## Monday, September 29, 2014

### 3 Ball Stress Calculator

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This calculates the stress applied in a 3 point ball loading configuration according to the formula:

$$\sigma = \frac{-3}{4 \pi} P \frac{\left( X - Y \right)}{b^2}$$

where
$$X = (1+\nu) \ln \left( \frac{r_2}{r_3}\right)^2 + \frac{1-\nu}{2} \left( \frac{r_2}{r_3}\right)^2,$$
$$Y = (1+\nu) \left(1+\ln \left( \frac{r_1}{r_3}\right)^2 \right) + (1-\nu) \left( \frac{r_1}{r_3}\right)^2,$$
$P$ is the applied load in Newtons,
$r_1$ is the radius of the support circle in mm,
$r_2$ is the radius of the loaded area in mm,
$r_3$ is radius of the specimen in mm,
$b$ is the sample thickness in mm,
and $\nu$ is Poisson's ratio.

The radius of the support circle is, as far as I can tell, the radius of the circle produced by the points of the 3 balls supporting the sample.  The radius of the loaded area is the radius of the piston on top pressing down on the sample.  The radius of the specimen is the radius of the circular sample.  From what I can tell, $r_2 < r_1 < r_3$ by definition.