Young's Modulus Calculator
With this Young's modulus calculator, you can obtain the modulus of elasticity of a material, given the strain produced by a known tensile or compressive stress. You can also calculate Young's modulus directly from a stress–strain curve by entering two points along its straight (elastic) portion. The tool supports both the US customary (Imperial) and the metric (SI) measurement systems, and automatically shows the result in both.
What is the modulus of elasticity?
Young's modulus, or modulus of elasticity, is a property of a material that tells us how difficult it is to stretch or compress the material in a given axis. The relation between the longitudinal strain and the stress that causes it is linear, so we can write Young's modulus as the quotient of the two terms.
This linear relation only holds up to a certain amount of stress. The region where the stress–strain proportionality remains constant is called the elastic region. If we remove the stress after stretching or compressing the material within this region, it returns to its original length. Because of that, Young's modulus is only defined within this elastic region.
Young's modulus equation
First, let's define the longitudinal strain ε — the relative change in length:
and the stress σ — the force F applied per unit cross-sectional area A:
Then the modulus of elasticity formula is simply:
- E — Young's modulus (Pa, MPa, GPa, or psi, ksi, Mpsi)
- σ — tensile / compressive stress
- ε — longitudinal strain (dimensionless)
- L₀ — original length, ΔL — change in length
How do I calculate Young's modulus?
- Determine the stress σ applied to the material (or compute it from force and area, σ = F/A).
- Measure the strain ε it produces (or compute it from the elongation, ε = ΔL/L₀).
- Divide the stress by the strain: E = σ / ε.
- Make sure the deformation stays within the elastic region, otherwise the result is not valid.
Example using the modulus of elasticity formula
Suppose a steel rod has an original length of L₀ = 2 m and a cross-sectional area of A = 100 mm² = 1 × 10⁻⁴ m². A tensile force of F = 10,000 N stretches it by ΔL = 1 mm = 0.001 m.
- Stress: σ = F / A = 10,000 / 0.0001 = 100,000,000 Pa = 100 MPa
- Strain: ε = ΔL / L₀ = 0.001 / 2 = 0.0005 (0.05%)
- Young's modulus: E = σ / ε = 100,000,000 / 0.0005 = 2 × 10¹¹ Pa = 200 GPa
That ≈ 200 GPa matches the typical value for steel — confirming our sample stayed within the elastic region.
How to calculate Young's modulus from a stress–strain curve
When you plot stress (y-axis) against strain (x-axis), the elastic region appears as a straight line that passes through the origin. Young's modulus is the slope of that line. Pick two points (σ₁, ε₁) and (σ₂, ε₂) on the straight portion and compute:
Select the "From stress–strain curve" method above, enter the two points, and the calculator returns the slope — your Young's modulus. The same approach is used by dedicated plotting software when fitting the elastic portion of experimental data.
Modulus of elasticity units
| System | Young's modulus units | Typical value for steel |
|---|---|---|
| SI (Metric) | Pa, kPa, MPa, GPa | E ≈ 200 GPa |
| US Customary (Imperial) | psi, ksi, Mpsi | E ≈ 29,000 ksi (29 Mpsi) |
Young's modulus of common materials
| Material | E (GPa) | E (ksi) |
|---|---|---|
| Rubber | ≈ 0.01–0.1 | ≈ 1.5–15 |
| Wood (oak) | ≈ 11 | ≈ 1,600 |
| Concrete | ≈ 30 | ≈ 4,350 |
| Aluminium | ≈ 69 | ≈ 10,000 |
| Brass | ≈ 110 | ≈ 16,000 |
| Steel | ≈ 200 | ≈ 29,000 |
| Tungsten | ≈ 411 | ≈ 59,600 |
| Diamond | ≈ 1,100 | ≈ 159,500 |
Frequently Asked Questions (FAQs)
A high Young's modulus means the material is stiff — it deforms very little under a given stress. Diamond (~1,100 GPa) and tungsten (~411 GPa) have very high moduli, while rubber (~0.05 GPa) is extremely compliant and stretches easily.
Among common engineering materials, diamond has one of the highest known Young's moduli (about 1,000–1,200 GPa). Carbyne and graphene exhibit even higher values in laboratory measurements.
Young's modulus has the same units as stress (pressure): the Pascal (Pa) in SI, usually expressed as MPa or GPa, and psi (or ksi, Mpsi) in the US customary system. Strain is dimensionless, so E inherits the units of σ. 1 GPa ≈ 145,038 psi ≈ 145 ksi.
No. Young's modulus is only defined within the elastic region, where stress and strain are proportional. Beyond the yield point the material deforms plastically and the stress–strain relationship is no longer linear, so E = σ/ε no longer applies.
For most metals and isotropic materials, Young's modulus is approximately the same in tension and compression. Some materials (such as concrete, cast iron, and composites) behave differently under tension and compression, so their moduli can differ.