Library / Foundations: Physics / FPH-18
Foundations: Physics
Young’s modulus from a test
Stiffness of a material from stress and strain.
- Ref
- FPH-18
- Rev
- v1
- Field
- Foundations: Physics
- Result
- E [GPa]
Use original area and length. Young’s modulus E is the stress–strain gradient in the linear elastic region, measured in Pa (shown in GPa).
A stiffer material needs more stress to stretch by the same fraction.
Calculator
- Stress σ200 MPa
- Strain ε1,000 µε
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Steel is about 200 GPa, aluminium 70 GPa, so aluminium stretches about three times as much under the same stress.
- E is the gradient of the straight part of the stress–strain graph.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (GPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Cross-section area | 1 to 1,000 mm² | 20 | 20,000 | 19,980 | +10% | −9.09 % |
| ΔL Extension | 0.1 to 20 mm | 20 | 4,000 | 3,980 | +10% | −9.09 % |
| F Force | 0.1 to 200 kN | 1 | 2,000 | 1,999 | +10% | +10 % |
| L₀ Original length | 0.5 to 10 m | 50 | 1,000 | 950 | +10% | +10 % |
Sweep of Extension
| Extension (mm) | Young’s modulus (GPa) |
|---|---|
| 0.1 | 4,000 |
| 0.1939 | 2,063 |
| 0.3761 | 1,064 |
| 0.7293 | 548.5 |
| 1.414 | 282.8 |
| 2.742 | 145.9 |
| 5.318 | 75.21 |
| 10.31 | 38.78 |
| 20 | 20 |
Sweep one variable
Variables and how each one contributes
Young’s modulus
What the equation gives you. Everything else feeds it.
Force
Tensile force. Directly proportional.
Cross-section area
Original area. Inversely proportional.
Extension
Measured extension. Inversely proportional: a stiffer wire stretches less.
Original length
Original length. Directly proportional.
Worked example with the values above
- Stress: σ = F/A = 20,000 N / 100 mm² = 200 MPa.
- Strain: ε = ΔL/L₀ = 0.002 / 2 = 0.001.
- E = σ/ε = 200 GPa.
Assumptions and limits
- Linear elastic region.
- Original area and length used.
Common mistakes
- Using mm² with N and forgetting the 10⁶ factor.
- Measuring beyond the elastic limit.
Statistics and uncertainty
Extension is tiny and hard to measure; a 0.1 mm error on 2 mm is 5% on E, usually the largest error in the test.
Where it is used
- Material testing.
- Selecting materials for stiffness.
Maths you need
- Stress and strain: force per area; extension per length
- Gradient: slope of a graph