Compaction Model for Highly Deformable Particle Assemblies

The compaction behavior of deformable grain assemblies beyond jamming remains bewildering, and existing models that seek to find the relationship between the confining pressure $P$ and solid fraction $\varphi$ end up settling for empirical strategies or fitting parameters. Using a coupled discrete-finite element method, we analyze assemblies of highly deformable frictional grains under compression. We show that the solid fraction evolves nonlinearly from the jamming point and asymptotically tends to unity. Based on the micromechanical definition of the granular stress tensor, we develop a theoretical model, free from ad hoc parameters, correctly mapping the evolution of $\varphi$ with $P$. Our approach unveils the fundamental features of the compaction process arising from the joint evolution of grain connectivity and the behavior of single representative grains. This theoretical framework also allows us to deduce a bulk modulus equation showing an excellent agreement with our numerical data.

Figure 1

Snapshots of the frictionless sample under a relative pressure (a) $P/E=5\times {10}^{-4}$ and (b) $P/E=5\times {10}^{-1}$. The color intensities are proportional to the volumetric deformation within the particles.

Figure 2

Evolution of solid fraction $\varphi$ as a function of $P/E$ for different value of coefficient of friction. We present our data along with four models set to best fit the frictionless case. Linear scale is shown in the inset.

Figure 3

The reduced coordination number $Z-Z_0$ as a function of the reduced solid fraction $\varphi -{\varphi }_0$ for all the coefficient of friction tested (log-log representation in the inset). The dashed black line displays the power law relation $Z-Z_0=k(\varphi -{\varphi }_0{)}^{\alpha }$.

Figure 4

Measured modulus $K$ normalized by $E$ (empty symbols) along with its microstructural origin proposed in Eq. (3) (full symbols). The gray line is the prediction using Eq. (2) for $\mu =0$, and dashed black and orange lines are predictions using Eq. (7) for $\mu =0$ and $\mu =0.8$, respectively.

Figure 5

Compaction curve of a single particle for different mesh resolutions. Red squares ($P_p$ fixed and E varied) and black circles ($E$ fixed and $P_p$ varied) are tests on a particle with 92 finite elements. For the other mesh resolutions, $E$ was fixed and $P_p$ varied. Snapshots for $N_e=968$ display the volumetric strain at different moments of the compression.

Figure 6

Compaction curve (same as Fig. 2) along with Eq. (6) for $\mu =0$ (black dashed line) and $\mu =0.8$ (orange dashed line). The inset presents the data in linear scale.