Hooke’s law and the stress-strain curve
Hooke’s law
Hang a long steel wire from a strong beam and add weights to it, one at a time. Each weight stretches the wire a little more.
Plot the stress, the force per unit area, against the strain, the fractional increase in length. For small loads the points fall on a straight line through the origin.
Double the stress and the strain doubles. Triple it and the strain triples. Stress is proportional to strain.
This was discovered by the English physicist Robert Hooke in the 1670s, and it is called Hooke’s law: for small deformations, stress is proportional to strain.
We write it as stress equals k times strain. The constant k is called the modulus of elasticity. Since strain has no unit, k has the same unit as stress, newtons per square metre.
A large modulus means a stiff material: you need a big stress for even a small strain. Each material, and each kind of strain, has its own modulus.
You have met this before. A spring obeys F equals minus k x. Hooke’s law for materials is the same idea, written per unit area and per unit length, so it does not depend on the size of the sample.
But Hooke’s law is an empirical law. It is found by experiment, it holds only for small deformations, and some materials never obey it at all. To see where it stops, we must keep pulling.
The stress-strain curve
In a tensile test, a machine pulls a metal wire or rod slowly and records the stress for every value of the strain, all the way until it breaks. Watch the graph being drawn.
From O to A the graph is a straight line. Here Hooke’s law holds and stress is proportional to strain. Point A is called the proportional limit.
From A to B the line starts to curve, so stress and strain are no longer proportional. But if we remove the load anywhere up to B, the wire still returns to its original length. It is still elastic.
Point B is the yield point, also called the elastic limit. The stress there is the yield strength, written σ y.
Beyond B the strain grows rapidly even for a small increase in stress. The material flows. This is the plastic region.
Stop at a point C in the plastic region and remove the load. The wire springs back along a line parallel to O A, but it does not reach zero. It is now permanently longer. This leftover strain is called the permanent set.
Keep pulling. The stress rises to a maximum at D, the ultimate tensile strength, σ u. Beyond D a narrow neck forms, and extra strain appears even when the force is reduced.
Finally, at point E, the wire breaks. This is the fracture point.
A common mistake is to think that every stretch below the breaking point is safe. In fact, anything past B leaves the wire permanently deformed. Engineers design to stay well inside the elastic region.
Ductile, brittle and elastomers
The stress-strain curve is different for every material. Look at where D and E sit on the curve.
For copper or mild steel, D and E are far apart. There is a long plastic region, so these metals can be drawn into thin wires or beaten into sheets. They are called ductile.
For glass or cast iron, D and E are close together, with almost no plastic region. Such a material snaps suddenly, with no warning stretch. It is called brittle. Think of a glass bangle that breaks without bending.
Now something completely different: rubber, or the elastic tissue of the aorta, the large blood vessel that carries blood from the heart.
Its stress-strain curve has no straight part at all, so it does not obey Hooke’s law. Yet it is elastic over a huge range: it can be stretched to several times its length and still return.
Substances like these, which can be stretched to cause large strains, are called elastomers.
Put the three curves side by side. Steel: a straight start, a yield point and a long plastic region. Glass: a straight line that ends suddenly. Rubber: a long, gentle S-shaped curve with no straight part at all.
So a material can be elastic without obeying Hooke’s law. Elastic means it comes back. Hooke’s law means stress is proportional to strain. Do not mix up the two.
Try it: wire-stretching tester
- Press Run test (or Space) and watch the curve pass A, B, D and E.
- The first half of the stretch slider is a close-up of the elastic region; the second half runs to fracture.
- Stop just past B and press Unload. The wire comes back along a line parallel to OA and keeps a permanent set.
- Change the diameter or length. The force and extension change, but the stress-strain curve does not.
- Try glass (brittle) and rubber (an elastomer) for contrast.
Past the yield point
A steel wire is stretched beyond its yield point B to a point C in the plastic region, and then the load is removed. What happens?
- It returns to its original length along the same curve
- It comes back along a line parallel to OA and stays permanently longer
- It stays at exactly the same length as at C
- It breaks, because it went past the elastic limit
Answer
B. Unloading is elastic again: the stress falls along a line parallel to OA (the same slope as Hooke’s law). That line reaches zero stress at a strain greater than zero, so the wire keeps a permanent set. It shrinks back a little, but not all the way. It breaks only at E.
Reading the graph
On the straight part of a metal’s stress-strain graph, a stress of 1.5 × 10⁸ N/m² produces a strain of 2.0 × 10⁻³. What is the modulus of elasticity?
- 3.0 × 10⁵ N/m²
- 7.5 × 10¹⁰ N/m²
- 1.3 × 10⁻¹¹ N/m²
- 7.5 × 10⁷ N/m²
Answer
B. In the Hooke’s law region, modulus = stress ÷ strain = 1.5 × 10⁸ ÷ 2.0 × 10⁻³ = 7.5 × 10¹⁰ N/m². That is the slope of the line OA. Multiplying instead of dividing gives 3.0 × 10⁵; dividing the wrong way round gives 1.3 × 10⁻¹¹. (This value is close to aluminium’s.)
Summary
- Hooke’s law:
stress = k × strain - Modulus of elasticity:
k = stress / strain (N/m²)
- Hooke’s law: for small deformations, stress is proportional to strain. It is an experimental law.
- On the stress-strain curve: A proportional limit, B yield point (elastic limit, σy), D ultimate tensile strength (σu), E fracture.
- Beyond B the deformation is plastic: on unloading the wire keeps a permanent set.
- D and E far apart: ductile (copper, mild steel). Close together: brittle (glass).
- Elastomers like rubber and the aorta are elastic over huge strains but do not obey Hooke’s law.