Role of anisotropy in the elastoplastic response of a polygonal packing

We study the effect of the anisotropy induced by loading on the elastoplastic response of a two dimensional discrete element model granular material. The anisotropy of the contact network leads to a breakdown of the linear isotropic elasticity. We report on a linear dependence of the Young moduli and Poisson ratios on the fabric coefficients, measuring the anisotropy of the contact network. The resulting nonassociated plastic flow rule and the linear relationship between dilatancy and stress ratio are discussed in terms of several existing models. We propose a paradigm for understanding soil plasticity, based on the correlation between the plastic flow rule and the induced anisotropy on the subnetwork of sliding contacts.

Figure 1

Left: Voronoi construction used to generate the convex polygons. The dots indicate the point used in the tessellation. Periodic boundary conditions were used. Right: Typical texture of marble. (Courtesy of Royer-Carfagni [17].)

Figure 2

Contact points $C_i$ before (left) and after the formation of a pathological contact (right). The vector denotes the contact line. $t$ represents the number of time steps.

Figure 3

Deviatoric stress and volumetric strain versus axial strain for different values of $p∕k_n$, where $p$ is the lateral pressure. $e>0$ represents compression of the sample.

Figure 4

Stress-strain relation resulting from the load-unload test. Grey solid lines are the paths in the stress and strain spaces. Grey dash-dotted lines represent the yield direction (upper) and the flow direction (lower). Dashed line shows the strain envelope response and the solid line is the plastic envelope response.

Figure 5

Elastic strain envelope responses $d{\widetilde{ϵ}}^e\left(\theta \right)$. They are calculated for a pressure $p=0.001k_n$ and taking deviatoric stresses with $q=0.0p$ (inner), $0.1p,\dots ,0.7p$ (outer).

Figure 6

The lines show the branch vectors of the contact network for ${\sigma }_1={\sigma }_3=0.001k_n$ (a) and ${\sigma }_1=0.65\times {10}^{-3}{k}_n$ and ${\sigma }_3=0.35\times {10}^{-3}{k}_n$ (b). The branch vectors of the subnetwork of sliding contacts are shown for the isotropic (c) and the anisotropic (d) case. The width of the lines represents the normal force.

Figure 7

Left: force distribution for the stress ratios $q∕p=0.1$, 0.35, and 0.65. Here $f_t$ and $f_n$ are the tangential and normal components of the force. They are normalized by the mean value of $f_n$. Right: orientational distribution of the contacts $\Omega \left(\phi \right)$ (outer) and of the sliding contacts ${\Omega }^s\left(\phi \right)$ (inner). $\phi$ represents the orientation of the branch vector.

Figure 8

Fabric coefficients of the contact network. They are defined in Eq. (27). The lines show the spline interpolation of the data.

Figure 9

Fabric coefficients of the distribution of the branch vectors of the sliding contacts. They are defined in Eq. (28). The lines show the spline interpolation of the data.

Figure 10

Young moduli. The lines are the linear fits of $E\left(n_i\right)$ according to Eq. (33).

Figure 11

Poisson ratios. The lines are the linear approximation of $\nu \left(n_i\right)$. See Eq. (33).

Figure 12

Plastic components ${\kappa }_1\left(\theta \right)$ (circles) and ${\kappa }_2\left(\theta \right)$ (dots) given by Eq. (38). The results for different stress values have been superposed. The solid line represents the truncated cosine function.

Figure 13

Dilatancy versus the stress ratio. The solid curve represents a fit with the Gutierrez and Ishihara model. The dashed curve represents the relation given by the Nova and Wood model.

Figure 14

Plastic envelope response resulting from isotropically compressed samples with a pressure $p=0.001k_n$.

Figure 15

The flow direction and the yield direction of the plastic response. Solid curves represent a fit using Eqs. (46, 47).

Figure 16

Inverse hardening modulus $h$ versus fraction of sliding contacts $n_s$. Different stress values with $q=0.01p,0.1p,\dots 0.7p$ and $p∕k_n=0.001$, 0.002, 0.004, and 0.008. The lowest value of $n_s$ corresponds to $q=0.01p$.