Anomalous stress fluctuations in athermal two-dimensional amorphous solids

We numerically study the local stress distribution within athermal, isotropically stressed, mechanically stable, packings of bidisperse frictionless disks above the jamming transition in two dimensions. Considering the Fourier transform of the local stress, we find evidence for algebraically increasing fluctuations in both isotropic and anisotropic components of the stress tensor at small wave numbers, contrary to recent theoretical predictions. Such increasing fluctuations imply a lack of self-averaging of the stress on large length scales. The crossover to these increasing fluctuations defines a length scale ${\ell }_0$, however, it appears that ${\ell }_0$ does not vary much with packing fraction $\varphi$, nor does ${\ell }_0$ seem to be diverging as $\varphi$ approaches the jamming ${\varphi }_J$. We also find similar large length scale fluctuations of stress in the inherent states of a quenched Lennard-Jones liquid, leading us to speculate that such fluctuations may be a general property of amorphous solids in two dimensions.

Figure 1

Geometry of our system box. $L_x$ and $L_y$ are the lengths in the $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ directions, and $\gamma$ is the skew ratio. Lees-Edwards boundary conditions are used.

Figure 2

Fluctuation of the isotropic part of the stress $C_{\mathrm{\Gamma }}\left(\mathbf{q}\right)=\langle {\mathrm{\Gamma }}_{\mathbf{q}}{\mathrm{\Gamma }}_{-\mathbf{q}}\rangle /V$ vs $q$, at wave vectors $\mathbf{q}=q\widehat{\mathbf{y}}$ (solid symbols) and $\mathbf{q}=q{\widehat{\mathbf{e}}}_{x+y}$ (open symbols), at stress per particle $\tilde{p}=\mathrm{\Gamma }/N=0.01831$ above jamming. Here $\langle \varphi \rangle =0.8857$, compared to ${\varphi }_J\approx 0.8416$. Results are shown for systems with different number of particles $N$. Solid line at small $q$ has slope $-1.3$.

Figure 3

Fluctuation of the isotropic part of the stress normalized by the stress per particle $\tilde{p}$ squared, $C_{\mathrm{\Gamma }}\left(\mathbf{q}\right)/{\tilde{p}}^2$, at wave vectors $\mathbf{q}=q\widehat{\mathbf{y}}$. Curves are for different values of $\tilde{p}=\mathrm{\Gamma }/N$, going from $\tilde{p}=0.00014$ on top to $0.01831$ on bottom. System has $N=65536$ particles. Vertical dashed lines delimit the range of $q$ where $C_{\mathrm{\Gamma }}\left(\mathbf{q}\right)$ is roughly constant.

Figure 4

Fluctuation of the anisotropic part of the stress $C_{{\mathrm{\Sigma }}_{xy}}\left(\mathbf{q}\right)=\langle {\mathrm{\Sigma }}_{xy\mathbf{q}}{\mathrm{\Sigma }}_{xy-\mathbf{q}}\rangle /V$ at wave vectors $\mathbf{q}=q{\widehat{\mathbf{e}}}_{x+y}$ (open symbols) and $C_{\delta \mathrm{\Gamma }}\left(\mathbf{q}\right)=\langle \delta {\mathrm{\Gamma }}_{\mathbf{q}}\delta {\mathrm{\Gamma }}_{-\mathbf{q}}\rangle /V$ at wave vectors $\mathbf{q}=q\widehat{\mathbf{y}}$ (solid symbols). Stress per particle is $\tilde{p}=\mathrm{\Gamma }/N=0.01831$ above jamming. Results are shown for systems with different number of particles $N$.

Figure 5

Fluctuation of the anisotropic part of the stress normalized by the stress per particle $\tilde{p}$ squared $C_{\delta \mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)/{\tilde{p}}^2$ vs $q$. Curves are for different values of $\tilde{p}=\mathrm{\Gamma }/N$, going from $\tilde{p}=0.00014$ on top to $0.01831$ on bottom. System has $N=65536$ particles. Vertical dashed lines delimit the range of $q$ where $C_{\delta \mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)$ is roughly constant.

