Modulus of Resilience Calculator – Formula, Units & Worked Example
Modulus of Resilience Calculator
Calculate the maximum elastic strain energy per unit volume a material can absorb before permanent (plastic) deformation begins — from yield strength and Young's modulus, from yield stress and strain, or from your own stress–strain data.
Calculator
Results update automatically as you type. All units convert internally — no manual conversion needed.
Show formula & calculation steps
Advanced: Total elastic energy for a volume (proof resilience)
Modulus of resilience is energy per unit volume. Enter a volume to get total recoverable elastic energy: \( R_p = U_r \times V \).
What modulus of resilience means
The modulus of resilience is the maximum energy per unit volume a material can store elastically — meaning it fully recovers, like a spring — without taking permanent (plastic) set. Geometrically, it's the shaded triangular area under the stress–strain curve, from zero strain up to the yield point.
How to calculate it
For a linear-elastic material, \( U_r = \sigma_y^2 / 2E \), which is equivalent to \( U_r = \tfrac{1}{2}\sigma_y\varepsilon_y \) because \( \varepsilon_y = \sigma_y/E \). The factor of one-half is essential — it comes from the triangular (not rectangular) area under the curve. For real, non-linear stress–strain data, the general definition is the integral \( U_r = \int_0^{\varepsilon_y}\sigma\,d\varepsilon \), evaluated here by trapezoidal numerical integration.
Units and conversions
Because stress and energy density share the same dimensions, \( 1\ \text{Pa} = 1\ \text{J/m}^3 \). This calculator lets you enter yield strength and Young's modulus in independent units (e.g. MPa and GPa) and displays the result in J/m³, kJ/m³, MJ/m³, or an equivalent pressure unit.
Worked example
For σy = 250 MPa and E = 200 GPa: yield strain εy = 250×10⁶ / 200×10⁹ = 0.00125. Then Ur = (250×10⁶)² / (2 × 200×10⁹) = 156,250 J/m³ = 156.25 kJ/m³.
Difference between resilience and toughness
Modulus of resilience covers only the elastic region up to yield. Modulus of toughness covers the full area under the curve up to fracture, including plastic deformation. A higher modulus of resilience means more elastic energy storage per unit volume — it does not mean the material is tougher overall.
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 and polymer grades) are often characterized using the 0.2% offset method. Preset material values are illustrative references, not certified design data — use certified material data sheets for engineering design and verification.
Frequently asked questions
What is the modulus of resilience?
It's the maximum strain energy per unit volume a material can absorb and still fully recover elastically, without permanent deformation.
What is the formula for modulus of resilience?
\( U_r = \sigma_y^2/2E = \tfrac{1}{2}\sigma_y\varepsilon_y \) for linear-elastic materials, or \( U_r = \int_0^{\varepsilon_y}\sigma\,d\varepsilon \) generally.
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 (modulus of resilience × volume).
What is the difference between resilience and toughness?
Resilience covers only the elastic region up to yield. Toughness covers the entire curve up to fracture, including the plastic region.
Can I calculate it from yield stress and strain?
Yes — use the "Yield Strength + Strain" mode above, which applies \( U_r = \tfrac{1}{2}\sigma_y\varepsilon_y \) directly.
What happens if the material does not have a clear yield point?
Use the 0.2% offset yield strength (a common convention for metals like aluminum) as your σy input, or use the stress–strain data mode to define the yield point manually.
Engineering disclaimer: this calculator is provided for educational and preliminary-design purposes only. Results should not replace certified material data, laboratory testing, or professional engineering verification for final design.
