Mechanical Properties of Solids

In this chapter: elasticity and plasticity, the three kinds of stress and strain, Hooke's law, the stress-strain curve, Young's modulus, shear modulus, bulk modulus, Poisson's ratio, elastic potential energy, and engineering uses of elastic behaviour.

Elasticity and plasticity

When a force deforms a solid, the atoms are pushed slightly out of their equilibrium positions and interatomic forces try to pull them back. This restoring force is why a body returns to its original shape when the force is removed. The property is called elasticity. Bodies that do not regain their shape, such as putty or mud, are plastic. No real material is perfectly elastic or perfectly plastic.

Stress and strain

Stress is the restoring force per unit area, equal in magnitude to the applied force per unit area: stress = F/A. Its SI unit is N m⁻² or pascal (Pa), with dimensions [ML⁻¹T⁻²]. Strain is the fractional change in size or shape and has no unit.

Hooke's law

For small deformations, stress is proportional to strain: stress = k × strain, where the constant k is the modulus of elasticity. Since strain has no unit, the modulus has the same unit as stress (Pa). Hooke's law is an empirical law that holds for most materials only up to a limit.

The stress-strain curve

If a metal wire is stretched steadily and the stress is plotted against the strain, the curve has the following parts.

• O to A: a straight line. Hooke's law holds, and the slope is Young's modulus. A is the proportional limit.
• A to B: stress and strain are no longer proportional, but the wire still returns to its original length when the load is removed. B is the yield point or elastic limit, and the stress there is the yield strength σ_y.
• Beyond B: strain increases rapidly even for a small change in stress. If the load is removed at a point C, the wire does not return to zero strain; the leftover deformation is called the permanent set. This is plastic behaviour.
• D is the maximum stress the wire can withstand, the ultimate tensile strength σ_u. Beyond D the wire thins at some point and breaks at the fracture point E, even if the stress is reduced.

If D and E are far apart, the material is ductile (it can be drawn into wires). If they are close together, it is brittle. Substances like rubber and the tissue of the aorta can be stretched to large strains, have no well-defined linear region, and still return to their original length; they are called elastomers.

Elastic moduli

Young's modulus

Young's modulus is a property of the material; it does not depend on the length or thickness of the wire. Metals have large values, so they need a large force for a small change in length. Steel is preferred in heavy-duty machines and buildings for this reason. Young's modulus is defined for solids only.

Worked example: A structural steel rod of radius 10 mm and length 1.0 m is stretched by a force of 100 kN along its length. Find the stress, the elongation and the strain. (Y for steel = 2.0 × 10¹¹ Pa)
Solution: A = πr² = 3.14 × (10⁻²)² = 3.14 × 10⁻⁴ m². Stress = 100 × 10³/(3.14 × 10⁻⁴) = 3.18 × 10⁸ Pa. Elongation ΔL = (stress × L)/Y = (3.18 × 10⁸ × 1)/(2 × 10¹¹) = 1.59 × 10⁻³ m = 1.59 mm. Strain = ΔL/L = 1.59 × 10⁻³, that is 0.16%.

Shear modulus

Bulk modulus

Solids are the least compressible, liquids more so, and gases the most compressible. For water, B ≈ 2.2 × 10⁹ Pa. A pressure of 1.0 × 10⁷ Pa (about 100 atmospheres) changes the volume of water by only 10⁷/(2.2 × 10⁹) ≈ 0.45%. Unlike Young's and shear moduli, the bulk modulus is defined for solids, liquids and gases.

Poisson's ratio

A wire that is stretched also becomes slightly thinner. The ratio of lateral strain to longitudinal strain is Poisson's ratio, σ = (Δd/d)/(ΔL/L). It has no unit and no dimensions, and for most metals it lies between about 0.25 and 0.35.

Elastic potential energy

The work done in stretching a wire is stored in it as elastic potential energy:

Applications of elastic behaviour

• Crane ropes: the rope must be thick enough that the stress under the maximum load stays well below the yield strength, with a safety factor. A single thick wire would be stiff, so ropes are made of many thin wires braided together.
• Beams and bridges: a beam of length l, breadth b and depth d, supported at its ends and loaded by W at the middle, sags by δ = Wl³/(4bd³Y). Since δ ∝ 1/d³, increasing the depth is the most effective way to reduce bending. An I-shaped girder gives a large depth with less material and weight.
• Mountains: the rock at the base must not flow under the weight of the mountain above it. This limits the height of mountains on Earth to about 10 km.

Common mistakes: (1) Thinking a thicker wire of the same material has a larger Young's modulus; it only stretches less, because the area is larger. (2) Forgetting to convert the diameter to radius, or mm² to m², when finding the area. (3) Mixing up the yield point and the ultimate tensile strength. (4) Leaving out the minus sign in the bulk modulus and then getting a negative B. (5) Assuming that a body with a larger strain for the same stress is "more elastic"; in physics, steel is more elastic than rubber because it has a larger modulus.

JEE and NEET focus

• Elongation of a wire, and ratios of elongation for wires of different length and radius.
• Reading the stress-strain curve and comparing materials from their graphs.
• Bulk modulus and compressibility, including change in volume or density under pressure.
• Elastic energy stored in a stretched wire and energy density.
• Combinations: wires in series and parallel, and a composite wire of two materials.

Practice questions