Differences in plasticity between hard and soft spheres

Contact changes in packings of sheared hard spheres invariably trigger instabilities and irreversible rearrangements, providing an archetypal scenario for plasticity of disordered media. Here we show that the plasticity of jammed soft spheres at any finite pressure follows a different scenario, with only 14% of contact changes leading to irreversible rearrangements, irrespective of pressure, size, dimension or interaction potential. Moreover, we find that for sheared soft spheres, the nonlinear quantities associated with either contact changes or irreversible events exhibit the same finite-size scaling with pressure and system size as linear response quantities such as the shear modulus, suggesting an unexpected connection between curvature and saddle points in the potential energy landscape. Together our results indicate that soft spheres at finite pressure are not a smooth perturbation away from hard spheres, and that the nonlinear response of soft spheres is singular at zero pressure.

Figure 1

(a) Typical stress strain curve for 2D Hertzian spheres with $N=128$ and $p={10}^{-2}$. Network events are labeled with blue dots and rearrangements are labeled with red triangles. Examples of particle positions and forces during (b) a rearrangement and (c) a network event, where grey disks represent positions before and solid disks represent positions after the event. Red lines represent new contacts and dashed blue lines represent broken contacts. A metric of reversibility versus the stress drop for 2D. (d) Hertzian and (e) Hookean disks distinguishes network events (lower left) from rearrangements (upper right) for all system sizes and pressures, where the colors represent $N$ = 32 (red), 64, 128, 256, 512, and 1024 (blue) with an even gradient, and pressures are not distinguished.

Figure 2

The average stress drop associated with rearrangements for each system size and pressure collapses onto a master curve when we plot it versus $N^{\beta }p$ for 2D. (a) Hertzian disks ($\beta =3$) and (b) Hookean disks ($\beta =2$). (Insets) The metric of reversibility associated with rearrangements also collapses onto a master curve when plotted versus $N^{\beta }p$. Dashed lines have slope unity and are shown to guide the eye. Error bars represent the middle 60% of each distribution, and colors are the same as in Fig. 1.

Figure 3

Strain steps between rearrangements (circles) and network events (crosses) for (a) 2D Hertzian disks and (b) 2D Hookean disks. Colors and error bars are the same as Fig. 2. Dashed lines with slope 1 are added as a guide to the eye. (Insets) Simple regression confirms that these two curves have the same slope, implying a constant fraction of rearrangements across system size and pressure.

Figure 4

The minimum force before a rearrangement in (a) 2D Hertzian and (b) 3D Hertzian systems. Dashed lines represent a slope of 1, 2, and 3. Our data is consistent with $f_{\min }\sim p^2/N$. Color and error bars same as Fig. 2.