View source for Angle of repose

{{Short description|Steepest angle at which granular materials can be piled before slumping}}
{{for-multi|the Wallace Stegner novel|Angle of Repose|the angle of friction between two solid objects|Friction#Angle of friction}}

[[File:Angleofrepose.png|thumb|Angle of repose of a heap of sand]]
[[File:Sandpile Matemateca 22.webm|thumb|Sandpile from the [[Matemateca]] ([[Institute of Mathematics and Statistics, University of São Paulo|IME-USP]]) collection]]

The '''angle of repose''', or '''critical angle of repose''',<ref>
{{cite journal
 |last1=Mehta |first1=A.
 |last2=Barker |first2=G. C.
 |year=1994
 |title=The dynamics of sand
 |journal=[[Reports on Progress in Physics]]
 |volume=57 |issue= 4|page=383
 |bibcode=1994RPPh...57..383M
 |doi=10.1088/0034-4885/57/4/002
|s2cid=250898376
 }}</ref> of a [[granular material]] is the steepest [[angle]] of descent or [[Strike and dip|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|sand grains]] cannot be piled as steeply as can rough, [[Shear strength (soil)|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 increases the angle of repose, and related quantities such as the [[bearing capacity|soil strength]].

When bulk granular materials are poured onto a horizontal surface, a [[cone (geometry)|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''<sub>s</sub> between the surfaces.

== Applications of theory ==
[[File:TalusConesIsfjorden.jpg|thumb|280px|Talus cones on north shore of [[Isfjord (Svalbard)|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 [[chute (gravity)|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 [[scree|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 [[slope stability|stability]] in vessels.

It is also commonly used by [[Mountaineering|mountaineers]] as a factor in analysing [[avalanche]] danger in mountainous areas.{{citation needed |date=December 2016}}

==Formulation==

If the [[coefficient of static friction]] ''μ''<sub>s</sub> 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.<ref>
{{cite book
 |last1=Nichols |first1=E. L.
 |last2=Franklin |first2=W. S.
 |year=1898
 |title=The Elements of Physics
 |url=https://books.google.com/books?id=8IlCAAAAIAAJ
 |volume=1 |page=101
 |publisher=[[Macmillan Publishers (United States)|Macmillan]]
 |lccn=03027633
}}</ref>

:<math>\tan{(\theta)} \approx \mu_\mathrm{s}\,</math>

where <math>\theta</math> is the angle of repose.

[[File:Free Body Diagram (Angle of Repose).png|thumb|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 resist collapse, the frictional forces must be equivalent to the horizontal component of the [[Gravity|gravitational force]] <math>m g \sin\theta</math>, where <math>m</math> is the mass of the material, <math>g</math> is the gravitational acceleration and <math>\theta</math> is the slope angle:

:<math>m g \sin\theta = f</math>

The frictional force <math>f</math> is equivalent to the multiplication product of the coefficient of static friction <math>\mu</math> and the Normal Force <math>N</math> or <math>mg\cos\theta</math>:

:<math>m g \sin\theta = N \mu </math>

:<math>m g \sin\theta = \mu m g \cos\theta </math>

:<math>\left ( \frac{\sin \theta}{cos\theta} \right ) = \mu </math>

:<math>\theta_R = \arctan(\mu) </math>

Where <math>\theta_R  </math> is the angle of repose, or the angle at which the slope fails under regular conditions, and <math> \mu </math> 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.

=== Tilting box method ===
This method is appropriate for fine-grained, non-cohesive materials with individual particle size less than 10&nbsp;mm. The material is placed within a box with a transparent side to observe the granular test material. It should initially be level and parallel to the base of the box. The box is slowly tilted until the material begins to slide in bulk, and the angle of the tilt is measured.

