Class 11 Physics · Chapter 8 NotesMechanical Properties of Solids

When external forces deform a body, internal restoring forces develop. The restoring force per unit area is called stress. Its SI unit is N m⁻² or pascal (Pa). Strain is the fractional change in dimension. There are three types of stress: tensile or compressive (longitudinal) stress, shearing stress, and hydraulic stress. Longitudinal strain is the change in length divided by original length. Shearing strain is the ratio of relative displacement of faces to the perpendicular distance, approximately equal to the angle θ. Volume strain is the change in volume divided by original volume. Strain has no units.

• Stress = F/A
• Longitudinal strain = ΔL/L
• Shearing strain = Δx/L = tan θ ≈ θ
• Volume strain = ΔV/V

For small deformations, stress is directly proportional to strain. This is known as Hooke's law. The constant of proportionality is called the modulus of elasticity. Hooke's law is an empirical law and is valid for many materials within the elastic limit. However, some materials like elastomers do not obey Hooke's law over most of the region. The modulus of elasticity is a characteristic of the material and is independent of the dimensions of the body.

• Stress ∝ Strain
• Stress = k × Strain, where k is modulus of elasticity

A typical stress-strain curve for a metal shows different regions. From O to A, stress is proportional to strain (Hooke's law). Point A is the proportional limit. From A to B, stress and strain are not proportional, but the body still returns to its original shape when the load is removed. Point B is the yield point or elastic limit, and the corresponding stress is the yield strength. Beyond B, plastic deformation occurs. Point D is the ultimate tensile strength, and fracture occurs at point E. If D and E are close, the material is brittle; if they are far apart, it is ductile.

• Proportional limitpoint A
• Yield point (elastic limit)point B
• Ultimate tensile strengthpoint D
• Fracture pointpoint E

Young's modulus (Y) is the ratio of tensile (or compressive) stress to longitudinal strain. It is a measure of the stiffness of a material. A material with a large Young's modulus requires a large force to produce a small change in length. For example, steel has a larger Young's modulus than copper, brass, and aluminium, so steel is more elastic in the sense that it stretches less for a given load. Young's modulus is relevant only for solids. Its SI unit is N m⁻² or Pa.

• Y = (F/A) / (ΔL/L) = (F × L) / (A × ΔL)
• UnitN m⁻² or Pa
• Steel is more elastic than copper, brass, and aluminium.

The ratio of shearing stress to shearing strain is called the shear modulus (G), also known as the modulus of rigidity. It is defined by G = (F/A) / θ, where θ is the angular deformation. Shear modulus is relevant only for solids because only solids have a definite shape. For most materials, G is less than Y, and approximately G ≈ Y/3. The SI unit is N m⁻² or Pa.

• G = (F/A) / θ = (F × L) / (A × Δx)
• UnitN m⁻² or Pa
• G ≈ Y/3 for most materials

The ratio of hydraulic stress to volume strain is called the bulk modulus (B). It is defined as B = –p / (ΔV/V), where p is the pressure and ΔV/V is the volume strain. The negative sign indicates that volume decreases with increase in pressure. The reciprocal of bulk modulus is compressibility (k). Bulk modulus is relevant for solids, liquids, and gases. Solids have the largest bulk modulus, gases the smallest. For example, the bulk modulus of steel is about 160 GPa, while that of air at STP is about 1.0 × 10⁻⁴ GPa.

• B = –p / (ΔV/V)
• Compressibility k = 1/B
• UnitN m⁻² or Pa

When a wire is stretched, its length increases and its diameter decreases. The strain perpendicular to the applied force is called lateral strain, and the strain along the force is longitudinal strain. Within the elastic limit, lateral strain is directly proportional to longitudinal strain. The ratio of lateral strain to longitudinal strain is called Poisson's ratio (σ). It is a pure number with no units. For steels, its value is between 0.28 and 0.30, and for aluminium alloys, it is about 0.33.

• Poisson's ratio σ = (Δd/d) / (ΔL/L)
• No units or dimensions
• Typical valuessteel 0.28–0.30, aluminium alloys ~0.33

When a wire is stretched, work is done against inter-atomic forces. This work is stored as elastic potential energy. For a wire of length L and area A, stretched by a small amount l, the work done is W = ½ × Y × (l/L)² × A × L. This can also be written as W = ½ × stress × strain × volume of the wire. The elastic potential energy per unit volume is u = ½ × stress × strain.

• W = ½ × Y × (strain)² × volume
• u = ½ × stress × strain

Knowledge of elastic properties is essential in engineering design. For example, the thickness of a crane rope is determined by ensuring that the stress does not exceed the yield strength of the material. Beams used in bridges and buildings are often given an I-shaped cross-section to reduce bending and buckling while minimising weight. The maximum height of a mountain on Earth is limited by the elastic properties of rocks; if the shear stress at the base exceeds the elastic limit, rocks flow. These examples show how the concepts of stress, strain, and elastic moduli are used in real-world applications.

• Crane rope thicknessA ≥ W/σ_y
• Beam bendingδ = W l³ / (4 b d³ Y)
• Mountain height limithρg = elastic limit of rock