Effect of thermal expansion on normal strain of a bar
I am doing some back-of-the-envelope calculations for some thermomechanics applications. I came across the following scenario, which gave me an unintuitive answer: A bar of length L and cross-sectional area A, made of a material with Young's modulus E and coefficient of thermal expansion a, is loaded in the normal direction with a force F. The change in length of the bar can then be expressed as:
u = (F * L)/(E * A).
So far so good. However, now heat the bar by some amount T. Intuitively, I would guess that u would increase, or at the very least remain unchanged; however, if thermal expansion is applied to both L and A, this does not seem to be the case:
u = (L * F * (1 + a * T))/(E * A * (1 + a * T)^2)
=> u = (L * F )/(E * A * (1 + a * T)).
This implies to me that the bar becomes stiffer as temperature increases. Is this conclusion correct, or am I overlooking something? I could imagine that E decreases as temperature increases, or something similar. I'd also be grateful for a source on the subject.