MEC-22

Hertz sphere contact

a = (3 F R / 4 E*)^{1/3}, p0 = 3F /(2π a²). Elastic spheres.

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ContactHertz 1882

Governing equation

a=\left(\dfrac{3FR}{4E^*}\right)^{1/3},\quad p_0=\dfrac{3F}{2\pi a^2}

$F \in \left[1, 8000\right]$

where

F: Load (N)
R: Reduced radius (mm)
E^*: Reduced modulus (GPa)
a: Contact radius (mm)
p_0: Peak pressure (MPa)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

a=\left(\dfrac{3FR}{4E^*}\right)^{1/3},\quad p_0=\dfrac{3F}{2\pi a^2}

$F \in \left[1, 8000\right]$

$F = 1$ $a = 0.041$

Along the curve

②

Domain of the variable — blue

F_{\min} = 1,\quad F_{\max} = 8000

$F \in \left[1, 8000\right]$

$F = 1$ $F = 1$

Along the curve

③

Response curve — green

a=\left(\dfrac{3FR}{4E^*}\right)^{1/3},\quad p_0=\dfrac{3F}{2\pi a^2}

$F \in \left[1, 8000\right]$

$F = 1$ $a = 0.041$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}a}{\mathrm{d}F}

$F \in \left[1, 8000\right]$

$F = 51.31$ $a^{\prime} = 0.001$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-22 — Hertz sphere contact) is the form associated with Hertz 1882. Working symbols: $F$ , $R$ , $E^*$ $\rightarrow$ $a$ , $p_0$ . Hertz inverted the half-space under a hemispherical pressure. 1/R = 1/R1+1/R2.

Purpose

Purpose: compute

a

p_0

from

F

R

E^*

in Mechanical via

a=\left(\dfrac{3FR}{4E^*}\right)^{1/3},\quad p_0=\dfrac{3F}{2\pi a^2}

a = (3 F R / 4 E*)^{1/3}, p0 = 3F /(2π a²). Elastic spheres. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

F = 200.000\,\mathrm{N}

R = 10.000\,\mathrm{mm}

E^* = 110.000\,\mathrm{GPa}

, the governing relation

a=\left(\dfrac{3FR}{4E^*}\right)^{1/3},\quad p_0=\dfrac{3F}{2\pi a^2}

yields

a = 0.2389\,\mathrm{mm}

p_0 = 1673.04\,\mathrm{MPa}

. Two spheres, a contact disk, a peak pressure. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Load F200 NReduced radius R10 mmReduced modulus E^*110 GPa

Outputs

Contact radius a0.2389 mm
Peak pressure p_01673.04 MPa

Reading speed

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Narration of this film

Two spheres, a contact disk, a peak pressure.

Hertz inverted the half-space under a hemispherical pressure. 1/R = 1/R1+1/R2.

Reading speed

Watch on YouTube