Stress profile in a two-dimensional silo: Effects induced by friction mobilization

The effects of friction mobilization on the stress profile within a two-dimensional silo are investigated via simulations of discrete elements. Friction mobilization is driven by cyclic vertical displacement of the sidewalls. Two regimes have been observed for small filling height, with stress profiles identified as saturated (Janssen's profile) and exponentially growing. The transition between these regimes is denoted by an almost linear stress profile, similar to that of a hydrostatic system, with a significantly greater characteristic height compared to the height of the column of grains. For tall columns, the process of friction inversion is more complex. A partial inversion of friction mobilization is observed when the motion is reversed from upward to downward, which results in two coexisting zones of opposite mobilization. These zones are separated by a wide compaction front with a gradual upward progression sustained by the displacement of the walls. Conversely, if the motion is reversed, the two opposing friction mobilization zones retract, the transition zone becomes smooth, and the system rapidly transforms from two coexisting mobilization states to a Janssen-like regime. In both regimes, the general characteristics from the resulting stress profiles are depicted by generalizing Janssen's equation to include partial mobilization through the varying effective friction coefficient along the silo walls.

Figure 1

(a) Schematic of stress components and coordinate system. $z_w$ is the relative displacement between the sidewalls and the base, fixed at $z=0$. (b) Scheme for the model of linear compression and Mohr-Coulomb plasticity without cohesion. $k_n$ and $k_s$ are normal and shear stiffness and ${\mu }_p$ is the friction coefficient of particles.

Figure 2

Typical behavior of forces and packing, averaged over 20 realizations, as a function of time. Normal force on the sidewalls: left (a), right (b), and the base (c). Shear force on the sidewalls: left (d), right (e), and the base (f). The dashed lines indicate the weight of the column of grains $P_c\sim 2.3\mathrm{N}$. (g) The packing fraction $\varphi$. Insets: Relative grain displacement fields; 1: at walls position $P_2$ with respect to $P_1$; 2: at $P_4$ with respect to $P_3$; and 3: at $P_6$ with respect to $P_5$. Positions $P_1–P_6$ are indicated in (h). (h) Walls move following a triangular waveform: both sidewalls move upward and then downward in an alternate motion at a constant speed $v_w=9\mathrm{mm}/\mathrm{s}$. The horizontal axis is common to all graphs.

Figure 3

(a) Average vertical stress profiles ${\sigma }_{zz}$ obtained at the center of the column of grains, normalized to $\rho g\varphi$ which correspond to three different positions of the sidewalls $z_w\sim 0.60d$ (curve $A$), $z_w\sim 0.75d$ (curve $B$), and $z_w\sim 0.60d$ (curve $C$), and their respective fitted curves (continuous lines), calculated from Janssen's analysis. The dashed line with slope $\rho g\varphi$ represents the stress profile in the hydrostatic regime. (b) The characteristic length $l$ obtained by fitting Janssen's solution to the vertical normalized stress profiles. The vertical dashed lines indicate the point of transition from exponentially growing to saturating regime and vice versa. (c) The relative position of the sidewalls $z_w$ with respect to the position of the base after ninth cycle. (b) and (c) share the horizontal axis.

Figure 4

Characteristic curves for a large filling height, downward motion case. (a) $\mathrm{\Psi }=(\varphi -{\varphi }_{\mathrm{ref}})/{\varphi }_{\mathrm{ref}}\times 100$; ${\varphi }_{\mathrm{ref}}$ is the reference packing fraction at $z_w=9\mathrm{mm}$ (the Janssen's state). (b) $\kappa ={\sigma }_{xx}/{\sigma }_{zz}$. (c) Friction $\mu$ at the sidewalls and (d) ${\sigma }_{zz}$. Continuous curves in (a), (b), and (c) are guiding curves and do not correspond to a fit. In (d) the continuous curves are obtained by integrating Eq. (4) using $\mu$ as input. (Inset) Zoom of curves for very small displacement respect to the reference state. The dashed line represents the hydrostatic limit. The symbols indicate the height of the sidewalls: $z_w\sim 9\mathrm{mm}$ ($◯$), $z_w\sim 7\mathrm{mm}$ ($▽$), $z_w\sim 5\mathrm{mm}$ ($\square$), $z_w\sim 3\mathrm{mm}$ ($⊳$), $z_w\sim 2\mathrm{mm}$ ($\diamond$), and $z_w\sim 0\mathrm{mm}$ ($⊲$).The horizontal axis is common to all graphs. Arrows indicate profiles for decreasing positions on the wall.

Figure 5

Characteristic curves for a large filling height, upward motion case. (a) $\mathrm{\Psi }=(\varphi -{\varphi }_{\mathrm{ref}})/{\varphi }_{\mathrm{ref}}\times 100$; ${\varphi }_{\mathrm{ref}}$ is the reference packing fraction at $z_w\sim 9\mathrm{mm}$ (Janssen's state). (b) $\kappa ={\sigma }_{xx}/{\sigma }_{zz}$. (c) Friction $\mu$ at the sidewalls and (d) ${\sigma }_{zz}$. Continuous curves in (a), (b), and (c) are guiding curves and do not correspond to a fit. In (d) the continuous curves are obtained by integrating Eq. (4) using $\mu$ as an input. (Inset) Zoom of curves for very small displacement respect to the reference state. The dashed line represents the hydrostatic limit. The symbols indicate the relative displacement of the sidewalls: $z_w\sim 0\mathrm{mm}$ ($◯$), $z_w\sim 2\mathrm{mm}$ ($▽$), $z_w\sim 5\mathrm{mm}$ ($\square$), $z_w\sim 6\mathrm{mm}$ ($⊳$), $z_w\sim 7\mathrm{mm}$ ($\diamond$), and $z_w\sim 9\mathrm{mm}$ ($⊲$). The horizontal axis is common to all graphs. Arrows indicate profiles for increasing positions on the wall.

Figure 6

Effect of the aspect ratio on ${\sigma }_{zz}$: (a) in the unloading and (b) loading cycles. $R_a\sim 0.6$ ($◯$), $R_a\sim 3.2$ ($⊳$), $R_a\sim 6.4$ ($\square$), and $R_a\sim 9.6$ ($\diamond$). The $z$ axis is shared by (a) and (b).

Figure 7

$\sigma$ components as a function of $x$ for different positions of the sidewalls and $R_a\sim 9.6$. (a) ${\sigma }_{xx}$, (b) ${\sigma }_{zz}$, and (c) ${\sigma }_{xz}$. $z_w\sim 0,0.2,0.6,0.8,1$ ($◯, ⊳, \square , ▽, \diamond$). (a), (b), and (c) share the $x$ axis.