Formula of Strain: ε = ΔL/L, Types and Worked Examples

The formula of strain is $\varepsilon = \frac{\Delta L}{L}$ : the change in length of a body divided by the length it had before the load went on. Since both are lengths, strain is a pure ratio with no unit. A strain of 0.001 means the body stretched by one thousandth of its length, or 0.1 percent.

Strain measures how much a solid deforms; stress measures the load that causes it. Their ratio is a modulus of elasticity, which is why strain shows up in every elasticity row of the mechanical properties summary. The ε symbol is common in US and engineering books; some Indian texts simply write "strain = ΔL / L" in words.

Quick facts

Formula
$\varepsilon = \frac{\Delta L}{L}$
Unit
none (a ratio); often given as a percentage
Dimensional formula
[M⁰ L⁰ T⁰]
Volume strain
$\frac{\Delta V}{V}$
Shear strain
$\theta \approx \frac{\Delta x}{L}$ , in radians
Typical size
below about 0.001 for metals in the elastic range

Strain formula in physics: the symbols

Quantities in ε = ΔL / L
Symbol	Meaning	Unit
$\varepsilon$	Longitudinal (tensile or compressive) strain	none
$\Delta L$	New length minus original length	m
$L$	Original, unloaded length	m

ΔL is positive when the body stretches and negative when it is squashed, though questions usually ask for the size. Always divide by the original length, not the new one. For the tiny changes in metals the difference is small, but for rubber it matters.

Three kinds of strain

Strain types and their formulas
Type	Formula	Matching modulus
Longitudinal	$\frac{\Delta L}{L}$	Young's modulus
Volumetric	$\frac{\Delta V}{V}$	bulk modulus
Shear	$\theta = \frac{\Delta x}{L}$	modulus of rigidity

In shear strain, Δx is how far the top face slides sideways and L is the height between the fixed and moving faces. Their ratio is tan θ, which equals θ in radians for small angles. Stretching a wire also makes it slightly thinner; the ratio of that lateral strain to the longitudinal strain is Poisson's ratio, about 0.3 for most metals.

Worked examples

Example 1: a stretched rod

A metal rod 2.5 m long stretches by 1.5 mm under load. Find the strain.

Convert: ΔL = 1.5 × 10⁻³ m.

$\varepsilon = \frac{1.5\times10^{-3}}{2.5} = 6.0 \times 10^{-4}$

As a percentage this is 0.060 percent.

Example 2: extension from a known strain

A lift cable 40 m long is working at a strain of 5.0 × 10⁻⁴. By how much has it stretched?

$\Delta L = \varepsilon L = 5.0\times10^{-4} \times 40 = 0.020\ \text{m} = 2.0\ \text{cm}$

Example 3: volume and shear strain

(a) A 2.0 L sample of liquid shrinks by 4.0 cm³ under pressure. (b) The top of a 5.0 cm thick rubber block slides 0.10 mm sideways. Find each strain.

(a) $\frac{\Delta V}{V} = \frac{4.0\ \text{cm}^3}{2000\ \text{cm}^3} = 2.0\times10^{-3}$

(b) $\theta = \frac{0.10\ \text{mm}}{50\ \text{mm}} = 2.0\times10^{-3}\ \text{rad}$

Strain needs both lengths in the same unit, but not necessarily metres. mm over mm works just as well, and the units cancel.

Strain on the stress strain graph

Plot stress up the side and strain along the bottom for a metal wire. The first part is straight and passes through the origin: stress is proportional to strain, which is Hooke's law, and the slope is the modulus of elasticity. Beyond the elastic limit the wire no longer returns to its original length when unloaded, and the leftover strain is called permanent set. Rubber behaves differently: it can reach strains of several hundred percent and still spring back, but its graph is curved, not straight. The formula of strain holds for every material; only the stress it needs changes.

Slips that cost marks with the formula of strain

Giving strain a unit. Write 6.0 × 10⁻⁴, not 6.0 × 10⁻⁴ m.
Dividing by the final length. Use the original length.
Mixing mm on top with m below. Convert one so both match.
Giving shear strain in degrees. The small angle formula gives radians.

Frequently asked questions

What is the formula of strain?

Longitudinal strain = change in length / original length, ε = ΔL / L. Volume strain is ΔV / V and shear strain is the angle θ ≈ Δx / L.

Does strain have a unit?

Strain is unitless, being one length divided by another (or one volume by another). Its bracket form: [M⁰ L⁰ T⁰].

How is strain different from stress?

Stress is force per unit area in pascals and is the cause. Strain is the fractional change in shape or size and is the effect.

Can strain be greater than 1?

For rubber and some plastics, yes: a strain of 2 means the band has stretched to three times its length. Metals break long before that.