[SOFTBITES] Elastogranularity and how soil shapes the roots of plants + Addendum

One of my articles was recently published by the great people at softbites.org!

We are Ph.D. students and postdocs from all over the world. Writing for Softbites is a way of sharing our passion for soft matter. We would especially love to attract younger students to research by explaining fascinating research papers, which are often technical and intimidating for people outside the field.

-Softbites team

I wrote about a really cool paper from the Doug Holmes lab. You can read it by clicking the picture below (or here).

This post is a more technical article and is an addendum to my article on Softbites. I will be diving into some specifics of the analysis of the experiment, and is intended for those who have read the article on Softbites (here) and want more details. I encourage you to please go and read the Soft bites article before continuing below.

Exploring the difference in buckling as a function of packing fraction

Recall the difference in buckling behavior for different packing fraction $\phi_{0}$, demonstrated in Figure 1:

Figure 1. An elastic beam in inserted into a rigid box filled with beads. Depending on the packing fraction,  $\phi_{0}$, of the beads, the beam exhibits one or two buckles. The curvature of these buckles are affected by $\phi_{0}$.

To quantify the difference in buckling that happens at different packing fractions, the increase in amplitude relative to wavelength of the buckles, $A_{0}/\lambda$, is plotted as a function of the length of insertion relative to initial beam length, $\sqrt{\Delta/L}$. For small values of $\Delta$, the curvature of the buckle is small and can be approximated as a triangle with height $A_{0}$ and base $\lambda = 2L_{0}$. As $\Delta$ increases, the height of the buckle increases according to $A_{0}/\lambda \propto \sqrt{\Delta/L}$. This is shown by the black dashed line in Figure 4, and experiments at both high and low $\phi_{0}$ (red and blue dots respectively) follow this relationship for small $\Delta$. As $\Delta$ increases, even in the absence of beads, the analytic solution for the beam’s buckling (shown by the blue curve) begins to deviate from this approximation. For intermediate values of $\Delta$, experiments at both high and low $\phi_{0}$ are shown to follow the analytic solution of a beam bending in the absence of beads. This suggests the beads are not yet confining the beam in any meaningful way. At even higher $\Delta$, low $\phi_{0}$ experiments continue following the behavior of a free beam but high $\phi_{0}$ begins to deviate rapidly. This deviation shows that confinement due to the beads has a strong effect on the geometry of the buckles in these cases, and the results are consistent with the numerically calculated solution for buckling in the presence of beads (red line).

Figure 2.  Increase in relative amplitude $A_{0}/\lambda$ as a function of $\sqrt{\Delta/L}$. Solutions for small buckles (dashed line), buckling in the absence of beads (blue line), and buckling accounting for beads (red line) are shown. Experimental data for small $\phi_{0}$ (blue dots) and large $\phi_{0}$ (red dots) show good agreement with each of their corresponding solutions.

So far we have seen that the nature of buckling is highly dependent on the bead packing fraction $\phi_{0}$. At low $\phi_{0}$, the beam buckles as if the beads were not present. At high $\phi_{0}$, the buckles become highly confined. Confining the buckles forces them to take on a higher curvature – the extra length of beam has less area to spread out in. This forces the beam into an unfavorable shape and effectively stores excess energy in the beam like compressing a spring would store energy within the coils of the spring. Like a spring, if you were to suddenly remove beads from the box (or the compression on the spring), the beam would suddenly pop into a more straight, less curved shape. What we can take away from this observation is that the higher the local curvature of the beam, the more stored energy there is within the beam, and the more the beam is pushing back on the beads in the area of high curvature. Eventually the pressure on the beads will be too great, and one will be forced to pop out of plane, on top of the rest of the beads.
The plot in Figure 3 shows maximum curvature $\kappa$ multiplied by beam thickness $h$ ($\kappa$ ~ compression of the spring, $h$ ~ stiffness of the spring) plotted against $\phi_{0}$ for higher and lower insertion lengths $\Delta/L$ (light blue and dark blue dots, respectively). The plot shows that higher $\phi_{0}$ leads to higher maximum curvature in the beam, as does larger insertion lengths. The plot also shows experiments where a bead becomes dislodged (red squares). This happens more frequently for highly curved and thick beams. The additional images in Figure 3 shows examples of bead movement for systems at various points on the plot. Longer arrows represent larger bead movement, and red circles represent a bead which dislodges from the rest of the beads. Also, notice the direction the beads move in Figure 3 (I). The arrows show how the beads would be pressing into the beam, and help explain why the buckles would move together at high packing fractions.
Figure 3. Beam curvature $\kappa$ normalized by beam thickness$h$ as a function of packing fraction $\phi_{0}$ for normalized insertion lengths $\Delta/L = 0.1$ (light blue) and $\Delta/L = 0.4$. $\kappa h$ is shown to increase for larger $\phi_{0}$. To the right of the plot are examples of bead movement (arrows) for experiments corresponding to regions within the plot. Dislocated beads are shown in red.