Modulus of Resilience Calculator – Formula, Units & Worked Example
SteelSolver · Engineering Calculators
Modulus of Resilience Calculator
Find the maximum elastic strain energy per unit volume a material can absorb — like a spring — before it takes permanent set.
Preset values are typical reference values only. Use certified material data for design.
Units convert automatically — MPa and GPa can be mixed freely.
Formula & substitution
Assumption: linear-elastic behavior up to the yield point (Hooke's law). Educational estimate — not a substitute for certified material data.
Stress–strain view
The shaded triangle is the modulus of resilience — the elastic region only, up to yield. It is not the full area to fracture (that is modulus of toughness).
Proof resilience — total elastic energy for a component
Total recoverable elastic energy \( R_p = U_r \times V \), where V is the component volume.
Uncertainty analysis
First-order propagation: \( \Delta U_r/U_r \approx 2\Delta\sigma_y/\sigma_y + \Delta E/E \) (simple sum) or the root-sum-square form for independent random uncertainties.
Compare materials
Add up to five materials to compare elastic energy-storage capacity.
| Material | σy (MPa) | E (GPa) | εy | Ur (kJ/m³) |
|---|
Sensitivity
Enter values above to see sensitivity.
What modulus of resilience means
Modulus of resilience is the maximum elastic strain energy a material can absorb per unit volume before it begins to deform permanently. Picture a steel spring: load it within its elastic range and it snaps back to its original shape, releasing the stored energy. Push it past yield and some of that shape change becomes permanent. Modulus of resilience quantifies how much energy the spring can store while staying fully recoverable.
How to calculate it
For a linear-elastic material, the area under the stress–strain curve up to yield is a triangle, so \( U_r = \tfrac12 \sigma_y \varepsilon_y \). Since \( \varepsilon_y = \sigma_y/E \), this simplifies to \( U_r = \sigma_y^2/2E \). For materials that are not perfectly linear, or when you have real test data, the general definition is the integral \( U_r = \int_0^{\varepsilon_y}\sigma\,d\varepsilon \), evaluated numerically with the trapezoidal rule.
Units and conversions
| Quantity | Common units | Note |
|---|---|---|
| Yield strength, E | Pa, MPa, GPa, psi, ksi | Independent unit selectors, converted automatically |
| Modulus of resilience | J/m³, kJ/m³, MJ/m³, Pa, MPa, psi | 1 J/m³ = 1 Pa dimensionally |
| Yield strain | decimal, %, µε | 0.2% is converted internally to 0.002 |
Worked example
For σy = 250 MPa and E = 200 GPa: εy = 250×10⁶ / 200×10⁹ = 0.00125. Then Ur = (250×10⁶)² / (2 × 200×10⁹) = 156,250 J/m³ = 156.25 kJ/m³. The material can theoretically store about 156.25 kJ of elastic strain energy per cubic metre before reaching that yield strength, assuming linear-elastic behavior.
Difference between resilience and toughness
Modulus of resilience covers only the elastic region — the shaded triangle up to yield in the diagram above. Modulus of toughness covers the entire area under the stress–strain curve, including the plastic region, all the way to fracture. A material can have high resilience but low toughness, or the reverse; they describe different aspects of mechanical behavior.
Assumptions and limitations
The σy²/2E formula assumes linear-elastic behavior up to a well-defined yield point. Materials without a sharp yield point (many aluminum alloys, polymers, cast iron, concrete, elastomers) are usually characterized using a 0.2% offset yield strength instead — use the Stress–Strain Data mode for those. All material preset values are illustrative references, not design values; actual properties depend on grade, heat treatment, temperature, loading direction, strain rate, and the applicable test standard.
Material-selection applications
Modulus of resilience is most relevant to springs, shafts, fasteners, impact buffers that must not take a permanent set, and any component subjected to repeated elastic loading. A higher modulus of resilience means the material can store more elastic energy per unit volume before yielding — useful for comparing candidate materials in energy-storage or shock-absorbing applications.
Frequently asked questions
What is the modulus of resilience?
It is the maximum elastic strain energy per unit volume a material can absorb before it starts to yield permanently — the area under the elastic part of the stress–strain curve.
What is the formula for modulus of resilience?
\( U_r = \sigma_y^2/2E \), equivalently \( U_r = \tfrac12\sigma_y\varepsilon_y \), or the integral \( \int_0^{\varepsilon_y}\sigma\,d\varepsilon \) for non-linear data.
What is the SI unit?
Joules per cubic metre (J/m³), which is dimensionally identical to the pascal (Pa).
Is modulus of resilience the same as proof resilience?
No. Modulus of resilience is energy per unit volume; proof resilience is the total elastic energy for a specific component, equal to modulus of resilience multiplied by volume.
What is the difference between resilience and toughness?
Resilience covers only the elastic region up to yield. Toughness covers the full stress–strain curve up to fracture, including the plastic region.
Can I calculate it from yield stress and strain?
Yes — use the "Strength + Strain" mode above with \( U_r = \tfrac12\sigma_y\varepsilon_y \).
Why is the result also dimensionally expressed in pascals?
Because 1 J/m³ = 1 Pa dimensionally. The energy-per-volume interpretation is the physically meaningful one, but the pascal-equivalent numeric value is identical.
Can this calculator use psi and ksi?
Yes, psi and ksi are available for stress, modulus, and result units throughout the calculator.
What happens if the material does not have a clear yield point?
Use the Stress–Strain Data mode with the 0.2% offset method, which estimates the yield point from the initial elastic slope and a 0.2% strain offset.
Can I calculate total elastic energy for a component?
Yes — open "Proof resilience" below the diagram and enter a volume or simple geometry to get total recoverable elastic energy in joules.
