Deformation of solids

When forces are applied to a solid object, they can change its shape. This change in shape is called deformation. There are two main types of forces to consider. Tensile forces are pairs of forces acting outwards from the centre of an object, tending to stretch it and increase its length. Compressive forces are pairs of forces acting inwards, tending to squash the object and decrease its length. The change in length is called extension (for tensile forces) or compression (for compressive forces). For these forces to cause deformation, they must act in balanced pairs; a single unopposed force would simply cause the object to accelerate.

When you apply a force to a spring, it stretches. For many materials, the extension is directly proportional to the applied force, as long as the force isn't too large. This relationship is known as Hooke's Law. We can write this as F = kx, where F is the applied force, x is the extension, and k is a constant called the spring constant. The spring constant (or force constant) is a measure of the stiffness of the spring; a stiffer spring has a larger k. If you plot a graph of force (y-axis) against extension (x-axis), the gradient of the straight-line section is the spring constant k. The point where the graph stops being a straight line is the limit of proportionality. Up to a certain force, called the elastic limit, the material will return to its original shape when the force is removed (elastic deformation). If you stretch it beyond the elastic limit, it will be permanently stretched (plastic deformation).

To compare the 'stretchiness' of materials without worrying about their shape and size, we use the concepts of stress and strain. Stress (σ) is a measure of how much force is applied per unit of area. It's defined as the force acting perpendicular to the cross-sectional area, A. So, stress σ = F/A. Its unit is the pascal (Pa) or N m⁻². Strain (ε) is a measure of the fractional change in length. It is the extension, x, divided by the original length, L. So, strain ε = x/L. Since it's a ratio of two lengths, strain has no units. For a material obeying Hooke's law, stress is proportional to strain. The constant of proportionality is called the Young modulus, E. It is a measure of the material's stiffness. The Young modulus E = stress / strain. A material with a high Young modulus, like steel, is very stiff and deforms very little under stress.

To measure the Young modulus of a material, typically in the form of a wire, a specific experiment is performed. A long sample wire (e.g., 2 metres) is hung from a rigid support with a measurement scale (e.g., a vernier scale) attached. A control wire is often hung nearby to compensate for temperature changes. First, the original length (L) of the wire is measured with a tape measure. The diameter(d) is measured at several points along the wire using a micrometer screw gauge and an average is taken; this is used to calculate the cross-sectional area (A = π(d/2)²). Then, known weights are added incrementally to apply a force (F = mg). For each weight, the new position on the vernier scale is read to find the extension (x). A graph of stress (σ = F/A) on the y-axis against strain (ε = x/L) on the x-axis is plotted. The gradient of the straight-line portion of this graph gives the Young modulus, E.

When you stretch or compress a material that obeys Hooke's law, you do work on it. This work is stored as energy within the material, called elastic potential energy or strain energy. As long as the deformation is elastic, this energy can be fully recovered when the force is removed. The work done in stretching a spring is not simply force × extension, because the force is not constant; it increases from zero to its final value. The work done is equal to the area under the force-extension graph. For a material that obeys Hooke's Law, this graph is a triangle. The area of a triangle is ½ × base × height, so the energy stored is E_p = ½Fx. Since F = kx, we can also write the stored energy as E_p = ½k(x)x = ½kx². If a material is stretched beyond its elastic limit, some of the work done goes into permanently rearranging the atoms (plastic deformation), and this energy is not recovered as elastic potential energy; it is mostly dissipated as heat.