Young's Modulus Calculator (from Test Data)

Background.

Hang about 510 kg from a steel bar two metres long and 200 mm² in section and it stretches by a quarter of a millimetre. Young's modulus is the material property that predicts that stretch, and it is found the other way round: load a specimen, measure how far it extends, and divide the stress by the strain. That is the calculation done here, with E as the unknown worked back from test data. If you already know E from a handbook and want to predict how far a part will stretch, the companion stress–strain calculator runs the same equation in the other direction.

Stress is the force spread over the cross-section, F/A, in pascals. Strain is the stretch as a fraction of the original length, ΔL/L₀, a pure number. Dividing one by the other cancels the specimen's size. A bar twice as thick needs twice the force to reach the same strain, and a bar twice as long stretches twice as far under the same stress, yet both give the same E. That is why the modulus can be tabulated for a material rather than for a particular part: OpenStax's University Physics lists about 200 GPa for steel, 110 GPa for copper and 70 GPa for aluminium.

The ratio means something only while the material is behaving elastically, in the straight opening stretch of its stress–strain curve where removing the load returns the specimen to its original length. Beyond that proportional limit part of the stretch becomes permanent, strain grows faster than stress, and a modulus worked from those readings comes out too low. Nothing in the arithmetic can tell which side of the limit your numbers came from, which is why the stress is reported alongside E: compare it with the yield strength of the material you tested.

What is young's modulus calculator?

Young's modulus, also called the elastic modulus or modulus of elasticity, is the ratio of tensile or compressive stress to the strain it produces, measured within the elastic range. On a stress–strain graph it is the slope of the initial straight portion. Its SI unit is the pascal, but values for engineering materials run into the billions, so tables quote gigapascals. It measures stiffness, not strength: E says how much a part deflects under a given load, while the yield and tensile strengths say when it gives way. Aluminium at about 70 GPa stretches nearly three times as far as steel at 200 GPa under the same stress, even where the two alloys are similarly strong.

How to use this calculator.

1. 01Enter the axial force in newtons: positive if the specimen was pulled, negative if it was squashed. A machine reading in kilonewtons needs multiplying by 1,000.
2. 02Enter the cross-sectional area in square metres, measured across the line of the force. Divide mm² by 1,000,000; a round bar of 16 mm diameter has π × 8² ≈ 201 mm², or about 0.000201 m².
3. 03Enter the original length over which the extension was measured. With a clip-on extensometer that is its gauge length (50 mm is 0.05 m), not the full length of the specimen between the grips.
4. 04Enter the measured change in length with the same sign as the force. Zero is not accepted, because a specimen that did not deform gives no strain to divide by.
5. 05Read E in pascals and gigapascals, then check the stress figure against the yield strength of your material before trusting the modulus.

The formula.

E = σ / ε = (F / A) / (ΔL / L₀), which rearranges to E = F × L₀ / (A × ΔL). F is the axial force in newtons, A the cross-sectional area in square metres, L₀ the unloaded length and ΔL the change in length. Stress σ = F/A comes out in pascals (N/m²); strain ε = ΔL/L₀ is dimensionless, so E also comes out in pascals, and dividing by 10⁹ gives the gigapascal figure. These are engineering stress and strain, based on the original area and length, which is the convention modulus tables use; at elastic strains of a fraction of a percent the difference from true stress and strain is negligible. The formula also assumes the force acts along the axis so that stress is uniform across the section. Because E is inversely proportional to both A and ΔL, the smallest measurements dominate the uncertainty. In the example below, misreading a 0.25 mm extension by just 0.01 mm moves E from 200 GPa to anywhere between 192.3 and 208.3 GPa, and a 1% error in a bar's measured diameter becomes about a 2% error in area and therefore in E. If force and extension are entered with opposite signs the result is negative, which signals a sign error rather than a property of the material.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax. University Physics Volume 1, Section 12.3 'Stress, Strain, and Elastic Modulus' — defines Young's modulus Y = tensile stress/tensile strain = (F⊥/A)/(ΔL/L₀), valid in the linear (elastic) regime.
2. [2]National Institute of Standards and Technology (NIST). CODATA-adjacent SI unit definitions — the pascal (Pa = N/m²) as the SI-derived unit of stress and pressure.
3. [3]Gere, J. M., & Goodno, B. J. (2013). Mechanics of Materials, 8th edition, Chapter 1 (Tension, Compression, and Shear) — the tensile test, Hooke's law, and the elastic-limit assumption underlying E = stress/strain. Cengage Learning.