Figure 6

Comparison of $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)/{\tilde{p}}^2$ and $C_{\delta \mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)/{\tilde{p}}^2$, vs $q$, for our smallest and largest stress per particle $\tilde{p}=0.00014$ and $0.01831$, respectively. At both values of $\tilde{p}$, the two correlations become exactly equal at small $q\lesssim 1$.

Figure 7

Fluctuation of the isotropic part of the stress averaged over a circular window of radius $R$ [see Eq. (18)], ${\mathrm{\Delta }}_{\mathrm{\Gamma }}\left(R\right)$ vs $R$, for systems with different number of particles $N$ at a total stress per particle $\tilde{p}=0.01831$.

Figure 8

Fluctuation of the isotropic part of the stress averaged over a circular window of radius $R, {\mathrm{\Delta }}_{\mathrm{\Gamma }}\left(R\right)/{\tilde{p}}^2$ vs $R$. Curves are for different values of the total system stress per particle $\tilde{p}=\mathrm{\Gamma }/N$, going from $\tilde{p}=0.00014$ on top to $0.01831$ on bottom. System has $N=65536$ particles.

Figure 9

Fluctuation of the anisotropic part of the stress $\delta \mathrm{\Gamma }$ averaged over a circular window of radius $R$ [see Eq. (18)], ${\mathrm{\Delta }}_{\delta \mathrm{\Gamma }}\left(R\right)$ vs $R$, for systems with different number of particles $N$ at a total stress per particle $\tilde{p}=0.01831$.

Figure 10

Fluctuation of the anisotropic part of the stress ${\mathrm{\Sigma }}_{xy}$ averaged over a circular window of radius $R$ [see Eq. (18)], ${\mathrm{\Delta }}_{{\mathrm{\Sigma }}_{xy}}\left(R\right)$ vs $R$, for systems with different number of particles $N$ at a total stress per particle $\tilde{p}=0.01831$.

Figure 11

Fluctuation of the anisotropic parts of the system stress averaged over a circular window of radius $R, {\mathrm{\Delta }}_{\delta \mathrm{\Gamma }}\left(R\right)/{\tilde{p}}^2$ and ${\mathrm{\Delta }}_{{\mathrm{\Sigma }}_{xy}}\left(R\right)/{\tilde{p}}^2$ vs $R$, for three different values of the total system stress per particle $\tilde{p}$. System has $N=65536$ particles.

Figure 12

Difference $[{\mathrm{\Delta }}_{\delta \mathrm{\Gamma }}\left(R\right)-{\mathrm{\Delta }}_{{\mathrm{\Sigma }}_{xy}}\left(R\right)]/{\tilde{p}}^2$ vs $R$ for three different values of the total system stress per particle $\tilde{p}$. System has $N=65536$ particles. Solid line has slope 2.

Figure 13

Fluctuation of the isotropic part of the stress $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)/{\tilde{p}}^2$ vs $q$, comparing results obtained when quenching from initial random configurations with ${\varphi }_{\mathrm{init}}=0.84$ with those obtained from ${\varphi }_{\mathrm{init}}=0.50$. Results are shown for final configurations at two different values of the stress per particle $\tilde{p}=\mathrm{\Gamma }/N=0.00014$ and $0.01831$. The system has $N=32768$ particles and results are averaged over roughly 1000 independent initial configurations.

Figure 14

Fluctuation of the (a) isotropic part of the stress $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)$ and (b) anisotropic part of the stress $C_{\delta \mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)$ vs $q$. Circles are for an ensemble of $N=65536$ particles at fixed stress, with $\tilde{p}=\mathrm{\Gamma }/N=0.01831$ and $\langle \varphi \rangle =0.8857$. Squares are for an ensemble of $N=77523$ particles at fixed square volume of side length $L=320$, with $\langle \tilde{p}\rangle =0.018$ and $\varphi =0.88$.

