What this calculator does
Elastic modulus, also called Young’s modulus, measures how stiff a material is: how much it resists stretching or compressing under a given load before it would start to deform permanently. A high elastic modulus, such as steel’s, means the material barely stretches under load. A low one, such as rubber’s, means it stretches noticeably for the same force.
The elastic modulus formula divides stress, the force spread over the cross-sectional area carrying it, by strain, the fractional change in length that force produces. Both quantities have to be measured within the material’s elastic range, meaning it springs back to its original length once the force is removed, which is the regime this calculator assumes.
The formula
Stress is the applied force divided by the cross-sectional area it acts over. Strain is the change in length divided by the original length, a dimensionless ratio. Elastic modulus is stress divided by strain, so it comes out in the same units as stress, typically expressed in gigapascals for structural materials.
| Term | Meaning |
|---|---|
| E | Elastic modulus (Young’s modulus), typically in GPa. |
| F | Force applied along the length of the sample, in newtons. |
| A | Cross-sectional area carrying that force. |
| L₀ | Original, unloaded length of the sample. |
| ΔL | Change in length under the applied force. |
The inputs explained
| Field | What to enter |
|---|---|
| Applied force (N) | The force pulling on or compressing the sample along its length. |
| Cross-sectional area (mm²) | The cross-sectional area the force passes through, at right angles to the length. |
| Original length (m) | The original length of the sample before any force is applied. |
| Change in length (mm) | How much the sample stretches (or shortens, entered as negative) under that force. |
When to use it
Checking the elastic modulus formula against a known material
Entering a test result for a sample of known dimensions and comparing the calculated modulus against published values for that material, such as roughly 200 GPa for structural steel, is a quick sanity check on the numbers.
Comparing two candidate materials
For the same applied force and cross-section, a material that stretches less under load has a higher elastic modulus and will generally be the stiffer, less flexible choice for a load-bearing part.
Working backwards from a specification
If a datasheet gives an elastic modulus and a maximum allowable stress, this calculator’s formula can be rearranged by hand to check what stretch to expect under a proposed load.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does stretch affect the calculated elastic modulus?
A 5,000 N force on a 50 mm² sample, 2 m long, with the amount of stretch increasing.
| Change in length | Elastic modulus | Strain |
|---|---|---|
| 0.5 mm | 400.000 GPa | 0.000250 |
| 1 mm | 200.000 GPa | 0.000500 |
| 1.5 mm | 133.333 GPa | 0.000750 |
| 2 mm | 100.000 GPa | 0.001000 |
| 2.5 mm | 80.000 GPa | 0.001250 |
How does the applied force affect stress and strain, for a fixed stretch?
Holding the geometry and the observed stretch fixed while the applied force increases.
| Applied force | Elastic modulus | Stress |
|---|---|---|
| 5,000 N | 133.333 GPa | 100.0 MPa |
| 10,000 N | 266.667 GPa | 200.0 MPa |
| 15,000 N | 400.000 GPa | 300.0 MPa |
| 20,000 N | 533.333 GPa | 400.0 MPa |
Questions
What is the elastic modulus formula?
Elastic modulus (E) equals stress divided by strain: E = (F/A) / (ΔL/L₀), where F is the applied force, A the cross-sectional area, L₀ the original length and ΔL the change in length.
What is the difference between elastic modulus and Young’s modulus?
They are the same thing. Young’s modulus is the more specific name, describing elastic modulus measured under simple tension or compression along one axis, which is the case this calculator handles.
What units is elastic modulus normally given in?
Pascals, though because the numbers involved are large for most solids, gigapascals (GPa) are the usual practical unit. Steel is around 200 GPa, aluminium around 69 GPa and rubber under 0.1 GPa.
Does this formula work for compression as well as stretching?
Yes, provided the material behaves elastically in compression too. Enter the change in length as the amount it shortens; the arithmetic is identical, only the physical direction of the deformation differs.
For force and stretch in a spring rather than a solid material sample, see the Hooke’s law and spring energy calculator. For pressure from a force spread over an area more generally, see the pressure calculator.