Hooke's law and the stress-strain curve

Hooke's law: for small deformations, stress is proportional to strain. Stress = k × strain, where the constant k is the modulus of elasticity.

Hooke's law is empirical. It holds for most materials over small strains, but some materials never show a straight-line relation.

In a tensile test a wire or test cylinder is loaded in steps and its strain recorded; stress (applied force per unit area) is then plotted against strain.

O to A: a straight line. Hooke's law holds and the body recovers fully when unloaded, so it behaves elastically.

A to B: stress and strain are no longer proportional, yet the body still recovers on unloading. B is the yield point, also called the elastic limit, and the stress there is the yield strength σy.

Beyond B the strain grows fast for small increases in stress. If the load is removed at a point C between B and D, the strain does not return to zero: the body has a permanent set, a plastic deformation.

D marks the ultimate tensile strength σu. Past D the strain keeps growing even under a smaller force, and the wire fractures at E.

D and E close together: the material is brittle. D and E far apart: the material is ductile.

Elastomers such as rubber and the elastic tissue of the aorta can be stretched to large strains and still recover. Their elastic region is huge, yet Hooke's law fails over most of it and there is no well-defined plastic region.