# Contact stress two sphere

## Values for calculation

$F$ $\mathrm{N}$
$D_1$ $\mathrm{mm}$
$E_1$ $\mathrm{MPa}$
$ν_1$
$HB_1$ $\mathrm{HB}$
$S_{y-1}$ $\mathrm{MPa}$
$D_2$ $\mathrm{mm}$
$E_2$ $\mathrm{MPa}$
$ν_2$
$HB_2$ $\mathrm{HB}$
$S_{y-2}$ $\mathrm{MPa}$
$C_c$
$S_F$

## Calculation

### Allowable Hertz pressure

$\text{if }\ \text{Hardened }$$\text{material }$$\text{1}= \text{yes}\wedge\text{Hardened }$$\text{material }$$\text{2}= \text{yes}$
$$σ_H=\min\left(\cfrac{4.2\cdot S_{y-1}}{S_F}\cdot C_c, \cfrac{4.2\cdot S_{y-2}}{S_F}\cdot C_c\right)$$
$\text{else if }\ \text{Hardened }$$\text{material }$$\text{1}= \text{yes}\wedge\text{Hardened }$$\text{material }$$\text{2}= \text{no}$
$$σ_H=\min\left(\cfrac{4.2\cdot S_{y-1}}{S_F}\cdot C_c, \cfrac{7\cdot HB_2}{S_F}\cdot C_c\right)$$
$\text{else if }\ \text{Hardened }$$\text{material }$$\text{1}= \text{no}\wedge\text{Hardened }$$\text{material }$$\text{2}= \text{no}$
$$σ_H=\min\left(\cfrac{7\cdot HB_1}{S_F}\cdot C_c, \cfrac{7\cdot HB_2}{S_F}\cdot C_c\right)$$
$\text{else}$
$$σ_H=\min\left(\cfrac{7\cdot HB_1}{S_F}\cdot C_c, \cfrac{4.2\cdot S_{y-2}}{S_F}\cdot C_c\right)$$

### Material coefficient

$$C_E=\cfrac{1-ν_1^2}{E_1}+\cfrac{1-ν_2^2}{E_2}$$

### Dimensional coefficient

$$K_D=\cfrac{D_1\cdot D_2}{D_1+ D_2}$$

### Diameter of sphere contact area

$$b=1.442\cdot\sqrt[3]{F\cdot K_D \cdot C_E}$$

### Contact stress

$$σ_c=0.918\cdot\sqrt[3]{\cfrac{F}{K_D^2\cdot C_E^2}}$$

$$σ_c\le σ_H$$