Chapter 8: Mechanical Properties of Solids | Summary

For small deformations, stress is directly proportional to strain. This empirical relationship is known as Hooke's law: stress = k × strain, where k is the modulus of elasticity. Hooke's law holds for most materials but not all; some materials exhibit a non-linear stress-strain relationship even within their elastic range.

The constant of proportionality k (modulus of elasticity) is a characteristic property of the material. It quantifies the stiffness of a material — a higher modulus means a stiffer material that requires greater force to deform by the same amount. Three distinct elastic moduli — Young's modulus, shear modulus, and bulk modulus — describe different modes of deformation.

A stress-strain curve is obtained experimentally by gradually increasing the tensile load on a wire or cylinder and recording the corresponding strain. In the region O to A (proportional limit), the curve is linear and Hooke's law is obeyed. From A to B, the relationship is no longer proportional, but the material still returns to its original dimension upon removal of load. Point B is the yield point (elastic limit) with corresponding yield strength σ_y.

Beyond the yield point, plastic deformation occurs. At point C (between B and D), even after stress is removed, a permanent set remains. Point D represents the ultimate tensile strength (σ_u), beyond which fracture occurs at point E. If D and E are close, the material is brittle; if far apart, it is ductile. Elastomers like rubber and aortic tissue can be stretched to very large strains but do not obey Hooke's law over most of their elastic region and have no well-defined plastic region.

Young's modulus (Y) is the ratio of tensile or compressive stress to longitudinal strain: Y = (F/A)/(ΔL/L) = (F×L)/(A×ΔL). Its SI unit is N m⁻² or Pa. Metals have large Young's moduli — steel (200×10⁹ N m⁻²) is stiffer than copper (110×10⁹ N m⁻²), aluminium (70×10⁹ N m⁻²), and brass. This is why steel is preferred in heavy-duty machines and structural designs. Wood, bone, concrete, and glass have comparatively smaller Young's moduli.

Shear modulus (G), also called modulus of rigidity, is the ratio of shearing stress to shearing strain: G = (F×L)/(A×Δx) = F/(A×θ). It is generally less than Young's modulus, and for most materials G ≈ Y/3. Bulk modulus (B) is the ratio of hydraulic stress to volume strain: B = −p/(ΔV/V). The negative sign reflects that pressure increase causes volume decrease. Solids have much larger bulk moduli than liquids, which in turn are far larger than those of gases. The reciprocal of bulk modulus is compressibility k = 1/B. Gases are about a million times more compressible than solids.

Poisson's ratio is the ratio of lateral strain (Δd/d) to longitudinal strain (ΔL/L). It is a pure dimensionless number depending only on the material. For steel, it lies between 0.28 and 0.30; for aluminium alloys it is about 0.33. Elastic potential energy per unit volume stored in a stretched wire is u = ½σε, where σ is stress and ε is strain. This energy is stored as work done against inter-atomic forces during elongation.

Engineering applications of elasticity include the design of crane ropes, bridges, beams, columns, and buildings. For a crane of 10-tonne capacity, the minimum cross-sectional area of a mild steel rope is calculated using the yield strength (≈300×10⁶ N m⁻²), giving a radius of about 1 cm; a safety margin of factor ten means a 3 cm radius rope is recommended. Since a single wire of this radius would be too rigid, ropes are made of many thin braided wires for flexibility and strength.

The bending of a beam of length l, breadth b, and depth d under a central load W is given by δ = Wl³/(4bd³Y). To reduce bending, one should maximise Y, and increase d rather than b (since δ ∝ d⁻³ but only ∝ b⁻¹). However, a very deep bar may buckle under off-centre loads. The I-shaped cross-section is a compromise that provides a large load-bearing surface and adequate depth while minimising weight and cost. Pillars with distributed ends support more load than those with rounded ends. The maximum height of a mountain (~10 km) can be estimated by equating the shearing stress at the base (hρg) to the elastic limit of rock (~30×10⁷ N m⁻²).

Chapter 8 establishes that solid bodies are not perfectly rigid and respond to deforming forces through stress and strain. Hooke's law governs elastic behaviour for small deformations, and three elastic moduli — Young's modulus, shear modulus, and bulk modulus — quantify a material's resistance to longitudinal, shear, and volumetric changes respectively. The stress-strain curve reveals key material properties such as yield strength, ultimate tensile strength, ductility, and brittleness. Poisson's ratio and elastic potential energy further enrich the understanding of how materials store and release energy under load.

From the exam perspective, this chapter is highly important for CBSE Class 11 board examinations. Students must be adept at applying formulas for Young's modulus, shear modulus, and bulk modulus to solve numerical problems, interpreting stress-strain curves, and explaining engineering applications such as the I-beam shape, the design of crane ropes, and the estimation of mountain height. Questions on elastomers, Poisson's ratio, compressibility, and the comparison of elastic moduli across materials are also frequently tested.