Stress-Strain and Young's Modulus
Calculate stress, strain, and Young's modulus for materials under tension or compression.
Covers elastic limit, yield strength, and deformation examples.
The Formulas
Strain (ε) = Change in Length / Original Length = ΔL / L₀
Young's Modulus (E) = Stress / Strain = σ / ε
Stress measures the internal force per unit area in a material. Strain measures how much the material deforms relative to its original size. Young's modulus describes how stiff a material is.
Variables
| Symbol | Meaning | Units |
|---|---|---|
| σ | Stress | Pascals (Pa) or N/m² |
| F | Applied force | Newtons (N) |
| A | Cross-sectional area | Square meters (m²) |
| ε | Strain (dimensionless) | No units (ratio) |
| ΔL | Change in length | Meters (m) |
| L₀ | Original length | Meters (m) |
| E | Young's modulus | Pascals (Pa) or GPa |
Common Young's Modulus Values
| Material | Young's Modulus (GPa) |
|---|---|
| Steel | 200 |
| Aluminum | 69 |
| Copper | 117 |
| Concrete | 30 |
| Wood (along grain) | 11 |
| Rubber | 0.01–0.1 |
Example 1
A steel rod with cross-section 0.0001 m² is pulled with 20,000 N. What is the stress?
σ = F / A = 20,000 / 0.0001
σ = 200,000,000 Pa = 200 MPa
Example 2
A 2 m aluminum rod stretches by 0.58 mm under a stress of 20 MPa. Verify the Young's modulus.
ε = ΔL / L₀ = 0.00058 / 2 = 0.00029
E = σ / ε = 20,000,000 / 0.00029
E ≈ 69 GPa (matches the known value for aluminum)
When to Use It
Use stress-strain formulas in structural and materials engineering:
- Designing beams, columns, and structural elements
- Selecting appropriate materials for a given load
- Predicting how much a part will stretch or compress under force
- Ensuring components stay within safe stress limits
Key Notes
- The stress-strain curve tells the material's story: Starting from the origin, the curve rises linearly (elastic region), reaches the proportional limit, then the yield point. Beyond yield: plastic deformation (permanent). The peak is the ultimate tensile strength (UTS). Fracture ends the curve.
- Young's modulus E = σ/ε (slope of the elastic region): The steeper the slope, the stiffer the material. Steel: E ≈ 200 GPa; aluminum: ≈ 70 GPa; rubber: ≈ 0.01–0.1 GPa. In the elastic region, strain is fully reversible — remove the stress, the material returns to its original shape.
- Poisson's ratio ν = −ε_lateral / ε_axial: When a material is stretched axially, it contracts laterally. Poisson's ratio is typically 0.25–0.35 for metals, ≈ 0.5 for rubber (nearly incompressible). It links the three elastic constants: E, G (shear modulus), and K (bulk modulus).
- Ductile vs brittle fracture: Ductile materials (mild steel, copper) have a long plastic region — they deform visibly before fracture, giving warning. Brittle materials (glass, cast iron, ceramics) fracture at or near the yield point with little plastic deformation and no warning.
- Applications: Stress-strain analysis is used in material selection for structural components, failure analysis investigations, finite element modeling (FEM), fatigue life prediction, pressure vessel design, and quality control of manufactured metal parts (tensile testing).