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Modulus of Resilience
Quantifies the elastic strain energy a material can absorb per unit volume up to its yield point, without permanent deformation.
Interpretation
U_r = σ_y²/(2E). Energy per unit volume that a material can absorb up to yield. Indicates ability to resist elastic deformation. Used in impact and spring design.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| U_r | Modulus of resilience | MJ/m3 |
| sigma_y | Yield strength | MPa |
| E | Young's modulus | GPa |
What it means
The modulus of resilience (U_r) is the strain energy per unit volume that a material can absorb without permanent deformation. It is calculated from the yield strength σ_y and Young’s modulus E: U_r = σ_y²/(2E). This represents the area under the elastic portion of the stress‑strain curve. Materials with high resilience are good for springs and components that need to store elastic energy, such as in shock absorbers and bumpers. Resilience is different from toughness (which includes plastic deformation). Understanding resilience is important for selecting materials for energy‑absorbing applications and for evaluating the service life of elastic components.
Worked example
Modulus of Resilience – Two Examples
Real‑World| Parameter | Value |
|---|---|
| σ_y | 250 MPa |
| E | 200 GPa |
| Parameter | Value |
|---|---|
| σ_y | 900 MPa |
| E | 200 GPa |
Common mistakes
- Modulus of resilience U_r: The strain energy per unit volume stored in a material up to the yield point.
- Yield strength σ_y: In Pa – use the same units as E.
- Young’s modulus E: In Pa.
- Units: U_r in J/m³ (same as Pa).
- Interpretation: A measure of the material’s ability to absorb energy in the elastic region – higher U_r means better resilience.
Applications
Modulus of resilience (U_r = σ_y²/(2E)) is the strain energy per unit volume that a material can absorb without permanent deformation. It represents the material's ability to absorb energy elastically. Engineers use it to compare materials for applications where elastic energy absorption is important, such as in springs, shock absorbers, and trampolines. High resilience materials (like spring steels) are chosen for cyclic loading where they must return energy. By calculating U_r, materials scientists can screen materials for energy‑storage applications and optimise designs for impact resistance.
- Spring and elastic component design
- Energy‑absorbing materials in automotive and safety systems
- Vibration damping and isolation
- Design of resilient structures and shock absorbers
- Material selection for energy‑efficient systems
Frequently Asked Questions
The modulus of resilience is the energy per unit volume that a material can absorb without permanent deformation. It is calculated from the area under the elastic portion of the stress‑strain curve: U_r = σ_y² / (2E), where σ_y is the yield strength and E is Young's modulus.
Confusing it with toughness. Toughness includes the energy absorbed up to fracture (including plastic deformation), while resilience is only the elastic energy.
Since it is energy per unit volume, the units are J/m³ (or MPa). It can also be expressed in MPa.
Materials with high resilience can absorb more elastic energy, making them suitable for springs, shock absorbers, and applications requiring repeated loading without fatigue.
- Structural steel: ~0.1‑0.3 MJ/m³.
- Spring steel: ~1‑3 MJ/m³.
- Rubber: ~0.1‑0.5 MJ/m³ (but non‑linear).
- Aluminium: ~0.05‑0.1 MJ/m³.
It is the integral of σ·dε from zero to the elastic limit. For linear elastic materials, it is simply the area of the triangle: ½·σ_y·ε_y, which simplifies to σ_y²/(2E).
Heat treatment can increase the yield strength (σ_y) without significantly changing E, thus increasing the modulus of resilience. This is why hardened steels are used for springs.
Resilience is the ability to absorb elastic energy; toughness is the ability to absorb energy up to fracture (including plastic deformation). Toughness is always greater than resilience.
- Springs and shock absorbers.
- Bows and crossbows.
- Energy‑absorbing structures.
- Elastic bands.
- Only accounts for elastic deformation.
- Assumes linear elastic behaviour.
- Does not consider strain‑rate effects or fatigue.