Stress and Strain
A steel cable holding up a footbridge and a steel paperclip are made of exactly the same material, yet one carries tonnes of load and the other barely holds two sheets of paper together. What actually determines how a loaded component behaves isn't the raw force on it, or how far it stretches — it's stress and strain, the two quantities that strip away size and shape and get down to the material itself.
Why force and extension aren't enough
Suppose you apply the same force to two rods made of the same material — one thin, one thick. The thin rod stretches more. Apply that same force to a short rod and a long rod of identical cross-section — the longer one stretches further in absolute terms, even though proportionally it may be behaving identically to the shorter one. Force and extension, on their own, tangle together two separate things: how the material itself responds to loading, and the specific geometry — cross-sectional area and original length — of the particular sample being tested.
For an engineer choosing a material for a bridge cable, an aircraft wing spar, or a bolt, what actually matters is the underlying material behaviour, not the arbitrary dimensions of whatever test sample happened to be pulled in a lab. That's the entire motivation for defining two new quantities — stress and strain — that normalise out size and shape, so that different specimens, and different materials, can be compared on equal terms.
Stress: force normalised by area
Stress, given the symbol $\sigma$, is defined as the applied force divided by the cross-sectional area over which it acts:
where $F$ is the force in newtons (N) and $A$ is the original cross-sectional area in square metres (m²). Stress therefore has units of newtons per square metre (N m⁻²), which is given the name pascal (Pa) — the same unit used for pressure, since stress is dimensionally identical to pressure, though physically the two describe different situations. In engineering contexts, stresses are often large enough that megapascals (MPa, $10^6$ Pa) or gigapascals (GPa, $10^9$ Pa) are more convenient units.
Dividing by area is what makes stress a property of the loading condition rather than of the specific sample size: double the cross-sectional area of a rod carrying the same force, and the stress halves, even though the force hasn't changed at all.
Tensile and compressive stress
Loading can act in different directions, and this lesson (and the rest of this topic) mostly builds on two basic types:
- Tensile stress — the load stretches the material, pulling it apart, as in a cable supporting a hanging load or a rod being pulled from both ends.
- Compressive stress — the load squeezes the material, pushing it together, as in a column supporting a roof or a foundation pillar under a building's weight.
The defining equation $\sigma = F/A$ applies to both; the distinction is simply the direction the force acts relative to the material, and correspondingly whether the material tends to lengthen or shorten under the load.
Strain: extension normalised by original length
Strain, given the symbol $\varepsilon$, is defined as the extension (or compression) produced by a stress, divided by the material's original length:
where $\Delta L$ is the change in length and $L_0$ is the original, unstressed length, both measured in the same units (typically metres). Because strain is a ratio of two lengths, it is dimensionless — it has no units. Strain is often quoted as a plain decimal (for example, $\varepsilon = 0.002$) or, equivalently, as a percentage.
Just as with stress, dividing by the original length removes the dependence on the specific size of the sample: a longer specimen naturally stretches further in absolute terms for the same underlying material response, but its strain — the fractional extension — is unaffected by how long it started out.
Correspondingly, tensile strain describes fractional lengthening under tensile stress, and compressive strain describes fractional shortening under compressive stress.
Stress and strain are defined precisely so that they describe the material, not the specimen. Two rods of the same steel but different sizes will stretch by different absolute amounts under the same force — but if you compute stress and strain for each, you get identical values. That's what makes stress and strain the quantities engineers actually use to characterise, compare, and select materials, rather than raw force and extension.
Watch out for the deliberately similar names: the Greek letter sigma ($\sigma$) is used for stress, and epsilon ($\varepsilon$) for strain. It's easy to mix them up under exam pressure — stress has units (Pa); strain does not.
Worked example
These two quantities — stress and strain — are the foundation for everything else in this topic. The next lesson introduces the Young modulus, which links them together and gives a single number describing how stiff a material is.
Key takeaways
- Stress $\sigma = F/A$ is force per unit cross-sectional area, measured in pascals (Pa) or N m⁻²; it normalises force by the specimen's cross-section.
- Strain $\varepsilon = \Delta L / L_0$ is the fractional change in length; it is dimensionless, normalising extension by the specimen's original length.
- Stress and strain describe the material's response independent of the specific size or shape of a given sample, unlike raw force and extension.
- Tensile stress/strain arise from stretching loads; compressive stress/strain arise from squeezing loads.
- Two differently sized specimens of the same material under equivalent loading conditions give the same stress and strain, even though their absolute forces and extensions differ.
- Stress and strain are the quantities engineers use to characterise, compare, and select materials for a given application.