Critical scaling near the yielding transition in granular media

We show that the yielding transition in granular media displays second-order critical-point scaling behavior. We carry out discrete element simulations in the low-inertial-number limit for frictionless, purely repulsive spherical grains undergoing simple shear at fixed nondimensional shear stress $\mathrm{\Sigma }$ in two and three spatial dimensions. To find a mechanically stable (MS) packing that can support the applied $\mathrm{\Sigma }$, isotropically prepared states with size $L$ must undergo a total strain ${\gamma }_{\mathrm{ms}}(\mathrm{\Sigma },L)$. The number density of MS packings ($\propto {\gamma }_{\mathrm{ms}}^{-1}$) vanishes for $\mathrm{\Sigma }>{\mathrm{\Sigma }}_c\approx 0.11$ according to a critical scaling form with a length scale $\xi \propto |\mathrm{\Sigma }-{\mathrm{\Sigma }}_c{|}^{-\nu }$, where $\nu \approx 1.7–1.8$. Above the yield stress ($\mathrm{\Sigma }>{\mathrm{\Sigma }}_c$), no MS packings that can support $\mathrm{\Sigma }$ exist in the large-system limit $L/\xi \gg 1$. MS packings generated via shear possess anisotropic force and contact networks, suggesting that ${\mathrm{\Sigma }}_c$ is associated with an upper limit in the degree to which these networks can be deformed away from those for isotropic packings.

Figure 1

Schematics of the simulation procedure for (a) 3D simple shear and (b) 2D simple shear. In panels (a) and (b), MS packings are first created under only a fixed normal force per area $-p\widehat{y}$. We then apply a shear force per area $\tau \widehat{x}$ and search for an MS packing at a given $\mathrm{\Sigma }=\tau /p$ and length $L$ in units of the small grain diameter $D$. (c) A schematic of the simulation procedure for the river-bed-like geometry in 2D. MS packings are first created via sedimentation under gravity $-g\widehat{y}$, then driven by a fluid-like drag force in the $x$ direction.

Figure 2

Distributions $P\left({\gamma }_{\mathrm{ms}}\right)$ of the strain ${\gamma }_{\mathrm{ms}}$ in 3D simple shear between the initial and final MS packings for (a) $\mathrm{\Sigma }<{\mathrm{\Sigma }}_c$, (b) $\mathrm{\Sigma }>{\mathrm{\Sigma }}_c$, and several $L$. (c) The mean strain $\langle {\gamma }_{\mathrm{ms}}\rangle$ in 3D simple shear between initial and final MS packings plotted versus system size $N=L^3$ for several values of applied stress $\mathrm{\Sigma }$. Solid (dashed) lines correspond to $|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|/{\mathrm{\Sigma }}_c$ less (greater) than 0.5. Error bars are the standard error of the mean, given by the standard deviation within the sample divided by the square root of the number of trials. (d) The data from panel (c), plus additional data for more $\mathrm{\Sigma }$, collapses when using the scaled variables ${\langle {\gamma }_{\mathrm{ms}}\rangle }^{-1}/{|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|}^{\beta }$ and $L^{-1}/{|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|}^{\nu }$, where $\beta /\nu =0.56, \nu =1.7$, and ${\mathrm{\Sigma }}_c=0.109$. The inset shows ${\langle {\gamma }_{\mathrm{ms}}(\mathrm{\Sigma },L)\rangle }^{-1}$ versus $\mathrm{\Sigma }$ for different $L$. The dashed line gives the large-system limit implied by Eq. (7) and the main plot in panel (d). The data in the inset are at constant $L$, not constant $L^{-1}/{|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|}^{\nu }$; see text and Appendix pp3 for further discussion.

Figure 3

(a) The mean strain $\langle {\gamma }_{\mathrm{ms}}\rangle$ in 2D simple shear between initial and final MS packings plotted versus system size $N=L^2$ for several values of applied stress $\mathrm{\Sigma }$. Solid (dashed) lines correspond to $|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|/{\mathrm{\Sigma }}_c$ less (greater) than 0.5. Error bars are the standard error of the mean, given by the standard deviation within the sample divided by the square root of the number of trials. (b) The data from panel (a), plus additional data for more $\mathrm{\Sigma }$, collapses when using the scaled variables ${\langle {\gamma }_{\mathrm{ms}}(\mathrm{\Sigma },L)\rangle }^{-1}/{|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|}^{\beta }$ and $L^{-1}/{|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|}^{\nu }$. The collapse shown uses $\beta /\nu =0.57, \nu =1.8$, and ${\mathrm{\Sigma }}_c=0.111$.

Figure 4

The mean grain displacement $\langle {\delta }_{\mathrm{ms}}\rangle$ in the 2D river-bed-like geometry between initial and final MS packings plotted versus system size $N=5L$ for several values of applied stress ${\mathrm{\Sigma }}^{\prime }$. Solid (dashed) lines correspond to $|{\mathrm{\Sigma }}^{\prime }-{\mathrm{\Sigma }}_c^{\prime }|/{\mathrm{\Sigma }}_c$ less (greater) than 0.5. Error bars are the standard error of the mean, given by the standard deviation within the sample divided by the square root of the number of trials. (b) The data from panel (a) collapses when using the scaled variables ${\langle {\delta }_{\mathrm{ms}}({\mathrm{\Sigma }}^{\prime },L)\rangle }^{-1}/{|{\mathrm{\Sigma }}^{\prime }-{\mathrm{\Sigma }}_c^{\prime }|}^{{\beta }^{\prime }}$ and $L^{-1}/{|{\mathrm{\Sigma }}^{\prime }-{\mathrm{\Sigma }}_c^{\prime }|}^{\nu }$. The collapse shown uses ${\beta }^{\prime }/\nu =1.7, \nu =1.75$, and ${\mathrm{\Sigma }}_c^{\prime }=0.41$.

