Mechanical Properties of Solids
'The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them.' — William Lawrence Bragg
1. Chapter Overview
Why does a rubber band stretch? Why does a steel bridge not collapse under its own weight? This chapter explores how SOLIDS respond to external forces. You will learn about STRESS, STRAIN, HOOKE'S LAW, and the three ELASTIC MODULI that characterise a material's stiffness. The chapter also covers ELASTIC POTENTIAL ENERGY and some practical applications.
2. Stress and Strain
Stress (σ)
- Definition: Restoring force per unit area SET UP INSIDE the body
- σ = F/A
- SI unit: N/m² or pascal (Pa)
- Tensile Stress: When forces tend to STRETCH the body
- Compressive Stress: When forces tend to SQUEEZE the body
- Shear Stress: When forces act PARALLEL to the surface
Strain (ε)
- Definition: Ratio of CHANGE in dimension to ORIGINAL dimension
- ε = ΔL/L (longitudinal strain), ΔV/V (volume strain), Δθ (shear strain)
- Strain is DIMENSIONLESS (no unit)
Types of Stress and Strain
| Type | Stress | Strain | Example |
|---|---|---|---|
| Longitudinal | Tensile/Compressive | ΔL/L | Stretching a wire |
| Volume (Hydraulic) | Pressure (ΔP) | ΔV/V | Compression under water |
| Shear | Tangential force/Area | Angle of shear θ | Rivet in shear loading |
3. Hooke's Law and Elastic Limit
- Hooke's Law: Within the ELASTIC LIMIT, stress ∝ strain
- σ = E × ε (where E is the elastic modulus)
- Elastic Limit: The maximum stress after which the material does NOT return to its original shape
- Beyond the elastic limit → PERMANENT deformation (plastic behaviour)
Stress-Strain Curve
- Region OA: Proportional limit (Hooke's law holds)
- Region AB: Elastic limit (material still recovers fully)
- Region BC: Yielding (large strain for small stress increase)
- Point C: Yield point
- Region CD: Plastic flow (material deforms permanently)
- Point D: Ultimate tensile strength (maximum stress)
- Region DE: Necking (cross-section reduces)
- Point E: Fracture point
4. Elastic Moduli
Young's Modulus (Y)
- Definition: Ratio of longitudinal stress to longitudinal strain
- Y = (F/A) / (ΔL/L) = FL/AΔL
- Measures a material's RESISTANCE to stretching/compression
- Typical values: Steel ≈ 2×10¹¹ Pa, Aluminium ≈ 7×10¹⁰ Pa
Bulk Modulus (B)
- Definition: Ratio of volumetric stress to volumetric strain
- B = -ΔP/(ΔV/V) (negative sign because volume decreases with pressure increase)
- Compressibility k = 1/B
- Solids have HIGHER bulk modulus than liquids and gases
Shear Modulus (G or η)
- Definition: Ratio of shear stress to shear strain
- G = (F/A)/θ = (F/A)/(Δx/L)
- Also called MODULUS OF RIGIDITY
Comparison
| Modulus | Symbol | What it resists | Expression |
|---|---|---|---|
| Young's | Y | Linear deformation | FL/AΔL |
| Bulk | B | Volume change | -VΔP/ΔV |
| Shear | G | Shape change | (F/A)/θ |
5. Elastic Potential Energy
- When a wire is stretched, work done is stored as ELASTIC POTENTIAL ENERGY
- U = ½ × Stress × Strain × Volume
- U = ½ × (F/A) × (ΔL/L) × AL = ½ × F × ΔL = ½ × (YA/L) × (ΔL)²
- SI unit: J (joule)
Worked Problem
Q: A steel wire of length 2 m and cross-section area 1 mm² is stretched by 1 mm. Find energy stored. (Y = 2×10¹¹ Pa) A: U = ½ × (YA/L) × (ΔL)² = ½ × (2×10¹¹×10⁻⁶/2) × (0.001)² = ½ × 10⁵ × 10⁻⁶ = 0.05 J.
6. Poisson's Ratio (σ or μ)
- When a wire is stretched longitudinally, it CONTRACTS laterally
- Poisson's ratio σ = (Lateral strain) / (Longitudinal strain) = -(ΔD/D)/(ΔL/L)
- Range: -1 < σ < 0.5 (typically 0.2 — 0.4 for most materials)
- Cork: σ ≈ 0 (no lateral contraction)
- Rubber: σ ≈ 0.5 (nearly incompressible)
7. Common Mistakes
- Confusing stress and pressure: Stress is RESTORING force/area INSIDE material; pressure is EXTERNAL force/area
- Hooke's law applies ONLY within elastic limit: Beyond yield point, a small force causes large deformation
- Young's modulus is NOT the same for all wires: It's a MATERIAL property, not a wire property
- Bulk modulus sign stays positive: The negative sign in B = -ΔP/(ΔV/V) accounts for volume decreasing
- Stress is NOT force: Stress = Force/Area; doubling area halves stress for same force
8. CBSE Exam Focus
- Stress-strain curve interpretation (3-mark)
- Young's modulus numericals (wire stretching problems)
- Bulk and shear modulus numerical problems
- Elastic potential energy derivation (3-mark)
- Poisson's ratio definition and applications
9. Key Formulas
- σ = F/A (Stress)
- ε = ΔL/L (Longitudinal strain)
- Y = FL/AΔL
- B = -ΔP/(ΔV/V)
- G = (F/A)/θ
- U = ½ × Stress × Strain × Volume = ½ × F × ΔL
- σ (Poisson's) = -Lateral strain/Longitudinal strain
10. Self-Test (5+ Q&A)
Q1: A 3 m long steel wire (Y = 2×10¹¹ Pa) of area 2 mm² is stretched by a 100 kg load. Find elongation (g = 10 m/s²). A: F = mg = 1000 N. ΔL = FL/AY = 1000×3/(2×10⁻⁶×2×10¹¹) = 3000/(4×10⁵) = 7.5×10⁻³ m = 7.5 mm.
Q2: What is the elastic limit? A: The MAXIMUM stress a material can withstand without permanent deformation. Beyond this, Hooke's law fails.
Q3: Which has higher Young's modulus — rubber or steel? A: Steel (≈ 2×10¹¹ Pa). Rubber has much lower Y (≈ 10⁶ Pa), meaning it stretches more for the same stress.
Q4: Water in a lake is compressed by 1%. Find the pressure increase. (B = 2.2×10⁹ Pa) A: B = -ΔP/(ΔV/V) → ΔP = B × |ΔV/V| = 2.2×10⁹ × 0.01 = 2.2×10⁷ Pa = 22 MPa.
Q5: Why does a wire become thinner when stretched? A: Due to POISSON'S EFFECT: longitudinal strain causes a lateral contraction. The ratio of lateral to longitudinal strain is Poisson's ratio (σ).
11. Conclusion
The mechanical properties of solids determine how materials behave under load — critical knowledge for engineering and construction. Hooke's law provides a LINEAR model of elasticity. The three moduli (Young's, Bulk, Shear) quantify a material's stiffness in different deformation modes. Elastic potential energy shows that stretched materials STORE energy, explaining everything from catapults to seismic waves.
