Energy density from stress, strain, and volume. Pick units, enter properties, and view computed strains. Download clean reports for design checks and documentation today.
Choose a method and loading state. Inputs accept decimals and scientific notation (e.g., 2.1e5). Stresses and modulus share the same unit selector.
| Case | Inputs (typical) | Energy density u (approx.) |
|---|---|---|
| Uniaxial steel | σ = 250 MPa, E = 200 GPa | 156250 Pa (0.15625 MPa) |
| Pure shear | τ = 80 MPa, G ≈ 77 GPa | 41558 Pa (0.04156 MPa) |
| Plane stress | σx=120 MPa, σy=60 MPa, τxy=30 MPa, E=210 GPa, ν=0.30 | 38143 Pa (0.03814 MPa) |
For linear elastic behavior, the strain energy density u is:
Common special cases:
Strain energy density, u, is stored elastic energy per unit volume. In linear elasticity it equals work done by stresses while strains develop. Since 1 Pa = 1 J/m³, energy density can be reported in stress units or in J/m³.
For a steel bar at 250 MPa with E ≈ 200 GPa, u = σ²/(2E) ≈ 1.56×105 J/m³ (0.156 MPa). Values from 104 to 106 J/m³ are common for elastic machine parts. For quick reading, 1 kJ/m³ equals 1000 Pa, and 1 MJ/m³ equals 1 MPa.
If σ = 120 MPa and E = 210 GPa, the elastic strain is ε ≈ 5.71×10-4 (571 microstrain). Then u ≈ 3.43×104 J/m³. If your output is far higher, confirm that E matches the selected stress unit.
For isotropic materials, G = E/(2(1+ν)). With E = 210 GPa and ν = 0.30, G ≈ 80.8 GPa. For τ = 80 MPa, γ ≈ 9.90×10-4, and u = τ²/(2G) ≈ 3.96×104 J/m³. In torsion, this often dominates fatigue hot-spots.
In thin plates, σz ≈ 0 and only σx, σy, and τxy contribute. For σx=120 MPa, σy=60 MPa, τxy=30 MPa, E=210 GPa, ν=0.30, u is about 3.8×104 J/m³. Mixed loading increases u, so it is a useful scalar for comparing different stress states.
In principal coordinates, shear terms vanish and u depends on σ1, σ2, σ3 with Poisson coupling. This helps compare materials: steel (E≈200 GPa, ν≈0.30) and aluminum (E≈69 GPa, ν≈0.33). Lower E typically produces higher u for the same stress level, which matters in spring-like designs.
Multiply u by volume to estimate total stored energy U. For u = 4.0×104 J/m³ and V = 2.0×10-3 m³, U ≈ 80 J. As a comparison, lifting an 8 kg mass by 1 meter stores about 78 J, so even “small” elastic volumes can store meaningful energy.
Keep stresses and modulus in the same unit family (MPa with MPa, psi with psi). Use microstrain for strain-gauge data, and choose “Direct stress & strain” when strains come from testing or FEA. Near ν → 0.5, small ν errors noticeably change G and shear energy.
Q1. What is the output unit for strain energy density?
In SI, u is J/m³, which is numerically equal to Pa. The calculator also displays u in your selected stress unit (MPa, psi, etc.) plus kJ/m³ and MJ/m³ for easier scaling.
Q2. Do I need to enter strain values?
Only in Direct mode. In isotropic mode, strains are computed from stress, E, and ν using linear elastic relations. Use Direct mode when strains come from gauges, experiments, or FEA results.
Q3. Why is Poisson’s ratio limited below 0.5?
For stable isotropic elasticity, ν must be less than 0.5. As ν approaches 0.5, the material becomes nearly incompressible and small ν errors can strongly affect G and shear-related energy terms.
Q4. How should I enter E when using MPa?
Select MPa as the stress unit, then enter E in MPa too (for steel, about 200000 MPa). If you enter 200 GPa while MPa is selected, the value will be off by a factor of 1000.
Q5. Can the result be negative?
u should be non‑negative for physically consistent linear elastic states. If you enter stress and strain with opposite signs in Direct mode, individual terms can reduce the sum, indicating inconsistent sign conventions.
Q6. What volume should I use for total energy?
Use the stressed volume of the part region you are analyzing. For a uniform bar, V = cross‑section area × length. For nonuniform stress, a local or element volume from FEA is more appropriate.
Q7. Is this valid for plastic deformation?
No. The formulas assume linear elasticity and small strains. Once yielding or nonlinear material behavior occurs, energy must be computed from the full stress–strain curve, not σ²/(2E) or τ²/(2G).
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.