Extent of force indeterminacy in packings of frictional rigid disks

Static packings of frictional rigid particles are investigated by means of discrete element simulations. We explore the ensemble of allowed force realizations in the space of contact forces for a given packing structure. We estimate the extent of force indeterminacy with different methods. The indeterminacy exhibits a nonmonotonic dependence on the interparticle friction coefficient. We verify directly that larger force indeterminacy is accompanied by a more robust behavior against local perturbations. We also investigate the local indeterminacy of individual contact forces. The probability distribution of local indeterminacy changes its shape depending on friction. We find that local indeterminacy tends to be larger on force chains for intermediate friction. This correlation disappears in the large friction limit.

Figure 1

Force indeterminacy $\eta$, quantified with different methods that are explained in the text, in terms of the friction coefficient $\mu$. Inset displays the same plot in log-log scale. The average coordination number $z$ as a function of $\mu$ is also shown (triangles).

Figure 2

The average critical force $⟨F_{\text{crit}}⟩$ scaled by the average normal contact force $⟨F_{\text{cont}}⟩$ (left) and the decay exponent $\alpha$ (right) as functions of the force indeterminacy ${\eta }_2$ for packings that are constructed with different friction coefficients.

Figure 3

The possible values of the normal contact forces $F_n$ (denoted by dots and mostly merged to intervals), scaled by the normal component of the original contact force $F_n^{\text{orig}}$, are shown for each contact of the packing. Each plot corresponds to the packing with the given friction coefficient. The fluctuations are small for small and large $\mu$ and become larger at intermediate $\mu$.

Figure 4

The probability distribution of the interval of possible normal contact forces for several friction coefficients $\mu$. Insets display semilogarithmic probability distributions for three different $\mu$.

Figure 5

The position of the contacts with original normal force $F_n$ larger than $2⟨F_n⟩$ (●) and the position of the contacts with force indeterminacy $\Delta F_n$ larger than $⟨\Delta F_n⟩$ $(\odot )$ in the packing with (a) $\mu =0.1$ and (b) $\mu =10$.

Figure 6

The correlation between the normal component of the original contact force $F_n$ and the force indeterminacy $\Delta F_n$ in terms of the friction coefficient $\mu$. Error bars display the maximum possible change in the correlation values when an arbitrary force network is chosen.