Deformation of solids

Syllabus points

• Understand that deformation is caused by tensile or compressive forces.
• State Hooke's law and define the spring constant.
• Define and use stress, strain and the Young modulus.
• Distinguish between elastic and plastic deformation.
• Determine the elastic potential energy from the area under a force-extension graph.

Tension and compression

A tensile force stretches a body; a compressive force squashes it. Both are described by the same quantities, and a question about a strut in compression uses exactly the equations below.

Hooke's law

For a material obeying Hooke's law, the extension is proportional to the force applied:

F = kx

where k is the spring constant or force constant, in N m⁻¹, and x is the extension, not the total length. Subtracting the natural length is the first step of most calculations here and the first thing people forget.

Hooke's law holds only up to the limit of proportionality. Beyond that the force-extension graph curves away from the straight line.

Springs in series and parallel

• In series, each spring carries the full load, so each extends by x and the total extension is doubled. Two identical springs in series have a combined spring constant of k/2, so the system is easier to stretch.
• In parallel, the load is shared, so each carries F/2 and extends half as much. The combined spring constant is 2k.

The reasoning is worth carrying rather than the result: work out what force each spring actually carries, and the rest follows.

The four points on the graph

Along a force-extension graph, in order:

1. Limit of proportionality — the end of the straight line. Beyond it, F is no longer proportional to x.
2. Elastic limit — the last point from which the material returns to its original length when the load is removed. It is at or just past the limit of proportionality.
3. Yield point — where the material extends with little or no extra force.
4. Breaking point — where it fractures.

The distinction between the first two is a favourite question. Proportionality is about the shape of the graph; the elastic limit is about whether the material returns to its original length. A material can be elastic without obeying Hooke's law: rubber does exactly that.

Elastic and plastic

• Elastic deformation: the body returns to its original shape when the load is removed. No permanent extension.
• Plastic deformation: a permanent extension remains after the load is removed.

Past the elastic limit, unloading follows a line parallel to the original loading line but displaced along the extension axis. The intercept on the extension axis is the permanent deformation. The area enclosed between the loading and unloading curves is the energy that has become internal energy in the material rather than being recovered.

Stress, strain and the Young modulus

The spring constant describes a particular object. To describe the material itself, independently of its dimensions, use:

Stress σ = F / A, in pascals. A is the cross-sectional area, so for a wire of diameter d it is πd²/4, not πd². Halving the radius quarters the area, and questions exploit that.

Strain ε = x / L, where L is the original length. Strain has no unit, being a ratio of two lengths, and is often given as a percentage.

Young modulus E = stress / strain = FL / Ax, in pascals.

E is a property of the material. A thicker wire of the same material has the same Young modulus but a larger spring constant, and separating those two statements is the point of the topic.

Worked example. A wire of length 2.00 m and diameter 0.50 mm is stretched by 1.2 mm under a load of 25 N. Find the Young modulus.

A = π(0.25 × 10⁻³)² = 1.963 × 10⁻⁷ m².

Stress = 25 / (1.963 × 10⁻⁷) = 1.27 × 10⁸ Pa.

Strain = 1.2 × 10⁻³ / 2.00 = 6.0 × 10⁻⁴.

E = (1.27 × 10⁸) / (6.0 × 10⁻⁴) = 2.1 × 10¹¹ Pa.

The radius is half the diameter, and forgetting that alone multiplies the answer by four.

The Young modulus is found in practice from the gradient of a stress-strain graph in its straight-line region. On a force-extension graph the gradient is the spring constant instead.

Elastic potential energy

The work done stretching a material is stored as elastic potential energy, and it is the area under the force-extension graph.

For a material obeying Hooke's law the graph is a triangle:

E = ½Fx = ½kx²

The ½ is not optional. Using Fx gives twice the right answer, and it comes from forgetting that the force starts at zero and builds up: the average force during the stretch is F/2.

Because of the square, doubling the extension quadruples the stored energy.

Beyond the limit of proportionality the graph is no longer a triangle, so the ½kx² formula fails and the area must be counted from the graph, usually by counting squares.

Ductile, brittle and polymeric

• A ductile material, such as copper, has a long plastic region and can be drawn into a wire.
• A brittle material, such as glass, breaks at the elastic limit with no plastic deformation at all.
• A polymeric material, such as rubber, has a curved loading line and does not obey Hooke's law, but is still elastic over a large extension.

Common mistakes

• Using the total length rather than the extension in F = kx.
• Confusing the limit of proportionality with the elastic limit.
• Using the diameter instead of the radius when finding the cross-sectional area.
• Giving strain a unit.
• Saying a thicker wire of the same material has a larger Young modulus.
• Omitting the ½ in the elastic potential energy.
• Reading the Young modulus off a force-extension graph instead of a stress-strain graph.
• Saying a material that does not obey Hooke's law cannot be elastic.