Microstructural Origin of Propagating Compaction Patterns in Porous Media

Porous rocks, foams, cereals, and snow display a diverse set of common compaction patterns, including propagating or stationary bands. Although this commonality across distinct media has been widely noted, the patterns’ origin remains debated—current models employ empirical laws for material-specific processes. Here, using a generic model of inelastic structured porous geometries, we show that the previously observed patterns can be attributed to a universal process of pore collapse. Furthermore, the pattern diversity can be mapped in a phase space of only two dimensionless numbers describing material strength and loading rate.

Figure 1

Typical compaction patterns emanating from a uniaxially loaded confined two-dimensional structured porous media, here with a solid area fraction of $\varphi =0.40$ and the following parameters: (a) $S=4\times {10}^{-3}$, $R={10}^{-5}$, (b) $S=4\times {10}^{-3}$, $R=4\times {10}^{-4}$, (c) $S=4\times {10}^{-2}$, $R=2\times {10}^{-2}$, (d) $S=1.32\times {10}^{-3}$, $R=4\times {10}^{-3}$, (e) $S={10}^{-1}$, $R={10}^{-5}$, (f) $S=4\times {10}^{-3}$, $R={10}^{-1}$. Contour plots show the spatiotemporal values of plastic Hencky strain rate ${\dot{\overline{ϵ}}}^P=\sqrt{{\dot{\mathit{ϵ}}}^P:{\dot{\mathit{ϵ}}}^P}$ as the samples are compressed until fully dense from the top under a growing nominally imposed strain $\epsilon =Vt/h_0$. Values on those plots are obtained by averaging over the horizontal direction, neglecting 10% of the area closest to the smooth sidewalls. A snapshot of the structure at $\epsilon =0.2$ is shown to the left of each case, colored in terms of the local ${\dot{\overline{ϵ}}}^P$. An enlarged view is shown below, thus highlighting the state of deformation. Supplemental Movie 1 displays the evolution of the plastic strain rate in the six structures until $\epsilon =0.6$. The stress $\sigma$ at the bottom and top plates are plotted below the corresponding contour plots.

Figure 2

Compaction patterns classified in the space of $S$ and $R$ for a structure with a solid area fraction $\varphi =0.40$. Note the separation of class (b) based on the number of reflections $N_r$ the compaction band displays. The background colors represent the type of deformation mechanism on the microscale.

Figure 3

Speed $c_b$ of the (prereflection) compaction bands, normalized by the imposed compressive speed $V$, as a function of $q_y/(\rho V^2)$ in log-log scale. The data points are colored according to the number of reflections $N_r$ of the compaction band. Circular and square markers correspond to regular and random structures, respectively.