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The density of a material is a measure of the mass per unit volume. It is given the symbol, ρ. Density is a useful quantity because it allows us to compare different materials

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the extension was proportional to the force applied.
extension ∝ force
Δl ∝ F

We can therefore write Hooke’s law as:
F = kΔl
where
F is the applied force in N
k is the spring constant in N m−1
Δl is extension in m

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A brittle material is one that shows little, or
no, plastic deformation before breaking.

Some materials do not show plastic behaviour but are brittle and break when the elastic limit is exceeded. Cast iron and glass are two examples of brittle materials. Figure 12.13 shows a typical force–extension graph for high-carbon steel, which is also a brittle material. The material fractures and breaks. It does not show plastic behaviour.

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Ductile materials can be formed into wires
by stretching them. They show ductility.

Wires obey Hooke’s law because the bonds between the metal atoms act like springs. When the wire is stretched the bonds lengthen slightly. When the force is removed, the bonds return to their original length.

However, if the force applied is too great, and the elastic limit exceeded, then the metal atoms will be able to move past one another and the wire lengthens. This is known as ductility, and is a very useful property as it allows metals to be formed into thin wires.

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The way in which ductile and brittle materials fracture is also different. Ina ductile material, the sample of material will elongate and ‘neck’ before itbreaks. On a force–extension graph, necking occurs in the plastic regionof the graph. In a brittle material there is no change in the shape of thematerial because it does not undergo plastic behaviour. A straight breakin the material is seen.

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elastic strain energy = 1/2 × F × Δl