What is Young's modulus, how is it calculated, why is steel so much stiffer than aluminium, and how do you measure it in the lab? Formulas, unit conversions, a calculator, a materials table and the full Searle's apparatus experiment.
When you pull on a metal wire, it stretches a tiny amount and springs back when you let go. Young's modulus tells you how hard you have to pull to get a given stretch. It is named after the English scientist Thomas Young and is the most commonly quoted measure of a material's stiffness.
The idea builds directly on Hooke's law. Hooke's law says stress is proportional to strain within the elastic limit; Young's modulus is the constant of proportionality for stretching or compressing a rod or wire.
A high Young's modulus means a material is stiff — it barely stretches under load. It does not mean the material is strong. Strength is described separately by yield strength and tensile strength (Section 7). Glass is stiff but brittle; rubber is flexible but can survive large strains.
Figure 1: A wire of original length L and cross-sectional area A is stretched by force F. The extension ΔL is tiny in real metals, so it is exaggerated here.
Young's modulus is built from two simpler quantities. Get these right and the rest follows.
Stress is the restoring force per unit cross-sectional area inside a deformed body. It measures how concentrated the internal force is.
Strain is the fractional change in a dimension caused by stress. It compares the change with the original size, so it has no unit.
| Quantity | Formula | SI Unit | Dimensional Formula |
|---|---|---|---|
| Stress (σ) | σ = F / A | pascal (Pa) = N/m² | [M L⁻¹ T⁻²] |
| Longitudinal strain (ε) | ε = ΔL / L | None (dimensionless ratio) | [M⁰ L⁰ T⁰] |
| Young's modulus (E) | E = σ / ε | pascal (Pa) | [M L⁻¹ T⁻²] |
Strain has no units because it is a ratio of two lengths (mm ÷ mm or m ÷ m). It is sometimes written as a percentage or as "microstrain" (µε = 10⁻⁶), but these are just scaling conventions. Stress units, by contrast, are always force per area: Pa, MPa, or N/mm².
E = Young's modulus (Pa) | F = applied force (N) | A = cross-sectional area (m²) | L = original length (m) | ΔL = extension (m)
When the load is a hanging mass M, F = Mg | This is the working formula used in the Searle's apparatus experiment
The SI unit of Young's modulus is the pascal (Pa), which equals one newton per square metre. Because engineering materials have enormous values, you will almost always see it written in gigapascals (GPa) or megapascals (MPa). In imperial units it appears as psi or ksi.
Because strain is dimensionless, Young's modulus has the same dimensions as stress and pressure:
The same dimensional formula applies to stress, pressure, shear modulus and bulk modulus
The most searched question about this topic is simply the value for a specific material. Typical room-temperature values are in the table below. Real values vary a little with alloy, grain structure, temperature and heat treatment, so treat these as engineering approximations.
| Material | Young's Modulus (GPa) | Young's Modulus (Pa) | Approx. psi |
|---|---|---|---|
| Steel (structural / mild steel) | ≈ 200 (190–210) | ≈ 2.0 × 10¹¹ | ≈ 29 × 10⁶ |
| Copper | ≈ 117 (110–130) | ≈ 1.2 × 10¹¹ | ≈ 17 × 10⁶ |
| Brass | ≈ 100 (100–125) | ≈ 1.0 × 10¹¹ | ≈ 15 × 10⁶ |
| Glass (typical) | ≈ 70 | ≈ 7.0 × 10¹⁰ | ≈ 10 × 10⁶ |
| Aluminium (aluminum) | ≈ 69–70 | ≈ 6.9 × 10¹⁰ | ≈ 10 × 10⁶ |
| Concrete (normal strength) | ≈ 30 (25–40) | ≈ 3.0 × 10¹⁰ | ≈ 4.4 × 10⁶ |
| Wood (along the grain) | ≈ 10–12 | ≈ 1.1 × 10¹⁰ | ≈ 1.6 × 10⁶ |
| Rubber | ≈ 0.01–0.1 | ≈ 10⁷–10⁸ | ≈ 1.5 × 10³ – 1.5 × 10⁴ |
Figure 2: Approximate Young's modulus on a linear scale. Steel is about three times stiffer than aluminium; rubber is so soft that its bar is almost invisible on this scale.
With E ≈ 200 GPa versus ≈ 69 GPa, steel is roughly 3 times stiffer than aluminium. Under the same stress, an aluminium bar stretches about three times as much as an identical steel bar. Aluminium's advantage is its much lower density (about one third of steel), so for equal weight the two metals are closer in stiffness than the raw modulus suggests.
Figure 3: Schematic stress-strain lines in the elastic region. Young's modulus is the slope: the steeper the line, the stiffer the material. Slopes are illustrative, not to scale.
