Young's Modulus Formula
Calculate Young's modulus (elastic modulus) using E = σ/ε.
Understand material stiffness and the relationship between stress and strain.
The Formula
Young's modulus measures a material's stiffness.
It tells you how much stress is needed to produce a given amount of strain.
A higher value means the material is stiffer and resists deformation more.
Variables
| Symbol | Meaning |
|---|---|
| E | Young's modulus (Pascals, Pa) |
| σ | Stress (Pascals, Pa) |
| ε | Strain (dimensionless) |
Common Values
| Material | Young's Modulus (GPa) |
|---|---|
| Steel | 200 |
| Aluminium | 69 |
| Copper | 117 |
| Concrete | 30 |
| Wood (along grain) | 11 |
| Rubber | 0.01 - 0.1 |
Example 1
A steel bar has a stress of 400 MPa and a strain of 0.002. What is Young's modulus?
E = σ / ε
E = 400,000,000 Pa / 0.002
E = 200,000,000,000 Pa = 200 GPa
Example 2
An aluminium rod experiences 138 MPa of stress and stretches with a strain of 0.002. Find the modulus.
E = σ / ε
E = 138,000,000 Pa / 0.002
E = 69,000,000,000 Pa = 69 GPa
When to Use It
Use Young's modulus when you need to:
- Compare the stiffness of different materials
- Predict how much a component will deform under a known load
- Select materials for structural applications
- Design springs, beams, and other elastic components
This formula only applies within the elastic region of a material.
Beyond the yield point, the relationship between stress and strain is no longer linear.
Key Notes
- Formula: E = σ / ε = (F/A) / (ΔL/L₀): E is Young's modulus (Pa or GPa), σ is stress (force per area), and ε is strain (fractional length change). A higher E means a stiffer material that deforms less under the same load.
- Material-specific constant: Young's modulus is a property of the material, not the shape of the object. Steel: ~200 GPa; aluminum: ~69 GPa; rubber: ~0.01–0.1 GPa. Temperature affects E — materials generally become less stiff when heated.
- Only valid within the elastic region: Young's modulus applies only below the yield strength of the material. Beyond the yield point, the stress-strain relationship becomes nonlinear and permanent deformation occurs.
- Three elastic moduli: Young's modulus (tension/compression), shear modulus G (shear deformation), and bulk modulus K (volumetric compression under pressure) are the three fundamental elastic constants. They are related through Poisson's ratio.
- Engineering applications: Young's modulus is used to design beams, select materials for springs, predict deflection in bridges, and ensure structural components don't deform excessively under load.