Modulus of Elasticity Formula: E = Stress/Strain, Types

The modulus of elasticity formula is $E = \frac{\sigma}{\varepsilon}$ : stress divided by the strain it produces, as long as the material stays within its elastic limit. It is Hooke's law written as a ratio. Because strain has no unit, the modulus has the same unit as stress, the pascal.

There is not just one modulus. Pulling a rod, squeezing a block from all sides and twisting a sheet each give a different ratio, so a material has a Young's modulus, a bulk modulus and a modulus of rigidity. In engineering, \"modulus of elasticity\" with no other word nearly always means Young's modulus. All three are tabulated in the elasticity part of the fluids and matter sheet.

Quick facts

General formula
$E = \frac{\text{stress}}{\text{strain}}$
SI unit
pascal (Pa); GPa in data tables
Dimensional formula
[M¹ L⁻¹ T⁻²]
Three kinds
Young's (Y), bulk (K) and rigidity or shear (G or η)
Law behind it
Hooke's law: stress ∝ strain
Steel
E ≈ 2.0 × 10¹¹ Pa (Young's modulus)

The modulus of elasticity formula, term by term

Terms in E = σ / ε
Symbol	Meaning	Unit
$E$	Modulus of elasticity	Pa
$\sigma$	Stress, force per unit area (stress formula)	Pa
$\varepsilon$	Strain, fractional change in size (strain formula)	none

A large modulus means a stiff material: a big stress is needed for a small strain. A small modulus means a soft or springy one. The ratio is a constant only on the straight part of the stress strain graph. Past the elastic limit, strain grows faster than stress and the formula no longer gives a single value.

The three elastic moduli compared

Which modulus for which kind of deformation
Modulus	Formula	Deformation	Applies to
Young's modulus Y	$\frac{\sigma}{\Delta L / L}$	change in length	solids only
Bulk modulus K	$-\frac{\Delta P}{\Delta V/V}$	change in volume	solids, liquids and gases
Modulus of rigidity G	$\frac{F/A}{\theta}$	change in shape	solids only

For an isotropic material these are not independent. With Poisson's ratio ν, they are linked by $Y = 2G(1 + \nu)$ and $Y = 3K(1 - 2\nu)$ . Since ν is about 0.3 for most metals, G comes out a bit under 40 percent of Y.

Worked examples

Example 1: from stress and strain

A metal sample under a stress of 2.0 × 10⁸ Pa shows a strain of 1.0 × 10⁻³. Find its modulus of elasticity.

$E = \frac{2.0\times10^8}{1.0\times10^{-3}} = 2.0\times10^{11}\ \text{Pa}$

That matches steel.

Example 2: a concrete column, starting from the load

A column with a 0.30 m × 0.30 m section carries 3.0 × 10⁵ N and shortens with a strain of 1.1 × 10⁻⁴. Find the modulus of elasticity of the concrete.

Stress: $\sigma = \frac{3.0\times10^5}{0.090} = 3.33\times10^6\ \text{Pa}$

$E = \frac{3.33\times10^6}{1.1\times10^{-4}} = 3.0\times10^{10}\ \text{Pa} = 30\ \text{GPa}$

Example 3: strain from a known modulus

An aluminium bar (E = 7.0 × 10¹⁰ Pa) carries a tensile stress of 35 MPa. What strain does it have?

$\varepsilon = \frac{\sigma}{E} = \frac{3.5\times10^7}{7.0\times10^{10}} = 5.0\times10^{-4}$

1 MPa = 10⁶ Pa and 1 GPa = 10⁹ Pa. Convert before dividing so the powers of ten line up.

Units, dimensions and the graph

Stress has dimensions [M L⁻¹ T⁻²] and strain is dimensionless, so every modulus is [M¹ L⁻¹ T⁻²], the same as pressure. On a stress strain graph, the modulus is the slope of the straight section starting at the origin. A steeper line means a stiffer material. The area under the line, per unit volume, is the elastic energy stored, equal to ½ × stress × strain.

Common mistakes with the formula for modulus of elasticity

Inverting the ratio. It is stress over strain, never strain over stress.
Giving strain as a percentage in the formula. Convert 0.05 percent to 5.0 × 10⁻⁴ first.
Picking the wrong modulus. Length change uses Y, volume change uses K, shape change uses G.
Using it beyond the elastic limit. The ratio is only constant on the straight part of the graph.

Frequently asked questions

What is the formula of modulus of elasticity?

Modulus of elasticity = stress / strain, E = σ / ε, valid within the elastic limit. Its three forms are Young's modulus, bulk modulus and modulus of rigidity.

What is the SI unit of modulus of elasticity?

The pascal (N/m²), because strain has no unit. Values are usually quoted in GPa.

Is modulus of elasticity the same as Young's modulus?

Often, yes. In engineering the phrase usually means Young's modulus. In physics it is the general name covering Young's, bulk and shear moduli.

Does a higher modulus mean a stronger material?

Not necessarily. A high modulus means stiffness, small strain for a given stress. Strength is about the stress at which it breaks, which is a separate property.