Strain Energy Density⚠ unverified
Mechanical / Stress Analysis · Compute the elastic strain energy density for a uniaxial stress state
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| sigma | σ | Pa | 1.0 | Applied normal stress |
| E | E | Pa | 1.0 | Young's modulus of the material |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | u | J/m**3 | Strain energy density, 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.