# Relating internal stress of an o-ring due to thermal expansion to amount of free volume in a gland

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.

• I'm not sure why you deleted this; if you found a solution, you could help others by undeleting this and writing up your own answer.
– Air
Jan 29 '16 at 23:16