Young's modulus

Young's modulus answers a plain question: how stiff is this material — how hard is it to stretch? It is the resistance to elastic (springy) stretching, so a high modulus means a lot of stress buys only a little strain. Steel has a big modulus and barely gives; rubber has a tiny one and stretches easily. Note this is stiffness, which is a different thing from strength (how much stress before it yields) — a bendy fishing rod can be strong, a stiff biscuit can be weak.

It is defined as E = stress / strain in the elastic region, the slope of the straight part of the stress-strain curve, and is quoted in gigapascals. Steel is about 200 GPa, aluminium about 70 GPa (so aluminium stretches roughly three times as much as steel under the same stress), titanium about 110 GPa, and many polymers only 1 to 4 GPa. Worked example: a 200 MPa stress on steel gives strain = 200 MPa / 200000 MPa = 0.001, while the same stress on aluminium gives 200 / 70000 = 0.0029, nearly three times the stretch.

The deep point is that Young's modulus is set by the stiffness of the atomic bonds — the curvature of the bonding-energy curve at the equilibrium spacing — so it is a near-fixed property of a given material. Crucially, heat treatment, cold work, and small alloying additions barely change it: a spring steel and a soft mild steel have almost the same modulus even though their strengths differ enormously. If you need something stiffer, you must change the bonding, meaning change the material (or make a composite), not just harden it.