Angle of repose

The angle of repose, or critical angle of repose,^[1] of a granular material is the steepest angle of descent or dip relative to the horizontal plane on which the material can be piled without slumping. At this angle, the material on the slope face is on the verge of sliding. The angle of repose can range from 0° to 90°. The morphology of the material affects the angle of repose; smooth, rounded sand grains cannot be piled as steeply as can rough, interlocking sands. The angle of repose can also be affected by additions of solvents. If a small amount of water is able to bridge the gaps between particles, electrostatic attraction of the water to mineral surfaces increase the angle of repose, and related quantities such as the soil strength.

When bulk granular materials are poured onto a horizontal surface, a conical pile forms. The internal angle between the surface of the pile and the horizontal surface is known as the angle of repose and is related to the density, surface area and shapes of the particles, and the coefficient of friction of the material. Material with a low angle of repose forms flatter piles than material with a high angle of repose.

The term has a related usage in mechanics, where it refers to the maximum angle at which an object can rest on an inclined plane without sliding down. This angle is equal to the arctangent of the coefficient of static friction μ_s between the surfaces.

Applications of theory

Talus cones on north shore of Isfjord, Svalbard, Norway, showing angle of repose for coarse sediment

The angle of repose is sometimes used in the design of equipment for the processing of particulate solids. For example, it may be used to design an appropriate hopper or silo to store the material, or to size a conveyor belt for transporting the material. It can also be used in determining whether or not a slope (of a stockpile, or uncompacted gravel bank, for example) would likely collapse; the talus slope is derived from angle of repose and represents the steepest slope a pile of granular material can take. This angle of repose is also crucial in correctly calculating stability in vessels.

It is also commonly used by mountaineers as a factor in analysing avalanche danger in mountainous areas.

Formulation

If the coefficient of static friction μ_s is known of a material, then a good approximation of the angle of repose can be made with the following function. This function is somewhat accurate for piles where individual objects in the pile are minuscule and piled in random order.^[2]

${\displaystyle \tan {(\theta )}\approx \mu _{\mathrm {s} }\,}$

where ${\displaystyle \theta }$ is the angle of repose.

This free body diagram demonstrates the relationship between angle of repose and material on the slope.

A simple free body diagram can be used to understand the relationship between the angle of repose and the stability of the material on the slope. For the heaped material to collapse, the frictional forces must be equivalent to the horizontal component of the gravitational force ${\displaystyle mg\sin \theta }$, where ${\displaystyle m}$ is the mass of the material, ${\displaystyle g}$ is the gravitational acceleration and ${\displaystyle \theta }$ is the slope angle:

${\displaystyle mg\sin \theta =f}$

The frictional force ${\displaystyle f}$ is equivalent to the multiplication product of the coefficient of static friction ${\displaystyle \mu }$ and the Normal Force ${\displaystyle N}$ or ${\displaystyle mg\cos \theta }$:

${\displaystyle mg\sin \theta =N\mu }$

${\displaystyle mg\sin \theta =\mu mg\cos \theta }$

${\displaystyle \left({\frac {\sin \theta }{cos\theta }}\right)=\mu }$

${\displaystyle \theta _{R}=\arctan(\mu )}$

Where ${\displaystyle \theta _{R}}$ is the angle of repose, or the angle at which the slope fails under regular conditions, and ${\displaystyle \mu }$ is the coefficient of static friction of the material on the slope.

Measurement

There are numerous methods for measuring angle of repose and each produces slightly different results. Results are also sensitive to the exact methodology of the experimenter. As a result, data from different labs are not always comparable. One method is the triaxial shear test, another is the direct shear test.

The measured angle of repose may vary with the method used, as described below.

Of various materials

This pile of corn has a low angle of repose

Here is a list of various materials and their angle of repose.^[3] All measurements are approximated.

With different supports

Different supports modify the shape of the pile, in the illustrations below sand piles, although angles of repose remain the same.^[6][7]

Support format	Support	Angle of repose
Rectangle
Circle
Square
Triangle
Double fork
Oval
One pit
Double pit
Multiple pit
Random format

Exploitation by antlion and wormlion (Vermileonidae) larvae

Sand pit trap of the antlion

The larvae of the antlions and the unrelated wormlions Vermileonidae trap small insects such as ants by digging conical pits in loose sand, such that the slope of the walls is effectively at the critical angle of repose for the sand.^[8] They achieve this by flinging the loose sand out of the pit and permitting the sand to settle at its critical angle of repose as it falls back. Thus, when a small insect, commonly an ant, blunders into the pit, its weight causes the sand to collapse below it, drawing the victim toward the center where the predator that dug the pit lies in wait under a thin layer of loose sand. The larva assists this process by vigorously flicking sand out from the center of the pit when it detects a disturbance. This undermines the pit walls and causes them to collapse toward the center. The sand that the larva flings also pelts the prey with loose rolling material that prevents it from getting any foothold on the easier slopes that the initial collapse of the slope has presented. The combined effect is to bring the prey down to within grasp of the larva, which then can inject venom and digestive fluids.

In geotechnics

See also

The angle of repose plays a part in several topics of technology and science, including:

References