Mechanical Properties of Solids

Every solid has a definite shape and size because of the strong interatomic and intermolecular forces holding its atoms at their equilibrium positions. Pushing, pulling or twisting a solid applies a deforming force, displacing atoms slightly; this produces an internal restoring force that tries to bring the body back to its original shape once the force is removed. This tendency to regain original shape and size is called elasticity.

No material is perfectly elastic or perfectly plastic, but both are used as idealised limits. A body that regains its original configuration completely and instantly is perfectly elastic (quartz, phosphor bronze). A body with no tendency to recover, staying deformed, is perfectly plastic (putty, wet clay). Most engineering materials — steel, copper, aluminium — lie between these extremes: elastic up to a certain load, and permanently deformed beyond it.

Elastic behaviour comes entirely from interatomic forces. Every atom normally sits at its position of minimum potential energy; a deforming force shifts atoms from this position, and the interatomic force acts as a restoring force pulling them back — exactly like a compressed or stretched spring connecting neighbouring atoms.

When a deforming force acts on a body, an internal restoring force develops that is equal and opposite to it in equilibrium. Stress is this restoring force per unit area, measured on any cross-section of the body.

Stress = Restoring force/Area. Its SI unit is N/m² (pascal, Pa), the same as pressure, and its dimensional formula is [ML⁻¹T⁻²]. Stress is not exactly a vector since it does not add like a force, but for board purposes it is enough to know it is a force per unit area with the units of pressure.

Strain is the fractional change in a body's dimension (length, shape or volume) due to the deforming force: Strain = change in dimension/original dimension. Being a ratio of two lengths (or volumes), strain has no unit and no dimensions — one of the most commonly tested facts in this chapter.

A common trap: writing a unit for strain, or forgetting to square the radius when finding area from a diameter given in mm. Always convert lengths to metres, and remember A = πr² uses the radius, not the diameter.

Depending on how the deforming force is applied, stress and strain are classified into three kinds.

1. Longitudinal (tensile/compressive) stress and strain. When two equal and opposite forces act along a rod's length, normal to its cross-section, and change its length, the stress is longitudinal stress, and the strain — ΔL/L — is longitudinal strain. Length increasing is tensile; length decreasing (compression) is compressive.

2. Shearing stress and strain. When equal and opposite tangential forces act on the top and bottom faces of a body, keeping the bottom fixed, its shape changes but its volume stays the same — one face slides relative to the other. The tangential force per unit area is the shearing stress; the resulting angle of shear θ (≈ Δx/L for small deformations) is the shearing strain.

3. Hydraulic (volume) stress and strain. When a body is subjected to uniform pressure from all sides, its volume changes without any change in shape. The pressure applied is hydraulic (volume) stress, and ΔV/V is the volume strain — the only kind of stress a fluid at rest can support, since it cannot resist a shearing force.

Robert Hooke found experimentally that, for small deformations, the stress developed in a solid is directly proportional to the strain produced, provided the strain stays small: stress ∝ strain. This is Hooke's law, and it holds only within a certain range called the elastic limit — beyond this limit, stress and strain are no longer proportional, though the body may still be elastic for a little longer before permanent deformation sets in.

The constant of proportionality connecting stress and strain for a given type of deformation is called the modulus of elasticity for that deformation. It has the same unit as stress (Pa), because strain itself has no unit. Three moduli are defined for the three kinds of strain described above — Young's modulus (for longitudinal strain), shear modulus (for shearing strain) and bulk modulus (for volume strain) — and these are studied in detail later in the chapter.

It is worth being precise in an exam answer: Hooke's law is an experimental law with a restricted range of validity, not a universal one. Every real material eventually departs from this linear behaviour as the load increases, and this departure is exactly what the stress-strain curve in the next section is designed to show.

A stress-strain curve plots the stress developed in a test wire against the strain produced, as the load is increased gradually until it breaks. For a ductile metal such as copper or mild steel, the curve has these labelled regions:

• Proportional limit (O to A): a straight line through the origin; stress ∝ strain, Hooke's law holds, and the slope equals the Young's modulus.
• Elastic limit (near A/B): the highest stress up to which the material fully returns to its original dimensions on unloading.
• Yield point (B): beyond this, strain increases rapidly for a small stress increase, and permanent deformation begins.
• Plastic region (B to D): permanent deformation; the body will not return to its original shape.
• Ultimate tensile strength (D): the maximum stress before the cross-section necks down rapidly.
• Fracture point (E): where the wire finally breaks.

Materials behave very differently here. Ductile materials (copper, mild steel) show a long plastic region and can be drawn into wires. Brittle materials (cast iron, glass) fracture soon after the elastic limit, with almost no plastic region. Elastomers (rubber) show large strain for small stress and a markedly non-linear curve, with no proportional or plastic region — yet remain elastic because they still return to their original length.