Unit mistakes are the most common reason Young's modulus calculations go wrong. Memorise the table below.
| Conversion | Rule | Example |
|---|---|---|
| GPa → Pa | × 10⁹ | 200 GPa = 2 × 10¹¹ Pa |
| GPa → MPa | × 1,000 | 200 GPa = 200,000 MPa |
| MPa → GPa | ÷ 1,000 | 69,000 MPa = 69 GPa |
| Pa → GPa | ÷ 10⁹ | 7 × 10¹⁰ Pa = 70 GPa |
| MPa ↔ N/mm² | 1 MPa = 1 N/mm² | 1 GPa = 1,000 N/mm² |
| GPa → psi | × ≈ 145,000 | 200 GPa ≈ 29 × 10⁶ psi |
| mm → m | ÷ 1,000 | 0.39 mm = 3.9 × 10⁻⁴ m |
| m → mm | × 1,000 | 2.0 m = 2,000 mm |
| mm² → m² | × 10⁻⁶ | 0.503 mm² = 5.03 × 10⁻⁷ m² |
If you enter the wire's diameter in millimetres but the length in metres without converting, your answer will be off by a factor of a million. Convert everything to SI (metres, newtons, square metres) before substituting into E = FL / (AΔL).
Use the calculator below to find Young's modulus from a wire stretching measurement. The default values come from the sample experiment later in this article (a 2 m steel wire, 0.8 mm diameter, loaded with 2 kg).
Enter the measurements in the units shown. Results update when you press Calculate.
A steel wire 2.0 m long and 0.80 mm in diameter stretches by 0.39 mm when a 2.0 kg mass hangs from it. Find Young's modulus (g = 9.8 m/s²).
SolutionStep 1 — Force: F = mg = 2.0 × 9.8 = 19.6 N
Step 2 — Area: A = π d² / 4 = π (0.80 × 10⁻³)² / 4 = 5.03 × 10⁻⁷ m²
Step 3 — Stress: σ = F / A = 19.6 / 5.03 × 10⁻⁷ = 3.90 × 10⁷ Pa
Step 4 — Strain: ε = ΔL / L = 0.39 × 10⁻³ / 2.0 = 1.95 × 10⁻⁴
Step 5 — Modulus: E = σ / ε = 3.90 × 10⁷ / 1.95 × 10⁻⁴ = 2.0 × 10¹¹ Pa = 200 GPa
Young's modulus only describes the first, straight part of the stress-strain curve. To describe what happens when a material is loaded further, engineers use two more properties.
Figure 4: The slope of the initial straight line is Young's modulus. Yield strength marks where permanent deformation begins; tensile strength is the highest stress reached before the material fails.
| Property | What It Measures | Type |
|---|---|---|
| Young's modulus (E) | Resistance to elastic stretching (slope of the straight line) | Stiffness |
| Yield strength (σᵧ) | Stress at which permanent (plastic) deformation begins | Strength |
| Tensile strength (UTS) | Maximum stress the material can withstand before necking and fracture | Strength |
Young's modulus describes stretching, but solids can also be twisted or squeezed. Each type of deformation has its own elastic modulus. Together with Poisson's ratio, they describe the elastic behaviour of an isotropic solid.
Tensile stress ÷ longitudinal strain. Describes change in length.
Shear stress ÷ shear strain. Describes change in shape, like twisting a rod.
Pressure change ÷ volumetric strain. Describes change in volume under uniform pressure.
Negative ratio of lateral strain to axial strain. Describes how much a stretched rod thins.
For most materials ν ≈ 0.2–0.5 (steel ≈ 0.3, rubber ≈ 0.5), so G is roughly E/2.6 and B is roughly E for steel
The most commonly performed experiment for measuring Young's modulus in school and first-year university labs uses Searle's apparatus. It is popular because it compares an experimental wire against an identical reference wire, which cancels out errors from temperature changes and from sagging of the support.
To determine Young's modulus of the material of a wire (such as steel or brass) by measuring its extension under increasing loads using Searle's apparatus.
Two identical wires (reference wire A and experimental wire B) hanging from a rigid support, joined to two metal frames.
A spirit level rests on the two frames; a micrometer screw on one frame restores the level after each load.
A fixed dead load on the reference wire and slotted masses (0.5 kg steps) for the experimental wire.
A screw gauge to measure wire diameter; a metre scale for the length from clamp to frame.
A load M hangs from the experimental wire of length L and radius r, producing an extension l. The tensile stress is Mg/πr² and the longitudinal strain is l/L, so:
M = load on the wire (kg) | g = 9.8 m/s² | L = length of wire | r = radius of wire | l = extension for load M
Figure 5: Searle's apparatus. The reference wire A carries a constant dead load; the experimental wire B carries the variable load. Re-levelling the spirit level with the micrometer screw measures the extension of wire B.
