Energy past the elastic limit: stored or lost?

8 minAS Level6.2.26.2.36.2.4

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can separate the work done stretching an object from the energy recovered when it unloads, and calculate the energy dissipated when the elastic limit is passed.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$E_P$	elastic potential energy

The area still measures work — but not all of it comes back

Area under the force–extension graph always measures the work done by the stretching force. That does not change when the graph curves.

What changes is where the energy goes.

Within the elastic limit, the material stores the work as elastic potential energy, and gives all of it back when unloaded.
Beyond the elastic limit, some of the work permanently rearranges the material. That part is dissipated — it becomes internal energy, so the material warms slightly.

The energy that does come back is the area under the unloading curve, because that is the work the material does on whatever is releasing it.

Bending a paper clip until it is warm

Bend a metal paper clip back and forth a few times and touch the bend. It feels warm.

You did work on the metal each time. Because the bending was plastic, not all of that work was stored elastically — some became internal energy, and the clip's temperature rose.

The same idea explains why a car crumple zone absorbs energy in a crash: the metal deforms plastically and does not spring back, so the kinetic energy is not returned to the car.

Worked example: Worked example: work in, energy out

A wire is stretched in two stages:

the force rises steadily from $0$ to $12\ \text{N}$ while the extension goes from $0$ to $0.030\ \text{m}$ ;
the force then stays at $12\ \text{N}$ while the extension increases from $0.030\ \text{m}$ to $0.050\ \text{m}$ .

The load is then removed, and the wire unloads along a line parallel to the first stage.

Find (a) the work done stretching the wire, (b) the energy recovered, and (c) the energy dissipated.

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