Granular elasticity: Stress distributions in silos and under point loads

An elastic-strain–stress relation, the result of granular elasticity as introduced in the preceding paper, is employed here to calculate the stress distribution (a) in cylindrical silos and (b) under point loads assuming uniform density. In silos, the ratio $k_J$ between the horizontal and vertical stress is found to be constant (as conjectured by Janssen) and given as $k_J=1-\sin \phi$ (with $\phi$ the Coulomb yield angle), in agreement with a construction industry standard usually referred to as the Jaky formula. Next, the stress distribution at the bottom of a granular layer exposed to a point force at its top is calculated. The results include both vertical and oblique point forces, which agree well with simulations and experiments using rainlike preparation. Moreover, the stress distribution of a sheared granular layer exposed to the same point force is calculated and again found in agreement with given data.

Figure 1

Stress components calculated using Eq. (3) and the force balance. The upper figure shows the vertical stress ${\sigma }_{zz}$; the middle figure shows Janssen’s ratio, $k_J={\sigma }_{rr}∕{\sigma }_{zz}$ (note the amplified vertical scale); and the lowest figure gives ${\sigma }_{rz}$ at different radii—all as functions of the depth $z$.

Figure 2

(a) The Janssen ratio $k_J$ increases only slightly with the wall friction ${\mu }_f$ at constant $\xi$, and ends at the boundary (thick curve) determined by the yield condition. The dotted line is given by the IF model, with $\phi =30°$. (b) Variation of $k_J$ with the yield angle $\phi$, given, respectively, by the Rankine states (solid lines), the Jaky formula (dotted line), and the present theory (circles).

Figure 3

(a) Bold line shows the normal stress $C$ at the bottom calculated from granular elasticity assuming glued grains at the bottom. “LE” stands for the same calculation, though employing isotropic linear elasticity, and “Boussinesq-Cerruti” for the analytic solution of an infinite half-space of a linearly elastic body. (b) The same three curves renormalized by their values at $r=0$; symbols are the measured data of Ref. 5. Inset shows the geometry.

Figure 4

Symbols: Variations of the half-width with the rescaled gravity constant for the two boundary conditions discussed in the text, as computed using granular elasticity. The experimental half-width of 0.94 was measured for $\tilde{G}\equiv \rho hG∕P_1$ varying between 1 and 3.2, as indicated.

Figure 5

Computed bottom pressure (full lines) for localized top force, either vertical (upper figure) or inclined by 45° (lower figure). The dashed lines are 2D simulation, as taken from Fig. 10 of Ref. 13, for rainlike pouring.

Figure 6

Sketch of the 3D shear box. The stress ${\sigma }_{ij}$ is diagonal in the coordinate system 1, 2, 3.

Figure 7

$1∕t\equiv E_1∕E_2$ versus $u\equiv E_2∕G_{12}$, as calculated employing granular elasticity (solid line), and as found by Atman et al., Fig. 7 from Ref. 14 (dashed line), for the 3D shear box. The dashed line represents all points of a linear anisotropic elastic theory compatible with the measured tilt angle of $\beta =8^{\circ }$. The solid line takes $F_s$ as a parameter, with which both $u$ and $t$ decrease monotonically. The lower right point of the solid line, $1∕t=1.38,u=2.18$, corresponds to $F_s=0$, and the upper left-hand point, $1∕t=5.31,u=0.1$ corresponds to yield, $F_s=F_s^{\max }$. Both lines intersect at $1∕t=1.61$, $u=1.83$, which corresponds to $F_s=0.54\times F_s^{\max }$.

Figure 8

The dimensionless Young moduli $E_i^{\star }$, ${\mu }^{\star }$ and the Poisson ratios ${\nu }_{ik}$ as functions of $F_s∕F_s^{\max }$, where $E_i=E_i^{\star }\mathcal{A}{(P_0∕\mathcal{A})}^{1∕3}$ and $\mu ={\mu }^{\star }\mathcal{A}{(P_0∕\mathcal{A})}^{1∕3}$.

Figure 9

Normalized pressure distribution ${\sigma }_{xx}(x=0,y)∕{\sigma }_{xx}(x=0,y=0)$ at the bottom of the sheared box. Solid curve, results of numerical calculations. Dashed line is plotted through the maximum of the solid curve. It is shifted by $0.05y∕H$ along the shear direction. Symbols, the experimental data extracted from Ref. 13.