Board exam focus — Mechanical Properties of Solids (CBSE & HBSE)

Mechanical Properties of Solids studies how solids deform under applied forces and recover their shape. It defines elasticity and plasticity, then quantifies deformation through stress and strain of various types (longitudinal, volumetric and shear). Hooke's law links stress and strain in the elastic region, leading to the three elastic moduli — Young's modulus, bulk modulus and shear (rigidity) modulus — and to Poisson's ratio. The stress–strain curve introduces proportional limit, elastic limit, yield point, plastic deformation and fracture. The chapter ends with elastic potential energy stored in a stretched wire. CBSE and HBSE set numericals on Young's modulus, elongation, and energy stored, plus conceptual questions on the stress–strain curve.

Elasticity, Stress, Strain and Hooke's Law

Elasticity and Plasticity

• Elasticity: the property by which a body regains its original shape and size after the removal of the deforming force (e.g. steel, rubber within limits).
• Plasticity: the property of a body that does not regain its original configuration after the force is removed (e.g. putty, clay).
• A perfectly elastic body recovers fully; a perfectly plastic body shows no recovery. No real body is perfectly elastic, though quartz fibre comes close.

Stress

Stress is the internal restoring force per unit area developed in a body under deformation:

$$\text{Stress} = \frac{F}{A}$$

SI unit: N/m² (pascal, Pa); dimensions [ML⁻¹T⁻²].

Strain

Strain is the ratio of change in configuration to the original configuration — it is a dimensionless quantity.

• Longitudinal strain = ΔL / L
• Volumetric strain = ΔV / V
• Shear strain = θ (angle of shear, in radians)

Hooke's Law

Within the elastic limit, stress is directly proportional to strain:

$$\text{Stress} \propto \text{Strain} \quad\Rightarrow\quad \frac{\text{Stress}}{\text{Strain}} = E$$

where E is the modulus of elasticity (a constant for a given material). It has the same units as stress (Pa).

Hooke's law holds only up to the proportional limit; beyond it the stress–strain relation becomes non-linear.

Stress–Strain Curve and the Three Elastic Moduli

The Stress–Strain Curve

For a ductile material like mild steel, the stress–strain graph shows distinct regions:

1. O to A (Proportional limit): Stress ∝ strain; Hooke's law obeyed; graph is a straight line.
2. A to B (Elastic limit / yield point): Slightly beyond A, the body still returns to original length but not perfectly linearly. B is the yield point.
3. B to D (Plastic region): Permanent deformation; the wire does not recover.
4. D (Ultimate tensile strength): Maximum stress the material can bear.
5. E (Fracture point): The wire breaks.

The region between the elastic limit and fracture point determines the ductility of a material. A large plastic region means the material is ductile.

Young's Modulus (Y)

Ratio of longitudinal stress to longitudinal strain:

$$Y = \frac{F/A}{\Delta L/L} = \frac{FL}{A\,\Delta L}$$

Applies to solids (wires, rods). Higher Y means more rigid material; steel (~2 × 10¹¹ Pa) is stiffer than copper.

Bulk Modulus (K)

Ratio of volumetric (hydraulic) stress to volumetric strain:

$$K = \frac{-\Delta P}{\Delta V/V}$$

The negative sign shows volume decreases as pressure increases. Compressibility = 1/K. Applies to solids, liquids and gases.

Shear (Rigidity) Modulus (η or G)

Ratio of shearing (tangential) stress to shearing strain:

$$\eta = \frac{F/A}{\theta}$$

For most materials Y > η, and a smaller modulus value generally indicates the material yields more easily to that type of stress.

Poisson's Ratio and Elastic Potential Energy

Poisson's Ratio (σ)

When a wire is stretched, it elongates longitudinally and contracts laterally. Poisson's ratio is the ratio of lateral strain to longitudinal strain:

$$\sigma = \frac{\text{Lateral strain}}{\text{Longitudinal strain}} = -\frac{\Delta d/d}{\Delta L/L}$$

• It is a dimensionless constant.
• Theoretical limits: −1 to 0.5; practical values lie between 0.2 and 0.4 for most materials.

A positive Poisson's ratio means the material gets thinner when stretched — the usual behaviour of metals and wires.

Elastic Potential Energy in a Stretched Wire

Work done in stretching a wire is stored as elastic potential energy. The average force during stretching is (½)F, so:

$$U = \frac{1}{2} \times \text{Force} \times \text{Extension} = \frac{1}{2}F\,\Delta L$$

In terms of stress and strain, the energy per unit volume (energy density):

$$u = \frac{1}{2} \times \text{stress} \times \text{strain} = \frac{1}{2}\,Y\,(\text{strain})^2$$

Important Applications and Concepts

• Why are girders I-shaped? To reduce bending (depression) while saving material — bending depends on the cube of the depth.
• Why is steel preferred over copper for construction? Steel has a higher Young's modulus, so it deforms less under the same load.
• Maximum height of a mountain is limited by the elastic (shear) strength of rock at the base.

Elastic energy density (½ × stress × strain) is a frequently asked CBSE/HBSE derivation — remember it equals the area under the stress–strain curve in the elastic region.

Frequently asked questions

Are these Mechanical Properties of Solids notes free?

Yes — the Mechanical Properties of Solids notes for Physics (Class 11) on Siksha Sarovar are completely free to read, with no account required.

Do these notes follow CBSE and HBSE?

Yes. The Mechanical Properties of Solids notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.

What does the Mechanical Properties of Solids chapter cover?

Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Mechanical Properties of Solids.