Young's Modulus Calculator (Modulus of Elasticity)
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/6/2026
Young's modulus is calculated with E = σ ÷ ε, the ratio between mechanical stress (σ = F ÷ A) and relative strain (ε = ΔL ÷ L₀). A steel sample subjected to 10,000 N over a 50 mm² cross-section and a 200 mm length, with a 0.2 mm elongation, gives a Young's modulus of 200,000 MPa (200 GPa), the typical value for steel.
Explanation
Young's modulus measures a material's intrinsic stiffness: its resistance to elastic deformation under a given stress. This calculator chains three steps: first the mechanical stress applied to the cross-section (σ = F ÷ A, in megapascals), then the relative strain that stress causes (ε = ΔL ÷ L₀, a unitless number, the elongation relative to the initial length), and finally their ratio, Young's modulus itself. This relationship, valid within the material's elastic range — meaning as long as the deformation stays reversible, before any permanent damage —, is a generalization of Hooke's law: just as a spring resists stretching in proportion to its stiffness k, a material resists deformation in proportion to its Young's modulus E, an intrinsic property of the material independent of the shape or size of the sample tested. A high Young's modulus (steel, around 200 GPa) signals a rigid material that deforms little under load; a lower modulus (rubber, on the order of just a few megapascals) signals a material that stretches easily — this is the quantity that allows an objective comparison of the stiffness of very different materials, independent of the dimensions of the sample tested.
Example: a steel sample, F=10,000 N, A=50 mm², L₀=200 mm, ΔL=0.2 mm
Inputs
Force: 10,000 N. Cross-sectional area: 50 mm². Initial length: 200 mm. Elongation: 0.2 mm.
Calculation
σ = 10,000 ÷ 50 = 200 MPa. ε = 0.2 ÷ 200 = 0.001. E = 200 ÷ 0.001 = 200,000 MPa, i.e. 200 GPa.
Result
This material's Young's modulus is 200,000 MPa (200 GPa), the typical value for steel.
Frequently asked questions
What happens beyond the elastic range?
The relationship E = σ ÷ ε no longer applies: beyond a stress called the yield strength, the material deforms permanently (plastic deformation) and no longer returns to its original shape once the load is removed. This calculator assumes the measured strain stays within the elastic, reversible range of the material tested.
Why doesn't Young's modulus depend on the size of the sample?
Because it relates two quantities already normalized by geometry: stress divides force by cross-sectional area, and strain divides elongation by initial length. A sample twice as long, subjected to a force twice as large over a cross-section twice as wide, would give exactly the same Young's modulus — this is precisely what makes it an intrinsic property of the material, comparable from one test to another.
What is the difference with a spring's stiffness?
A spring's stiffness, in Hooke's law, depends on both the material AND the geometry of the object (a thicker or shorter spring is stiffer, for the same material). Young's modulus, on the other hand, is a property of the material alone, independent of the shape of the part tested — in fact, it's the material's Young's modulus which, combined with the spring's precise geometry, determines its stiffness.