Mechanical Properties of Solids — NEET Physics Notes

This chapter covers elasticity, deformation of materials under stress, and the mechanical behaviour of solids — a fundamental topic in mechanics with consistent weightage in NEET Physics.

Quick Revision

• Elasticity: Property of a body to regain its original shape/size after removal of deforming force
• Stress: Internal restoring force per unit area = F/A, unit: N/m² or Pascal
• Strain: Relative change in dimension = ΔL/L, dimensionless
• Hooke’s Law: Stress ∝ Strain (within elastic limit)
• Young’s Modulus (Y): Y = (F/A) / (ΔL/L) = FL/AΔL
• Bulk Modulus (B): B = −P / (ΔV/V)
• Shear Modulus (G): G = F/A (shear stress / shear strain)
• Poisson’s Ratio (σ): σ = (lateral strain) / (longitudinal strain)
• Elastic Potential Energy: U = (½) × stress × strain × volume

Standard Study

Types of Stress

1. Tensile Stress: Pulling force per unit area — length increases
2. Compressive Stress: Pushing force per unit area — length decreases
3. Shear Stress: Tangential force per unit area — shape changes

Types of Strain

1. Longitudinal Strain: Change in length / original length
2. Volume Strain: Change in volume / original volume
3. Shear Strain: Angular deformation θ where tan θ ≈ θ

Hooke’s Law and Moduli

Stress-Strain Curve:

• Proportional Limit: Up to point P — stress ∝ strain (Hooke’s law holds)
• Elastic Limit: Up to point E — body regains original shape after removal
• Yield Point (Y): Beyond this, permanent deformation occurs
• Fracture Point (F): Material breaks here

Young’s Modulus (Y)

• Most commonly tested modulus in NEET
• Y = (FL) / (AΔL)
• Higher Y → stiffer material (harder to stretch/compress)
• Example: Steel Y ≈ 200 GPa, Rubber Y ≈ 0.05 GPa

Bulk Modulus (B)

• B = −P / (ΔV/V)
• Negative sign because pressure increases when volume decreases
• Compressibility = 1/B

Shear Modulus (G)

• G = (F/A) / (x/h) where x = parallel displacement, h = distance between layers
• Also called Modulus of Rigidity

Poisson’s Ratio

• σ = (lateral strain) / (longitudinal strain)
• For most metals: 0.25 ≤ σ ≤ 0.33
• Theoretical range: −1 ≤ σ ≤ 0.5 (for isotropic materials, σ ≤ 0.5)

Relation Between Moduli

For isotropic elastic materials, the three moduli and Poisson’s ratio are related:

• Y = 2G(1 + σ)
• B = Y / (3(1 − 2σ))
• Y = 3B(1 − 2σ)

Elastic Potential Energy

• When a wire is stretched, work done is stored as elastic PE
• U = (½) × Stress × Strain × Volume
• U = (½) × Y × (strain)² × V
• Energy density (U/V) = (½) × stress × strain = (½) × Y × (strain)²

Deep Study

Stress-Strain Behaviour

Ductile Materials (e.g., copper, aluminium):

• Large plastic deformation before fracture
• Large region between elastic limit and fracture
• Used for making wires and sheets

Brittle Materials (e.g., glass, ceramics):

• Very little plastic deformation before fracture
• Breaking occurs close to elastic limit
• Example: Cast iron, concrete

Elastomers (e.g., rubber)::

• Large elastic strain (up to several hundred percent)
• Stress-strain curve is non-linear
• No well-defined yield point

Cantilever and Beam Bending

• When a beam is loaded at one end (cantilever), it bends
• Depression at free end: δ = (WL³) / (3YI)
• W = load, L = length, Y = Young’s modulus, I = moment of inertia

Torsion

• When a cylinder is twisted, shear stress is produced
• Angle of twist θ = (TL) / (GJ)
• T = torque, J = polar moment of inertia, G = shear modulus

Thermal Stress

• When temperature changes in a constrained rod, thermal stress develops
• Thermal stress = Y × α × ΔT
• α = coefficient of linear expansion
• ΔT = change in temperature

Exam Tips

1. Hooke’s Law is valid ONLY within proportional/elastic limit
2. Young’s Modulus is the most frequently asked — know its formula and unit
3. Stress has unit N/m² (Pascal), same as pressure
4. Bulk modulus applies to liquids/gases under pressure — compression
5. Wire stretching: work done = ½ × F × ΔL = ½ × Y × A × (ΔL)²/L
6. Thermal stress formula YαΔT is commonly used in problems
7. Poisson’s ratio must be dimensionless — watch for unit conversion errors
8. Elastic PE per unit volume = ½ × stress × strain

Common Pitfalls

• Confusing stress with pressure (while dimensionally same, stress has directional nature)
• Forgetting that stress ∝ strain only within elastic limit — outside this relationship breaks
• Not converting units properly — use consistent SI units (Pa, m, m²)
• Confusing shear modulus with Young’s modulus in torsion problems
• Forgetting negative sign in Bulk Modulus formula (B = −PΔV/V)
• Not realising that for same force, thinnest wire experiences highest stress

Suggested Study Order

1. Basic concepts: elasticity, stress, strain definitions
2. Hooke’s Law and Young’s Modulus
3. Stress-Strain curve analysis
4. Bulk modulus and shear modulus
5. Poisson’s ratio and relations between moduli
6. Elastic potential energy
7. Thermal stress problems
8. Cantilever and beam bending