Limit of proportionality (P)
Materials - OCR A-Level Physics
Limit of proportionality (P)
The point up to which stress is directly proportional to strain. Hooke's lawThe extension of a spring is directly proportional to the applied force, provided the limit of proportionality is not exceeded. applies, and the stress-strain graph is a straight line through the origin.
Elastic limit (E)
The maximum stress a material can take and still return fully to its original length when the load is removed. Loading beyond E causes permanent (plastic) deformation. E is usually just past P on the curve.
Yield point (Y), UTS, fracture (F)
Yield point: stress at which the material starts to flow plastically; a small extra stress causes a large extra strain. Ultimate tensile stress (UTS): the maximum stress the material can carry, the peak of the stress-strain curve. Fracture point: the stress at which the material finally breaks.
- Order along the curve: P first, then E (very close to P), then Y, then UTS at the peak, then F where the material snaps. The order is fixed and examiners check it.
- Between P and E the curve only just starts to bend; beyond E the material no longer obeys $F = kx$ on the spring or $\sigma = E\varepsilon$ on the wire.
- Between UTS and F the material 'necks' (thins at one spot), so true stress in the neck keeps rising even though the load drops.
- The area under a stress-strain graph gives the energy stored per unit volume of material (units: $\text{J m}^{-3}$). Below P, that area is recoverable elastic potential energy.
Common Mistake
MEDIUM
Wrong: Treating P and E as the same point, or putting Y before E. Defining UTS as the stress at which the wire breaks.
Right: P is where the line stops being straight. E is slightly past P, where deformation becomes permanent. UTS is the peak stress on the curve, not the breaking stress. Fracture (F) usually happens at a lower stress than UTS because the wire is necking.
Right: P is where the line stops being straight. E is slightly past P, where deformation becomes permanent. UTS is the peak stress on the curve, not the breaking stress. Fracture (F) usually happens at a lower stress than UTS because the wire is necking.
Examiner Tips and Tricks
- When labelling a stress-strain graph, place P, E, Y, UTS and F in that order along the curve. Mark schemes deduct for swapped letters.
- The phrase 'limit of proportionality' is mark-scheme language for 'the line is no longer straight'.