M5 Locomotion in Granular Media

Topic

Locomotion through granular substrates, such as sand or soil, involves two types of physical resistance: the internal friction of the granular medium itself—which scales with depth and contact area—and inertial pressure, which scales with the medium's density multiplied by the square of the animal's movement speed.

This same granular medium behaves very differently depending on the speed of the disturbance: under a slow, sustained load, it acts like a solid, resisting deformation; under a rapid impact, however, it behaves like a fluid, yielding and flowing around the disturbing object. This transition between solid-like and fluid-like behavior, dictated by the speed of the disturbance, determines the resistance the animal encounters while moving through the medium.

To cope with this resistance, burrowing mammals have evolved specialized morphologies—such as shovel-like limbs and cylindrical bodies—and movement strategies designed to minimize the energy cost of digging. Even with these adaptations, the energy cost of burrowing remains immense: between five hundred and two thousand times higher than the cost of traveling the same distance across the surface.

Digging is not only energetically costly; it also creates a critical physiological challenge within the tunnel itself: air renewal. Airflow through a narrow passage decreases in proportion to the fourth power of its radius; consequently, halving the tunnel's radius reduces available airflow sixteen-fold, rendering narrow tunnels virtually impermeable to air. Some species of naked mole-rat have evolved adaptations that allow them to tolerate carbon dioxide concentrations of up to ten percent of the air they breathe—and oxygen concentrations below eight percent—within their tunnels. How close a granular medium is—at any given moment—to behaving like a solid or a fluid can be characterized by a dimensionless number comparing the rate of deformation to the internal pressure maintaining the medium's compactness: when this number is much less than one, the medium behaves quasi-statically, like a solid; when it approaches one, the medium enters an inertial regime, behaving much more like a fluid.

The animal's forward speed while burrowing is determined by the ratio of its available metabolic power to the resistive force exerted by the granular medium in that regime: the more compacted the substrate—or the deeper the animal burrows—the greater that resistive force becomes, and the lower the forward speed that the same metabolic power can sustain.

In the aforementioned quasi-static regime, the energy cost per unit volume excavated is independent of forward speed: digging slowly versus quickly requires exactly the same total energy per meter of completed tunnel, even though rapid digging obviously demands higher sustained metabolic power over a shorter duration. This independence of cost from speed is, ultimately, a direct consequence of the combination of internal friction and inertial pressure with which this description of locomotion in granular media began.

ID:2613