=== Fixed funnel method ===
The material is poured through a funnel to form a cone. The tip of the funnel should be held close to the growing cone and slowly raised as the pile grows, to minimize the impact of falling particles. Stop pouring the material when the pile reaches a predetermined height or the base a predetermined width. Rather than attempt to measure the angle of the resulting cone directly, divide the height by half the width of the base of the cone. The inverse tangent of this ratio is the angle of repose.

=== Revolving cylinder method ===
The material is placed within a cylinder with at least one transparent end. The cylinder is rotated at a fixed speed, and the observer watches the material move within it. The effect is similar to watching clothes tumble over one another in a slowly rotating clothes dryer. The granular material assumes a certain angle as it flows within the rotating cylinder. This method is recommended for obtaining the dynamic angle of repose, which may vary from the static angle of repose measured by other methods.

== Of various materials ==
[[File:Cornpiled2017.jpg|thumb|This pile of [[maize|corn]] has a low angle of repose]]
Here is a list of various materials and their angle of repose.<ref>
{{cite book
 |last=Glover |first=T. J.
 |year=1995
 |title=Pocket Ref
 |publisher=[[Sequoia Publishing]]
 |isbn=978-1885071002
}}</ref> All measurements are approximated.

{| class="wikitable sortable"  style="text-align:left;"
|-
! Material (condition) !! Angle of Repose (degrees)
|-
| [[Wood ash|Ashes]] || 40°
|-
| [[Asphalt concrete|Asphalt]] (crushed) || 30–45°
|-
| [[Bark (botany)|Bark]] (wood refuse) || 45°
|-
| [[Bran]] || 30–45°
|-
| [[Chalk]] || 45°
|-
| [[Clay]] (dry lump) || 25–40°
|-
| [[Clay]] (wet excavated) || 15°
|-
| [[Clover|Clover seed]] || 28°
|-
| [[Coconut]] (shredded) || 45°
|-
| [[Coffee bean]] (fresh) || 35–45°
|-
| Flour (corn) || 30–40°
|-
| Flour (wheat) || 45°
|-
| [[Granite]] || 35–40°
|-
| [[Gravel]] ([[crushed stone]]) || 45°
|-
| [[Gravel]] (natural w/ sand) || 25–30°
|-
| [[Malt]] || 30–45°
|-
| [[Sand]] (dry) || 34°
|-
| [[Sand]] (water filled) || 15–30°
|-
| [[Sand]] (wet) || 45°
|-
| [[Snow]] || 38°<ref name="Telluride Magazine">{{cite web| url= http://www.telluridemagazine.com/?page_id=160| title= Anatomy of an Avalanche| last1= Rikkers| first1= Mark| last2= Rodriguez| first2= Aaron| website= Telluridemagazine.com| date= 23 June 2009| publisher= Telluride Publishing| access-date= 3 October 2016| url-status= live| archive-url= https://web.archive.org/web/20160819092433/http://www.telluridemagazine.com/?page_id=160| archive-date= 19 August 2016}}</ref>
|-
| [[Soil]] || 30–45°
|-
| [[Urea]] (Granular) || 27° <ref>{{cite web |url=http://www.potashcorp.com/media/POT_SS_FER_URGRAN_TRI.pdf |title=Urea Granular Agricultural Grade MSDS |access-date=2013-04-05 |url-status=dead |archive-url=https://web.archive.org/web/20120412003104/http://www.potashcorp.com/media/POT_SS_FER_URGRAN_TRI.pdf |archive-date=2012-04-12 |publisher=PCS Sales (USA), Inc. |year=2008}}<!--Title derived from https://web.archive.org/web/20120417231121/http://www.potashcorp.com/customers/products/fertilizer/--></ref>
|-
| [[Wheat]] || 27°
|}

