Stress Chain Solutions in Two-Dimensional Isostatic Granular Systems: Fabric-Dependent Paths, Leakage, and Branching

The stress field equations for two-dimensional disordered isostatic granular materials are reformulated, giving new results beyond the commonly accepted force chains. Localized loads give rise to exactly determinable cones of influence, bounded by stress chains. Disorder couples same-source chains, attenuates stresses along chains, causes stress leakage from chains into the cone, and gives rise to branching. Chains from spatially separate sources do not interact. The formulation is convenient for computation, and several numerical solutions are presented.

Figure 1

The full numerical solution for the stress field in a semi-infinite plane, whose fabric tensor is given by (11), under a narrow Gaussian-shaped source of ${\sigma }_{xy}$ applied to the boundary $x=0$. We show contour plots of (a) ${\sigma }_{xy}$ and (b) ${\sigma }_{xx}$. Contour colors give the stress magnitude. Note that ${\sigma }_{xx}$ is nonzero even with no boundary load in this component.

Figure 2

The full numerical solutions for the stresses in a semi-infinite plane, when $P$ is given by (14) and a narrow Gaussian-shaped ${\sigma }_{xy}$ is applied to the boundary. Contour colors give stress magnitudes. (a) For ${ϵ}_{ij}=0.1$, the stress paths deviate only slightly from straight, and the stresses along them hardly attenuate or leak. (b) For ${ϵ}_{ij}=0.5$, the cone of influence is still well defined, but both attenuation and leakage increase significantly. Because of the coupling, the stress at any point within the cone, e.g., $A$, is determined by the two sections of the main stress paths that extend between the source and points $B_1$ and $B_2$, respectively.

Figure 3

Applying a narrow Gaussian ${\sigma }_{xx}$ at the boundary point $(0,0)$ of a system with a strong gradient in $P$ within a thin midlayer between the dashed lines, sandwiched between two regions of (different) constant $P$, results in branching of the stress paths. The strength of a branch depends on the gradients in $P$ and the characteristic variables along the branch. In this case, only the lower path leads to a noticeable branch. The colored contours give the magnitude of ${\sigma }_{xy}$.

Figure 4

An example of a stress solution, developing in an annulus of constant fabric tensor under shear. Three localized load sources of ${\sigma }_{r\theta }$ are applied along the inner boundary, giving rise to three pairs of stress paths, each forming a region of influence that resembles the cones of the semi-infinite plane. The colored contours give the magnitude of the displayed stresses.