Sample values for a steel wire with L = 2.00 m and diameter 0.80 mm (r = 0.40 mm, A = 5.03 × 10⁻⁷ m²), g = 9.8 m/s². Replace with your own readings.
| S.No. | Load M (kg) | Force Mg (N) | Extension l (mm) | Extension l (m) | Y = MgL/(Al) (× 10¹¹ Pa) |
|---|---|---|---|---|---|
| 1 | 0.5 | 4.9 | 0.10 | 1.0 × 10⁻⁴ | 1.95 |
| 2 | 1.0 | 9.8 | 0.19 | 1.9 × 10⁻⁴ | 2.05 |
| 3 | 1.5 | 14.7 | 0.29 | 2.9 × 10⁻⁴ | 2.02 |
| 4 | 2.0 | 19.6 | 0.39 | 3.9 × 10⁻⁴ | 2.00 |
| 5 | 2.5 | 24.5 | 0.49 | 4.9 × 10⁻⁴ | 1.99 |
| 6 | 3.0 | 29.4 | 0.58 | 5.8 × 10⁻⁴ | 2.02 |
Y ≈ 200 GPa, in good agreement with the accepted value for steel
Plot load M (kg) on the y-axis against extension l (mm) on the x-axis. The graph is a straight line through the origin, confirming that extension is proportional to load (Hooke's law). Young's modulus follows from the slope: Y = (g L / π r²) × (M / l).
Figure 6: The six points lie on a straight line through the origin, so extension is proportional to load. The slope (about 5.2 kg per mm, i.e. 5.2 × 10³ kg/m) gives Young's modulus ≈ 2.0 × 10¹¹ Pa.
The load-extension graph is a straight line through the origin, so Hooke's law is obeyed within the loads used. Young's modulus of the material of the wire is Y ≈ 2.0 × 10¹¹ N/m² (≈ 200 GPa), consistent with steel.
| Source of Error | Effect | How to Reduce It |
|---|---|---|
| Non-uniform wire diameter | Wrong area, since r² enters the formula | Measure diameter at many points; take the mean |
| Error in length measurement | Direct error in L | Measure from the clamp to the frame carefully |
| Temperature change and support sag | False extension | Use the reference wire (it is the whole purpose of Searle's design) |
| Elastic after-effect | Wire does not return exactly on unloading | Wait between readings; use the mean of loading and unloading |
| Exceeding the elastic limit | Permanent stretch, non-linear graph | Keep to a modest maximum load; check unloading readings |
Young's modulus, also called the modulus of elasticity or elastic modulus, is a measure of the stiffness of a solid material. It is defined as the ratio of tensile (or compressive) stress to the corresponding longitudinal strain within the elastic limit: E = stress / strain. A higher Young's modulus means the material is stiffer and deforms less under the same load.
Young's modulus E = stress / strain = (F/A) / (ΔL/L) = FL / (AΔL). Here F is the applied force, A is the cross-sectional area, L is the original length and ΔL is the extension. For a wire of radius r, A = πr².
The SI unit of Young's modulus is the pascal (Pa), equal to one newton per square metre (N/m²). Because common materials have very large values, it is usually quoted in gigapascals (GPa) or megapascals (MPa). Its dimensional formula is [M L⁻¹ T⁻²], the same as pressure and stress, because strain is dimensionless.
The Young's modulus of steel is about 200 GPa (2 × 10¹¹ Pa), with structural and mild steels typically between 190 and 210 GPa. It varies only slightly with alloy and heat treatment, which is why almost all steels are treated as having the same stiffness in engineering design.
The Young's modulus of aluminium (aluminum) is about 69 to 70 GPa, roughly one third that of steel. This means an aluminium part of the same size stretches about three times more than a steel part under the same load, even though aluminium is also much lighter.
Calculate stress as force divided by cross-sectional area (σ = F/A), calculate strain as extension divided by original length (ε = ΔL/L), and then divide stress by strain. In the lab, plot stress against strain (or load against extension) and find E from the slope of the straight-line portion.
They are three elastic moduli for three kinds of deformation. Young's modulus relates tensile stress to longitudinal strain, shear modulus relates shear stress to shear strain (change in shape), and bulk modulus relates pressure to volumetric strain (change in volume). For an isotropic solid they are linked through Poisson's ratio: E = 2G(1 + ν) and E = 3B(1 − 2ν).
Young's modulus measures stiffness, meaning how much a material stretches elastically under a given stress. Tensile strength measures strength, meaning the maximum stress a material can bear before breaking, and yield strength is the stress at which permanent deformation begins. A material can be stiff but weak, or strong but flexible.
1 GPa = 1,000 MPa = 1,000,000,000 Pa (10⁹ Pa). To convert GPa to Pa multiply by 10⁹, to convert GPa to MPa multiply by 1,000, and to convert MPa to GPa divide by 1,000. For example, 200 GPa = 200,000 MPa = 2 × 10¹¹ Pa.
