Elastogranularity and how soil may shape the roots of plants

Original paper: Elastogranular Mechanics: Buckling, Jamming, and Structure Formation

How an elastic beam deforms under load has been a question for as long as there have been engineers to ask it. In some cases, the force on a beam is approximated as a single point. For example, if a diving board is large enough, a diver at the end can be treated as a point mass on the beam. Another common approximation is to consider the force to be a continuous pressure along its length. Treating wind that bends a tree branch as a continuous pressure along the branch’s length is much simpler than adding up the force from every molecule of air on the branch. However, consider the case of a root growing into a granular material like soil. As the root burrows through the soil it will bend due to varying point-like forces along its length. The result is a branching and twisting root system that tries to grow along the path of least resistance. An example of the diversity in plant root morphologies is shown in Figure 1 and gives a sense of how complicated and interesting the physics behind this growth can be.  

Figure 1. An example of the complex and diverse morphology of several types of plants which all live in the same ecosystem [1]

With this in mind, David J. Schunter Jr. et al. from the Holmes group at Boston University have developed a beautiful experiment to study what they call “elastogranular” phenomena. In their experiment, an elastic beam is inserted into a box which is filled at a particular density with uniform beads. An image of a typical experiment is shown in Figure 2. Once the beam reaches the end of the box it will not be able to penetrate any further, and trying to push more of the beam into the box will cause a compression along the beam’s length. This resembles a classic experiment where a beam is compressed in the same manner, but in the absence of beads. Compressing the beam against the end of the box becomes increasingly difficult until eventually the beam “pops” into one large buckle. In this simpler case, the beam was free to buckle with no restrictions. Introducing beads to the system reinforces the beam non-uniformly and constrains the shapes it can take on when it buckles. This complicates the buckling event and leads to interesting new behaviors.

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Figure 2. An elastic beam is inserted into a box of length L_{0} and width 2W_{0} filled with beads at an initial packing fraction of \phi_{0} = 0.89. After a length of beam is inserted equal to L_{0}, inserting additional length \Delta results in buckling. In this experiment the beam takes on two buckles with wavelength \lambda, and amplitudes of A_{0} and A_{1} respectively.

In the experiment performed by the Holmes group, a beam is compressed against the end of the box until the beam buckles. If the packing fraction \phi_{0} (the fraction of space within the box that is covered in beads) is low, it buckles much like one would expect in the absence of beads—one large buckle, as in Figure 3i. If \phi_{0} is higher at the beginning of the experiment, like in Figures 3ii and 3iii, the buckling behavior becomes more complicated. The beam will form one large buckle as before, and as the buckle grows it will take up more area on one side of the box. This forces the beads to reorganize themselves, and the beads on the compressed side of the box become very tightly packed. At this point, they are in a hexagonal arrangement, and they are said to have crystallized [2]. As the beads in one side crystalize, that side becomes stiffer and suppresses further growth of the amplitude of the first buckle, A_{0}. In order to accommodate the extra length being inserted, \Delta, an additional buckle forms. The difference in buckling behavior for three values of \phi_{0} are shown in Figure 3. [3]

Figure 3. Shape profiles of the inserted rod for various packing fractions (\phi_{0}) for the same normalized inserted length \Delta/L_{0}. i) At low \phi_{0} the beam forms one large buckle. ii) As \phi_{0} increases, the amplitude of the single buckle is suppressed, leading to a second buckle forming on the other side. iii) At even higher \phi_{0}, the buckles rotate and grow toward each other.

Figures 3 shows that not only are the number and amplitude of buckles significantly affected by the beads that surround the beam, but the orientations of the buckles are changed as well. When the experiment begins at a high \phi_{0}, the beam finds both sides of the box to be stiff and hard to penetrate. The initial buckle does not grow very much before the second buckle forms, and at high enough \phi_{0} both buckles occur nearly simultaneously, forming a twin buckle. Figure 4 shows two systems with different \phi_{0} forming twin buckles. In the top sequence where \phi_{0} is lower, the buckles maintain a constant distance between each other as they grow since there are plenty of uncrystallized areas (light blue circles) ahead of the buckles into which they can grow. The lower sequence shows that, at higher \phi_{0}, the majority of uncrystallized beads are found in the wake of a buckle so the buckles instead grow into these regions, as demonstrated by the red lines.

Figure 5. Twin buckles increase in amplitude as \Delta increases (left to right) growing into less-dense, uncrystallized regions (light blue circles). For lower \phi_{0} (top sequence), uncrystallized beads in front of the buckles can be pushed aside, crystallizing beads away from the buckles (dark blue and yellow circles). At higher \phi_{0} (lower sequence), the uncrystallized sections occur behind the buckles, causing the two buckles to grow closer together which is shown by the red lines.

This system bears a striking resemblance to that of plant roots growing into the soil and could be useful in understanding how environmental pressures cause plant root systems to evolve. For example, cacti need to absorb as much water as they can from their environment. One way of accomplishing this is to increase the surface area of the root system by growing wide and close to the surface, rather than deep, in order to collect water from a larger area. By developing thin roots that buckle before they can deeply penetrate the soil, many cacti are able to produce the shallow, wide-reaching roots system they need to find water.

David J. Schunter Jr. and coworkers have shown that combining two well-understood problems—buckling of a beam and reorganization of beads—can lead to unique and interesting bending dynamics. By confining a beam to a box of beads, the buckling of the beam becomes strongly influenced by the packing fraction and reorientation of the beads. This particular system shows a strong resemblance to plant root growth, but also be informative for synthetic applications involving the insertion of flexible filaments into deformable materials.

[1] Michigan Natural Shore Partnership, http://www.mishorelinepartnership.org/plants-for-inland-lakes.html

[2] When spheres crystallize in two dimensions, the hexagonal lattice is the closest possible packing with an area fraction of \phi = 0.9069. Interestingly, Figure 3iii shows a packing fraction of 0.91, which is higher than this maximum value. This is because the beads are able to pop out of the plane at very high compressions, which can lead to a calculated packing fraction larger than that of the hexagonal lattice. For more information on hexagonal packing, see Wikipedia.

[3] For more information about the specifics of how the deformation of the beam is quantified, a summary of the analysis is available here.

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