Elasticity, Stress, Strain and Hooke's Law: Exam Notes

Every solid changes shape a little when a force acts on it. Elasticity is the property by which a body returns to its original shape and size once the deforming force is removed, and plasticity is the opposite property, in which the change stays after the force is withdrawn. A perfectly elastic body would recover completely and a perfectly plastic body would not recover at all; quartz fibre comes closest to the first and putty to the second. This chapter supplies a steady two or three questions in science papers of the SSC, Railway and Defence examinations, and almost all of them turn on definitions, units and the stress-strain curve.

Stress and its kinds

When an external force deforms a body, the internal forces between its particles set up an equal and opposite restoring force. Stress is this restoring force measured per unit area of the surface on which it acts, so stress equals force divided by area. Its SI unit is the newton per square metre, which is given the name pascal, and its dimensional formula is the same as that of pressure. Three kinds are distinguished. Longitudinal or normal stress acts along the length and is called tensile when it stretches and compressive when it squeezes. Volume or hydraulic stress acts equally on all faces, as when a body is placed in a fluid under pressure. Shear or tangential stress acts along a surface and tends to slide one layer over another.

Strain and its kinds

Strain is the measure of the deformation produced. It is the change in a dimension divided by the original value of that dimension, so it is a pure ratio with no unit and no dimensions. Longitudinal strain is the change in length divided by the original length, volume strain is the change in volume divided by the original volume, and shear strain is the angle through which a face is tilted, measured in radians. Each kind of stress has its own kind of strain, and each pair gives its own modulus of elasticity.

Hooke's law and the three moduli

Robert Hooke stated in the seventeenth century that within the elastic limit the stress produced in a body is directly proportional to the strain. The constant of proportionality, the ratio of stress to strain, is called the modulus of elasticity, and because strain has no unit the modulus carries the unit of stress, the pascal. Three moduli follow from the three pairs.

The reciprocal of the bulk modulus is called compressibility, so a gas, which is easy to squeeze, has a very small bulk modulus and a very large compressibility. Young's modulus and the modulus of rigidity are defined only for solids, while the bulk modulus is defined for solids, liquids and gases alike. A larger modulus means a stiffer material: the Young's modulus of steel is about two hundred thousand million newtons per square metre, roughly three times that of copper and enormously greater than that of rubber, which is why steel, not rubber, is called the more elastic of the two in physics.

The stress-strain curve

If a wire is loaded steadily and the stress is plotted against the strain, the graph tells the whole story of the material. The first part is a straight line through the origin, and the stress at its upper end is the proportional limit, the farthest point at which Hooke's law holds exactly. A little beyond it lies the elastic limit or yield point, up to which the wire still returns to its original length. Past the yield point the material flows plastically and a small increase in load gives a large extension. The highest stress the material can bear is the ultimate tensile strength, after which the wire develops a local thinning called necking, the apparent stress falls, and the wire snaps at the fracture point. A material that shows a long plastic stretch before breaking is ductile, such as copper, and one that breaks soon after the elastic limit is brittle, such as glass or cast iron. Rubber and other elastomers return to shape but do not obey Hooke's law at all and their loading and unloading curves do not coincide, an effect called elastic hysteresis.

Poisson's ratio, energy and uses

When a wire is stretched it also grows thinner. The ratio of the lateral strain to the longitudinal strain is called Poisson's ratio; it has no unit, its theoretical limits are minus one and one half, and for most common materials it lies between about 0.2 and 0.4. Work done in stretching a body is stored as elastic potential energy, and the energy stored per unit volume equals one half of the product of stress and strain. These ideas govern a great deal of engineering: the depression of a beam loaded at its centre varies inversely with the cube of its depth, which is why girders are given the shape of the letter I, and the thickness of the ropes of a crane is fixed by the breaking stress of steel, a quantity that depends on the material alone and not on the length or the thickness of the sample.