The physics
A testing machine pulls a rod and plots stress against strain. At first the line is straight. Its slope is Young's modulus Y.
Up to the elastic limit, the rod springs back to its old length. Past it, some stretch stays. Far past it, the rod breaks.
Fmax = σlimit × A
A slow pull stores energy in the wire. It is the area under the F–ΔL line, a triangle. Per cubic metre, it is the area under the stress–strain line.
U = ½ F ΔL = ½ (AY/L) ΔL² u = ½ Y ε²
Heat a rod and it grows by LαΔT. Clamps stop it, so the rod pushes on them. A cooled wire between clamps gets pulled tight in the same way. The length cancels.
stress = Y α ΔT F = Y A α ΔT
Some graphs put strain up the side and stress along the bottom. Then the slope is 1/Y. There the flatter line is the stiffer wire.