Force-Extension and Force-Compression Graphs

A force-extension graph shows how a material deforms when a force is applied.

Reading the graph

Straight section through the origin: the material obeys Hooke’s law. Extension is proportional to force.

$k = \frac{F}{x} = \text{gradient of the straight section}$

Limit of proportionality: the point where the graph begins to curve. Beyond this, $F$ is no longer proportional to $x$.

Elastic limit: the maximum force that can be applied and still allow the material to return to its original length when unloaded. This may be at or slightly beyond the limit of proportionality.

Elastic vs plastic deformation

• Elastic deformation: the object returns to its original shape when the force is removed. No permanent change.
• Plastic deformation: the object does not return to its original shape. Permanent deformation has occurred.

Loading and unloading curves

If the unloading curve follows the same path as the loading curve, the deformation is elastic. If the unloading curve is below the loading curve, energy has been dissipated and some deformation is permanent.

Energy stored in a spring

The energy stored in a stretched spring equals the area under the force-extension graph. For the linear section:

$E = \frac{1}{2}Fx = \frac{1}{2}kx^2$

Common errors and how to correct them

• Reading the total length instead of the extension from the graph.
• Confusing the limit of proportionality with the elastic limit (they are close but not always identical).

How to apply this in an exam

Calculate $k$ from the gradient of the straight section. Identify the limit of proportionality by finding where the curve starts. State whether behaviour is elastic or plastic.