Contact changes near jamming

We probe the onset and effect of contact changes in soft harmonic particle packings which are sheared quasistatically. We find that the first contact changes are the creation or breaking of contacts on a single particle. We characterize the critical strain, statistics of breaking versus making a contact, and ratio of shear modulus before and after such events, and explain their finite size scaling relations. For large systems at finite pressure, the critical strain vanishes but the ratio of shear modulus before and after a contact change approaches one: linear response remains relevant in large systems. For finite systems close to jamming the critical strain also vanishes, but here linear response already breaks down after a single contact change.

Figure 1

(a) The first contact change in a sheared packing ($N=64,P={10}^{-6}$) occurs at a strain ${\gamma }^{*}=9.003851\left(2\right)\times {10}^{-7}$, when the two marked particles lose their contact. (b) The corresponding stress-strain curve remains continuous but exhibits a sharp kink; we define $G_0$ as the shear modulus of the undeformed packing, and $G_1$ as the shear modulus of the packing just above ${\gamma }^{*}$.

Figure 2

(a) Complementary cumulative distribution function (ccdf) of ${\gamma }^{*}/\langle {\gamma }^{*}\rangle$ for $N=16,P={10}^{-6} (\times )$; $N=16,P={10}^{-2} (+)$; $N=1024,P={10}^{-6} (\square )$; and $N=16,P={10}^{-2} (\circ )$. The black line is the ccdf for an exponential distribution with unity mean. (b) Scaling of $N\Delta z/2$ as a function of $N^{2}P$ (data from earlier simulations [13]). The arrows indicate volumetric strains corresponding to a single contact change.

Figure 3

Scaling of ${\gamma }^{\mathrm{cc}},{\gamma }^{\mathrm{bk}}$, and ${\gamma }^{\mathrm{mk}}$. (a) Scaling of the strain at first contact change. (b) Logarithmic corrections improve the collapse. In (a) and (b), symbols indicate packing sizes: $⊲(N\le 32),\square (32<N\le 1024)$, and $⊳(N>1024)$. (c) Scaling of the strains for contact making $(△)$ and contact breaking $(▽)$. (c) Inset: PDF of ${\gamma }_{\mathrm{LR}}/{\gamma }_{\mathrm{QS}}$ for $P={10}^{-2}$ and various $N$, to compare the characteristic strains in strict linear response, ${\gamma }_{\mathrm{LR}}$, to those from quasistatic shear simulations, ${\gamma }_{\mathrm{QS}}$. Curves have been offset for clarity. (d) Again, logarithmic corrections improve the collapse.

Figure 4

The probability distribution functions of $G_1/G_0$ for a range of values of $N$ and $P$ become narrowly peaked when $N^{2}P$ becomes large. We offset curves for different $N^{2}P$ for clarity. Inset: The standard deviation $\sigma$ of the distribution of $G_1/G_0$ vanishes as ${\left(N^{2}P\right)}^{-\beta }$, with $\beta =0.35\pm 0.01$, as indicated by the fitted line.