Eshelby inclusions in granular matter: Theory and simulations

We present a numerical implementation of an active inclusion in a granular material submitted to a biaxial test. We discuss the dependence of the response to this perturbation on two parameters: the intragranular friction coefficient on one hand, and the degree of the loading on the other hand. We compare the numerical results to theoretical predictions taking into account the change of volume of the inclusion as well as the anisotropy of the elastic matrix.

Figure 1

Setup for the numerical simulations. $N$ grains are contained within four walls; the bottom and left walls are fixed, and the upper wall moves downward with constant velocity. The right wall is mobile and subjected to a constant pressure $p_0$. Throughout the paper, the coordinate axes $x$ and $y$ are oriented as shown.

Figure 2

Stress-strain curves for the simulations analyzed in this paper.

Figure 3

Volumetric strain ${\varepsilon }_{xx}+{\varepsilon }_{yy}$ for the tests studied in this paper.

Figure 4

Polar representation of the function $f\left(\theta \right)$ [Eq. (5)]: (a) contractant event, $e_{xx}^{*}=0.3$ and $e_{yy}^{*}=-0.7$; (b) dilatant event, $e_{xx}^{*}=0.7$ and $e_{yy}^{*}=-0.3$.

Figure 5

Two contacts inside the region $\mathbb{V}$ where the stress is being calculated: upper contact ($\alpha =1$), both grains lie within $\mathbb{V}$, and ${\mathbf{r}}_1$ is the line of centers; lower contact ($\alpha =2$), only one grain lies within $\mathbb{V}$, and ${\mathbf{r}}_2$ extends from this grain, along the line of centers, to the boundary of $\mathbb{V}$.

Figure 6

Examples of generated Eshelbies, $N=256\times 256$ grains. Top row, ${\mu }_m=2$; middle row, ${\mu }_m=0.6$; bottom row, ${\mu }_m=0.2$. ${\mathbf{\sigma }}^{*}$ is coaxial with the load; white grains correspond to a reduction in deviatoric stress. The value of the global strain is given under each panel.

Figure 7

An estimate of the angular dependence of the perturbation stress $f\left(\theta \right)$. The stress on grains is averaged over radial bins and then rescaled to estimate $f\left(\theta \right)$. This example is for ${\mu }_m=0.6, {\varepsilon }_{yy}={10}^{-5}$.

Figure 8

Angle of diagonal bands for simulations with different friction coefficients. For each value of the strain, four angles are averaged together to obtain a mean. The error bars give an idea of the scatter of the different angles measured. The horizontal line indicates an angle of ${45}^{\circ }$.

Figure 9

Harmonic components obtained from the fits obtained using Eq. (17): (a) ${\mu }_m=2$, (b) ${\mu }_m=0.6$, and (c) ${\mu }_m=0.2$.