C130 Low level

While differential thermal expansion (what you are talking about) does cause thermal stress in static parts, and cycle counts are part of the equation, the main thing that damages things in the hot section is time at temperature. Low cycle fatigue is not much of an issue in transport aircraft since they pretty much sit a cruise power for most of their lives. LCF is a much bigger issue in fighters because of the much larger number of throttle excursions.

Material at high temperature obviously isn't as strong as it is at room temperature and above about 1300 degrees F the amount of stress a part can withstand is based on time at temperature. The amount of stress that a part can withstand for 1,000 hours at a temperature like 1800F (typical metal temperature of a turbine vane) is a lot less than you might think. For example a typical cast blade or vane material like Inco 713 at 1800 F has an ultimate strength of 68,000 psi, but if you put only 13,000 psi on it for 1,000 hours it will break due to stress rupture. That's roughly 1/5 the stress that the material can withstand for a short time.

This is what really causes hot section parts to crack and become distressed, the simple reason is that after a long time at temperature it doesn't take much stress to cause things to fail due to creep ruptre. Lots of parts creep when hot and then when the cool they crack, but the amount of creep is a time at temperature function, and less of number of thermal cycles issue. Turbine vanes are an example of this. Long periods of hot time cause creep rupture and the vanes crack on the surface and eventually fail, but it wasn't the cyclic count that killed them it was the amount of time at temperature.