# 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.

#### 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:
$E\approx2.2\cdot\frac{E1\cdot E2}{E1+E2}$

$R_i$ 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

$r_i$ 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 $C_{local}=\pi\cdot\sigma_{max}\cdot r$

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