Elastic potential energy is the energy stored in a body when it is stretched, compressed, bent, or otherwise
deformed elastically. If the deformation is elastic, the body can give this energy back when the load is removed.
1. Linear elastic force-deformation relation
For many small deformations, force is proportional to deformation:
\[
F=kx.
\]
Here, \(k\) is stiffness and \(x\) is extension, compression, or deflection.
2. Energy stored in a linear spring or structure
The stored energy is the area under the force-deformation graph:
\[
U=\int_0^x F\,dx.
\]
If \(F=kx\), then
\[
\boxed{
U=\frac12kx^2
}.
\]
Since the final force is \(F=kx\), the same result can be written as
\[
\boxed{
U=\frac12Fx
}.
\]
The factor \(1/2\) appears because the force grows from zero to its final value as the object deforms.
3. Wire or rod in tension
For an elastic wire or rod,
\[
\varepsilon=\frac{\Delta L}{L},
\qquad
\sigma=E\varepsilon.
\]
The force is
\[
F=\sigma A.
\]
Combining these gives
\[
F=\frac{EA\Delta L}{L}.
\]
So the axial stiffness is
\[
\boxed{
k=\frac{EA}{L}
}.
\]
4. Strain energy in a wire from extension
Starting from \(U=\frac12F\Delta L\) and \(F=EA\Delta L/L\),
\[
\boxed{
U=\frac{EA(\Delta L)^2}{2L}
}.
\]
5. Strain energy in a wire from force
If the force is known, then
\[
\Delta L=\frac{FL}{EA}.
\]
Substituting into \(U=\frac12F\Delta L\) gives
\[
\boxed{
U=\frac{F^2L}{2EA}
}.
\]
6. Energy density from stress and strain
Strain energy density is stored energy per unit volume:
\[
u=\frac{U}{V}.
\]
For a linear elastic material,
\[
\boxed{
u=\frac12\sigma\varepsilon
}.
\]
Since \(\sigma=E\varepsilon\), this can also be written as
\[
\boxed{
u=\frac12E\varepsilon^2
}.
\]
7. General stress-strain curve
If the stress-strain curve is not perfectly linear, the stored or absorbed energy density is the area under the
curve:
\[
\boxed{
u=\int \sigma\,d\varepsilon
}.
\]
For discrete data points, the area can be estimated with trapezoids:
\[
u\approx
\sum_i
\frac{\sigma_i+\sigma_{i+1}}{2}
(\varepsilon_{i+1}-\varepsilon_i).
\]
Stress has units of \(\mathrm{Pa}\), and strain is dimensionless, so
\[
\mathrm{Pa}=\mathrm{N/m^2}=\mathrm{J/m^3}.
\]
Therefore, the stress-strain area has units of energy per volume.
8. Total energy from energy density
Once energy density is known, total stored energy is
\[
\boxed{
U=uV
}.
\]
9. Springs, wires, and beams
| Case |
Main formula |
Graph interpretation |
| Spring |
\(U=\frac12kx^2\) |
Area under \(F=kx\) |
| Wire from extension |
\(U=\frac{EA(\Delta L)^2}{2L}\) |
Area under force-extension graph |
| Wire from force |
\(U=\frac{F^2L}{2EA}\) |
Area under force-extension graph |
| Linear beam deflection |
\(U=\frac12F\delta\) |
Area under load-deflection graph |
| General stress-strain curve |
\(u=\int\sigma d\varepsilon\) |
Area under stress-strain graph |
10. Worked example: wire stretched by a force
Suppose a wire is stretched with
\[
F=500\ \mathrm{N},
\qquad
\Delta L=4\ \mathrm{mm}.
\]
Convert extension to metres:
\[
\Delta L=4\ \mathrm{mm}=0.004\ \mathrm{m}.
\]
The stored elastic energy is
\[
U=\frac12F\Delta L.
\]
\[
U=\frac12(500)(0.004).
\]
\[
\boxed{
U=1.0\ \mathrm{J}
}.
\]
This means one joule of elastic potential energy is stored in the wire, assuming the force-extension behavior is
linear.
11. Important limitations
These formulas assume elastic behavior. If the material yields, cracks, or permanently deforms, not all of the
input work is recoverable elastic potential energy.
Key idea: elastic potential energy is the area under the loading curve. For a linear elastic object, that area is a triangle.
