Which parameter is the deciding criteria in order to estimate possible structural failure in the below case?

Let's say we have a rod of steel material. When this rod is put to two different temperatures at the two opposite ends, My question is when we have only this thermal load applied to the rod, what would be the parameter which is of prime importance which dictates the structural failure of the material? To put it clear, I am interested to know how can we find out what must be the maximum temperature which can be applied to one end keeping the other end at the same temperature to avoid structural failure only due to thermal load and which parameter is the deciding criteria for this calculation.