Microscopic Origins of Shear Jamming for 2D Frictional Grains

Shear jamming (SJ) occurs for frictional granular materials with packing fractions $\varphi$ in ${\varphi }_S<\varphi <{\varphi }_J^0$, when the material is subject to shear strain $\gamma$ starting from a force-free state. Here, ${\varphi }_J^{\mu }$ is the isotropic jamming point for particles with a friction coefficient $\mu$. SJ states have mechanically stable anisotropic force networks, e.g., force chains. Here, we investigate the origins of SJ by considering small-scale structures—trimers and branches—whose response to shear leads to SJ. Trimers are any three grains where the two outer grains contact a center one. Branches occur where three or more quasilinear force chain segments intersect. Certain trimers respond to shear by compressing and bending; bending is a nonlinear symmetry-breaking process that can push particles in the dilation direction faster than the affine dilation. We identify these structures in physical experiments on systems of two-dimensional frictional discs, and verify their role in SJ. Trimer bending and branch creation both increase $Z$ above $Z_{\mathrm{iso}}\simeq 3$ needed for jamming 2D frictional grains, and grow the strong force network, leading to SJ.

Figure 1

(a) Sketch of simple shear with solid lines showing the principal strain directions, and neutral lines (dashed) along which there is neither compression nor dilation. (b) Sketch of pure shear showing principal directions and neutral lines similar to those of (a). In (a) and (b), the shaded regions undergo compression. (c) A sequence of three frictional grains (solid circles with numbers indicating indices) forms a trimer. $\theta$ characterizes the straightness of the trimer. $\alpha$ indicates the angle between the principal strain axis, and the line through the contacts between particles 1 and 2 and particles 2 and 3. In (a)–(c), C and D indicate the directions of maximum compression and dilation, respectively.

Figure 2

Contact data vs $f_{\mathrm{NR}}$ for one run with $\varphi =0.805$. (a) Blue symbols: fraction of NR particles with the indicated normalized number of contacts. Red circles: coordination number, $Z$. (b) $N_i$, the total number of particles with $i$ strong ($f\ge \overline{f}$) contacts (and possibly other weaker contacts), normalized by $N$, the total number of particles in the system. The color bar gives the average $Z$, including moderate and weak contacts, for a given symbol.

Figure 3

Displacements (blue arrows) for each particle in units of the small particle radius $R$, for $\varphi =0.805$, just at shear jamming when $\gamma =0.0999$. Red arrows indicate the direction of simple shear.

Figure 4

Representative example of trimer response to shear. (a) $P$ for particle 2 and photoelastic images of the trimer (particles 1-2-3) vs $\gamma$. $Z$ for particle 2 is indicated by different symbols. At $\gamma =0.1161$, 2 contacts 4, and $P$ on 2 starts to rise quickly. When $Z=4$, 2 makes weak contact with 5. In the lab frame (inset) the coordinates are $x\text{−}y$, with shear along $y$. The compression direction is along $\sim 45°$. (b) Opening angle, $\theta$ (black triangles), and orientation $\alpha$ (red circles) of the trimer. (b) Inset: distance from particle 2 to the line connecting the centers of 1 and 3 (blue circles) and the corresponding distance if every particle moved affinely with the shear (red squares). (c) A broader region around the trimer of (a) for $\gamma =0.108$.

Figure 5

(a) $\overline{O}$, for trimers in the force network at SJ for five runs with $\varphi =0.805$, with the black crosses indicating when $Z$ just reaches 3. (b) $\overline{O}$ values averaged over five runs for each density, blue squares: $\gamma =0$; red squares: ${\gamma }_{\mathrm{SJ}}$. (c) Two-dimensional PDF of $O$ vs $-({\widehat{b}}_i·{\widehat{b}}_j-c_{ij})/(1+c_{ij})$ and $\cos (2\alpha )$ for run 1 in (a) when $Z$ just reaches 3. (d) Rescaled pressure $P_r$ for the trimers with $O\ge 0$ vs $f_{\mathrm{NR}}$, for five $\varphi$’s. Black dashed line: fit for $P_r$ for trimers with $O<0$ (see the Supplemental Material for more details [27]).