Confined thermal stress with changing modulus

[cindmp]

Hello,
I would like to compare rods made of different materials based on the amount of force or stress they would generate when fixed at both ends. normally, I would just compare the product of the coefficient of thermal expansion ($\Alpha$), change in temperature ($\Delta T$) and modulus (E). However, I have taken data to get modulus vs. temperature and free deflection vs temperature using DMA and TMA and the modulus and rate of expansion and contraction are not constant.

If, starting from room temperature and increasing in temperature, the modulus decreases, I would expect the stress just to be
$$\begin{equation*} \sigma &= E(T) * \Delta L(T)/L \end{equation*}$$
where Delta L(T) is the amount of unconstrained extension due to the change in temperature and L is the original length. On the other hand, as temperature decreases, modulus is increasing and I would expect the stress generated to be the sum of the incremental changes in length/original length multiplied by the modulus at each temperature. I would think that I would use the same calculation whether it is getting colder and stiffer or warmer and softer, but maybe this is not the case?

Thanks in advance for your help.

 

[Mapes]

Hi cindmp, welcome to PF.

I would think that I would use the same calculation whether it is getting colder and stiffer or warmer and softer, but maybe this is not the case?

No, that should be the case. The stress should always be $\sigma=-\alpha(T)E(T)\Delta T$, where $\alpha(T)=L(T)/(L_0\Delta T)$ is the temperature-dependent coefficient of thermal expansion that you've measured.