Sphericity

Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and rounding (horizontal).

Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape. Defined by Wadell in 1935, the sphericity, [math]\displaystyle{ \Psi }[/math], of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:

[math]\displaystyle{ \Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} }[/math]

where [math]\displaystyle{ V_p }[/math] is volume of the object and [math]\displaystyle{ A_p }[/math] is the surface area. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any shape which is not a sphere will have sphericity less than 1.

Sphericity applies in three dimensions; its analogue in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft, is called roundness.

Ellipsoidal objects

The sphericity, [math]\displaystyle{ \Psi }[/math], of an oblate spheroid (similar to the shape of the planet Earth) is:

[math]\displaystyle{ \Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} = \frac{2\sqrt[3]{ab^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln{\left(\frac{a+\sqrt{a^2-b^2}}b\right)}}, }[/math]

where a and b are the semi-major and semi-minor axes respectively.

Derivation

Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.

First we need to write surface area of the sphere, [math]\displaystyle{ A_s }[/math] in terms of the volume of the object being measured, [math]\displaystyle{ V_p }[/math]

[math]\displaystyle{ A_{s}^3 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac{4^2 \pi^2}{3^2} r^6\right) = 36 \pi \left(\frac{4 \pi}{3} r^3\right)^2 = 36\,\pi V_{p}^2 }[/math]

therefore

[math]\displaystyle{ A_{s} = \left(36\,\pi V_{p}^2\right)^{\frac{1}{3}} = 36^{\frac{1}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = 6^{\frac{2}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }[/math]

hence we define [math]\displaystyle{ \Psi }[/math] as:

[math]\displaystyle{ \Psi = \frac{A_s}{A_p} = \frac{ \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }{A_{p}} }[/math]

Sphericity of common objects

Name	Picture	Volume	Surface area	Sphericity
Sphere		[math]\displaystyle{ \frac{4}{3}\pi r^3 }[/math]	[math]\displaystyle{ 4\pi\,r^2 }[/math]	1
Disdyakis triacontahedron		[math]\displaystyle{ \frac{180}{11}\left(5+4\sqrt{5}\right)\,s^3 }[/math]	[math]\displaystyle{ \frac{180}{11}\sqrt{179-24\sqrt{5}}\,s^2 }[/math]	[math]\displaystyle{ \frac{\left(\left(5+4\sqrt{5}\right)^{2}\frac{11\pi}{5}\right)^{\frac{1}{3}}}{\sqrt{179-24\sqrt{5}}}\approx0.9857 }[/math]
Rhombic triacontahedron		[math]\displaystyle{ 4\sqrt{5+2\sqrt{5}}\,s^3 }[/math]	[math]\displaystyle{ 12\sqrt{5}\,s^2 }[/math]	[math]\displaystyle{ \frac{\pi^{\frac{1}{3}}\left(24\sqrt{5+2\sqrt{5}}\right)^{\frac{2}{3}}}{12\sqrt{5}}\approx0.9609 }[/math]
Icosahedron		[math]\displaystyle{ \frac{5}{12}\left(3+\sqrt{5}\right)\,s^3 }[/math]	[math]\displaystyle{ 5\sqrt{3}\,s^2 }[/math]	[math]\displaystyle{ \left(\frac{\left(3+\sqrt{5}\right)^2\pi}{60\sqrt{3}}\right)^{\frac{1}{3}}\approx0.939 }[/math]
Dodecahedron		[math]\displaystyle{ \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3 }[/math]	[math]\displaystyle{ 3 \sqrt{25 + 10\sqrt{5}}\, s^2 }[/math]	[math]\displaystyle{ \left(\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}\right)^{\frac{1}{3}}\approx0.910 }[/math]
Ideal torus [math]\displaystyle{ (R=r) }[/math]		[math]\displaystyle{ 2\pi^2Rr^2=2\pi^2\,r^3 }[/math]	[math]\displaystyle{ 4\pi^2Rr=4\pi^2\,r^2 }[/math]	[math]\displaystyle{ \left(\frac{9}{4 \pi}\right)^{\frac{1}{3}}\approx0.894 }[/math]
Ideal cylinder [math]\displaystyle{ (h=2\,r) }[/math]		[math]\displaystyle{ \pi\,r^2h=2\pi\,r^3 }[/math]	[math]\displaystyle{ 2\pi\,r(r+h)=6\pi\,r^2 }[/math]	[math]\displaystyle{ \left(\frac{2}{3}\right)^{\frac{1}{3}}\approx0.874 }[/math]
Octahedron		[math]\displaystyle{ \frac{1}{3}\sqrt{2}\,s^3 }[/math]	[math]\displaystyle{ 2 \sqrt{3}\,s^2 }[/math]	[math]\displaystyle{ \left(\frac{\pi}{3\sqrt{3}}\right)^{\frac{1}{3}}\approx0.846 }[/math]
Hemisphere (half sphere)		[math]\displaystyle{ \frac{2}{3}\pi\,r^3 }[/math]	[math]\displaystyle{ 3\pi\,r^2 }[/math]	[math]\displaystyle{ \left(\frac{16}{27}\right)^{\frac{1}{3}}\approx0.840 }[/math]
Cube (hexahedron)		[math]\displaystyle{ \,s^3 }[/math]	[math]\displaystyle{ 6\,s^2 }[/math]	[math]\displaystyle{ \left(\frac{\pi}{6}\right)^{\frac{1}{3}}\approx0.806 }[/math]
Ideal cone [math]\displaystyle{ (h=2\sqrt{2}r) }[/math]		[math]\displaystyle{ \frac{1}{3}\pi\,r^2h=\frac{2\sqrt{2}}{3}\pi\,r^3 }[/math]	[math]\displaystyle{ \pi\,r(r+\sqrt{r^2+h^2})=4\pi\,r^2 }[/math]	[math]\displaystyle{ \left(\frac{1}{2}\right)^{\frac{1}{3}}\approx0.794 }[/math]
Tetrahedron		[math]\displaystyle{ \frac{\sqrt{2}}{12}\,s^3 }[/math]	[math]\displaystyle{ \sqrt{3}\,s^2 }[/math]	[math]\displaystyle{ \left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}}\approx0.671 }[/math]

See also

• Equivalent spherical diameter
• Flattening
• Isoperimetric ratio
• Rounding (sediment)
• Roundness
• Willmore energy

External links

• Grain Morphology: Roundness, Surface Features, and Sphericity of Grains