Hertz contact: Universal point

Engineering Fundamentals

Introduction

Universal calculation of Hertz point contact between two arbitrarily curved bodies under load F and each body has radii in x and y direction of which the largest radius is called Ri and the smallest is called ri.

Hertz contact - Universal point

Symbols

Young’s Modulus, Poisson ratio, small radius, large radius per body: $\ E,\nu,\ r,\ R$

Equivalent Young’s modulus

$E=\left(\frac{1-\nu_1^2}{2E_1}+\frac{1-\nu_2^2}{2E_2}\right)^{-1} $
Note: for metals $\nu\approx0.3$, so the equivalent Young’s modulus for metal on metal contact is:
$E\approx2.2\cdot\frac{E1\cdot E2}{E1+E2}$

Equivalent large radius

$R_i$ are the large radii of both bodies.
$R=\frac{R_1\cdot R_2}{R_1+R_2} $
Note: $R_i=\infty$ for flat surface; $R_i<0$ for hollow surface

Equivalent small radius

$r_i$ are the small radii of both bodies.
$r=\frac{r_1\cdot r_2}{r_1+r_2} $
Note: $r_i=\infty$ for flat surface; $r_i<0$ for hollow surface

Curvature ratio

Always ≥ 1.
$\omega=\frac{R}{r}$

Half long axis of contact ellipse

If 1 ≤ ω < 25:
$a=2^{-\frac{1}{3}}\cdot\omega^\frac{11}{24}\cdot\left(\frac{3Fr}{E}\right)^\frac{1}{3}$

if 25 < ω < 1e5:
$a=\frac{2^\frac{5}{3}}{\pi}\cdot\omega^\frac{8}{24}\cdot\left(\frac{3Fr}{E}\right)^\frac{1}{3}$

Half short axis of contact ellipse

$b= 2^{-\frac{1}{3}}\cdot\omega^\frac{-4}{24}\cdot\left(\frac{3Fr}{E}\right)^\frac{1}{3}$

Approach of bodies

$\delta=\frac{a^2}{2R}+\frac{b^2}{2r}$

Average stress, Maximum stress

$\sigma_{av}=\frac{F}{\pi ab}$, $\sigma_{max}=\frac{3}{2}\frac{F}{\pi ab}$

Average stiffness

Only valid when $\omega=1$
$C_{av}=\frac{F}{\delta}=\sqrt[3]{\frac{4}{9}R\cdot F\cdot E^2}$

Local (maximum) stiffness

Only valid when $\omega=1$
$C_{local}=\frac{dF}{d\delta}=\frac{3}{2}C_{av}$ or $C_{local}=\pi\cdot\sigma_{max}\cdot r$ 

Tech Support

Please submit a message and we will come back to you on short notice.

Precision Point sheet download

Please fill in your details to receive the requested Precision Point sheet.

We use cookies to ensure to give you the best experience on our website. If you continue to use this site we will assume that you are okay with it.