== With different supports ==
Different supports modify the shape of the pile, in the illustrations below sand piles, although angles of repose remain the same.<ref>Ileleji, K. E.. (2008-10-28). "The angle of repose of bulk corn stover particles". Powder Technology 187 (2): 110–118. {{doi|10.1016/j.powtec.2008.01.029}}.</ref><ref>Lobo-Guerrero, Sebastian. (2007-03-23). "Influence of pile shape and pile interaction on the crushable behavior of granular materials around driven piles: DEM analyses" (em en). Granular Matter 9 (3–4): 241. {{doi|10.1007/s10035-007-0037-3}}. {{ISSN|1434-5021}}.</ref>

{{Multiple image|width=110px|align=center|image1=Sandpile Matemateca (1).webm|image2=Sandpile Matemateca (2).webm|image3=Sandpile Matemateca (4).webm|image4=Sandpile Matemateca (5).webm}}

{| class="wikitable sortable"  style="text-align:left;"
|-
! Support format !! Support !! Angle of repose
|-
| Rectangle || [[File:Sandpile Matemateca 01.jpg|85px]] ||[[File:Sandpile Matemateca 02.jpg|85px]]
|-
| Circle || [[File:Sandpile Matemateca 03.jpg|85px]] ||[[File:Sandpile Matemateca 04.jpg|85px]]
|-
| Square || [[File:Sandpile Matemateca 05.jpg|85px]] ||[[File:Sandpile Matemateca 06.jpg|left|65px]] [[File:Sandpile Matemateca 07.jpg|65px]]
|-
| Triangle || [[File:Sandpile Matemateca 08.jpg|85px]] ||[[File:Sandpile Matemateca 09.jpg|85px]]
|-
| Double fork || [[File:Sandpile Matemateca 13.jpg|85px]] ||[[File:Sandpile Matemateca 14.jpg|85px]]
|-
| Oval || [[File:Sandpile Matemateca 17.jpg|85px]] ||[[File:Sandpile Matemateca 18.jpg|85px]]
|-
| One pit || [[File:Sandpile Matemateca 10.jpg|85px]] ||[[File:Sandpile Matemateca 11.jpg|left|65px]] [[File:Sandpile Matemateca 12.jpg|65px]]
|-
| Double pit || [[File:Sandpile Matemateca 15.jpg|85px]] ||[[File:Sandpile Matemateca 16.jpg|85px]]
|-
| Multiple pit || [[File:Sandpile Matemateca 20.jpg|85px]] ||[[File:Sandpile Matemateca 19.jpg|85px]]
|-
| Random format || ||[[File:Sandpile Matemateca 21.jpg|85px]]
|}

== Exploitation by antlion and wormlion (Vermileonidae) larvae ==
[[File:Antlion trap.jpg|thumb|280px|right|Sand pit trap of the antlion]]

The larvae of the [[antlion]]s 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.<ref>
{{cite journal
 |last1=Botz |first1=J. T.
 |last2=Loudon |first2=C.
 |last3=Barger |first3=J. B.
 |last4=Olafsen |first4=J. S.
 |last5=Steeples |first5=D. W.
 |year=2003
 |title=Effects of slope and particle size on ant locomotion: Implications for choice of substrate by antlions
 |journal=[[Journal of the Kansas Entomological Society]]
 |volume=76 |issue=3 |pages=426–435
}}</ref> 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 ==
{{Excerpt|Slope stability#Angle of repose}}

== See also ==
The angle of repose plays a part in several topics of technology and science, including:
{{col div|colwidth=20em}}
* [[Aeolian processes]]
* [[Barchan]]
* [[Bulk cargo]]
* [[Concrete slump test]]
* [[Grade (slope)]]
* [[Mass wasting]]
* [[Oceanic trench]]
* [[Retaining wall]]
* [[Rotary kiln]]
* [[Sand volcano]]
{{colend}}

== References ==
{{Reflist}}

{{Geotechnical engineering}}
{{Authority control}}

{{DEFAULTSORT:Angle Of Repose}}
[[Category:Particulates]]
[[Category:Shear strength]]
[[Category:Soil mechanics]]
[[Category:Engineering concepts]]