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, $\mathrm{\Psi }$, of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:

$\mathrm{\Psi }=\frac{{\pi }^{\frac{1}{3}}(6V_p{)}^{\frac{2}{3}}}{A_p}$

where $V_p$ is volume of the object and $A_p$ 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, $\mathrm{\Psi }$, of an oblate spheroid (similar to the shape of the planet Earth) is:

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

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, $A_s$ in terms of the volume of the object being measured, $V_p$

$A_s^3={\left(4\pi r^2\right)}^3=4^3{\pi }^3{r}^6=4\pi \left(4^2{\pi }^2{r}^6\right)=4\pi \cdot 3^2\left(\frac{4^2{\pi }^2}{3^2}{r}^6\right)=36\pi {\left(\frac{4\pi }{3}{r}^3\right)}^2=36\pi V_p^2$

therefore

$A_s={\left(36\pi V_p^2\right)}^{\frac{1}{3}}=36^{\frac{1}{3}}{\pi }^{\frac{1}{3}}{V}_p^{\frac{2}{3}}=6^{\frac{2}{3}}{\pi }^{\frac{1}{3}}{V}_p^{\frac{2}{3}}={\pi }^{\frac{1}{3}}{\left(6V_p\right)}^{\frac{2}{3}}$

hence we define $\mathrm{\Psi }$ as:

$\mathrm{\Psi }=\frac{A_s}{A_p}=\frac{{\pi }^{\frac{1}{3}}{\left(6V_p\right)}^{\frac{2}{3}}}{A_p}$

Sphericity of common objects

Name	Picture	Volume	Surface area	Sphericity
Sphere		$\frac{4}{3}\pi r^3$	$4\pi r^2$	1
Disdyakis triacontahedron		$\frac{180}{11}(5+4\sqrt{5})s^3$	$\frac{180}{11}\sqrt{179-24\sqrt{5}}{s}^2$	$\frac{{\left({(5+4\sqrt{5})}^2\frac{11\pi }{5}\right)}^{\frac{1}{3}}}{\sqrt{179-24\sqrt{5}}}\approx 0.9857$
Rhombic triacontahedron		$4\sqrt{5+2\sqrt{5}}{s}^3$	$12\sqrt{5}{s}^2$	$\frac{{\pi }^{\frac{1}{3}}{\left(24\sqrt{5+2\sqrt{5}}\right)}^{\frac{2}{3}}}{12\sqrt{5}}\approx 0.9609$
Icosahedron		$\frac{5}{12}(3+\sqrt{5})s^3$	$5\sqrt{3}{s}^2$	${\left(\frac{{(3+\sqrt{5})}^2\pi }{60\sqrt{3}}\right)}^{\frac{1}{3}}\approx 0.939$
Dodecahedron		$\frac{1}{4}(15+7\sqrt{5})s^3$	$3\sqrt{25+10\sqrt{5}}{s}^2$	${\left(\frac{{(15+7\sqrt{5})}^2\pi }{12{(25+10\sqrt{5})}^{\frac{3}{2}}}\right)}^{\frac{1}{3}}\approx 0.910$
Ideal torus $(R=r)$		$2{\pi }^{2}R{r}^2=2{\pi }^2{r}^3$	$4{\pi }^{2}Rr=4{\pi }^2{r}^2$	${\left(\frac{9}{4\pi }\right)}^{\frac{1}{3}}\approx 0.894$
Ideal cylinder $(h=2r)$		$\pi r^{2}h=2\pi r^3$	$2\pi r(r+h)=6\pi r^2$	${\left(\frac{2}{3}\right)}^{\frac{1}{3}}\approx 0.874$
Octahedron		$\frac{1}{3}\sqrt{2}{s}^3$	$2\sqrt{3}{s}^2$	${\left(\frac{\pi }{3\sqrt{3}}\right)}^{\frac{1}{3}}\approx 0.846$
Hemisphere (half sphere)		$\frac{2}{3}\pi r^3$	$3\pi r^2$	${\left(\frac{16}{27}\right)}^{\frac{1}{3}}\approx 0.840$
Cube (hexahedron)		$s^3$	$6s^2$	${\left(\frac{\pi }{6}\right)}^{\frac{1}{3}}\approx 0.806$
Ideal cone $(h=2\sqrt{2}r)$		$\frac{1}{3}\pi r^{2}h=\frac{2\sqrt{2}}{3}\pi r^3$	$\pi r(r+\sqrt{r^2+h^2})=4\pi r^2$	${\left(\frac{1}{2}\right)}^{\frac{1}{3}}\approx 0.794$
Tetrahedron		$\frac{\sqrt{2}}{12}{s}^3$	$\sqrt{3}{s}^2$	${\left(\frac{\pi }{6\sqrt{3}}\right)}^{\frac{1}{3}}\approx 0.671$

See also

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

External links

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