§ 6.1 Stress and strain
Key ideas
- Hooke's law: F = kx up to the limit of proportionality; k is the spring constant in N m⁻¹.
- Tensile stress σ = F/A (Pa); tensile strain ε = x/L, a ratio with no unit.
- The Young modulus E = σ/ε is a property of the material, independent of the sample's shape or size.
- Measure E for a wire by recording extension against load, with the original length and diameter measured separately.
Equations
σ = F / Atensile stress = force ÷ cross-sectional areaPa
ε = x / Ltensile strain = extension ÷ original lengthno unit
E = σ / εYoung modulus = stress ÷ strainPa
Fig. 1 · Stress against strain: Hooke's law holds up to P; between P and E the material is still elastic but no longer proportional; beyond E the deformation is permanent.
Watch out: strain has no unit and is usually tiny, so the Young modulus comes out enormous (steel ~2 × 10¹¹ Pa). A "reasonable-looking" small E almost always means the area or length went in wrong.
§ 6.2 Elastic and plastic behaviour
Key ideas
- Elastic deformation fully recovers when the load is removed; plastic deformation is permanent.
- The elastic limit marks the boundary: beyond it the sample keeps a permanent extension.
- Strain energy (elastic potential energy) is the area under the force-extension graph; for a Hooke's-law spring it is the triangle W = ½Fx = ½kx².
Equations
F = k xHooke's law, up to the limit of proportionalityN
W = ½ F x = ½ k x²strain energy stored = area under the F-x graphJ
Fig. 2 · The work done stretching the spring is stored as strain energy: the shaded triangle, ½ × final force × extension.
Watch out: ½Fx and ½kx² are only valid while the graph is a straight line. Beyond the limit of proportionality the stored energy is still the area under the curve, found by counting squares, not by the triangle formula.