Figure 15

Fluctuation of the isotropic part of the stress $C_{\mathrm{\Gamma }}\left(\mathbf{q}\right)$ vs $q$ for $\mathbf{q}=q\widehat{\mathbf{y}}$ (squares) and for $\mathbf{q}=q{\widehat{\mathbf{e}}}_{x+y}$ (circles), for a binary Lennard-Jones system with effective packing fractions $\varphi =0.699$ and 0.80, in a fixed volume ensemble. Solid symbols are for a system of length $L=320$, open symbols are for $L=640$. Lengths are measured in units where small particles have diameter $d_s=1$, while big particles have $d_b=1.4$.

Figure 16

Plot of $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)=\langle {\mathrm{\Gamma }}_{\mathbf{q}}{\mathrm{\Gamma }}_{-\mathbf{q}}\rangle /V$ vs (a) $q$, (b) $q/{\tilde{p}}^{1/2}\sim q{\ell }_L$, and (c) $q/{\tilde{p}}^{1/4}\sim q{\ell }_T$, for different values of the total stress per particle $\tilde{p}=\mathrm{\Gamma }/N$ for a system with $N=65536$ particles. Curves are for $\tilde{p}=0.01831$ to $0.00014$ going top to bottom. The high $q$ data, where one sees the peak in Fig. 3, have been truncated since if there is critical scaling it would apply only to the long length scale, and so small $q$, region of the data. Horizontal solid lines extrapolate through the region where the curves $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)$ are approximately constant; the solid vertical line denotes the approximate point where $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)$ departs from this horizontal line, for the smallest $\tilde{p}=0.00014$.

Figure 17

Plot of $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)/{\tilde{p}}^2$ vs (a) $q$ and (b) $q/{\tilde{p}}^{1/4}\sim q{\ell }_T$, for our four smallest values of the total stress per particle $\tilde{p}=\mathrm{\Gamma }/N$ for a system with $N=65536$ particles. Only data for $q$ values below the peak (see Fig. 3) are shown. We see that the data collapse looks distinctly better when plotting vs $q$ than vs $q/p^{1/4}$.

Figure 18

Stress correlations (a) $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)$, (b) $C_{\delta \mathrm{\Gamma }}\left(q\widehat{\mathbf{q}}\right)$, and (c) $C_{{\mathrm{\Sigma }}_{xy}}\left(q\widehat{\mathbf{y}}\right)$ vs $q$ for $\tilde{p}=0.01831$ and $N=65536$ particles. Results are shown for different values of the minimization convergence parameter $\varepsilon ={10}^{-5}$ to ${10}^{-11}$.

Figure 19

Stress correlations (a) $C_{\mathrm{\Gamma }}\left(q\widehat{\mathbf{y}}\right)$, (b) $C_{\delta \mathrm{\Gamma }}\left(q\widehat{\mathbf{q}}\right)$, and (c) $C_{{\mathrm{\Sigma }}_{xy}}\left(q\widehat{\mathbf{y}}\right)$ vs $q$ for $\tilde{p}=0.00014$ and $N=65536$ particles. Results are shown for different values of the minimization convergence parameter $\varepsilon ={10}^{-5}$ to ${10}^{-10}$.

Figure 20

Stress correlations $C_{{\mathrm{\Sigma }}_{xy}}\left(q\widehat{\mathbf{y}}\right)$ and $C_{\delta \mathrm{\Gamma }}\left(q{\widehat{\mathbf{e}}}_{x+y}\right)$ vs $q$ for a constant volume ensemble at fixed packing fraction $\varphi =0.88$, corresponding to an average stress per particle $\langle \tilde{p}\rangle =0.018$. The system box has length $L=320$ and there are $N=77523$ particles.