Materials

What materials covers

What happens inside a solid when you pull on it, and how to separate a property of an object from a property of the material. The unit also covers pressure in a fluid, upthrust and viscous drag. It is short, heavily graph-based, and its measurement of the Young modulus is a standard practical question.

The main ideas

• Density, and getting the conversion between g cm^-3 and kg m^-3 the right way round.
• Hooke's law while it holds, with the limit of proportionality and the elastic limit as two separate points.
• Energy stored as the area under a force-extension graph, and the energy that does not return after plastic deformation.
• Tensile stress, tensile strain and the Young modulus, taken from the gradient of the straight part of a stress-strain graph.
• Yield, breaking stress and brittle failure read off a curve, and the ductile, brittle and polymeric shapes told apart.
• Pressure in a fluid, upthrust as the weight of fluid displaced, and Stokes' law with the terminal velocity of a small sphere.

The equations it turns on

ρ = mV$\rho = \frac{m}{V}$ρ = m/V

density

F = kΔL$F = k\Delta L$F = kΔL

Hooke's law, up to the limit of proportionality

energy stored = 12FΔL$\text{energy stored} = \tfrac{1}{2}F\Delta L$energy stored = (1/2)FΔL

the triangle under a force-extension line

stress = FA      strain = ΔLL$\text{stress} = \frac{F}{A} \qquad \text{strain} = \frac{\Delta L}{L}$stress = (F/A) strain = ΔL/L

the two quantities the Young modulus compares

E = stress/strain$E = \text{stress}/\text{strain}$E = stress/ strain

the Young modulus, assembled as FL / (A delta L) when needed

p = ρgh     F = 6πηrv$p = \rho gh \qquad F = 6\pi\eta rv$p = ρgh F = 6πηrv

pressure with depth, and viscous drag on a sphere

Where it usually goes wrong

• The spring constant belongs to the object and the Young modulus to the material. Cut a wire in half and k doubles while E is unchanged, which is why both quantities exist.
• The limit of proportionality and the elastic limit are different points. Past the first the line bends; past the second the extension does not fully recover.
• Most Young modulus calculations start from a diameter, so the area is pi d^2 / 4 and the millimetres must become metres before anything else.
• Floating depends on density, not weight: the weight of fluid displaced has to equal the weight of the object.

Where to start

Density and Hooke's law first, because the Young modulus lesson is the argument for why the spring constant is not enough. Take fluids last, and check your specification before spending time on it, since the boards differ on how much they require.