The strain formula is strain (ε) = change in length / original length = ΔL / L. Strain measures how much a body deforms compared with its original size. Because it is a length divided by a length, it has no units: it is written as a plain number (0.0008), as a percentage (0.08%) or in microstrain (800 µε, where 1 µε = 10-6).
What is strain?
When a load acts on a bar, the bar stretches, shortens or distorts. The actual change in length depends on how long the bar was, so a 1 mm stretch means very different things for a 10 mm bolt and a 10 m cable. Strain removes that size effect by dividing the change by the original dimension. That is why engineers compare materials by strain, not by elongation in millimetres.
Strain is the response; stress (force per unit area) is the cause. The two are connected through the elastic constants, covered below.
Strain formula and units
ε = ΔL / L = (Lfinal – L) / L
- ΔL = change in length (final length minus original length)
- L = original length, in the same unit as ΔL
- Tensile strain (stretching) is taken as positive; compressive strain (shortening) as negative.
Units: none. You will still see “mm/mm” or “m/m” written to show what was divided by what. To convert: 0.001 = 0.1% = 1000 µε.
Types of strain
1. Tensile and compressive (normal) strain
Normal strain acts along the direction of the load. A tie rod under pull has tensile strain; a column under load has compressive strain. Both use ε = ΔL / L.
2. Lateral strain and Poisson’s ratio
A bar that is stretched also gets thinner. The strain across the load direction is the lateral strain = Δd / d. For a given material, within the elastic range, the ratio of lateral to longitudinal strain is constant. This is Poisson’s ratio:
ν = – (lateral strain) / (longitudinal strain)
The minus sign makes ν positive, because the two strains have opposite signs. Typical values: steel about 0.3, aluminium about 0.33, concrete about 0.1-0.2, rubber close to 0.5, cork close to 0 (which is why a cork slides easily into a bottle neck).
3. Shear strain
Shear strain is the change in angle between two lines that were originally at right angles. If the top face of a block of height h moves sideways by x relative to the bottom face:
γ = tan θ = x / h ≈ θ (in radians, for small angles)
Shear strain is linked to shear stress by the modulus of rigidity: τ = Gγ. See shear modulus.
4. Volumetric strain
Volumetric strain is the change in volume per unit original volume:
εv = ΔV / V = εx + εy + εz (for small strains)
For a bar under a single axial load, the two lateral strains are each -νε, so εv = ε(1 – 2ν). Under equal pressure from all sides, εv = 3ε. Derivations and more problems are on our volumetric strain page.
5. Thermal strain
Heating makes a free body expand even with no load:
εth = α × ΔT
where α is the coefficient of linear thermal expansion (per °C). If the body is not free to expand, this strain turns into thermal stress, σ = EαΔT. You can check numbers with the thermal expansion calculator.
Stress-strain relation: Hooke’s law
Up to the limit of proportionality, stress is proportional to strain:
E = stress / strain = σ / ε, so ε = σ / E and ΔL = PL / (AE)
E is Young’s modulus: about 200 GPa for steel, about 70 GPa for aluminium. Because strain is dimensionless, E has the same units as stress. For the full set of elastic constants and how they relate (E = 2G(1 + ν), E = 3K(1 – 2ν)), see stress and strain, Hooke’s law and elastic constants and Young’s modulus. Background on E from a materials angle: Young’s modulus (NCBI).
Elastic strain vs plastic strain
Elastic strain disappears when the load is removed. Plastic strain stays: the part is permanently bent or stretched. For mild steel with a yield stress of 250 MPa, yielding starts at a strain of only 250 / 200,000 = 0.00125 (0.125%), yet the same steel can stretch more than 20% before it breaks. Almost all of that is plastic strain.
How the curve behaves between yield and fracture, and what ductility and toughness mean, is covered point by point in mechanical properties of materials. Plastic strain is also what strain hardening in cold working of steel is built on.
Engineering strain vs true strain
Engineering strain divides the change in length by the original length every time: e = ΔL / L0. True strain adds up each small increment divided by the length at that instant, which integrates to:
εtrue = ln(L / L0) = ln(1 + e)
For small strains the two are nearly equal. At large strains, such as in forging, rolling and wire drawing, true strain is the right one to use, because true strains from successive stages can simply be added, while engineering strains cannot.
Solved examples on the strain formula
Example 1: Steel bar under tension
A steel bar 2 m long and 20 mm in diameter carries an axial pull of 50 kN. E = 200 GPa, ν = 0.3. Find the stress, strain, elongation, change in diameter and change in volume.
