Hookes law and deformation

Key Definition Hooke’s law states that the force applied to an elastic object is directly proportional to its extension, provided the limit of proportionality is not exceeded.

F = kx

The constant of proportionality, k, is the stiffness (also called the spring constant or force constant), measured in N m⁻¹.

To compare the behaviour of materials independently of sample size, three quantities are used:

• Stress — the force applied per unit cross-sectional area (measured in Pa, i.e. N m⁻²)
• Strain — the extension expressed as a fraction of the original length (a ratio, so it has no units)
• The Young modulus — the ratio of stress to strain in the linear (Hooke’s law) region (measured in Pa)

stress, σ = F / A strain, ε = x / L Young modulus, E = σ / ε

The Young modulus is a property of the material, while stiffness is a property of the object.

A force-extension graph or stress-strain graph reveals key behaviour points:

• The limit of proportionality marks where Hooke’s law fails, so stress is no longer proportional to strain.
• The elastic limit marks the point beyond which deformation becomes permanent.
• Beyond the yield point, the material extends rapidly with little extra force.
• The breaking stress is the maximum stress the material can withstand before it fractures.

Brittle materials fail with little plastic deformation, while ductile materials show a long plastic region before breaking.

The gradient of the linear region of a stress-strain graph gives the Young modulus, allowing the same data to be used both to identify a material and to predict whether it will fail under a given load.