Undulating compression and multistage relaxation in a granular column consisting of dust particles or glass beads

For fundamentally characterizing the effect of hierarchical structure in granular matter, a set of compression-relaxation tests for dust particles and glass beads confined in a cylindrical cell was performed. The typical diameter of both grains is approximately 1 mm. However, dust particles are produced by binding tiny ($\sim 5\mu \mathrm{m}$) glass beads. The granular columns were compressed with a piston until reaching a maximum load force of 20 N with a constant compression rate $v$ ($0.17\le v\le 2000\mu \mathrm{m}{\mathrm{s}}^{-1}$). After that, the piston was stopped and the relaxation process was quantified. From the experimental results, we found that the compression force $F$ nonlinearly increases with the increase of compression stroke $z$ depending on particles. Besides, periodic undulation and sudden force drops were observed on $F\left(z\right)$ in dust particles and glass beads, respectively. The relaxation process was characterized by an exponential decay of stress followed by a logarithmic dependence one in both kinds of particles. These experimental findings are the main point in this study. To understand the underlying physics governing the compression mechanics, we assumed empirical forms of $F\left(z\right)$; $F\propto z^{\alpha }$ for dust particles and $F\propto exp(z/z_G)$ for glass beads ($\alpha =2.4$ and $z_G=70\mu \mathrm{m}$). Then, we found that the growing manners of periodic undulation and force drops were identical to those of mean compression forces, i.e., power law in dust particles and exponential in glass beads. In addition, the undulation amplitude and wavelength decreased as $v$ increased in dust-particle compression. On the basis of experimental results and the difference between dust particles and glass beads, we also discuss the origin of undulation and the physical meaning of granular-compression models used in engineering fields.

Figure 1

(a) Experimental setup. (b) Compression of an individual dust particle showing the appearance of a fracture. (c) Snapshots showing the compression of a granular column composed of dust particles.

Figure 2

Compression force $F$ as a function of time $t$ measured at different compression rates followed by system relaxation for (a) dust particles and (b) glass beads. In these plots, $t=0$ corresponds to the moment at which the maximum compression load $F_{\max }=20$ N is attained by increasing the stroke $z$ from 0 to $z_{\max }$. After $t=0$, the stroke is fixed and the force is measured as a function of time $t$.

Figure 3

Force $F$ vs stroke $z$ for a dust-particle-column compression. Color in (a) indicates the compression rate $v$ as denoted in the legend. The inset of (a) displays the magnified view of periodic undulations. Some data are not shown to clearly show the $v$-dependent amplitude. A typical example of $F\left(z\right)$ ($v=83\mu \mathrm{m}{\mathrm{s}}^{-1}$) is presented with its power-law fit (red dashed curve) in (b). The inset of (b) shows the development of the periodic undulation with a power-law envelope fit (black dotted curve).

Figure 4

Compression rate $v$ dependencies of the parameters characterizing $F\left(z\right)$ curves in dust-particle compression. (a) Power-law exponents for bulk compression and periodic undulation ($\alpha$ and ${\alpha }_p$) are almost independent of $v$; $\alpha \simeq {\alpha }_p=2.4\pm 0.7$. (b) Force coefficient for bulk compression $F_{*}$ shows large fluctuation around $F_{*}\simeq {10}^{-2}$ N. (c) Force coefficient characterizing periodic undulation $F_{*p}$ shows power-law decay depending on $v$; $F_{*p}\propto v^{-\beta }$ with $\beta =0.4\pm 0.1$. (d) Dominant wavelength of periodic undulation $\lambda$ logarithmically decreases with increasing $v$; $\lambda ={\lambda }_{*}ln(v_{\mathrm{up}}/v)$, where ${\lambda }_{*}=0.09\pm 0.01$ mm and $v_{\mathrm{up}}=24\pm 6$ mm ${\mathrm{s}}^{-1}$. The inset in (d) shows the power spectrum of the undulation shown in the inset of Fig. 3. Error bars indicate the fitting uncertainty. Error bars of red data points in (a) and (b) are omitted because the errors are smaller than the size of symbols.

