MEC-22

Hertz sphere contact

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

Reading speed

ContactHertz 1882

Governing equation

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

where

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

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

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

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