Young's modulus (Y) is the ratio of longitudinal stress to longitudinal strain, within the elastic limit. Since strain is dimensionless, Y carries the unit of stress, Pa, and dimensional formula [ML⁻¹T⁻²].

A larger Y means a stiffer material. Steel (Y ≈ 2.0 × 10¹¹ Pa) is about 3 times stiffer than copper (≈ 1.1–1.3 × 10¹¹ Pa) and aluminium (≈ 7.0 × 10¹⁰ Pa) — exactly why steel is used for load-bearing structures.

Determining Y (Searle's apparatus). A reference wire and an experimental wire hang side by side from a rigid support, joined at the bottom by a frame with a spirit level and vernier scale. The reference wire, kept taut by a fixed weight, cancels out effects of temperature change or support yielding, since both wires are affected equally. A small initial load removes kinks in the experimental wire before the zero reading; the load is then increased in steps, and elongation ΔL read off each time. With length L and area A = πr² (r from a screw gauge), Y = FL/(AΔL) is found for each load. A stress-strain graph gives a straight line through the origin whose slope is the most reliable value of Y.

The shear modulus (G), also called the modulus of rigidity, is the ratio of shearing stress to shearing strain, within the elastic limit. Its SI unit is again the pascal, since shearing strain (the angle θ) is dimensionless.

For a rectangular block of height L, if a tangential force F is applied on the top face of area A while the bottom is fixed, and the top face shifts sideways by Δx, then the shearing strain is θ = Δx/L (for small θ), and G = (F/A)/θ.

For most solids, G is smaller than Y — typically G is roughly a third of Y for common metals — because it is generally easier to change a solid's shape (slide layers past each other) than to stretch its bonds along a line. This is also why a solid loses its shape (shears) more easily than it loses its volume.

Only solids possess a meaningful shear modulus. Fluids (liquids and gases) at rest cannot sustain a shearing stress at all — they simply flow to relieve it — so their shear modulus is taken as zero. This single fact is one of the most repeated one-mark/assertion-reason questions from this section.

The bulk modulus (B) is the modulus of elasticity for volume strain: the ratio of hydraulic (volume) stress to volume strain, within the elastic limit. Since a pressure increase always decreases volume, ΔV is negative when p is positive; the negative sign is included purely so B itself comes out positive.

Solids have the largest bulk modulus, liquids intermediate values, and gases very small ones — this is why gases compress far more easily than liquids, and liquids more than solids. Water's B (≈ 2.2 × 10⁹ Pa) is roughly a thousand times that of air, and steel's (≈ 1.6 × 10¹¹ Pa) roughly a hundred times that of water. The reciprocal of B is compressibility, k = 1/B — a large B means small compressibility, i.e. hard to compress; a common mix-up is reading a large B as 'compresses easily', which is the opposite of the truth.

When a wire stretches, it also gets very slightly thinner. The ratio of this lateral strain to the longitudinal strain is Poisson's ratio (σ) — dimensionless, typically 0.2–0.4 for common materials.

Elastic potential energy in a stretched wire. Work done stretching a wire is stored as elastic potential energy. For a wire of length L, area A, obeying Hooke's law, tension at extension x is F = YAx/L, so W = ∫₀^ΔL F dx = ½ Y A (ΔL)²/L. Using stress = YΔL/L and volume = AL, this becomes U = ½ × stress × strain × volume, so energy stored per unit volume is u = ½ × stress × strain = ½ Y (strain)².

The elastic moduli directly decide how engineers design cables, columns, pillars and beams.

Cranes and lifts. Steel ropes are chosen thick enough that the maximum stress stays well below the elastic limit of steel, with a safety margin. Steel is preferred here specifically for its very high Young's modulus — for the same load, it stretches far less than copper, aluminium or a fibre rope, so the load barely moves the cable.

Columns and pillars. A column under a compressive load can fail not just by crushing but by bending/buckling sideways if it is too thin for its height. Engineers use thick pillars, or more efficiently, hollow cylindrical columns: for the same mass as a solid column, spreading the material to a larger radius resists bending much better, because bending resistance depends strongly on how far material sits from the central axis, not just on the cross-sectional area. This is why many concrete pillars and bridge legs are hollow or ring-shaped rather than solid.

Beams and girders. A beam of length l, breadth b and depth d, supported at both ends and loaded with weight W at the middle, sags by δ = Wl³/(4bd³Y). Since depression falls as d³, increasing depth reduces sag far more effectively than increasing breadth, for the same material. This is why girders and rails use an I-shaped cross-section: material is concentrated in the top and bottom flanges (farthest from the neutral axis, where bending stress is largest), while the thin web near the neutral axis (where bending stress is nearly zero) saves weight without sacrificing strength.