Elasticity

Elasticity deals with the behaviour of solids when external forces cause deformation and how the body regains its original shape after the forces are removed.

1. Intermolecular / Interatomic Forces

• In a solid at equilibrium, atoms or molecules are separated by an equilibrium distance r₀.
• At r₀, the force between atoms is zero and potential energy is minimum.
• If distance < r₀ → repulsive force acts.
• If distance > r₀ → attractive force acts.
• Atoms in solids behave like they are connected by stiff springs.

2. Elasticity

• Elasticity is the property of a material by which it regains its original shape and size after removal of deforming forces.
• Deforming force: External force causing change in shape or size.
• Restoring force: Internal force opposing deformation.
• If a body does not regain original shape, it is called plastic.

3. Stress

Stress is the restoring force per unit area.

Stress = F / A

• Normal stress → force perpendicular to area
• Tangential stress → force parallel to area
• Stress is a scalar quantity

Unit: N m⁻² (Pascal)

4. Strain

Strain is the fractional change in dimension.

Strain = Change in dimension / Original dimension

• Strain has no unit and is dimensionless
• It is a scalar quantity

Types of Strain

• Longitudinal Strain: ΔL / L
• Volumetric Strain: ΔV / V
• Shearing Strain: tan θ ≈ θ

5. Hooke’s Law

Within elastic limit:

Stress ∝ Strain

Stress = E × Strain

• The range where Hooke’s law is valid is called the proportional limit.
• Beyond elastic limit, permanent deformation occurs.

6. Moduli of Elasticity

(a) Young’s Modulus (Y)

Y = (F / A) / (ΔL / L)

Measures resistance to change in length.

(b) Bulk Modulus (B)

B = −ΔP / (ΔV / V)

Measures resistance to change in volume.

(c) Modulus of Rigidity (η)

η = Tangential stress / Shearing strain

Measures resistance to change in shape.

7. Poisson’s Ratio (σ)

σ = Lateral strain / Longitudinal strain

• Value lies between 0.1 and 0.3 for most materials
• Negative sign indicates decrease in radius on stretching

Relation between elastic constants:

Y = 3B(1 − 2σ) = 2η(1 + σ)

8. Elastic Hysteresis

• Difference in loading and unloading stress–strain curves
• Area of loop represents energy lost
• Materials with high hysteresis are used as vibration absorbers

9. Elastic Potential Energy

Energy stored in a stretched wire:

U = ½ × stress × strain × volume

U = ½ Y × strain² × volume

Energy density:

u = U / V = ½ Y × strain²

10. Important Applications

• Change in density with pressure: ρ = ρ₀ (1 + CP)
• Angle of twist: Lφ = rθ
• Twisting couple: C = πηr⁴ / 2L
• Bending of beam (rectangular): δ = MgL³ / 4bd³Y
• Bending of beam (circular): δ = MgL³ / 12πr⁴Y