Elastic Strain Energy Density
Elastic Strain Energy Density
When a material is loaded elastically, it stores energy the way a spring does, and this stored energy per unit volume — the strain energy density — is exactly the area under the linear portion of the stress-strain curve. It is the basis for a material property called modulus of resilience, used to compare how much energy different materials can absorb and still fully recover. Because this quantity depends on stress squared but only linearly on the inverse of stiffness, a material with high strength but modest stiffness can actually store more recoverable energy than a much stiffer but weaker one — an important and sometimes counterintuitive consideration when selecting materials for springs or energy-absorbing components.
Elastic strain energy density is u = sigma^2/(2E). where sigma_u is the applied uniaxial stress and E_u is the material's elastic (Young's) modulus.
Inputs
sigma_u := 150 MPa=150 MPa
E_u := 200 GPa=200000 MPa
This is just the triangular area under a linear stress-strain curve up to the applied stress, expressed directly in terms of stress and modulus rather than stress and strain.
u_energy := sigma_u^2/(2·E_u)=56250 Pa
Results
The resulting energy density here is on the order of tens of kilojoules per cubic metre, a modest but physically meaningful value for a moderately loaded steel component. Evaluating this at the material's yield strength instead gives the modulus of resilience, a standard property for comparing spring materials — since it grows with the square of yield strength, high-strength steels can store dramatically more elastic energy per unit volume than mild steel even at similar stiffness.
Assumptions & limitations
- Linear elastic material behavior — not valid once the material yields.
- Uniaxial stress state assumed (no multiaxial correction).
- Represents recoverable (elastic) energy only, not the total work done including plastic deformation.
- Assumes isotropic, homogeneous material properties.