Stress, strain and Hooke's law

Pull a rubber band and let go: it snaps back. Pull a piece of chewing gum and it stays stretched. A body that returns to its shape when the force is removed is elastic; one that stays deformed is plastic (putty, wax). Nothing is perfectly elastic; a quartz fibre comes closest.

When you pull or squash a body, its particles push back with an internal restoring force. Stress is this restoring force per unit area, F/A. Its unit is N m⁻², called the pascal (Pa).

There are different kinds of stress, depending on how the force acts:

• Tensile stress stretches a wire; compressive stress squashes a pillar. Both act along the length (longitudinal stress).
• Shearing stress acts along a surface, like pushing the top of a book sideways: a rectangle becomes a parallelogram.
• Hydrostatic stress presses equally from all sides, like water on a sunken stone. It changes only the volume.

Strain measures the deformation as a fraction of the original size, so it has no unit:

• Longitudinal strain = ΔL/L.
• Shear strain = Δx/L, the sideways shift ÷ the height, which equals the (small) angle θ the side turns through.
• Volume strain = ΔV/V.

Hooke's law: for small deformations, stress is proportional to strain. Their ratio is a constant for the material, called a modulus of elasticity.

The stress–strain curve of a metal wire tells its whole story:

If D and E are far apart, the metal is ductile (it can be drawn into wires); if close, it is brittle. Elastomers such as rubber, and the tissue of the aorta, stretch a great deal and spring back, but their curve is never straight, so they do not obey Hooke's law.