What Is Modulus of Resilience?
Modulus of resilience measures how much energy a material can absorb elastically — that is, without any permanent damage — before it reaches its yield point. Stretch or compress a material within this range and it springs straight back to its original shape once the load is removed, releasing the stored energy rather than deforming.
Engineers rely on this number whenever a part has to absorb sudden loads and bounce back cleanly: springs, shock absorbers, energy-storing components, and anything exposed to repeated impacts or vibration.
The Modulus of Resilience Formula
For a material that behaves elastically (obeys Hooke's law) up to its yield point, the modulus of resilience is:
[
U_r = \frac{\sigma_y^2}{2E}
]
Where:
- U_r is the modulus of resilience, in energy per unit volume (e.g. J/m³ or kJ/m³)
- σᵧ is the yield stress — the stress at which the material stops behaving elastically
- E is Young's modulus — the material's stiffness
This comes directly from the elastic potential energy stored under a linear stress-strain curve: the area of a right triangle with base (strain at yield) and height (yield stress). Since a linear elastic material obeys σ = Eε, the strain at yield is εᵧ = σᵧ / E. Substituting into the triangle's area, ½·σᵧ·εᵧ, gives ½·σᵧ·(σᵧ/E) = σᵧ²/(2E) — which is exactly why the stress is squared.
Worked Example: Structural Steel
Take a common structural steel with:
- Yield stress (σᵧ): 250 MPa
- Young's modulus (E): 200 GPa
Converting both to pascals (250 × 10⁶ Pa and 200 × 10⁹ Pa) and applying the formula:
[
U_r = \frac{(250 \times 10^6)^2}{2 \times 200 \times 10^9} = 156{,}250 \text{ Pa} = 156.25 \text{ kJ/m}^3
]
That means every cubic metre of this steel can absorb about 156 kilojoules of energy elastically before any part of it yields — a figure that lands right in the middle of steel's typical range.
Modulus of Resilience for Common Materials
| Material | Approximate Value (kJ/m³) |
|---|---|
| Steel | 100–200 |
| Aluminum | 50–100 |
| Rubber | 5,000–10,000 |
| Glass | 1–5 |
| Wood | 50–150 |
Rubber sits far above the metals here — not because it's stronger, but because its Young's modulus is so low that it can stretch enormously before anything resembling a yield point, storing huge amounts of elastic energy per unit volume in the process.
Modulus of Resilience vs Modulus of Toughness
It's easy to confuse these two related-sounding properties, but they describe different parts of the same stress-strain curve:
| Property | Region of the curve | Describes |
|---|---|---|
| Modulus of resilience | Elastic region only, up to yield | Energy fully recovered on unloading |
| Modulus of toughness | Entire curve, up to fracture | Total energy absorbed before breaking, including permanent deformation |
A material can be resilient without being tough (glass stores a little elastic energy but shatters immediately after yielding), and tough without being especially resilient (mild steel deforms a lot plastically before it finally fractures).
Quick Recap
- U_r = σᵧ² / (2E) — modulus of resilience depends on yield stress squared, divided by twice the stiffness.
- It only measures the elastic region: energy that comes back when the load is removed.
- Materials with high yield strength and low stiffness — like spring steel and rubber — store the most energy per volume.
- Use the calculator above to check any material's modulus of resilience once you know its yield stress and Young's modulus.
If you're comparing how materials store and release energy, the elastic potential energy calculator is a natural next step.