Young's modulus from stress and strain:
how stiff a material really is

Stiffness as the slope of the line

Load a steel bar gently, record the stress and the strain at several points and plot them. For the first part of the test the points sit on a straight line through the origin. The slope of that line is Young's modulus, written E.

A steeper line means a stiffer material: it takes a lot of stress to produce a small strain. Rubber has a shallow slope and steel a very steep one. Importantly, modulus describes stiffness, not strength. A material can be stiff yet brittle, or flexible yet strong.

The straight-line region is called the elastic range. In it, removing the load returns the material to its original length, and Hooke's law applies. The formula below is valid only there.

The formula and its units

Common units are pascals (Pa), megapascals (MPa) and gigapascals (GPa). 1 GPa equals 1,000 MPa. Steel is usually listed as 200 to 210 GPa; the same value appears as 200,000 MPa when stress is in MPa.

The relation also gives extension for a given load: ΔL = (F × L₀) ÷ (A × E). Rearranging the definition this way is how designers predict how much a rod, cable or column will stretch.

Worked example: 200 MPa and a strain of 0.001

In a tensile test a steel specimen shows an axial stress of 200 MPa when its strain is 0.001, which is 0.1 percent. Find the modulus.

Use it forward: a 2 m long bar of this material at 200 MPa extends by strain × length = 0.001 × 2,000 mm = 2 mm. At half the stress, 100 MPa, the extension is 1 mm, because the relationship is linear.

Check plausibility: 200 GPa matches the commonly quoted modulus of steel. If a calculation for steel returns 20 GPa or 2,000 GPa, there is probably a missing factor of ten in the strain or stress.

Typical modulus values for comparison

Steel is about three times stiffer than aluminium. A beam of the same shape in aluminium would bend roughly three times as much under the same load, even though aluminium is much lighter. That trade-off sits behind many design choices in aircraft and bicycles.

Values depend on alloy, temperature and manufacturing. Use the figure from the datasheet for a specific grade whenever the extension or deflection matters.

Getting a reliable modulus from test data

1. 1Compute stress for each reading: load in newtons divided by the original cross-section in mm².
2. 2Compute strain for each reading: change in gauge length divided by original gauge length.
3. 3Plot stress against strain and fit a line through the initial straight portion, ignoring the first part where grips settle.
4. 4Divide the rise by the run of that line to get E, or compute σ ÷ ε at a point within the linear range.

Single-point calculations are sensitive to small errors in strain, which is tiny for metals. A slope fitted to several points is more reliable than one division.

Using E to predict the stretch of a wire

Once the modulus is known, any member of that material can be analysed. Take a steel wire 2 m long and 2 mm in diameter carrying a 100 N pull, a load of just over 10 kg.

The result shows why steel feels rigid: a ten-kilogram hanging load stretches a two-metre wire by about a third of a millimetre. It also shows the combination that controls stiffness: material (E), cross-section (A) and length (L).

Replace the wire with aluminium of E = 70,000 MPa and the same load stretches it by 0.91 mm, about 2.9 times as much. To match the steel wire's stiffness, you would need a diameter about 1.7 times larger.

Modulus versus the stiffness of a structure

E belongs to the material; the stiffness of a particular member also depends on its shape and length. A bar loaded along its length has axial stiffness E × A ÷ L, so doubling the area doubles it, while doubling the length halves it.

For bending, the geometry enters through the second moment of area I, and stiffness is proportional to E × I. That is why an I-beam or a tube resists bending far better than a solid bar of the same weight, even though the modulus of the steel is identical.

In practice, designers ask for a material with high modulus for stiffness-critical parts such as machine frames, and use the geometry to gain more stiffness cheaply. Replacing steel with aluminium without increasing the section loses roughly two thirds of the stiffness.

Composite materials add a wrinkle: carbon-fibre laminates may reach 70 to 150 GPa along the fibres and far less across them, so a single modulus is not enough and the direction must be specified.

Whenever you compare two materials for stiffness per unit of weight, divide E by density. Steel and aluminium come out nearly equal on that measure, which surprises many people.

Where the simple modulus stops helping

• Past the elastic limit. Yielding bends the curve; stress divided by strain then gives a lower 'secant' value that is not the modulus.
• Non-linear materials. Concrete, rubber and soft tissue do not have one constant slope, so tangent or secant moduli are quoted instead.
• Direction-dependent materials. Timber and fibre composites are stiffer along the fibres than across them.
• Temperature and rate. Polymers soften noticeably when warm and stiffen when loaded fast.