Hooke’s Law and Elastic Energy

For many materials, the extension is proportional to the applied force, provided the $\text{limit of proportionality}$ is not exceeded: $F=k\Delta x$. The constant $k$ is the $\text{stiffness}$ (spring constant), in $\mathrm{N}{\mathrm{m}}^{-1}$.

On a force–extension graph this is a straight line through the origin with gradient $k$. A stiffer spring has a steeper line — it takes more force for the same extension.

Beyond the limit of proportionality the graph curves. The material may still return to its original length up to the $\text{elastic limit}$, but past that point it is permanently deformed.

Two identical springs side by side in $\text{parallel}$ share the load, so each extends half as much and the combination is twice as stiff: $k=k_1+k_2$.

Two springs end to end in $\text{series}$ each carry the full load, so each extends fully and the total extension adds. The combination is less stiff: $\frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}$.

Note these rules are the reverse of those for resistors — parallel springs add directly, series springs add reciprocally.

$\text{Elastic}$ deformation is fully reversible: remove the load and the material returns to its original shape. At the atomic scale, bonds stretch but no atoms change place.

$\text{Plastic}$ deformation is permanent. Beyond the elastic limit, layers of atoms slide past one another, so removing the load leaves a permanent extension.

On a loading–unloading graph, elastic behaviour retraces the same line; plastic behaviour unloads along a line parallel to the original but offset, meeting the extension axis at the permanent extension.

Work done stretching a spring is stored as $\text{elastic strain energy}$. Because the force increases from zero, the work is the $\text{area under the force–extension graph}$, not simply force times extension.

Within Hooke’s law that area is a triangle of base $\Delta x$ and height $F$, giving $E=\frac{1}{2}F\Delta x$. Substituting $F=k\Delta x$ gives the alternative $E=\frac{1}{2}k\Delta x^2$.

Beyond the limit of proportionality the graph is curved, so the formula no longer applies — the area must be estimated, for instance by counting squares.