Young modulus and elastic strain energy

Key Definition The Young modulus describes the stiffness of a material and equals the ratio of stress to strain within the linear (Hooke's law) region of behaviour.

Key Definition Stress is the force acting per unit cross-sectional area, while strain is the extension per unit original length. Stress is measured in pascals (Pa), and strain has no units.

Core Practical 3

Core Practical 3 determines the Young modulus by stretching a long, thin wire laid horizontally over a pulley and loaded with known masses (weights).

The original length is measured with a metre ruler, and the diameter is measured at several points along the wire using a micrometer screw gauge so that an average can be taken to reduce random error and allow for any variation in thickness.

Plotting stress against strain produces a straight line in the elastic region, and the gradient of this line gives the Young modulus directly.

Elastic strain energy

Key Definition Elastic strain energy is the work done in deforming a material elastically, stored as elastic potential energy in the material.

For materials that obey Hooke's law, the energy stored is given by

$E_{\mathrm{el}}$ = ½Fx

or equivalently

$E_{\mathrm{el}}$ = ½kx²

On any force–extension graph, the elastic strain energy equals the area under the curve up to the final extension.

Linear graphs give a triangular area, while non-linear graphs (such as those for rubber) require the area to be estimated by counting squares or summing trapeziums.

Doubling the extension within the linear region quadruples the energy stored, because the energy is proportional to the square of the extension.