Figure 5

Raw data set for glass-bead-column compression. $F\left(z\right)$ curves with various compression rates are plotted in (a). (b) shows the log-linear plot of the same $F\left(z\right)$ data. $F\left(z\right)$ curve with $v=17\mu \mathrm{m}{\mathrm{s}}^{-1}$ is presented in (c). The red dashed curve indicates the exponential fit. The inset in (c) shows the fluctuation part of the compression force with an exponential envelope fit (black dotted curve).

Figure 6

The values of fitting parameters for glass-bead-column compression tests are presented as functions of the compression rate $v$. (a) Characteristic length scales ($z_G$ and $z_{Gp}$) for exponential growths of bulk compression force and fluctuation amplitude are assumed to be roughly identical values independent of $v$; $z_G\simeq z_{Gp}=0.07\pm 0.01$ mm. (b) The force coefficient value $F_{*G}$ is plotted as a function of $v$. Very large scattering of $F_{*G}$ (over 6 orders of magnitude) can be confirmed. (c) The dominant wavelength ${\lambda }_G$ for the fluctuation part is also independent of $v$; ${\lambda }_G=0.04\pm 0.02$ mm. The inset of (c) shows the power spectrum of the fluctuation shown in the inset of Fig. 5. Errors computed from the fitting uncertainty are shown if the scale of error is greater than the symbol's size.

Figure 7

Snapshots obtained from DIC analysis applied to consecutive frames of videos of dust particles compressed at (a) $v=17$ and (b) $2000\mu \mathrm{m}{\mathrm{s}}^{-1}$. The snapshots were taken at 0.073 frames ${\mathrm{s}}^{-1}$ and 5 frames ${\mathrm{s}}^{-1}$ in (a) and (b), respectively, and the scale color bar indicates the displacement in mm ranging from 0 to 0.25 mm of each vector (represented by a color arrow). The corresponding maximum velocity is 18 and 1250 $\mu \mathrm{m}{\mathrm{s}}^{-1}$, respectively. A three-step iterative analysis was applied considering interrogation window sizes of 512, 256, and 128 pixels in each case.

Figure 8

(a) Direct comparison of $F/F_{\max }$ vs $t$ for dust particles and glass beads after being compressed until $F_{\max }=20$ N at $v=17\mu \mathrm{m}{\mathrm{s}}^{-1}$. [(b) and (c)] $F/F_{\max }$ as a function of time for dust particles and glass beads, respectively. The black lines correspond to the data fit given by Eq. (3) with the parameters reported in Table 1. Gray dotted curves show the fitting with ${\tau }_1=\infty$.

Figure 9

Stress-strain relations for single dust particle compression tests. Particle size (diameter) was measured with a vernier caliper. Color indicates the particle size as labeled in the legend. Compression rate is basically fixed to $v=1.7\mu \mathrm{m}{\mathrm{s}}^{-1}$ except for 6.0 mm size particle case ($v=17\mu \mathrm{m}{\mathrm{s}}^{-1}$).

Figure 10

Stress-strain relations for periodic undulations in dust-particle-column compression. The color code used is identical to Fig. 3. Definition of strain is same as the single particle compression.

Figure 11

Plot for evaluating the compression index model. Data from (a) dust-particle compression and (b) glass-bead compression are presented. The data are identical to those shown in Figs. 3 and 5. Dotted lines are displayed as guides to the eye. The steep angle in small $p$ of (b) comes from the subtlety in the initial stage of glass-bead compression (Fig. 5).

Figure 12

Heckel model plot of (a) dust-particle compression and (b) glass-bead compression. The data are identical to those shown in Figs. 3 and 5. Dotted lines are shown as guides to the eye.

Figure 13

Force curve of the compression test of monomers. Tiny ($5\mu \mathrm{m}$) glass beads were used as monomer particles (${\varphi }_{\mathrm{bulk}}\simeq 0.3$). Monomers are simply poured into the cylindrical cell without granulation process. Neither periodic undulation nor sudden force drops are observed during the compression. The inset shows the logarithmic-linear plot of the same data.