1. Stress and Strain
Stress and strain
Hooke's law and the Young modulus
A pair of tensile forces stretches an object, producing an extension; compressive forces produce a compression. The force applied is the load.
Hooke's law: the extension of an object is proportional to the load applied, provided the limit of proportionality is not exceeded. The constant is the spring constant k = F/x (N m−1), a property of that particular object.
To describe a material, divide out the dimensions. Stress = force per unit cross-sectional area; strain = extension per unit original length (no unit); Young modulus = stress/strain (Pa), constant in the proportional region.
Definitions
F = kx σ = F/A ε = x/L E = σ/ε = FL/(Ax)
Worked Example
A steel wire of original length 2.00 m and diameter 0.50 mm extends by 4.0 mm under a load of 80 N. Find the stress, the strain and the Young modulus.
Practice Question
A spring extends by 4.0 cm when a load of 6.0 N is hung from it. Find its spring constant and the extension for a load of 9.0 N (within the limit of proportionality).
Experiment: Young modulus of a wire
Clamp a long (about 2 m), thin wire at one end and pass it over a pulley with a load hanger at the other. Measure the original length L to a marker with a metre rule, and the diameter at several points and orientations with a micrometer (average them). Add loads in steps and record the extension from the marker moving along a fixed scale (or a vernier). Plot load F against extension x: the gradient in the straight region gives F/x, so E = (gradient) × L/A. A long, thin wire gives an extension large enough to measure accurately.
2. Elastic and Plastic Behaviour
Elastic and plastic
Energy stored in a deformed material
Elastic deformation
Plastic deformation
elastic limit
area under a force–extension graph
Elastic potential energy (within the limit of proportionality)
EP = ½Fx = ½kx2