- Area A = π/4 × 202 = 314.16 mm2
- Stress σ = 50,000 N / 314.16 mm2 = 159.15 N/mm2 = 159.15 MPa
- Strain ε = σ / E = 159.15 / 200,000 = 7.96 × 10-4 (about 796 µε, or 0.0796%)
- Elongation ΔL = ε × L = 7.958 × 10-4 × 2000 = 1.59 mm
- Lateral strain = -νε = -0.3 × 7.958 × 10-4 = -2.39 × 10-4; change in diameter = -2.387 × 10-4 × 20 = -0.0048 mm (the bar gets thinner)
- Volumetric strain = ε(1 – 2ν) = 7.958 × 10-4 × 0.4 = 3.18 × 10-4; V = 314.16 × 2000 = 628,319 mm3; ΔV = 3.183 × 10-4 × 628,319 = 200 mm3 (increase)
Check: ΔV = PL(1 – 2ν)/E = 50,000 × 2000 × 0.4 / 200,000 = 200 mm3. Note how small the diameter change is compared with the elongation.
Example 2: Shear strain in a rubber pad
A rubber bearing pad 20 mm thick is fixed at the bottom. A horizontal force moves its top face 0.5 mm sideways. Find the shear strain.
- γ = tan θ = x / h = 0.5 / 20 = 0.025 rad
- θ = tan-1(0.025) = 0.02499 rad = 1.43°, so the small-angle approximation γ ≈ θ is accurate here to about 0.02%.
Example 3: Engineering strain vs true strain
A tensile specimen with a 50 mm gauge length is stretched to 60 mm. Then a second specimen is compressed from 50 mm to 40 mm.
- Tension: e = (60 – 50) / 50 = 0.200; true strain = ln(60/50) = ln 1.2 = 0.182. Engineering strain overstates it by about 10%.
- Compression: e = (40 – 50) / 50 = -0.200; true strain = ln(40/50) = ln 0.8 = -0.223.
- At small strain, 50 mm to 50.5 mm: e = 0.0100 and true strain = ln 1.01 = 0.00995, a difference of only 0.5%.
The equal and opposite engineering strains (+0.2 and -0.2) give unequal true strains. True strain is symmetric: stretching 50 to 60 mm and then compressing 60 back to 50 mm gives +0.182 and -0.182, which add to zero, as they should.
Example 4: Thermal strain in a steel rail
A 12 m steel rail heats up by 30 °C. Take α = 12 × 10-6 per °C (a typical value for steel) and E = 200 GPa.
- Thermal strain = αΔT = 12 × 10-6 × 30 = 3.6 × 10-4 (360 µε)
- If free to expand: ΔL = 3.6 × 10-4 × 12,000 mm = 4.32 mm
- If fully restrained (no gap): σ = EαΔT = 200,000 × 3.6 × 10-4 = 72 MPa compressive
This is why jointed track has expansion gaps, and why continuously welded rail must be laid and fastened with its temperature stresses in mind.
How is strain measured? Strain gauges
Strains of a few hundred microstrain are far too small to see, so they are measured with an electrical strain gauge: a thin foil grid glued to the surface. When the surface stretches, the grid gets longer and thinner and its resistance rises. The gauge factor links the two:
GF = (ΔR / R) / ε
Metal foil gauges typically have GF of about 2, and common resistances are 120 Ω and 350 Ω. For the bar in Example 1 (796 µε), a 120 Ω gauge with GF = 2 changes by ΔR = 2 × 7.96 × 10-4 × 120 = 0.19 Ω. That tiny change is why gauges are read with a Wheatstone bridge and an amplifier.
Common mistakes with the strain formula
- Mixing units: ΔL in mm and L in m gives a strain 1000 times too small. Use the same unit for both.
- Dividing by the final length instead of the original length (that is neither engineering nor true strain).
- Forgetting the sign: compressive strain is negative, and it matters when adding strains for volumetric strain.
- Using shear strain in degrees. γ must be in radians.
- Applying ε = σ/E beyond the elastic limit, where Hooke’s law no longer holds.
Practise with the stress-strain calculator and the stress and strain MCQs. The strength of materials syllabus these topics sit in follows the AICTE model curriculum.
FAQs
What is the formula of strain?
Strain = change in length / original length, written ε = ΔL / L. For shear, strain is γ = x / h (the change in angle in radians), and volumetric strain is ΔV / V.
What is the SI unit of strain?
Strain has no unit because it is a ratio of two lengths. It is written as a plain number, a percentage, or in microstrain (1 µε = 10-6).
How are stress and strain related?
Within the elastic limit, stress is proportional to strain (Hooke’s law): stress = E × strain, where E is Young’s modulus. For shear, τ = Gγ.
What is the difference between engineering strain and true strain?
Engineering strain divides the extension by the original length, e = ΔL / L0. True strain uses the instantaneous length and equals ln(1 + e). They are almost equal below about 1% strain but differ a lot at large strains.
What is the relation between volumetric strain and linear strain?
For small strains, volumetric strain equals the sum of the three normal strains, εx + εy + εz. For a bar in simple tension it is ε(1 – 2ν), and for a cube under equal pressure on all sides it is 3ε.