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:
$E\approx2.2\cdot\frac{E1\cdot E2}{E1+E2}$
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 $C_{local}=\pi\cdot\sigma_{max}\cdot r$