Strain Energy Density⚠ unverified
Physics / Materials · Compute elastic strain energy density
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| stress | stress | Pa | 1.0 | Normal stress |
| strain | strain | — | 1.0 | Corresponding strain, dimensionless |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | U | J/m^3 | Strain energy per unit volume, in joules per cubic metre (J/m^3) |
The science & history
Understanding the Parameters
- $\sigma$, $\varepsilon$ — must be the conjugate elastic pair on the same uniaxial path.
- $U$ — area under the stress–strain curve to the current point (triangle for linear elastic).
Derivation (Approaching a Proof)
Work density $dU = \sigma\,d\varepsilon$. Integrate from 0 to $\varepsilon$ with $\sigma = E\varepsilon$: $U = \int_0^{\varepsilon} E\varepsilon'\,d\varepsilon' = \tfrac12 E\varepsilon^{2} = \tfrac12\sigma\varepsilon$.
History
Strain energy methods (Castigliano, etc.) are classical structural analysis tools.
Related Concepts: Hooke's Law strain, Hookes Law Strain Stress, Von Mises Stress
Notes: Registry calculator materials-strain-energy-density (unverified). Uniaxial linear elastic.