Stress-Strain Formula
Calculate stress, strain, and Young's modulus for materials under load.
Essential for structural analysis and material selection.
The Formulas
Strain (epsilon) = deltaL / L₀
Young's Modulus: E = sigma / epsilon
Stress measures the internal force per unit area within a material. Strain measures how much the material deforms relative to its original length.
Young's modulus (E) links stress and strain in the elastic region. A higher Young's modulus means the material is stiffer and resists deformation more strongly.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| sigma (stress) | Force per unit area | pascals (Pa) or N/m² |
| F | Applied force | newtons (N) |
| A | Cross-sectional area | m² |
| epsilon (strain) | Relative change in length | dimensionless |
| deltaL | Change in length | m |
| L₀ | Original length | m |
| E | Young's modulus (modulus of elasticity) | Pa or GPa |
Common Young's Modulus Values
| Material | Young's Modulus | Category |
|---|---|---|
| Steel | 200 GPa | Very stiff |
| Aluminum | 69 GPa | Stiff |
| Copper | 117 GPa | Stiff |
| Concrete | 30 GPa | Moderate |
| Wood (oak) | 12 GPa | Flexible |
| Rubber | 0.01-0.1 GPa | Very flexible |
Example 1 — Steel Rod Under Tension
A steel rod has a cross-sectional area of 0.001 m² and is pulled with a force of 50,000 N. Find the stress and the elongation if the rod is 2 m long. (E for steel = 200 GPa)
Step 1: Stress = F / A = 50,000 / 0.001 = 50,000,000 Pa = 50 MPa
Step 2: Strain = stress / E = 50,000,000 / 200,000,000,000 = 0.00025
Step 3: deltaL = strain x L₀ = 0.00025 x 2 = 0.0005 m
Stress = 50 MPa, Elongation = 0.5 mm
Example 2 — Comparing Materials
An aluminum wire and a steel wire have the same dimensions (1 mm² cross-section, 1 m long). Both carry a 100 N load. Compare their elongation.
Stress = F / A = 100 / 0.000001 = 100 MPa (same for both)
Steel: strain = 100,000,000 / 200,000,000,000 = 0.0005 -> deltaL = 0.5 mm
Aluminum: strain = 100,000,000 / 69,000,000,000 = 0.00145 -> deltaL = 1.45 mm
The aluminum wire stretches about 2.9 times more than the steel wire
When to Use It
Use the stress-strain formula when:
- Selecting materials for structural components that must support specific loads
- Calculating how much a beam, rod, or cable will stretch or compress
- Determining if a material will stay within its elastic limit under a given force
- Comparing stiffness of different materials for engineering design
- Analyzing failure modes in mechanical and civil engineering