I am trying to come up with an analytical way to determine the optimal gland size that an elastomer o-ring will sit in. The amount of free space in the gland is determined by the thickness of two backup rings that surround it. Thus, the thicker the back-ups, the less free space to allow for thermal expansion.
My first step was to determine the internal stress on the o-ring due to temperature:
Coefficient of Thermal Expansion, $\alpha = 1.6\times10^{-4}\ \mathrm{\frac{in}{in-°F}}$
Modulus, $E = 1900\ \mathrm{\frac{lbf}{in^2}}$
Temperature, $\Delta T = 400 \mathrm{°F}$
Stress: $$\begin{align} \sigma=E\epsilon=E\alpha\Delta T &= (1900\ \mathrm{\frac{lbf}{in^2}})(1.6\times10^{-4}\ \mathrm{\frac{in}{in-°F}})(400\ \mathrm{°F}) \\ &=122\ \mathrm{\frac{lbf}{in^2}} \\ \end{align}$$
This where I am stuck relating this back to the dimensions of the gland. How can I move forward in order to relate this to the amount of free space in the gland? Assume worst case when the o-ring is fully constrained with no room for expansion.