Figure 5

(a) The packing fraction ${\varphi }_{\mathrm{ms}}$ of MS packings at jamming onset as a function of applied shear stress $\mathrm{\Sigma }$. Different colors represent different system sizes. ${\varphi }_{\mathrm{ms}}$ is independent of $\mathrm{\Sigma }$, and approaches ${\varphi }_J\approx 0.643$ for large system sizes. (b) The same data from panel (a) plotted instead as a function of system size $L$. The different symbols represent different values of $\mathrm{\Sigma }$, but the curves lie one top of one another. The inset to panel (b) shows ${\varphi }_J-{\varphi }_{\mathrm{ms}}$ versus $L$ plotted on a double logarithmic scale. The solid black line has slope $-1.2$, implying that ${\nu }_J\approx 0.8$ if $({\varphi }_J-{\varphi }_{\mathrm{ms}})\sim L^{-1/\nu }$.

Figure 6

Close-up of MS packings in 2D illustrating features of the (a) stress and (e) fabric tensors for the central grain. ${\mathbf{\sigma }}_1, {\mathbf{\sigma }}_2$ and ${\mathbf{R}}_1, {\mathbf{R}}_2$ denote the eigenvalue-eigenvector pairs from the sum over the contacts (blue and green arrows) for the center grain $i$ [Eqs. (8) and (9)]. The magnitudes of the arrows are proportional to the eigenvalues and the directions are along the eigenvectors. ${\theta }^{\prime }$ is the angle between the larger eigenvector and the compressive direction. (b), (f) Ensemble averages of ${\sigma }_{xy}/{\sigma }_{yy}$ and $R_{xy}/R_{yy}$ for MS packings are plotted versus $\mathrm{\Sigma }$ for varying $L$, showing ${\sigma }_{xy}/{\sigma }_{yy}=-\mathrm{\Sigma }$ and $R_{xy}/R_{yy}\approx -0.4\mathrm{\Sigma }$ for all $L$. Ensemble averages of the normal anisotropies in the $x$ direction in the (c) stress tensor, ${\sigma }_{xx}/{\sigma }_{yy}-1\equiv {\lambda }_x$, and (g) fabric tensor, $R_{xx}/R_{yy}-1\equiv {\rho }_x$, are plotted versus $\mathrm{\Sigma }$ for varying $L$. Ensemble averages of the normal anisotropies in the $z$ direction in the (d) stress tensor, ${\sigma }_{zz}/{\sigma }_{yy}-1\equiv {\lambda }_z$, and (h) fabric tensor, $R_{zz}/R_{yy}-1\equiv {\rho }_z$, are plotted versus $\mathrm{\Sigma }$ for varying $L$.

Figure 7

Data for the scaled ${\langle {\gamma }_{\mathrm{ms}}\rangle }^{-1}$ versus scaled $(\mathrm{\Sigma }-{\mathrm{\Sigma }}_c)$ using the scaling function in Eq. (B1) for (a) 3D simple shear and (b) 2D river-bed geometries.

Figure 8

The critical values (a) $\nu$, (b) ${\mathrm{\Sigma }}_c$, (c) $\beta /\nu$ from the Levenberg–Marquardt method for 3D simple shear are plotted versus the minimum system length for various intervals about ${\mathrm{\Sigma }}_c$. Error bars represent one standard deviation. The ${\chi }^2/n$ values for each fit are plotted in panel (d).

Figure 9

The critical values (a) $\nu$, (b) ${\mathrm{\Sigma }}_c$, (c) $\beta /\nu$ from the Levenberg–Marquardt method for the 2D river-bed-like geometry are plotted versus the minimum system length for various intervals about ${\mathrm{\Sigma }}_c$. Error bars represent one standard deviation. The ${\chi }^2/n$ values for each fit are plotted in panel (d).

Figure 10

A plot of ${\langle {\gamma }_{\mathrm{ms}}(\mathrm{\Sigma },L)\rangle }^{-1}$ versus $\mathrm{\Sigma }$ for different values of ${\tilde{L}}^{-1}\equiv L^{-1}/{|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|}^{\nu }$. Data shown are within 10% of the relevant value of ${\tilde{L}}^{-1}$ (e.g., data for ${\tilde{L}}^{-1}=100$ include $90<{\tilde{L}}^{-1}<110$). The solid curves are ${\langle {\gamma }_{\mathrm{ms}}(\mathrm{\Sigma },L)\rangle }^{-1}=A{|\mathrm{\Sigma }-{\mathrm{\Sigma }}_c|}^{\beta }$, where $A$ is equal to the value of $f_{+}$ (for $\mathrm{\Sigma }>{\mathrm{\Sigma }}_c$) or $f_{-}$ (for $\mathrm{\Sigma }<{\mathrm{\Sigma }}_c$) at that particular value of ${\tilde{L}}^{-1}$. The values of $A$ can be determined from the scaling collapse in Fig. 2.