Strain: Longitudinal, Shearing and Volumetric Explained

While stress tells us how hard a material is being pulled internally, strain tells us how much the material actually deforms in response. Strain is the fractional change in a dimension: how much longer, how much more tilted, or how much smaller in volume the body becomes, compared with its original size.

Strain matters because it is the visible half of elasticity. Engineers measure strain with sensors on bridges and aircraft wings to detect dangerous overloads, and the ratio of stress to strain reveals a material's stiffness — its modulus of elasticity.

Key Concepts

• Longitudinal strain: change in length divided by original length (ΔL ÷ L). A stretched wire has positive strain; a compressed pillar has negative strain.
• Shearing strain: the sideways shift of the top face divided by the height (Δx ÷ h), equal to the tilt angle in radians for small shifts.
• Volumetric strain: change in volume divided by original volume (ΔV ÷ V), negative when the body is compressed.
• Dimensionless: strain has no units — it is a pure ratio, often quoted as a percentage.
• Small-strain approximation: the textbook formulas assume deformations are tiny compared with the object's size, which is true for metals and concrete in normal use.

Formulas

Longitudinal strain ε = ΔL ÷ L, where ΔL is the change in length and L the original length. Shearing strain = Δx ÷ h, where Δx is the sideways displacement of the top surface and h is the height of the block. Volumetric strain = ΔV ÷ V. All three are pure numbers with no SI unit. They pair with the matching stresses: longitudinal strain with longitudinal stress (giving Young's modulus), shearing strain with shearing stress (shear modulus), and volumetric strain with hydraulic stress (bulk modulus).

Worked Example

A steel rod 2 m long stretches by 3 mm under a heavy load. Convert: 3 mm = 0.003 m. The longitudinal strain is ε = ΔL ÷ L = 0.003 ÷ 2 = 0.0015, or 0.15%. Tiny — yet in a 2-metre rod that is a visible 3 mm stretch, enough to be measured with a ruler. As a second check, imagine a rubber block 0.05 m tall whose top face slides sideways by 0.002 m (2 mm). The shearing strain is 0.002 ÷ 0.05 = 0.04, a tilt of about 2.3 degrees. The same formulas work for any material; only the resulting numbers differ.

Common Misconceptions

• "Strain is measured in metres or pascals." Strain is a ratio of two lengths (or volumes), so the units cancel — it is dimensionless, though people often write it as a percentage.
• "A large strain always means a large stress." Rubber shows huge strain under small stress while steel shows tiny strain under large stress; the stress–strain link depends on the material's modulus.
• "Strain means permanent damage." Elastic strain vanishes completely when the load is removed — only strain beyond the elastic limit stays permanent.

Real-World Applications

• Strain gauges: tiny sensors glued to bridges, dams and aircraft wings measure strain electrically and warn of overload.
• Expansion joints: bridges and railway tracks leave gaps because temperature changes produce thermal strain that must go somewhere.
• Weighing scales: electronic scales measure the microscopic strain in a metal element to compute your weight.
• Wire manufacturing: drawing metal through dies is controlled plastic strain, shaping wire without breaking it.

Strain is simply fractional deformation — stretch over length, slide over height, or volume change over volume — and it carries no units. Paired with stress, it unlocks the moduli that let engineers predict exactly how much any structure will bend, stretch or squeeze.