero pressure, $z$ approaches $z^{\infty ,0}=3.00$ [2], while ${\varphi }_{+\mathrm{R}}$ approaches ${\varphi }_{+\mathrm{R}}^{\infty ,0}=0.758$ [10]. The deviations from these limiting values also appear to be related by a scaling relation of the form $(z-z^{\infty ,0})\sim {({\varphi }_{+\mathrm{R}}-{\varphi }_{+\mathrm{R}}^{\infty ,0})}^{1.7\left[2\right]}$.